# Non-mechanical steering of the optical beam in spectral-domain Optical
Coherence Tomography

Daniel F. Urrego<EMAIL_ADDRESS>ICFO – Institut de Ciencies
Fotoniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels, Barcelona, Spain.  Gerard J. Machado  Department of Physics,
Imperial College London, Prince Consort Road, SW7 2AZ London, United Kingdom.
Juan P. Torres  ICFO – Institut de Ciencies Fotoniques, The Barcelona
Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain.
Department of Signal Theory and Communications, Universitat Politecnica de
Catalunya, 08034 Barcelona, Spain

(August 28, 2024)

###### Abstract

We demonstrate in a proof-of-concept experiment spectral-domain optical
coherence tomography where steering of the optical beam that probes the sample
in a transverse scan does not make use of any mechanical element. Steering is
done with the help of a spatial light modulator, that introduces a spatially-
dependent phase between the two orthogonal polarization components of an
optical beam, and some optical elements that control the polarization of
light. We demonstrate that making use of the non-mechanical steering system
considered here, we can reproduce the main traits of imaging with standard OCT
that makes use of mechanical-assisted optical beam steering.

##  I Introduction

Optical coherence tomography (OCT) is a three-dimensional imaging technique
introduced in 1991 that makes use of a Michelson interferometer, where light
in one arm illuminates the sample and light in the other arm serves as a
reference Huang  _et al._ (1991); Dresel  _et al._ (1992). The axial and
transverse resolutions of OCT are independent. To obtain information in the
transverse direction (plane perpendicular to the beam propagation), OCT
focuses light to a small spot that is scanned over the sample. To obtain
information in the axial direction (along the beam propagation), OCT uses a
source of light with short coherence length that allows optical sectioning of
the sample. OCT is nowadays an active area of research as well as a mature
technnology Brezinski (2006); Drexler and Fujimoto (2015); Bouma and Ternay
(2019) that finds applications in many areas of science and technology, from
medicine, particularly ophthalmology Drexler and Fujimoto (2008); Schuman  _et
al._ (2006); Haeusler and Lindner (1998) to art conservation studies Adler
_et al._ (2007); Liang  _et al._ (2007).

The first OCT schemes used time-domain OCT, where axial sectioning of the
sample was done by moving a mirror in the reference arm with the help of a
fast-moving mechanical platform, such as a Galvo scanner or a 2D translation
stage. The translation of the mirror in the reference arm severely limited the
imaging speed achievable. Stratus OCT, introduced in 2002, achieved speeds of
$\sim 400$ A-scans per second Fujimoto and Swanson (2016). In 1995, Spectral
or Fourier-domain OCT was a major breakthrough for the development and rapid
expansion of OCT Fercher  _et al._ (1995). In spectral-domain OCT, axial
sectioning of the sample is done measuring the spectrum of the light back-
scattered from the sample with a spectrometer, eliminating the need to move
the position of the mirror in the reference arm. OCT can also be done with a
laser that changes fast its frequency (swept-source OCT) Chinn  _et al._
(1997), and both spectral-domain and swept-source OCT schemes provides a
better axial resolution than time-domain OCT Choma  _et al._ (2003); Leitgeb
_et al._ (2003a). One advantage of swept-source OCT is that a spectrometer is
not required, only a broadband detector. Spectral-domain OCT is currently the
standard for ophthalmic instruments, with imaging speeds in excess of $\sim
25,000$ A-scans per second Fujimoto and Swanson (2016).

In order to obtain an image of the sample with good transverse resolution, all
the OCT schemes considered still require the use of a fast-moving mechanical
device to scan a focused spot of light across the sample. We propose a
significant innovation, demonstrated through a proof-of-concept experiment,
which introduces a method for performing transverse scans of the sample
without requiring a mechanical platform. The mechanical element is replaced by
a spatial light modulator (SLM) in combination with some simple optical
elements to modify the polarization of light Moreno  _et al._ (2007). We will
show that by engineering the phase introduced by groups of pixels of the SLM,
one can perform the transverse scan of the light beam without any mechanical
scan needed.

