1. Introduction

It is now known that several diseases in cells and tissues, such as cancer and brain abnormalities, involve intracellular structural alterations at the nano to sub-micron scales originating from the rearrangements of macromolecules within the cells [1,2]. This demands more accurate probes and analysis of the cells/tissues to quantify disease progressions. Since cellular material are good refractive index media, optical methods are ideally suited for this purpose. Biological applications of optical spectroscopy aim to identify structural alterations in the early stages that are histologically indistinguishable. Recent advances in cellular optics have shown that as an alternative to using costly high resolution or direct (phase) imaging techniques, analyzing the intensity of the reflected light within the mesoscopic wave-scattering framework can provide valuable statistical information of the structural changes on a range of lengthscales including those below the diffraction limit [3].

Mesoscopic physics describes the effects of the superposition of multiply scattered coherent waves. In particular, it is known that scattered waves in a disordered medium interfere constructively to enhance the reflected signal [4], with the enhancement effects being more profound in lower dimensions [5]. Partial wave spectroscopy (PWS), for example, is a well established technique that takes full advantage of the increased interference effects in one dimension [1,6]. The technique uses diffraction-limited coherent light and a low-NA collection system of the reflected light in far-field to virtually divide the cell into a collection of optically independent disordered one-dimensional (1D) channels. Most PWS analyses are, however, performed taking into account only the induced RI fluctuations by the intracellular macromolecular structures while neglecting the mean RI of the cellular medium and the mounting slides, i.e., the RI-mismatch between the mean sample RI and the outer media is assumed to be zero.

In the present work, motivated by the PWS setup we formulate and solve analytically the reflectance from a 1D stochastic model that explicitly takes into account the interplay between the fluctuating RI and a finite background RI-mismatch. We provide numerical support to show how the cellular RI and the morphological parameters can be accurately determined using realistic values appropriate for cells. A new decorrelation lengthscale originating from the interplay of disorder scattering and thin-film interference is identified.

The principles of mesoscopic physics evolved from the study of electronic transport at very low temperatures where the wave nature of the electrons manifests itself [7]. The expectation that electrons in a static fluctuating potential undergo diffusive motion was upended by Anderson [8] who showed that when the wave nature of the electrons is considered the coherent superposition of the scattered matter-waves can arrest the diffusive motion. In long 1D channels, the interference completely dominates so that all states are exponentially localized with a localization length proportional to the electron energy $E$ and inversely proportional to the strength of the fluctuations $\langle V^2\rangle$, i.e., the inverse localization length $\xi _k^{-1}\sim \langle V^2\rangle /E$ [9,10].

In a finite 1D system of length $L$, the manifestation of localization physics was elucidated by Landauer [11] by expressing the average resistance $\langle \rho \rangle =\langle r/(1-r)\rangle$ in terms of the reflectance $r$. It was shown that while diffusive behavior is restored at small lengths in the form of Ohm’s law $\langle \rho \rangle \sim L/\xi _k$, localization effects accumulate when $L\gg \xi _k$ leading to an exponential growth of the resistance $\langle \rho \rangle \sim e^{L/\xi _k}$. Since $\xi _k$ decreases with $E_k$, it is difficult to access the diffusive regime in 1D electronic systems [5]. Nevertheless, the fact that a single scale $\xi _k$ controls the behavior for all $L$ makes the reflectance sensitive to changes in $l_c$ and $\langle V^2\rangle$ at the smallest scales. Furthermore, because coherence extends over a large portion of the disordered system, the relative fluctuations are strongly non-self-averaging and non-additive [12,13]; even in thin samples, where the behavior $\langle \rho \rangle \approx \langle r\rangle =L/\xi _k$ is nominally ohmic, the ratio $\sqrt {\langle (\Delta r)^2\rangle }/\langle r\rangle \rightarrow 1$ does not vanish. Thus the same scale $\xi _k$ controls both the mean and the variance.

