Issue 
J. Eur. Opt. SocietyRapid Publ.
Volume 20, Number 1, 2024



Article Number  13  
Number of page(s)  10  
DOI  https://doi.org/10.1051/jeos/2024010  
Published online  18 April 2024 
Research Article
Effects of refractive index mismatch between sample and immersion medium in linefield confocal optical coherence tomography
Université ParisSaclay, Institut d’Optique Graduate School, CNRS, Laboratoire Charles Fabry, 2 av. Augustin Fresnel, 91127 Palaiseau Cedex, France
^{*} Corresponding author: arnaud.dubois@institutoptique.fr
Received:
6
February
2024
Accepted:
14
March
2024
Linefield confocal optical coherence tomography (LCOCT) is an optical technique based on lowcoherence interference microscopy with line illumination, designed for tomographic imaging of semitransparent samples with micrometerscale spatial resolution. A theoretical model of the signal acquired in LCOCT is presented. The model shows that a refractive index mismatch between the sample and the immersion medium causes a dissociation of the coherence plane and the focal plane, leading to a decrease in the signal amplitude and a degradation of the image’s lateral resolution. Measurements are performed to validate and illustrate the theoretical predictions. A mathematical condition linking various experimental parameters is established to ensure that the degradation of image quality is negligible. This condition is tested experimentally by imaging a phantom. It is verified theoretically in the case of skin imaging, using experimental parameters corresponding to those of the commercially available LCOCT device.
Key words: Optical coherence tomography / Interference microscopy / Biomedical imaging
© The Author(s), published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Linefield confocal optical coherence tomography (LCOCT) is a recent imaging technique based on lowcoherence interference microscopy [1–4]. LCOCT uses line illumination with broadband spatially coherent light and detection with a line camera to acquire crosssectional images of semitransparent samples, including skin tissues [5–13]. The focus is dynamically adjusted during the scan of the sample depth, allowing the use of a microscope objective to image with high lateral resolution. By using a supercontinuum laser as a broadband light source, the axial resolution can be similar to the lateral resolution, i.e. ~1 μm at a central wavelength of ~750 nm, measured at the surface of the sample [1, 2]. The usual experimental arrangement of LCOCT is based on a twobeam interference microscope in the Linnik configuration (see Fig. 1). The sample to image is placed in one arm of the interferometer in contact with a glass window under a microscope objective. A reference surface of low reflectivity is placed in the other arm of the interferometer under an identical microscope objective. The Linnik interferometer is mounted on a piezoelectricactuated (PZT) translation stage for scanning the sample depth (see Fig. 1). Immersion microscope objectives are used with an immersion medium whose refractive index is as close as possible to the refractive index of the sample. The equality of the refractive indices ensures the preservation of the symmetry of the interferometer during the depth scan, using a single motorized displacement. This avoids defect of focus [14–18] and dispersion mismatch between the interferometer arms [19–24].
Fig. 1 Linnik interferometer implemented in LCOCT. The components in the red dashed frames are mounted on a piezoelectric (PZT) translation stage for scanning the sample depth (motion indicated by the double red arrow). 
In practice, however, it is generally impossible for the refractive indices of the immersion medium and the sample to be strictly identical regardless of the depth probed, at least because of inhomogeneities in the sample [25–27]. In skin tissues, for example, the refractive index varies with depth. The superficial layer of the skin (stratum corneum) has a refractive index in the near infrared around 1.54, while that of the epidermis is around 1.41 and that of the dermis around 1.38 [28–30].
The aim of this paper is to study the main consequences on the quality of the LCOCT images of a difference in refractive index between the sample and the immersion medium. Theoretical analyses are carried out to understand the physical phenomena that occur and to see the influence of experimental parameters such as spectral characteristics of the light detected, beam focusing and depth in the sample. Measurements are performed to validate and illustrate the theoretical predictions. A mathematical condition dependent on the value of experimental parameters is established to avoid significant degradation of image quality. This condition is tested using two different immersion media, both experimentally by imaging a phantom and theoretically in the case of skin imaging.
2. LCOCT signal modeling
This section presents a theoretical model of the signal acquired in LCOCT as a function of depth in the sample. This model will allow us to understand and predict effects of a difference in refractive index between the sample and the immersion medium. The dependence of the refractive indices with the optical frequency is ignored. This assumption is reasonable since the effect of chromatic dispersion up to first order in media of practical interest at depths less than a millimeter is not significant. In water, for example, the standard deviation of the refractive index is less than 0.002 (0.14%) over the wavelength range 0.6–1.0 μm [31].
2.1 Focal plane and coherence plane
In lowcoherence interferometry, interference can be observed when the optical path difference in the interferometer does not exceed the temporal coherence length of the light source, with maximum contrast when the two optical paths are identical. In LCOCT, the optical length of the reference arm of the interferometer determines the position of a plane in the sample arm, called the coherence plane, which corresponds to the equality of the optical paths. Interference is detected when a reflective structure in the sample is located in the coherence plane with an uncertainty of depth equal to the coherence length.
