oct

Optical Coherence TomographySean C. Crosby

Supervisor: Associate Professor Ann Roberts

Submitted in partial fulfilment of the requirementsof the degree of Bachelor of Science(Honours)

Honours Report, 2003.

Copyright c�

2003 Sean Crosby. This material may be distributed only subject to the terms and conditions setforth in the Open Publication License, version 1.0 or later. The latest version of the Open Publication Licenseis currently available at http://opencontent.org/openpub/

Signature ............................................

Abstract

Beam splitting interferometry yields high visibility interference fringes when the path length difference (PLD)of the two beams is an integer multiple of the wavelength of light used, with the maximum PLD over whichfringes are observed being the coherence length of the source. Low coherence interferometry enables high axialresolution scans to be taken, as the coherence length of the source (a super-luminescent diode) is relativelysmall (10 - 20 �� ). This project aims to use Optical Coherence Tomography (OCT) in the form of beamsplitting interferometry with a low coherence light source, to produce detailed axial scans of objects. As anatural consequence of this, a frequency spectrum of the sources used will be produced, and compared against,those measured using an optical spectrum analyser.

i

AcknowledgmentsI would like to thank my supervisor, Ann Roberts, for putting me in my place when I thought the projectwas sailing along nicely. She made me work harder, and that made me better understand the physics behindOCT simply by doing more experiments.

Michael Pianta of the Department of Optometry and Vision Sciences was helpful in supplying informationon the OCT system used there.

I would also like to thank my officemates, Cath, Foti, Clare and Benedicta. Cath and Clare have beenespecially helpful to me in locating formula relating to refractive indices, and simply for making my insanityseem normal. And the endless Buffy watching always made me go home with a smile on my face.

Thanks go to Simon, for his help and conversations in both the coursework section of my Honours degreeand the writeup of my thesis, and to Daniel, who I can always bounce Futurama and Simpsons quotes off, andhe knows where they come from. It makes me feel less nerdy.

And finally, thanks must also go to Max Roberts, who kindly donated my tooth specimen.

Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

1 Introduction 11.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Recovering the Phase and Amplitude of the Coherence Function . . . . . . . . . . . . . . . . 5

1.2.1 Calibration Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Simulations 82.1 Fringe Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Effect of noise on the retrieved coherence function amplitude . . . . . . . . . . . . . . . . . . 92.3 Unequal arm intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Experimental Setup 11

4 Power Spectral Density Function 14

5 Experimental Linewidths, Coherence lengths and Coherence Functions 165.1 Extending the principles to more complex objects . . . . . . . . . . . . . . . . . . . . . . . . 19

5.1.1 Two Coverslips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.1.2 Two coverslips with air between them . . . . . . . . . . . . . . . . . . . . . . . . . . 215.1.3 Silicon Grease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.1.4 Teeth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 Further Work 246.1 Heterodyne Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.2 2D Scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Conclusion 25

Appendices 26

A Manufacturer’s Specifications for SLDs 27

B Takeda Method for Isolating Phase and Amplitude of the Coherence Function 28

C IDL Simulation Code 30

Bibliography 35

Chapter 1

Introduction

Optical Coherence Tomography (OCT) has many uses in medicine and biology. It is a non-invasive

technique of looking at the layered structures of tissue like skin, the retina and teeth. The scanning of

retinas to reveal defects in the underlying layers has been done extensively and stand-alone units are

now sold for this purpose. Other proven uses of OCT are the scanning of hearts for defects, in both

animals [1] and humans, and the scanning of the digestive tract [2]. The outcomes of these studies,

where a spatial resolution approximately ten times that of ultrasound was achieved, has prompted

investigations into the feasibility of extending OCT into other ultrasound dominated areas. There

are currently trials to check the feasibility of measuring blood glucose levels [3] with OCT, and the

imaging of arteries [4]. The aim of this project was to build an OCT system with a view to using it

to investigate paint layers in artworks and other materials applications.

OCT has its roots in white-light interferometry, especially in the form of a Michelson interferom-

eter (see Figure 1.1).

Figure 1.1: Experimental Setup: A modified Michelson interferometer

2

With the advent of optical fibers, and with it, the need to find faults in them, the method of optical

coherence-domain reflectometry (OCDR) was invented, which is basically the 1D form of what we

now refer to as OCT. It used the coherent properties of laser light, and the fact that interference

only occurs when the reference beam and the object beam (see Figure 1.1) are matched to within

the coherence length of the light source used, to obtain an approximate distance to where the fault

in the optical fiber lies. OCDR was required to scan only in one direction (along the length of the

optical fiber), but its 2D and 3D uses were recognised when the ability of OCDR to perform ranging

measurements of the structure of the retina was found. OCDR was extended to 2 and 3 dimensions

by D. Huang et al [5] in 1991, and thus the method of OCT was founded.

Since the first OCT system was developed, work has been done to improve three major areas of

the system [6]:

� The light source.

The original light source was a 830nm super luminescent diode (SLD) with a coherence length

of 17 �� in air. The SLD has a high irradiance and low cost, which makes it a good choice

to image simple objects such as the eye, but the coherence length is too long to image single

cells and nuclei in cells. One way to overcome this limitation is to change the source to a

femtosecond pulse laser [7], which has a very large linewidth, and a coherence length of the

order of 1 �� . Most recently the femtosecond laser has been coupled into a micro-structured

optical fiber to further increase this linewidth by producing a continuum of wavelengths with

the central wavelength unchanged [8]. This allows the use of wavelengths as high as 1300nm,

which provides a long penetration depth in biological tissue, with a coherence length of 1.3 �� .

� The interferometer and detector.

Even though OCDR and OCT were based on a free space Michelson interferometer, most

implementations of OCT are fully contained in optical fiber, including the first OCT setup.

The problems with this implementation though are that it’s more susceptible to thermal drifts

(see Chapter 5), which introduces noise, and is harder to set up. To overcome this, heterodyne

detection (see Section 6.1) is commonly used to increase the signal-to-noise ratio (SNR) [9].

� The scanning instruments.

