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The Pennsylvania State University
The Graduate School
Department of Electrical Engineering
COMPUTATIONAL METHODS AND APPLICATIONS
IN OPTICAL IMAGING AND SPECTROSCOPY
A Dissertation in
Electrical Engineering
by
Nikhil Mehta
2016 Nikhil Mehta
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
August 2016
The dissertation of Nikhil Mehta was reviewed and approved* by the following:
Zhiwen Liu
Professor of Electrical Engineering
Dissertation Advisor
Chair of Committee
Venkatraman Gopalan
Professor, Materials Science and Engineering
Associate Director, Center for Optical Technologies
Iam-Choon Khoo
William E. Leonhard Professor of Electrical Engineering
Timothy Kane
Professor of Electrical Engineering
*Signatures are on file in the Graduate School
Victor Pasko
Professor of Electrical Engineering
Graduate Program Coordinator
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ABSTRACT
This dissertation presents my work in application of computational techniques and the
resulting enhancements to several non-linear and ultrafast optical imaging and spectroscopy
modalities. The importance of novel computational optical imaging schemes which aim to
overcome the limitations of conventional imaging techniques by leveraging the availability of
computational resources and the vast body of literature in computational signal processing is
emphasized. In particular the computational techniques of compressive sensing and two
dimensional phase retrieval are introduced in the broad context as inversion techniques suitable
for optical applications including coherent anti-Stokes Raman holography, non-linear
spectroscopy, and ultrashort pulse characterization. It is shown that both computational
techniques seek to improve key signal metrics such as higher signal to noise ratio (SNR) and
better resolution than can be obtained traditionally within the specific imaging modality.
Coherent anti-Stokes Raman scattering (CARS) holography is a novel imaging modality
which combines the principles of coherent anti-Stokes Raman scattering and holography to
provide label-free, chemical selective, scanning-free, and single shot 3D imaging modality.
Compressive CARS holography is introduced as a sparsity constrained holographic image
reconstruction technique to enhance the optical sectioning capability of CARS holography by
suppressing out-of-focus background noise inherent in 3D images processed from a typical single
2D hologram. The advantages of compressive sensing guided signal acquisition strategy in
optical spectroscopy is presented by proposing ‘compressive multi-heterodyne optical
spectroscopy’ as a novel technique for ultra-high resolution frequency comb spectroscopy. Using
numerical simulations, our proposed compressive frequency comb spectroscopy technique is
shown to be well-suited for recording narrow line spectra at ultra-high sampling over broad
spectral range by leveraging sparsity inherent in such spectra.
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We next present applications of phase retrieval in optical imaging and spectroscopy. In
particular we use 2D phase retrieval technique to enhance the resolution of sum frequency
generation vibrational spectroscopy (SFG-VS) whose unique surface selectivity enables
qualitative and quantitative study of chemical species at surfaces/interfaces. Specifically, our key
contribution is that we show that our 2D phase retrieval based inversion algorithm enables
measurement of characteristic molecular vibrational spectra of air/dimethyl sulfoxide interface at
resolutions significantly better than that achievable in conventional SFG-VS acquisition system.
Lastly, we address the limitation of the commonly used pulse characterization technique:
frequency resolved optical gating (FROG) to spatio-temporally characterize the ultrafast pulse.
Using a simple spectral holographic recording technique, we present a modified 2D phase
retrieval based algorithm to measure the spectral phase at every spatial location in the vicinity of
focus of an objective and thereby track the spatio-temporal evolution of the pulse along its optical
axis.
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TABLE OF CONTENTS
List of Figures .......................................................................................................................... v
List of Tables ........................................................................................................................... vi
Acknowledgements .................................................................................................................. vii
Chapter 1 Introduction and Motivation .................................................................................... 1
Organization ..................................................................................................................... 3
Chapter 2 Computational Methods in Optics ........................................................................... 5
Introduction ...................................................................................................................... 5 Compressive Sensing ....................................................................................................... 6
Motivation ................................................................................................................ 6 Description and Formulation .................................................................................... 7 Reconstruction using TwIST .................................................................................... 10
Two Dimensional Phase Retrieval ................................................................................... 13 Motivation ................................................................................................................ 13 Uniqueness ............................................................................................................... 14 Solution using Generalized Gerchberg-Saxton Algorithm....................................... 16
Chapter 3 Applications of Compressive Sensing ..................................................................... 19
Introduction ...................................................................................................................... 19 Compressive off-axis coherent anti-Stokes Raman scattering holography ...................... 19
Introduction .............................................................................................................. 19 Formulation .............................................................................................................. 22 Numerical Results and Discussion ........................................................................... 24 Conclusion ................................................................................................................ 28
Compressive Frequency Comb Spectroscopy .................................................................. 29 Introduction and Motivation ..................................................................................... 29 Formulation .............................................................................................................. 32 Spectrum Estimation Using Compressive Sensing .................................................. 36 Numerical Results .................................................................................................... 37 Conclusion ................................................................................................................ 43
Chapter 4 Applications of Two Dimensional Phase Retrieval................................................. 44
Introduction ...................................................................................................................... 44 Computational sum frequency generation vibrational spectroscopy ............................... 44
Introduction and Motivation ..................................................................................... 44 Formulation of 2D PR based SFG spectroscopy ...................................................... 48 Numerical Simulation .............................................................................................. 52 Experiment ............................................................................................................... 55 Results and Discussion ............................................................................................. 58
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Spatio-temporal characterization of ultrashort optical pulse ............................................ 61 Introduction and motivation ..................................................................................... 61 Spatio-temporal characterization of ultrafast pulse .................................................. 63 Numerical Simulation .............................................................................................. 69 Experiment ............................................................................................................... 71 Results and Discussion ............................................................................................. 73 Conclusion ................................................................................................................ 75
Chapter 5 Summary and Future Prospects ............................................................................... 77
Appendix A Opto-electronic nano-probe ................................................................................. 79
Introduction and Motivation ..................................................................................... 79 Selective filling of PCF ............................................................................................ 81 Nano-tube attachment .............................................................................................. 84 Device characterization ............................................................................................ 88 Conclusion ................................................................................................................ 96
Bibliography ............................................................................................................................ 98
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LIST OF FIGURES
Figure 2-1: Illustration of why 1 norm promotes sparsity. 1 norm regularizer results in
higher penalty for smaller components of signal than its larger components. ................. 11
Figure 2-2: Generalized Gerchberg-Saxton algorithm: error reduction algorithm .................. 17
Figure 2-3: Generalized projection method description of GS based error reduction
algorithm .......................................................................................................................... 18
Figure 3-1: Schematic of CARS holography (a) Schematic diagram of CARS holography
(b) Central part (256x256 pixels) of a recorded CARS hologram; (c) DC filtered two
dimensional Fourier transform (amplitude) of the hologram in (b); (d) A digitally
filtered and re-centered sideband; (e) Reconstructed CARS field (amplitude) at the
recording plane. ................................................................................................................ 25
Figure 3-2: Image reconstruction using conventional digital propagation. The nine
polystyrene microspheres are visible, but the image quality is affected by out-of-
focus background contributions. ...................................................................................... 26
Figure 3-3: Compressive sensing based image reconstruction using the TwIST algorithm.
The diffraction noise is suppressed and the spheres signal can clearly be seen to be
localized at different depth positions. Red arrow indicates spheres used for SNR
analysis in Fig. 4. ............................................................................................................. 26
Figure 3-4: Axial localization using compressive CARS holography. (a) and (b) show
SNR for two spheres (indicated by red arrows in Fig. 3 along the z depth position
using conventional digital back-propagation algorithm and compressive
reconstruction, respectively.............................................................................................. 27
Figure 3-5: (a) Illustrative schematic for CMOS. The frequency comb is first filtered by
weighting its comb modes with a digitally controlled SLM and then combined with
the signal from the sample. The balanced detector measures the RF beat signal
between the reference comb and signal. The RF signal is further filtered by a Low
Pass Filter (LPF) and detected by a spectrum analyzer or a D-FFT block. The
broadband quadrature phase-shifter is used to generate a / 2 phase shifted copy of
the comb for the coherent case. (b) Frequency axis indicating relative value of
frequency variables. D is the characteristic response time of the fast detector.
(*diagram is not to scale) ................................................................................................. 34
Figure 3-6: Estimation of hypothetical coherent spectrum using CMOS-TwIST:
Comparison of (a) absorption spectrum and (b) refractive index ( n ) for K=100;
(c) absorption spectrum and (d) refractive index ( )n for K=200. The reduction in
inversion noise as K is increased can be seen by visual inspection. ................................ 40
Figure 3-7: Estimation of hypothetical incoherent spectrum using CMOS-TwIST:
Comparison of absorption spectrum for (a) K=100 and (b) K=200 measurements.
Inset highlights FWHM linewidth of simulated spectrum. .............................................. 42
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Figure 4-1: generalized Gerchberg-Saxton algorithm for BB-SFG-VS .................................. 50
Figure 4-2: Pulse profile of simulated IR and NIR pulses and hypothetical (2) spectrum
used in numerical simulation, (b) simulated SFG spectrogram. 2nd
panel in (b) plots
the line trace at zero delay in spectrogram, which illustrates the loss of resolution
typically inherent in BB-SFG-VS. ................................................................................... 52
Figure 4-3: Simulation of high resolution SFG spectroscopy using 2D PR. (a) SFG
spectrogram of hypothetical transform limited NIR probe pulse and transform
limited IR pump pulse, (b) Estimate of (2) and comparison with simulated
(2)
(c) Error Estimation of spectrogram and response spectrum and the error between
simulated and estimated spectrogram. ............................................................................. 53
Figure 4-4: (a) Simulated SFG spectrogram using hypothetical (2) and chirped NIR
probe pulse. (b) Numerically computed SFG spectrogram using 2D PR, (c) Estimate
of (2) (black dotted curve) computed using 2D PR, hypothetical
(2) (green
curve) and trace of spectrogram through 0 delay (See dotted white line in (b). Notice
that spectrogram trace does not resolve adjacent peaks at 2915 cm-1
and 2920 cm-1
.
(d) Simulated and computed spectral amplitude and phase of chirped NIR probe. ......... 54
Figure 4-5: Schematic of experimental setup (ssp configuration) ........................................... 57
Figure 4-6: Match between experimentally recorded and computationally retrieved SFG
spectrogram of air/DMSO interface. (a) Recorded BB-SFG SFG spectrogram of the
air/DMSO interface taken in the range of ±6.4 ps delay between the 2 ps probe pulse
and broadband pump pulse at ~3.4 μm. (b) Numerically estimated BB SFG
spectrogram using 2D PR. (c) Trace of spectrograms when both pump and probe
overlap with no delay. (d) Trace of both spectrograms along peak of the spectrum. ...... 58
Figure 4-7: (a) RMS error between measured and computed spectrograms. (b) Estimated (2) spectrum of air/DMSO interface ............................................................................. 60
Figure 4-8: Schematic depicting change in fs pulse due to its interaction within a
hypothetical process. Ultrafast pulse characterization can accurately measure
amplitude and phase of fs pulse. ...................................................................................... 61
Figure 4-9: Numerical simulation. (a) Schematic of simulated experiment. (b) Typical
simulated HcFROG trace. (c) 2D F.T. of HcFROG traces at locations ‘P1’ and ‘P2’.
r is the spatial separation between the two locations and gv is the group velocity
through the hypothetical medium. (d) Relative FROG phasor of the pulse at ‘P2’
with respect to ‘P1’. ......................................................................................................... 67
Figure 4-10: Results of numerical simulation using HCFROG algorithm (a) retrieved and
theoretical complex pulse profile. FWHM is marked for pulse at each location. (b)
computed spatial separation between the 3 locations compared with actual
separation used for simulation. The table lists the actual values of spatial separation
as used in the numerical experiment and retrieved using method described. (c)
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comparison of retrieved and theoretical pulse spectrum at each location. FWHM is
marked for the pulse spectrum. (d) comparison of retrieved and theoretical SH
spectrum of pulse at each location. FWHM is marked for the pulse spectrum. ............... 70
Figure 4-11: (a) Schematic of our experimental setup. (b) The inset shows SEM image of
the BaTiO3 micro-cluster used for generating FROG signal. (c) Typical recorded
HcFROG trace. The fringe structure can be seen along both delay and wavelength
dimension. ........................................................................................................................ 73
Figure 4-12: Pulse characterization at multiple locations using FROG holography. (a)
Retrieved pulse profile (amplitude and phase, FWHM ~89 fs) at each position; (b)
Comparison of both fundamental and SH power spectrum of the retrieved pulse at
each position with measured power spectra (black squares) (FWHM ~12 nm) (c)
Comparison between retrieved and experimentally recorded spatial separation
between each location with respect to pivot. The table in inset lists actual values in
m . ................................................................................................................................ 74
Figure A-1: Schematic illustrating patterning of PCF for selective filling .............................. 82
Figure A-2 Pumping molten alloy through patterned PCF. (a) Schematic of oven showing
a PCF held in a chamber. The inset shows cutaway of the chamber in which
patterned pcf is held using PDMS and sunk in bath of alloy. (b) Filled PCF under
500 psi showing alloy leaking out as shiny blob from the other end held outside the
oven at room temperature. (c) SEM image of a pcf with three holes filled with alloy,
(d) Setup for checking electrical continuity and typical resistance of filled electrode
(1657 ohms). .................................................................................................................... 85
Figure A-3: Laser welding process for CNT attachment. (a) Side view image of CNT just
prior to being laser welded to alloy filled pcf. Inset shows the schematic top view
illustrating the fiber taper used for delivery. (b) Laser welded CNT. .............................. 87
Figure A-4: FIB assisted CNT attachment. (a) CNT lying on lacie carbon mesh is bonded
to 500 µm tungsten probe tip via Pt deposition. (b) CNT liftoff. Part of carbon mesh
can be seen adhered to the CNT. Individual synapses need to be milled for ‘clean’
liftoff. (c) Sequential targeted attachment of CNT via e-beam assisted tungsten
deposition. (d) Ion beam image of a three-arm device. .................................................... 88
Figure A-5: (a) The average refractive index and extinction coefficients of Ostalloy 158
obtained via ellipsometry of the inset sample, (b) simulation of the typical optical
mode field for the unfilled PCF, (c) the mode field when three of the most center air
holes are filled with Ostalloy ........................................................................................... 89
Figure A-6: Group velocity dispersion for nano-probe with 3 filled electrodes. (a) Mach-
Zehnder setup for spectral holographic measurement. (b) Typical recorded spectral
hologram (c) Measured relative phase, and 3rd
order polynomial fit, (d) estimated
dispersion parameter for unfilled pcf (red dotted) and fabricated device (blue
dotted). Corresponding COMSOL simulated dispersion for unfilled pcf (blue red
line) and modeled device (solid blue line) are also plotted for comparison. The black
curve is dispersion parameter for bare fiber as specified by manufacturer. ..................... 92
x
Figure A-7: Electro-mechanical deflection of fabricated nano-probe. (a) Montage of 3
sequential frames from a video recording CNT deflection. The dotted line shows the
trace along which intensity is plotted in (b) and (c). The markers in (c) indicate pixel
positions of the ‘tip’ of the lower CNT in each frame. .................................................... 94
Figure A-8: Numerical simulation results. (a) charge distribution at the tip of a CNT
when a -20V ramp wave is applied to the terminal, (b) charge distribution along two
CNTs with ±20V applied. The deflection of the CNT is indicated by the red line
which marks the original position, (c) the tip displacement as a function of voltage
applied to the electrodes within the PCF. ......................................................................... 96
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LIST OF TABLES
Table 3-1: Coherent Spectrum Estimation: Comparision between theoretical and
numerical results of TwIST applied to Eq. (0.10) ............................................................ 41
Table 3-2: Incoherent Spectrum Estimation: Comparison between theoretical and
numerical results of TwIST applied to Eq. (0.11) ............................................................ 42
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ACKNOWLEDGEMENT
I take this opportunity to express my heartfelt gratitude to my advisor Prof. Zhiwen Liu
for guiding me during the course of my PhD program. Prof. Liu is a teacher par excellence and
has the rare ability to provide insight into the subject matter by drawing parallels between
concepts in different fields. I acknowledge his academic mentorship and his lasting influence on
my approach to assimilate new material and creativity towards problem solving.
My workspace was adjacent to Prof. Gopalan who was on my PhD committee and I thank
him for several insightful academic discussions, and especially for showing faith and confidence
in my ability to teach. Prof. Kane and Prof. Khoo equipped me with the knowledge of inversion
techniques and laser fundamentals that was necessary for my PhD projects and I thank them for
their support and encouragement during my search of career opportunities.
I recall joining the lab of Prof. Liu with no experience in an optics lab. Therefore, my
creativity and hands-on skills of working in lab are testament to the crucial support and
contribution of all my labmates. Kebin, Heifeng and Qian showed by example what it takes to
convert a good PhD thesis into a better PhD thesis. Perry set the bar on dexterity in the lab and I
enjoyed many engaging discussions with him, even on topics outside of optics. Chuang and Ding
showed me how not to make common mistakes in lab by actually doing them and always had
joke at hand to fix all problems! I also had the pleasure of working with Corey Janisch, Chenji
Zhang, Yizhu Chen, Alexander Cocking, Atriya Ghosh and Rong Tang and mentoring them when
needed. I have learned that it is because of my labmates that everything works and we all know
why!
I would like to thank my collaborators Christopher Lee and Prof. Seong Kim from Penn
State, and Prof. Yong Xu from Vriginia Tech, with whom I have enjoyed working towards
fruitful research. It is important to also highlight the key contribution of MRI staff, namely
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Trevor Clark and Joshua Maier for their unerring support and patience when I was learning to use
FIB and of Maria and Julie for support when I was the lab manager. I am particularly thankful to
the administrative staff at EE department including SherryDawn Jackson and MaryAnn
Henderson among others for always having my back regardless of whether I had a missing
signature or deadline.
The time of my life that I spent in Happy Valley with my friends in ‘709’ and ‘241D’ is
an integral part of me. I have very fond and special memories with each and every one on them.
Kadappan and Vishesh for being there always no matter what; Himanshu and Aditya for showing
how everything can be fun; Kishen and Ashish for the delightful discussions over tea; Vivek,
Arun, Apoorva, Niddhi, Amruta, Deepti, Nandhini, Piyush, Divya, Adarsh, Aseem, all have
added a unique flavor and I am grateful for their company.
My wife Prachi has always been my best critic. I will remain thankful for her rock steady
support during the uncertain times at later stages of my PhD. It is often difficult to describe the
contribution of your loved ones in the space of few words. I am indebted to my wife, my parents,
and my brother for all their contributions to my success that are best acknowledged with a wink
and for their presence in my life.
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Chapter 1
Introduction and Motivation
In a general sense, optical sensing is mapping (or encoding) of the optical information of
interest in either time, frequency, space or angular spectrum, or some parameter space specified
by a combination of these domains. A conventional optical image such as a photograph produced
by an imaging lens of a camera is an isomorphic measurement of the light scattering from the
elements in the field of view. Similarly a conventional dispersive spectrometer isomorphically
detects the optical spectrum of the incident light. Optical sensing in general can be modeled as the
detection of the signal of interest f mapped to a set of measurements ( ) ( )i im K q f q dq
where ( )iK q is the sensing system model (which may be isomorphic or anisomorphic, that is an
indirect measurement which needs to be processed to retrieve the signal) and q is a (possibly
multi-dimensional) quantity which parameterizes the basis space, such as time or frequency.