There exists a well established connection between quantum and classical wave propagation [14,15], which follows from the similarity of the Maxwell’s wave equation and the Schrödinger equation. In 1D, the time-independent wave equation for the amplitude of the electric field $\mathcal {E}$ of a monochromatic electromagnetic wave in an optical medium with weakly varying RI reads as: $-d^2\mathcal {E}/dx^2-2\delta n(x) k^2\mathcal {E}=\bar {n}^2k^2\mathcal {E}$. (Here, $n(x)=\bar {n}+\delta n(x)$ is expressed in terms of the mean background RI, $\bar {n}$, and a weakly fluctuating component, $|\delta n/\bar {n}|\ll 1$; $k=\omega /c$ is the dispersion in vacuum.) It can be brought into the form of the Schrödinger equation, $-d^2\psi /dx^2+V(x)\psi =E\psi$, by identifying the potential $V(x)\rightarrow -2\delta n(x) k^2$ and the energy $E\rightarrow \bar {n}^2k^2$. (In units of $\hbar ^2/2m=1$.) The crucial difference is the presence of the $k^2$ prefactor for classical waves compared to the electronic case in $V(x)$. The effective light-matter coupling is therefore suppressed at low energies [16], resulting in the classical light localization length to increase with decreasing photon energy; the inverse is equal to $\xi _k^{-1} \sim k^2 \langle (\delta n)^2\rangle /\bar {n}^2$. Thus, compared to the electronic case, light scattering involving thin samples, such as biological cells, naturally fall in the domain $L\ll \xi _k$, which we call the diffusive regime from here on.

For 1D systems (single channel), we show that a perturbative analysis of the invariant-imbedding equation [17,18] captures quite accurately the reflectance statistics in the diffusive regime. Details of the linearized model are introduced and justified in Sec. 2. The statistical properties of the reflectance are derived in Sec. 3. Finally, in Sec. 4. we discuss how the analytical results may be applied to a PWS setup to extract the cellular parameters by analyzing numerically generated data with realistic inputs.

2. Model

A typical light-scattering setup involves a three-medium wave problem where the disordered sample with mean RI $\langle n(z)\rangle =\bar {n}$ is sandwiched by uniform media on either side with different RIs, say, $n_i$ (incident side) and $n_t$ (transmitted side). Since the absorption regions in the visible and near-infrared spectrum in biological cells are few and weak [3], we consider only elastic scattering, i.e., $\epsilon (z)=n^2(z)$ is assumed to be real.

A direct first-principles way to study the reflection and transmission coefficients from a layered medium is to use the invariant-imbedding method. The method converts a two-point boundary value problem to an initial value problem. When applied to light-scattering, the method directly accesses the change in, for example, the reflection coefficient, $R(z)$, as the sample length $z$ (imbedding parameter) is varied. Integrating the evolution equation from $z=0$ to $L$ gives $R(L)$ with the initial value imposed on $R(0)$. A pedagogical introduction can be found in Ref. [19], where the evolution equation for $R$ in a three-medium problem is given as

Here $k$ is the wavenumber (in vacuum) of the incident light. Eq. (1) takes into account the local fluctuation in the sample RI through $n(z)$ with mean $\langle n(z)\rangle =\bar {n}$. In the following we consider the case where the sample is sandwiched between identical media with known uniform RIs $n_0$, i.e., $n_i=n_t\equiv n_0$ (see Fig. 1). Thus, a single constant $\Delta n=\bar {n}-n_0$ parameterizes the RI-mismatch between the mean sample RI and the outside media.

 

Fig. 1. Schematics of the three-medium wave problem: a disordered sample (e.g., a biological cell/tissue) with mean RI, $\bar {n}=\langle n(z)\rangle$, extending between $0\leq z\leq L$ is shown sandwiched between identical media (say, glass slides) with uniform RIs, $n_0$. The fluctuations are drawn to scale with respect to the parameters chosen in Sec. 4.2.

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2.1 Clean limit

We first analyze Eq. (1) in the clean limit $n(z)=\bar {n}$ (for all $0\leq z\leq L$), i.e., without disorder. The solution with initial value $R(0)=0$ (cf. Eq. (1b)) can be written as:

Here $\bar {k}=\bar {n}k$ is the wavenumber inside the medium. This well-known result reproduces the optical interference of multiple reflections from the boundaries of a dielectric slab of length $L$.

