1. University Of Manchester
School of Physics and Astronomy
MPhys Project Report
Optical Coherence Tomography
- Fourier Domain
Author:
Riccardo William
Monfardini
Student ID: 7484334
Supervisor:
Dr. Mark Dickinson
Collaborator:
Alicia Burn
January 24, 2014

2. Abstract
A Fourier Domain Optical Coherence Tomography apparatus was
built and used to determine the thickness of a microscope cover glass;
it was measured to be t = 154.33±4.70µm. A green super-luminescent
LED was used as a broadband light source, with a calculated coherence
length of lc = 4.7266 ± 0.0059µm. The intensity of the system was
observed to decline with depth faster than the theoretical prediction,
resulting in only one surface being seen at one time; correcting the
linear relationship between sample movement and depth measured
allowed the thickness to be found indirectly.
1 Introduction
The modern era has seen the development and the wide use of a new range
of imaging techniques in a variety of scientific disciplines, with a particular
interest shown by the biomedical sector [1]; Optical Coherence Tomography
(OCT) technology is one of these techniques. The first investigation of OCT
was performed in 1991 by Huang and Fujimoto [2], who coined the name,
and since then it has been, and will be, used extensively [3, 4].
The OCT technique resolution ranges from 1-15µm with a penetration
of 2-3mm [5]. A comparison of OCT with ultrasound and microscopy, two
other imaging methods, is shown in Figure 1; from this graph it is possible to
see that ultrasound has a deeper image penetration (10cm), and microscopy
has a slightly better resolution (< 1µm). OCT manages to bridge the gap
between the two by attaining high resolution deep images. Because of these
properties OCT has become the standard technique for imaging the retina
of the eye, where other techniques fail to produce the high resolution that it
can achieve.
In essence OCT works just like a Michelson Interferometer using a low-
coherent light source. The resolution is due to the small coherence length
where interference is seen. Time Domain is the most common method used
for OCT, where a scanning mirror is used to find an interference pattern and
thus match the depth of the interface of the tissue. However, for the purpose
of this report, Fourier Domain OCT has been used. This has the advantage of
not requiring the mirror to move, because a full depth profile can be obtained
with a single scan. This study proceeded by using the apparatus of a previous
study [6, 7] with the aim of improving the results previously obtained and
measuring the thickness of a glass cover of the order of a 100µm.
1

3. Figure 1: Resolution and image penetration comparison between Ultrasound,
OCT and Microscopy imaging techniques. Normal clinical Ultrasound can scan
deep structures; higher resolution can be achieved using higher frequencies, at
the cost of penetration due to attenuation of the signal. Microscopy gives the
highest sub-micron resolution, but the optical scattering does not allow more than
superficial scans. OCT can achieve similar resolutions, since it is an optical
method, and by using IR light is it possible to reach deeper depths. [5]
2 Theory
As previously mentioned, OCT apparatus is essentially a Michelson Interfer-
ometer (MI) using a broadband light source. Therefore, a short overview of
how this works is provided.
2.1 Michelson Interferometer
The particle-wave duality of light means that electromagnetic waves can in-
terfere with each other. The output will be the superimposed sum of all
waves, with constructive or destructive interference. This is what happens
in the MI apparatus, as shown in Figure 2.
2

4. Figure 2: Diagram of a basic Michelson Interferometer. Light from a source S
hits a beam splitter that sends it towards two mirrors, M1 and M2. The reflected
beams recombine to give output B. Because M2 is at a distance d further than
M1, the light beam has to travel 2∆d more than its counterpart. This causes the
two beams to be out of phase thus creating an interference pattern in B. If M2
is movable then a scan across various depths can be performed to measure B.
Modified from [8].
By moving a mirror by an extra amount ∆d the light beam in that arm
has to travel an amount 2∆d, introducing a phase difference
∆φ = 2π
(2∆d)
λ
, (1)
and output intensity is
I(∆d) =
S
2
(1 + cos (∆φ)) , (2)
where S is the original intensity of the source and λ is the wavelength of
the light source. Therefore consecutive intensity maxima are separated by a
distance λ/2 [9].
3

