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102 NATURE PHOTONICS | VOL 7 | FEBRUARY 2013 | www.nature.com/naturephotonics

The real-time measurement of fast non-repetitive events is arguably the most challenging problem in the field of instru-mentation and measurement1–4. These instruments are

needed for investigating rapid transient phenomena such as chemi-cal reactions, phase transitions, protein dynamics in living cells and impairments in data networks. Optical spectrometers are the basic instrument for performing sensing and detection in chemi-cal and biological applications1–6. Unfortunately, the scan rate of a spectrometer is often too long compared with the timescale of the physical processes of interest. In terms of conventional optical spec-troscopy, this temporal mismatch means that the instrument is too slow to perform real-time single-shot spectroscopic measurements because it either employs a moving component, such as a trans-lating slit, or relies on a detector array, such as a charge-coupled device (CCD), with limited refresh rate (typically up to ~10 kHz)1–6. Single-shot measurement tools7–15 such as frequency-resolved opti-cal gating (FROG)8–13 and spectral phase interferometry for direct electric-field reconstruction (SPIDER)14,15 are, although powerful, therefore unable to perform pulse-resolved spectral measurements in real time. Although pump–probe methods offer the ability to perform time-resolved spectroscopic analysis with extremely fine temporal resolutions of less than 1 ps, they are based on the stro-boscopic measurement technique and hence do not operate in real time16–19, thus making them unable to capture non-repetitive and rare events such as those found in complex physical, chemical and biological systems.

Dispersive Fourier transformation (DFT)20–23 — also known as real-time Fourier transformation24–27 — is a powerful method that overcomes the speed limitation of traditional spectrometers and hence enables fast real-time spectroscopic measurements. DFT is an example of the analogy between paraxial diffraction (that is, Fraunhofer diffraction) and temporal dispersion28,29. This analogy, known as space–time duality, emerges from Maxwell’s equations and has been employed to produce a wide variety of elegant methods (including DFT) for high-speed all-optical signal processing such as Fourier optics30,31 and temporal imaging (the temporal equivalent of

Dispersive Fourier transformation for fast continuous single-shot measurementsK. Goda1,2,3,4* & B. Jalali2,3,4,5,6

Dispersive Fourier transformation is an emerging measurement technique that overcomes the speed limitations of traditional optical instruments and enables fast continuous single-shot measurements in optical sensing, spectroscopy and imaging. Using chromatic dispersion, dispersive Fourier transformation maps the spectrum of an optical pulse to a temporal waveform whose intensity mimics the spectrum, thus allowing a single-pixel photodetector to capture the spectrum at a scan rate significantly beyond what is possible with conventional space-domain spectrometers. Over the past decade, this method has brought us a new class of real-time instruments that permit the capture of rare events such as optical rogue waves and rare cancer cells in blood, which would otherwise be missed using conventional instruments. In this Review, we discuss the principle of dispersive Fourier transformation and its implementation across a wide range of diverse applications.

a classical lens)32–35. Temporal imaging has been applied to ultrafast waveform measurement36–39, waveform compression40 and temporal cloaking (the temporal analogue of optical cloaking)41. In the appli-cation of space–time duality to DFT, the concept that diffraction in the far-field regime causes the spatial frequency spectrum of light to appear as an intensity image is extended to the time domain. Specifically, using chromatic dispersion, DFT maps the temporal fre-quency spectrum of a pulse to a temporal waveform whose intensity envelope mimics the spectrum20–27. This occurs when the pulse prop-agates inside a dispersive medium with group-velocity dispersion (GVD). For this to occur, the pulse must propagate sufficiently to sat-isfy the temporal equivalent of the far-field condition in diffraction.

The ability of DFT to perform fast continuous single-shot meas-urements is made possible by replacing the diffraction grating and detector array in traditional spectrometers with a temporal dis-persive element and single-pixel photodiode. In other words, this approach converts the spatial rainbow pattern into a temporal waveform that resembles a communication signal. The tempo-ral waveform is stretched in time (bandwidth compressed) so that it is slow enough to be captured by a photodetector and real-time digitizer. Simultaneously, the waveform can be optically amplified to overcome the thermal noise floor of the high-speed photodiode. Because the time-stretched waveform is sufficiently slow, DFT allows the optical spectrum to be measured directly in the time domain. By sampling the temporal waveform, a real-time analog-to-digital converter (ADC) samples the optical spectrum at a scan rate signifi-cantly beyond that of a conventional grating-based spectrometer20–27.

