# Low-coherence interferometric measurement of the spatial frequency dependence of the light field backscattered by optical interfaces

M. Lequime, I. Khan, A. Bolliand, M. Zerrad, C. Amra

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M. Lequime, I. Khan, A. Bolliand, M. Zerrad, C. Amra. Low-coherence interferometric measurement of the spatial frequency dependence of the light field backscattered by optical interfaces. Applied Physics Letters, 2023, 122 (19), pp.1911003. 10.1063/5.0139822. hal-04281272

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Distributed under a Creative Commons Attribution 4.0 International License
Low-coherence interferometric measurement
of the spatial frequency dependence of the light
field backscattered by optical interfaces

Cite as: Appl. Phys. Lett. **122**, 191103 (2023); doi: 10.1063/5.0139822

Submitted: 23 December 2022 · Accepted: 22 April 2023 ·

Published Online: 9 May 2023

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M. Lequime, ID, I. Khan, ID, A. Bolliand, ID, M. Zerrad, a) ID and C. Amra ID

AFFILIATIONS

Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel, Marseille, France

Note: This paper is part of the APL Special Collection on Gravitational Wave Detectors.

a) Author to whom correspondence should be addressed: myriam.zerrad@fresnel.fr

ABSTRACT

In this paper, we show how the combined use of low-coherence interferometry, balanced detection, and data processing comparable to that
used in Fourier transform spectrometry allows us to characterize with ultimate resolutions (sub-parts per million in level, 0.2 nm in wave-
length, and 25 mdeg in angle) the retro-reflection and retro-scattering response of both sides of a 2 mm thick silica wafer.

© 2023 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0139822

Scattered light is one of the most important disturbance factors that can limit the ultimate sensitivity of giant laser interferometers developed for gravitational wave detection. ¹,² This is especially true when this scattered light is superimposed on the main beam with a time-varying phase term, as the resulting spurious signal, if falls within the operating frequency band of the interferometer, can degrade the detection quality of a signal generated by the passage of a gravitational wave. ³

Therefore, it is very important to be able to not only estimate theo-
retically the intensity and angular distribution of this scattered light
inside the interferometer⁴⁻⁶ but also to measure experimentally the frac-
tion of this scattered light that couples coherently to the main beam, first
at the level of elementary components,⁷,⁸ and then within the instru-
ment itself.⁹ In this paper, we describe an interferometric method that
allows the recording of such a measurement not only from elementary
components but also from more complex subsystems.

This paper is organized as follows: After describing the experi-
mental setup used for the acquisition of the data related to this mea-
surement, we detail the digital processing implemented to improve the
signal-to-noise ratio of these raw data and to determine the spectral
dependence of the amplitude of the field retro-propagated by the sam-
ple. We then present the result of these measurements when the sam-
ple is a silica wafer with, on one side, a V-shaped antireflection coating
centered at 1055 nm. In conclusion, we make a critical analysis of these
first results and consider possible ways to improve this new measure-
ment method and to extend its use to more complex systems.

A schematic of our experimental setup, named BARRITON (for
BAck-scattering Retro-Reflection by InterferomeTry with IOw
cohereNce) is given in Fig. 1(a).

The linearly polarized divergent light beam provided by a fibered
superluminescent diode (SLD) centered at 1050 nm (light power about
70 mW, spectral bandwidth about 80 nm for a maximum driving cur-
rent of 1000 mA) is collimated by an off-axis parabolic mirror of 7 mm
focal length. The orientation of the polarization direction of this weakly
divergent Gaussian light beam (w<sub>f</sub> = 0.67 mm and θ<sub>f</sub> ~ 0.03°) is
finely tuned with respect to the vertical transmission axis of a LP<sub>0</sub> nano-
particle polarizer using a rotating half-wave plate (HWP). This polarized
beam is divided into two perpendicular sub-beams by the non-
polarizing cube splitter BS<sub>1</sub>: the reflected sub-beam is directed toward
the sample and the light flux retro-reflected or back-scattered by the
sample is transmitted by the same cube splitter BS<sub>1</sub> (SIG arm). The
transmitted sub-beam (REF arm) is simultaneously retro-reflected and
laterally shifted by a hollow retro-reflector (HRR) mounted on a motor-
ized translation stage whose axis of movement is coincident with the
direction of the REF beam (delay line). This beam then passes through a
right angle prism (RAP) and is combined with the SIG beam through
the non-polarizing cube splitter BS<sub>2</sub>. The two output channels of this
cube splitter are finally detected by the two photodiodes (PD1 and PD2)
of a balanced receiver (Nirvana).<sup>10</sup>