One word of caution might be appropriate here. We put forward a scheme, that
in combination with spectral-domain OCT, allows to perform OCT fully in the
optical domain without the need of any mechanical steering of the optical beam
that probes the sample. Since we use a slow-response spatial light modulator
(Hamamatsu X10468), our proof-of-principle demonstration does not achieve the
scan speeds of state-of-the-art OCT systems. In spite of this, it can help
usher in the development of alternative OCT schemes with potentially faster
scan speeds than current OCT systems by avoiding the use of any kind of
mechanical platforms.

Non-mechanical laser beam steering, for applications such as beam steering in
free-space laser communications, has been proposed and demonstrated using
liquid crystal optical phased arrays Wang  _et al._ (2018). Most commonly used
liquid crystal devices share that they show a slow response, due to the slow
rotation of the liquid crystal molecule. However, there is on-going research
aimed at speeding up agile beam steering, developing new materials and
arquitectures that allows faster change of molecules orientation Fan  _et al._
(2004); Chen  _et al._ (2013); Peng  _et al._ (2015). Recently, a spatial
light modulator architecture has been proposed that can achieve two-
dimensional phase-only modulation at high speed in excess of GHz. It makes use
of a tunable two-dimensional array of vertically oriented, one-sided
microcavities that are tuned through an electro-optic material such as barium
titanate Peng  _et al._ (2019).

Figure 1: Experimental setup for non-mechanical transverse steering of an
optical beam in optical coherence tomography. A superluminescent diode
illuminates an SLM that is the key element of the non-mechanical scanning
system. The SLM introduces a different phase on a pixel-by-pixel basis. The
output beam, with a new spatial shape, enters a polarization-sensitive
Michelson interferometer. A Liquid Crystal Variable Retarder introduces a
controllable phase difference between the light beams coming from the sample
and reference arms, that are recombined at the outport port of the Michelson
interferometer. Two interferograms are acquired employing a spectrometer with
a phase difference $\Delta\phi=0,\pi$. Several methods (see main text) are
used to filter out the meaningless peaks of the Fourier transform signal, and
enhance the sought-after peaks that carry information about the longitudinal
structure of the sample.

##  II Theory and Experimental setup

Figure 1 shows a detailed overview of the experimental setup. It consist of a
polarization-sensitive Michelson interferometer, in combination with a spatial
light modulator (SLM) and some optical elements that control the polarization
of light. A superluminescent diode (SLD) generates light with central
wavelength $810\leavevmode\nobreak\ \text{nm}$ and a bandwidth of $\sim
29.5\leavevmode\nobreak\ \text{nm}$. The polarization of the light generated
by the SLD is selected to be horizontal with the help of a polarizer. The
spatial shape of the light beam is Gaussian, and its beam width is magnified
by a telescope with lenses $\text{L}_{1}$ and $\text{L}_{2}$, reaching a beam
waist of $\text{w}_{0}=1.26\leavevmode\nobreak\ \text{mm}$. This is the light
beam that will enter the non-mechanical transverse steering system of the OCT
configuration.

###  II.1 Non-mechanical steering of the optical beam

We aim at generating a Gaussian-like beam with a small cross section in one
transverse direction, that can be steered in the transverse plane, without the
use of any mechanical platform, and still cover the whole area of the sample
to be scanned. For achieving this goal we employ a half-wave plate
($\text{HWP}_{1}$), a spatial light modulator (SLM, Hamamatsu X10468, 1060
$\times$ 970 pixels), a second half-wave plate ($\text{HWP}_{2}$) and a
polarizing beam splitter ($\text{PBS}_{1}$). Half-wave plate $\text{HWP}_{1}$
is rotated an angle $\theta$, so it generates a light beam with polarization

$\hat{e}_{in}=\cos 2\theta\,\hat{\bf e}_{H}+\sin 2\theta\,\hat{\bf e}_{V},$
(1)

where $\hat{\bf e}_{H}$ and $\hat{\bf e}_{V}$ designate horizontal and
vertical polarization, respectively.