With its unique ability to perform ultrafast real-time spectro-scopic measurements, DFT has been employed for a diverse range of sensing and detection applications. The first and most obvious appli-cation of DFT is spectroscopy21,23,42–48. DFT enables us to identify the dynamics of optical spectra in absorption and Raman spectroscopy that would otherwise be invisible using conventional space-domain spectrometers23,46–48. It has hence led to the discovery of optical rogue waves — an optical counterpart of rogue waves, oceanic waves in the open sea with extremely large amplitude46, and the demonstration

1Department of Chemistry, University of Tokyo, Tokyo 113-0033, Japan. 2Department of Electrical Engineering, University of California, Los Angeles, California 90095, USA. 3California NanoSystems Institute, Los Angeles, California 90095, USA. 4Department of Bioengineering, University of California, Los Angeles, California 90095, USA. 5Department of Surgery, David Geffen School of Medicine, University of California, Los Angeles, California 90095, USA. 6Eli and Edythe Broad Center of Regenerative Medicine and Stem Cell Research, David Geffen School of Medicine, University of California, Los Angeles, California 90095, USA. *e-mail: [email protected]

REVIEW ARTICLESPUBLISHED ONLINE: 31 JANUARY 2013 | DOI: 10.1038/NPHOTON.2012.359


NATURE PHOTONICS | VOL 7 | FEBRUARY 2013 | www.nature.com/naturephotonics 103

of stimulated supercontinuum generation — a highly stable type of coherent broadband (white) light49,50. Another application of DFT is fast continuous imaging51–59, which uses DFT as a means of simul-taneously performing image serialization, time dilation and image amplification, all in the optical domain. This imaging method has recently been combined with microfluidics and high-speed digital signal processing to provide high-throughput automated image cytometry and real-time detection of rare cancer cells in blood57. DFT can also be converted to an inertia-free laser scanner for per-forming high-speed surface vibrometry60 and optical coherence tomography (OCT)61–65. Moreover, DFT has been applied to analog-to-digital conversion to achieve much higher sampling rates than those of conventional electronic digitizers66–71.

In this Review, we discuss the principle of DFT and its broad util-ity in scientific research, industry and biomedicine. Specifically, we discuss the theory, methods and applications of DFT. First, we cover the theory of DFT, including the requirements for the spectrum-to-time mapping and the specifications of DFT, such as spectral resolution, scan rate and sensitivity (optical amplification). We then discuss various methods for the implementation of DFT and pro-vide a standard recipe for building a DFT system. Finally, we cover the various applications of DFT, which include spectroscopy, imag-ing, laser scanning and analog-to-digital conversion.

Theory of DFTPrinciple. DFT is conceptually depicted in Fig. 1a. The disper-sive Fourier transformer consists of a dispersive element with a large GVD (for example, a dispersive fibre or chirped fibre Bragg

grating) and a photodetector (for example, a photodiode or ava-lanche photodetector)20–27,42–48. When a train of optical pulses enters the dispersive element, the spectrum of each pulse is mapped to a temporal waveform by the large GVD in the disper-sive element. In other words, the resulting time-domain waveform of each pulse is the temporal analogue of the far-field (Fraunhofer) diffraction pattern in the spatial domain. The temporally dispersed pulse train is then detected by the photodetector, digitized by an ADC, and subject to digital signal processing for pulse-by-pulse spectral analysis. The modulation signal is also stretched in the time domain. This analog bandwidth compression enables us to digitize the fast single-shot spectra at high scan rates in real time. Because DFT operates in the far-field temporal dispersion, it relies on the far-field approximation being satisfied. As discussed later in this Review, the achievable spectral resolution improves with the extent of the far-field propagation and hence with an increasing amount of GVD.

Optical amplification. The definite advantage of DFT over con-ventional grating-based spectrometers is its ability to use the dispersive element as a gain medium for simultaneous opti-cal amplification of the spectrum23,51–57. With distributed optical amplification in the dispersive fibre (Fig. 1b), this process, known as ‘amplified’ DFT, overcomes the fundamental three-way trade-off in spectroscopic measurements between sensitivity, speed and spectral resolution (that is, trade-off relations between sen-sitivity and speed, between speed and spectral resolution, and between spectral resolution and sensitivity) — a predicament

Figure 1 | Dispersive Fourier transformer. a, Dispersive Fourier transformer. When a train of optical pulses enters a dispersive element with a large GVD, the spectrum of each pulse is mapped to a temporal waveform by the dispersive element and then captured by a single-pixel photodetector. b, Amplified dispersive Fourier transformer. Distributed optical amplification in the dispersive medium enables simultaneous signal amplification. c, Simulation of the evolution of an optical pulse in DFT. Top, spectrum of the pulse in the frequency (wavelength) domain. Bottom, waveform of the pulse in the time domain for various GVD values: (i) −120 ps nm–1; (ii) −360 ps nm–1; and (iii) −1,080 ps nm–1. Figure c reproduced with permission from ref. 22 © 2009 APS.