Research in computational optical sensing based on modeling and manipulation of light has
resulted in advances in the overall capability of optical sensing systems by reducing noise,
improving resolution, enhancing measurement sensitivity and dynamic range, enabling higher
dimensional imaging using lower dimensional measurement and allowing for image compression.
For instance, stochastic optical reconstruction microscopy or photo-activated localization
microscopy [1]–[3] achieves significantly higher resolution than the Abbe diffraction limit by
computationally post-processing multiple low resolution images taken using conventional
fluorescence microscope. The field of astronomical imaging can be credited with the invention of
several innovative techniques such as aperture synthesis [4] which enabled high resolution
imaging by electronically correlating signals from individual receivers to synthesize apertures so
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large that they would be impossible to construct in practice; and wavefront sensing [5] which
advanced earth based imaging by allowing to correct for atmospheric turbulence adaptively
among other applications. Spectral imaging which recognizes the rich information content of the
electromagnetic spectrum combines 2D imaging and spectroscopy and has important application
such as remote sensing and object identification. The data intensive spectral imaging
measurement, which seeks to capture the optical spectrum at every spatial location in image,
benefits from signal acquisition and computational processing techniques, such as tomographic
spectral imager [6] and coded aperture snapshot spectral imager [7], which can retrieve the 3D
image of interest from a lower dimensional simplified measurement. In 1948 Gabor proposed
holography [8], a novel imaging modality which is capable of recording and reconstructing a 3D
representation of an object by detecting the complex optical field originating from the object
instead of only the image intensity, and laid the foundation of optical holography. In fact digital
holography, pioneered by Goodman [9] which involves the use of a digital recording medium and
allows for numerical reconstruction of the object was an early demonstration of the use of
computational techniques for image formation. These examples serve to illustrate that
measurement schemes which involve innovation both in acquisition and signal-processing can
improve existing imaging modalities and potentially overcome limitations imposed by
conventional hardware. Indeed one may observe that the invention of laser and advances in
detector technology such as charge-coupled device (CCD) technology and the complementary
metal oxide semiconductor (CMOS) technology has revolutionized the field of optics by
providing both versatile optics sources and powerful electronic detectors for storage and access.
In this context one can say that the continuous accelerated advances and increasingly cheaper
access to computing resources for optical sensing is similarly having a transformational impact in
optics by enabling modeling of the optics between (and including) the source and detector and the
optical phenomena itself. With the increasing availability of computational resources it has now
3
become possible to abstract physical hardware by developing novel computational techniques
which not only model their functionality but also overcome their limitations. Moreover, access to
computational resources has driven the application of this research outside the laboratory, as
evidenced by familiar medical imaging innovation such as “computer aided tomography”.
Therefore there is clear motivation to explore novel measurement schemes and augment existing
optical sensing methods by leveraging computational techniques developed in other fields of
study. The presented work reflects this trend by focusing on several emerging areas of imaging,
including coherent anti-Stokes Raman holography, high-resolution non-linear spectroscopy, and
spatio-temporal imaging of ultrashort pulse.
Organization
This dissertation presents projects involving application of computational methods and
development of prototype optical measurement systems for overcoming exiting limitations in
several non-linear and ultrafast optical imaging and spectroscopy techniques.
In particular, chapter 2 introduces the computational techniques of compressive sensing
and two-dimensional phase retrieval. Compressive sensing is a novel signal processing paradigm
which seeks to minimize the number of measurements and reconstruct the signal by leveraging
sparsity inherent in the signal. 2D phase retrieval is an iterative algorithm used for obtaining the
phase of a complex-valued field from some intensity measurement involving the field. Both
computational techniques seek to improve key signal metrics such as higher signal to noise ratio
(SNR) and better resolution than can be obtained directly.
In chapter 3, we discuss the application of compressive sensing to imaging and
spectroscopy. In particular, we develop compressive coherent anti-Stokes Raman holographic
imaging to enhance the apparent optical sectioning capability of coherent Raman holography by
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minimizing out-of-focus background noise. We propose compressive frequency comb
spectroscopy as a novel technique for ultra-high resolution multi-heterodyne optical spectroscopy
and demonstrate its potential over broad spectral range using numerical simulations to study
acquisition of sparse line spectra.
Chapter 4 concerns with the application of 2D phase retrieval to optical imaging and
spectroscopy. We present original demonstration of 2D phase retrieval based technique for high-
resolution sum frequency generation vibrational spectroscopy (SFG-VS) by retrieving the
vibrational spectrum of molecules with significantly improved resolution than can be achieved in
conventional experimental setup. Lastly, as another application of 2D phase retrieval, we
demonstrate coupled spatio-temporal characterization (imaging) of ultrafast optical pulse using
‘holographic frequency resolved optical gating’ (HCFROG) in which the spectral hologram of the
conventional FROG trace is used to map the pulse in space-time with micrometer spatial and
femtosecond temporal resolution.
Finally chapter 5 summarizes the work presented in this thesis and postulates future work
in further leveraging computational techniques towards even more exciting optical imaging and
sensing technologies.
5
Chapter 2
Computational Methods in Optics
Introduction
In this chapter we will review the concept, theory and the application of two
computational methods, namely ‘compressive sensing’ (CS) and ‘two dimensional (2D) phase
retrieval’ in optical data analysis. Essentially both CS and 2D phase retrieval are inversion
techniques in which the signal of interest is retrieved from measurements based on a model of the
signal generation and acquisition system. CS is a theoretical framework in signal processing
paradigm and outlines an optimum sampling strategy by specifying the conditions under which a
signal of interest may be acquired such as to significantly minimize the number of measurements
required for its reconstruction. Phase retrieval is a sub-class of a general class of problems in
which only part of the information in the signal can be measured experimentally and yet the
missing information can be estimated from this partial knowledge by imposing the experimental
and physical constraints in a self-consistent manner using a suitable computational algorithm. In
particular, in phase retrieval an estimate of the missing phase of the object field can be obtained
from a measurement of its magnitude only. Thus arguably both CS and phase retrieval seek to
estimate a signal from an incomplete measurement set that is often times an anisomorphic
representation of the signal of interest.
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Compressive Sensing
Motivation
It is well known that the Nyquist-Shannon sampling theorem [10] prescribes that any
bandlimited signal can be reconstructed accurately if it is sampled at a rate at least twice as fast as
its largest frequency. By sampling at the data rate, the sampling theorem ensures that information
is not obscured or lost via aliasing which is an artifact of under-sampling. However, it is
commonly observed in practice that signals retain the information content even if they are under-
sampled. For example, the image recorded by a conventional digital camera of a typical outdoor
scene is sampled at the native pixel pitch of the detector and can be huge in terms of data
captured. However, image compression algorithms (e.g. JPEG, PNG, etc.) are practically always
used to reduce the size of digital images for subsequent storage, processing and transmission
despite being potentially lossy because they can retain the essential information using
significantly less data. This suggests that signals of interest can be sampled at the ‘information
rate’ rather than ‘data rate’, or in other words that the signal can be specified using fewer degrees
of freedom than implied by its bandwidth, as in the Nyquist sampling strategy. [11] The question
of efficient sampling of signals such that their reconstruction is accurate is especially important in
optical imaging and spectroscopy. For example, in the field of medical imaging and astronomical
imaging, images are often studied to identify interesting edge-like, point-like or similar localized
features scattered throughout the scene. Clearly the Nyquist sampling criteria is inefficient (and
even prohibitively costly) in such cases since such high contrast features always have very large
spatial bandwidth. Similarly the problem of recording 3D imagery (such as hyperspectral
imagery) can be highly data intensive and experimentally challenging because detectors
inherently capture only 2D data. Even in optical spectroscopy, the ideal requirements of wide
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scanning range, very fine spectral resolution, high SNR and minimum acquisition time pose
competing challenges to the design parameters of the device.
Description and Formulation
Compressive sensing is the flagship mathematical framework which definitively answers
the question of whether sampling and reconstruction strategies superior to conventional wisdom
exist. The central idea of compressive sensing is to exploit the sparse structure inherent in typical
signal of interest to design efficient sensing and compressing strategy such that an accurate, and
possibly exact recovery of signal is possible. CS leverages two basic principles: sparsity and
incoherence. Sparsity embodies the feature of compressibility of a signal and expresses the fact a
signal may be specified by its ‘information rate’ which can have much fewer degrees of freedom
than implied by its Fourier transform. [11] In other words, it is possible to find concise
representation i.e. a sparse transformation of the signal in which the signal can be expressed in
terms of only few non-zero (or significant) coefficients. Incoherence is the characteristic of a
measurement (encoding) system which allows efficient representation of the sparse signal in the
sense that the information embedded in the sparse signal can be condensed into few
measurements. Formally, we can express the acquisition process in terms of the linear model
y f where y is the set of measurements of signal f and is the system model. Let
denotes a sparse transformation of f into x : f x such that x has only S significant
components (we say x is S sparse) and suppose that a coherence measure can be defined as
1 ,( , ) ( )·max | , |jk
k j nn
. Then low coherence refers to the fact that the sensing matrix
is not correlated with the signal representation matrix . As a familiar example, the model of
conventional Nyquist-rate sampling is a special case of CS in which the Fourier basis, i.e. the set
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of complex exponential harmonics, which is used for sensing the signal is clearly uncorrelated
with the delta comb which is used to sample the signal (say in time or space). Thus it may be said
that incoherence embodies a more general form of the ‘time-frequency’ duality. [11] This insight
helps in understanding the particularly counter-intuitive aspect of CS which is that it prescribes
the use of random matrices, i.e. matrices which are entirely non-adaptive to the structure of the
signal to guarantee accurate reconstruction of the sparse signal from the significantly incomplete
measurement set. In fact, random matrices with independent identically distributed samples of a
Gaussian or Bernoulli ( binary values) random variable are known to exhibit very low
coherence with any sparsifying basis [12]. Given a low coherence pair of system matrix and
sparsifying basis , the problem of reconstructing x from the measurements y is stated as
0ˆ such that arg min || ||
x
xx x xy f A in which the zero norm
0|| #{ | }|| 0ii xx is simply the total number of non-zero elements in x and A is the overall
sensing matrix. However, it is well known that the 0 search problem is 1 mathematically and
computationally intractable, or ‘NP-hard’. [13], [14] Instead, researchers Daubechies et al., and
Candes, Tao, Romberg and Donoho [15]–[18] proposed a computationally simpler convex
constrained optimization problem of minimization, namely:
1ˆ such that arg min | (P1)| ||
x
x yx Ax
where 1 | ||| || xx to leverage the rich literature available on solving (P1). [19] In particular, it
has been established that the solution x is optimal in the sense that exact recovery is provable for
exactly sparse signals, if the number of measurements is at least
2( , )· ·logM c S N (0.1)
9
where N is the length of the sparse signal x . [17], [18] The best estimate of the original signal f
can then be easily constructed from the sparse coefficients of x using ˆ ˆf x . The condition on
minimum measurements M stated in terms of the coherence measure is effectively a property
required by the system matrix and hence A to enable stable reconstruction. It is well known
that the existence of the null space of under-constrained system matrix A , as is the case in CS
(M<N), allows infinitely many solutions to the linear problem y Ax , i.e. the solution is ill-
posed in general. However the apriori knowledge of sparsity (only S non-zero elements) in x
intuitively suggests that M NS measurements should allow us to overcome this inherent
ambiguity, if the sub-matrix A composed of the columns of A indexed by the support of the S
non-zero elements of x preserves the sparsity. This leads us to define ‘restricted isometry
property’ (RIP) of the system matrix. [17] It states that a matrix A satisfies RIP of order S if
there exists a small constant (0,1) such that for any S-sparse vector x
2
2
1|| ||
1|| ||
A x
x
(0.2)
In other words, RIP as defined in Eq. (0.2) requires that every sub-matrix A formed by choosing
any set of S columns of A be an approximately orthonormal matrix (preserves the length and
structure) of corresponding S-sparse vectors). Since CS assumes no prior knowledge other than
sparsity of the signal, any matrix which satisfies the RIP property describes a linear and non-
adaptive measurement of the signal and can be used for inversion. We now present a brief
discussion of the particular reconstruction technique we have used for our CS–based applications
presented in this dissertation.
10
Reconstruction using TwIST
We noted earlier that the CS based acquisition using the formulation based on
constrained 1 minimization (P1) simplifies a computationally difficult problem by making it
into a convex optimization problem. An important variant of (P1) is the combined 1 2
unconstrained optimization problem [20]
2
2 1
1ˆ || || || || (P2)
2arg min ( ) arg min
x x
x y Ax xh x
which is useful to account for noise in real world measurements via the 2 norm of the error. The
form (P2) is similar to the well-known regularized least squares except that here the convex
objective function h is penalized by the 1 norm regularizer whose penalty is controlled by the
regularization parameter . It is easy to illustrate the fact that the 1 norm penalty promotes the
search for the sparsest solution, which is the aim in CS based reconstruction. Figure 2-1 plots the
values of the 1 and 2 terms in (P2) for a general vector x as a function of the value of an
element ix in that vector. As ix increases, the value of the 2 term 2
2||·|| grows quadratically
whereas the value of the corresponding 1 norm 1||·|| grows linearly.
11
Figure 2-1: Illustration of why 1 norm promotes sparsity. 1 norm regularizer results in higher
penalty for smaller components of signal than its larger components.
This dependence of the 1 and 2 norm implies that the small component in signal are penalized
more than its large components by the 1 norm. In other words, the 1 regularizer favors
significant components in the signal and thus promotes sparsity in CS based signal reconstruction.
In fact, (P2) describes the general form of many problems which arise in signal processing in a
variety of applications such as signal denoising, deblurring, interpolation and compression [20]
apart from compressed sensing. As such, huge body of literature has been developed prior to the
discovery of CS theory to computationally solve the problems of the form (P2), notable amongst
which are the basis pursuit [17], [18], [21] and LASSO [22] techniques. One of the approaches is
the iterative shrinkage thresholding (IST) family of algorithms. Donoho et al. [23] pointed out
early on the connection between shrinkage and 1 norm penalty in context of superresolution of
nearly sparse astronomical image recovery. Donoho and Johnstone also demonstrated the idea of
‘thresholding’ as a suitable technique for reconstructing signals from noisy measurements in the
context of wavelet based image deconvolution. [24], [25] In recent years many researchers have
studied different aspects of iterative shrinkage thresholding algorithms. In essence, the IST
12
algorithm operates independently on each component of the sparse vector x by either shrinking
it or setting it zero, based on its value compared to the preset regularization parameter . In fact,
this shrinkage/thresholding operator perturbs the solution obtained by the standard gradient
descent method in every iteration. This nonlinear shrinkage thresholding operator )·( is given
by,
1 1(1 ) ( ) ( ( ))
( [ ]) ( [ ]) max{0, [ ] }; [ ]
( [ ]) max{0, }; [ ]
·
T
n n n n n
i i
x x x x A y Ax
x n sign x n x n x n
x n e r x n re
(0.3)
The flavor of IST that we have implemented in this work is a two-step algorithm called ‘TwIST’,
which gets its ‘two-step’ description from the fact that the next iteration 1nx depends on
previous two iterations nx and 1nx , as can be seen from first sub-equation in Eq. (0.3). [26] (We
refer the interested reader to Ref. [26] for the definition of parameters and and for
additional details.) In this dissertation, we use TwIST algorithm to demonstrate application of CS
for optical imaging and spectroscopy. In particular, we demonstrate significant reduction in noise
and hence apparently enhanced optical sectioning capability of coherent Raman holographic
imaging modality, and propose a novel compressive optical spectroscopy scheme based on the
use of frequency combs
13
Two Dimensional Phase Retrieval
Motivation
The problem of 2D phase retrieval belongs to the general class of problems encountered
in several areas of optics, namely the determination of a complex function from a limited
knowledge of its Fourier transform. Mathematically, if we consider an object ( )f q and its
Fourier transform ( )( ) ( ) iF R e u
u u which are related via the standard definition:
( ) ( )ex ·21
2p( )F f i d
u qu q q and ( ) ( )ex
1·p( 2 )
2f F i d
u qq u u , then we are
interested in the solution to the problem of obtaining the phase ( ) u from a measurement of
2| ( ) |R u subject to constraints on ( )f q based on its specific characteristics. Here q (and
hence u ) is a vector whose elements specify the multi-dimensional basis for the support of the
object. Thus in the typical phase retrieval problem in optics, the goal is to obtain the phase of a
complex object from a measurement of the squared modulus of its Fourier transform. Phase
retrieval problems are commonly found in several areas in optics. For example in x-ray
crystallography, information of the electron density function characterizing the crystal structure is
contained in both the amplitude and phase of the observed diffraction pattern, however only the
intensity of the diffraction pattern can be measured experimentally. The technique of wavefront
sensing for compensating optical aberrations in an imaging system relies on obtaining an estimate
of the complex optical transfer function from its measured point spread function which is related
via the Fourier transform. Indeed such a technique was successfully applied to design the
compensating optics of the famous Hubble Space Telescope [27], [28] and resulted in spectacular
improvement of its imaging capability. In these and several other problems in optics, the
measured signal is related to the magnitude of Fourier transform representation of the signal of
14
interest. The importance of the problem of phase retrieval lies in the fact that in general the
magnitude of Fourier transform of the signal alone is not sufficient to uniquely specify the
complex signal. For example if the unknown signal is filtered with an all-pass filter then the
measured signal has the same spectral amplitude but different spectral phase so that signal
recovery may become impossible in the absence of additional constraints on the measurement
model or signal itself. The importance of the knowledge of phase for complete determination of
the signal has been demonstrated by the work of Srinivasan [29], Oppenheim [30], and other
investigators. In fact these researchers have shown that a signal synthesized by using only its true
phase function in combination with some amplitude function (even if the amplitude function is
unrelated) can bear close resemblance to the actual signal.