The original instruments to scan the reference beam were stepper motors. This limits the speed

at which data can be taken, and thus if OCT can be used for live patients, as a long scan time is

more susceptible to motion by the patient (e.g. eye blink). The current technology to scan the

reference beam is piezo controlled fiber optic stretchers, which does not have this limitation on

the scan time, as it does not introduce as much noise.

In the 10 years since OCT had its first biomedical application [5], OCT has become commer-

1.1 Principles 3

cialised, and has been modelled for many different, and distinct purposes.

Standalone units, such as the Zeiss STRATUSoct, used by the Victorian College of Optometry for

retinal scans, are becoming more advanced and used in the diagnosis of disease. The STRATUSoct

can be fully controlled by a physician who does not necessarily know exactly how OCT works.

Internal scans have also been accomplished with the advent of a fiber optic catheter (see Figure

1.2). This catheter is essentially the object arm fully encased in optical fiber, with only the end

outside to image the object. It scans perpendicular to the catheter and is flexible enough to allow for

bending during passage through the contours of internal channels, such as blood vessels.

Figure 1.2: OCT catheter-endoscope inside an artery. Taken from [4]

The experimental setup used for this project is essentially a free space version of the first imple-

mentation of OCT.

1.1 Principles

The principle of OCT is based on temporal coherence and interference at a point. The interference

of two sources at a point is given by

�� ! #"%$'&)(* ,+-�� .�/&102 ,+3 5476 (1.1)

where�

is the intensity of the beam at the interference point, subscripts 8 and 9 refer to 8;:=< ?>A@ and

9 ?B� 9 �CD>� respectively,�

is the temporal separation of the two beams,&E,+3

is the electric field at

time@, and

�=$F&)GH4

where$!4

refers to the time average [10]. The I superscript is to signify that

the electric field may not be the same after interacting with the specimen in the object arm.

1.1 Principles 4

Assuming that both arms of the interferometer act like perfect mirrors (i.e. the quasi-monochromatic

nature of the light is not destroyed by interacting with the specimen), the interference term

Re"%$'& (* ,+5�J�� .�/& 02 ,+K 7456 can be rewritten as

�! L"%$'& (* ,+5�� .�/&102 ,+3 74-6�NMPOE�� MRQ�SUT;WVKXDYKZ��[�'\]�� ^ (1.2)

whereOE��

is a complex temporal coherence function,Y#Z

is the peak frequency, and\]��

is the rela-

tive phase between the two arms. Using the Weiner-Khintchine theorem,OE��

= _a`Zcb WY� = �dUe]=fgVKXihRY� Rj#Y ,

where b WY� is the power spectral density of the source as a function of frequency. Thus scanning the

reference mirror will produce varying intensities oscillating around�

=��k�l�J��

.

To demonstrate the production of a fringe pattern, a simulated result is shown in Figure 1.3. A

Gaussian power spectral density function, with properties shown in Table 1.1, of the form

b WY� ��nm�oqp]=f-r�sutDWVU WYvfwYKZ? Rxy Y x (1.3)

wherey Y

is the frequency linewidth, was assumed.

z�{ (x |A}3~�� Hz) � { (nm) � z (x |A}3~�� Hz)4.41 6810 8.009

Table 1.1: Power Spectral Distribution Function properties for obtaining Figure 1.3

This translates to havingOE��

centered on�

=0 with a FWHM of 1.103 x � �� s.

Figure 1.3: Fringe pattern using a gaussian as the power spectral density

As can be seen in Figure 1.3, there exists both negative and positive temporal separations. A

1.2 Recovering the Phase and Amplitude of the Coherence Function 5

common convention in the literature is to designate positive temporal separation when the reference

arm$

object arm. This convention will be followed in this report.

The FWHM of the fringe pattern in position space is twice the coherence length, which for a

Gaussian spectral density function can be written as �� =x^��K�P��x�K�]� (See [11]). This is the maximum

spatial axial resolution that can be achieved using a simple OCT system: The smallest spatial sepa-

ration in the z direction of features must be greater than the coherence length of the source in order

to be resolved.

The goal of OCT is the extraction of the amplitude and phase of the coherence function from a

fringe pattern such as that in Figure 1.3.

1.2 Recovering the Phase and Amplitude of the Coherence Function

To measure thicknesses of objects placed in the object arm, and to calibrate the movement of the

stepper motors, it is necessary to determine the phase and amplitude ofOE��

. We considered two

methods for extracting the amplitude and phase ofOE��

from a fringe pattern such as Figure 1.3:

The method of Takeda [12], and the Hilbert transform method [11]. The method of Takeda was not

eventually used for any of the results presented in this report and is discussed in Appendix B.

The method described in [11] uses Hilbert transforms to find the complex degree of coherence.

Setting the interference term, Re"%$F& (* ,+5�� .�/& 02 ,+3 7476 in Eqn 1.2 to be Re[ � �� ], and using ana-

lytic continuation, we find that

� �� Vi� �� hV�� " � �� 6�� (1.4)

where F(�

) is the measured interference term (i.e. the measured intensity, minus the mean, so that it

is centered around I=0), and HT is the Hilbert Transform.

Thus, by assuming � ,@� = A(t)m�o�p],h=\�,@� ^

, the amplitude (A(t)) is found to be

� ,@� �� V�� ?� ,@^ � x � �?�¡� " � ,@^ �6W� xL¢ (1.5)

and its phase (\],@�

) is

\],@� ��£K¤ Q�¥�£;t � �� " � ,@� �6� ,@^ �U�

(1.6)

The phase,\],@�

, shown in Figure 1.5, is related to absolute displacement, s(x), by s(x) =\],@� �¦ � .

This can be used to create calibration graphs for the stepper motors (see Section 1.2.1).

The phase calculated using Eqn 1.6 has a range of [-� x ,� x ], so it requires ”unwrapping” to get the

actual phase.


The amplitude found using Eqn 1.5 andOE��

used originally in Eqn 1.2 to generate the fringe

pattern is plotted in Figure 1.4.

Figure 1.4: Retrieved Amplitude and Original Coherence Function. Both Amplitudes are exactly thesame.

Figure 1.5: Retrieved Phase

As seen by Figure 1.4, the amplitude extracted using the Hilbert transform method coincides

perfectly with the original coherence function.