The sample is placed in a piezoelectric gimbal mount with two
axes that each rotate its front side around a gimbal axis, holding the
<figure><img src="image_5.png" /><figcaption>FIG. 1. (a) Schematic diagram of the BARRITON setup. (b) Structure of the [ARC-UNC] test sample.</figcaption></figure>

center of the sample in a fixed position. Each axis is capable of scanning an angular range of 1.5° with a resolution of 3 µdeg. This system allows the front side of the sample to be positioned perpendicular to the incident beam. The variation of the angle of incidence (AOI) of the light beam on the sample is ensured by a motorized rotation stage with vertical axis (Newport RGV100BL-S), characterized by a minimum incremental motion of 0.1 mdeg. A manual horizontal translation stage, located between the gimbal mount and the rotation stage, allows to adjust the axial position of the front face of the sample so that it contains the rotation axis of this motorized stage.

The balanced receiver delivers a voltage V proportional to the difference between the electrical currents I₁ and I₂ of the two photodiodes, and which corresponds to the following expression:¹¹,¹²

$$V = G\mathcal{T}\Re\left\{\int_{0}^{\infty} S(f)\mathcal{P}(f)\rho(f)e^{-ik\Delta L} df\right\}, \quad (1)$$

where f is the frequency of the optical field, S(f) is the spectral dependence of the photodiode responsivity, P(f) is the power spectral density of the light source, ρ(f) is the coherent coefficient of retro-reflection/backscattering of the sample,¹¹,¹³ k is the wave vector, ΔL is the optical path difference between SIG and REF channels, G is the transimpedance gain of the balanced receiver, and T is a global transmission factor characteristic of the setup.¹¹

This voltage V is digitized over 16 bits with a sampling rate F_s of 1 MHz, while the hollow retro-reflector (HRR) is translated at a constant speed v along the z axis [ΔL = 2vt]. Figure 2(a) shows the time dependence of this digitized voltage when the front face of the sample corresponds to the coated face of a 2 mm thick silica wafer (0° AOI, v = 0.5 mm/s). Figure 2(b) [respectively, 2(c)] shows a magnified view of the echoes associated with each of the faces of this silica wafer (in red, the coated front face; in blue, the uncoated back face), in retro-reflection for an AOI of 0° (respectively, in backscattering, for an AOI of 0.3°). The digitizing range (10, 1, and 0.1 V) is automatically adapted to the amplitude of the strongest echo.

The two graphs of Figs. 2(b) and 2(c) show that the signal-to-noise ratio of the interferogram decreases very rapidly with the tilt angle of the sample, because the level of the backscattered flux is much

lower than that of the retroreflected flux. Moreover, the profile of each of these two echoes does not give any information about the spectral dependence of the light back-scattered by each interface of the sample. To overcome these two limitations, we use a data processing transposed from the one implemented in Fourier transform spectrometry.¹²

We first isolate in the interferogram recorded at θ₀² two temporal windows centered on each of the two echoes and with a full width ΔT. Then, we perform a discrete Fourier transform (DFT) of the corresponding dataset,

$$S_m(F_l, \theta_0^i) = \sum_{k=-N/2}^{k=N/2-1} V_m(t_k, \theta_0^i) e^{-2i\pi F_l t_k} dt, \quad (2)$$

where V_m(t_k, θ_0^i) is the time dependence of the voltage V(t) in the time window corresponding to the echo m (m = 1 or 2), sampled at t = t_k, defined by

$$t_k = t_m + k \cdot dt; \quad dt = \frac{1}{F_s} = \frac{\Delta T}{N-1}, \quad (3)$$

while F_l is the frequency dataset, conjugate of t_k, defined by

$$F_l = l \cdot dF; \quad dF = \frac{1}{(N-1)dt} = \frac{1}{\Delta T}. \quad (4)$$

Notice that, by comparing Eqs. (1) and (2), we also have

$$k\Delta L(t_k) = 2\pi \frac{2\nu}{c} f_l t_k = 2\pi F_l t_k \Rightarrow F_l = \frac{2\nu}{c} f_l, \quad (5)$$

where f_l is the optical frequency associated with F_l.