Uniqueness
It is well known that the one dimensional (1D) phase retrieval problem has no unique
solution and it is in fact plagued by the existence of many nontrivial solutions [31]. On the
contrary, solution of the 2D phase retrieval problem in which the support of ( )f q is two
dimensional, for example the x and y coordinates of an image, is overwhelmingly likely to be
unique in practice aside from the trivial ambiguities of mirror image (i.e. inversion in origin) and
a constant shift. The important question of uniqueness of the 1D and 2D phase retrieval has been
investigated by several researchers; see for example [32], [33], [34]. The key argument which
explains the uniqueness criteria of the solution of the general phase retrieval problem hinges on
reducibility (or ‘factorability’) of a polynomial in one or more dimensions. While the complete
mathematical arguments discussing uniqueness are beyond the scope of this dissertation, we
intend to present only a distilled and simplified discussion to highlight how the ability to factorize
polynomials results in very different non-unique solutions to the 1D phase retrieval problem, but
15
enables practically unique solution to the 2D problem. The analytic continuation of the Fourier
transform of the signal of interest via the change of variable 'i u u u z is given by
( ) ( )ex ·21
2p( )F f i d
q zz q q . Under the assumption that ( )f q has finite support and is
of finite energy, it is known that ( )F z can be expressed as the Hadamard product
1(1 /( ) )kF
kz zz [35] where kz are complex zeros of the analytic continuation. This
implies that any 1D signal can be specified uniquely by the set of the complex zeros of its
analytic continuation in the complex plane. In many optics experiments the squared modulus of
the Fourier transform 2( ) | ( ) |I Fu u is the measurable and relates to the autocorrelation of the
signal via the Parseval’s theorem. The analytic continuation of ( )I u is given by
1( )() )( kI
*
k kz z z zz so that ( )I z is characterized not only by kz but also by the
set of complex conjugate zeros *
kz which are the zeros of * *( )F z . Now it can be shown that the
so-called ‘Blaschke factor’ ( ) / ( ) k
*
kz z z z allows synthesis of a different function
1( ) ( )[( ) / ( )]F F *
kkz z z z z z which has the same squared modulus 1( ) ( )I Iu u but which
is specified by *
kz , i.e. zeros that are ‘flipped’ across the real axis. In other words, only the
knowledge of the intensity measurement ( )I u does not allow unique determination of the
complex functions ( )F z or 1( )F z . Thus the reducibility of a 1D polynomial into N complex
zeros results in 12N
-fold ambiguity (not counting the trivial ambiguity resulting from flipping
all zeros that forms the set of mirror image functions) of whether the zero or its complex
conjugate informs the observable quantity ( )I u . This mathematical argument in Fourier domain
has a very familiar manifestation in the signal domain. We note that it is common knowledge that
two signals with significantly different distribution can have visually indistinguishable
16
autocorrelation shapes. [36] However, the story changes completely in the case of functions of
more than one variable. In fact, the fundamental theorem of algebra does not guarantee such
reducibility for nearly all polynomials of interest in two variables, which is the situation in the 2D
phase retrieval problem. Therefore the polynomial reducibility argument presented above does
not hold for the 2D case. This mathematical fact essentially allows the existence of unique
solutions for the 2D phase retrieval problem in practice, (except the trivial ones of inversion in
origin and constant shift noted above), and motivates investigation into techniques for finding the
unique solution.
Solution using Generalized Gerchberg-Saxton Algorithm
One early technique for phase retrieval proposed initially by Wolf [37] and developed
later by Mehta [38] involved use of exponential filters to effectively measure the modulus of
analytic continuation of its transform and use mathematical techniques to reconstruct its phase
from this measurement. Other techniques demonstrated include an interferometric measurement
of the correlation function with known reference field [39]–[41], a maximum-likelihood estimator
for the phase constrained by the measured modulus [42]–[44] and a host of analytical and
experimental techniques which are based on applying the constraints imposed by the analytic
properties of the correlation functions [see list in Ref 4 of [45]]. A notable numerical technique
was the Gerchberg-Saxton iterative algorithm (GS) [46], [47] for estimating phase from two
intensity constraints in the object and its Fourier domain. By far the most successful and
commonly used technique for solving for the estimate of the 2D phase retrieval problem is the
error reduction algorithm due to Fineup [48] who demonstrated “dramatic” efficacy of retrieved
objects by modifying and generalizing the constraints of the GS algorithm.
17
Figure 2-2: Generalized Gerchberg-Saxton algorithm: error reduction algorithm
It is depicted pictorially in Figure 2-2, where we have used two variables explicitly to emphasize
the 2D phase retrieval problem in particular. As can be seen, the generalized GS algorithm is used
to solve the 2D phase retrieval problem by iteratively transforming between the object and
Fourier domains while applying the constraints in each domain dependent on the model of
generation of object field (i.e. signal of interest) until the solution converges as determined by a
particular error metric. The error reduction algorithm by Fineup has been treated by the method of
generalized projections by Levi and Stark [49], [33]. In this approach, each constraint applied in
the object and the Fourier domain defines a set of permissible functions, and imposing the
constraint is equivalent to finding the projection of the solution on that set. In this sense, the error
reduction algorithm operates by projecting the current estimate on each constraint set sequentially
and iteratively to seek a point in that set closest to the estimate such that a certain error metric is
minimized. The generalized projection implementation is depicted pictorially in Figure 2-3 which
also shows the ‘seed’ for the iterative algorithm. The arrows depict the progression of iterative
algorithm and convergence is indicated by the fact that the solution moves progressively ‘closer’
to both constraint sets simultaneously.
18
Figure 2-3: Generalized projection method description of GS based error reduction algorithm
Thus solving the phase retrieval problem using the generalized projections method is akin
to finding a member of the intersection of all constraint sets. The method of generalized
projections allows at least one of the sets to be non-convex, as indicated in Figure 2-3
which is true in case of phase retrieval since the measured Fourier modulus imposes a
non-linear constraint. Convergence is reached when the point being sought lies in the
intersection of both constraint sets and the overall error is lowest.
In this dissertation, we will show that the problem of spatio-temporal
characterization of ultrafast laser pulse and the problem of high resolution of nonlinear
spectroscopy can be stated as 2D phase retrieval problems. Our solutions obtained by
applying the error reduction algorithm demonstrate enhancements to overcome
limitations with existing techniques in both optical imaging and spectroscopy in line with
the theme of this work.
19
Chapter 3
Applications of Compressive Sensing
Introduction
We overviewed the concept, theory and reconstruction techniques of compressive sensing
in chapter 2. It was seen that CS is a novel approach in signal processing at ‘information-rate’
with the key advantage of minimizing the number of measurements by leveraging sparsity
inherent in typical signals of interest. In this chapter we will demonstrate the impact of this
distinguishing advantage of CS in optical imaging and spectroscopy by considering two important
applications: label-free, non-scanning 3D microscopy, and very high resolution frequency comb
spectroscopy. We will lay out the importance of both applications, motivate enhancements to
overcome existing limitations and formulate the optical problem in the appropriate framework to
demonstrate the enhancements using CS based signal recovery.
Compressive off-axis coherent anti-Stokes Raman scattering holography
Introduction
Optical microscopy based on fluorescence imaging is highly mature and standard
imaging modality for observing biological samples and is an indispensable tool in a typical
scientific laboratory. It is well-known that fluorescence microscopy requires staining the sample
using specific fluorophores to enable discrimination between different chemical species in the
sample. This sample preparation step has the adverse effects of photo-bleaching in which the
fluorophores lose their ability to fluoresce, severely limiting the exposure time, and photo-
20
toxicity, hindering the long-term imaging of live specimens. Additionally traditional focal-plane
imaging techniques (such as conventional fluorescence microscopy) have poor axial resolution
due to the lack of optical sectioning and generally their image quality suffers from a relatively
strong unfocused background. Confocal microscopy [50]–[52] successfully eliminates out of
focus fluorescence signal and significantly improves axial resolution to enable 3D imaging. Two-
photon fluorescence is another novel imaging technique which not only suppresses the
background (due to it being a non-linear process) but also penetrates deeper and minimizes
scattering in the sample. However in these techniques the resulting 3D image is acquired by
scanning the sample in all three spatial dimensions or scanning the excitation beam. The principle
of holographic imaging invented by Gabor [8] allows single-shot multi-focal plane imaging
without the need for axial scanning. Holography is capable of 3D reconstruction of the source by
recording the amplitude and phase of the complex optical field and hence measuring its (limited)
angular spectrum. With the inception of digital holography [9], the ability to digitally record the
optical field has enabled digital algorithms based holographic reconstruction using the diffraction
theory (i.e., digital focusing). This suggests that if a suitable imaging technique which can
discriminate chemical species without staining biological samples were to exist, its combination
with holography can potentially enable a powerful and live 3D imaging capability. In fact, Raman
scattering is indeed such a phenomena whose optical spectrum is characteristic of the intrinsic
vibrational response of the molecular species. The Raman response of a molecule therefore
provides a characteristic vibrational spectral ‘fingerprint’ for detection. This spectral ‘fingerprint’
can be exploited as a chemically selective ‘label free’ contrast mechanism. In contrast to
spontaneous Raman emission, coherent Raman scattering features orders of magnitude improved
sensitivity since the molecular response is driven coherently by tuning a pair of coherent
excitation sources to excite the vibrational mode(s). In this work, we restrict our attention to
coherent anti-Stokes Raman scattering (CARS) to integrate with holography. CARS was
21
originally observed by Maker and Terhune as coherent, directional and blue-shifted radiation
which is free from one-photon induced fluorescence [53]. After the demonstration of scanning
CARS microscopy by Duncan [54], Zumbusch et al. [55] re-ignited the field by demonstration of
their 3D sectioning capable scanning CARS microscope using tightly focused beams. Digital
CARS holography [56]–[59] on the other hand uses holographic imaging principle to capture the
complex anti-Stokes field produced in a sample upon excitation with a pump/probe and a Stokes
beam, and therefore features the desired label-free, chemical selective, and scanning-free, single
shot 3D imaging capability. However traditional digital propagation based reconstruction using
CARS holograms results in images whose quality suffers due to presence of out-of-focus
background. It can be shown that since the angular spectrum of the source distribution is
effectively low-pass filtered (/band-limited) in holographic measurement, the problem of
retrieving the exact source distribution using a single hologram does not permit unique solution
[60]. Numerical inversion techniques using 1 norm as regularizer have been proposed for
holographic reconstruction [61]–[65] to assist in the solution of the under-constrained problem of
estimating 3D distribution from 2D image. In particular, D. Brady et al. [66] has recently
demonstrated that compressive sensing based algorithms can estimate a sparse volumetric sample
distribution from a single two-dimensional hologram thereby reducing the number of
measurements required as compared to diffraction tomography. Compressive holographic
reconstruction has also been applied to inline CARS holography [58]. Here we present
compressive off-axis CARS holography to enable noise-suppressed inference of volumetric
source distribution of (3) from measured complex anti-Stokes field. In particular, using
compressive CARS holography, we show the significant reduction of out-of-focus background
noise while estimating the sparse volumetric distribution from a single CARS hologram.
Complex valued numerical model of the holographic reconstruction method, including the
22
complex valued measurement as in our case, improves the performance of compressive sensing
based inversion algorithms by allowing the system matrix to be well-conditioned.
Formulation
Model of signal generation
The model of compressive CARS holography can be understood as the linearization of the
equation governing the formation of a CARS hologram from a given (3) ( , , )x y z source
distribution. The pump field pik r
p pE A e
and Stokes field sik r
s sE A e
excite molecular
vibrational modes in a sample of thickness L. The probe field is assumed to be the same as the
pump. The resultant anti-Stokes field is given by · ˆ, , asik z
s
z
sa aE A x y z e ( z is the unit vector of
ask ). The amplitude of the anti-Stokes field in the frequency domain is given by [59]:
2 2
( )( ) / ( 22 * (3)
0
)
( , , ) ( , , ) x asy
L
i L z ki kz
as p s
k k
A A du v L A e u v z ez
(0.4),
where the tilde represents the two dimensional Fourier transform with respect to x and y, and
ˆ(2 ·)p s as
kz k k k zz is the phase mismatch. Eq. (0.4) can also be written in the spatial domain,
i.e.,
2 2( )
(3)
0
( )( , , ) ( , , ) / ( )as
L x yi kz
as a
L z
s
i
A x y L x y z i Lze e zd
, where as is the anti-Stokes
wavelength. In other words, the output field is simply the summation of the diffracted anti-Stokes
field generated at each slice in the volume distribution. This observation motivates the application
of digital back-propagation for retrieving the unknown volume source distribution, provided that
23
the complex anti-Stokes field at the exit plane can be digitally captured. To accomplish this goal,
an off-axis digital CARS hologram can be recorded using a reference beam at the same frequency
and incident at a small angle with respect to the anti-Stokes field. The recorded hologram can
be written as2 *
2( ) ( ) ( )*2
e | |a as assik sin ik sin ik sin
as r as
x x x
r asr as rI A A A A A A e A A e
, where sin( )asik
r
xA e
denotes the reference field at the appropriate anti-Stokes frequency. The sideband of interest
( )* asik sin
as
x
rA A e can be digitally separated from the other three terms and re-centered (see Figure
3-1(d)) to obtain the desired carrier-free complex anti-Stokes field.
Model of inversion algorithm based on compressive sensing
We define (3) 2 * (3 . . .) (3)
( , ( ,, ( , )) , ) ,p s
I F T
A x y z u v zx y z A X , so that
2 2
(3)
0
(3) ( ) )( )2 (
,
( , , ) or,
, ,
( , ) ( , , )
( , ) ( ) as
L
as
un x vm y i kli L l vz i z u
as
l n m
A dzH u v z X
A X x m y
u v u v z
u v n eel ez
(0.5)
where 2 2
( )( )( , , ) as
i L z u vi kzH u v z e e
is the transfer function of the medium multiplied by the
phase mismatch factor and ‘I.F.T.’ refers to the two-dimensional inverse Fourier transform. Let
us denote by 0( , )[ ] ( , ,· · . ·]) .[LK u v H u v z F T the discretized sensing matrix operator, (which
includes the 2D Fourier transform and the diffraction transfer function) which operates upon each
slice of the volume distribution 0
(3) ( , , )x y z throughout the depth L and generates the recorded
complex CARS field ( , )asA u v . In this notation, Eq. (0.5) is essentially analogous to a linear
system of the form (3)
as KA . With a priori knowledge that the (3) distribution is sparse,
this under-constrained linear problem of estimating the 3D distribution from its 2D hologram can
24
be solved using the method of compressive sensing. In particular, we use the two-step iterative
shrinkage thresholding (TwIST) algorithm [26] to solve for the source 3D distribution by
minimizing the objective function 1
(3) (3) 2 (3)
2( ) 0.5 || || || ||asf A K where is the
regularization parameter. In light of the fact that the measurement is complex valued (because the
hologram records both amplitude and phase), the 1 norm constrained objective function can be
minimized using a complex thresholding operator to iteratively update the next estimate of (3)
using the TwIST algorithm (see remark 2.5 in [15]). The results of the volume distribution
retrieval using the single shot recorded CARS hologram are discussed in the next section.
Numerical Results and Discussion
CARS holograms were recorded using the experimental setup detailed in Ref.[59],
whereby a pulsed nanosecond laser supplying the pump/probe beam and a type II OPO was used
to generate the Stokes and reference beams. The pump/probe and Stokes beam are incident on a
sample consisting of multiple polystyrene spheres which are sandwiched between two glass slides
and immobilized using UV curable glue. When the Stokes wavelength is tuned to match one of
the molecular vibrational frequencies of polystyrene (Raman resonance of polystyrene at 3060
cm-1
) a strong CARS signal can be observed. This anti-Stokes signal was then combined with a
reference beam at the identical wavelength and propagating at a small off-axis relative angle of
approximately 1.7o and the resultant hologram was digitally recorded on a CCD camera as shown
in the schematic in Figure 3-1(a). The recorded hologram of multiple polystyrene microspheres
used in this study is shown in Figure 3-1(b). The two-dimensional Fourier transform shown in
Figure 3-1(c) reveals the expected sidebands for this off-axis recorded hologram, of which we
digitally retain only one sideband Figure 3-1(d). The inverse Fourier transform of the re-centered
25
sideband is the complex anti-Stokes field recorded in the hologram. Figure 3-1(e) shows the
amplitude of the complex diffracted CARS field at the detector plane. It shows the ring-like
features associated with the diffraction of the signal originating from each polystyrene sphere.
The complex field is used as the input for our algorithm to estimate the source 3D distribution.
Figure 3-1: Schematic of CARS holography (a) Schematic diagram of CARS holography (b)
Central part (256x256 pixels) of a recorded CARS hologram; (c) DC filtered two dimensional
Fourier transform (amplitude) of the hologram in (b); (d) A digitally filtered and re-centered
sideband; (e) Reconstructed CARS field (amplitude) at the recording plane.
Figure 3-2 shows typical images of the distribution of the microspheres in a series of ‘z’
slices, ranging from propagation distance of 5 μm to 14 μm, in steps of 1 μm. The out-of-
focus background noise is clearly seen affecting the image quality in each slice. We then
use the TwIST algorithm to iteratively minimize the 1 constrained objective function to
retrieve the image in each slice. The regularization parameter is chosen by trial and
error, i.e., by running the algorithm as varies from 510 to
410. A typical
reconstructed distribution of microspheres in each of the 10 ‘z’ slices is shown in Figure
3-3. We notice that the image in each slice is visually seen to have reduced noise and
sharper images.
26
Figure 3-2: Image reconstruction using conventional digital propagation. The nine polystyrene
microspheres are visible, but the image quality is affected by out-of-focus background
contributions.
Figure 3-3: Compressive sensing based image reconstruction using the TwIST algorithm. The
diffraction noise is suppressed and the spheres signal can clearly be seen to be localized at
different depth positions. Red arrow indicates spheres used for SNR analysis in Fig. 4.
27
To help better appreciate the improvement of image quality and removal of out-of-focus noise, in
Figure 3-4 (a-b) we plot the signal-to-noise ratio (SNR) for two spheres (indicated by arrows in
Figure 3-3) as function of the z depth position. The ‘signal’ is computed as the mean value of a
small 5x5 pixel region around the sphere center, and ‘noise’ is computed as mean value of the
background in the immediate neighborhood of the sphere, within a 101x101 pixel region. The
dotted curves are Gaussian fit to the SNR. It is clearly seen that as compared to the conventional
digital propagation algorithm, our compressive CARS holography technique improves the
localization of the spheres along z, because it minimizes the out-of-focus background noise. This
is further corroborated by Figure 3-4(c-d), which shows an axial slice of the reconstruction of the
lower sphere. The apparent improved “sectioning” occurs because the 1 constrained inversion
favors the strong signal in the image plane and shrinks the weak signal (noise) based on an
empirically chosen threshold constant. Since compressive sensing algorithms search for the
sparsest image, the apparent optical “sectioning” is ultimately limited by the inherent sparsity of
the sample.
Figure 3-4: Axial localization using compressive CARS holography. (a) and (b) show SNR for
two spheres (indicated by red arrows in Fig. 3 along the z depth position using conventional
digital back-propagation algorithm and compressive reconstruction, respectively.
28
Conclusion
We have demonstrated the efficacy of using compressive sensing for 3D label-free
chemically selective volumetric estimation through off-axis compressive CARS holography.
Although off-axis CARS hologram helps remove the twin image noise associated with inline
holograms, the traditional digital propagation based retrieval still suffers from out-of-focus
background noise due to the limited axial resolution, which is inherent in the fact that the
hologram only samples a very small subset in the ‘k-space’ describing the volume distribution of
the sample. It is well known that diffraction tomography can help improve axial resolution
through sampling a larger subset of the object ‘k-space’. If the volumetric distribution of interest
has sparse features, compressive off-axis CARS holography can emulate the higher axial
resolution using only a single shot CARS hologram compared to multiple field measurements
and/or rotation of the sample required for diffraction tomography.