The phase is not what is expected. At the beginning of the motion, it does not follow a straight

line. This is due to end effects resulting from performing a discrete Hilbert transform, and not a

continuous one. These effects get smaller as the step size is decreased. But as the motion continues,


it behaves as a straight line, which is expected for our simulated data.

1.2.1 Calibration Graphs

Calibration graphs are helpful to know approximately how much the displacement recorded by the

scanning software differs from the real displacement. But it is only an approximate measurement.

The displacement error depends on the exact absolute position of the Nanomover stage. Displace-

ment errors are due to imperfections in the rod (found inside the Nanomover) that holds the stage at

the required position, and how much pressure is being applied on the rod by the stage. Logically, the

displacement error is smallest in the middle of the Nanomovers absolute movement, and largest at the

start and finish of its absolute movement. Each time the absolute displacement of the Nanomovers is

changed, a new calibration graph would have to be found.

The calibration for certain parts of the Nanomovers motion (not necessarily the same Nanomover)

has been completed by Stephen Rhodes [13].

Chapter 2

Simulations

By assuming a Gaussian profile for S(Y

) for the SLDs, and knowing that the Fourier Transform of a

Gaussian is still a Gaussian,OE��

in Eqn 1.2 was chosen to be a Gaussian with FWHM in position

space being twice the coherence length found in Table 4.1.

Fringe patterns, phase profiles and the auto-correlation function were generated using a program

written in IDL.

2.1 Fringe Patterns

To simulate reflection off perfect mirrors, the intensities from the object and reference arms were set

to be equal. With�?�

and��

being both 0.5 andY;Z

= §��¨ =r��Pr � x � �q� ¦ Hz, the fringe pattern shown in

Figure 2.1 was calculated for a 100 � m scan with a 100nm step size using Eqns 1.1 and 1.2.

Figure 2.1: Simulated fringe pattern for a 100 © m scan

This is an ideal fringe pattern: there is no noise and the intensities are equal in both arms. When

measuring a real sample, this does not actually occur: Objects other than a perfect mirror in the

2.2 Effect of noise on the retrieved coherence function amplitude 9

object arm cause the intensity coming from that arm to be ª reference arm.

Eqn 1.2 is altered somewhat to include a random Poisson distribution of numbers with element 9 ,and mean of zero, to simulate noise.

� ��VU l«¬MPOE�� MRQ�SUT;WVKXDYKZ��[�'\]�� ^ 7® � � 9 (2.1)

The resulting fringe pattern is shown in Figure 2.2.

Figure 2.2: Simulated fringe pattern for a 100 © m scan with random noise and equal arm intensities

2.2 Effect of noise on the retrieved coherence function amplitude

After using Eqn 2.1 to generate a noisy fringe pattern, the temporal coherence function amplitude

was retrieved using Eqn 1.5 and the Hilbert transform function built into IDL. It is shown, with the

original coherence function, in Figure 2.3.

After applying a Gaussian fit to this data, the coherence length was calculated to be 8.34 �� ,

compared to the original coherence length of 8.43 �� used to generate this coherence function.

Thus, all noisy measurements will result in coherence lengths that are shorter than in reality.

The noise also changes the central position of this peak from 0 to 1.5 nm. Thus, all measurements

of thickness from noisy data will also be shorter than the actual value.

2.3 Unequal arm intensities 10

Figure 2.3: Retrieved coherence function amplitude from a noisy fringe pattern (dashed), and originalcoherence function (solid)

2.3 Unequal arm intensities

To simulate unequal intensities in the arms,�k�

was set to 0.1 and��

to 1 -��

= 0.9 to keep the pattern

centered on 0.5. The result is shown in Figure 2.4. As can be seen in Figure 2.4, fringe visibility

Figure 2.4: Simulated fringe pattern for a 100 © m scan with random noise and with unequal arm inten-sities

has been reduced compared to Figure 2.2. Thus, as reflectivity is reduced in the object arm, neutral

density filters need to be put in the reference arm to approximately equalise the intensities to increase

the fringe visibility. The need for filters can also be reduced or eliminated through synchronous

detection (see Chapter 6).

Chapter 3

Experimental Setup

Figure 3.1: Experimental setup

Three sources, each coupled to an optical fiber, were used in this experiment:

� 670nm Melles-Griot S1959 laser diode

� 681nm SLD (Super Luminescent Diode)

� 980nm IR (Infra-red) SLD

The manufacturer’s specifications for the two SLDs are contained in Table A.1.

Two different beam splitters were used, as two distinct wavelengths each require a different coat-

ing on the beam splitter to minimise multiple reflections, and to ensure an approximate 50/50 split.

Since both the SLDs have a very short coherence length (see Table 4.1), the outputs of the laser

diode and the 681nm SLD were coupled together into one fiber using an approximately 50/50 fiber

12

coupler to help with the finding of fringes for the SLD (see Figure 3.2): The interference region of

the SLD is in the region of maximum fringe visibility for the laser diode. Since the latter can be

found very easily as a consequence of the relatively long coherence length of the laser diode, and the

alignment does not have to be disturbed when the outputs are coupled together to change between the

laser diode and the SLD, the coherence region for the SLD can be found quite easily. This coupler

was removed once fringes were found and before data was taken, because there was a large loss of

output from the 681nm SLD when put through the coupler, which made it difficult to get adequate

fringe visibility.

Figure 3.2: Schematic diagram of experimental setup

The detection system consisted of:

� A PIN diode to convert intensity to a voltage

� An amplifier (1000x) to amplify this voltage

� A 16-bit analogue to digital converter (ADC) with dynamic voltage range of [-5.00V, 5.00V]

The amplifier was seen to introduce a minimal amount of noise, and significantly less than 0.1V.

To move the mirrors on both the reference and object arms, ¯±° C 83�¡83² 9U³3´iµ (Melles-Griot) stepper

motors were used. The calibration of these stepper motors is discussed in Section 1.2.1.

