# Review of interferometric spectroscopy of scattered light for the quantification of subdiffractional structure of biomaterials

**Lusik Cherkezyan**,

**Di Zhang**,

**Hariharan Subramanian**,

**Ilker Capoglu**,

**Vadim Backman**

Northwestern University, Department of Biomedical Engineering, Evanston, Illinois, United States

**Allen Taflove**

Northwestern University, Department of Electrical Engineering, Evanston, Illinois, United States

*J. Biomed. Opt*. 22(3), 030901 (Mar 14, 2017). doi:10.1117/1.JBO.22.3.030901

#### Open Access

## Abstract

**Abstract.**
Optical microscopy is the staple technique in the examination of microscale material structure in basic science and applied research. Of particular importance to biology and medical research is the visualization and analysis of the weakly scattering biological cells and tissues. However, the resolution of optical microscopy is limited to $\u2265200\u2009\u2009nm$ due to the fundamental diffraction limit of light. We review one distinct form of the spectroscopic microscopy (SM) method, which is founded in the analysis of the second-order spectral statistic of a wavelength-dependent bright-field far-zone reflected-light microscope image. This technique offers clear advantages for biomedical research by alleviating two notorious challenges of the optical evaluation of biomaterials: the diffraction limit of light and the lack of sensitivity to biological, optically transparent structures. Addressing the first issue, it has been shown that the spectroscopic content of a bright-field microscope image quantifies structural composition of samples at arbitrarily small length scales, limited by the signal-to-noise ratio of the detector, without necessarily resolving them. Addressing the second issue, SM utilizes a reference arm, sample arm interference scheme, which allows us to elevate the weak scattering signal from biomaterials above the instrument noise floor.

Structure quantification and imaging at submicrometer scales is paramount in research fields from materials science to biology and medical diagnostics. At the molecular level, ultrasmall angle X-ray scattering, often in combination with neutron scattering and nuclear magnetic resonance and computation-heavy molecular modeling, has been very successful in characterizing isolated single molecule structures in solutions. Electron microscopy can provide information about much more complex structural organization such as that of biological cells or tissues with nanometer-resolution imaging. However, it is extremely time, labor, and resource intensive, most often requiring contrast agents, and the extensive sample processing alters the native structure of biomaterials. Optical microscopy techniques are key for imaging materials at the microscale due to their ease of real-time operation and the nondestructive nature of the visible light. The difficulties associated with light microscopy investigation of biological materials are the diffraction limit of resolution ($\u2265200\u2009\u2009nm$) and the optically transparent nature of the biological samples. To characterize structural properties that are indiscernible in microscope images, various techniques have coupled microscopic imaging with spectroscopic quantification.

In particular, spectral or angular properties of light scattering are utilized in techniques such as confocal light absorption and scattering spectroscopic^{1} microscopy or spectral encoding of spatial frequency,^{2}^{–}^{4} for quantifying sizes of structures within the samples. In addition, a multitude of quantitative phase imaging techniques utilizes the spectral interference profiles of wavelength-dependent bright-field microscope images for accurately extracting phase information in a spatially resolved manner.^{5}^{–}^{8} Conventionally, the light scattering-based techniques quantify the inhomogeneities within the studied samples, and the phase-quantification techniques focus on more cumulative characteristics such as mass and thickness. At the same time, some studies use the spatial distribution or slight changes in a sample’s optical path length to quantify its internal organization at unresolvable scales.^{8}^{–}^{10} While some of the above techniques can sense changes occurring at the nanoscale, the rigorous analytical link between exact sample structure and the measured quantity is either unclear or involves strong assumptions about sample structure.