5. 2.2 Low coherence interferometry
An interference pattern will be visible only if the waves are coherent, i.e. if
the phase relationships are constant. The coherence length lc for a Gaussian
spectrum define has the length at which the fringe visibility drops to 50% is
lc =
2 ln 2
nπ
λ2
∆λ
, (3)
where n is the refractive index of the medium, λ is the peak wavelength and
∆λ is the full width half maximum (FWHM) of the light source intensity
which is assumed to have a Gaussian shape [9].
OCT works on these principles, but rather than using a highly coherent
light source, like a laser, it instead uses a broadband source. The short
coherence length means that interference is seen in a narrow depth window
∆z, inversely proportional to the bandwidth of the source,
∆z =
1
2
lc =
ln 2
nπ
λ2
∆λ
, (4)
where the 1/2 comes from the double pass. The length ∆z is the resolution
of the OCT apparatus. If one of the mirrors is the sample, interference is
seen when the reference mirror is within ∆z from the sample source. Time
Domain OCT works by scanning the reference mirror across a depth z and
recording the intensity output.
2.3 Fourier Domain OCT
It is possible to detect the depth of the sample with a single scan by using
a Fourier Domain OCT apparatus, as shown in Figure 3, using spectral
interferometry [9].
A depth profile I(z) can be obtained by Fourier transforming the signal
spectrum I(k),
I(z) = FT{I(k)}. (5)
The output signal is the sum of the wave from the reference mirror, with
amplitude ar and the elementary waves propagating from different depths z
of our sample with amplitude a(z). The spectrometer divides and measures
them in different wavenumber channels.
4

6. Figure 3: Diagram of a basic Fourier Domain OCT apparatus. The light source
is split into two beams and recombined as in Figure 2, but the diffraction grating
separates the different wavelengths of the output. These are detected by a CCD
detector array. Modified from[1].
The interference signal can be written as
I(k) = S(k) arei2kr
+
∞
−∞
a(z)exp{i2k[r + n(z)z]}dz
2
, (6)
where 2r and 2(r + n(z)z) are the lengths the light has to travel in the
reference and sample arm respectively, n(z) is the refractive index and S(k)
is the source spectrum. By setting r = 0, because the interest lies in the
difference between the two path lengths, and assuming constant refractive
index the intensity in equation 6 becomes
I(k) = S(k) 1 +
∞
−∞
a(z)ei2knz
dz
2
= S(k) 1 +
∞
−∞
a(z) cos(2knz)dz +
∞
−∞
a(z)a(z )ei2kn(z−z )
dzdz .
(7)
5

7. In the brackets, the first term is a constant offset while the third is a term
describing the mutual interference between the elementary waves coming
from the sample; the second term encapsulates the information about the
depth. This can be extracted taking a Fourier transform. Assuming that
a(z) is symmetrical, the substitution ˆa(z) = a(z) + a(−z) is made giving
I(k) = S(k) 1 +
∞
−∞
ˆa(z)eiknz
dz +
1
4
∞
−∞
AC[ˆa(z)]e−iknz
dz , (8)
where AC[ˆa(z)] is the autocorrelation.
Therefore the depth profile is given by the convolution
I(x) = FT{S(k)} ⊗ [δ(z)] +
1
2
ˆa(z) +
1
8
AC[ˆa(z)] . (9)
The first term in the brackets can be avoided by subtracting a reference
spectrum, S(k), when no interference is seen, and the third term is centred
around z = 0. Thus the depth is given by the convolution of the second term
with the Fourier transform of the spectrum [9].
In the case where the sample consists of a mirror, as described later in
3.2, equation 7 becomes
I(k) = S(k) cos(2knz), (10)
because the reference spectrum is subtracted and there is only one reflecting
surface at z = 2∆d; and thus
I(∆d) = FT{S(k)} ⊗
1
2
(δ(k) + δ(−k)) . (11)
2.3.1 From Fourier bin to Depth
The Fourier transform of the signal gives the frequency (bin number) of the
cosine function in equation 7. If instead of I(k) the spectrum is measured
with respect to wavelength I(λ) the bin number is
Fbin =
Range
∆λ
, (12)
where Range is the range of frequencies and ∆λ is the wavelength difference
between successive intensity maxima. It is understood that a constructive
interference for a particular wavelength λ happens when
mλm = 2∆d;
6