Each pixel, or group of pixels, of the SLM introduce a spatially-dependent
phase $\Phi(x,y)$ only to the vertical polarization component, so the
polarization of the light beam after reflection from the SLM is

$\hat{e}_{SLM}(x,y)=r_{H}\,cos2\theta\,\hat{e}_{\text{H}}+r_{V}\,\sin
2\theta\,\exp\left[i\Phi(x,y)\right]\hat{e}_{\text{V}},$ (2)

where we have considered that the reflection coefficients for the horizontal
and vertical polarizations ($r_{\text{H}}$ and $r_{\text{V}}$) can be
different.

Half-wave plate $\text{HWP}_{2}$ is rotated an angle $\pi/8$ so that the
polarization of the optical beam just before PBS1 is

$\displaystyle\hat{e}_{out}(x,y)=\frac{1}{\sqrt{2}}\Big{\\{}r_{H}\cos
2\theta+r_{V}\sin 2\theta\,\exp\left[i\Phi(x,y)\right]\Big{\\}}\hat{\bf
e}_{H}$ $\displaystyle+\frac{1}{\sqrt{2}}\left\\{r_{H}\cos 2\theta-r_{V}\sin
2\theta\,\exp\left[i\Phi(x,y)\right]\right\\}\hat{\bf e}_{v}.$ (3)

We can choose the spatial dependence of $\Phi(x,y)$ so as to tailor the
polarization of the light reflected by groups of pixels. In the ideal case of
equal reflectivities ($r_{H}=r_{V}$), we can see from Eq.(3) that for
$\theta=\pi/8$, a phase $\Phi(x,y)=0$ would generate a reflected beam with
horizontal polarization. PBS1 will transmit this section of the beam with
horizontal polarization, that will illuminate the sample in the optical
coherence tomography scheme. For all other pixels, we can choose a phase
$\Phi(x,y)=\pi$, so that this section of the light beam shows vertical
polarization, which is reflected by PBS1 and consequently discarded.

However, the reflection coefficients in the SLM for both polarizations are
different. We measured the values $r_{\text{H}}=0.91$ and $r_{\text{V}}=0.7$.
We can tune the orientation angle $\theta$ of HWP1 to minimize the intensity
associated with the unwanted polarization components of the output beam, i.e.,
$I_{\text{V}}(\Phi=0)$, that is equivalent to minimizing
$I_{\text{H}}(\Phi=\pi)$. The intensity of the horizontal component of the
output beam, before PBS1 is

$I_{\text{H}}(x,y)=\frac{1}{2}\,\Big{|}r_{H}\,\cos 2\theta+r_{V}\,\sin
2\theta\,\exp\left[i\Phi(x,y)\right]\Big{|}^{2}.$ (4)

The maximum and minimum values of the intensity of the horizontal polarization
component are obtained for $\Phi(x,y)=0,\pi$, so that
$I_{\text{H}}^{max,min}(x,y)=\frac{1}{2}\Big{[}|r_{H}|^{2}\,\cos^{2}2\theta+|r_{V}|^{2}\,\sin^{2}2\theta\pm
2\,|r_{H}|\,|r_{V}|\,\cos 2\theta\,\sin 2\theta\Big{]},$ (5) where the
positive sign corresponds to the phase $\Phi(x,y)=0$ and the negative sign to
$\Phi(x,y)=\pi$. The visibility
$V=[I_{\text{H}}^{max}-I_{\text{H}}^{min}]/[I_{\text{H}}^{max}+I_{\text{H}}^{min}]$
is

$V=\frac{2|r_{H}|\,|r_{V}|\,\cos 2\theta\,\sin
2\theta}{|r_{H}|^{2}\,\cos^{2}2\theta+|r_{V}|^{2}\,\sin^{2}2\theta}.$ (6)

Taking in account the experimental values of $r_{\text{H}}=0.91$ and
$r_{\text{V}}=0.70$, we get the maximum visibility for $\theta\approx
26.2^{\circ}$. This is the value of $\theta$ that we will use in the
experiments.