29
Compressive Frequency Comb Spectroscopy
Introduction and Motivation
High resolution optical spectroscopy is a vital tool for research in fundamental science as
well as for enabling technological applications. The pioneering work of Fraunhofer in 1814 and
later Kirchoff and Bunsen in 1859 (among other luminaries) in showing that each atom and
molecule can be identified by its characteristic spectrum established optical spectroscopy as a
preeminent scientific tool for optical interrogation of matter. For centuries, optical spectroscopy
has thus been pivotal in enabling astrophysicists to probe the chemical make-up and evolution of
our vast universe. A state of the art spectrometer featuring resolution of 150 kHz at 500 THz can
facilitate Doppler shift measurements (~ 101/ ~ 0ev c ) to enable detection of candidate Earth-
sized planets [67]. Indeed, a very high resolution spectrometer can make it possible to observe the
expansion of the universe in real-time by tracking the galactic Doppler shifts. In physics, the
capability to measure frequencies with extremely high resolution and Hz-level accuracy allows
for measurement and tracking of slow drifts in fundamental physical constants. For example, the
Hz-level measurement [68] of the 2466.061 Thz 1S-2S transition in atomic hydrogen with a line
width of a 1.3 Hz can allow determining Rydberg constant. The significance of the ability to
precisely measure the Rydberg constant is evident in the fact that it helps define bounds on the
values of other physical constants [69]. The capability to measure optical spectra with high
precision and stability also has important technological applications. For example, identification
of drugs [70] and biological species [71] has implications in medical, health and security industry.
Similarly, molecular fingerprinting of gas molecules [72] and characterization of optical devices
(such as high-Q micro-cavities [73]) can lead to design of optical sensors with unprecedented
sensitivity with applications such as non-invasive sensing of oil and gas.
30
Compressive Frequency Comb Spectroscopy strategy
A desirable spectrometer can be described as having wide spectral coverage, high
resolution throughout its spectral range, fast acquisition, high throughput (large signal-to-noise
(SNR)) and high sensitivity. The operating principle of most conventional optical spectrometers
falls in either of two categories: dispersive or interferometric. Dispersive spectrometers typically
use a grating to disperse incident light and the resolution is dependent on the groove density, size
of grating, and the size of entry and exit slits. Fabry-Perot interferometer can achieve higher
spectral resolution but typically have a limited free spectral range. Fourier transform spectroscopy
(FTIR) is the predominant method of measuring optical spectra in the IR wavelength range whose
operating principle is based on the Wiener-Khinchin theorem. In addition to high throughput,
FTIR spectrometers feature higher resolution and allow multiplex measurement of optical
spectrum, but they employ an optical delay line which requires scanning the reference path. These
conventional spectrometer designs can provide wide coverage over UV, visible and infrared, but
are typically limited to a resolution of about 0.01 nm in the visible (~=10 GHz), which can be
improved if used in conjunction with a virtually imaged phased array [74].
The advent of octave spanning frequency comb using a mode-locked laser [75], [76], [77]
has ushered in a new era of ultra-high resolution optical spectroscopy with orders of magnitude
improved resolution potentially limited only by the Hz-level linewidth of the comb mode. The
principle of an optical frequency comb was described by Hansch et al. in 1977 [78] who showed
that the evenly spaced train of narrow pulses from a stabilized mode-locked laser corresponds to a
comb in the frequency domain. The defining feature of the frequency comb is the highly stable
and strictly periodic arrangement of its modes over a large optical bandwidth thereby providing a
reliable phase-coherent link between the RF and optical domains, not available hitherto. The
frequency comb has been utilized to calibrate astronomical spectrometers beyond state-of-the-art
31
accuracy of 6 MHz (~9m/s Doppler shift) [79]. The frequency comb lends itself naturally to a
massively parallel heterodyne scheme using its modes as the local oscillators (LO).
Optical spectroscopy using the heterodyne technique has been demonstrated [80]–[85] by
mixing a reference frequency comb with a dedicated signal comb which has a slightly different
repetition rate. Continuous frequency coverage may be achieved at the expense of the acquisition
speed by actively manipulating the repetition rate of one of the combs [86], or by scanning the
offset frequency. However we note that optical spectra of interest sometimes consist of only few
characteristic narrowband lines which may be widely separated, for example corresponding to the
energy levels of an inhomogeneous (multi-species) sample. In other words optical spectra are
often ‘sparse’ signals. Efficient measurement of such sparse signals is the focus of large body of
literature in information theory. Recently, compressive sensing is being extensively developed as
general signal processing paradigm for efficient acquisition of sparse signals such as above
mentioned optical spectra, in order to minimize the number of measurements and accurately
reconstruct the signal. By exploiting the unique characteristic of optical frequency comb as a
source of precise and stable optical LO, with the simple yet powerful signal acquisition and
reconstruction strategies based on compressive sensing, here we present ‘compressive multi-
heterodyne optical spectroscopy (CMOS)’ which is a novel technique for ultra-high resolution
optical spectroscopy. We propose multiplexed acquisition of optical spectrum by combining it
with a dynamically encoded frequency comb to map the multi-THz optical band to few GHz-wide
RF baseband. We leverage the sparsity exhibited by optical spectra of interest by compressive
sensing based acquisition strategy [17], [18] to minimize the number of RF measurements
required to reconstruct the optical spectrum with high fidelity. The spectral resolution achievable
by our method is not dependent on the mode spacing of the comb, but rather determined by the
RF instrument used for spectrum measurement. The accuracy of our technique is potentially
32
limited only by the Hz-level uncertainty [87] of the frequency comb, since the accuracy of
heterodyne detection is ultimately determined by the stability of the LO.
Formulation
Our formulation for acquisition of the optical spectrum involves modeling the heterodyne
mixing of the frequency comb and the signal. We begin by analyzing the signal generated in a
photo-detector by combining a real optical signal ( )s t with a real frequency comb ( )c t whose
repetition rate is rf ( 2r rf ). The photo-detector output is proportional to the interference
term in the time averaged intensity ( )· of the incident field
2 2 2 2( ( ) ) ( ( ) ( )) 2 ( ) ( ) ( ) ( ) .e t s t c t s t c t s t c t By using a balanced detection
scheme [88] to retain only the interference term, we can analyze the output m(t) as,
*
* *
0 0
( ) ( )
1( ) ( ) . .
( )
2
n nj t j t
n n
n
j t j t
n n n n
n
s t c e c e
S c e d S c e d c
m t
c
(0.6)
where ( ) F.T.[ ( )]S s t is the Fourier transform of ( )s t and nc is the complex Fourier series
coefficient corresponding to the thn comb mode. It is assumed that the detector is fast enough to
measure only the difference frequencies | | .n Equation (0.6) shows that the spectrum of
the RF signal includes contributions from the ‘image’ pair n as well as .n A
low pass filter (LPF) with a cutoff frequency of / 2LPF rf can be used to define the
spectral coverage of the RF spectrum F.T.[ (( )]) mM t (see Figure 3-5(a)). The different
frequency variables are indicated on the frequency axis shown in Figure 3-5(b). Then the RF
spectrum is
33
* *( ) ( )) ;( n n n n LPF
n
M S c S c (0.7)
By mapping each spectral sub-band 1,( )n n to the same RF window, Eq. (0.7)represents a
particular multiplex measurement ( )M of the entire optical spectrum. Multiple such
measurements may be recorded by dynamically encoding the weight of the comb modes; and the
optical spectrum can be reconstructed by solving the resulting system of linear equations. To do
this, as shown in Figure 3-5(a), the frequency comb can be filtered by using a pulse shaper
consisting of a programmable spatial light modulator (SLM) [89] to digitally control the
reflection or transmission coefficient of each SLM pixel. The encoded comb can then be mixed
with the signal in the balanced detector. The balanced detector removes the d.c. term from the
beat signal and also helps to increase the SNR by common-mode rejection of noise. The RF
interference signal is finally filtered through the LPF and measured by using either a digitizer and
Fourier transform block (D-FFT) (complex amplitude spectrum) or a spectrum analyzer (SA)
(power spectrum).
34
Figure 3-5: (a) Illustrative schematic for CMOS. The frequency comb is first filtered by
weighting its comb modes with a digitally controlled SLM and then combined with the signal
from the sample. The balanced detector measures the RF beat signal between the reference comb
and signal. The RF signal is further filtered by a Low Pass Filter (LPF) and detected by a
spectrum analyzer or a D-FFT block. The broadband quadrature phase-shifter is used to generate
a / 2 phase shifted copy of the comb for the coherent case. (b) Frequency axis indicating
relative value of frequency variables. D is the characteristic response time of the fast detector.
(*diagram is not to scale)
35
Heterodyne measurement of coherent optical signal
The RF spectrum )(M contains contributions from both spectral components
)( nS and )( nS via the thn mode nc . For an ideal coherent optical signal, we can
devise a scheme [90] to resolve the ‘image’ components by additionally mixing it with a / 2
phase-shifted comb, ( ).jc t This scheme may be implemented by sequentially selecting the two
orthogonal linearly polarized components from a circularly polarized comb (prepared by using a
broadband quarter-wave plate, e.g., Fresnel rhomb), and projecting them to the same linear
polarization state by using a 45o-tilted linear polarizer. Figure 3-5 schematically shows the
broadband quadrature phase-shifter to choose between the two copies of the comb. The resulting
model describing the RF spectrum of coherent optical spectrum can be simplified as
*( ) ( ) ;c n n LPF
n
S cM (0.8)
Heterodyne Measurement of incoherent optical signal
In the case of incoherent optical signal, for stationary, ergodic random processes the
spectral density is well defined and can be measured. Mixing of mutually incoherent signals has
been used, for instance, in the determination of laser line-width by measuring the power spectrum
of the beating signal resulted from mixing the laser field with its significantly time-delayed hence
incoherent copy [91]. Note that *( ) ( ) ( ) ( ),S S P where ( )P is the spectral
density of the incoherent signal [see sec 2.4 in [92]]. The model for spectral density of the
measured RF signal is therefore given by:
2( ) | | ( ( ) ( ));P n n n LPF
n
M c P P (0.9)
36
Spectrum Estimation Using Compressive Sensing
Our measurement strategy involves the superposition of a very large number of optical
bands (up to 4~ 10 ) weighted by the encoded comb modes. We noted in chapter 2 that one of the
characteristics of a good system sampling matrix is that it should be maximally incoherent with
the signal representation basis. In our case the signal is the sparse spectrum, so that the signal
representation basis is the identity matrix. Therefore a random matrix appears to be good choice
for the system sampling matrix. We propose to encode the frequency comb with a binary random
matrix consisting of independent identically distributed samples of a Gaussian random variable
which is also known to satisfy the RIP condition [12]. Below we describe how the system model
can be modified to account for the selectively weighted frequency comb (which may be
implemented for example by using a programmable SLM. Then a sparse optical spectrum may be
retrieved from several multiplexed measurements of the form given by Eq. (0.8) or Eq. (0.9).
Coherent spectrum estimation
To represent the dynamically weighted modes of frequency comb as described before, the
coefficient of the mode *
nc should be replaced by *( ) ( ) * ,k k
n n nf w c where ( )k
nw are the
programmable weights during the thk measurement. Accordingly, we can now define the system
matrix as ( )
,
k
k n nf and re-write Eq. (0.8) in matrix form as
[ 1] [ ] [ 1]c K K N N LPFM S (0.10)
A digitizer followed by FFT block can be used to record the complex spectrum ( ).cM
37
Incoherent spectrum estimation
In this case, we define ( ) ( ) * 2=|w |k k
n n ncf . Since ,LPF r each spectral component
0 1 0( ; )n nP in Eq. (0.9) contributes to only two beat notes
0 1 0( ), (( ,))P n P nMM with the unique pair of weights ( ) ( )
1 ,( ).k k
n nf f (Note
that .r ) To solve for the family of spectral components associated with 0( ),P we can
solve the combined system of equations of the form given by Eq. (0.11)corresponding to the K
measurements at both the complimentary beat notes. Hence the incoherent measurement can be
written in matrix form as,
[2 2 ] [2 1]
[2 1]
( )
( )
P P
K N N
P K
MP
M
(0.11)
Here, the unknown vector P and the matrix P
are explicitly of the form,
1 1[ ( ), ( )... ( ), ( ')' ]N N
TP P P P P and
(1) (1) (1) (1) (1) (1)
1 2 2 1
( ) ( ) ( ) ( ) ( ) ( )
1 2 2 1
(1) (1) (1) (1) (1) (1)
2 1 3 1 1
( ) ( ) ( ) ( ) ( ) ( )
2 1 3 1 1
N N N
K K K K K K
P N N N
N N N
K K K K K K
N N N
f f f f f f
f f f f f f
f f f f f f
f f f f f f
(0.12)
Numerical Results
We validate our formulation of the problem of spectrum retrieval using CMOS by using
Two Step Iterative Shrinkage Thresholding, “TwIST” algorithm [26] as described in chapter 2.
We have simulated hypothetical optical spectrum with bandwidth of 42.66nm at 800 nm, or
38
equivalently 20 THz optical band at cf = 375 THz with a sampling interval of 100 MHz
(0.213pm). We have assumed a hypothetical uniform frequency comb ( 1nc n ) with a
repetition rate of rf = 50 GHz. We note the recent demonstration of 25 GHz laser frequency
combs generated from a conventional Ti:Sapphire mode-locked laser using Fabry-Perot cavity
filtering scheme [93]. Thus the simulated frequency comb has 400N modes. Lastly, we have
assumed an encoding scheme for the frequency comb by defining a random binary matrix based
on the Gaussian ensemble. For example, the matrix entries may be considered to be reflection
coefficients nw of an SLM which can be ideally set to either 0 (no reflection) or 1 (maximum
reflection).
In order to evaluate the performance of TwIST applied to our problem, we use the ‘phase
diagram’ for 1 2 minimization [94]. Briefly speaking, the phase diagram plots the normalized
sparsity /p K of a p-sparse signal against a given system indeterminacy / ,K N where
K is the number of measurements. Accordingly, given a K N system matrix which describes
the K (<<N) measurements of given N length sparse signal, the phase diagram estimates the
number of significant components of the sparse signal that can be recovered using an algorithm
that solves the 1 2 optimization problem. To adapt the definition of normalized sparsity
obtained from the phase diagram to our case, we assume a fictitious threshold th to discriminate
between weak and significant components in the spectrum. The value of the threshold th is set
such that the number of significant spectral components equals that obtained from the phase
diagram. On the other hand, we quantify the error ( )e in the estimated spectrum, due to the
inversion algorithm in terms of its root mean square [RMS] value
1/2
21( )
L
i
i
eL
and
39
compare this noise level with the fictitious threshold th . One can see that a good comparison
between these two values indicates that our CMOS technique is capable of reconstructing the
sparse optical spectrum accurately as guaranteed in theory. Below we discuss the numerical
results for the specific case of hypothetical coherent and incoherent spectrum estimation for three
examples: K = 100, 150 and 200. These examples correspond respectively to 25%, 37.5% and
50% reduction (indeterminacy) compared to total number of spectral components in the
hypothetical spectrum.
Coherent spectrum estimation
We simulate the hypothetical spectrum of a sample whose dielectric function is based on the
Lorentz model 2
1,22
0
) 1( 1 ,;p
i i
r r
i
i
an
j
where21 131.78 ,10p s
10 20370 THz; 380 THzf f and 12 1 12 1
1 20.13 ; 0.410 1 .0s s In this case the
light source (e.g. another frequency comb) is assumed to be coherent and its phase is pre-
calibrated. The sample thickness is assumed to be l = 0.5 mm. We digitally subtract the simulated
measurement of the spectrum of the ambient medium (without the sample) from that in the
presence of the sample. Thus the complex sparse spectrum of interest to be retrieved using TwIST
is ( )
( )rj l j l
c cTs e e
and the corresponding error is ˆ| ( ) | |( () ) | .T Ts se Typically,
the optimum value of regularization parameter in (P2) is chosen to be that which yields
smallest error found by running several numerical experiments. We found that the 2 norm of the
inversion error in the amplitude ˆ ( )Ts of the estimated spectrum is lowest for 0.008 . Figure
3-6 plots both the simulated and the estimated absorption spectrum and the index n for the
40
cases K=100 and K=200 using 0.008 . Visual inspection of the absorption plots shows
reduced inversion noise for the case K=200 as compared to the case K=100. We also note the
FWHM linewidths of the two simulated Lorentzian lines (see insets in Figure 3-6) to highlight the
fact that narrow spectral features favor sparsity and aid CS-based inversion.
Figure 3-6: Estimation of hypothetical coherent spectrum using CMOS-TwIST: Comparison of
(a) absorption spectrum and (b) refractive index ( n ) for K=100; (c) absorption spectrum and
(d) refractive index ( )n for K=200. The reduction in inversion noise as K is increased can be
seen by visual inspection.
Table I summarizes the performance of CMOS-TwIST applied to Eq. (0.10) for our three test
cases: K = 100, K = 150 and K = 200 by comparing the numerical error in the reconstructed
signal with the fictitious threshold obtained from theory as described earlier. We observe that in
all three cases the noise level in reconstructed spectrum is lower than the threshold. This implies
that for even for the case of K=100 (which corresponds to only 25% measurements) all the
41
significant components that define the spectrum can be retrieved accurately. Clearly the noise
decreases as the number of measurements increase to K=200.
Table 3-1: Coherent Spectrum Estimation: Comparision between theoretical and numerical results
of TwIST applied to Eq. (0.10)
K = 150 K = 200 K = 250
thσ (theory) 2.445e-3 1.575e-3 1.085e-3
RMS error σ(simulation)
1.45e-3 1.23e-3 1.084e-3.
Incoherent spectrum estimation
In the incoherent case, we consider a similar hypothetical sample ( )
2( ) | |r
cj l
s e
with Lorentzian permittivity r as before, where the sample thickness is again l 0.5 mm. The
sample is assumed to be illuminated by an incoherent source. We digitally subtract the
background before using TwIST to estimate the power spectrum, based on Eq. (7). Thus, in this
case, the spectrum to be estimated using TwIST is ( ) 1 ( ).Ts s In this case, we observed
that value of regularization parameter 43 10 yields optimum de-noising performance.
Figure 3-7 compares the absorption spectrum retrieved using CMOS-TwIST for the two cases: K
= 100 and K = 200 with the simulated absorption spectrum. It can be seen that the two spectra are
essentially indistinguishable from each other.
42
Figure 3-7: Estimation of hypothetical incoherent spectrum using CMOS-TwIST: Comparison of
absorption spectrum for (a) K=100 and (b) K=200 measurements. Inset highlights FWHM
linewidth of simulated spectrum.
To complete the summary, Table II compares the rms error in the numerically retrieved spectrum
by applying TwIST to Eq. (0.11) with the threshold estimated from the phase diagram.