On the other hand, a reflective structure in the sample imaged by LCOCT is in focus when it is located in the focal plane of the microscope objective placed in the sample arm of the interferometer, with an uncertainty of depth related to the depth of field.
In practice, the LCOCT device is adjusted so that the coherence plane and the focal plane coincide at the surface of the sample pressed against a glass window (see Fig. 2a). In order to image in depth in the sample, the distance between the sample and the interferometer is reduced. After relative displacement of the interferometer a distance z closer to the sample (see Fig. 2a), the coherence plane is then at a distance z_{coh} below the sample surface and the focal plane at a distance z_{foc} below the sample surface.
Fig. 2 Mismatch of the coherence plane and focal plane in LCOCT. The plain red lines represent the beam in the plane of the figure, whereas the dotted red lines represent the beam in the direction orthogonal to the plane of the figure. (a) The interferometer is adjusted so that the coherence plane and the focal plane coincide at the surface of the sample. (b) When the interferometer is lowered by a distance z, the coherence plane is at a distance z_{coh} below the sample surface and the focal planes at a distance z_{foc} (here n > n_{im}). 
The position of the coherence plane can be found by considering that the difference between the optical path lengths in the sample and reference arms of the interferometer is zero for the normally incident light beam:$${z}_{\mathrm{coh}}=\frac{{n}_{\mathrm{im}}}{n}z,$$(1)where the quantities n and n_{im} are the refractive indices of the sample and immersion medium, respectively.
The position of the focal plane can be calculated using geometrical optics [18, 32, 33] under the paraxial approximation:$${z}_{\mathrm{foc}}=\frac{n}{{n}_{\mathrm{im}}}z.$$(2)
The paraxial approximation is applicable to microscope objectives with small numerical apertures (NA). Due to the refractive index mismatch between the sample and the immersion medium, the rays focused by the objective converge at different depths depending on their inclination. This creates uncertainty about the position of the focal plane, known as spherical aberration, which increases with the NA of the objective. LCOCT typically employs a NA of 0.5. The formula provided in reference [27] and generalized in reference [34] can be used to calculate the distance between the marginal focus (corresponding to the most inclined rays) and the paraxial focus. For a NA of 0.5, refractive index values of n_{im} = 1.33 and n = 1.40, and a depth of 500 μm in the sample (typical maximum depth probed in LCOCT), the distance between the two foci is less than 5 μm. This value typically corresponds to the depth of field, rendering spherical aberration insignificant and the paraxial calculation (Eq. (2)) satisfactory. Immersion microscope objectives with large apertures are equipped with a collar to correct for spherical aberration caused by the cover glass thickness variations and refractive index mismatch. In such cases, the focal plane’s position corresponds to the paraxial focus’s position and can thus be determined through paraxial calculation. However, the correction collar is designed only for small thicknesses and refractive index mismatches.
The comparison of equations (1) and (2) shows that the coherence plane and the focal plane do not coincide in the sample (z_{coh} ≠ z_{foc}) if the refractive indices of the sample and the immersion medium are not equal (n ≠ n_{im}) [15, 16]. In a homogeneous sample, the distance between the two planes (z_{foc} − z_{coh}) increases linearly with z. The consequences of the mismatch of the coherence plane and the focal plane on the LCOCT images will be studied in the following of this paper.
2.2 Theoretical signal
Let us consider a reflective structure of the sample located at depth z_{s} below the sample surface. The optical intensity (or irradiance) on the detector, as a function of the depth scan (variable z, as shown in Fig. 2), is proportional to$$I\left(z\right)=\underset{0}{\overset{+\infty}{\int}}{\left{A}_{\mathrm{Ref}}\left(\omega ,z\right)+{A}_{\mathrm{S}}\left(\omega ,z\right)\right}^{2}\mathrm{d}\omega ={\int}_{0}^{\infty}{\left{A}_{\mathrm{Ref}}\left(\omega ,z\right)\right}^{2}\mathrm{d}\omega +{\int}_{0}^{\infty}{\left{A}_{\mathrm{S}}\left(\omega ,z\right)\right}^{2}\mathrm{d}\omega +2{\int}_{0}^{\infty}\left{A}_{\mathrm{Ref}}\left(\omega ,z\right)\right\times \left{A}_{\mathrm{S}}\left(\omega ,z\right)\right\mathrm{cos}\left[\varphi \left(\omega ,z\right)\right]\mathrm{d}\omega .$$(3)
A_{Ref}(ω, z) and A_{S}(ω, z) represent the complex amplitudes of light at angular frequency ω, returning from the reference and sample arms of the interferometer, respectively. ϕ(ω, z) is the phase difference between waves A_{Ref}(ω, z) and A_{S}(ω, z). It can be expressed as$$\varphi \left(\omega ,z\right)=2n\left({z}_{\mathrm{coh}}{z}_{\mathrm{s}}\right)\omega /c+{\varphi}_{0}=2\left({n}_{\mathrm{im}}zn{z}_{\mathrm{s}}\right)\omega /c+{\varphi}_{0}$$(4)with ϕ_{0} a constant phase difference.