Here, we review one distinct form of spectroscopic microscopy (SM) founded on the interferometric spectroscopy of scattered light, which utilizes the second-order spectral statistic $\Sigma \u02dc2$ of wavelength-resolved bright-field far-zone microscope images to characterize complex, weakly scattering, label-free media at subdiffraction scales. We also review the three-dimensional (3-D) light transport theory behind the technology with the explicit expression relating $\Sigma \u02dc$ to the statistics of refractive index (RI) fluctuations inside the weakly scattering label-free sample with an arbitrary form of RI distribution. SM’s sensitivity to subtle structural changes is widely applicable in fields from semiconductors and material science to biology and medical diagnostics. In particular, SM-based partial-wave spectroscopic (PWS) microscopy has facilitated the development of screening techniques for multiple early stage human cancers^{11}^{–}^{17} as well as label-free imaging of the native, living cellular nanoarchitecture.^{18}

Here, we review and emphasize the most important theoretical aspects behind this form of SM. Section 2 provides a basic introduction to SM theory, establishing the physics phenomenon behind its nanoscale sensitivity. Section 3 discusses the length scale (LS) sensitivity of SM in detail from various physically meaningful perspectives. Section 4 presents the main approaches to extracting the internal-only structural information of biological cell structure. Specifically, we will discuss how to design an SM instrument that is perfectly suited to the type of samples studied by the user, methods for measuring structure inside rough media, and comparisons between samples with different thicknesses. Section 5 presents alternative approaches to structural quantification using the SM signal including real-time, whole-slide imaging, temporally resolved quantification, explicit measures of sample structure, etc. Finally, Sec. 6 summarizes and discusses future directions in SM.

One of the approaches to subdiffraction-scale analysis of material structure is based on the notion of statistical nanosensing, which postulates that structural properties at LSs below a certain limit of resolution can be extracted from the sample organization statistics at larger LSs. In short, when true structure cannot be resolved with absolute precision due to fundamental resolution limits, an optical instrument effectively senses an RI distribution that is blurred or smeared in space. Notwithstanding, when the spatial correlation function (SCF) of this locally averaged RI distribution is quantified by scanning the sample in lateral (as in Ref. ^{19}) or axial directions (as in Ref. ^{20}), the SCF of the original perfect resolution RI can be reconstructed, yielding structural information about LSs far below the resolution limit.^{21}^{,}^{22} Inverse spectroscopic optical coherence tomography is one example technique founded on statistical nanosensing, and it focuses on evaluating the decay rate of the locally averaged RI SCF.^{20} The herein reviewed SM technique, in turn, focuses on measuring the area under the effective SCF,^{19} as discussed in greater detail in Sec. 2.3.

The SM instrument is a white-light epi-illumination, bright-field far-zone microscope with spectrally resolved image acquisition, small numerical aperture ($NAi$) of illumination ($NAi<0.3$), moderate-to-large NA of collection ($NA>0.3$), and with a pixel size of microscope image corresponding to an area in sample space that is smaller than the diffraction limit of light. In turn, the requirements to sample geometry include: (i) a weakly scattering sample of interest, (ii) sample thickness not greater than the microscope’s depth of field (for most setups, 5 to $15\u2009\u2009\mu m$), (iii) in the axial dimension, the sample should be RI-matched on one side (substrate in Fig. 1) and has a strong RI mismatch on the other (air in Fig. 1). Below, we thoroughly review the theoretical principles of the method, explaining the physical basis behind the above requirements.

**F1 :**

Sample: RI of the middle layer is random, RIs of the top and bottom layers are constant; RI as a function of depth is shown in gray. Coherent sum of $U(r)$ and $U(s)$ is detected. Reflection from the substrate (glass slide) is negligible as its thickness (1 mm) is much larger than the microscope’s depth of field. Reproduced with permission from Ref. ^{23}, courtesy of J. Biomed. Opt.