To automate the moving of the Nanomovers, and the taking of voltage measurements, a 486 com-

puter running a program 1dnano (Stephen Rhodes/Duncan Butler) was used. This program does not

13

facilitate the taking of 2D/3D measurements. It is planned for this program to be altered to allow

such measurements to take place. (See Section 6.2)

Chapter 4

Power Spectral Density Function

To know if the experimental coherence lengths are correct, and thus if OCT produces valid and

accurate results, frequency spectra of the three lasers (see Chapter 3) were taken using an AGILENT

86140B Spectrum Analyser, and Gaussians of the form of Eqn 1.3 were fitted to them to find the

central wavelength ( ¶ Z ), and the FWHM (y ¶ ), which can be related to coherence length.

Figure 4.1 shows the spectra for the 681nm SLD.

Figure 4.1: 681nm SLD Frequency Spectra (solid) and its Gaussian fit (dashed)

The spectra for the other two lasers are shown in Figures 4.2 and 4.3.

Using the fitted gaussian, ¶ Z = 682.25nm,y ¶ = 24.37nm,

YUZ= §��¨ =

r��·;¸U¹x � �q� ¦ Hz,

y Y=

§� �¨ y ¶ = � ��U¹ � x � � �� Hz, �5� =x^��¨ �P��x�K�]� = º �PrL· �� .

This data is also presented, with the data for the other two lasers, in Table 4.1, with statistical

errors shown.

15

Figure 4.2: 980nm IR SLD Frequency Spectra (solid) and its Gaussian fit (dashed)

Figure 4.3: 670nm Laser Diode Frequency Spectra

The secondary peaks in Figure 4.3 looks like they are cavity modes.

Manufacturer’s Data

Laser » ¨ (nm) ¼L» (nm) » ¨ (nm) ¼L» (nm) ½ ¨ (x ¾ ¨ ¾u¿ Hz) ¼L½ (x ¾ ¨ ¾ÁÀ Hz) Â�Ã ( Ä^Å )681nm SLD 681.0 23.8 682.25 24.37 Æ 0.19 4.397 1.571 Æ 0.122 8.43 Æ 0.06980nm IR SLD 982 54.6 985.88 54.65 Æ 0.45 3.043 1.687 Æ 0.140 7.85 Æ 0.07670nm Laser Diode 676.04 0.1979 Æ 0.0094 4.438 0.013 Æ 0.0006 1019 Æ 48

Table 4.1: Physical properties of the three lasers used.

All three frequency spectra seem to have a Gaussian shape, and our assumption of this shape is

vindicated by noting that the values ofy ¶ found for the SLDs agree very well with the values given

by the manufacturer, shown in Table A.1, and in the above table.

Chapter 5

Experimental Linewidths, Coherencelengths and Coherence Functions

Scans using the two SLDs, with the properties shown in Table 5.1, were performed with a near-

perfect mirror. The resulting fringe patterns are shown in Figures 5.1 and 5.2.

Laser Scan Size ( �� ) Step Size (nm) Perspex Box On?681nm SLD 100 100 Yes980nm IR SLD 200 100 Yes

Table 5.1: Scan Types for gaining Coherence Functions

Figure 5.1: 681nm SLD 100 ©LÇ scan Fringe Pattern

It is thought that thermal currents in the air produce phase changes in certain sections of the

apparatus by changing the refractive index of one or both of the arms, thereby slowing the laser light

down.

A 100 �� scan with 100 nm step was performed without the perspex box on. The box was then

17

Figure 5.2: 980nm IR SLD 200 ©LÇ scan Fringe Pattern

placed over the entire apparatus, and before the ’Box On’ measurement was taken, the air inside

the box was allowed to settle for 10 minutes. This was done to try to minimise the phase changes

introduced by the air.

Figure 5.3 shows the fringe pattern from the ’Box Off’ case, and Figure 5.4 shows the fringe

pattern in the same region for the ’Box On Case’. As can be seen from these fringe patterns, fringe

visibility gradually increases in a linear fashion for the ’Box On Case’, and has more ’jumps’ and

unexpected changes in the ’Box Off Case’.

Figure 5.3: Fringe pattern forthe ’Box Off Case’

Figure 5.4: Fringe pattern forthe ’Box On Case’

The fringe patterns found in Figures 5.1, 5.2 and for the ’Perspex Box Off’ case were converted

back to a Coherence Function using the method described in Section 1.2.

The resulting amplitudes of the temporal coherence functions, together with their Gaussian fits,

are shown in Figures 5.5 and 5.6. The experimental coherence times and lengths are found in Table

5.2.

An effect seen in the experimental data is that the FWHM becomes larger, and further away from

the expected value seen in Table 4.1 when the box is not in place. Because of this, whenever a data

set is to be analysed, a scan with the box in place will be used.

18

Figure 5.5: 681nm SLD 100 ©LÇ scan Coherence Function. Green=Minimum Fit, Red=Maximum Fitand Blue=Least Squares Fit

Figure 5.6: 980nm IR SLD 200 ©LÇ scan Coherence Function

Comparing the data contained in Table 5.2 to the data shown in Table 4.1 shows that the expected

coherence lengths (found in Chapter 4) are well within the bounds set by Table 5.2. The agreement

between both results could be strengthened by removing some of the background noise manually,

and thus not having a large gap between maximum and minimum fits.

5.1 Extending the principles to more complex objects 19

Coherence Lengths ( �� )Laser Coherence Time (x |A}UÈ ~�É s) Box On Box Off Minimum Maximum681nm SLD 2956.5 8.87 Ê 0.06 9.46 Ê 0.06 8.25 9.75980nm SLD 2736.45 8.21 Ê 0.07 7.5 11.25

Table 5.2: Data from Gaussian fitting

5.1 Extending the principles to more complex objects

As complex objects go, a near-perfect mirror is not one of them. It has only one reflecting layer, and

the topology does not change across the surface. The idea of this section is to extend the concepts and

principles demonstrated before to a more complex object: a microscope coverslip. A glass coverslip

has two reflecting faces: the front and the back surfaces.

Figure 5.7 shows the setup of two coverslips in the object arm. For the preliminary measurements,

only the back coverslip was used. The double sided tape, and the front coverslip were not present.