The solution to linear order in $|\Delta n/n_0|\ll 1$, which we denote by $R(L)\equiv R_\Delta ^<(L)$, gives the thin-film interference contribution

Note that $R_\Delta ^<(L)$ solves the linearized equation obtained by expanding Eq. (1) to $\mathcal {O}(\Delta n/n_0)^2$

2.2 Weak disorder

In a weakly disordered medium, the local RI fluctuations $\delta n(z)=n(z)-\bar {n}$ are typically treated to linear order in $\delta n/\bar {n}\ll 1$ by expanding

Substituting (5) into Eq. (1) results in the addition of a non-linear stochastic term to Eq. (4)

The relevant known results for $\Delta n=0$ (RI matched case) are reviewed below: The evolution of the probability density function (PDF), $W(r)$, is obtained by averaging Eq. (8) over the $\delta$-correlated disorder (7); it is generally assumed that the phase angle $\theta$ of the complex reflection coefficient, $R=\sqrt {r}e^{i\theta }$, is distributed uniformly between $0$ and $2\pi$, which is justified when $\xi _k\gg 1/k$. The Fokker-Planck equation for the evolution of the PDF in the uniform-phase approximation is expressed as a function of the reduced length $r_L=L/\xi _k$ as [18,23]:

The exact solution of Eq. (9) is known for the resistance variable $\rho =r/(1-r)$ [13]; for our purpose, the integral solution is reproduced here in terms of $r$

On the other hand, we posit that a similar analysis at shorter lengths $r_L\ll 1$ produces a reliable approximation for the diffusive regime. The linearized Fokker-Planck equation obtained by setting $r=0$ where possible in Eq. (9) gives the approximate Fokker-Planck equation

A comment about the applicability of the uniform-phase distribution: The assumption was re-examined (for $\Delta n=0$) in short diffusive samples in Refs. [25,26] using a perturbative approach. (The approach is equivalent to starting with the linearized equation derived in Eq. (13) below.) The advantage of such a treatment is that it makes no assumption about the distribution of the phase. It was found [25] that although the resistance moments calculated perturbatively captured certain fast coherent oscillations not present in the uniform-phase approximation, they appear as powers of $\textrm {sinc}(kL)=\sin (kL)/(kL)$. It was therefore concluded that the uniform-phase approximation is valid when $1/k \ll L$, provided $1/k\ll \xi _k$.

2.3 Linearized stochastic model

The approximate Fokker-Planck Eq. (11) for $\langle r\rangle \ll 1$, can be arrived at by setting the non-linear term $(1+R)^2\approx 1$ in Eq. (8), leading to the linearized model:

3. Results

The starting point is the solution to Eq. (13) with boundary condition $R(0)=0$:

The oscillating term $r_\Delta$ in Eq. (17) is the thin-film contribution [26,27]

The reflectance $r(L)$ for a given disorder realization is shown in Fig. 2(a) (details are given in Sec. 4.2). In the following subsections, we use the relations derived above to calculate the relevant statistical properties of $r(L)$.

 

Fig. 2. (a) The reflectance $r(\lambda )=R^*R$ for a typical disorder realization is generated using Eq. (15) and plotted (dots) for $\lambda =500-670$ nm. (b) $N=500$ realizations of the reflectance traces are averaged to produce $\langle r(\lambda )\rangle$ (dots) and compared (not fitted) with the solution in Eq. (24) (solid line). At the minima the boundary reflections interfere destructively to give $r_\Delta (\lambda _\textrm {min})=0$ and the dashed (blue) line joining the minima traces $r_L=L/\xi _k$ (cf. Eq. (24)). The straight line can be inverted to obtain the disorder parameter $\tilde {L}_d$ by using Eq. (45). (c) The PDF of $r$ is generated from $N=500$ realizations (dots) for a fixed $\lambda =585$ nm and compared with the analytical result for $W_\Delta ^<(r)$ (solid line) derived in Eq. (43). The dashed (blue) line marks the theoretically determined location of the maxima (cf. Eq. (44b)), which can be used to determine $\bar {n}$ using Eq. (46). (d) The linear plots of $\log [C_k(\Delta k)/C_k(0)]$ vs $(\Delta k)^2$ derived in Eqs. (34a) (solid orange line) and (34b) (dashed orange line) are compared with the numerically computed correlations at specific representative $k$ values corresponding to $\lambda _\textrm {max}$ (dots) and $\lambda _\textrm {min}$ (squares), respectively, to highlight the spectral dependence of the correlation function in thin samples satisfying $L\ll L_\Delta ^*$. The lengthscale $L_\Delta ^*$ is defined in Eq. (31) and is estimated to be $\sim 12\,\mu$m. Substituting the value of $\bar {n}$ combined with the measured slope (solid orange line) into Eq. (34a) provides an estimate of the cell thickness $L$. The extracted parameters when substituted into Eq. (45) allows the disorder parameter $\tilde {L}_d$ to be determined.