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REVIEW ARTICLESNATURE PHOTONICS DOI: 10.1038/NPHOTON.2012.359



that exists in space-domain spectroscopy. Although loss in the dispersive fibre can be compensated for by employing discrete optical amplifiers such as rare-earth-doped fibre amplifiers and semiconductor optical amplifiers, distributed optical amplifica-tion via stimulated Raman scattering within the dispersive fibre is superior because it maintains a relatively constant signal level throughout the frequency-to-time mapping process. This property is critical because it maximizes the signal-to-noise-and-distortion ratio by keeping the signal power away from low power (noisy) and high power (nonlinear) regimes. This advantage of distributed Raman amplification over the use of discrete amplifiers (includ-ing lumped Raman amplifiers) is known in long-haul fibre-optic communication links72,73. In addition to a lower noise figure, the intrinsic properties of Raman amplification, such as its naturally broadband gain spectrum due to the amorphous nature of opti-cal glass fibres72–75, are also favourable for amplified DFT. The gain bandwidth can be further broadened by using multiwavelength pump lasers. In addition, extremely broadband gain spectra can be realized by employing incoherent pump sources. Raman-amplified dispersive elements also eliminate the need for high-power optical sources, which can potentially damage or modify the sample under study for spectroscopic measurements75.

Mathematical formalism. From the nonlinear Schrödinger equa-tion with linear gain and loss, the propagation of an optical pulse through a GVD element with up to the second-order dispersion coefficients (that is, negligible higher-order dispersion coefficients within the bandwidth of interest) is described as22,73,75,76:

(1)β2zβ2z2T

-∞

∞ ~ 2 2

u(0, ω − ω0)ei ∫2

2π1|u(z,T)|2 = e(g − α)z ω − ω0 dω ,

where u is the field amplitude of the optical pulse, ω0/2π is the centre frequency of the pulse, g is the gain coefficient (for amplified DFT), α is the absorption coefficient, z is the propagation distance (equiva-lent to the GVD length), β2 is the second-order dispersion coeffi-cient and T is the time in the reference frame of the pulse that propagates at the group velocity given by T = t − β1z. Here, β1 is the first-order dispersion coefficient and is also the reciprocal of the group velocity. In equation (1), for large values of GVD (that is, β2z), the integrand oscillates rapidly in the frequency domain except at the frequency at which the phase vanishes, which is given by:

(2)β2zT ω = ω0 + .

This criterion, also known as the saddle-point approximation, cre-ates a one-to-one mapping between the optical frequency and time, which can be better understood by rewriting equation (2) in the form T(ω) = β2z(ω − ω0). From equation (1), the intensity pro-file of the temporally dispersed pulse then becomes propor-tional to12:

β2zT

πβ2z2|u (z,T)|2 = e(g − α)z u 0,~ ( )

,

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which indicates the equivalence between the time-domain inten-sity modulation of the signal and its own optical spectrum (that is, the frequency-domain signal) with a proportionality constant of 2exp[(g−α)z]/(πβ2z) and the frequency-to-time mapping relation given by equation (2). The frequency-to-time transformation pro-cess is simulated and shown in Fig. 1c. The linear mapping is a property of the linear dependence of the group delay on the fre-quency. Higher-order frequency dependency results in nonlinear frequency-to-time mappings known as time warps77 and requires

corrections to the mapping relation, with general results in the far-field approximation being22:

βm + 1zT (ω) = ∑ (ω − ω ) , m = 1

∞

m! 0m

where βm are the higher-order dispersion coefficients.The basic parameters of DFT, including spectral resolution,

scan rate, number of data points on the spectrum and their inter-relations, are obtained as follows. First, the scan rate of DFT, R, is equivalent to the pulse repetition rate of the laser. Second, the mapping relation between wavelength and time is given by Δτ = |D|zΔλ, where Δλ is the bandwidth of the laser, D is the GVD of the dispersive element per unit length (typically expressed in units of ps nm–1 km–1) and Δτ is the time duration into which the laser spectrum is mapped. Here, to avoid overlaps between consecutive pulses that are temporally stretched, the upper limit on Δτ (and hence on D) is imposed by the scan rate (pulse rep-etition rate) such that Δτ < R–1. Third, the number of sampled points over the mapped spectrum in the time domain is given by N = fΔτ = f |D|zΔλ, where f is the sampling rate of the electronic digitizer. The sampling rate of the digitizer limits the spectrometer resolution, δλdig, according to δλdig = (f |D|z)–1. Likewise, another limit on the spectral resolution is imposed by the analog band-width of the detection system (consisting of the photodetector and digitizer), giving δλdet = (B|D|z)–1. In an ideal digitizer, B and f are related by the Nyquist criterion f = 2B, although in practice B < f/2. Furthermore, the GVD of the dispersive element also lim-its the spectral resolution. This is the temporal equivalent of the far-field spectral resolution in spatial diffraction and is given by δλGVD = λ0√(2(c|D|z)–1), where λ0 is the centre wavelength of the laser and c is the speed of light in vacuum. Combining this with the upper limit on Δτ and the number of sampled points yields a limit on the spectral resolution as a function of the pulse repetition rate and optical bandwidth: δλGVD > λ0√(2RΔλc–1). Finally, combining all the limiting spectral resolutions, the overall limit of DFT on the spectral resolution is found to be:

(3)δλ = max (δλdig, δλdet, δλGVD).