R_{ref}(ω) denoting the reflectivity of the reference surface in the reference arm of the interferometer, and S(ω) the spectral intensity of the light source, we can write, ignoring a multiplication factor$${\left{A}_{\mathrm{Ref}}\left(\omega ,z\right)\right}^{2}={R}_{\mathrm{ref}}\left(\omega \right)\mathrm{S}\left(\omega \right).$$(5)
Assuming that the reflectivity of the reference surface has a low dependence with ω over the spectral domain of the detector sensitivity, we have$$\underset{0}{\overset{+\infty}{\int}}{\left{A}_{\mathrm{Ref}}\left(\omega ,z\right)\right}^{2}\mathrm{d}\omega ={R}_{\mathrm{ref}}{I}_{0},$$(6)where$${I}_{0}=\underset{0}{\overset{+\infty}{\int}}S\left(\omega \right)\mathrm{d}\omega ,$$(7)is the optical intensity delivered by the light source.
Similarly, we write:$${\left{A}_{\mathrm{S}}\left(\omega ,z\right)\right}^{2}={{\gamma}_{\mathrm{foc}}}^{2}\left(\omega ,z\right){R}_{\mathrm{S}}\left(\omega \right)S\left(\omega \right),$$(8)where R_{ S }(ω) is the reflectivity of the reflective structure considered as a planar reflector located at depth z_{S} in the sample. Function γ_{foc}(ω, z), called focus function, describes the normalized distribution of light amplitude incident into the sample as a function of depth. According to the theory of Gaussian beams, the axial distribution of light amplitude of a radially symmetrical Gaussian beam is inversely proportional to the beam width [35]. The beam radius varies with axial coordinate Z as$$w\left(Z\right)={{w}_{0}\left[1+{\left(\frac{Z}{{Z}_{R}}\right)}^{2}\right]}^{1/2}.$$(9)
The beam waist, defined as the minimal value of the beam radius, is related to the NA of the Gaussian beam and the optical wavelength in vacuum λ_{0} as [35]$${w}_{0}=\frac{{\lambda}_{0}}{\pi \mathrm{NA}}.$$(10)
The quantity Z_{R}, called Rayleigh length, can be written as [35]$${Z}_{R}=\frac{n{\lambda}_{0}}{\pi {\mathrm{NA}}^{2}},$$(11) n being the refractive index of the medium in which the beam propagates.
In LCOCT, the laser beam is focused in only one transverse direction to illuminate the sample with a line of light (see Fig. 2). Function γ_{foc}(ω, z) is therefore inversely proportional to the square root of the beam width in the direction perpendicular to the illumination line. The maximum of γ_{foc}(ω, z) is reached when the beam is focused in the plane of the reflective structure, i.e. when z_{foc} = z_{S}, which corresponds to z = (n_{im}/n) z_{S}. The focus function can therefore be expressed as$${\gamma}_{\mathrm{foc}}\left(z\right)=\sqrt{\frac{{w}_{0}}{w\left(Z={z}_{\mathrm{foc}}{z}_{\mathrm{S}}\right)}}.$$(12)
Using the expression of the beam waist radius given in equation (9), an expression of the focus function γ_{foc} at central angular frequency ω_{0} = 2πc/λ_{0} can be written as$${\gamma}_{\mathrm{foc}}\left(z\right){=\left\{1+64{\left[\frac{\left(n/{n}_{\mathrm{im}}\right)z{z}_{\mathrm{S}}}{\u2206{z}_{\mathrm{foc}}}\right]}^{2}\right\}}^{1/4},$$(13)with ∆z_{foc} the fullwidthathalfmaximum (FWHM) of γ_{foc}(z) being ∆z_{foc} ≈ 8Z_{R}.
Assuming that the NA of the microscope objectives matches the NA of the laser beam in the plane perpendicular to the illumination line, the width of the focus function can be written according to equation (11) as$$\u2206{z}_{\mathrm{foc}}\approx \frac{8}{\pi}\frac{n{\lambda}_{0}}{{\mathrm{NA}}^{2}}.$$(14)
Assuming that the reflectivity of the structure has a low dependence with ω over the spectral domain of the camera sensitivity, the optical intensity returning from the sample arm can be written as$$\underset{\infty}{\overset{+\infty}{\int}}{\left{A}_{\mathrm{S}}\left(\omega ,z\right)\right}^{2}\mathrm{d}\omega ={{\gamma}_{\mathrm{foc}}}^{2}\left(z\right){R}_{\mathrm{S}}{I}_{0}.$$(15)
The detected LCOCT signal, as a function of depth in the sample (Eq. (3)) can now be written as$$I\left(z\right)={I}_{0}\left\{{R}_{\mathrm{ref}}+{R}_{\mathrm{S}}{{\gamma}_{\mathrm{foc}}}^{2}\left(z\right)\right\}+2\sqrt{{R}_{\mathrm{ref}}{R}_{\mathrm{S}}}{\gamma}_{\mathrm{foc}}\left(z\right)\underset{0}{\overset{+\mathrm{\infty}}{\int}}S\left(\omega \right)\mathrm{cos}\varphi \left(\omega ,z\right)\mathrm{d}\omega .$$(16)
Using complex analysis, and substituting ϕ(ω, z) by its expression given in equation (4), the integral in the previous equation can be written as$${\int}_{0}^{\infty}S\left(\omega \right)\mathrm{cos}\left[\varphi \left(\omega ,z\right)\right]\mathrm{d}\omega =\mathfrak{R}e\left\{{\int}_{\infty}^{\infty}S\left(\omega \right)\mathrm{exp}\left[\mathrm{i\varphi}\left(\omega ,z\right)\right]\mathrm{d}\omega \right\}=\mathfrak{R}e\left\{\mathrm{exp}\left[i{\varphi}_{0}\right]\widehat{S}\left(\xi \right)\right\},$$(17)where $\mathfrak{R}e\left\{\right\}$ denotes the real part of a complex number and $\widehat{S}\left(\xi \right)$ the Fourier transform of S(ω) with ξ = (n_{im} z − nz_{s})/πc.