The sample geometry utilized by the SM technique is as follows: a spatially inhomogeneous sample with RI distribution $n1[1+n\Delta (r)]$ as a function of location $r$ is placed on a microscopy slide and exposed to air. Thus, the SM requirements are satisfied as the sample is sandwiched between two semi-infinite homogeneous media (Fig. 1), one of which has a strong RI mismatch with the sample (air $n0=1$), and the other is RI-matched (substrate RI denoted as $n2$). We assume $n1=n2=1.53$, mimicking the case of fixed biological samples on a glass slide, where $n1$ was evaluated using the Gladstone–Dale relation $n=nw+\alpha \rho $, where $nw$ is the refractive index of water, $\alpha $ is the specific refractive increment ($0.18\u2009\u2009ml/g$), and $\rho $ is the cell dry density which was approximated as that of stratum mucosum ($1.15\u2009\u2009g/ml$).^{24}^{–}^{26}

To describe light propagation through SM sample, an accurate model of the sample’s internal organization is required. Electron microscopy-based observations of nanoscale material distribution indicate that mass-density distribution inside biological media is best described as continuous random media rather than a multitude of discrete particles. A versatile mathematical approach for modeling light propagation through such media is based on the Whittle–Matern (WM) family of SCF $Bn\Delta (r)$.^{27}^{,}^{28} The flexible WM correlation family $Bn\Delta (r)$ is expressed as

^{29}

^{–}

^{32}stretch-exponential at $D\u2208(3,4)$,

^{32}

^{,}

^{33}Heyney–Greenstein at $D=3$,

^{34}exponential at $D=4$,

^{21}and Gaussian at $D\u2192\u221e$.

In terms of scaling the SCF along the vertical axis, several approaches for the normalization coefficient $An$ have been discussed,^{27} with the major issue lying in the fact that, mathematically, a power-law SCF is infinite at separation distances $r\u21920$. At the same time, physically, SCF must equal the variance of RI $\sigma n\Delta 2$ at $r=0$. Thus, we follow a normalization approach, in which a smallest structural LS $rmin$ is introduced, and $An$ is defined so that $Bn\Delta (rmin)=\sigma n\Delta 2$ is satisfied^{27}^{,}^{35}

^{35}The value of $rmin$ being equal to the size of biological monomers ensures that the macroscopic view of matter applies to all the length scales $r>rmin$ considered.

In terms of scaling the SCF along the horizontal axis, the width of SCF is regulated by parameters $D$ and $Ln$, with $D$ changing the functional form of SCF (in turn affecting its decay rate) and $Ln$ denoting the LS after which RI correlation decays exponentially. Hence, the “width” of SCF is determined by an interplay of $Ln$ and $D$, rather than either of them independently. Therefore, we introduce a more intuitive and universal measure of the SCF width, the effective correlation length $lceff$, which is the LS at which RI correlation decreases by a factor of $e$ from its value at $rmin$ (Fig. 2).

Finally, we note that in a biologically relevant range of sample properties, the physical size of the sample is comparable in magnitude to the characteristic LS of its internal structure (e.g., size of the nucleus as well as chromatin aggregates is comparable to the size of a cell). Therefore, the true SCF of $n\Delta (r)$ is an anisotropic function which depends on both the internal organization and the sample thickness $L$. Hence, we define $Ln$ and $D$ as the statistical properties of an unbounded medium $n\Delta \u221e(r)$, and the sample as a horizontal slice of $n\Delta \u221e(r)$ with thickness $L$

Importantly, despite the sample being weakly scattering, the Born approximation in its traditional form does not apply due to the required strong RI mismatch at sample–air interface.^{36}^{,}^{37} As has been validated by numerical full-vector solutions of Maxwell’s equations,^{19} the Born approximation can still be utilized for calculation of the scattered field inside the weakly scattering object, and ray optics can be used to describe propagation of the incident and the scattered fields across high RI-mismatch interfaces.

Thus, as a unit-amplitude plane wave with a wave vector $ki$ is normally incident onto the sample, its reflection from and transmission into the sample is described by ray optics. Then, the transmitted field with amplitude $t01=2n0/(n0+n1)$ is scattered from RI fluctuations inside the sample as described by the Born approximation: the far-zone scattering amplitude of the scattered field $U(s)$ with wave vector $ko$ is $fs(ks)=t01\u222bk22\pi n\Delta (r\u2032)e\u2212iks\xb7r\u2032d3r\u2032$, where $ks=ko\u2212ki$ is the scattering wave vector (inside the sample).^{36} When the scattered field leaves the sample, its transmission amplitude through the top interface is described again by ray optics $t10=2n1/(n0+n1)$. Finally, the field that reaches the image plane of an epi-illumination bright-field microscope is a result of optical interference between (i) the field reflected from the sample’s top surface [referred to as reference arm $U(r)$, amplitude $r01=(n0\u2212n1)/(n0+n1)$] and (ii) the field scattered from its internal fluctuations [sample arm $U(s)$, Fig. 1], with only the waves propagating at solid angles within the NA of the objective being collected. Thus, for a microscope with magnification $M$, moderate NA ($kz\u2248k$), $U(s)$ focused at a point ($x\u2032,y\u2032$) in the image plane is^{38}