Methods for DFTThe principal requirement for DFT is a sufficiently large and lin-ear GVD such that the frequency-to-time mapping process occurs without distortion. Various methods for DFT have been proposed and demonstrated over the past decade (Fig. 2a–c) to cover different spectral bands (centred about 800 nm, 1,000 nm and 1,550 nm with ~100 nm optical bandwidth). A typical experimental demonstration of DFT is shown in Fig. 2d, which indicates a one-to-one mapping between the optical wavelength (frequency) and time. The most common approach is the use of standard single-mode fibres such as long-haul transmission fibres59,78,79, dispersion-compensation fibres (DCFs)42–44,46–48,51–57 and small-core fibres60. Other methods for DCF include the use of chirped fibre Bragg gratings (CFBGs)25–27,80, large spatial chirps81–83, and chromo-modal dispersion (CMD)84,85. In this section, we discuss the working principles of these different tech-niques for DFT, as well as their advantages and disadvantages.

Single-mode fibres. The simplest and most popular method for DFT is the use of standard single-mode fibres (Fig. 2a). However, the type of single-mode fibre used varies depending on the spectral range and requirements of the set-up. Chromatic dispersion for a single-mode fibre is a function of wavelength and the sum of material and wave-guide dispersions73,75. Material dispersion arises from the change

REVIEW ARTICLES NATURE PHOTONICS DOI: 10.1038/NPHOTON.2012.359



in a material’s refractive index with wavelength, which changes the propagation velocity of light as a function of wavelength. Waveguide dispersion is a separate effect that arises from the dependence of mode size relative to waveguide dimensions with wavelength. For example, decreasing the wavelength increases the relative waveguide dimensions, which causes a change in the distribution of light in the cladding and core. Because material and waveguide dispersions are wavelength dependent, the dispersion is a function of wavelength and hence the dispersion slope can be either positive or negative.

The most popular type of long-haul transmission fibres is Corning’s SMF-28 fibre86, which is often employed for long-haul fibre-optic communication. The SMF-28 fibre provides moder-ate GVD in the spectral band of 1,200–1,600 nm and its GVD is characterized by the equation D(λ) ≈ S0(λ − λ0

4/λ3)/4 ps nm–1 km–1, where S0 is the zero dispersion slope (≤0.092 ps nm–2 km–1), λ0 is the zero dispersion wavelength (1,302 nm ≤ λ ≤ 1,322 nm), and λ is the operating wavelength. The dispersion slope is relatively linear and smooth for an optical bandwidth of ~200 nm centred about 1,550 nm. Based on the equation, the GVD at 1,550 nm is 17.4 ps nm–1 km–1. The attenuation of the SMF-28 fibre is excellent (0.34 dB km–1 at 1,310 nm, and 0.20 dB km–1 at 1,550 nm) because it is designed to minimize the inherent optical loss in the fibre for long-haul transmissions. Recently, a single-mode fibre with specifi-cations similar to those of the SMF-28 fibre was shown to be effec-tive for DFT at 1 μm (ref. 59).

The DCF — a class of optical fibre designed specifically for long-haul communications — compensates for unwanted disper-sion in the transmission path to minimize the total dispersion. A negative dispersion slope allows dispersion to be cancelled over a larger wavelength range because the dispersion slope of the standard

SMF-28 fibre is usually positive. The DCF has a smaller effective mode area (about 20 μm2), a larger GVD (about −120 ps nm–1 km–1) and larger loss (around 0.4 dB km–1) than the SMF-28 fibre (80 μm2, 17.4 ps nm–1 km–1 and 0.2 dB km–1, respectively). Similar to the SMF-28 fibre, the dispersion slope of the DCF is smooth for an optical bandwidth of more than 200 nm. Although the DCF is excellent for DFT in terms of dispersion-to-loss ratio and bandwidth, its major downside is that it is available only in the fibre-optic communication band around 1,550 nm.

The small-core fibre provides sufficient GVD in spectral bands out-side the fibre-optic communication band centred around 1,550 nm. DFT has previously been restricted to the fibre-optic communication band owing to the commercial unavailability of dispersive fibres with high dispersion-to-loss ratio outside the 1,550 nm band. However, shorter wavelengths are desirable for applications such as biomedi-cal imaging and spectroscopy. For example, the spectral band around 800 nm is suitable for biomedical applications because it enables larger penetration depths in tissue and a reduction in autofluorescence. For DFT at around 800 nm, researchers have used a single-mode fibre whose effective mode field diameter is smaller (4 μm) than that of con-ventional fibres (5–6 μm) in the operating wavelength range. The fibre therefore provides an increased contribution of waveguide dispersion to the total chromatic dispersion of the fibre, resulting in a GVD of approximately −120 ps nm–1 km–1 at 800 nm. Although the small-core fibre is effective for DFT in spectral bands other than 1,550 nm, its major downside is its high loss (around 3 dB km–1) due to the large contribution of Rayleigh scattering at shorter wavelengths.