Figure 5.7: Setup of Two Coverslips in the Object arm

An OD (optical density) 0.5 filter was placed in the reference arm to approximately equalise the

intensities in each arm. The refractive index of the glass was known to be 1.45. The coverslip width

was known to be 170 Ë 1 �� , measured using a micrometer. Therefore, the approximate optical

thickness was Ì 170 x 1.45 = 246 �� . This suggested performing a 450 �� scan, with 150 nm step

size with the 681 nm SLD. The resulting fringe pattern is shown in Figure 5.8.

The amplitude of the envelope function was calculated using the method of [11] yielding Figure

5.9.

The envelope functions are asymmetric due to one side of the function propagating through glass,

and the other side propagating through air.

Even though this asymmetry was noticed, the peaks were still fitted with Gaussians, as the central

position of these peaks needs to be known to calculate the distance between the two. From fitting the

peaks with Gaussians, the following data was obtained for the central positions of these Gaussians.

Peak 1: 1.91 Ë 0.02 �� Peak 2: 224.71 Ë 0.10 ��The errors calculated are statistical errors, and hence are very low.

Therefore, the optical width of the coverslip is 222.80 Ë 0.12 �� . Assuming the refractive index


Figure 5.8: Resulting fringe pattern from a coverslip in the object arm

Figure 5.9: Retrieved envelope function using [11]

of the glass in the coverslip to be 1.45, this translates to the coverslip having a physical thickness

of 153.66 Ë 0.07 �� . The physical thickness results aren’t very close to the expected value of 170

�� . This is hypothesised to be because of the error in the calibration of the Nanomovers, the result

of noise in the fringe patterns (see Section 2.2) and variations in the coverslip thickness1. Since the

distance over which the scan is taken is relatively large, a small error in each step accumulates over

Ì 2000 steps, and this maybe the source of the experimental error.

A second analysis, using the 980nm IR SLD, was performed on the same coverslip setup. The

results are shown in Table 5.3.1The coverslip measured was not actually the coverslip used for the scan


Laser Coverslip Optical Width ( �� ) Coverslip Physical Width ( �� )681nm SLD 223.0 Ê 0.1 154.00 Ê 0.07980nm SLD 220.0 Ê 0.1 151.00 Ê 0.07

Table 5.3: Coverslip Width Results for the two SLDs

5.1.1 Two Coverslips

One of the original motivations for assembling the OCT apparatus was to image paint artworks,

where multiple reflecting layers are present, for example, varnish, paint, primer, canvas, with each

layer reflecting less light than the previous one.

To demonstrate the imaging of more complex an object, two coverslips were put together, in the

same setup as Figure 5.7.

Figure 5.10: The reflections off two coverslips

A problem was encountered that severely limits the range of objects that can be scanned using

our simple OCT setup. It was discovered that if the distance between the two coverslips shown in

Figure 5.10 was too large, the two reflections off the coverslips would not be on top of one another,

and as such, a scan could not be taken which contained phase information for both the front and back

coverslips. If this occurs, we cannot get an accurate idea of the distance between two reflecting layers,

because this information would be contained in two completely different scans. The reflections off

a coverslip are specular reflections, and as such, this problem does not occur in the simple case.

However, for a diffuse object, the only reflection that is recorded by the detector is the backscatter,

but each backscatter may not have the same directionality as the others, and as such, the reflections

do not overlap at the detector.

5.1.2 Two coverslips with air between them

A 1mm scan with 222 nm step size was taken of two coverslips with an air gap between them. The

resulting envelope function, extracted using the method described in [11], is shown in Figure 5.11

with their fitted Gaussians.


Figure 5.11: The retrieved coherence function from two coverslips

First Coverslip Optical Thickness ( �� ) Air Gap ( �� ) Second Coverslip Optical Thickness ( �� )218.82 76.46 219.61

Table 5.4: Retrieved thicknesses of the two coverslips and the air gap

The extra, small, peaks seen in Figure 5.11 before the back coverslip, and between the front

and back faces of the back coverslip, are thought to be multiple reflections from the beamsplitter.

False coherence (i.e. something that looked like a fringe pattern) was seen on a CCD camera when

setting up the IR SLD for use with the near-perfect mirror. This false coherence behaved like a real

coherence pattern (i.e. disappeared after approx 40 �� , which is what it should do because of the

coherence length of the SLD), but had very poor fringe visibility. As the only thing that changed

when switching between the visible SLD and the IR SLD was the beamsplitter, and there are 4 of

these false peaks, which is how many there should be if it’s another reflection, my hypothesis is that

these false coherence regions are due to multiple reflections off the beamsplitter.

The results from the Gaussian fitting is shown in Table 5.4.

The widths of the coverslips compare very well with the results found using only one coverslip,

shown in Table 5.3.

The micrometer measured value of 85 Ë 1 �� for the thickness of the double sided tape does not

compare well with the value found in Table 5.4 for the Air Gap though. But, as with the thickness of

the coverslips, the experimentally measured value is less than the actual value, which still suggests

that it could be a result of noise (see Section 2.2) or calibration errors.


5.1.3 Silicon Grease

Silicon vacuum grease was placed between the two coverslips, to simulate a region that is turbid, and

to thus decrease the intensity hitting the back coverslip.

As explained above, two reflections were seen, and thus an accurate measure of the thickness of

this murky layer could not be calculated. This problem could be overcome using a smaller beam

size, by using a microscope objective to focus the beam down to a very small point. Another way

to overcome this problem is to increase the wavelength: The higher the wavelength, the smaller the

scatter.

5.1.4 Teeth

A tooth was placed in the reference arm, and an attempt was made to setup the apparatus. The tooth

was chosen as it has multiple reflecting layers, and has more in common with an artwork than two

coverslips. Even though the IR SLD was striking the tooth, no reflection could be seen. A possible

explanation for this is given in [14]. Scattering in tissue occurs in many angles. The angle that is

measured by our detector is the backscatter. All others do not reach the detector. The backscatter

cross-section must be low for a tooth, and thus we do not detect anything with our detector. It is

anticipated that the introduction of synchronous detection would permit the detection of the signal

reflected from the enamel and underlying layers.

Chapter 6

Further Work

Some improvements can be made to my simple OCT setup to increase the number of samples that

can be scanned, and the resolution obtained.