8. thus, rearranging and partially differentiating, it is derived that successive
maxima (∆m = 1) are separated by
∆λ =
λ2
2∆d
, (13)
substituting in equation 12 and rearranging
∆d =
λ2
2Range
Fbin, (14)
thus giving a constant of proportionality between the Fourier bin and depth
[10].
2.3.2 Sensitivity
The sensitivity of a Fourier Domain OCT apparatus will decrease as a func-
tion of depth. This is due to the finite resolution of our spectrometer; thus
it becomes more difficult for the Fourier transform to detect the higher fre-
quencies (small ∆λ) due to this limited window. The magnitude of decrease
is given by
R(z) = sinc(ζ)2
· exp −
w2
2 ln 2
ζ2
, (15)
where ζ = π
2
z
zmax
with the maximum measurable depth given by zmax = λ2
4λpixel
and λpixel is the wavelength spacing between pixels, and w = FWHM/2λpixel
[11].
7

9. 3 Method
Figure 4: The apparatus as described in 3.3. In the experimental apparatus
described in 3.2 the Sample was a mirror identical to the Reference Mirror, and
the ND filter was missing; in 3.1 it was the same but with a screen rather than a
spectrometer, the Pin Hole open, and without the lenses or the Green LED.
3.1 Set-up of the Michelson Interferometer
A Michelson interferometer was built as shown in Figure 4. Light from a
HeNe laser passed through a 50:50 beam splitter which directed two beams
to a fixed reference mirror and a sample mirror; the latter was placed on
a NanoMax stage which could be controlled by a stepper motor for fine
displacements of a few microns, installed on a moving platform for manual
controlled displacement of several mm. The mirrors were adjusted in order
to reflect the beams back to the beam splitter and recombine; thus they over-
lapped and interfered with each other. The high coherence of the HeNe laser
meant it was possible to see interference when the path difference between
the two arms was substantial; projecting the output onto a screen, the fringes
8

10. appeared as concentric circles. This set-up was used to detect an approxi-
mate position for the zero path length difference (PLD), ∆d = 0: by moving
the sample mirror across the zero point, this is determined by recording when
the fringes swap from moving inwards to outwards. The imperfect alignment
meant that the mirrors were repeatedly adjusted in order to see this change
of direction.
3.2 Introducing a low coherence source
A green super-luminescent LED was introduced instead of the laser. Light
from the LED passed through lens 1 with focal length 25.4mm which was
used to collect as much light as possible; spatial coherence was enhanced
by filtering the light through a pin hole. The light passed through lens 2
placed at its focal length, 50.0mm from the pin hole, to redirect the light
into parallel plane waves towards the beam splitter. The beams reflected
back from the reference and sample mirrors recombined and passed through
lens 3 which was used to concentrate the beam into a spectrometer; this was
placed at the focal length of the lens, 75.0mm. By opening the pin hole and
placing a piece of paper in front of it, it was possible to adjust the platform
position until fringes were visible to the naked eye, when looking directly
into the beam splitter. Measurements of the spectrum were then performed
at various positions or depths of the sample mirror. This was achieved by
fine movement of the NanoMax stage using the stepper motor controlled by
a computer. The spectrometer was a Ocean Optics USB4000+, with 3648
pixels across the visible range, from 350 to 1040 nm [12].
3.3 Microscope cover glass sample
The apparatus was used to measure the thickness of a microscope cover glass,
which substituted the sample mirror used previously. Its thickness ranged
from 130 to 160µm and refractive index nD = 1.5230 (nD is the refractive
index of a medium at λ = 589.8nm) and reflectivity of 8.3% [13]. Due to the
lower reflectivity of the glass, the beam from the reference mirror was dimmed
using a natural density (ND) filter of optical density 0.6 positioned at an angle
so that its reflection did not affect the interference output. This ensured
that the two beams were of approximately equal intensity and thus that the
interference was maximised. The increased path length of the reference beam
due to the higher refractive index of the ND filter moved the zero point, which
had to be found again in order to see the interference coming from the sample.
Measurements were taken using the stepper motor at various positions, as
before.
9