CFBGs. The CFBG is a type of distributed Bragg reflector con-structed in a short segment of an optical fibre that reflects

Figure 2 | Methods for DFT. a, Standard single-mode fibre. This may be a standard fibre, DCF or small-core fibre, depending on the requirements for the spectral band, bandwidth and loss. b, Chirped fibre Bragg grating. The sign of the GVD can be chosen depending on the injection port. c, Chromo-modal dispersion. Different wavelengths of light experience different mode profiles in the multimode fibre. d, Typical temporal waveform obtained with DFT. The inset shows the one-to-one mapping between the temporal waveform of a pulse captured by an oscilloscope and the spectrum of the pulse measured by a conventional optical spectrum analyser. Figure d reproduced with permission from ref. 60 © 2012 NPG.

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particular wavelengths of light and transmits all others (Fig. 2b). This is achieved by creating a periodic variation in the refractive index of the fibre core, which generates a wavelength-specific die-lectric mirror. The refractive index profile of the grating is chirped so that different wavelengths reflected from the grating undergo dif-ferent time delays, thus leading to a large GVD. The advantages of the CFBG for DFT are its short length and ability to customize the amount of GVD easily, while its disadvantages include group-veloc-ity ripples that are converted to fast temporal modulations after DFT and its inability to offer internal optical amplification, which is possible with dispersive fibres. It is also important to note that the product of the total GVD and the optical bandwidth must be less than the repetition period of the laser. Hence, for a given laser repetition rate and large bandwidth, there is an upper limit on the amount of GVD.

CMD. Chromo-modal dispersion is a relatively new type of disper-sive device that exploits spatial chirping by combining a diffraction grating and a multimode fibre (Fig. 2c). It leverages the large modal dispersion of a multimode waveguide in combination with the angular dispersion of diffraction gratings to create chromatic dis-persion. Specifically, spatially dispersed broadband light is focused onto a multimode fibre via a lens that provides angular dispersion. The input facet of the multimode fibre is placed at the lens focus such that the various spectral components of the light are coupled into different fibre modes. The CMD can be tuned throughout both the anomalous and normal dispersive regimes by adjusting the alignment of the lens and fibre relative to the spatially dispersed spectrum. The advantages of CMD are its ability to provide a large GVD in any spectral band and to tune the amount and sign of GVD, while the disadvantages are its nonlinear dispersion slope and its challenge to perform optical amplification in the multimode fibre.

Recipe for building a dispersive Fourier transformerFirst, choose the spectral band of interest, as the method for DFT varies depending on the spectral band. Roughly speaking, the DCF, SMF-28 fibre and CFBG are all ideal choices for implementing DFT in the ~1,550 nm spectral band, whereas the small-core fibre, CMD and CFBG are ideal for shorter-wavelength bands of 800–1,000 nm. Second, choose the repetition rate of the optical pulses; that is, the scan rate of the dispersive Fourier transformer required for the par-ticular application of interest. This choice selects the optical band-width of interest not only because of the mapping relation between wavelength (frequency) and time, but also because of the constraint on the total GVD to avoid overlaps between consecutive pulses that are temporally stretched. Third, design the detection circuitry care-fully to achieve the desired spectral resolution. It is important to take equation (3) into account when choosing the photodetector and digitizer, given the dispersive element. Because the limiting fac-tors are the bandwidth and sampling rate of the digitizer (typically ~10 GHz and ~20 GS s–1, respectively), the bandwidth of the photo-detector (typically ~10 GHz), and the spectral resolution imposed by the GVD of the dispersive element itself (typically ~500 ps nm–1 or larger), the values for the parameters need to be calculated with care before building the DFT system, so as not to waste high speci-fications on any one of the limiting components. Fourth, identify whether simultaneous optical amplification is required. This can be estimated by comparing the optical signal strength to the ther-mal noise level of the detection circuitry. If optical amplification is needed to compensate for signal loss or simply to achieve a higher signal level, it is important to choose the right pump lasers in terms of both intensity and wavelength for stimulated Raman scattering, in which the difference between the Stokes and pump wavelengths is wavelength dependent (while it is constant in the frequency domain). This difference depends on the wavelength squared; for example, it is around 100 nm at 1,550 nm, but approximately 25 nm

at 800 nm. Alternatively, if discrete amplifiers (such as rare-earth-doped fibre amplifiers) are available in the spectral band, they can concurrently be used before or after the distributed Raman ampli-fier in the dispersive medium. Finally, develop a computational tool with proper algorithms to decompose the temporal data into a series of optical spectra at the pulse repetition rate. The goal of the computational tool is to interleave the measured consecutive pulses into a series of individual pulses that represent optical spectra. It is critical to identify the timing of each pulse carefully because there are fluctuations in the arrival time of the pulses arising from the laser source and other optical components. Figure 2d shows a typi-cal waveform obtained by DFT.

Applications of DFTOver the past decade, DFT has been applied to a variety of sci-entific, industrial and biomedical applications in the form of spectroscopy21,23,42–48,87–89, imaging51–59, laser scanning60–65 and analog-to-digital technology66–71. In this section, we discuss how DFT is implemented in these settings to realize fast continuous sin-gle-shot measurements that are not achievable using conventional methods. We also provide examples of using such measurements to capture dynamical processes as well as rare, non-repetitive and transient events.

