Review Papers

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

[+] Author Affiliations
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
History: Received August 13, 2016; Accepted February 20, 2017
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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 200  nm 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 (200  nm) 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 spectroscopic1 microscopy or spectral encoding of spatial frequency,24 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.58 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.810 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 Σ˜2 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 Σ˜ 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 cancers1117 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  μm), (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.

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Fig. 1
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.

Sample Structure

The sample geometry utilized by the SM technique is as follows: a spatially inhomogeneous sample with RI distribution n1[1+nΔ(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+αρ, where nw is the refractive index of water, α is the specific refractive increment (0.18  ml/g), and ρ is the cell dry density which was approximated as that of stratum mucosum (1.15  g/ml).2426

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 BnΔ(r).27,28 The flexible WM correlation family BnΔ(r) is expressed as Display Formula

BnΔ(r)=An·(rLn)D32·KD32(rLn),(1)
where r is the separation distance, Kν(·) is the modified Bessel function of the second kind with order ν, An is the fluctuation strength of RI, Ln is the characteristic length of heterogeneity (also representing a transition point beyond which the function decays exponentially), and D determines the functional form of the distribution at r<Ln. This three-parameter model covers a wide range of RI correlation functional forms, including those commonly used to describe the structure and light scattering from biological cells and tissues: power-law at D<3,2932 stretch-exponential at D(3,4),32,33 Heyney–Greenstein at D=3,34 exponential at D=4,21 and Gaussian at D.

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 r0. At the same time, physically, SCF must equal the variance of RI σnΔ2 at r=0. Thus, we follow a normalization approach, in which a smallest structural LS rmin is introduced, and An is defined so that BnΔ(rmin)=σnΔ2 is satisfied27,35Display Formula

An=σnΔ2(rminLn)3D2/KD32(rminLn),(2)
where rmin is the minimum structural LS of the sample’s internal structure. Specifically for the study of structural composition of biomaterials, we define rmin=2  nm, approximating the size of such biological monomers as amino acids, monosaccharides, B-form DNA, etc.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).

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Fig. 2
F2 :

BnΔ/σnΔ2 versus D for Ln=0.5  μm (dashed lines) and Ln=1.5  μm (solid lines). Horizontal black line indicates the level at which correlation decays by a factor of e.

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 nΔ(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 nΔ(r), and the sample as a horizontal slice of nΔ(r) with thickness LDisplay Formula

nΔ(r)=TLnΔ(r),(3)
where TL is a windowing function along the vertical axis with width L.

Light Propagation

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)=t01k22πnΔ(r)eiks·rd3r, where ks=koki 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=(n0n1)/(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 (kzk), U(s) focused at a point (x,y) in the image plane is38Display Formula

Uim(s)(x,y,k)=kt10i2π|M|TkNAfsei(kxx+kyy)dkxkdkyk,(4)
where TkNA is the microscope’s pupil function—a cone in the spatial-frequency space with a radius kNA [Fig. 3(a)]. As seen in Eq. (4), the objective performs low-pass transverse-plane spatial frequency filtering, with the cutoff corresponding to the spatial coherence length.

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Fig. 3
F3 :

Spatial-frequency space with kz-axis antiparallel to ki. (a) Cross section of TΔks, 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 kks [Fig. 3(a)], Uim(s) is Display Formula

Uim(s)(x,y,k)=T10T01i|M|kn1D(r)ei2kzdz,(5)
where r is the location (xM,yM,z) inside the sample, and n1D is the nΔ(r) convolved () with the unitary Fourier transform (F) of TkNATks in the transverse plane (xy, ), n1D(r)=F{TkNATks}nΔ(r).

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

I(x,y,k)=R012RI{+kn1D(r)ei2kzdz},(6)
where I denotes the imaginary part of a complex number, R01=r012 is the intensity reflectance coefficient, and R=R01t012t102. Since the sample is weakly scattering, O(nΔ2) terms are neglected here.

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(nΔ2) to O(nΔ). 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.

Quantification of Subdiffraction Length Scales Using Spectroscopic Microscopy

There can be a range of approaches for the quantification of nΔ(r) from the SM signal. One notable optical measure of nanoscale sample structure is Σ2: the spectral variance of the image intensity within the detector bandwidth Δk.19,21 Since the expectation of the spectrally averaged image intensity equals R012, Σ2(x,y) is defined as Display Formula

Σ2(x,y)=Δk[I(x,y,k)R012]2dk/Δk.(7)
For convenience, we introduce a windowing function TΔks that is a unity at k=ks for all ki with magnitudes within the Δk of the system and is zero elsewhere [|ki| between k1 and k2 in Fig. 3(a)]. Denoting kc as the value of the central wavenumber of illumination bandwidth inside the sample, Σ2(x,y) equals19Display Formula
Σ2(x,y)=R2kc2Δk|F{TΔksTkNA}nΔ(r)|2dz.(8)
Physically, TΔks accounts for the limited bandwidth of illumination and serves as a band-pass longitudinal spatial-frequency filter of RI distribution with its width related to the temporal coherence length lτ=2π/Δk. The interception of the two frequency filters associated with the spatial and temporal coherence TkNA and TΔks signifies the frequency-space coherence volume centered at kz=2kc: T3D=TΔksTkNA [Fig. 3(a)].