We suppose the spectral intensity of the light source to be described by a Gaussianshaped function of width (FWHM) ∆ω, centered at ω = ω_{0}:$$S\left(\omega \right)={S}_{0}\mathrm{exp}\left[4\mathrm{ln}2{\left(\frac{\omega {\omega}_{0}}{\u2206\omega}\right)}^{2}\right].$$(18)
The Fourier transform of S(ω) is$$\widehat{S}\left(\xi \right)={I}_{0}\mathrm{exp}\left[{\left(\frac{\pi \u2206\omega}{2\sqrt{\mathrm{ln}2}}\xi \right)}^{2}\right]\mathrm{exp}\left[i{2\pi \omega}_{0}\xi \right],$$(19)the intensity of the light source being here$${I}_{0}=\underset{\infty}{\overset{+\infty}{\int}}S\left(\omega \right)\mathrm{d}\omega ={S}_{0}\frac{\u2206\omega}{2}\sqrt{\frac{\pi}{\mathrm{ln}2}}.$$(20)
The integral in equation (16) can finally be calculated as$$\underset{0}{\overset{+\infty}{\int}}S\left(\omega \right)\mathrm{cos}\varphi \mathrm{d}\omega ={I}_{0}{\gamma}_{\mathrm{coh}}\left(z\right)\mathrm{cos}\left[2\left({n}_{\mathrm{im}}zn{z}_{\mathrm{s}}\right){\omega}_{0}/c+{\varphi}_{0}\right],$$(21)with$${\gamma}_{\mathrm{coh}}\left(z\right)=\mathrm{exp}\left\{4\mathrm{ln}2{\left[\frac{\left({n}_{\mathrm{im}}/n\right)z{z}_{\mathrm{s}}}{\u2206{z}_{\mathrm{coh}}}\right]}^{2}\right\},$$(22)and$$\u2206{z}_{\mathrm{coh}}=\frac{4c\mathrm{ln}2}{n\Delta \omega}.$$(23)
The Gaussian function γ_{coh}(z) represents the temporal coherence function. It is maximum when z = (n/n_{im}) z_{S}, i.e. when z_{coh} = z_{S} (the reflective structure is then located in the coherence plane). ∆z_{coh} is the FWHM of γ_{coh}(z), which can be approximated by$$\u2206{z}_{\mathrm{coh}}\approx \frac{2\mathrm{ln}2}{\mathrm{n\pi}}\left(\frac{{\lambda}_{0}^{2}}{\u2206\lambda}\right),$$(24)where ∆λ represents the FWHM of the source spectral intensity expressed as a function of wavelength, and λ_{0} the central optical wavelength.
Finally, the LCOCT axial signal given by a planar reflector of reflectivity R_{S} located at depth z_{S} can be written as$$I\left(z\right)={I}_{0}\left\{{R}_{\mathrm{ref}}+{R}_{\mathrm{S}}{{\gamma}_{\mathrm{foc}}}^{2}\left(z\right)+2\gamma \left(z\right)\sqrt{{R}_{\mathrm{ref}}{R}_{\mathrm{S}}}\mathrm{cos}\left[4\pi \left({n}_{\mathrm{im}}z{\mathrm{nz}}_{\mathrm{s}}\right)/{\lambda}_{0}+{\varphi}_{0}\right]\right\},$$(25)with$$\gamma \left(z\right)={\gamma}_{\mathrm{foc}}\left(z\right){\times \gamma}_{\mathrm{coh}}\left(z\right).$$(26)
The signal expressed in equation (25) is the sum of three terms. The first term (R_{ref} I_{0}) is the intensity of light reflected by the reference surface (constant term). The second term (R_{S} γ_{foc} ^{2}(z)I_{0}) corresponds to the intensity of light reflected by the reflective structure in the sample. The third term is identified as the interferometric signal, which consists of a sinusoidal modulation of period λ_{0}/(2n_{im}) comprised in an envelop $2{I}_{0}\gamma \left(z\right)\sqrt{{R}_{\mathrm{ref}}{R}_{\mathrm{S}}}$. Function γ(z) is the product of the focus function γ_{foc}(z) and the coherence function γ_{coh}(z), given by equations (13) and (22), respectively. Plots of equation (25) are shown in Figure 3 for a reflective structure located at different depths z_{S} when n ≠ n_{im}. The signal consists of a peak and interference fringes, which correspond in the simulations to R_{S} γ_{foc} ^{2}(z) and $2\gamma \left(z\right)\sqrt{{R}_{\mathrm{ref}}{R}_{\mathrm{S}}}\mathrm{cos}\left[4\pi ({n}_{\mathrm{im}}z{\mathrm{nz}}_{\mathrm{s}})/{\lambda}_{0}\right]$, respectively. When z_{S} = 0, the focus function and the fringe envelope coincide at z = 0 (γ_{foc}(z) and γ(z) are centered at z = 0 when z_{S} = 0). When the depth z_{S} of the reflective structure increases, the distance between the focus function and the fringe envelope increases, whereas the amplitude of the interference fringes decreases.