Spectroscopy. DFT’s first and most obvious application is real-time spectroscopy. Real-time spectroscopy provides invaluable informa-tion about the evolution of dynamical processes such as non-repet-itive phenomena, but the continuous acquisition of rapidly varying spectra represents an extremely difficult challenge. DFT circum-vents the limited scan rate of conventional spectrometers (that is, up to ~100 kHz) and permits continuous single-shot measurements of rapidly evolving or fluctuating spectra at a scan rate equivalent to the pulse repetition rate of the laser. DFT has been applied to the real-time sensing of gas and combustion via absorption spec-troscopy, identification and control of optical rogue waves through continuous single-shot intensity measurements of repetitive pulses and the stimulated Raman spectroscopy of silicon.

DFT-based spectroscopy is realized by using DFT as a fast sin-gle-shot spectrometer, and has been broadly employed for vari-ous types of spectroscopic measurements. The first example is the broadband absorption spectroscopy of a gas mixture, in which the broad 600 nm bandwidth covered 1,100–1,700 nm for simultane-ous identification of CH4 and H2O (Fig. 3a)43. Here, the nonlinearity of the GVD in the dispersive fibre was corrected to calibrate the wavelength-to-time mapping process. The second example is the continuous single-shot acquisition of the rapidly evolving Raman spectrum of a silicon waveguide (Fig. 3b)23. The Raman spectra are plotted using the GVD parameter to calibrate the wavenumber scale. DFT allows many thousands of such spectra to be captured in one continuous measurement, thereby allowing a dynamical pro-cess to be monitored in real time. The third example is the experi-mental discovery of optical rogue waves (Fig. 3c)46, in which DFT was used to monitor the single-shot spectra of a large number of repetitive pulses that transmit through a highly nonlinear fibre and undergo supercontinuum generation. The ability to capture large data sets — even for rare events — is central to the development of models that demystify such events1. To this end, the generation of optical rogue waves was shown to be controllable by optimizing the initial conditions of noise-sensitive soliton-fission supercontinuum generation49. Recently, real-time fluctuations across the full band-width of a fibre supercontinuum have been studied to provide fur-ther insights into correlations between spectral components of the supercontinuum (Fig. 3d) and hence into simultaneous sideband generation, pump depletion and soliton-like pump dynamics48. The fourth example is the real-time capturing of the single-shot spectra of modulation instability, which reveals previously unknown and




surprising patterns in modulation instability formation47. The final example is the demonstration of a biologically inspired DFT-based vector spectrum analyser for measuring the amplitude and phase of rare optical signals in real time89.

Imaging. Fast real-time optical imaging is an indispensable tool for studying dynamical events such as shockwaves, chemical dynam-ics in living cells, neural activity, laser surgery and microfluidics. However, conventional imagers such as CCDs90,91 and comple-mentary metal–oxide–semiconductor92,93 devices are incapable of capturing fast dynamical processes with high sensitivity and tem-poral resolution. This is due in part to a technological limitation: it takes time to read out the data from the sensor array. There is also a fundamental trade-off between sensitivity and speed: at high frame rates, fewer photons are collected during each frame. This is a problem that affects nearly all optical imaging systems. In scientific applications, time-resolved imaging of the dynamics of fast events is often achieved using the pump–probe technique, but this requires the event to be repetitive94–99. Because this approach does not oper-ate in real time, it is unable to capture non-repetitive random events that occur only once or do not occur at regular intervals. The detec-tion of such events requires an imaging technology with fast, con-tinuous and real-time capability.

DFT has proved useful for high-speed imaging at record-high frame rates and shutter speeds. This technique — known as serial time-encoded amplified imaging or microscopy (STEAM)51–59 — is

made possible by employing amplified DFT for simultaneous opti-cal image serialization and amplification. Specifically, as shown in Fig. 4a, an image of the object is encoded into the spectrum of a broadband pulse that is reflected from or transmitted through the object. The image-encoded pulse is then temporally stretched or serialized by DFT and simultaneously amplified via stimulated Raman scattering in the dispersive fibre for optical image amplifi-cation. The optical image amplification is critical because it over-comes the fundamental trade-off between sensitivity and speed, and the time stretch makes it possible to digitize the image in real time. Figure 4b shows the dynamics of laser ablation captured by STEAM with a shutter speed of 440 ps, a frame rate of 6.1 MHz and an optical image gain of 25 dB (ref. 51). Figure 4c compares STEAM with conventional cameras in terms of imaging performance, indi-cating that STEAM’s superior shutter speed and optical image gain enable the capture of cells and microparticles in high-speed flow without motion blur. Recently, STEAM has been integrated with a flow cytometer and high-speed digital signal processing to per-form high-throughput automated image cytometry at a record-high throughput of 100,000 cells per second. The technique was used to identify rare cancer cells in a large heterogeneous population of blood cells with an unprecedented low false-positive rate of one in a million57. Figure 4d shows the screening process of the automated image cytometry system that screened around 100,000,000 images to identify rare (~10) cancer cells with a custom real-time digital image processor. This method is effective for the high-throughput

Figure 3 | Spectroscopy with DFT. a, Broadband absorption spectrum of a gas mixture (CH4 and H2O), covering an optical bandwidth of 1,100–1,700 nm. The absorption lines of the gas mixture are imprinted onto the spectrum of each optical pulse. b, Dynamics of the Raman spectrum of silicon captured by amplified DFT. c, Discovery of optical rogue waves with DFT. The occurrence probability and intensity distribution of optical rogue waves can be controlled, depending on the initial conditions. d, Study of the correlations between separated spectral components of a fibre supercontinuum. Figure reproduced with permission from: a, ref. 43 © 2007 OSA; b, ref. 23 © 2008 NPG; c, ref. 46 © 2007 NPG; d, ref. 48 © 2012 NPG.