Figure 2: Characteristics of the non-mechanical steering system. (a) Intensity
of the output light as a function of the grayscale displayed in the SLM. The
intensity is measured after $\text{PBS}_{1}$. The graph indicates the
intensity change induced by the SLM. The maximum normalized standard error
obtained in the measurement is $\sigma_{E}=6.6\times 10^{-5}$. (b) Three
different configurations of the phases introduced by groups of pixels on the
screen of the SLM. The phase introduced by each pixel is selected using the
corresponding value of the grayscale. (c) Pictures taken in the sample plane
using the configuration depicted in (b). (d) Intensity distribution that
arrives at the sample plane as a function of the translation of the bar
displayed on the screen of the SLM. The maximum normalized standard error
obtained in the measurement is $\sigma_{E}=2.47\times 10^{-2}$.

The control parameter that allows to tune the phase introduced by each pixel
of the SLM is the grayscale displayed on each pixel of the SLM’s screen.
Figure 2(a) shows the intensity of the light beam after $\text{PBS}_{1}$ as a
function of the grayscale displayed on each pixel of the SLM’s screen. From
this data, one can obtain the values of grayscale that maximize and minimize
the intensity transmitted through $\text{PBS}_{1}$. These values correspond to
the introduction of phases $0$ and $\pi$, respectively, and they are
designated as states on/off of the pixels of the SLM. In our experimental set-
up, the values of grayscales are 31 and 158, respectively. Fig. 2(b) shows
three different arrays of on/off of pixels. The blue regions correspond to
pixels in state off, and the yellow regions correspond to the pixels in state
on. For the sake of example, we consider the case of the pixels forming a bar.
The bar measures $80\leavevmode\nobreak\ \mu\text{m}$ in width and has a
length exceeding $15\leavevmode\nobreak\ \text{mm}$.

Fig. 2(c) shows the shape of the output light beam generated by the
corresponding distribution of on/off pixels, displayed in Fig. 2(b). The
spatial profile of the beam is measured employing a CCD camera in the sample’s
plane. The bar can be displaced by changing the sets of pixels in the screen
that are on/off, as shown in Fig. 2(b). The shape and intensity of the light
beam that illuminates the sample depends on the characteristics of the input
beam that illuminates the SLM and the position of the pixels that constitute
the bar in the SLM. Fig. 2(d) shows the maximum possible intensity that can
reach the sample when the scan is done employing a bar with a width of
$80\leavevmode\nobreak\ \mu\text{m}$. It is important to clarify that the
measured intensity is acquired when all the incoming light to the
interferometer is directed toward the sample. The maximum intensity is reached
when the bar is in the middle of the Gaussian input beam, and its value is
$\text{I}_{\text{max}}=180\leavevmode\nobreak\ \mu\text{W}$.

###  II.2 The polarization-sensitive Michelson interferometer

A polarization-sensitive Michelson interferometer is the core of the OCT
configuration. A $4f-\text{system}$ with focal lengths
$\text{L}_{3}=\text{L}_{4}=400\leavevmode\nobreak\ \text{mm}$, allows the
diffraction-less propagation of the light beam from the plane of the SLM’s to
the sample’s plane. The polarizing beam splitter $\text{PBS}_{2}$ splits the
input light into the reference and sample arms. $\text{HWP}_{3}$ changes the
polarization of the light beam coming from PBS1, which is horizontal, and is
used to unbalance the intensities between both arms, since the object in the
sample arm will show in general a low reflectivity. In the reference arm,
there is a quarter-wave plate (QWP) oriented at $45^{\circ}$ and a mirror
mounted on a motorized translation stage. In the sample arm, there is a QWP at
$45^{\circ}$ and the reflective sample to be probed. The light beams coming
from the two arms of the interferometer are recombined in PBS2, and due to the
presence of the two QWP’s, the output beam leaves the interferometer through
the output port.

Figure 3: (a) and (b): Spectrum measured with an spectrometer for two values
of the phase introduced by the Liquid Crystal Variable Retarder
($\Phi=0$,$\pi$); (d) and (e): Signals obtained after Fourier-transform of the
corresponding spectrum; (c) Signal resulting from subtracting the spectra
shown in (a) and (b); (f) Signal obtained after Fourier-transform of the
subtraction of the spectra shown in (c).