If the width ΔT of the time window is less than or equal to the time interval ΔT_c between the two echoes, it can be shown¹² that the DFT associated with the echo m is proportional to the complex conjugate of the corresponding retro-propagation coefficient ρ_m, i.e.,

$$S_m(F_l, \theta_0^i) = G\mathcal{T}\Re\frac{c}{4\nu}S(f_l)\mathcal{P}(f_l)\rho_m^*(f_l, \theta_0^i), \quad m = 1, 2. \quad (6)$$

Figure 2(d) shows the frequency dependence of |S_m(F_l)|² for the two echoes recorded at 0.3° when the DFT is calculated with two different processing modalities: HiRes (ΔT = ΔT_c) and LoRes (ΔT = ΔT_c/10).
<figure><img src="image_6.png" /><figcaption>FIG. 2. Examples of interferograms recorded with BARRITON when the front face of the sample corresponds to the antireflective coated face of a 2 mm thick silica wafer (translation speed v = 0.5 mm/s). (a) Full interferogram recorded at AOI 0° (digitizing range of 10 V). (b) Magnified view of the two echoes recorded at AOI 0° (coated face in red, uncoated face in blue, digitizing range of 10 V). (c) Magnified view of the two echoes recorded at AOI 0.3° (coated face in red, uncoated face in blue, digitizing range of 0.1 V). (d) Square modulus of the DFTs of the interferogram recorded at AOI 0.3° (high resolution and low resolution modes).</figcaption></figure>

Note the significant improvement of the signal-to-noise ratio resulting from the calculation of these DFTs (the white noise of quantum origin that affects the interferograms is dispersed over the whole spectrum, and its importance is, therefore, automatically reduced). It is also important to underline the additional gain brought by the use of LoRes mode, but equally its negative impact on the spectral resolution. This one is indeed inversely proportional to ΔT and equal to 0.2 nm in HiRes mode and 2 nm in LoRes mode.

For example, let us suppose that the front face of the sample is the uncoated one. Under these conditions, we have¹²

* at zero AOI ($\theta_0^i = 0$),

$$R_2(f_i) = \frac{R_1(f_i) |S_2(f_i, 0)|^2}{[1 - R_1(f_i)]^2 |S_1(f_i, 0)|^2}, \quad (7)$$

where $R_1(f_i)$ is the spectral dependence of the reflection coefficient of the front face, here uncoated, and $R_2(f_i)$ is the spectral dependence of the reflection coefficient of the second (coated) face.

* For non-zero AOI,

$$|\rho_m(f_i, \theta_0^i)|^2 = R_m(f_i) \frac{|S_m(f_i, \theta_0^i)|^2}{|S_m(f_i, 0)|^2}, \quad m = 1, 2, \quad (8)$$

where we have assumed that the angle dependence of the front face transmittance is negligible.

Similar relationships can be established for the case where the front side of the sample is the coated side.

The sample we used for this experimental demonstration is a high quality component purchased from LIGHT MACHINERY,¹⁴ consisting of a 2 mm thick, 25 mm diameter, 7980A grade fused silica window with a wedge angle of 0.2 mdeg, which is one side coated with a V-shaped antireflective coating centered at 1055 nm.

The measurements are performed with the following operating conditions: SLD driving current I of 190 mA, translation speed v of 0.5 mm/s, sampling frequency $F_s$ of 1 MHz, angle pitch of 10 mdeg, and angular range from 0° to 10°. Consider for example the [UNC-ARC] configuration, in which the front face of the sample is not coated. The coefficient $R_1$ has then the following expression:

$$R_1(f_i) = \left[ \frac{1 - n_s(f_i)}{1 + n_s(f_i)} \right]^2, \quad (9)$$

where $n_s(f_i)$ is the refractive index of the silica substrate. The spectral dependence of this refractive index is perfectly known, which allows the value of $R_1(f_i)$ to be calculated at any frequency $f_i$. The use of Eq. (7) allows us to deduce $R_2(f_i)$ using the results of our measurements at zero incidence [see Fig. 3(a)].
<figure><img src="image_7.png" /><figcaption>FIG. 3. Results of the experimental measurement of the retroreflection and backscattering properties of the silica wafer sample (front face uncoated, rear face coated). (a) Spectral dependence of the retroreflection coefficient of each of the two faces. (b) Evolution of the ratios $|S_m(\nu)|^2/|S_m(0)|^2$ ($m=1, 2$) as a function of the spatial frequency $\nu = 2\nu_i$ (front face, magenta dots, and magenta solid line; rear face, cyan dots, and cyan solid line).</figcaption></figure>