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3.1 Average reflectance

To compute $\langle r\rangle$, the square of the stochastic terms (21a) and (21b) are averaged over the short-ranged $\delta$-correlated distribution defined in Eq. (7)

Adding the contributions from Eqs. (18) and (22), and keeping in mind that $\langle r_{\Delta \delta }\rangle =0$ in Eq. (19) vanishes, it follows from Eq. (17) that

It has been demonstrated that the spectral oscillations originating from the RI-mismatch can be used to extract particle sizes of homogeneous micrometer size scatterers with sub-diffractional accuracy [28]. We show in Sec. 4.2 how the sample thickness of an inhomogeneous system may be extracted similarly.

3.2 Correlation function

The spectral correlation function at two different wavenumbers $k_+=\bar {k}+q/2$ and $k_-=\bar {k}-q/2$ is derived with both waves seeing the same disorder configurations. It is defined in the standard way as (the $k$-dependence is explicitly shown in this section where necessary)

The disorder averaging is easily done using the short-ranged $\delta$-correlated distribution defined in Eq. (7). From Eq. (20) the net contribution to $C_\delta$ takes the form

Note that the first term in Eq. (29) originates from disorder scattering only. Since the weak spectral dependence of $r_L$ on $k$ is ignored when $q\ll k$, the decorrelation lengthscale in the diffusive limit, $L\ll \xi _k$, with $\Delta n=0$, is simply controlled by the optical length $L_\textrm {opt}=1/q$, which is understood in the standard way as the distance over which the phases of two waves separated by momentum $q$ become significantly different.

The second term in Eq. (29) highlights the interplay of disorder scattering and thin-film interference. The latter is given by $c_\Delta (\bar {k},q)\equiv R_\Delta (k_+)R_\Delta (k_-)$ and is equal to

To further explore the interplay between the thin-film interference and the disorder, we expand the ratio $C_k(q)/C_k(0)$ to second order in the limit $qL\ll 1$:

It is worth noting that in the localized regime, $L\gg \xi _k$, the increased scattering due to the localization of the field gives rise to similar non-analyticities leading to linear-$L$ corrections [29]. On the other hand, the field enhancement in the diffusive regime within the sample is driven by the destructive thin-film interference of the reflection coefficient near $k\approx k_\textrm {min}$. This synergistic (non-additive) effect is the origin of the non-analytic dependence on $L$.

3.3 Probability density function

It is seen by comparing the variance $C_k(0)=\langle (\Delta r)^2\rangle$ with $\langle r\rangle ^2$ (cf. Eqs. (24) and (25)) that a finite RI-mismatch suppresses the mesoscopic fluctuations:

In Eq. (16), $r(L)$ was expressed as the sum of squares of two stochastic trigonometric variables $a_c$ and $a_s$. For notational convenience, we rewrite them collectively as $r(L)=\sum _{t=c,s}a_t^2$, where:

The key point is that the covariance of $a_c$ and $a_s$ vanishes when $\bar {k}L\gg 1$

To this end, we first note from the linear relation in (37) that the vanishing of the covariance in Eq. (38) also implies that the $t_\delta$’s are mutually independent. We use the result that they are both gaussian-distributed and equal to [25]