The recombined beam traverses a Liquid Crystal Vaiable Retarder (LCVR) that
adds an extra phase to the light coming from the reference arm (see Appendix A
to see details of the relationship between the voltage and the phase
introduced by the Liquid Crystal Variable Retarder). $\text{HPW}_{4}$ oriented
at $\pi/8$ in combination with $\text{PBS}_{3}$ allows to separate the light
beams coming from the two arms of the interferometer into diagonal and anti-
diagonal orthogonal polarizations. Light transmitted by $\text{PBS}_{3}$ is
sent to a spectrometer (Ocean Optics, HR4000) utilizing a multimode fiber
(MMF), and the spectrum measured is analyzed and processed in order to
retrieve the sought-after information of the sample. Light reflected from PBS3
is sent to a photodiode (PD) with the help of a collecting lens with focal
length $\text{L}_{5}=75\leavevmode\nobreak\ \text{mm}$.

Figure 4: Measurement of the location of a moving mirror, using spectral-
domain OCT and applying post-processing corrections. (a): Intensity of the
signal resulting from the Fourier-transform of the subtraction of two spectra
with phase difference $\pi$, for some selected positions of the moving mirror.
Every signal (different colors) corresponds to a position of the mirror along
the reference arm. (b): The dots corresponds to the amplitudes of the peaks
measured and shown in (a). The solid line is a fit to this data, and is the
key element for doing the sensitivity decay correction in all experiments
shown in this paper. (c): Signal obtained after applying the sensitivity decay
correction. (d): Path length difference measured by Optical Coherence
Tomography as function of the path length difference induced by the motorized
translation stage. The dots are the experimental data, and the solid line is a
fit. The slope of the line matches the expected path length difference
generated, $m_{\text{theo}}=2$.

###  II.3 Stepped Phase shifting interferometry

For any axial scan, i.e. for a specific transverse position, two spectra with
a phase difference $\Delta\phi=\pi$, introduced by the Liquid Crystal Variable
Retarder, are measured and subsequently subtracted. The resulting signal is
Fourier transformed with the help of a standard Fast-Fourier transform (FFT)
routine (see Appendix B for further details about how to perform the Fourier-
transform of the signal measured with the spectrometer). Without subtraction,
the Fourier-transformed signal, contains a DC peak, and some auto-correlation
peaks that appear due to interference of multiple reflections in the sample.
The presence of these peaks make it difficult the characterization of the
longitudinal structure of the sample. By subtracting the two spectra with a
phase difference of $\pi$, all of these peaks are filtered out. The remaining
peaks, that are enhanced, are the cross-correlations ones, the ones which bear
relevant information about the structure of the sample Wojtkowski  _et al._
(2002).

Figure 5: (a) and (b): Spectrum measured with an spectrometer for two values
of the phase introduced by the Liquid Crystal Variable Retarder
($\Phi=0$,$\pi$); (d) and (e): corresponding signals obtained after Fourier-
transform of the spectrum; (c) Signal resulting from subtracting the spectra
shown in (a) and (b); (f) Signal obtained after Fourier-transform of the
subtraction of spectra shown in (c). Dots: experimental data. Solid lines:
best fit to the experimental data.

##  III Demonstration of Optical Coherence Tomography with a non-mechanical
steering system

###  III.1 First example: location of the position of a single-layer sample

The Fourier-domain OCT apparatus is first characterized by employing a mirror
instead of a multi-layered sample. This is equivalent to considering a single-
layer sample with a reflection coefficient of about $\sim 0.97$. For the sake
of convenience, the characterization is made by changing the position of the
mirror in the reference arm. The motorized translation stage controls the path
difference between the two arms of the interferometer.

Figures 3(a) and 3(b) show an example of the experimental spectra obtained for
the case of a single-layer sample (mirror) with $\phi=0$ and $\phi=\pi$,
respectively. Figures 3(d) and 3(e) show the signal obtained after Fourier
transform of the spectrum measured and shown in Figs. 3(a) and (b). One can
see the presence of a strong DC peak. Figure 3(c) is the result of the
subtraction of signals shown in Figs. 3(a) and (b), and Fig. 3(f) is the
corresponding Fourier transform. The DC peak has been filtered out, and the
only remaining peak allows to retrieve the position of the mirror.