Table 3-2: Incoherent Spectrum Estimation: Comparison between theoretical and numerical
results of TwIST applied to Eq. (0.11)
K = 150 K = 200 K = 250
thσ (theory) 3.2e-3 1.325e-3 6.35e-4
RMS error σ(simulation)
7.87e-4 6.944e-4 5.937e-4
In this case, we again observe that the error from CMOS-TwIST is consistently small enough to
retrieve all the significant spectral components even with as few as 25% measurements. We point
out that TwIST has significantly improved performance when applied to Eq. (0.10) which solves
for a real vector than to Eq. (0.11) which solves for an unknown complex vector. We postulate
that the reason for this different performance is that the objective function in (P2) does not
express any explicit constraints on the phase as it does on the amplitude of the signal, although
this has been investigated before. [95], [96]. In our simulations we have used an all zero signal as
the initial seed required for TwIST algorithm. However we observed that the error in the
estimated spectrum is not sensitive to the choice of the seed as was reported earlier. [26]
43
Conclusion
We have presented a general scheme for high resolution multi-heterodyne optical
spectroscopy and its implementation based on the theory of compressive sensing. Our numerical
simulations present a strong case for leveraging sparsity in optical spectra of interest in ultra-high
resolution optical spectroscopy, which can reduce the number of measurements traditionally
required under the Shannon-Nyquist sampling scenario. We find that as few as 25%
measurements (K/N=0.25) and potentially fewer are enough to retrieve the optical spectrum of
incoherent signal from its multiplexed RF measurement. For the case of complex spectrum, the
noise due to digital inversion is higher; however the complex spectrum can still be retrieved in
both amplitude and phase as predicted by the phase diagram. In summary, our compressive
frequency comb optical spectroscopy technique presents a novel and viable pathfinder scheme to
exploit the exciting potential of recent advances in both the experimental generation of optical
frequency comb and numerical techniques for efficient signal acquisition and processing based on
compressive sensing.
44
Chapter 4
Applications of Two Dimensional Phase Retrieval
Introduction
The problem of retrieval of phase of the signal from its measured Fourier transform
modulus is commonly encountered in many optical applications as was discussed in chapter 2.
We saw that the special case of two dimensional phase retrieval (2D PR) overwhelmingly
possesses a unique solution in practical applications of interest and that the generalized GS
algorithm due to Fineup can converge iteratively towards that solution. This chapter focuses on
the applications of 2D PR to nonlinear optical spectroscopy and characterization of ultrafast
optical pulse. We will show that 2D PR enables high resolution retrieval of the 2nd
order optical
susceptibility spectrum using sum frequency generation vibrational spectroscopy than can be
obtained using conventional experimental scheme. For the next application we will show that the
technique of frequency resolved optical gating which is based on 2D PR can be empowered to
enable spatio-temporal characterization of femtosecond laser pulse.
Computational sum frequency generation vibrational spectroscopy
Introduction and Motivation
Investigation of interface between different media which may be of same or different
phase such as gas/liquid or liquid/liquid is of significant scientific and technological interest. It
provides key insights such as structure of chemical bonds formed at the interface, molecular
orientation, inter-molecular forces and molecular dynamics such as vibrational modes of
45
molecular motion [97]. Such molecular properties are unique to the specific interface and differ
from the properties found in corresponding bulk media. This is because they result from the
asymmetry of environment of the molecule at the interface, whose thickness is typically limited
to a few molecular layers. Commonly used techniques for surface studies such as low energy
electron diffraction provide access to the surface crystallography. Other techniques such as Auger
electron spectroscopy and x-ray photoemission spectroscopy allow chemical analysis of the
surface. However these techniques require the sample surface to be probed under ultra-high
vacuum conditions which can potentially damage the surface especially if it is not solid.
Additionally, these electron scattering and reflection based techniques cannot probe an interface
buried within dense media. To overcome these limitations, Shen, et al. [98]–[100] pioneered the
studies of interfaces via the non-linear optical phenomenon of sum frequency generation (SFG).
SFG phenomenon exhibits intrinsic interface selectivity and monolayer sensitivity because the
underlying mechanism of generation of the sum frequency signal forbids 2nd
order optical
interaction in centro-symmetric or random media (such as within bulk liquid, amorphous solid, or
crystals with inversion symmetry) under the dipole approximation. Over the decades, SFG
spectroscopy has been established as the ideal non-invasive optical sensing technique of choice
for probing variety of surfaces and interfaces owing to its several attractive features. In fact, SFG
vibrational spectroscopy (SFG-VS) provides label-free chemical selectivity via spectroscopic
detection of vibrational resonance enhanced SFG signal and enables qualitative and quantitative
analysis of molecular species at the interface. Since it is a laser-excited coherent process, the SFG
signal is highly directional and sub-picosecond pulses can be used to monitor fast vibrational
dynamics and surface reactions in situ. Use of lasers excitation further endows the SFG signal
with inherently high spatial and spectral resolution allowing in situ mapping of molecular
arrangement and chemical composition of the surface. Polarization analysis of the SFG signal
reveals specific information of the molecular orientation and conformation to the surface as has
46
been investigated in thorough detail by Hirose and co [101], Shen et al. [102] and later by Wang
et al. [103]. These attractive features have fueled the rapid adoption of SFG-VS technique for
studying a wide variety of surfaces and interfaces, particularly for aqueous solution interfaces
[104]–[108]; for polymer surfaces and interfaces [109]–[113]; for solid interfaces [114], [115];
solid gas interfaces [116] and solid liquid interfaces [117]–[120]. We point the interested reader
to the multitude of excellent and comprehensive reviews of SFG-VS, notable amongst which are
[121], [97], [107], [122] and [123] and references therein.
Existing approaches
SFG-VS experiments have been traditionally performed using two main schemes
depending on the pulse lengths of the excitation lasers. One scheme utilizes nanosecond (ns) or
pico-second (ps) tunable scanning infra-red (IR) laser and fixed wavelength near IR (NIR) lasers
because of the fine spectral resolution achievable using relatively long pulses (~1 cm-1
for ns
pulses and ~5-20 cm-1
for ps pulses). In this scheme, the IR source is scanned across wavelength
region of interest and the resultant SFG signal is recorded using spectrometer. Broadband SFG-
VS (BB-SFG-VS) has been developed as a scanning-free technique [124], [125] to overcome
experimental stability issues such as laser fluctuation and drift during the time-consuming
scanning procedure. In BB-SFG-VS an 800 nm femtosecond (fs) source laser is used to generate
an ultrafast broadband (~100 fs) mid-IR pulse and a short (~7 ps) NIR pulse (~2-3 ps NIR pulses
are also used commonly). The principle of BB-SFG-VS can be understood by observing that the
spectrally broad IR pulse simultaneously excites many vibrational modes coherently in the
molecular ensemble. The ps NIR pulse is long enough to probe the evolution of these excited
vibrational modes which happens on a much slower time scale than the almost instantaneous non-
resonant electronic background. Although this technique avoids scanning it suffers from poor
47
spectral resolution, typically ~10 cm-1
or more due to the spectrally wider NIR pulse and efforts
have been made to improve the resolution for example by pulse shaping the mid-IR pulse [126]
and better modeling the spectral broadening. [127] Fourier transform SFG is a promising
technique to achieve higher spectral resolution limited only by the optical path difference
between the two interferometer arms and not dependent on the pulse widths [128], however it has
been shown to have limited signal-to-noise performance [129]. Very recently Wang and co-
workers demonstrated high resolution ‘HR-BB-SFG-VS’ technique [130], in which a temporally
much longer (~90 ps) NIR pulse is obtained independently from a separate laser which is
synchronized with the ultrafast IR to achieve sub-wavenumber spectral resolution. This technique
requires two lasers working in tandem and is experimentally more involved than the setup of the
conventional BB-SFG-VS.
Computational high resolution SFG spectroscopy
Here we introduce a novel computational technique for achieving high resolution SFG
spectroscopy by showing that the problem of retrieving the high resolution SFG spectrum (and
hence the (2) spectrum) from its SFG spectrogram is equivalent to the mathematical problem of
estimating a complex function from a two dimensional mapping of the intensity of its Fourier
transform. Our computational approach for high resolution SFG spectroscopy is based on the
iterative two dimensional phase retrieval technique and can have spectral resolution potentially
comparable with that obtained in HR-BB-SFG-VS, using the experimental setup of conventional
BB-SFG-VS. In our technique, the SFG spectrum of vapor/DMSO interface is measured by
combining ~100 fs ultrafast mid-IR pulse with a (~2 ps) NIR pulses (800 nm) with a relative time
delay, and a two dimensional SFG spectrogram is recorded by varying the delay between them.
We illustrate our computational approach by proof-of-concept numerical simulation to
48
demonstrate retrieval of a hypothetical (2) spectrum from its SFG spectrogram. We then apply
our technique to demonstrate high resolution retrieval of vapor/DMSO spectrum using the
experimentally recorded spectrogram.
Formulation of 2D PR based SFG spectroscopy
Sum frequency generation is a 2nd
order non-linear optical process in which two lasers,
typically a NIR pulse ( )v vE and an IR pulse ( )IR IRE interact coherently to generate a signal at
the sum frequency v IR . The source of 2nd
order nonlinearity that drives this process is
attributed to the breakdown of inversion symmetry and is manifested in the presence of strong 2nd
order nonlinear optical susceptibility(2) , as is the situation in the case of molecules at interface
of two different media possessing inversion symmetry. In this process, the IR laser induces a
transient polarization in the molecules at the interface and a (non-resonant) NIR laser ‘probes’ the
IR-induced molecular vibrational mode(s) via the sum frequency generation process. The
resultant vibrational-resonance enhanced SFG signal can be detected spectroscopically to reveal
the characteristic vibrational response of the ensemble of molecules at the interface.
We can model the generation of SFG-VS using an equivalent non-linear polarization
source (2) ( ; ) ( )( ) ( )NL IR v IR IR v vP E E . Assuming that the IR signal is spectrally
broad such that it can induce multiplexed excitation of multiple vibrational modes, we can
assume ( ) ~ constantIRE over the wavelength region of interest. (We assume here that the IR
pulse is transform limited, and note that in principle this assumption is not essential and can be
relaxed for the analysis.) The total NLP can be accounted for by summing the contribution of all
frequencies in the broadband IR pump as (2) ( )) ) ((NL v v v vP E d . In our technique,
49
a relatively delayed NIR pulse ( ) I.F.T [ ( ) ( )]v v vvt E exp iE is overlapped with the IR
pump pulse and the resulting SFG signal is detected spectroscopically at varying delays. Here
I.F.T. denotes the inverse Fourier transform and is the relative time delay between NIR and IR
pulses. The corresponding nonlinear polarization source can be written as,
(2), ( ) ( )exp(( ))NL v v v v vE iP d (0.13)
The nonlinear source in (0.13) represents a time-frequency signature of the SFG process. The
intensity of SFG spectrum recorded by the spectrometer is proportional to the nonlinear
polarization source 2
E nlI P . Thus the SFG spectrogram, which is a two-dimensional mapping
of the intensity spectrum of the SFG signal can be written as,
2
( ) ( ), , exp( )E v v vi dI A (0.14) where,
(2) (( , ) ) ( )v v v vA E (0.15),
and is some proportionality constant. ,( )NLP in (0.13) is essentially the I.F.T along v
dimension of the function ( ), vA which resembles the spectral cross-correlation between the
NIR signal and the non-linear susceptibility. Thus, the function ( ), vA must satisfy two
distinct constraints: 1) Eq. (0.14): the squared magnitude of its F.T. must equal the experimentally
measured ( , )EI , and 2) Eq. (0.15): it belongs to the set of all functions whose unique
mathematical relationship to the unknown susceptibility is based on the physical model of the
non-linear process.
Thus the problem of estimating (2) ( ) from its SFG spectrogram is equivalent to the
general two dimensional phase retrieval problem, which seeks to estimate a complex function that
simultaneously satisfies constraints in the object domain and its Fourier domain. The generalized
Gerchberg-Saxton (GS) algorithm is used to solve the 2D phase retrieval problem by iteratively
50
transforming between the object and Fourier domains while applying the constraints in each
domain until the solution converges as determined by a particular error metric. It is depicted
pictorially in Figure 4-1. The key step in the algorithm shown in Figure 4-1 is the error
minimization step to solve for the next guess of(2) . Our technique for solving for the next
guess follows along the lines of [131]. We note from Eq. (0.15) that ( ), vA can be uniquely
mapped to a matrix (2)( ) (, ( )· )T
v v vO E which is the outer product of two vectors.
Figure 4-1: generalized Gerchberg-Saxton algorithm for BB-SFG-VS
Hence to find the next guess we seek a factorization of ( ), vO into its basis vector pairs, in
other words, its singular value decomposition (SVD).
*· ·O VU (0.16)
Here U and V are unitary matrices such that their columns represent the left and right singular
vectors. is a rectangular matrix whose diagonal entries are the sorted singular values that
determine the relative contribution of the vector pairs to the outer product. It can be shown that
only the principal component, that is the singular vector with the largest singular value, of the
generalized projection of O survives after successive iterations. Hence, we set the next guess for
(2) ( ) as the most dominant left singular vector, which is the first singular vector
51
(2)
1( ) (1) (:,1)n U since the singular values are sorted with the largest value first. Similarly
the next guess for the NIR pulse is updated as (2) *
1( ) (1) (:,1)n v VE . Such an algorithm in
which the guess for both the vectors is updated in each iteration is also called double blind
decomposition, because it does not utilize the information of the measured NIR pulse. This is
especially crucial in situations when only the NIR spectrum can be measured but it is
experimentally difficult to directly measure the complex profile of the NIR pulse shape.
Following this step, the algorithm continues to iterate by updating the outer product with this
guess. Our implementation of the generalized GS algorithm can be summarized in the following
steps:
1) (2)ˆ ˆ, [ ( , )( ) ( ) ([ · ])] T
v v vn nA O E
2) ˆ ˆ( , ) I.F.T. [ ( , )]vS A
3) 1
ˆ( , )ˆ , ( , )·ˆ| (
(, ) |
)n E
SS I
S
4) 1 1
1 1 1[ˆ ˆ ˆ( , ) ( , ) F.T. [ ( , )[ ]]]n v n v nO A S
5) 1
ˆ · · T
nO U V
6)
(2)
1
(2) *
1
( ) (:,1)
( ) (1) (:,1)
n
n vE
U
V
In our implementation [ ]O A is a unique map from the outer product form O to the function
A and it consists a series of matrix row transformations. The details of the exact implementation
of can be found in [131].
52
Numerical Simulation
We present a proof-of-concept numerical simulation of high resolution SFG-VS
measurement of a hypothetical (2) response function based on our algorithm. The hypothetical
(2) ( ) spectrum is modeled as sum of three Lorentzian lines centered at 2850 cm-1
, 2915 cm-1
and 2920 cm-1
whose homogenous broadening is characterized by full width half maximum
(FWHM) linewidth parameters of 30 cm-1
, 8.80 cm-1
and 4.4 cm-1
respectively. The simulated
SFG-VS signal is generated by using an assumed transform-limited 100 fs IR laser and 2 ps NIR
laser, which is similar to the conventional BB-SFG-VS setup (see Figure 4-2(a)).
Figure 4-2: Pulse profile of simulated IR and NIR pulses and hypothetical (2) spectrum used in
numerical simulation, (b) simulated SFG spectrogram. 2nd
panel in (b) plots the line trace at zero
delay in spectrogram, which illustrates the loss of resolution typically inherent in BB-SFG-VS.
53
Figure 4-2(b) shows the simulated SFG spectrogram. The loss of resolution using 2 ps NIR pulse
is clearly seen in the 2nd
panel in Figure 4-2(b), where the closely spaced lines at 2915 cm-1
and
2920 cm-1
are not resolved. Using our implementation of the generalized GS algorithm, the
simulated spectrogram is almost exactly recovered within 40 iterations. Figure 4-3(b) also shows
the retrieved spectral response function, which is in excellent agreement with the simulated
spectrum. The error between the computed and simulated spectrograms plotted in Figure 4-3(c).
Figure 4-3: Simulation of high resolution SFG spectroscopy using 2D PR. (a) SFG spectrogram
of hypothetical transform limited NIR probe pulse and transform limited IR pump pulse, (b)
Estimate of (2) and comparison with simulated
(2) (c) Error Estimation of spectrogram and
response spectrum and the error between simulated and estimated spectrogram.
We mentioned earlier that a unique feature of our algorithm is that it is double blind, and can
guess not only the susceptibility but also the complex spectrum of the NIR pulse using the
measured 2D SFG spectrogram. Such a capability is particularly crucial in situations in which it is
not experimentally feasible to measure both amplitude and phase of the NIR pulse. To validate
this feature, we simulate a hypothetical chirped NIR pulse and show that both the NIR pulse and
the(2) spectrum can be estimated from its 2D SFG spectrogram. In particular, effect of chirped
NIR pulse on the spectrogram is twofold, as seen in Figure 4-4(a). Firstly, it causes one half of
the spectrogram to become narrow along frequency (wavenumber) axis and correspondingly
elongate along the delay axis. Notice the asymmetry in the spectrogram shape. This reciprocal
change in the spectrogram shape along the delay and frequency axis is consistent with the fact
that a pulse which is broader in time has narrower bandwidth. Secondly, the spectrogram shows a
54
distinct skew towards lower wavenumber. The skew is dependent on the sign of chirp, and
positive chirp will cause the spectrogram to skew towards larger wavenumber. We note that we
have observed very similar features in the experimentally recorded spectrograms that were
obtained in this project.
Figure 4-4: (a) Simulated SFG spectrogram using hypothetical (2) and chirped NIR probe pulse.
(b) Numerically computed SFG spectrogram using 2D PR, (c) Estimate of (2) (black dotted
curve) computed using 2D PR, hypothetical (2) (green curve) and trace of spectrogram through
0 delay (See dotted white line in (b). Notice that spectrogram trace does not resolve adjacent
peaks at 2915 cm-1
and 2920 cm-1
. (d) Simulated and computed spectral amplitude and phase of
chirped NIR probe.
We then used the 2D phase retrieval algorithm to retrieve both the (2) and the complex NIR
spectrum from the simulated spectrogram shown in Figure 4-4(a). Figure 4-4 (b), (c) and (d)
55
demonstrate that we can retrieve even the asymmetric spectrogram and (2) spectrum as well as
chirped NIR pulse with high fidelity. Again, the two closely spaced peaks are clearly resolved
and we observe robust agreement between the numerically retrieved and simulated spectra of
(2) . It is clear from these simulations that our algorithm can enable high resolution SFG
vibrational spectroscopy using the conventional broadband SFG-VS. Indeed this is not surprising
and can be understood by noting the underlying physical significance of the data encoded in the
spectrogram. The spectrogram records, spectrally, the dynamics of the evolution of the
vibrational mode(s) being probed by using the relatively delayed NIR pulse, as is typically done
in time-resolved spectroscopy. This 2D dataset is partially equivalent to what would be measured
if one synthesizes an effectively much longer NIR pulse by combining the delayed copies of the
actual NIR pulse used in the experiment. However, the phase of the effectively synthesized NIR
pulse is not available with such measurement. This missing piece of information is what is
precisely estimated using the 2D PR computational technique. Thus we can not only leverage the
temporal resolution inherent in broadband SFG spectroscopy, but also recover the spectral
resolution computationally. This is the most unique feature of our technique which is in addition
to the fact that the conventional broadband SFG spectroscopy is experimentally easier to perform
than the more involved high resolution SFG spectroscopy using sophisticate pump and probe
laser systems.
Experiment
The full details of the SFG experimental setup for the air/DMSO interface studies were
previously published. [132] Briefly, the Ti:Sapphire oscillator and amplifier (Libra, Coherent,
Inc.) generates ~85 fs 800 nm pulses at 2 kHz repetition rate (2.4 mJ/pulse) and 12 nm full width
56
half maximum (fwhm). The ps NIR pulse is obtained from the 800 nm pulse by broadening it.