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screening of an extremely small number (~10 per millilitre of blood) of circulating tumour cells — precursors to cancer metastasis that cause 90% of cancer deaths. It can analyse 10 mL of lysed blood in less than 15 min and holds promise for the evaluation of chemo-therapy and the early, non-invasive, low-cost detection of cancer.

Laser scanning. Laser scanning technology is an integral part of scientific research, manufacturing, defence and biomedicine100–103. In many applications, high-speed scanning capability is essential for scanning a large area in a short period of time and the mul-tidimensional sensing of moving objects and dynamical processes with fine temporal resolution. Such applications include the identi-fication of aircraft and automobiles using light detection and rang-ing4,100–104, the non-destructive evaluation of structural dynamics and microelectromechanical systems through laser-scanning sur-face vibrometry101–104 and the observation of biomechanical motility and neural activity using laser-scanning confocal and multiphoton microscopy105,106. Unfortunately, conventional laser scanners such as galvanometric mirrors and acousto-optic deflectors are often too

slow, thus providing limited precision and utility. Here, the princi-pal requirement for sensing and imaging such events in real time is a temporal resolution shorter than the timescale of changes in the dynamical process.

DFT can be used in conjunction with spatial dispersers to form a laser scanner that enables ultrafast inertia-free scans at speeds 1,000 times higher than those of conventional laser scanners60. This device — known as the hybrid dispersion laser scanner (HDLS) — is based on the spatial dispersion of a broadband optical pulse onto the target after DFT, with a temporal dispersion that ensures each frequency component of the pulse arrives at a different set of spatial coordinates on the target at a different time (Fig. 5a). The HDLS is therefore somewhat analogous to STEAM, which exploits spa-tiotemporal dispersion. The HDLS’s ability to scan without using mechanical and active electronic components eliminates the speed bottleneck that exists in galvanometric mirrors and other tech-niques, thus enabling ultrafast scanning at rates equivalent to the pulse repetition rate of the laser (typically 10–100 MHz). Figure 5b compares a CCD image of the test target to a digitally reconstructed

Figure 4 | Imaging with DFT. a, Schematic of STEAM. The spatial information of the object is encoded into the spectrum of each pulse and then decoded by the amplified dispersive Fourier transformer, all in the optical domain. b, Observation of laser ablation dynamics captured by STEAM operating at a frame rate of 6.1 MHz (corresponding to a temporal resolution of 163 ns). c, Comparison of STEAM with conventional cameras (shutter speeds of 17 ms (CCD) and 1 μs (CMOS)); optical image gains for imaging of cells and microparticles in high-speed flow are 0 dB (CCD), 0 dB (CMOS) and 30 dB (STEAM). d, Screening process of the automated image cytometry system to identify extremely rare cancer cells in blood. Figure reproduced with permission from: b, ref. 51 © 2009 NPG; c & d, ref. 57 © 2012 NAS, USA.

CCD CMOS STEAM

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image of the target scanned by the HDLS, with 27,000 resolvable points and a dwell time of 51 ps at a one-dimensional scan rate of 90.8 MHz and a two-dimensional scan rate of 105.4 kHz. Figure 5d shows a frame sequence of a 1 kHz nanomechanical surface vibra-tion captured by the HDLS in an interferometric configuration with a subnanometre axial resolution of 0.4 nm at a frame rate of 105.4 kHz (corresponding to a temporal resolution of 9.5 μs).

DFT-based laser scanning is also useful in OCT61–65. DFT has been used to perform temporal laser scanning and wavelength sweeping — both of which can significantly increase the axial scan rate of OCT. A train of temporally dispersed pulses behaves as a pas-sive wavelength-swept source, thus enabling OCT at an axial scan rate equivalent to the pulse repetition rate of the laser. Figure 5c shows laser ablation dynamics captured by DFT-based OCT that operates at record-high axial and lateral scan rates of 90.9 MHz and 50 kHz, respectively65 — a few orders of magnitude faster than con-ventional OCT methods. This approach is expected to be valuable for industrial applications in which high axial and lateral scan rates are required, such as material characterization, microfluidics, and manufacturing and process control.