6.1 Heterodyne Detection

Synchronous detection (heterodyne) involves introducing a phase modulation in the reference arm at

a known frequency. This modulated signal can be extracted using a lock-in amplifier. The electronic

system isolates the non-interacting laser frequency only, completely removing the phase changes due

to noise etc. It also removes the background intensity, which centers the fringe pattern around I=0.

This greatly increases fringe visibility without the need of filters to equalise the beam intensities.

Also, because the noise from the Nanomovers are frequency based, these stepper motors can be

made to operate faster, and thus reducing the scan time, because the noise can be removed using this

system of heterodyne detection.

6.2 2D Scanning

As can be seen in this report, no 2D scans were taken. This was not by choice, but because of software

restrictions. Currently, the program (1dnano) cannot deal with scanning in both dimensions, but it is

anticipated that when the source code is found, and it does exist, that it should be elementary to code

this functionality into the program.

Once this functionality has been accomplished, scans of both the single and double coverslips

should be performed to see if the setup can detect a tilt in the coverslip setup. If it can, this could be

used to automatically align the apparatus to achieve maximum fringe visibility.

Conclusion

A low coherence interferometer was constructed as part of this project.

Simulations were used to show how noise affected the determination of the coherence lengths

and distance between reflecting layers, thus affecting the accuracy of simple OCT. It was shown that

these effects can be reduced or eliminated by using a perspex box to reduce phase changes caused by

thermal currents in the air.

This project also showed that the coherence lengths of the laser sources used could be calculated

using OCT, and are extremely similar to that found using an optical spectrum analyser.

The ability for OCT to scan simple objects such as a coverslip was demonstrated, with a value of

153.66 Ë 0.07 �� obtained for the thickness of the coverslip.

The OCT process was extended to objects more complex, with mixed results. The thickness of

double sided tape was found to be 76.46 �� . But using the same two coverslip setup, no experimental

data could be found when silicon grease was placed between the two coverslips. A tooth was also

tried as a sample, but no reflection could be seen. Heterodyne detection was suggested as a way

of overcoming this limitation on samples able to be scanned using a simple OCT setup, as similar

experiments using this detection method have been able to detect the layers in teeth.

Further work on this project would include introducing heterodyne detection, and implementing

2D scanning in the software used.

Appendices

Appendix A

Manufacturer’s Specifications for SLDs

For operating conditions ofV;�LÍ

C and 150mA current

Manufacturer Model � { (nm) FWHM (nm)SuperLum Diodes (Russia) SLD-261-MP-DIL-SM-PD 681.0 23.8SuperLum Diodes (Russia) SLD-481-MP-DIL-SM-PD 982.0 54.6

Table A.1: Manufacturer’s Specifications for SLDs used in this experiment

Appendix B

Takeda Method for Isolating Phase andAmplitude of the Coherence Function

The method used in [12] yields better results when multiple reflections aren’t seen (see Section 5.1)

as it yields less noise, but this method requires manual noise reduction of the Fourier Transform to

obtain realistic and expected results. It is also dependent on experimentally found values such as

the wavelength of light used, and a precise measurement of the step size, which introduces more

errors. The method employed in [11] uses Hilbert transforms and produces accurate results without

any manual noise reduction.

Using the method set out in [12], the fringe pattern is of the form

Î ,d ¢^Ï l� ° ,d ¢^Ï �� : ,d ¢^Ï qQ�SUTk"PVKXDYKZ�dE�'\],d ¢^Ï �6(B.1)

� ° ,d ¢^Ï D��>;,d ¢^Ï #m�o�p]WVKXihÐYKZ�d� ��> ( ,d ¢^Ï #m�oqpa=fgVKXihÐYKZ�d� (B.2)

where h(x,y) is the intensity measured, a(x,y) is the background intensity (i.e. the sum of the two

intensities plus noise),b(x,y) is the interference envelope (OE��

),\],d ¢^Ï

is the phase, and

>K,d ¢^Ï ��Ñ�V : ,d ¢^Ï #m�oqpa,hR\],d ¢^Ï ^ (B.3)

Taking the Fourier Transform of Eqn (B.2) yields

� WY ¢^Ï �� WY ¢^Ï D� � WYEf®YKZ ¢^Ï �� (KWY[�JYKZ ¢^Ï (B.4)

The resulting frequency spectrum is shown in Figure B.1

Shifting the positive frequency peak toYÒ�Ó�

by redefiningYÒ�ÓYHfÔY#Z

, and eliminating the zero

frequency offset and the negative frequency peak yields the equation

� WY[�JY Z ¢^Ï �� WY ¢^Ï .� (B.5)

29

Figure B.1: Simulated Frequency Spectrum

Eliminating both A(Y

, y) and � ( WY!��YKZ ¢^Ï is a difficult step which involves manually removing

data to the left and the right of the wanted C(YÕfJYLZ

,y) peak. Anything that is not removed, and is

not actually part of the C(YEfÖY Z

,y) peak contributes noise, and if too much noise is there, an output

that is not expected is seen.

Performing the Inverse Fourier Transform on Eqn (B.5) gives c(x,y) and, using Eqn (B.3),

sutD" >;,d ¢^Ï �6��sutD" �V : ,d ¢^Ï �6��whR\],d ¢^Ï (B.6)

the phase (\],d ¢^Ï

) and the amplitude (MPOE�� M

) of the coherence function can be obtained by taking

the real and imaginary parts of the complex natural logarithm of c(x,y).