Fig. 3 Theoretical axial signal acquired in LCOCT from a reflective structure located at different depths z_{S} (simulation based on Eq. (25)). The parameters of the simulation are: λ_{0} = 750 nm, ∆λ = 200 nm, n = 1.45, n_{im} = 1.40, NA = 0.5, R_{ref} = 0.19%, R_{S} = 0.1%, ϕ_{0} = 0. 
2.3 Experimental validation
This section compares the theoretical model presented above with measurements obtained using an LCOCT device available in the laboratory and described in reference [2]. Measurements of the axial signal acquired by LCOCT have been performed to check the validity of the theoretical expression given by equation (25). A simple sample was made by introducing a liquid of refractive index n between the glass window of the LCOCT device and a glass plate placed at a distance z_{S} below the glass window (see Fig. 4). The two interfaces between glass (fused silica) and the liquid have the same reflectivity R_{S} = (n − n_{glass})^{2}/(n + n_{glass})^{2}. This constitutes a sample of known refractive index n with two reflecting structures of reflectivity R_{S} located at depth z_{S} = 0 and z_{S} > 0. Three different samples were made with pure water (n = 1.33), oil (n = 1.51), and pure silicon oil (n = 1.40) placed between the glass window and the glass plate. The oil was an immersion oil for microscopy (SigmaAldrich, ref. 56822). The silicon oil was provided by SigmaAldrich (ref. 378399).
Fig. 4 Test sample with two reflecting structures: interface 1 located at z_{S} = 0, and interface 2 located at z_{S} > 0. The glass material is fused silica (n_{glass} = 1.45). The immersion medium is silicon oil (n_{im} = 1.40). The liquid placed between the two interfaces is water (n = 1.33), oil (n = 1.51) or silicon oil (n = 1.40). 
The signal acquired by a pixel of the LCOCT camera as a function of the depth scan is shown in Figure 5 (experiment and simulation). Due to the reflection of light on the two interfaces, the signal consists of the sum of two peaks and two packets of interference fringes. The LCOCT device is adjusted so that the coherence plane and the focal plane coincide at interface 1. Therefore, the interference fringes and the peak of light produced by the interface 1 coincide at z = 0 (γ(z) and γ_{foc}(z) are centered at z = 0 when z_{S} = 0). The interference fringes and the peak of light generated by light reflection on interface 2 located at z_{S} = 200 μm do not coincide when the refractive index of the sample differs from the refractive index of the immersion medium (n ≠ n_{im}), as predicted by the simulation. Depending on the value of n, the interference fringes are detected before or after the focus. Simulations are in good agreement with measurements. Note, however, that the amplitude of the interference fringes produced by interface 2 is slightly lower in the experiment. This can be explained by the transmission factor of interface 1 (T = 1 − R_{S}), not taken into account in the simulation, which reduces the amplitude of the detected signal from interface 2. The presence of chromatic dispersion mismatch in the interferometer when n ≠ n_{im}, not taken into account in the simulation either, also contributes to these discrepancies. Other factors can decrease interference contrast in practice, such as the existence of stray light that does not contribute to interference, defects in the beam splitter that is not perfectly 50/50 across the entire spectral range of detector sensitivity, and imperfections in the interferometer setting.
Fig. 5 Signal acquired in LCOCT from the test sample shown in Figure 4. The parameters are: λ_{0} = 750 nm, ∆λ = 200 nm, z_{S} = 200 μm, n_{im} = 1.40, n_{glass} = 1.45, NA = 0.5. R_{ref} = 0.19%. Three values of the refractive index n are considered (1.33, 1.40, 1.51). (a–c) Simulations based on equation (25). (d–f) Experiment. The peak corresponding to the focus on interface 2 coincides with the center of interference fringes only when n_{im} = n ((c) and (f)). 
3 Effects of refractive index mismatch between sample and immersion medium
The aim of this section is to study the consequences on LCOCT image quality of the shift between the focus function and the fringe envelop, resulting from a difference in refractive index between the sample and the immersion medium. A degradation of the signal amplitude and image resolution will be highlighted and quantified. Conditions to ensure that degradation is not significant will be established.