Figure 4(a) shows the signal obtained after Fourier-transform of the
subtraction of the two spectra with phase difference $\pi$, for some selected
positions of the mirror. Each peak of the Fourier Transform correspond to a
position of the moving mirror. One would expect to observe the same peak
amplitude for all positions of the mirror, since the reflectivity of the
object does not change, only its location. However, for increasing path length
differences between light reflected from the two arms of the Michelson
interferometer, the amplitude of the peaks of the signal diminishes. This
decay is due to the finite pixel size of the detector in the spectrometer
Leitgeb  _et al._ (2003b); Nassif  _et al._ (2004); Mujat  _et al._ (2007). To
correct for this effect in all experiments, we will apply the sensitivity
decay correction depicted in Fig. 4(b). After this, we obtain Fig. 4(c).
Considering the design of the Michelson interferometer, varying the position
of the sample results in a twofold increase in the path length difference.
Figure 4(d) shows the positions of the peaks of the signal as function of the
sample’s position, which is controlled by a motorized translation stage. The
solid line is a linear fit. The slope of the linear fit is $m_{exp}=1.98$, and
it agrees with the theoretically expected value of $m_{theo}=2$.

###  III.2 Second example: Three-dimensional image of a multilayer sample

In order to prove the feasibility of the OCT scheme with a non-mechanical
steering system implemented here, we do OCT with three different samples (see
Fig. 6): (1) a piece of glass (refractive index $n_{\text{glass}}=1.5$); (2) a
piece of glass with a film of gold of thickness $40\leavevmode\nobreak\
\text{nm}$ that covers the middle of the surface; and (3) a piece of glass
with a fringe of gold of thickness $40\leavevmode\nobreak\ \text{nm}$ and
width $1\leavevmode\nobreak\ \text{mm}$ that covers half of the surface of the
glass. The gold film works as a mirror and the glass as a two-layer sample.

Figure 6: OCT images of the samples. The axial scan is achieved by using
Fourier-domain OCT, and the transverse scan is done employing the non-
mechanical steering system described above. The samples shown in (b) and (c)
consist of a piece of glass with a certain area covered with a thin film of
gold, working as a mirror. For each sample, the three figures below show the
signal obtained for three different transverse positions along the sample. The
Fourier transform peaks indicate the positions of various layers within the
sample.

Figures 5(a) and 5(b) show an example of the experimental spectra obtained for
a bi-layer sample (a piece of glass with no gold on the surface) with $\phi=0$
and $\phi=\pi$, respectively. Figure 5(d) and 5(e) show the signal resulting
of the Fourier transform of the corresponding spectrum. Figure 5(c) is the
result of subtracting the two spectra with phase difference of $\pi$, and Fig.
5(f) is the signal after Fourier transform. We can see that the DC term, and
the autocorrelation terms present in Figs. 5(d) and (e) have been filtered
out.

Figures 6(a), (b) and (c) shows the experimental results obtained after doing
the transverse scans in the three samples using the light beam with the bar
shape shown in Fig. 2(b). Figures 6(a), (b) and (c) corresponds to the three
samples described above. The numbers in each image designate a specific
position of the transverse scan. The letters in each image designate a
specific longitudinal position in the sample. For the transverse locations
where there is no gold on top of the piece of glass, all insets show two peaks
given that we are observing a two-layer sample. The distance between the peaks
allows us to estimate the optical thickness of the piece of glass. In the
transverse locations where there is gold, we observe a single peak that
corresponds to a single-layer sample (a mirror).

##  IV Conclusions

We have demonstrated, in a proof-of-principle experiment, spectral-domain
Optical Coherence Tomography where beam steering of the optical beam to
perform transverse scans of the sample is done without using any mechanical
element. Instead we steer the optical beam with the help of a spatial light
modulator and some optical elements that control the polarization of light.

The standard SLMs that we use in our proof-oc-concept experiments can change
the phase map displayed by arrays of pixels with speeds of around $60$-$200$
Hz. Although our proof-of-principle demonstration does not achieve the scan
speeds of state-of-the-art OCT systems, it can help usher in the development
of alternative OCT schemes with potentially faster scan speeds than current
OCT systems by avoiding the use of any kind of mechanical platforms. Recent
proposals for phase-only two-dimensional modulators show that one can achieve
high-speed modulations in excess of a few GHz, that is one order of magnitude
faster that the speeds achieved nowadays with fast mechanical devices.