In the case of the quantities $|\rho_m(f_i, \theta_0^i)|^2$, which describe the spectral and angular dependence of the light flux backscattered by the $m$th interface, the presentation of our experimental results requires beforehand the statement of some theoretical considerations. The elementary light fields scattered by each interface $m$ toward the incident medium may be described by a general expression of the type¹³

$$ \vec{\mathcal{A}}_m^-(\vec{\nu}_d) = \sum_{j=0}^{p_m} C_{m,j}^- [\hat{h}_m \star \hat{s}_e]_{\vec{\nu}_d, \vec{\nu}_i} \vec{\mathcal{A}}_i, \quad (10) $$

where $\vec{\mathcal{A}}_i$ and $\vec{\mathcal{A}}_m^-$ are the incident field and the elementary field scattered by the side $m$, respectively, $p_m$ is the number of layers deposited on the interface $m$, $C_{m,j}^-$ are coefficients related to the optical structure of the component, $\hat{h}_m$ is the Fourier transform of the $m$ interface height profile (for simplicity, we assumed that the profiles of the layers are identical to the one of the interface on which they are deposited), $\hat{s}_e$ is the Fourier transform of the spatial distribution of the incident light field at the surface of the component, and $\vec{\nu}_d$ (respectively, $\vec{\nu}_i$) is the spatial frequency of the scattered (respectively, incident) field.

For the backscattered field, we have

$$ \vec{\nu}_d = -\vec{\nu}_i \text{ with } \vec{\nu}_i = (\nu_i, 0) \quad \text{and} \quad \nu_i = \frac{\sin \theta_0^i}{\lambda}. \quad (11) $$

For a weakly divergent Gaussian beam, $\hat{s}_e$ can be assimilated to a Dirac function $\delta(\vec{\nu}_d - \vec{\nu}_i)$, and the amplitude of the backscattered fields satisfies the following equation:

$$ \vec{\mathcal{A}}_m^-(\vec{\nu}_i) \propto \hat{h}_m(\vec{\nu}_d - \vec{\nu}_i) = \hat{h}_m(-2\vec{\nu}_i) = \hat{h}_m^*(2\vec{\nu}_i), \quad (12) $$

where the last equality is due to the Hermitian symmetry of $\hat{h}_m$ ($h_m$ is real). This means that the results of our measurements should be plotted not as a function of the angle of incidence $\theta_0^i$ or wavelength $\lambda$, but as a function of twice the spatial frequency $\nu_i$ associated with the incident wave. This is particularly important because it shows that the number of independent data we have for the measurement of the spatial frequency dependence of the quantities $\rho_m$ is in fact equal to the product of the number of angular values (here, 1000) by the number of spectral values (in our case, about 300 in HiRes mode and 30 in LoRes mode), which makes it possible to improve both the signal-to-noise ratio of the measurement and its resolution in spatial frequency.

Figure 3(b) shows the results obtained when the step $\delta\nu$ is equal to $10^{-4} \mu\text{m}^{-1}$. On these curves, we see the appearance of, beyond the specular response ($\nu \ge 2.5 \times 10^{-3} \mu\text{m}^{-1}$), a speckle phenomenon whose grain size is governed by both the divergence of the incident beam and the spectral resolution of the measurement.