There are two common methods used for broadening the femtosecond pulse. In one method, a set
of two Fabry-Pérot etalons is used in series and in the second method the pulse is spatially filtered
using a combination of grating, lens and slit. Approximately 30% of the fundamental broadband
800nm output is filtered through two Fabry-Pérot etalons (TecOptics) in series to produce the
spectrally narrow (~2.1 ps) pump pulses (bandwidth Δν =13.4 cm-1
). However the use of etalons
in series is known to broaden the pulse asymmetrically with a longer tail. This causes the shape of
the SFG spectrogram to be distorted and introduces errors in the obtaining the solution using the
2D phase retrieval technique. To avoid this problem the grating and slit method is used. This
setup consists of a grating (830 groves/mm, with blazing angle of 19.7o at 820 nm), length
cylindrical lens with focal length of 100 mm, an adjustable slit, and a mirror. The grating
spectrally disperses the pulse and the cylindrical lens focuses it into a line on the mirror. The
variable slit is placed right in front of the mirror and can be closed in order to spatially filter the
pulse spectrum, thereby broadening it. The spatially filtered reflected pulse is then picked off by
another mirror. In this way, we were able to generate pulses with variable band width by
controlling the slit width. The 800 nm power was reduced to 3mW (1.5 μJ/pulse) or 30mW (15
μJ/pulse) when the slit was opened by 2.5 or 7 μm, respectively. Polarization control was adjusted
using a combination of two half-wave plates and Glan-Thompson polarizing prism.
Approximately 70% of the broadband 800nm output is used to produce broad bandwidth mid-
infrared probe pulses tunable in the range (2.3‒10 μm), using a two stage unit (OPERA solo,
Coherent, Inc.) with optical parametric amplification (OPA) followed by difference frequency
generation (DFG) with AgGaS2 (Light Conversion, Vilnius, Lithuania). The IR pulse shape had a
Gaussian profile and the full width at half maximum (FWHM) = 130 cm-1
, which was
characterized by analysis of the SFG signal collected from the surface of α-quartz in reflection
mode. The measured output energy by pyroelectric detector of the broadband IR was 20μJ/pulse
57
in the CH stretching region (2800‒3000 cm-1
). The IR pulse was centered at 2925 cm-1
and 2940
cm-1
for the DMSO and α-pinene experiments, respectively. For sample mounting, neat liquid
solution of both DMSO and α-pinene was placed in a Teflon dish and for TIR-SFG experiments,
a 45° CaF2 window was placed covering the dish. The experimental schematic is shown in Figure
4-5.
Figure 4-5: Schematic of experimental setup (ssp configuration)
The pump and probe pulses were overlapped on the air/DMSO interface using a single CaF2
focusing lens and the broadband SFG signals were collected in the momentum conserved
direction. Thus the incidence angles for all beams were ~45° with respect to the surface normal.
This geometry gives simple alignment between the reflection and total internal reflection (TIR)
modes as well as simple alignment of the invisible SFG beam into the detector slit (50μm) using
the reflected 800nm beam. The polarization combination for all experiments was ssp, (s-polarized
SFG, s-polarized pump and p-polarized probe). For acceptable signal to noise ratio, each
spectrum was collected with a 10sec exposure time and 4 spectra were arithmetically averaged
together and normalized from the SFG signal taken from the surface of α-quartz. Collection of
spectra using different time delay between 2 ps and 85 fs pulses was performed by moving the
58
motorized delay stage at 100fs scanning steps in a total range of ±6.4ps. The SFG signals were
dispersed by a holographic imaging spectrograph, 2.1cm-1
/pixel (Holospec) and recorded with a
CCD camera (iDuS 420A, Andor) which has 1024x256 active pixels.
Results and Discussion
We now show the results of using two dimensional phase retrieval for retrieving the
(2) spectrum of air/DMSO interface from its SFG spectrogram. Figure 4-6(a) shows the
experimentally measured spectrogram acquired by recording the spectrum as the NIR probe is
delayed with respect to the IR pump by a range of delays from 6.4 ps to 6.4 ps.
Figure 4-6: Match between experimentally recorded and computationally retrieved SFG
spectrogram of air/DMSO interface. (a) Recorded BB-SFG SFG spectrogram of the air/DMSO
interface taken in the range of ±6.4 ps delay between the 2 ps probe pulse and broadband pump
59
pulse at ~3.4 μm. (b) Numerically estimated BB SFG spectrogram using 2D PR. (c) Trace of
spectrograms when both pump and probe overlap with no delay. (d) Trace of both spectrograms
along peak of the spectrum.
Using the recorded spectrogram and the mathematical model of SFG process as the two
constraints, the generalized GS algorithm begins with a generic guess for the spectral form of
(2) and iteratively updates the guess to converge to the estimated spectrum of (2) . The NIR
pulse spectrum is experimentally measured separately and is used in the numerical reconstruction
of the spectrogram in each step using the update guess for (2) spectrum. The mismatch between
the numerical reconstruction and the experimental measurement is defined as the error metric.
Figure 4-6(b) shows the spectrogram after the algorithm has converged. The excellent agreement
between the two spectrograms can be seen from Figure 4-6(c-d) which compare the line traces
through the center of both spectrograms along wavenumber and delay axis respectively. To
further verify that algorithm does indeed converge to the closest solution, Figure 4-7(a) tracks its
performance by plotting the error, and shows convergence in under 20 steps in the case of single
peak for air/DMSO interface. Figure 4-7(b) shows the estimated (2) spectrum, and a Gaussian
fit to it.
60
Figure 4-7: (a) RMS error between measured and computed spectrograms. (b) Estimated (2)
spectrum of air/DMSO interface
It shows a linewidth of ~8.7 cm-1
for the 2925 cm-1
DMSO line. This agrees very well with the
recently reported linewidth of ~8.8 cm-1
using the experimentally sophisticated HR-BB-SFG-VS
technique for the same line. In contrast to this, we also show the line trace of the measured
spectrogram along zero delay, i.e. the SFG spectrum when the pump and probe overlap with no
delay. This spectrum has linewidth of 10.95 cm-1
, in other words it is ~26% broadened than the
true linewidth due to the effect of the ~2 ps long NIR pulse. It is clear from these results that the
computational 2D PR technique is capable of recovering the high resolution spectrum from low
resolution measurements typical of conventional broadband SFG spectroscopy. This capability is
inherent in the fact that the 2D SFG spectrogram dataset is effectively equivalent to that obtained
by using a much longer probe which may be synthesized using the relatively delayed copies of
the actual probe. By recovering the phase information, our technique enables broadband SFG
vibrational spectroscopy to capture the vibrational dynamics with high temporal and high spectral
resolution simultaneously. In addition, our computational technique retains the experimental
simplicity of the conventional broadband SFG spectroscopy and thus potentially enriches the
existing research initiatives in the field of surface and interface studies using SFG spectroscopy.
61
Spatio-temporal characterization of ultrashort optical pulse
Introduction and motivation
Accurate characterization of ultrafast optical laser field is important for many
applications of scientific and technological interest since it can provide insight into the temporal
and even spatial dynamics of the pulse as it interacts with its environment. One can imagine that
such knowledge can be critical in many applications. For example, Figure 4-8 shows a
hypothetical scheme which depicts change in a femto-second (fs) pulse shape which may occur
due to some process such as light matter interaction. Precise knowledge of the amplitude and
phase of the fs pulse in both space and time before and after the process can help analyze the
process in terms of the change in the pulse behavior.
Figure 4-8: Schematic depicting change in fs pulse due to its interaction within a hypothetical
process. Ultrafast pulse characterization can accurately measure amplitude and phase of fs pulse.
The capability of characterization of ultrashort pulse is important in several fundamental studies
which involve dynamic pulse synthesis and pulse shaping to influence light matter interaction.
[133] It can aid in fundamental science by providing guidance for synthesis of ultrashort pulses to
enable spatio-temporal control of lattice vibrational waves [134], to influence molecular
dynamics by enabling single-molecule control and manipulation [135]–[137] and to realize
coherent control of photons [138] to achieve local field enhancement [139] and spatio-temporal
62
control on femtosecond time and nanometer length scale [140]. Similarly, knowledge of the
ultrafast optical pulse can aid laser performance diagnostics [141], [142] and help tailor the
excitation to optimize nonlinear microscopy [143]. Moreover we find applications in which the
spatial distribution of optical field is not the same for all frequency components, i.e.,
( , ) X( ) ( )E r r where ( , ) F.T. ( , ) ( , ) i tE r r tE E r t e dt and F.T. stands for
Fourier transform. As a simple example, coupled spatio-temporal evolution of ultrashort field can
be observed in diffraction and dispersion effects of lenses, and as a result the pulse can have a
complicated profile near the focus. Multiple scattering of coherent pulsed light through thick
strongly scattering media may also exhibit non-trivial wavelength dependent speckle pattern
[144]. Dynamic coherent control of surface plasmon polaritons in nanometer femtosecond regime
has been demonstrated by controlling the phase of the coupled spatio-temporal field [145]. It is
well-known that for plasmonic systems with both nanoscale modal distribution (i.e., hot spots) as
well as a large plasmonic resonance the spatio-temporal dependence of the ultrafast electric field
cannot be decoupled. Therefore it is necessary to develop techniques to characterize the spatio-
temporal evolution of ultrashort optical pulse.
Frequency resolved optical gating and 2D phase retrieval
The fundamental dilemma of characterization of the femtosecond pulse, which can be as
short as only a few cycles long, by sampling was a daunting challenge until recently because
ultrafast pulses are some of the fastest transient phenomena themselves. The Wiener-Khinctine
theorem relates the pulse spectrum to its autocorrelation via the Fourier transform, but it cannot
uniquely determine the phase; this is the 1D phase retrieval problem we have seen in chapter 3.
Similarly intensity auto-correlation only yields a measure of the pulse width but fails to reveal the
63
complex pulse shape. Trebino and co-workers pioneered the non-linear technique of frequency
resolved optical gating (FROG) [146] by combining the measurement of autocorrelation and
spectrum to generate the pulse spectrogram. FROG has been described as an auto-correlator type
measurement in which the auto-correlator signal beam is spectrally resolved. In fact the pulse
spectrogram is similar to that seen in sum frequency generation spectroscopy. If we take the
example of a very common FROG measurement modality, namely second harmonic generation
FROG, then the SHG-FROG signal is generated by gating the unknown pulse
0( )ex( ) pi t
EE t t
with itself at variable delay times in a 2nd
order non-linear optical crystal
and the generated second harmonic signal is recorded spectrally for each delay. Thus the SHG-
FROG spectrogram is given as 2
( ) ( , )exp(, )FROG sigI E t ti dt where
( , ) ( ) ( )sig E EE t t t . Thus if there exists a unique way to obtain ( )E t from ( , )sigE t , then
the 2D PR problem can be found readily applicable to the problem of pulse retrieval using the
FROG spectrogram. In particular, the functional form of ( , )sigE t and the measured intensity of
the FROG spectrogram act as the two constraints for the iterative GS algorithm, which upon
convergence can determine the ultrafast pulse uniquely up to some trivial ambiguities (which we
will describe in next paragraph). Thus FROG is a 2D PR based technique for measuring the
amplitude and phase of typical ultrafast pulses and indeed there is a sizeable body of work in this
field [36].
Spatio-temporal characterization of ultrafast pulse
The conventional FROG technique has a few important limitations. FROG obtains the
complex pulse shape of the “whole beam” by assuming that an ultrashort optical field can be
64
separated into a spatial distribution multiplied by an independent temporal evolution function
( ), ( ) ( )rE r . We have seen above that this assumption is clearly inconsistent with
several applications due to phenomena such as wavelength dependent dispersion, diffraction and
scattering. Since SHG-FROG trace is related to the pulse auto-correlation in which the pulse
gates itself, the inherent symmetry results is time ambiguity: i.e. the field
0( )
0 0( ) ( )i t t
t t A t t eE
for some arbitrary time shift 0t , and the time reversed field
( )( ) ( ) i tt A tE e will also generate the same FROG trace. To overcome these limitations
and enable coupled spatio-temporal characterization, several techniques based on the concept of
spectral interferometry have been proposed. Spatiotemporal characterization of ultrashort pulses
has been demonstrated by spatial-spectral interference (SSI) [147], [148]. SEA TADPOLE [149]
and STARFISH [150], [151] are experimentally simpler implementations of SSI technique using
optical fibers. Dorrer et al. [152] reported a technique combining SPIDER with spatial shearing
interferometry for measuring both the spectral and spatial phase. STRIPED FISH [153] is a
holographic method for single shot measurement of three-dimensional spatiotemporal electric
field by recording spatially separated array of quasi-monochromatic holograms. NSOM probes
may be used with SEA TADPOLE [154] or other techniques to improve spatial resolution.
However, existing NSOM probes have limitations as they require converting local non-
propagating field into radiating field traveling in free space or in the NSOM fiber, which could
potentially lead to perturbation of the local field itself and hence inaccuracy for measurement at
nanoscale. Nano-FROG [155] achieves high spatial resolution using individual second harmonic
(SH) nanocrystal clusters dispersed on a substrate; however the presence of the substrate inhibits
near field measurements. Multi-pulse Interferometric FROG (MI-FROG) which combines FROG
and spectral interferometry is capable of single shot characterization of electric field consisting of
multiple ultrashort pulses and obtaining the relative phase and the temporal offset between the
65
pulses [156]. Previously, our group has developed a new class of Second HARmonic nano-Probes
(SHARP), consisting of second harmonic nanocrystals attached to carbon nanotubes which are in
turn attached to tapered optical fibers [157]. Using SHARP-based collinear FROG (CFROG)
measurements, our group has demonstrated the capability to characterize ultrashort optical fields.
However, SHARP probes lack a phase reference to coherently link the optical pulse characterized
at multiple locations. Critical information such as phase velocity and group velocity therefore
cannot be retrieved.
Here, we introduce holographic FROG, a novel technique for spatio-temporal
characterization of ultrashort optical field by combining FROG and spectral holography [158]. In
FROG holography, the use of reference beam at the SH wavelength establishes a phase coherent
link which makes it possible to measure the spatio-spectral phase of the pulse at multiple
locations. In the following, we first discuss the theory of FROG holography. Next we present a
numerical simulation to illustrate pulse retrieval at each location from the holographic FROG
trace. Finally we present an experimental study to demonstrate the measurement of group delay
of the pulse, which is not possible using the conventional FROG technique.
Formulation
Our technique of holographic FROG is designed for spatiotemporal characterization of
the ultrashort optical field. To do this, we propose to interferometrically detect the collinear
FROG (CFROG) signal by using a reference beam at the SH frequency as a local oscillator. Let
us consider the field 0( , e( , ) ) xpi t
r E r tE t
at location r , where ( , )( , ) ( , ) exp i r tE r t I r t
is the complex amplitude, ( , )I r t is the intensity, and ( , )r t is the phase, and 0 is the center
angular frequency. Also, consider the SH reference pulse 02( ) ( )exp ,
tSHG SHG i
R RE t tE
where
66
( )SHG
RE t is its complex amplitude. The corresponding holographic CFROG (HCFROG) trace is
given by:
2
2,( ) F.T. ( ) ( ) ( ), , , SHG
HCFRO RG r r tI E Er tE t
(0.17)
where is a proportionality constant. Let 02 . We define the following:
2( , )F.T. ( ),,SHGE r t E r ( , ) ( , )F.T. 2 ( , ),FROGE r t E Er t r and
F.T. ( ) ( )SHG
R Rt EE . Eq. (0.17) can then be rewritten as
0
0
2
2
( ) | ( ) ( )
( ) (
,
)
,
|
, ,
, ,
ii
HcFROG SHG SHG
i
FROG R
I E E e e
E
r r r
e Er
(0.18)
The holographic term of interest in Eq. (0.18) is 0 *, , ,) ( ) ( ),( R
i
FROGr E rH e E .
Algorithm
Our algorithm to retrieve the ultrashort pulse profile from its HCFROG trace is designed
to track the phase as the pulse evolves spatially. We illustrate the algorithm by considering a
numerical simulation of ultrashort pulse retrieval as it propagates through a hypothetical
dispersive medium, a schematic of which is shown in Figure 4-9(a). A typical simulated
HCFROG trace and its 2D Fourier transform (F.T.) is shown in Figure 4-9(b) and Figure 4-9(c)
respectively. The characteristic fringes of HCFROG trace are clearly seen along both wavelength
and delay axis. The 2D F.T. clearly identifies the HCFROG term, i.e. the 2D F.T. of
0 *, , ,) ( ) ( ),( R
i
FROGr E rH e E , which can be digitally filtered.
67
Figure 4-9: Numerical simulation. (a) Schematic of simulated experiment. (b) Typical simulated
HcFROG trace. (c) 2D F.T. of HcFROG traces at locations ‘P1’ and ‘P2’. r is the spatial
separation between the two locations and gv is the group velocity through the hypothetical
medium. (d) Relative FROG phasor of the pulse at ‘P2’ with respect to ‘P1’.
In our simulation of the hypothetical dispersive medium, the two points P1 and P2 at which the
HCFROG trace is measured are separated by 5 um. Although their 2D F.T. traces appear similar,
we note that the time offset of the holographic sideband corresponds to the spatial separation of
the two locations, and hence the HCFROG trace encodes the spatial evolution of the ultrashort
pulse. Indeed, the ratio of ,( ),rH at ir and jr yields the relative FROG phasor
, (( ) ,, , )( , )/i FROGFROG ij ji r r i ri r
e e e
(0.19)
where ( , , )FROG r denotes the phase of , ,( )FROGE r at r . We note that the reference field
and its delay are maintained fixed and hence it does not need to be characterized independently.
We set ‘P1’ as “pivot”, pivotr and Eq. (0.19) is used to compute the relative FROG phasor at ‘P2’
(and every other location) with respect to pivotr (see Figure 4-9(d)). We obtain the standard
background-free SHG FROG spectrogram by filtering the d.c. term and apply the conventional
68
2D phase retrieval FROG algorithm to retrieve the pulse profile at pivotr . In particular, the
conventional FROG algorithm computes the time-domain FROG trace
( ,, ( , ( )) ) ,pivot pivot pivotsigE r E r t E r tt via 1D I.F.T. of the SHG FROG spectrogram. We
normalize it by noting that ( , ) ( , ) ( , , )sigE r t d E r t dt E r t dtd so that
( , )( , , )
, ( , ) e) p( , )
( x,
pivotsig pivoti r t
pivot pivot
sig pivot
E r t dE r t I r t
E r t dtd
(0.20)
is used as the guess for next iteration, until convergence is reached. At ‘P2’, the corresponding
SHG FROG spectrogram , ,( )FROGI r and the relative FROG phasor ,( )pivoti r r
e
(computed
using Eq. (4.7)) is used to obtain the complex spectrogram
( ), ,( , )( ), , pivot FROG pivoti r r i r
FROGI e er
. Then the pulse profile at ‘P2’ is retrieved by simply
following the prescription of Eq. (0.20). In this way the spatial and temporal characterization of
the ultrashort pulse is performed by holographic FROG technique. Amongst the salient features
of our technique, we note that ,( ),rH is linear in , ,( )FROGE r and the homodyne
detection increases sensitivity to ‘weak’ spots in the spectrogram. The iterative phase retrieval is
applied only to the “pivot” location (which may be arbitrarily chosen in principle) whereas pulse
retrieval at every other location involves a single F.T. and basic algebraic operations to make use
of the phase information inherent in the corresponding HCFROG trace. This adds minimal
computational overhead to the overall pulse retrieval algorithm.