Secondly, since $a_t=t_\delta +\Delta _t$ (cf. Eq. (37)), the PDF $P^{(t)}(a)=\langle \delta (a-a_t)\rangle =P_0(a-\Delta _t)$ is obtained by a simple shift of the argument in $P_0(v)$ by $v=a-\Delta _t$, i.e.,

Thirdly, changing the variable to $u=a^2$ gives the desired PDF $P^{(t)}_\Delta (u)$ defined in Eq. (39). The change of variables introduces two domains $a=\pm \sqrt {u}$ satisfying $u>0$ and a Jacobian:

Finally, substituting $P_\Delta ^{(t)}$ in Eq. (39) and using the identity $\Delta _c^2+\Delta _s^2=r_\Delta$ and the integral for the modified Bessel function [30] $I_0(\sqrt {a^2+b^2})=\tfrac {2}{\pi }\int _0^1 {dy} \cosh (a y)\cosh (b \sqrt {1-y^2})/{\sqrt {1-y^2}}$, we arrive at the desired result

The following properties are easily checked: (i) the mean $\langle r\rangle =\int _0^\infty dr\, rW_\Delta ^<(r)$ and the variance $C(0)=\int _0^\infty dr\, r^2W_\Delta ^<(r)-\langle r\rangle ^2$ reproduces the results derived in Eqs. (24) and (29); (ii) When the RI-mismatch $r_\Delta =0$, we recover the exponential distribution (12); (iii) In the opposite limit of a clean slab, i.e., $r_L=L/\xi _k\rightarrow 0$ keeping $L$ fixed, it can be shown using the asymptotic form, $I_0(y)\sim e^y/\sqrt {2\pi y}$ as $y\rightarrow \infty$, that $W_\Delta ^<(r)= \delta (r-r_\Delta )$ peaks at the reflectance of the thin-film as expected. It follows that the distribution evolves from an exponential when $r_\Delta \ll r_L$ to a non-monotonic distribution peaked at $r_\Delta$ when $r_\Delta \gg r_L$.

The exact location of the maximum is obtained by setting $\left.\partial _rW_\Delta ^<(r)\right |_{r=r_\textrm {max}}=0$ (cf. Eq. (43)), which results in the following transcendental equation for $r_\textrm {max}$

We emphasize that the peak in $W_\Delta ^<(r)$ (see Fig. 2(c); details are given in Sec. 4.2) originates from the RI-mismatch and is not to be confused with the non-monotonic behavior due to enhanced backscattering in the strongly localized regime $L\gg \xi _k$ as can be observed from the exact solution (for $\Delta n=0$) reproduced in Eq. (10).

4. Discussion and conclusions

4.1 Biological application

As an illustration of the usefulness of our results for biological applications, we focus on the PWS setup, which is a broadband spectroscopy technique capable of isolating partial 1D back-scattered waves in cells and tissues. For concreteness, we show that the parameter ranges described in Eq. (14) are generously satisfied in the PWS setups studied in Refs. [1,6] with the addition of realistic glass slides with known RI to the analysis.

The ranges of the experimental parameters are given as: the relevant broadband spectrum ranges from $\lambda \sim 500-670$ nm (in vacuum). For cell lines, the cell thickness ranges from $L\sim 1.5-4\,\mu$m and the mean cellular RI of normal cells varies from $\bar {n}\sim 1.35-1.4$ [31]. The RI of the surrounding medium equals $n_0=1$ for air to $1.52$ when glass slides are used. Thus the weak RI-mismatch condition, $(\Delta n/n_0)^2\ll 1$, and the uniform-phase approximation, $L\gg 1/\bar {k}$, are both justified; here, $\bar {k}=\bar {n}k$ is the wavenumber in the cell. The disorder parameter $L_d=\sigma _n^2l_c$ (cf. Eq. (7)) is assumed in a biological cell to be correlated with the intracellular morphology via (i) its dependence on the average size $l_c$ of the intracellular structures and (ii) the variance in the induced RI fluctuations $\sigma _n^2=\langle (\delta n)^2\rangle$. The key observation in Ref. [1] was that the nanoscale features associated with the early onset of carcinogenesis have significant measurable effects on scales as small as $L_d\sim 10^{-2}$ nm (corresponding to $l_c\sim 30$ nm $\ll \lambda$ and $\sigma _n\sim 0.02$), which is below the diffraction limit and therefore not visible in standard optical microscopy. Substituting $L_d=10^{-2}$ nm into Eq. (23), we get $\xi _k\gtrsim 150\,\mu$m $\gg 10 L$ at the lower end.