Given an infinite bandwidth, one could reconstruct the full 3-D RI from I(x,y,k). However, since Δk and kc are finite, Σ 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, nΔ(r)F{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: Display Formula

Bneff=BnΔBF{T3D}.(9)

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 Σ2 (denoted as Σ˜2), which is related to the RI distribution within the sample as Ref. 19Display Formula

Σ˜2=Rkc2LΔkT3DϕnΔ(k)d3k,(10)
where ϕnΔ=|F{nΔ(r)}|2 is the power spectral density (PSD) of the sample RI nΔ. PSD is a crucial parameter of sample organization: it fully quantifies the amplitude, size, and orientation of all RI fluctuations present within complex inhomogeneous samples, which cannot be otherwise measured by a single parameter of size or RI.

The general quadrature-form expression relating Σ˜2 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 n1n2, the expression for Σ˜2 has deterministic change in the prefactor and an additional offset value, both of which are defined by the sample geometry Display Formula

Σ˜2=R(2R12+(1+R12/R01)kc2LΔkT3DϕnΔdk),(11)
as derived in Ref. 19).

As follows from Eq. (10) in most general terms, Σ˜2 measures the integral of the tail of the PSD within T3-D. Several properties of Σ˜ are direct consequences of this relation:

  • Σ˜ is a linear function of the standard deviation of RI fluctuations σnΔ (which is ϕnΔ).
  • As described in more detail below, Σ˜ can be a monotonic function of the width of RI PSD.
  • Σ˜ 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 Σ˜. This phenomenon embodies the concept of statistical nanosensing.

Using Eq. (10), closed-form solutions for Σ˜2 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 Σ˜2 for a given experimentally obtained SCF that may not have an explicit, analytically defined functional form.

We here present an analytical solution for Σ˜2 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 Display Formula

ϕnΔ(ks)=AnLn3Γ(D2)π3/22(5D)/2·(1+ks2Ln2)D/2.(12)

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: Display Formula

ϕnΔ(k)=|F{TL}F{nΔ}|2.(13)
While the substitution of Eqs. (12) and (13) into Eq. (10) has no closed-form solution, it has been shown that Σ˜2 can also be calculated by independent computation of contributions from the light scattered from random RI variations within the sample (Σ˜R2), and the light reflection at the sample-substrate interface (Σ˜L2):19Display Formula
Σ˜2=Σ˜R2+Σ˜L2.(14)
Here, Σ˜L is fully described by the RI contrast at the bottom surface, and Σ˜R is defined by the PSD ϕnΔ which does not depend on sample thickness. Essentially, Σ˜L and Σ˜R perform two different measurements of sample internal structure as the former probes its statistics in two-dimensional (2-D) (along horizontal plane of z=L) and latter probes its statistics in 3-D (scattering from 3-D structures inside the sample).

Σ˜L is calculated using the fact that at z=L the RI contrast causing reflection of light is measured by the variance of n1D in the transverse plane Display Formula

σ2(n1D)=0BnΔ(ρ)kNAJ1(ρkNA)dρ,(15)
where ρ is the radial distance in polar coordinates.19 Substituting the expression for BnΔ(r) from Eq. (1) and introducing a unitless parameter of size with respect to wavelength x=kcLn, σ2(n1D) is found: Display Formula
σ2(n1D)=An2D52Γ(D32){1[1+(xNA)2]3D2}.(16)

Therefore, the corresponding contribution to spectral variance Σ˜L2=Rσ2(n1D)/4 becomes Display Formula

Σ˜L2=RAn2D92Γ(D32){1[1+(xNA)2]3D2},(17)
Σ˜R, in turn, is obtained by substituting the expression for the PSD of nΔ(r) from Eq. (12) into Eq. (10): Display Formula
Σ˜R2=An2RkcLπΓ(D2)x23D2(D2),{(1+4x2)1D/2[1+x2(4+NA2)]1D/2}.(18)
Finally, combining the above expressions for Σ˜R2 and Σ˜L2 as per Eq. (14), Σ˜2 is obtained: Display Formula
Σ˜2=RAn2D92Γ(D32){1[1+(xNA)2]3D2}+An2RkcLπ·Γ(D2)x23-D2(D2)×{(1+4x2)1D/2[1+x2(4+NA2)]1D/2}.(19)
Equation (19) is the closed-form analytical solution relating Σ˜2 measured from a wavelength-resolved microscope image to sample structure and accommodates a wide range of internal organization properties described by the general family of WM SCFs.