In the spatial frequency domain corresponding to retroreflection ($0 \le \nu \le 2.5 \times 10^{-3} \mu\text{m}^{-1}$), the analytical calculation can be completed, and shows that the spatial frequency dependence of the ratio $|\rho_m(\nu, f_i)|^2/R_m(f_i)$ is described by¹⁵

$$ \frac{|\rho_m(\nu, f_i)|^2}{R_m(f_i)} = e^{-\pi^2 w_m^2 \nu^2}, \quad \text{where} \quad w_m = \frac{w_s}{\sqrt{1 + \cos^2 2\theta_0^i}}. \quad (13) $$

The two green curves in Fig. 3(b) correspond to the best fits of the experimental data based on the relation (13) and allow to determine the values of $w_m$, i.e., $w_1 = 496$ and $w_2 = 469$ $\mu\text{m}$. We can then deduce the value of $w_s$ which corresponds to the half-size of the beam at the sample. The value obtained, $w_s = 682 \pm 50$ $\mu\text{m}$, is in very good agreement with the one ($w_s = 706$ $\mu\text{m}$) resulting from the application of the formulas of propagation of the Gaussian beams between the exit of the reflective collimator and the sample, that is,

$$ w_s = \sqrt{1 + \left(\frac{d_s}{z_R}\right)^2} \quad \text{with} \quad z_R = \frac{\pi w_f^2}{\lambda} \quad \text{and} \quad w_f = \frac{f \lambda}{\pi w_0}, \quad (14) $$

where $w_0 = 3.5 \ \mu\text{m}$ @ $\lambda = 1070$ nm, $f=7$, and $d_s = 375$ mm.

The knowledge of this quantity will allow us to transform the curves represented on the graph of Fig. 3(b) into BRDF $\cos \theta$. Indeed, we have⁸,¹¹

$$ \text{BRDF}_m \cos \theta = \text{ARS}_m(\lambda, \nu) = \frac{\pi w_s^2}{\lambda^2} R_m(\lambda) \frac{|S_m(\nu)|^2}{|S_m(0)|^2}. \quad (15) $$

For the uncoated face ($m=1$), all measurements are made with a satisfactory signal-to-noise ratio, demonstrating that our setup is capable of measuring ARS values as low as $10^{-4} \text{ sr}^{-1}$ over a spatial frequency range from 0 to $3.5 \times 10^{-1} \mu\text{m}^{-1}$. For the coated face ($m=2$), the highest spatial frequency is reduced to $5 \times 10^{-2} \mu\text{m}^{-1}$, due to the presence of the antireflective coating.
A theoretical analysis of the different noise sources¹² shows that it is possible to significantly improve the detection sensitivity of this setup, by optimizing different parameters, such as the translation speed $v$, the bandwidth $B$ of the balanced receiver, or the optical power corresponding to the saturation of the photodiodes. A gain of 500 seems quite accessible, which would allow the detection of ARS as low as $2 \times 10^{-7} \text{sr}^{-1}$.

With such detection sensitivity, it would also become possible to accurately measure the spectral dependence of the phase of the field backscattered from the first interface, using the following relation:

$$ \arg[\rho_1(f_l, \theta_0^i)] = -\arg[S_1(f_l, \theta_0^i)], \quad (16) $$

which paves the way for a reconstruction of the effective height profile of the optical interface under observation.

Finally, in the graph of Fig. 3(b), it can be seen that the scattered intensities from the two sides of the sample show a strong degree of correlation for spatial frequencies between $2 \times 10^{-3}$ and $2 \times 10^{-2} \mu\text{m}^{-1}$. This cannot be caused by detection crosstalk because it is canceled out by the use of low-coherence interferometry, as shown by the independent measurement of the reflection coefficients of the two sides of the optical window. This phenomenon, therefore, has a purely physical cause, and a possible explanation lies in a common source of deformation of the wavefronts backscattered by the two surfaces (for example, a flatness error of the front surface). Further theoretical description and numerical modeling are in progress to confirm this hypothesis.

This work is a part of the StrayLight Working Group for the Laser Instrument Group of the LISA Consortium. The authors would like to thank the AMIDex Talents Management Program, Aix Marseille University, and the French National Space Center (CNES) for their financial support, and the Virgo Collaboration for fruitful discussions.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## Author Contributions

**Michel Lequime:** Conceptualization (lead); Formal analysis (lead); Investigation (equal); Methodology (lead); Validation (lead); Writing – original draft (lead). **Imran Khan:** Investigation (equal); Methodology (equal); Resources (equal); Validation (equal); Writing – review & editing (equal). **Adrien Bolliand:** Investigation (equal); Resources (equal); Software (equal). **Myriam Zerrad:** Funding acquisition (lead); Methodology (equal); Project administration (lead); Supervision (lead); Validation (equal); Writing – review & editing (equal). **Claude Amra:** Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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