We therefore conclude that the linearized model (13) is a sufficient model to describe the diffuse 1D back-scattering spectrum from cell lines. We note, however, that $L\lesssim 0.1\xi _k$ may not be easily satisfied for tissue samples that can be tens of microns thick or when the disorder variations are much larger than $L_d\gg 10^{-2}$ nm; in such cases numerical solutions of the complete Riccati Eq. (1) may be required.

4.2 Extraction of the cell parameters from simulated data

The PWS setup treats each virtual 1D channel in a cell as a single realization of the disorder; a typical cell is effectively divided into hundreds of such channels [1]. In Fig. 2, we numerically generate and analyze $N=500$ realizations, which is a realistic number of channels per cell in the experiments, to illustrate how the cellular parameters can be reliably extracted by comparing with the analytical formulas.

To this end, Eq. (15) is used to generate $r=R^*R$ for different random RI realizations in the spectral range $\lambda =500-670$ nm. We assume the following experimentally relevant parameters: Sample thickness $L=2\,\mu$m; mean cellular RI $\bar {n}=1.38$; without loss of generality we assume the cells are enclosed between glass slides with RI $n_0=1.52$ on either side. The following definitions are used, namely, $k=2\pi /\lambda$ in vacuum and $\bar {k}=\bar {n}k$ is the wavenumber within the sample.

Different realizations are generated using the correlated disorder (6) with $l_c=30$ nm and $\sigma _n=0.02$, which translates to $L_d=1.2\times 10^{-2}$ nm. We note that the localization length in Eq. (23) was determined using the $\delta$-correlated disorder (7). It is easily shown by redoing the averages in Eq. (22) using correlated disorder (6) that the localization is longer by a factor $(1+(2\bar {k}l_c)^2)$ [23,26]. For our purpose, we absorb this factor into $\tilde {L}_d=L_d/(1+(2\bar {k}l_c)^2)$, so that the form of $\xi _k$ in (23) remains the same with $L_d$ replaced by $\tilde {L}_d$. As we will see below, only the renormalized parameter $\tilde {L}_d$ can be extracted experimentally, in other words, the microscopic scale $l_c$ cannot be determined. For reference, close to the center wavelength $\lambda _\textrm {mean}=585$ nm, the factor $1+(2\bar {k}_\textrm {mean}l_c)^2\approx 1.8$, and hence the localization length $\xi _k\approx 785\,\mu$m is almost twice longer making the regime $L/\xi _k\lesssim 0.1$ wider; practically this implies that our analysis can be applied to thicker cell/tissue samples for these parameters.

In Fig. 2(a), the reflectance data for a typical disorder realization is shown as a function of $\lambda$: the dots (black) are numerically generated. The disorder averaged reflectance $\langle r\rangle$ is shown in Fig. 2(b) and plotted (not fitted) against Eq. (24): the solid line (orange) is the plot of Eq. (24); the dashed (blue) line is the plot of $r_L(\bar {k})=L/\xi _k$. We emphasize that the locations of the minima are special in that they are exact: because $\sin (\bar {k}L)=0$ at these points, $R=R_\Delta ^<=0$ is independent of the linear approximation made in going from $R$ in Eq. (2) to $R_\Delta ^<$ in Eq. (3). Nevertheless, the total reflectance signal is not dark due to the residual impurity backscattering; joining the minima and comparing with $r_L$ allows the extraction of the disorder parameter $\tilde {L}_d$ in terms of the experimentally extracted values at any of the minima $\langle r(\lambda _\textrm {min})\rangle$ and its location $\lambda _\textrm {min}$:

Since the optical paths traverse the sample, the relevant length is the effective length $\bar {L}=\bar {n}L$. Unfortunately, knowing the location of the miminum does not solve $\sin (k\bar {L})=0$ uniquely for $\bar {L}$. Hence an alternate method is needed to extract both $L$ and $\bar {n}$ independently.