**F3 :**

Spatial-frequency space with $kz$-axis antiparallel to $ki$. (a) Cross section of $T\Delta ks$, $TkNA$, and their interception, $T3-D$; (b) PSD of the RI fluctuation (blue) and $T3-D$ (gray) when the sample can be considered infinite (i.e., the LSs of internal organization are much smaller than sample thickness $L$); and (c) when the sample is finite. Reproduced with permission from Ref. ^{19}, courtesy of Phys. Rev. Lett.

Substituting $fs$ into Eq. (4) and introducing a windowing function $Tks$ that equals one at $k=ks$ and zero at $k\u2260ks$ [Fig. 3(a)], $Uim(s)$ is

Finally, the wavelength-resolved reflectance intensity recorded in the microscope image, normalized by the image of the light source, is an interferogram

Equation (6) describes the explicit relation between the SM signal and the sample RI distribution. From the optics perspective, Eq. (6) has extended the traditional Born approximation to include high RI mismatch at sample boundaries as well as far-field microscope imaging. From the mathematics perspective, Eq. (6) has established that to describe a one-dimensional (1-D) SM signal, the 3-D problem of light propagation can be reduced to a 1-D problem where the RI is convolved with the Airy disk in the transverse plane.

One important observation to make from Eq. (6) is that due to the RI mismatch at sample interface, the amplitudes of weakly scattered waves are multiplied by the strong reference wave, as a result of which the measured scattering signal increases from $O(n\Delta 2)$ to $O(n\Delta )$. Thus, the experimental noise generated by background photon flux, shot noise, and dark current noise become negligible, and the remaining noise from temporal lamp intensity fluctuations, on- and off-chip camera noise (i) is much lower in magnitude and (ii) can be reduced even further by frame averaging and frequency-based signal processing. As a result, the signal-to-noise ratio (SNR) is significantly enhanced. A second advantage, as compared to traditional interferometric techniques, such as optical coherence tomography or microscopy, is the simplicity of instrumentation, wherein the reference arm originates at the sample plane, eliminating the need for building a separate optical path for the reference arm.

There can be a range of approaches for the quantification of $n\Delta (r)$ from the SM signal. One notable optical measure of nanoscale sample structure is $\Sigma 2$: the spectral variance of the image intensity within the detector bandwidth $\Delta k$.^{19}^{,}^{21} Since the expectation of the spectrally averaged image intensity equals $R012$, $\Sigma 2(x,y)$ is defined as

^{19}

Given an infinite bandwidth, one could reconstruct the full 3-D RI from $I(x\u2032,y\u2032,k)$. However, since $\Delta k$ and $kc$ are finite, $\Sigma $ detects the variance of an “effective RI distribution,” i.e., of the refractive index that has been smeared in space according to the degree of spatiotemporal coherence, $n\Delta (r)\u2297F{T3D}$ [Eq. (8)]. The SCF $Bneff$ of this locally averaged RI, in turn, is a convolution of the SCF of the true RI distribution with the autocorrelation of the underlying resolution-limiting spatial filter of RI:

As shown below, the height of this effective SCF or the variance of the spatially filtered RI distribution presents a measure of sample organization that is sensitive to arbitrarily small structural LSs.