69
Numerical Simulation
We demonstrate the validity of our algorithm by a simple numerical simulation based on
the same schematic shown in figure 2. We consider the propagation of transform limited
Gaussian pulse2 2
0/( )
tA e
at 850 nm through a hypothetical dispersive medium characterized
with a group delay dispersion parameter 31133 /0D ps nm km 2 2'' 51( 512 0 / )k s m .
The pulse has full width half maximum (FWHM) of 0t = 37 fs at ‘P1’, which is taken as the pivot
location in this simulation. The positively chirped pulse profile can be computed at any location
within the medium using 2 ' 2
0(1 ')( / )( , )
ia tA z e
where ' / da z L and
2'
0 0 1 / dt t z L [23].
For this simulation, ‘P2’ is 5 um and ‘P3’ is 15 um from ‘P1’.
70
Figure 4-10: Results of numerical simulation using HCFROG algorithm (a) retrieved and
theoretical complex pulse profile. FWHM is marked for pulse at each location. (b) computed
spatial separation between the 3 locations compared with actual separation used for simulation.
The table lists the actual values of spatial separation as used in the numerical experiment and
retrieved using method described. (c) comparison of retrieved and theoretical pulse spectrum at
each location. FWHM is marked for the pulse spectrum. (d) comparison of retrieved and
theoretical SH spectrum of pulse at each location. FWHM is marked for the pulse spectrum.
71
Figure 4-10 shows the comparison between the pulse profiles retrieved from the HCFROG trace
at each location by using the algorithm discussed above, with the corresponding theoretical pulse
profiles. The pulse amplitude (including FWHM) and quadratic phase profiles (due to positive
chirp) agree with their theoretical profile at each location Figure 4-10(a)). The minor deviation
from flat phase for the pulse at ‘P1’ is attributed to numerical errors due to discretization.
Similarly the retrieved pulse spectrum (Figure 4-10(c)) and its SH spectrum (Figure 4-10(d)) also
agree well with the theoretical profiles at each location. We obtain the relative spectral phase of
the pulse between each location , ,( , ; ) ( ) ( )i ji j r r using the retrieved pulse profile
at each location. The group delay of propagation through the medium can be obtained by
computing the gradient ( /, ; )g id dj of the relative spectral phase between any pair
of locations; and the separation between those locations is obtained by g gv . Figure 4-10(b)
plots the spatial separation thus computed numerically compared to the actual separation used for
the simulation, and the included table lists the actual values in microns. The comparison in all
result panels is very solid. This clearly shows that the HCFROG trace records the carrier phase
and group delay of pulse propagation, i.e. the first two terms in the Taylor series of the spectral
phase 2
0 0 0( , ) ( , ) ( , ) 0.5 ( ) ...,r r r r in addition to the O(2) terms
that are measured by conventional FROG (aside from an overall undetermined constant).
Experiment
As a proof-of-concept demonstration of our HCFROG technique, we in situ characterize
the pulse near the focus of an objective lens, by using a BaTiO3 micro-cluster air-dried on a cover
glass as the nonlinear probe and traversing through the focus of the objective along its axis. As
shown in the setup in Figure 4-11(a), a Michelson interferometer with a variable delay arm
72
generates two copies of the fundamental pulse which is obtained from a Ti:Sapphire laser
(KMLabs, central wavelength ~818nm, average output power ~400mW, repetition rate 88MHz).
The two pulses combined at the output of the interferometer first propagated through a short
length (~10cm) of photonic crystal fiber before being focused on the BaTiO3 micro-cluster by a
40 objective lens to generate the SH CFROG signal. An SEM image of the micro-cluster is
shown in Figure 4-11(b). We vary the delay in steps of 0.4 fs which is enough to satisfy the
Nyquist sampling rate for the CFROG signal, although it has been shown that the delay step may
be increased to speed up acquisition without affecting the final retrieved pulse [159]. In our setup
we have mounted the PCF output end and the collimating and focusing objectives on the same
translational stage (as indicated in Figure 4-11(a)). This arrangement effectively allows us to
probe the focused femtosecond beam at multiple locations along the objective axis without the
need to move the micro-cluster itself or the signal acquisition optics, thereby maintaining the
optical path of generated CFROG signal. Additionally, a SH reference pulse (~409nm) is
generated from the fundamental beam by using a Barium Borate (BBO) crystal, separated by a
dichroic filter, and appropriately delayed to interfere with the CFROG signal. The resulting
HCFROG trace is filtered by a band-pass filter (D400/70, Chroma Technology) and detected by a
spectrograph (PI Acton SP2500i with a liquid nitrogen cooled charge coupled device camera,
resolution: 0.03nm at 409nm). Figure 4-11(c) shows a typical recorded HCFROG trace which
exhibits the characteristic fringe structure along both the delay and wavelength dimensions. We
acquire such HCFROG traces at five locations as we traverse through the focus: ‘location 2’ at
9.43 μm, ‘location 3’ at 18.11 μm, ‘location 4’ at 27.37 μm and ‘location 5’ at 37.75 μm with
respect to the pivot ‘location 1’.
73
Figure 4-11: (a) Schematic of our experimental setup. (b) The inset shows SEM image of the
BaTiO3 micro-cluster used for generating FROG signal. (c) Typical recorded HcFROG trace.
The fringe structure can be seen along both delay and wavelength dimension.
Results and Discussion
We implement the procedure described in section 2 to retrieve the complex pulse profile
from the recorded HCFROG trace. In our experiment, among the five locations at which
HCFROG trace is acquired, the ‘pivot location’ is chosen to be closest to the focusing objective
along the axis. As noted in Figure 4-11(a), ‘location 4’ is nearest to the focus. First panel in
Figure 4-12(a) shows the intensity and phase of the pulse at pivot location retrieved by applying
the conventional FROG algorithm at that location. The full width at half maximum (FWHM) of
the retrieved pulse is 89 fs (at temporal resolution = 4 fs). At other locations, the relative FROG
phasor with respect to pivot is computed as prescribed by Eq.(0.20). Combining the relative
FROG phasor with the FROG trace (extracted digitally from the corresponding experimentally
measured HCFROG trace at that location), the resulting complex spectrogram can be simply
inverse Fourier transformed and processed to obtain pulse amplitude and phase at that location, as
described before.
74
Figure 4-12: Pulse characterization at multiple locations using FROG holography. (a) Retrieved
pulse profile (amplitude and phase, FWHM ~89 fs) at each position; (b) Comparison of both
fundamental and SH power spectrum of the retrieved pulse at each position with measured power
spectra (black squares) (FWHM ~12 nm) (c) Comparison between retrieved and experimentally
recorded spatial separation between each location with respect to pivot. The table in inset lists
actual values in m .
75
The panels in Figure 4-12(a) show pulses retrieved at each location. The pulse generally
maintains its amplitude profile, displays slightly different phase profile, and is shifted along the
time axis in accordance to the separation of the spatial locations as it travels along axis through
the focus. To check the validity of the retrieval process, in Figure 4-12(b) we compare the
measured laser spectra (both fundamental and at SH) with that computed from the retrieved pulse
profile at each location. The measured and computed spectra agree reasonably well and do not
vary significantly between the five locations. The FWHM bandwidth of the fundamental pulse
spectrum is ~12 nm. Further we computed the spatial separation between each of the locations
using the derivative of the relative spectral phase as was discussed in the simulation section.
Figure 4-12(c) and the table within show the comparison between numerically obtained and
experimentally measured distances from the pivot to every other location. The agreement not only
validates our retrieval procedure but also serves as proof-of-concept demonstration of the
potential of FROG holography to fully characterize the spatiotemporal evolution of an ultrafast
optical field.
Conclusion
In summary, our technique of FROG holography utilizes a reference beam to record
spectral holograms of conventional CFROG traces at multiple spatial locations. In contrast to
other existing techniques, our method does not require the reference beam to be characterized, as
long as it stays stable during the entire measurement. We have shown that it contains not only the
standard non-collinear FROG term but also the relative FROG phasor and thereby coherently
‘links’ the measurements at different spatial locations. We have discussed a method to retrieve
the complex pulse profile from the recorded HCFROG traces, and demonstrated the measurement
of group delay of the pulse in the vicinity of the focal point of an objective lens. In short, FROG
76
holography combines high sensitivity of spectral holography with FROG to characterize the
detailed temporal phase structure of the pulse as well as provide key information about group and
carrier velocities of pulse propagation. The improved sensitivity of homodyne detection is useful
for probing “weak” spots or for increasing the scanning speed by reducing the integration time of
the spectrometer. In principle, FROG holography makes it possible to measure the spatio-
temporal evolution of complicated ultrashort pulse that can be characterized by the general FROG
technique. In combination with nonlinear nanoprobes [157], FROG holography has the potential
to image the spatiotemporal evolution of ultrafast optical fields in femtosecond time scale and 3D
nanometer length scale, which could benefit emerging applications in biomedical imaging,
plasmonics, and metamaterials.
77
Chapter 5
Summary and Future Prospects
We have presented a few examples in imaging and spectroscopy in which the application
of computational methods has enabled enhancements and helped overcome limitations in existing
techniques. Compressive coherent anti-Stokes Raman holography was shown to enhance the
apparent optical sectioning capability of image reconstruction using digital holography. This
capability enriches the existing label-free chemical selectivity and non-scanning 3D imaging
features of coherent Raman holography. This can be further improved by using the beam
propagation method which accounts for non-uniform propagation medium and potentially extend
this capability for 3D imaging in the presence of scattering elements using single hologram. Our
proposed compressive frequency comb spectroscopy technique can in principle achieve sub-
picometer resolution across tens of nm of broad spectral coverage. Such ultra-high resolution can
enable molecular fingerprinting and accurate spectral assignment to help identification of
unknown species. The current research in development of stable and miniaturized frequency
comb will undoubtedly help the development of ultra-high resolution optical metrology in
general, and our spectroscopy technique provides an example of efficient optical sensing
applications based on compressive sensing. Non-linear optical spectroscopy such as sum-
frequency generation spectroscopy has advanced the science of surface chemistry by enabling
non-invasive, non-destructive optical probe with excellent intrinsic molecular specificity and
surface selectivity. Our computational technique based on two dimensional phase retrieval for
SFG-VS enhances the spectral resolution achievable using the most commonly implemented
conventional experimental procedure thereby providing simultaneous high temporal and spectral
resolution insight for analyzing different molecular species at the surface of media. All these
applications employ specialized pulsed laser sources, especially ultrafast femto-second lasers and
78
as such it can be important to characterize the spatio-temporal dynamics. We have developed a
sensitive technique which combines the existing FROG technique with spectral holography to
help overcome some key ambiguities in conventional FROG and enable characterization of the
spatio-temporal evolution of the ultrashort pulse. In particular, the capability to measure the
group delay could significantly enhance imaging modalities based on time of flight imaging by
providing femtosecond resolution. Additionally, the potential of our technique lies in its
capability to achieve 4D imaging of the ultrafast field ( , , , )E x y z t with nanometer spatial and
femtosecond temporal resolution, in essence nano-femto microscopy.
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Appendix A
Opto-electronic nano-probe
In this appendix, I report on a project which was done based on support of NSF (Grant#
1128587). It represents a substantial part of my overall output during PhD. Hence I have opted to
include a report of the project in the appendix to my thesis.
Introduction and Motivation
The field of fiber optics has witnessed significant advances owing to the pioneering work
of Russell and coworkers which has led to the development of micro-structured optical
fiber,[160]–[162] in particular photonic crystal fiber (PCF). An important advantage of PCF over
conventional optical fiber is the ability to precisely engineer the optical modal properties by
designing the shape, size and distribution of the air holes. The recent active scientific interest can
be attributed to the fact that the properties of PCF can be altered by incorporating special
materials in its air holes. Indeed, the choice of material(s) and selective filling of air holes offer
additional degrees of freedom which vastly expand the scope of PCF based optical devices and
their applications. Of particular interest here is the fact that electrically conductive media, such as
metals, may be filled in the PCF structure to design novel devices and sensors. For example,
Sazio et al. demonstrated unique all-in-fiber active photonic devices [163], [164] by filling silicon
and germanium within the air holes. Recently several applications based on filling metals in PCF
have been reported.[165]–[169] Furthermore, surface plasmon polariton sensors based on metallic
filling of PCF has also been the focus of active research.[170], [171]
Metal filled PCF devices feature the ability to conduct electricity as well as guide optical
waves. Furthermore, nanotubes or nanowires can be incorporated within metal filled PCF such
80
that each nanoprobe may be individually controlled. This points to the exciting potential of
developing an integrated opto-electronic fiber platform which is capable of simultaneous
electrical and optical interrogation and probing at the nanoscale. Key to developing such a
platform is the ability to repetitively and reliably fill metal into the air holes of the PCF
selectively without damaging its structural integrity or its optical wave-guiding functionality.
Towards that end goal, here we report on the fabrication and characterization of a novel fiber
based opto-electronic nanoprobe engineered by selectively penetrating metal electrodes in PCF
and by attaching nanotubes individually to each electrode. We present methods to create any
desired configuration of such electrodes and show the feasibility that the attached nanotubes can
be actuated through electrostatic force by controlling the voltages applied to the electrodes. In
other words, the PCF serves as scaffold for an electromechanical nanoprobe as well as an optical
waveguide. This unique aspect of our device differentiates it from existing single tip
nanomanipulators [172]–[175] and two arm nanotweezers. [176]–[179]
The report is organized as follows: we begin by presenting the techniques we have
developed for fabrication of the proposed opto-electronic nano-probe, including selective filling
of metal electrodes in PCF and precise attachment of individual nanotubes to them. We then
report results of optical characterization of our device using spectral holography to quantify the
group velocity dispersion of the metal-filled PCF. Lastly we discuss the potential
electromechanical functionality of the device by demonstrating the feasibility of electronic
actuation of the attached nanotubes, and comparing the observed movement with numerically
simulated deflection.
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Selective filling of PCF
Selective penetration of foreign material especially metals in air holes of PCF has led to
design of novel fiber devices optimized for particular applications. A wide variety of techniques
have been demonstrated to achieve selective filling. Some of the early techniques for infiltrating
metal involved carefully inserting thin metal wires directly into specific fiber air holes [180] and
using vacuum suction [181] to draw molten material into the fiber.
Huang et al. leveraged non-uniform air hole sizes in their multistep injection-cure-cleave
technique to demonstrate selective filling of larger diameter holes with a UV curable polymer by
applying external pressure.[182] Another technique based on varying size of air holes involves
use of fusion splicer to collapse smaller holes and leave open larger holes for filling. [183], [184]
Borrowing on the technologies developed in the field of microelectronics, a novel technique
based on use of high pressure chemical vapor deposition was developed to successfully
demonstrate filling of semiconductors and metals to form functionalized MOF fiber
devices.[163], [185] It has also been shown that the conventional stack and draw procedure can
be adapted to fabricate PCF with appropriately positioned stack of metal rods around the core,
instead of penetrating the PCF post fabrication. [186] Vieweg et al. have shown selective filling
of PCF in arbitrary pattern by targeting and sealing individual air holes using a focused pulsed
laser to polymerize a photo-resist covering those holes, so that the unsealed holes are available to
be filled. [185] Another method utilizes side access to the air holes through the fiber walls created
using focused ion beam, so that the process of selective filling does not occlude optical access to
the PCF facet. [187] In our technique detailed below, we selectively pattern the PCF facet by
targeted application of UV curable optical glue to individual air holes, followed by pumping
molten metal in the open holes under high pressure. In fact, variants of the technique of using
pressure differential to pump or draw molten metal into PCF has been reported by several
82
researchers in the past. [165], [166], [169], [188]–[190] Our procedure does not require the air
holes to be of varying size, and is not influenced significantly by fluidic properties of the liquid
phase metal. It does not involve milling/splicing/arcing or other steps which may cause damage
and alter the optical properties of the PCF. This ensures that the fiber can guide light and
electricity along its length and perform both electrical and optical probing as envisaged above.
We demonstrate our procedure, illustrated in Figure A-1, by filling multiple individual holes in
more than 15 inch long endlessly single mode PCF (SC-PCF, Blaze Photonics ESM-12-01) with
low melting point metal.
Figure A-1: Schematic illustrating patterning of PCF for selective filling
The fiber has a 12 µm solid silica core surrounded by hexagonal array of 3.68 µm air holes in
cladding region. The pitch of cladding holes is 8 µm. As shown in Figure A-1(1), initially the
three holes to be infiltrated are each sealed by local application of a drop of UV-curable optical
glue (NOA-81) under an optical microscope using a fiber taper maneuvered with a three-axis
stage. Upon curing, the selectively blocked facet of the PCF is immersed into NOA-81 which
draws the glue into all but the blocked holes at least 5-6 mm deep via capillary action (blue
83
column in Figure A-1 (2) and (3)). Further UV curing of the penetrated optical glue, followed by
cleaving the PCF close to the facet creates the required PCF template with only the three holes
exposed (Figure A-1(3)) as desired. The PCF template is then mounted within a home-made high
pressure pumping system for pumping molten metal through it. In particular, the patterned facet
of the fiber is held in place in a chamber using polydimethylsiloxane (PDMS) (Dow Corning
Sylgard 184 and curing agent 10:1 ratio). The chamber is topped off with Ostalloy 158 (eutectic
solder alloy of 50% bismuth, 26.7% lead, 13.3% tin and 10% cadmium, having melting point 70
°C) and is placed inside an oven which is maintained at a temperature above the melting point
(approximately 100 °C). The inset in Figure A-2(a) shows the cutout of the chamber with the
fiber facet sunk in the alloy. The other facet of the fiber is held outside the oven at room
temperature as shown in Figure A-2(a). After the temperature within the oven equilibrates,
compressed nitrogen gas at pressure of about 500 psi is used to pump the molten alloy through
the unblocked holes of the patterned fiber, until the molten alloy leaks through from the other end
which is seen as a shiny blob in the image in Figure A-2(b). This indicates penetration of the
metal along the entire fiber length and typically takes a few minutes for our nearly 15 inch long
fiber. It has been reported that pumping liquefied metal by applying pressure often causes air gaps
along the length of the filled metal due to the difference in the thermal expansion coefficient of
the metal and silica [165], [166] In contrast, our technique of maintaining the distal end of the
fiber at room temperature causes the leaking alloy to solidify into the shiny blob which then acts
as a plug to seal the channel and avoid any voids as the alloy stays compact while cooling down
under constant pressure. The selectively filled fiber is seen in the scanning electron microscope
(SEM) image in Figure A-2(c) with arrows pointing to alloy-filled holes. We have successfully
fabricated several dozens of such long selectively filled PCF containing multiple 3.68 µm sized
alloy electrodes with high yield. We further confirmed the electrical continuity of each filled
electrode by measuring the resistance between the two ends at the fiber facet by using a pair of
84
tungsten probes as illustrated in Figure A-2(d). A 4cm long piece of the filled fiber electrode
typically has measured d.c. resistance of 1657 . Based on the reported conductivity of the alloy
of 2.320 x 106 S/m, the computed resistance of 1621 agrees well with our measured d.c.
resistance. We note that following this method, we have also successfully patterned PCF with
indium electrode which melts at higher temperature of 156.6 °C.