3.1 Degradation of signal amplitude
The signal in the LCOCT images correspond to the envelop of the interference fringes, which is obtained by digital processing of the acquired signal I(z) expressed in equation (25) [1, 2, 36]. The processed LCOCT axial response given by a reflective structure of reflectivity R_{S} located at depth z_{S} in the sample is therefore proportional to $\sqrt{{R}_{\mathrm{S}}}\gamma \left(z\right)$. Theoretical plots of γ(z) (Eq. (26)) are shown in Figure 6 for different values of the structure depth z_{S} with a mismatch of the refractive index between the sample and the immersion medium. The deeper the structure, the weaker the signal amplitude.
Fig. 6 Simulation of γ(z) (Eq. (26)) for different values of the reflecting structure depth z_{S}. The parameters of the simulation are λ_{0} = 750 nm, ∆λ = 200 nm, n = 1.45, n_{im} = 1.40, NA = 0.5. 
We will now establish an explicit expression of the signal attenuation by simplifying the expression of function γ(z). Typical values of the experimental parameters in LCOCT are λ_{0} = 750 nm, ∆λ = 200 nm, NA = 0.5 and n_{im} = 1.40 [1–5]. Considering a sample with n = 1.45 yields ∆z_{foc} ≈ 10 μm (Eq. (14)) and ∆z_{coh} ≈ 1 μm (Eq. (24)). Since ∆z_{coh} ≪ ∆z_{foc}, function γ(z) can be approximated by the coherence function γ_{coh}(z) multiplied by the value of γ_{foc}(z) at the maximum of γ_{coh}(z), i.e. when z_{coh} = z_{S}, which corresponds to z = (n/n_{im}) z_{S}. The fringe envelope γ(z) can therefore be considered as a Gaussian function multiplied by an attenuation factor α < 1 given by$$\alpha ={\gamma}_{\mathrm{foc}}\left(z=\left(n/{n}_{\mathrm{im}}\right){z}_{\mathrm{S}}\right)={\left[1+{\left(8\epsilon \frac{{z}_{\mathrm{S}}}{\u2206{z}_{\mathrm{foc}}}\right)}^{2}\right]}^{1/4},$$(27)with$$\epsilon =1{\left(\frac{n}{{n}_{\mathrm{im}}}\right)}^{2}.$$(28)
Assuming that Δn = n − n_{im} ≪ 1, an approximate expression of the signal attenuation is$$\alpha ={\left[1+{\left(2\pi \frac{\Delta n{z}_{\mathrm{S}}{\mathrm{NA}}^{2}}{{{n}_{\mathrm{im}}}^{2}{\lambda}_{0}}\right)}^{2}\right]}^{1/4}.$$(29)
Equation (29) is plotted in Figure 7 as a function of the depth z_{S} of the reflective structure for different values of the refractive index mismatch Δn. As can be seen, the attenuation may be significant for values of Δn of only a few percent. With the following condition:$$\frac{{\mathrm{NA}}^{2}}{{\lambda}_{0}}\frac{\u2206n}{{{n}_{\mathrm{im}}}^{2}}{z}_{\mathrm{S}}\le 1,$$(30)the attenuation of the signal is less than 2.5 (α ≈ 0.4). Equation (30) can be seen as a condition for keeping signal attenuation low.
Fig. 7 Signal attenuation α as a function of depth in the sample for various values of Δn. Simulation based on equation (29) with λ_{0} = 750 nm, ∆λ = 200 nm, NA = 0.5, n_{im} = 1.4. 
3.2 Degradation of lateral resolution
If the focal plane of the microscope objective does not coincide with the coherence plane, the image of the structures in the vicinity of the coherence plane appears blurred because of the defect of focus, which degrades the lateral resolution. If n_{im} ≠ n, the degradation increases with depth and it is obviously all the more important as the depth of field of the objectives is small (i.e. NA is high). Inhomogeneities of the refractive index within the sample also induce a distortion of the optical wavefront, which introduces optical aberrations. Considered alone, this effect yields to an effective lateral resolution degraded compared to its diffractionlimited theoretical value. Although less significant than the effect of focus defect, those aberrations cannot be easily corrected.
As seen previously, a reflective structure located at z_{S} is detected with a maximum signal when it is located in the coherence plane, i.e. when z_{coh} = z_{S}, which corresponds to z = (n/n_{im})z_{S}. We consider that a defect of focus is not visible provided that the structure lies within the depth of field ∆z_{foc}, i.e. when the distance ∆z between the coherence plane and the focal plane is such that$$\u2206z=\left{z}_{\mathrm{coh}}{z}_{\mathrm{foc}}\right\le \u2206{z}_{\mathrm{foc}},$$(31)with ∆z_{foc} defined as the width of the focus function (Eq. (14)).