As described in Eq. (44b), $\bar {n}$ can be extracted from the location of the maximum of the PDF, $r_\textrm {max}=r_\Delta =2\bar {r}_\Delta \sin ^2(k\bar {L})$ (see also Eq. (18)). If the PDF is constructed judiciously for $\lambda =\lambda _\textrm {max}$, then the unknown $\sin (k_\textrm {max}\bar {L})\approx 1$ is fixed and we get

To determine $L$, we use the $q$-dependence of the correlation function $C_k(q)$ in Eq. (25) where $q=k_+-k_-$ is defined about a central wavenumber $\bar {k}$ so that $k_\pm =\bar {k}\pm q/2$. To separate the mean RI of the medium, we write $q=\bar {n}\Delta k$, where $\Delta k\approx k(\delta \lambda /\lambda )$ is the momentum difference in vacuum in the limit $\delta \lambda \ll \lambda$. For our purpose the total interval $\delta \lambda$ is sampled in steps of $\Delta \lambda =0.25$ nm. In Fig. 2(d), the numerically computed values for $\ln (C_k(\Delta k)/C_k(0))$ are shown for $\lambda _\textrm {max}$ (dots) and $\lambda _\textrm {min}$ (squares) and compared with Eq. (34a) (solid orange line) and Eq. (34b) (dashed orange line), respectively. The key point is that the slope at $\lambda _\textrm {max}$ is independent of the disorder parameter and can therefore yield the value of $L$ after substituting for $\bar {n}$ from Eq. (46).

Finally, substituting the extracted values of $L$ and $\bar {n}$ into Eq. (45) allows for $\tilde {L}_d$ to be determined. We emphasize that an independent check for the consistency and accuracy of the extracted parameters, $\bar {n},L$ and $\tilde {L}_d$, can be ascertained from the goodness of the fit of $\log (C_k(\Delta k)/C_k(0))$ at $\lambda =\lambda _\textrm {min}$ to Eq. (34b) (the dashed orange line in Fig. 2(d)).

The procedure outlined in this section provides a complementary method to that utilized in Refs. [1,6]. We remark that the validity of the linear-$L$ dependence of the correlation function assumed in these references is not generally valid in cells/thin-tissues, which are realistically mounted between glass slides (with a cover slip), due to the strong spectral dependence of $C_k(\Delta k)$ on $k$ originating from the RI-mistmatch, as clarified in our analysis.

In conclusion, in this work we introduced a linearized stochastic model (13) for which we have provided analytical and numerical support to show that the model is ideally suited to analyze the backscattered light from thin disordered optical media, such as biological cells/thin-tissues, surrounded by a uniform medium with a different RI. We derived exact closed-form formulas for the statistics and the wavelength-dependent correlations of the optical reflectance in the diffusive regime. The results are easily adapted to different forms of correlated noise with finite decay lengths. The formulas will be useful for experimentalist to monitor the changes of cellular parameters during disease progression such as in cancer or brain abnormalities. We have explicitly demonstrated how to extract (i) the cell thickness, (ii) the mean RI of the cell and (iii) the structural disorder parameter by fitting to simulated data with parameters typical for a PWS setup. The model works well for tissue samples as thick as $L\sim 0.1$ of the localization length; since the typical localization length for a tissue $\gtrsim 100\,\mu$m, $L\sim 10\,\mu$m. A strong spectral dependence of the auto-correlation function was found arising from the synergistic (non-additive) interaction between disorder scattering and thin-film interference leading to a new decorrelation lengthscale described in Eq. (31). We propose that the sensitivity of the lengthscale to the RI-mismatch can be taken advantage of to manipulate the spectral dependence of the reflectance by dipping the tissue/cell samples in media with high RI such as oil.