11. 4 Results & Discussion
The green super-luminescent LED spectrum is shown in Figure 5. Using
equation 3 the coherence length was calculated to be lc = 4.7266±0.0059µm.
As described in section 3.2, measurements of interference between the beams
400 500 600 700 800 900 1000
0
1
2
3
4
5
6
x 10
4
Wavelength [nm]
Intensity[a.u.]
Green Super Luminescent LED spectrum
λ
peak
= 534.28 ±1.02nm
FWHM = 26.65 ±0.14nm
Figure 5: Spectrum of the green super-luminescent LED light source.
reflected by the two mirrors were taken at various depths; a reference dark
spectrum was taken at a distance where no interference was noticed; this
was subtracted to avoid the autocorrelation of the reference spectrum of the
light source. The depth profiles were acquired using the MATLAB FFT
algorithm. As an example, one of the spectrum measurements is shown
with its depth profile in Figure 6. It can be seen in the top graph that
the measurement yielded in a Gaussian spectrum of the light source shown
in Figure 5 with a sinusoidal modulation minus the reference spectrum as
described by equation 10. The bottom graph of Figure 6 displays the Fourier
transform of the measurement, as described by equation 11. The depth is
given by the depth of the absolute of the FFT intensity peak; the uncertainty
is half the FWHM of this peak. The depth in Figure 5 was measured to be
∆d = 17.06 ± 2.21µm.
Using this method, an investigation of how the PLD changed with the
motor movement, was conducted. The maximum FFT intensity position is
plotted against the motor position with respect to an arbitrary zero point in
10

12. 400 500 600 700 800 900 1000
−2
−1
0
1
2
x 10
4 Spectrum from measurement N005
Wavelength [nm]
Intensity[a.u.]
0 50 100 150 200 250 300 350
−1.5
−1
−0.5
0
0.5
1
1.5
x 10
6 Fourier Transform of the output spectrum
Depth [um]
Intensity[a.u.]
Figure 6: The top graph shows the output spectrum after the reference spectrum
is subtracted.The bottom graph shows the FFT of this spectrum against depth,
where the half specular datapoints have been excluded. Both the real part (blue),
with the sinusoidal behaviour, and absolute (red) are plotted. The PLD is ∆d =
17.06 ± 2.21µm
Figure 7. From the investigation various features emerged:
• Within certain limits the proportionality between motor movement and
depth measured is linear as expected, but with a depth to motor pos-
ition ratio of 1.09 ± 0.02 rather than 1; this is shown by the bottom
right graph of Figure 7. A conclusive explanation of the cause of this
deviation has not been established; however, it is possible that the
spectrometer measures the spectrum in evenly spaced wavelength in-
crements rather than wavenumber. The Fourier transform explained in
section 2.3 links z to k; thus the non-linear relationship between k and
λ leads to a broadening of the depth resolution [14], to which must be
added a general misalignment of the apparatus.
• The linear relationship shifts by ∼10µm when the mirror moves across
certain positions, indicated by the arrows in the top graph in Figure
7. No conclusive investigation was conducted. The fact that the linear
relationship is preserved except for these points could suggest that the
NanoMax stage or the stepper motor slipped.
11

13. −400 −300 −200 −100 0 100 200 300 400
0
50
100
150
200
250
300
350
Linearity between motor movement and calculated depth
Mirror position (Motor movement) [um]
Depth(FFT)[um]
0 50 100 150
0
50
100
150
200
Absolute Mirror Position [um]
Depth[um]
y = 1.09*x − 12.68
±0.02 ±0.74
0 100 200 300 400
−2
0
2
x 10
4 FFT of measurement N380
Depth [um]
Intensity[a.u.]
Figure 7: The top graph shows the calculated depth by the FFT of the spectrum
against the mirror position with respect to an arbitrary zero. The arrows high-
light the measurement where the liner relationship seems to shift by 10µm in
depth. The circles highlight the measurements where no interference between the
mirrors’ reflected beams is seen; however, a constant frequency corresponding to
194.90±30.09µm was measured. The bottom left graph shows one of the measure-
ments demonstrating this feature. The bottom right graph illustrates the linear
relationship between motor position and the measured depth for a selected number
of measurements highlighted in the rectangular box; these are the measurement
that include the zero PLD but exclude any depth shift pointed by the arrows.
12

14. • The FFT of the measured spectra returned a constant frequency with
small intensity around a depth of 194.90 ± 30.09µm; when the FFT
intensity of the interference of two mirrors was weak, which happened
at greater depths, this frequency became predominant. All of these
measurements are displayed within circles in the top graph of Figure
7. One of these measurements, at mirror position −380µm is shown in
the bottom left graph. It is speculated that this feature is caused by
aliasing due to the finite size of the pixels, which is quoted as being
8µm × 200µm [12].
0 50 100 150 200 250 300 350 400
0
5
10
15
x 10
5 Intensity decline
Depth um
MaxIntensity[a.u.]
−400 −300 −200 −100 0 100 200 300 400
0
5
10
15
x 10
5
Mirror Position (Motor Movement) um
MaxIntensity[a.u.]
FFT max Intensity
1−R(z)
Error in the intensity is σ= ±3.09×10
3
[a.u.]
Figure 8: The top graph shows the decline of the FFT maximum Intensity in
function of depth compared with the theoretical decline in sensitivity R(z). The
bottom graph shows FFT maximum intensity in function of mirror position.
An investigation of the sensitivity of the experimental apparatus was
conducted. In Figure 8 the top graph shows the FFT maximum intensity
against depth, which is compared to the theoretical value given by equation
15; in the bottom graph of the same figure, the intensity against the mirror
position is displayed. These graphs show that intensity declines in function
of depth as expected, but a consistent discrepancy can be noted between
the theoretical curve and the measurements; and there is a different decline
13