Nano-tube attachment
As stated earlier, the proposed opto-electronic nanoprobe is designed to allow
electromechanical probing at the nanoscale in addition to functioning as an optical waveguide. It
is well known that carbon nanotubes (CNT) have several unique electronic and mechanical
properties which make them suitable for this purpose. Multi-walled CNT (MWCNT) may be
either metallic or semiconducting with a variable band-gap based on their chirality and diameter.
CNTs can carry high currents without heating and are stiff and exceptionally strong. [191] Owing
to their properties, CNT has been used in variety of applications as tweezers or probes. [175],
[177], [192]–[194] In this section, we discuss our results with two different techniques of
attaching CNT to the selectively filled PCF for fabricating our device.
85
Figure A-2 Pumping molten alloy through patterned PCF. (a) Schematic of oven showing a PCF
held in a chamber. The inset shows cutaway of the chamber in which patterned pcf is held using
PDMS and sunk in bath of alloy. (b) Filled PCF under 500 psi showing alloy leaking out as shiny
blob from the other end held outside the oven at room temperature. (c) SEM image of a pcf with
three holes filled with alloy, (d) Setup for checking electrical continuity and typical resistance of
filled electrode (1657 ohms).
We note that the non-planar, circular geometry and small, uneven cross section of the
patterned facet of the PCF precludes the use of conventional photolithography based techniques
for fabrication of our device. Additionally, the small dimension and high aspect ratio of CNT
poses unique challenges for achieving targeted attachment of individual CNT to each filled
electrode while ensuring their correct alignment. Hence, we explored and developed two separate
techniques for attaching individual CNT to each filled electrode of the fiber - (a) laser welding,
and (b) focused ion beam (FIB) and electron beam assisted deposition. In the laser welding
technique, a laser beam is steered to focus on the electrode using a 20x long working distance
objective. To prepare the CNT sample, we sonicate a dispersion of CNTs (PECVD MW-CNT
86
from NanoLab Inc. 200nm in diameter and about 20 µm long) in DI water in ultrasonic bath (Tru-
Sweep 275DA, Crest Ultrasonics), and air-dry a small drop of the dispersed solution on a glass
slide. A single CNT is then fished from the glass slide sample under a microscope using a fiber
taper and is maneuvered to a PCF electrode using multiple 3D stages while the laser is incident,
as shown in Figure A-3(a). The schematic in the inset illustrates the process from the top view.
The laser power is increased very carefully until the filled electrode melts only slightly due to
absorption of laser energy to reflow and locally weld the CNT. A gentle tug on the delivery taper
is enough to detach it from the welded CNT leaving it electrically welded with the electrode as
shown in Figure A-3(b). The same procedure is repeated for attaching each CNT to the electrode.
It is important to monitor and control the laser power carefully since the heat may melt an
adjacent electrode as well and may cause the CNT attached to it to re-align incorrectly or even
detach.
87
Figure A-3: Laser welding process for CNT attachment. (a) Side view image of CNT just prior to
being laser welded to alloy filled pcf. Inset shows the schematic top view illustrating the fiber
taper used for delivery. (b) Laser welded CNT.
The increased risk of device failure associated with the laser welding method prompted
us to investigate another technique for picking and targeted bonding of individual CNT using
FIB. In this method, the CNT dispersion is air-dried on a TEM lacie carbon grid (Electron
Microscopy Sciences LC-200 Cu). The CNT lying on the carbon mesh is attached using FIB
assisted platinum deposition (FEI Helios NanoLab 660) to a fine tungsten probe (tip size
~500nm) (7.7 pA at 30 kV) with the CNT aligned correctly (Figure A-4(a) and Figure A-4(b)). It
may be necessary to mill individual synapses of the carbon mesh which may be adhered to the
CNT as the nanotube is lifted off the grid. The tungsten probe carrying the CNT is then navigated
near the filled electrode and the CNT is bonded with the alloy using electron beam assisted
tungsten deposition (3.2 nA at 2 kV) (Figure A-4(c)). After CNT attachment the probe is milled at
its tip to release it from the CNT. We note that the fiber is coated with iridium (Emitech K575X)
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prior to using FIB to avoid charging under ion beam. In this way, we have fabricated many
devices by sequentially attaching multiple individual CNT symmetrically around the solid core of
the PCF to individual electrodes. Figure A-4(d) shows the ion beam image of such a typical
fabricated device. We found that the yield of device fabricaton was significantly improved using
FIB owing to the controlled assembly in our pick and bond technique.
Figure A-4: FIB assisted CNT attachment. (a) CNT lying on lacie carbon mesh is bonded to 500
µm tungsten probe tip via Pt deposition. (b) CNT liftoff. Part of carbon mesh can be seen adhered
to the CNT. Individual synapses need to be milled for ‘clean’ liftoff. (c) Sequential targeted
attachment of CNT via e-beam assisted tungsten deposition. (d) Ion beam image of a three-arm
device.
Device characterization
In order to investigate the performance of the proposed opto-electronic nano-probe we
performed both numerical simulation and experimental measurements to characterize the group
velocity dispersion and electromechanical actuation. We first measured the refractive index and
extinction coefficient of Ostalloy 158 by using a rotating compensator spectroscopic ellipsometer
89
(M-2000, J.A. Woollam Co.). The alloy sample is an approximately 200 µm thin disk (shown in
inset of Figure A-5(a)) that was prepared by melting a small drop of alloy over a glass slide and
letting it reflow as it solidified on the slide. The disk was then carefully flipped over so that its
smooth and un-oxidized underside is face-up and consistent ellipsometric data was measured in
reflection geometry at multiple angles and from several locations on the disk. Figure A-5(a) plots
the averaged refractive index and extinction coefficient over 600-1400 nm range obtained from
the ellipsometric measurement.
Figure A-5: (a) The average refractive index and extinction coefficients of Ostalloy 158 obtained
via ellipsometry of the inset sample, (b) simulation of the typical optical mode field for the
unfilled PCF, (c) the mode field when three of the most center air holes are filled with Ostalloy
90
Group velocity dispersion
Using the experimentally measured conductivity of Ostalloy and the dispersion for silica
modeled using its Sellmeier coefficients, we simulated the eigenmodes and effective mode index
( )effn of the selectively filled PCF in COMSOL. Figure A-5 (b-c) shows the simulated field
distribution of the fundamental mode within the core of the bare as well as selectively filled PCF.
The presence of the electrodes in close vicinity of the core along the entire length of the fiber
slightly distorts the mode field distribution as can be seen in Figure A-5(c). The group velocity
dispersion of the PCF can be computed using 2
2
( )( )
effd nD
c d
. In this way by simulating
the effective index for both the bare as well as selectively filled PCF, we numerically obtained the
dispersion curve for each from 1.1 μm to 1.3 μm.
We also used spectral holography to experimentally characterize the group velocity
dispersion of both the bare and selectively filled PCF. Spectral holography is a sensitive and
linear optical technique to measure the relative spectral phase of the complex optical field. [195]
For spectral holography a supercontinuum source with a broad spectrum (600-1600 nm) is
generated via four-wave mixing by propagating sub-ns pulses at 1064 nm in a nonlinear PCF
(SC-5.0-1040). The reference arm of the Mach-Zehnder interferometer shown in Figure A-6(a)
consists of the supercontinuum beam delayed using a delay line. The ‘object arm’ consists of the
PCF under test. By appropriately adjusting the delay line to obtain consistent and good quality
fringes over the 1.1μm -1.3μm wavelength region, we detect the spectral hologram of the field at
the output of the PCF using an optical spectrum analyzer (Ando AQ-6315E). The spectral
hologram can be modeled as 2 2 2( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )S R S R S R S R where
( )R is the reference field and )(S denotes the output field from the PCF. Typical recorded
spectral hologram is shown in Figure A-6(b). Since the spectral hologram contains information in
91
its interference fringes, the appropriate sideband of the inverse Fourier transform of the hologram
is digitally filtered, and the Fourier transform of the filtered sideband yields the complex
spectrum ( ) ( )S R . In order to remove the contribution due to the presence of other optical
components in the ‘object arm’, we additionally obtain the spectral hologram in the absence of
the fiber, and thereby calibrate the transfer function of the ‘object arm’. Taking the ratio of
processed sideband terms of these two measurements, we can acquire the complex transfer
function containing both the amplitude and the phase information of the selectively filled PCF.
Figure A-6(c) shows the typical relative spectral phase of the filled PCF. We believe that the
noise in the relative spectral phase can be attributed to the fact that the low melting point Ostalloy
is observed to exhibit plasticity during the measurement when exposed to focused laser light. This
may lead to variation in the optical path in the object arm of the interferometer. Since the group
velocity dispersion is obtained from the 2nd
order derivative of the relative spectral phase, we
smoothen the measured phase by fitting a curve with a 3rd
order polynomial in Matlab. Finally we
compute the group velocity dispersion values for D [ps/nm/km] by taking into account the length
of the fiber used in the measurement. Thus following this procedure we experimentally obtained
the dispersion parameter plot for both the bare (unfilled) and selectively filled PFC. Figure A-6(d)
shows the comparison of the numerically simulated (solid lines) and experimentally obtained
(dotted lines) curves for D for both such fibers. It also plots the manufacturer supplied data for D
for the unfilled fiber (black line). Our results show that the selective metal filling does not
damage the waveguiding properties of the unfilled PCF. In other words, the optical channel and
electric channel in our device can be accessed without affecting the performance of each other.
However we do note that the transmission efficiency of the selectively filled PCF can be
significantly lower than the unfilled fiber, especially when the holes being filled are immediately
adjacent to the central core. [169]
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Figure A-6: Group velocity dispersion for nano-probe with 3 filled electrodes. (a) Mach-Zehnder
setup for spectral holographic measurement. (b) Typical recorded spectral hologram (c) Measured
relative phase, and 3rd
order polynomial fit, (d) estimated dispersion parameter for unfilled pcf
(red dotted) and fabricated device (blue dotted). Corresponding COMSOL simulated dispersion
for unfilled pcf (blue red line) and modeled device (solid blue line) are also plotted for
comparison. The black curve is dispersion parameter for bare fiber as specified by manufacturer.
Electromechanical characterization
The ability to control the motion of multiple nanoprobe tips allows the possibility to
mechanically interact with the sample under study as well as interrogate the sample electrically
and optically. A basic requirement of such a nanoprobe device is the ability to actuate the probe
tips to open and close them. Our proposed opto-electronic nano-probe consists of multiple CNTs
attached to each filled electrode in the selectively filled PCF. To investigate this ability in our
device, we experimentally measured the deflection of the CNT tips by applying variable potential
93
difference of up to 40V between the pair of electrodes. In particular, a pair of tungsten electrodes
was positioned in place using three-dimensional precision stages in order to contact the two
electrodes exposed at one facet of the filled PCF (without CNTs). We highlight the fact that each
electrode was independently tested for its electrical continuity by measuring its d.c. resistance as
mentioned earlier (see Figure A-2(d)). The tungsten electrodes were connected to a signal
generator to apply a manually variable dual polarity dc voltage (±20V) or generate a pure tone
sine wave of ac amplitude 40V. Thus the CNTs connected at the other facet accumulated charges
of opposite polarity and varying density as the voltage was varied and bent towards each other in
response to attracting charges. We recorded a video of this motion of the CNTs on a CCD
camera. Figure A-7(a) shows a montage of three frames in sequence from the recorded video in
which the CNT deflection is visible. The observed range of nano-tube deflection was small. To
help visualize the deflection, Figure A-7(b) plots the line trace of the image along the line marked
in each frame. We observe a deflection of the tip of the nanotubes of 2 to 3 pixels (note the pixel
numbers in the magnified Figure A-7(c)). The deflection is more pronounced in the lower CNT
than the upper (in the frame). Accounting for the magnification of our imaging system, this
corresponds to approximately 260 nm to 392 nm deflection.
94
Figure A-7: Electro-mechanical deflection of fabricated nano-probe. (a) Montage of 3 sequential
frames from a video recording CNT deflection. The dotted line shows the trace along which
intensity is plotted in (b) and (c). The markers in (c) indicate pixel positions of the ‘tip’ of the
lower CNT in each frame.
To investigate the relatively small deflection of the nano-tubes, we numerically simulated
the bending of the CNTs in a model of the fabricated device. We utilized the MEMS module of
COMSOL Multiphysics to solve the coupled physical model based on structural mechanics and
electrostatics. This involves modeling the CNT as an electrically conductive and linearly elastic
material by specifying its physical parameters including density (ρ), Young's Modulus (E) and
Poisson's ratio (ν). With the exception of the resistivity, which was measured experimentally, the
physical properties of the CNT were chosen to be in between those of graphite and multi-walled
CNTs. These values are ρ=2300 kg/m3, E=1.28 GPa, and ν=0.35 respectively. The choice of
Young's Modulus was most critical as this parameter varies several orders of magnitude (TPa to
GPa) based on the number of walls and diameter of the CNT. [196], [197] We therefore chose
the values noted above as being between those of bulk graphite and small MWCNTs. The length
95
and diameter of the CNT in the simulations were chosen to be 20 µm by 200nm as an average of
those used in the experiments. The accumulation of charge on the tips of the CNT in response to
applied potential difference between the electrodes was verified through two different methods.
Two dimensional simulations were performed, in which CNTs are embedded in a silica substrate
and suspended in air. The first simulation was conducted in the time domain to verify the
accumulation of charges at the tip rather than at an arbitrary point along the length of the tubes.
Figure A-8(a) plots the change in the space charge density at the tip of the CNT for ±20V applied
to the electrodes. The charge accumulation at the tip follows the input triangle wave as expected.
While only the magnitude of the charge accumulation at the tips is plotted, it was verified along
the length of the probes to be greatest at the free ends. The second simulation was to verify that
the deflection is a function of the instantaneous electrostatic force acting on the CNT tips as a
result of charge accumulation. The bending of the CNT in response to the charge accumulation
though a MEMS simulation in the steady state condition is illustrated in Figure A-8(b-c). Figure
A-8(b) shows the CNT deflection as they are bent towards each other. The nonlinear
displacement at the tips of the CNTs as a function of applied voltage in the simulation verifies
that the deflection is due to electrostatic forces. Both the time dependent and steady state
simulations thus indicate that accumulation of charge density at the CNT tip drives the
electromechanical deflection of approximately 670 nm upon application of ±20V as shown in
Figure A-8(c). There is a reasonable agreement between the electro-mechanical deflection of the
tip observed experimentally and that obtained from simulation. We highlight that a key difference
between the PCF based opto-electronic nano-probe presented here and several devices reported in
the literature is that the CNTs in our fabricated device are spatially situated ~24 µm apart due to
the use of PCF with 12 µm core, whereas most devices reported in the literature feature a
separation of <2 µm. [192]–[194] This is the main reason why the deflection observed in the tips
of the CNTs is less compared to typical performance reported on nanotweezers. We verified this
96
inference by simulating more closely spaced CNTs with identical properties in COMSOL, which
indeed shows a significant bending effect as the CNT separation is reduced. We believe that using
a PCF with a smaller core size as the platform for filling and attaching the CNTs will result in
improved performance. Another approach to improve the CNT deflection is to taper the fiber at
one end so that the separation between the CNTs is much smaller.
Figure A-8: Numerical simulation results. (a) charge distribution at the tip of a CNT when a -20V
ramp wave is applied to the terminal, (b) charge distribution along two CNTs with ±20V applied.
The deflection of the CNT is indicated by the red line which marks the original position, (c) the
tip displacement as a function of voltage applied to the electrodes within the PCF.
Conclusion
We have introduced a novel hybrid optoelectronic nanoprobe device based on a
selectively filled PCF. Our selective filling method involves use of UV curable optical glue to
seal all PCF holes except the ones intended to be filled and then pumping molten metal under
high pressure. We have demonstrated the method by filling electrically continuous electrodes
97
made of a low melting point alloy as well as using indium metal. Furthermore, we presented two
techniques for precise and targeted attachment of individual carbon nanotubes to each filled
electrode, namely laser welding and pick and bond using FIB and electron beam assisted
deposition. Our device can guide light and conduct electricity to potentially allow both optical
and electronic probing capability to interrogate a nanoscale sample held in place by the attached
nanotubes. Towards this aim we characterized the optical waveguiding property and movement
control of the attached nanotubes by estimating the group velocity dispersion and the electrically
actuated deflection of the CNT tips both experimentally and numerically, which agree well with
each other. The concept of functionalizing a PCF to tune its properties by incorporating foreign
materials in the air holes has led to the evolution of optical fiber as platform for fiber-integrated
opto-electronics. On the other hand, development of electrically actuated nanotweezers has been
recognized as an important step towards assembling complex nanostructures, and as an enabling
technology for realizing nano-electro-mechanical and nano-robotic systems. While the device we
have fabricated and presented here is elementary, it embodies the concept of combining fiber-
based opo-electronic sensor with nanotweezers in a single package. We believe that this forward
looking hybrid feature makes our device unique and warrants further development and research.
98
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VITA
Nikhil Mehta
EDUCATION
PhD in Electrical Engineering, Pennsylvania State University, 2015
MS in Electrical Engineering, Pennsylvania State University, August 2010
SELECTED PUBLICATIONS
Refereed Journals
N. Mehta∗, C. Yang∗, Y. Xu, & Z. Liu, “Characterization of the spatio-temporal evolution of
ultrashort optical pulses using FROG holography”, Optics Express, 22(9) (2014) (* equal
contribution)
J., Corey, N. Mehta, D. Ma, A. Elas, N. Perea-Lpez, M. Terrones, & Z. Liu, “Ultrashort optical
pulse characterization using WS2 monolayers”, Optics Letters, 39(2) (2014)
J. Corey, Y. Wang, D. Ma, N. Mehta, A. Elas, N. Perea-Lpez, M. Terrones, V. Crespi, & Z. Liu,
“Extraordinary second harmonic generation in tungsten disulfide monolayers”, Scientific Reports,
4(5530) (2014)
Y. Maan, A. Deshpande, V. Chandrashekar, ... N. Mehta, et al. “RRI-GBT multi-band receiver:
motivation, design, and development”, The Astrophysical Journal Supplement Series, 204(1)
(2013)
N. Mehta, J. Chen, Z. Zhang, & Z. Liu, “Compressive multi-heterodyne optical spectroscopy”,
Optics Express, 20(27) (2012)
P. Edwards, N. Mehta, K. Shi, X. Qian, D. Psaltis & Z. Liu, “Coherent anti-Stokes Raman
scattering holography: theory and experiment, Journal of Nonlinear Optical Physics & Materials,
21(2) (2012)
K. Yoo, N. Mehta, & R. Mittra, “A new numerical technique for analysis of periodic structures,
Microwave and Optical Technology Letters, 53(10) (2011)
A. Cocking∗, N. Mehta∗, K. Shi, & Z. Liu, “Compressive coherent anti-Stokes Raman scattering
holography”, Optics Express, 23(19) (2015) (* equal contribution)
Pending
N. Mehta, D. Ma, C. Zhang, A. Cocking, H. Li & Z. Liu, “Development of opto-electronic fiber
device with multiple actuating nano-probes”, submitted to Nanotechnology.