Analog-to-digital conversion. The demand for higher-perfor-mance digitizers has become of paramount importance, owing to

the ever-increasing speeds of electronic circuits and data rates in communication systems. High-bandwidth digitizers are needed in defence applications such as radar and the detection of electro-magnetic pulses. In science, digitizers are a primary tool in particle accelerators, X-ray free-electron lasers and time-resolved fluores-cence microscopy. In telecommunications, fast digitizers are used for measuring eye diagrams to characterize impairments in high-speed serial links. All of these applications require high temporal resolution, thus making high-bandwidth digitizers invaluable.

To meet such needs, DFT has been employed to boost the bandwidth of electronic ADCs and enable the digitization of high-bandwidth signals that otherwise cannot be digitized with traditional electronic ADCs in real time66–71,107,108. This method — known as the photonic time-stretch ADC — slows down the analog signal in the time domain or compresses its bandwidth before it can be digitized by an electronic ADC, which would oth-erwise have insufficient bandwidth and sampling rate. As shown in Fig. 6a, the basic operating principle of the photonic time-stretch ADC involves time-stretching the original analog signal and then segmenting it using a time-stretch optical frontend. The segments are captured by conventional electronic ADCs, and the digitized samples are rearranged to obtain the digital representation of the original signal. Figure 6b shows the real-time digitization of

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t = 284 µs

200100

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200

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–1000

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200100

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Y (µm) X (µm)

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200100

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200

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200100

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200

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200100

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200100

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Figure 5 | Laser scanning with DFT. a, Schematic of HDLS. b, Image of the reflective target with the “UCLA” texture reconstructed by the laser scanner at a two-dimensional scan rate of 105.4 kHz (right), in comparison with the image of the target captured by a CCD camera (left). c, OCT with DFT. Left, one-dimensional laser ablation dynamics obtained by the OCT system at an axial scan rate of 90.9 MHz. Right, two-dimensional laser ablation dynamics obtained by the OCT system at a lateral scan rate of 50 kHz. d, Surface vibrometry of the nanomechanical vibration performed by the laser scanner in an interferometric configuration. Figure reproduced with permission from: a, b & d, ref. 60 © 2012 NPG; c, ref. 65 © 2012 OSA.




an ultrahigh-frequency radiofrequency signal with the photonic time-stretch ADC at a record-high sampling rate of 10 TS s–1 (cor-responding to a digitization resolution of 100 fs)69. In addition, an all-optical time-stretch ADC has recently been proposed and dem-onstrated. This approach combines four-wave mixing and the pho-tonic time-stretch ADC to record a 40 Gbit s–1 non-return-to-zero

on–off-keying optical data stream with a stretch factor of 54 and a back-end electronic bandwidth of 1.5 GHz at an effective sampling rate of 1.25 TS s–1 (Fig. 6c)71.

SummaryThis Review has covered the principle of DFT and its broad utility across a wide range of fields. High-speed instruments capable of capturing fast transient events are seeing increasing demand as the physical processes of interest become more complex. With its abil-ity to perform fast continuous measurements, DFT is expected to be useful for high-throughput screening and studying non-repeti-tive rare phenomena for which conventional pump–probe methods fall short due to their requirement for repetitive events. DFT has proved effective for a diverse range of applications that require fast continuous measurements, including the absorption spectroscopy of combustion engines, the characterization and control of opti-cal rogue waves, imaging laser ablation, the identification of rare cancer cells in blood, the surface vibrometry of microelectrome-chanical systems, high-throughput OCT for manufacturing and process control, and analog-to-digital conversion for wide-band-width communications in defence. Future work on DFT includes the use of dispersion engineering to develop low-loss dispersive elements for chip-scale DFT. Although it remains to be seen how this improvement will be realized, previous work on DFT and its applications by a number of research groups has already shown impressive performance and holds great promise for a novel class of real-time instruments.

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Figure 6 | Analog-to-digital conversion with DFT. a, Schematic of photonic time-stretch ADC. The signal-encoded spectrum is de-interleaved into four channels. The signal in each channel is time-stretched and digitized by a slow electronic digitizer. The digitized waveforms are combined in the digital domain to restore the original signal, but at a digitization rate four times higher than that of the electronic digitizer alone. b, Radiofrequency signal digitized in real time by the photonic time-stretch ADC with a record-high sampling rate of 10 TS s–1 (corresponding to a real-time digitization resolution of 100 fs). c, Eye diagram of a 40 Gbit s–1 optical data stream captured by the all-optical time-stretch ADC at an effective sampling rate of 1.25 TS s–1. Figure reproduced with permission from: b, ref. 69 © 2007 AIP; c, ref. 71 © 2012 AIP.

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–0.5

0

0.5

1.0

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Digitizer

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Digitizer

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b

c

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AcknowledgementsThe authors acknowledge support from the US Defense Advanced Research Projects Agency, the National Science Foundation, the National Institutes of Health and Congressionally Directed Medical Research Programs. K.G. is supported by the Burroughs Wellcome Fund Career Award at the Scientific Interface. The authors also thank C. Kaminski, D. Solli, A. Fard and E. Diebold for permission to use their figures.

Additional informationCorrespondence and requests for materials should be addressed to K.G.

Competing financial interestsThe authors declare no competing financial interests.


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