Since z_{foc} = (n/n_{im})z, ∆z can be expressed as a function of the refractive indices of the sample and immersion medium and depth in the sample as$$\u2206z=\left1{\left(\frac{n}{{n}_{\mathrm{im}}}\right)}^{2}\right{z}_{\mathrm{S}}.$$(32)
Assuming that Δn = n − n_{im} ≪ 1, a condition for a focus defect not to be noticeable is$$\frac{{\mathrm{NA}}^{2}}{{\lambda}_{0}}\frac{\u2206n}{{{n}_{\mathrm{im}}}^{2}}{z}_{\mathrm{S}}\le \frac{4}{\pi}\approx 1.$$(33)
It is interesting to notice that the conditions for no significant loss of signal (Eq. (30)) and no significant defect of focus (Eq. (33)) are identical. The maximum depth of a structure that can be imaged without significant signal loss and resolution degradation can therefore be estimated from equation (33) alone. Figure 8 shows the imaging depth that can be reached without degradation of image quality as a function of the refractive index mismatch Δn for different values of the microscope objective NA.
Fig. 8 Depth (z_{S}) in the sample that can be probed without significant loss of both signal amplitude and lateral resolution, as a function of refractive index mismatch. Simulation based on equation (33), with n_{im} = 1.40 and λ_{0} = 750 nm, for different values of the microscope objective NA. 
4. LCOCT imaging
In this final section, we propose to illustrate the image degradation due to a difference in refractive index between the sample and the immersion medium by imaging a phantom with a known refractive index using two different immersion media. Finally, we propose to see theoretically whether the condition of nondegradation of image quality (Eq. (33)) is verified for skin imaging using two different immersion media.
4.1 Image of a phantom
A phantom was fabricated by embedding titanium dioxide (TiO_{2}) particles in a polydimethylsiloxane (PDMS) matrix (Sylgard 184 silicone Elast omer Kit, Neyco, France) whose refractive index is 1.41 at the wavelength of 750 nm [37]. The PDMS matrix was fabricated using a ratio of 10:1 by weight of PDMS prepolymer and curing agent (Sylgard 184). The phantom was prepared by first mixing the curing agent with a powder of TiO_{2} particles (SigmaAldrich). The mixture was placed in an ultrasonic bath for 30 min to prevent particles aggregation, and then mixed with the PDMS prepolymer. The obtained mixture was poured on a 40mm diameter petri dish. Air bubbles were removed using a vacuum pump for 1 h, and the phantom mixture was finally cured for 1 h30 min at 80 °C. The concentration of TiO_{2} particles in the PDMS is 0.40% by weight, and their diameter is less than 5 μm. This results in a refractive index of the phantom that is similar to that of the PDMS matrix, i.e. n ~ 1.41.
LCOCT images of the phantom in vertical section were acquired, using two different immersion media: silicon oil and water. The images are shown in Figure 9. The refractive index of the phantom (n ~ 1.41) being close to that of the silicon oil (n_{im} = 1.40), focusing and signal amplitude are maintained throughout the scan depth when silicon oil is used as the immersion medium (Fig. 9a). In this case, equation (33) is fully satisfied since ∆n ~ 0. When water is used as the immersion medium (n_{im} = 1.33), the mismatch in refractive indices (∆n ~ 0.08) leads to a focusing defect and a signal loss that increases with depth (Fig. 9b). In this case, equation (30) is satisfied (i.e. image quality is not degraded) for a depth z_{S} < 70 μm (taking n = 1.41, NA = 0.5, λ_{0} = 750 nm and ∆λ = 200 nm), which seems to agree with the observation of the acquired image.
Fig. 9 LCOCT images (vertical sections) of TiO_{2} particles embedded in PDMS (n ~ 1.41), using an immersion medium of refractive index n_{im} = 1.40 (a) and n_{im} = 1.33 (b). 
4.2 Skin imaging
LCOCT has been designed so far for imaging of skin tissues [1, 5]. The superficial skin layer (stratum corneum) has a refractive index in the near infrared around n = 1.54, and a thickness of 10–30 μm. The rest of the epidermis has a mean refractive index around n = 1.41 and a thickness of 100–200 μm. The refractive index of the dermis, the deepest layer, is around n = 1.38 [28–30]. Due to variations in skin refractive index, a mismatch between the coherence plane and focal plane is unavoidable during depth scanning. In this section, we propose to calculate the defocus ∆z = z_{coh} − z_{foc} that occurs as a function of the imaging depth in skin tissues. The simulation was carried out using equation (32) with silicon oil (Fig. 10a) and water (Fig. 10b) as the immersion medium. The epithelium thickness was considered to be 120 μm, with a 20 μm stratum corneum. The tolerable defocus, considered to be depth of field ∆z_{foc}, was calculated using equation (14) with NA = 0.5 and λ_{0} = 750 nm. The simulation shows that image quality is maintained in all skin layers when silicone is used as the immersion medium (n_{im} = 1.40). This corresponds to the LCOCT device commercialized by Damae Medical. If water is used instead of silicon oil as the immersion medium (n_{im} = 1.33) for skin imaging, the condition for preserving image quality is not satisfied at depths larger than z_{S} ~ 70 μm. This depth corresponds to about half of the thickness of the epidermis in the skin model considered in this simulation.