Appendix C

IDL Simulation Code

31pro read_data,mydata,templ,filenmydata=read_ascii(filen,template=templ)end

pro calcwave,c,lambda print, ’ Enter frequency: ’ read, efreq elambda = c/efreq print, ’ The calculated wavelength is ’,elambda print, ’ It should be ’,lambda print, ’ The percentage error is ’, abs(100*(1−elambda/lambda))end

pro calcfwhm,cprint, ’ Enter FWHM in secs’read, efwhmedist = c*efwhmprint, ’ The coherence length is ’, edist

end

pro awd,numofwindowsi=1while i le numofwindows do begin

wdelete,ii=i+1

endend

beginmypath=’ c:\data’datafile = ’ seanir8.dat’restore,mypath + ’ \mys.sav’mydata=read_ascii(mypath + ’ \’ + datafile,template=mys)mydata.z = mydata.z/1d3 ;Because data output from MCA is in mm (Convert to m)n=size(mydata.z,/n_elements)c=3.d8lambda=981d−9f=c/lambdastepsize=222.d−9steptime=2*stepsize/cstepf=1/steptimestepf=stepf/nn21=n/2 − 1fourier=shift(fft(shift(mydata.v,−n21),−1),n21)frequency=stepf*findgen(n) − n21*stepftime=make_array([n],/double)time=2*mydata.z/c

meanv=mean(mydata.v)avv=mydata.v − meanv

avf=shift(fft(shift(avv,−n21),−1),n21)

;begin phase retrieval stuffrfpp=fix(f/stepf)k=fix(n21+(49d0/50d0)*rfpp)maxk=fix(n21+(51d0/50d0)*rfpp);maxk=n−1rightpeak=make_array([n],/double);j=n21+1j=k

Nov 05, 03 16:29 Page 1/3sean.pro;Hilbert transform phase and amplitudegamma=0.5d0*sqrt(avv^2 + (hilbert(avv))^2)hv = imaginary(hilbert(avv))nv = real_part(hilbert(avv))ratiov = hv/nvphi2 = atan(ratiov)diserror=phi2*lambda/(4*!pi);

;rightpeak is avfwhile j le maxk do begin rightpeak[j] = avf[j] j=j+1endrightpeak=shift(rightpeak,−rfpp)w=shift(fft(shift(rightpeak,n21),1),−n21)phi=imaginary(alog(w))diserror=phi*lambda/(4*!pi)

v=real_part(alog(w))v=2*exp(v)

window,1,title=’Frequency Spectrum Minus Mean’plot,frequency,avf,xrange=[3d14,7d14]

window,2,title=’Envelope Function Retrieved’plot,time,v

openw,1,’c:\blah.csv’i=0while i le (n−1) do begin

printf,1,mydata.z[i]*1d7,’ ’ ,real_part(gamma[i])i=i+1

endclose,1

window,3,title=’Right peak’plot,rightpeak

plotothers=1wn=9

myfit=make_array(n,/double)i=0maxi=n−1y0=0.0049d0x0=7340133.3396d−17b=5647.4962d−17a=1.0489d0while i le maxi do begin x=time[i] exponential=−0.5d0*((x − x0)/b)^2

myfit[i]=y0+a*exp(exponential)i=i+1

end

if plotothers eq 1 then beginwindow,4,title=’Normal Frequency Spectrum’plot,frequency,fourier

window,5,title=’Normal Fringe Pattern’plot,time,mydata.v

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experimentaldata

32window,6,title=’Fringe Pattern Minus Mean’plot,mydata.z,avv

window,7,title=’Phase Retrieved’plot,phi

window,8,title=’Hilbert Transform Envelope’plot,time,gamma

window,9,title=’Hilbert Transform Phase’plot,phi2

endif

endend

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Figure

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Code

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experimentaldata

33function computegaussian,position,clexponent=−4d0*alog(2d0)*(position/(2*cl))^2return,exp(exponent)end

function computegaussian2,position,clexponent=−0.5d0*(position/cl)^2amplitude=sqrt(1d0/(2*!pi))*(1/cl)return,amplitude*exp(exponent)end

;scalescale = 1d0;Wavelengthlambda = 682.25d−9*scale;speed of lightc=3.0d8;frequencyf=c/(lambda);Wave numberk=2*!pi/lambda;frequency FWHMsigma=1.571d13*(1d0/scale);wavelength FWHMsigmal=(sigma/f)*lambda;Coherence lengthcl=(2*alog(2)/!pi)*(lambda^2)/sigmalcl2=!pi/3d0;Coherence timect=cl/c;Step size (position)stepz=50.d−9;Step size ( time) (2* because you have round trip)stept=2*stepz/c;Step size (frequency)stepf=1/stept;Max and min zminz=−50.d−6maxz=50.d−6maxztime=maxz/c;Num of z stepsnumofzsteps=(maxz−minz)/stepz + 1stepf=stepf/numofzsteps;intensitiesi1=0.5d0i2=1−i1

gauss=make_array([numofzsteps],/double)gauss2=make_array([numofzsteps],/double)position=findgen(numofzsteps)*stepz − maxzposition2=make_array([numofzsteps],/double)time=findgen(numofzsteps)*stept − 2*maxztimej=minzic=0m=−2d0*!pistepm=4d0*!pi/numofzstepswhile j le maxz do begin

gauss[ic]=computegaussian(j,cl)position2[ic]=mgauss2[ic]=computegaussian2(m,cl2)j=j+stepzic=ic+1

Nov 06, 03 12:29 Page 1/3simulation.prom=m+stepm

endwhiletherand=randomn(seed,numofzsteps,poisson=0.5d0)therand=therand/max(therand)i= sqrt(i1*i2)*cos(2*!pi*f* time)*gaussirand=i*(1d0+therand)i = 0.5d0*(i1 +i2) + iirand = 0.5d0*(i1 +i2) + irandwindow,1,title=’ fringe pattern without noise’plot, time,i,xtitle=’ Temporal separation (s)’,ytitle=’ intensity (arbitrary)’window,2,title=’ fringe pattern with noise’plot, time,irand,xtitle=’ Temporal separation (s)’,ytitle=’ intensity (arbitrary)’;offset −> can minus away the mean of the intensity functionoffset=mean(i)offsetrand=mean(irand)meanmi=i−offsetmeanmirand=irand−offsetrand;Hilbert transform phase and amplitudegamma=0.5d0*sqrt(meanmi^2 + (hilbert(meanmi))^2)hv = hilbert(meanmi)nv = real_part(meanmi)ratiov = hv/nvphi2 = atan(ratiov);n21=numofzsteps/2 − 1fourier=shift(fft(shift(i,−n21),−1),n21)fourierrand=shift(fft(shift(irand,−n21),−1),n21)fouriermean=shift(fft(shift(meanmi,−n21),−1),n21)fouriermeanrand=shift(fft(shift(meanmirand,−n21),−1),n21)frequency=findgen(numofzsteps)*stepf − n21*stepfwindow,3,title=’ frequency’plot,frequency,fourierwindow,4,title=’ frequency with noise’plot,frequency,fourierrandrfpp=fix(f/stepf)+1k=fix(n21+rfpp/2d0)maxk=fix(n21+3d0*rfpp/2d0)numofksteps=maxk−krightpeak=make_array([numofzsteps],/double)rightpeakrand=make_array([numofzsteps],/double)balh=make_array([numofzsteps−1],/double)j=kwhile j le maxk do begin