In 2006, clinical OCT instruments scanned at a rate of $\sim 400$ axial scans
per second Brezinski (2006); Fujimoto and Swanson (2016). Nowadays, most
spectral-domain OCT systems can make axial scan at rates of tens of KHz
Fujimoto and Swanson (2016); Bhende  _et al._ (2017). The scheme proposed
here, in combination with the development of faster phase-only two-dimensional
modulators can contribute to continue the increase of the rate axial scans in
OCT. Moreover, the scheme demonstrated here can also be combined with the use
of Digital Micromirror Devices (DMD), that can reach pattern rates of $\sim
32$ KHz, comparable to current scan rates for spectral domain OCT.

## Acknowledgements

This work is part of the R$\&$D project CEX2019-000910-S, funded by the
Ministry of Science and innovation (MCIN/ AEI/10.13039/501100011033/). It has
also been funded by Fundació Cellex, Fundació Mir-Puig, and from Generalitat
de Catalunya through the CERCA program. We acknowledge suport from the project
20FUN02 “POLight”, that has received funding from the EMPIR programme co-
financed by the Participating States and from the European Union’s Horizon
2020 research and innovation programme. W also acknowledge financial support
from project QUISPAMOL (PID2020-112670GB-I00) funded by MCIN/AEI
/10.13039/501100011033. GJM was supported by the Secretaria d’Universitats i
Recerca del Departament d’Empresa i Coneixement de la Generalitat de Catalunya
and European Social Fund (FEDER).

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##  Appendix A Characterization of the Liquid Crystal Variable Retarder

A liquid crystal variable retarder (LCVR) is a cell filled with nematic liquid
crystal molecules with ordered orientation. A differential voltage applied to
the cell produces a tilt in the molecules’ orientation. The effect of this
tilt is the introduction of a phase in a polarization component, i.e.,
$\vec{E}_{\text{H}}+\vec{E}_{\text{V}}\Longrightarrow\vec{E}_{\text{H}}+\exp(i\varphi)\,\vec{E}_{\text{V}}$.

The characterization of the LCVR is performed by setting an elliptical
polarization to the input electric field and observing the intensity of
horizontal or vertical polarization when the induced voltage is changed. Fig.
7(a) shows the intensity of the horizontal component of the electric field as
function of the voltage applied. The intensity $I$ is

$I=\frac{I_{max}}{2}\Big{[}1-\cos\varphi\Big{]},$ (7)

where $I_{max}$ is the maximum intensity López-Téllez and Bruce (2014). As a
result, the phase introduced by LCVR can be obtained as

$\varphi=\cos^{-1}\Big{[}1-\frac{2I}{I_{max}}\Big{]}.$ (8)

It is necessary to perform the unwrapping method to guarantee the phase
continuity. Fig. 7(b) shows the phase (or retardance) as a function of the
voltage applied.

Figure 7: Characterization of the liquid crystal variable retarder. (a)
Intensity of the polarization component of the electric field as a function of
the differential voltage. Each point in the graph corresponds two the average
of 1000 samples. The voltage is varied in steps of $0.02\leavevmode\nobreak\
\text{mV}$. The maximum standard error obtained is $\sigma_{E}=9.6\times
10^{-4}$. (b) Retardance (or phase) recovered from the intensity curve.

##  Appendix B Calculation of the signal resulting from the Fourier transform
of the spectrum measured in an axial scann of Fourier-domain OCT

The spectrum $S(\lambda)$ is measured with a spectrometer. The spectrum is
rewritten as function of the wavenumber $k=2\pi/\lambda$. Considering the
Jacobian of the transformation, the k-spectrum $S(k)$ is

$S(k)=\frac{2\pi}{k^{2}}S(\lambda)$ (9)

The spectrum is re-sampled to obtain a function $S(k)$ with equally-spaced
$k$-values. The Fourier transform of the re-sampled spectrum is calculated as

${\cal F}[S(k)]=\int dk\,S(k)\,\exp\left(ikz\right)$ (10)

where $z$ is the axial position along the sample.