The statistical nanosensing, or the quantification of subdiffractional structural composition of the sample, is achieved by calculating the expected value of $\Sigma 2$ (denoted as $\Sigma \u02dc2$), which is related to the RI distribution within the sample as Ref. ^{19}

The general quadrature-form expression relating $\Sigma \u02dc2$ to PSD [Eq. (10)] is valid for an arbitrary weakly scattering sample, which is RI matched on one side and RI-mismatched on the other side (in the special case of $n1\u2260n2$, the expression for $\Sigma \u02dc2$ has deterministic change in the prefactor and an additional offset value, both of which are defined by the sample geometry

^{19}).

As follows from Eq. (10) in most general terms, $\Sigma \u02dc2$ measures the integral of the tail of the PSD within $T3-D$. Several properties of $\Sigma \u02dc$ are direct consequences of this relation:

- $\Sigma \u02dc$ is a linear function of the standard deviation of RI fluctuations $\sigma n\Delta $ (which is $\u221d\varphi n\Delta $).
- As described in more detail below, $\Sigma \u02dc$ can be a monotonic function of the width of RI PSD.
- $\Sigma \u02dc$ scales linearly with the deterministic sample-geometry parameter $R$.

Importantly, while $T3D$ does not include spatial frequencies above $2k$, the subdiffraction-scale structural alterations change the width of PSD and, therefore, the value of $\Sigma \u02dc$. This phenomenon embodies the concept of statistical nanosensing.

Using Eq. (10), closed-form solutions for $\Sigma \u02dc2$ for specific cases with any particular functional forms of RI SCF can be obtained. The general nature of Eq. (10) also allows numerical evaluation of $\Sigma \u02dc2$ for a given experimentally obtained SCF that may not have an explicit, analytically defined functional form.

We here present an analytical solution for $\Sigma \u02dc2$ for the general case when RI SCF is described by the versatile WM family widely applicable in the field of light scattering.^{27} The PSD of such sample with an infinite size is

For a finite sample, as defined in Eq. (3) and illustrated in Figs. 3(b) and 3(c), the PSD is an anisotropic function of $L$ along the $kz$ axis:

^{19}

$\Sigma \u02dcL$ is calculated using the fact that at $z=\u2212L$ the RI contrast causing reflection of light is measured by the variance of $n1D$ in the transverse plane

^{19}Substituting the expression for $Bn\Delta (r)$ from Eq. (1) and introducing a unitless parameter of size with respect to wavelength $x=kcLn$, $\sigma \u22a52(n1D)$ is found:

Therefore, the corresponding contribution to spectral variance $\Sigma \u02dcL2=R\sigma \u22a52(n1D)/4$ becomes

In practice, it is often advantageous to approximate the RI SCF within the sample as exponential, reducing the three-parameter model to two: the LS and the amplitude of RI fluctuations. The exponential approximation of SCF may not be as robust in terms of describing the nature of RI organization as the WM model does via $D$, but it can be useful from the experimental perspective. Furthermore, calculations based on electron microscopy images of biological cell nuclei have shown that the $\Sigma \u02dc2$ predicted based on the actual, experimentally measured RI distribution is in good agreement with that predicted based on a correlation length $lc$ value that assumes an exponential RI correlation.^{21} In this special case of exponential functional form of SCF with RI variance $\sigma n\Delta 2=Bn(0)$ (no $rmin$ is necessary in this case) and exponential correlation length $lc$, $\Sigma \u02dc2$ is found from Eqs. (10) and (14) as

^{19}

After establishing that $\Sigma \u02dc2$ quantifies the statistics of RI distribution inside weakly scattering media by analyzing the spectroscopic content of their microscope image [Eq. (10)], a question arises: what are the structural LSs sensed by $\Sigma \u02dc2$? At first, the answer seems quite simple: $\Sigma \u02dc$ senses sample structure with spatial frequencies contained within $T3D$. However, identification of a more comprehensive LS sensitivity range for $\Sigma \u02dc$, as well as other light scattering means of structure quantification, is remarkably nontrivial.