rightpeak[j] = fouriermean[j]rightpeakrand[j] = fouriermeanrand[j]j=j+1

endrightpeak=shift(rightpeak,−rfpp)rightpeakrand=shift(rightpeakrand,−rfpp)w=shift(fft(shift(rightpeak,n21),1),−n21)wrand=shift(fft(shift(rightpeakrand,n21),1),−n21)phi=imaginary(alog(w))phirand=imaginary(alog(wrand))v=real_part(alog(w))vrand=real_part(alog(wrand))v=2*exp(v)vrand=2*exp(vrand)window,0,title=’ envelope function supplied, and retrieved’plot, time,gauss,xtitle=’ temporal separation (s)’,ytitle=’ intensity (arbitrary)’oplot, time,(1d0/v[n21])*v,linestyle=2oplot, time,(1d0/real_part(gamma[n21]))*gamma,linestyle=9window,7,title=’ envelope function supplied, and retrieved with noise’

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Code

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ulatingresults

34plot, time,gaussoplot, time,(1d0/real_part(gamma[n21]))*gamma,linestyle=9

; write it out to file so that Sigmaplot can fit itopenw,1,’ c:\blah2.csv’ic=0n=numofzstepswhile ic le (n−1) do begin

printf,1, time[ic]*1d17,’ ’,real_part(v[ic])ic=ic+1

endclose,1

;My phase unwrapping subroutinej=0maxj=numofzsteps−2while j lt maxj do beginbalh[j] = phi[j+1] − phi[j]j=j+1endj=1m=0maxj=maxj+1condition=0.9d0*2*!pioldphi=phiturnon=0while j le maxj do begin

if balh[j−1] le 0d0 then sig=−1d0 else sig=1d0if abs(balh[j−1]) ge condition then begin

phi[j] = 2d0*!pi − balh[j−1] + phi[j−1]turnon=1

endif else beginif turnon eq 1 then phi[j] = balh[j−1] + phi[j−1] else phi[j] =

phi[j]endelsej=j+1

end

window,5,title=’ phase retrieved and unwrapped’plot,phi,xtitle=’ step number’,ytitle=’ phase (radians)’window,6,title=’ phase retrieved with noise’plot,phirandend

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Code

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Bibliography

[1] G.J. Tearney, B.E. Bouma, S.A. Boppart, M.E. Brezinski, J.F. Southern, E.A. Swanson, and

J.G. Fujimoto. Endoscopic Optical Coherence Tomography. Lasers and Electro-Optics Society

Annual Meeting, 1996. LEOS 96., IEEE, 1:18, 1996.

[2] M.D. Kulkarni, J.A. Izatt, K. Kobayashi, S. Yazdanfar, R.C.K. Wong, M.V. Sivak, J.K. Barton,

and A.J. Welch. Optical coherence tomography: a novel tool for micron-resolution biomedical

diagnostics. Engineering in Medicine and Biology society. Proceedings of the 19th Annual

International Conference of the IEEE , 30 Oct.-2 Nov. 1997, 2, 1997.

[3] K.V. Larin. Study into the feasibility of using OCT to monitor blood glucose levels.

www.ki.se/molmed/thediabetesnews/TDN-2002-12.pdf.

[4] G.J. Tearney, S.A. Boppart, B.E. Bouma, M.E. Brezinski, N.J. Weissman, J.F. Southern, and

J.G. Fujimoto. Scanning single-mode fiber optic catheter-endoscope for optical coherence to-

mography. Optics Letters, 21:543, 1996.

[5] D. Huang, E.A. Swanson, C.P. Lin, J.S. Schuman, W.G. Stinson, W. Chang, M.R. Hee, T. Flotte,

K. Gregory, C.A. Puliafito, and J.G. Fujimoto. Optical coherence tomography. Science,

254:1178, 1991.

[6] L. Thrane. Optical coherence tomography: Modeling and Applications. PhD Thesis, Riso

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[7] U. Morgner, W. Drexler, X. Li, F.X. Kartner, C. Pitris, S.A. Boppart, E.P. Ippen, and J.G. Fu-

jimoto. Spectroscopic optical coherence tomography. Quantum Electronics and Laser Science

Conference, 1999. Technical Digest. Summaries of Papers Presented at the, page 62, 1999.

[8] L. Hartl, X.D. Li, C. Chudoba, R.K. Ghanta, T.H. Ko, and J.G. Fujimoto. Ultrahigh-resolution

optical coherence tomography using continuum generation in an air-silica microstructure opti-

cal fiber. Optics Letters, 26:608, 2001.

BIBLIOGRAPHY 36

[9] Joseph A. Izatt, Michael R. Hee, Gabrielle M. Owen, Eric A. Swanson, and James G. Fujimoto.

Optical coherence microscopy in scattering media. Optics Letters, 19:590, 1994.

[10] E. Hecht. Optics. Addison-Wesley, Second edition, 1987.

[11] A.F. Fercher, W. Drexler, C.K. Hitzenberger, and T. Lasser. Optical Coherence Tomography -

Principles and applications. Rep. Prog. Phys., 66:239, 2003.

[12] M. Takeda, H. Ina, and S. Kobayashi. Fourier-transform method of fringe-pattern analysis for

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[13] S.K. Rhodes. High resolution studies of electomagnetic fields in focal regions. PhD Thesis,

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[14] J.M. Schmitt. Optical Coherence Tomography (OCT): A Review. IEEE Journal of Selected

Topics in Quantum Electronics, 5:1205, 1999.

oct

Documents

coherence lengths

coherence functions

low coherence interferometry

low coherence light

d form

d uses

whitelight interferometry

optical spectrum analyser