15. rate between the two sides of the zero PLD. It is believed that imperfect
alignment is the reason for this, which means that the output beam moved
with respect to the spectrometer as the sample mirror moved.
Finally, the thickness of a microscope cover glass was determined by sub-
stituting it with the sample mirror. It was not possible to see the interference
from both the front and back surfaces because of the decline in intensity dis-
cussed previously. However, in Figure 9 a plot of the linear relationship
between motor movement and depth is shown; it can be observed that both
surfaces create a linear relationship, describing a ‘W’ shape. Therefore, the
distance between the intercepts of the linear equations describing each ‘V’ is
the thickness of the cover glass, accounting for an increased refractive index.
With a refractive index of n = 1.52 [13] the thickness was calculated to
be t = 168.23 ± 4.70µm but if we take into account the depth-proportional
offset described by the 1.09 gradient in the bottom right graph of Figure 7 the
thickness is corrected to t = 154.32 ± 4.70µm; this is towards the higher end
but still within the manufacturer’s claim that the thickness of its microscope
cover glasses ranges between 130 to 160 µm [13].
−200 −100 0 100 200 300 400 500
0
20
40
60
80
100
120
140
160
Linearity between motor movement and calculated depth
Mirror position (Motor movement) [um]
Depth(FFT)[um]
±0.05 ±1.28
±0.08 ±3.12
±1.25
±0.09 ±3.03
y=1.09⋅x −9.69 ±0.04
y=−1.09⋅x +259.49
y=1.15⋅x −273.84
y=−1.05⋅x +11.25
Figure 9: Graph showing the calculated depth of the interference from the cover
glass against mirror position. Each surface is responsible for one of the ‘V’s; the
distance between them is nt, the thickness of the glass times the refractive index.
The thickness is t = 168.23 ± 4.70µm or t = 154.32 ± 4.70µm if compensated for
the depth proportional measurement offset.
14

16. 5 Conclusion
Using an Fourier Domain Optical Coherence Tomography apparatus it was
possible to determine the thickness of a microscope cover glass. This was
found to be t = 154.33 ± 4.70µm. To achieve this a green super-luminescent
LED light was used, with a measured coherence length of lc = 4.7266 ±
0.0059µm. Investigations of the apparatus used showed several features:
• Within some limits the linear relationship between mirror movement
and depth calculated was found to be 1.09±0.02. This was an improve-
ment from the value of 3.32 of the previous study [7, 6]. This offset was
considered in the thickness calculation. It was suggested that this was
due to misalignment of the apparatus and to a non-even wavenumber
spacing in the spectrum measurement.
• A shift in the measured depth of ∼10µm was observed at certain mirror
positions. It was suggested that this could be caused by a slip in the
stepper motor or the NanoMax stage.
• A constant interference was seen at a frequency corresponding to a
depth of 194.90±30.09µm. This was suggested to be due to an aliasing
effect of the pixel size of the spectrometer quoted as being 8µm×200µm
The apparatus sensitivity was shown to reduce more rapidly than the pre-
dicted theoretical amount as shown by the decline in the FFT maximum
intensity. Due to this the two surfaces were not seen at the same time, and
the thickness was measured indirectly by the linear relationship with the
motor movement.
Future studies are recommended to process the measurement of the spec-
trum evenly spaced in wavelength to produce a spectrum evenly spaced
in wavenumber by using an interpolating non-linear scaling algorithm[14].
This will improve sensitivity and resolution. Determination of the causes of
anomalous features found in the apparatus is advised.
15

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17

19. Acknowledgements
I would like to thank the previous students involved in this project Christo-
pher Cox, Jonathan Entwisle, for their initial help when handing over the
apparatus and in how to use the software. And a special thanks to our su-
pervisor Mark Dickinson for providing us with guidance and advice during
the course of the semester.
18