The difficulty in quantification of LS sensitivity is underlined by the fundamental difference between “resolution” and “sensitivity”: whereas resolution applies to imaging techniques and has a hard, purely instrument-dependent limit (e.g., diffraction limit), the ability to sense scattering events from certain structural sizes largely depends on the sample itself. First, it is simply a matter of which LSs and in what proportion is present inside the sample. The most illustrative example is the blue sky: even the human eye can sense light scattering by molecules smaller than 1 nm when larger ones are absent. Second, as seen in Eqs. (12) and (13), the shape of a sample’s PSD and, therefore, the sensitivity of $\Sigma \u02dc$ depend on $L$. Finally, due to the nonlinear relation between $n\Delta (r)$ and $\varphi n\Delta $, scattering contributions from different internal structures are not independent or linearly additive.

Thus, a universal LS sensitivity interval of $\Sigma \u02dc$ cannot exist, as it always depends on the sample structure. Below, we summarize approaches to assess various aspects of the LS sensitivity of $\Sigma \u02dc$, including (i) the functional dependence of $\Sigma \u02dc$ on the shape of RI SCF; (ii) fundamental limits to sizes of detectable structures; (iii) ranges of sizes predominantly detected by $\Sigma \u02dc$ within complex samples with various properties of internal organization, including those with analytically defined and experimentally obtained forms of RI SCF.

As a light scattering-based parameter for structure quantification, $\Sigma \u02dc$ is defined by the statistics of RI distribution inside the sample rather than the exact 3-D profile of RI. Therefore, we follow the common (in the field of light scattering^{37}) approach of characterizing structural sensitivities through the functional dependence of the measured marker on the width parameters of RI SCF (Fig. 4).

**F4 :**

$\Sigma \u02dc$ for $D\u2208(2,4)$ for samples with (a) $L=0.5\u2009\u2009\mu m$ and (b) $L=2\u2009\u2009\mu m$ shows a monotonic increase with $D$ and a negligible dependence on the correlation outer scale $Ln$. The dependence on $D$ for $Ln\u2208(0.5,1.5)\u2009\u2009\mu m$ explained in terms of the effective correlation length $lceff$ in case of (c) $L=0.5\u2009\u2009\mu m$ and (d) $L=2\u2009\u2009\mu m$.

For the general case of WM family of SCFs, Eq. (19) postulates that $\Sigma \u02dc$ increases monotonically with the shape parameter $D$ within the physiologically relevant range $D\u2208(2,4)$ and is weakly dependent on the outer LS $Ln$ [Figs. 4(a) and 4(b)]. Furthermore, for materials with fractal organization ($D$ between 2 and 3), $\Sigma \u02dc$ is an approximately linear function of fractal dimension $D$. Note that here $Ln>0.5\u2009\u2009\mu m$ to ensure that the functional form of SCF at separation distances below the diffraction limit of light is indeed determined by $D$ (SCF decays exponentially at $r>Ln$).

An alternative way to describe the dependence of $\Sigma \u02dc$ on the width of SCF is through a single parameter $lceff$, defined as the separation distance at which SCF decays by a factor of $e$, and referred to as “effective correlation length.” As illustrated in Figs. 4(c) and 4(d), $\Sigma \u02dc$ senses arbitrarily small, deeply subdiffractional RI correlation lengths.^{19}$\Sigma \u02dc(lceff)$ is a monotonically increasing function of the width of SCF, and its functional form at $lceff<200$ is well approximated to be $\u221dlceff$. At the same time, $\Sigma \u02dc$ is independent of $lceff$ for $lceff\u226b1/kc$, and therefore, the sensitivity of $\Sigma \u02dc$ to changes at smaller correlation lengths is not obscured by changes at larger scales.

We note that in the above two approaches to SCF parameterization, $lceff$ and $D$ measure the width of SCF in an interdependent manner [as seen in Figs. 4(c) and 4(d), $D$ is bound to increase with $lceff$], the use of $lceff$ can be advantageous in cases when the functional form of SCF is unknown, or is not well-represented by an analytical expression (e.g., when the SCF is measured directly from an experiment^{21}^{,}^{32}). At the same time, $D$ has a well-defined physical meaning and its use is preferred in cases when the SCF can be well described by $D$.