Fig. 10 Theoretical defocus (∆z = z_{coh} − z_{foc}) in the different skin layers with an immersion medium of refractive index n_{im} = 1.40 (a) and n_{im} = 1.33 (b) compared to the tolerable defocus (∆z_{foc}). Calculations based on equations (32) and (14) with NA = 0.5, λ_{0} = 750 nm and n = 1.54/1.41/1.38 (stratum corneum/epidermis/dermis). 
5 Conclusion
A theoretical model of the signal acquired in LCOCT has been described. The model shows that a difference in refractive index between the sample and the immersion medium causes dissociation between the coherence plane and the focal plane, resulting in a decrease in signal amplitude and a degradation of the image’s lateral resolution. Measurements were carried out to validate and illustrate the theoretical predictions. A mathematical condition (Eq. (33)) linking various experimental parameters was established to ensure that image quality degradation is insignificant. This condition was tested experimentally by imaging a phantom. It was verified theoretically in the case of skin imaging, with experimental parameters corresponding to those of the LCOCT device marketed by Damae Medical. The mathematical condition expressed in equation (33) can be used to define the experimental parameters of an LCOCT device, including the NA of the microscope objectives, the center optical wavelength, the maximum imaging depth and the refractive index of the immersion medium, in order to image with optimum quality a sample whose refractive index is known.
Acknowledgments
The author thanks Dr. Frederic Pain for preparing the PDMS samples. He is also grateful to the company DAMAE Medical for technical support.
Funding
This research did not receive any specific funding.
Conflicts of Interest
The author declares no conflicts of interest.
Data availability statement
This article has no associated data generated and/or analyzed.
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All Figures
Fig. 1 Linnik interferometer implemented in LCOCT. The components in the red dashed frames are mounted on a piezoelectric (PZT) translation stage for scanning the sample depth (motion indicated by the double red arrow). 

In the text 
Fig. 2 Mismatch of the coherence plane and focal plane in LCOCT. The plain red lines represent the beam in the plane of the figure, whereas the dotted red lines represent the beam in the direction orthogonal to the plane of the figure. (a) The interferometer is adjusted so that the coherence plane and the focal plane coincide at the surface of the sample. (b) When the interferometer is lowered by a distance z, the coherence plane is at a distance z_{coh} below the sample surface and the focal planes at a distance z_{foc} (here n > n_{im}). 

In the text 
Fig. 3 Theoretical axial signal acquired in LCOCT from a reflective structure located at different depths z_{S} (simulation based on Eq. (25)). The parameters of the simulation are: λ_{0} = 750 nm, ∆λ = 200 nm, n = 1.45, n_{im} = 1.40, NA = 0.5, R_{ref} = 0.19%, R_{S} = 0.1%, ϕ_{0} = 0. 

In the text 
Fig. 4 Test sample with two reflecting structures: interface 1 located at z_{S} = 0, and interface 2 located at z_{S} > 0. The glass material is fused silica (n_{glass} = 1.45). The immersion medium is silicon oil (n_{im} = 1.40). The liquid placed between the two interfaces is water (n = 1.33), oil (n = 1.51) or silicon oil (n = 1.40). 

In the text 
Fig. 5 Signal acquired in LCOCT from the test sample shown in Figure 4. The parameters are: λ_{0} = 750 nm, ∆λ = 200 nm, z_{S} = 200 μm, n_{im} = 1.40, n_{glass} = 1.45, NA = 0.5. R_{ref} = 0.19%. Three values of the refractive index n are considered (1.33, 1.40, 1.51). (a–c) Simulations based on equation (25). (d–f) Experiment. The peak corresponding to the focus on interface 2 coincides with the center of interference fringes only when n_{im} = n ((c) and (f)). 

In the text 
Fig. 6 Simulation of γ(z) (Eq. (26)) for different values of the reflecting structure depth z_{S}. The parameters of the simulation are λ_{0} = 750 nm, ∆λ = 200 nm, n = 1.45, n_{im} = 1.40, NA = 0.5. 

In the text 
Fig. 7 Signal attenuation α as a function of depth in the sample for various values of Δn. Simulation based on equation (29) with λ_{0} = 750 nm, ∆λ = 200 nm, NA = 0.5, n_{im} = 1.4. 

In the text 
Fig. 8 Depth (z_{S}) in the sample that can be probed without significant loss of both signal amplitude and lateral resolution, as a function of refractive index mismatch. Simulation based on equation (33), with n_{im} = 1.40 and λ_{0} = 750 nm, for different values of the microscope objective NA. 

In the text 
Fig. 9 LCOCT images (vertical sections) of TiO_{2} particles embedded in PDMS (n ~ 1.41), using an immersion medium of refractive index n_{im} = 1.40 (a) and n_{im} = 1.33 (b). 

In the text 
Fig. 10 Theoretical defocus (∆z = z_{coh} − z_{foc}) in the different skin layers with an immersion medium of refractive index n_{im} = 1.40 (a) and n_{im} = 1.33 (b) compared to the tolerable defocus (∆z_{foc}). Calculations based on equations (32) and (14) with NA = 0.5, λ_{0} = 750 nm and n = 1.54/1.41/1.38 (stratum corneum/epidermis/dermis). 

In the text 
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