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 Σ˜2 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 σnΔ2=Bn(0) (no rmin is necessary in this case) and exponential correlation length lc, Σ˜2 is found from Eqs. (10) and (14) as Display Formula

Σ˜2=2RσnΔ2πkcLx3NA2[1+x2(4+NA2)](1+4x2)+RσnΔ24(111+(xNA)2).(20)
The analytical relations relating Σ˜2 to the sample’s internal organization have been validated by numerical full-vector solutions of Maxwell’s equations. 19

After establishing that Σ˜2 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 Σ˜2? At first, the answer seems quite simple: Σ˜ senses sample structure with spatial frequencies contained within T3D. However, identification of a more comprehensive LS sensitivity range for Σ˜, 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 Σ˜ depend on L. Finally, due to the nonlinear relation between nΔ(r) and ϕnΔ, scattering contributions from different internal structures are not independent or linearly additive.

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

Functional Dependence on the Shape of Spatial Correlation Function

As a light scattering-based parameter for structure quantification, Σ˜ 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 scattering37) approach of characterizing structural sensitivities through the functional dependence of the measured marker on the width parameters of RI SCF (Fig. 4).

Graphic Jump Location
Fig. 4
F4 :

Σ˜ for D(2,4) for samples with (a) L=0.5  μm and (b) L=2  μm shows a monotonic increase with D and a negligible dependence on the correlation outer scale Ln. The dependence on D for Ln(0.5,1.5)  μm explained in terms of the effective correlation length lceff in case of (c) L=0.5  μm and (d) L=2  μm.

For the general case of WM family of SCFs, Eq. (19) postulates that Σ˜ increases monotonically with the shape parameter D within the physiologically relevant range D(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), Σ˜ is an approximately linear function of fractal dimension D. Note that here Ln>0.5  μm 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 Σ˜ 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), Σ˜ senses arbitrarily small, deeply subdiffractional RI correlation lengths.19Σ˜(lceff) is a monotonically increasing function of the width of SCF, and its functional form at lceff<200 is well approximated to be lceff. At the same time, Σ˜ is independent of lceff for lceff1/kc, and therefore, the sensitivity of Σ˜ 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 experiment21,32). 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.

To summarize, the combination of spectroscopy and microscopy achieves “quantification” of subdiffractional structure using spectroscopy and “visualization” of larger-scale structures using microscopy.

Fundamental Limits to Sizes of Detectable Structures

Next, we note that the width of SCF is a cumulative statistic of all LS present within a sample, and therefore, Σ˜(lceff) is a measure of light scattering from all LSs within the sample, including those larger than the diffraction limit of light. Thus, when quantifying the LS sensitivity of Σ˜ through a statistic of RI distribution, it remains ambiguous precisely that structural LSs within the sample are detected. One way to isolate the sensitivity to a given size, or spatial frequency of RI fluctuations, is to evaluate Σ˜ measured from a sample composed of structures with only that spatial frequency.21

Mathematically, unbounded media nΔ(r) composed of a single spatial frequency kLS (evaluated in vacuum) are characterized by expectation of PSD ϕnΔ in the spatial-frequency domain which is an infinitely thin spherical shell with a radius n1kLS centered at the origin, ϕnΔ=σnΔ2δ(kn1kLS) [illustrations in Figs. 5(a)5(d)]. The finite sample, following the definitions of the Eq. (3), is defined as a section of nΔ(r) with thickness L. It is important to note that the PSD of nΔ(r), due to its finite thickness, is no longer a spherical shell and is expressed as ϕnΔ=|σnΔδ(kn1kLS)F{TL}|2, as shown in Figs. 5(c) and 5(d).

Graphic Jump Location
Fig. 5
F5 :

Cross section of single spatial-frequency medium with 1/kLS of (a) 15 and (b) 20 nm (kLS evaluated in vacuum). Cross section of PSD of an infinite (red) and thin (L=0.5  μm) media with 1/kLS of (c) 15 and (d) 20 nm. The sensitivity of Σ˜ to periodic structures with subdiffractional frequencies illustrated in the clear difference in the value of Σ(x,y) corresponding to the media with 1/kLS of (e) 15 and (f) 20 nm. Note that the subdiffractional structures are not resolved in the diffraction-limited Σ(x,y) image.