1. Introduction

In the fields of life science and biomedicine, it is becoming increasingly important to develop an ability to optically image objects embedded within scattering media, particularly biological tissues, for monitoring the physiological states of living cells, tissues and organs in their native states. Even with the numerous technical advances in light sources, optical components, detectors and other electronics, the progress in addressing the problem of imaging targets inside scattering media has been relatively slow. This is due to the fundamental limitations imposed by the effect of multiple light scattering that distorts light waves traveling to and from the target objects. The unperturbed wave by the scattering medium is termed a single-scattered wave as it interacts with the target object only once and not at all with the scattering medium. Its intensity tends to decay exponentially with the length scale of the scattering mean free path (MFP, ${l_\textrm{s}}$). The remaining waves go through multiple scattering events and occupy the major fraction of the detector dynamic range. The fluctuations of these so-called multiple-scattered waves serve as noise, thereby obscuring the image information embedded in the single-scattered wave. With the increase of imaging depth, single scattering signal to multiple scattering noise ratio decreases exponentially, making optical imaging extremely near-sighted [1–3].

Over the past decades, a major direction to enhance the imaging depth is to reduce the multiple scattering noise. Various gating operations such as confocal gating [4,5], temporal (or coherence) gating [6,7], and polarization gating [8,9] have been developed to filter out multiple scattering noise based on the intrinsic properties of single-scattered waves. Recently, we also proposed a new gating method termed space gating where multiple scattering noise traveling outside of the acoustic focus is rejected by means of acousto-optic interaction [10]. Methods to exploit wave correlation properties of single scattering have been proposed to enhance single-scattering signal [11–17]. In particular, we developed a method called collective accumulation of single scattering (CASS) microscopy for preferentially sampling single-scattered waves in which both the time gating and the momentum conservation property of single-scattered waves were used to image deep inside scattering medium at high resolving power [12]. The momentum conservation property was used by coherently adding those waves that have the same momentum difference between the reflected waves and incident waves. With the combined action of the time-gated detection and momentum conservation, the imaging depth was increased in comparison with incoherent addition of time-gated images.

Even with the application of all these deep-tissue imaging methods, single scattering signal is eventually dominated by the remaining multiple scattering noise beyond a certain target depth [2,3]. When signal to noise ratio (SNR) is below a certain threshold, the resolving of single scattering signal is no longer possible. However, the loss of resolving power is not abrupt, but rather gradual with the increase of depth. This is because the magnitude of the optical transfer function of typical imaging systems is attenuated with respect to the increase of the spatial frequency [18]. With the increase of target depth, SNR of high spatial frequency components of a target object is lowered below a threshold, which results in a gradual decrease of the resolving power. There have been many studies concerning the image formation in the presence of scattering media [19–24]. However, little study has been conducted on the effect of multiple scattering noise to the optical transfer function in the context of time-gated coherent imaging, necessary for the quantitative assessment of resolving power reduction.

In this paper, we report our investigation on the spatial frequency spectra of multiple-scattered waves, as well as that of single-scattered waves in the CASS microscopy. The functional dependence of SNR on the spatial frequency was derived theoretically, and numerical and experimental studies were performed to support the theoretical derivation. In doing so, we provide a way to quantitatively assess the depth-dependent spatial resolution of CASS microscopy. The imaging performance of conventional incoherent imaging was also analyzed as a point of reference. This study provides a theoretical framework for understanding the high-resolution optical imaging in the presence of multiple light scattering.

This paper consists of seven sections. In section II, the singe- and multiple-scattered waves are described for an object embedded within a scattering medium. In sections III and IV, the optical transfer functions of incoherent imaging and CASS microscopy are derived, respectively, in the presence of multiple-scattered waves. The spatial frequency-dependent SNR for the two imaging modalities is compared, and their respective achievable imaging depth is derived in section V. Experimental results are provided in section VI, followed by a summary in section VII.

2. Description of single- and multiple-scattered waves

Let us first describe the contribution of single- and multiple-scattered waves to the reflectance mode of imaging (Fig. 1). Consider a unit-amplitude plane wave, ${E_{in}}({x,y,z = 0;{{\boldsymbol k}_{in}}} )= exp [{i({k_x^{in}x + k_y^{in}y} )} ]$, incident to a target object located at $z = 0$ on which a scattering layer of thickness L is placed. Here ${{\boldsymbol k}_{in}} = ({k_x^{in},k_y^{in}} )$ is the transverse wavevector of the incident wave. The total reflected wave ${E_{\textrm{tot}}}({x,y,\tau ;{{\boldsymbol k}_{in}}} )$ with a certain flight time $\tau $ is obtained for the incident wave ${E_{in}}({x,y,z = 0;{{\boldsymbol k}_{in}}} )$ at the output plane conjugate to the sample plane. Then, the 2D Fourier transform of ${E_{\textrm{tot}}}({x,y,\tau ;{{\boldsymbol k}_{in}}} )$ provides the angular spectrum of the total reflected wave ${\tilde{E}_{\textrm{tot}}}({{{\boldsymbol k}_o},\tau ;{{\boldsymbol k}_{in}}} )$, where ${{\boldsymbol k}_o} = ({{k_x},{k_y}} )$ is the transverse wavevector of the reflected waves measured at the detector. Note that we consider each thin 2D section within a 3D object because the temporal and focus gating attenuate the contribution of the out-of-focus part of the object. Since each 2D section gives rise to the change in the transverse wavevector of the incoming wave, the change in the transverse momentum allows us to reconstruct the object function. ${\tilde{E}_{\textrm{tot}}}({{{\boldsymbol k}_o},\tau ;{{\boldsymbol k}_{in}}} )$ consists of the spectrum of the single-scattered wave ${\tilde{E}_\textrm{S}}({{{\boldsymbol k}_o},{\tau_\textrm{S}};{{\boldsymbol k}_{in}}} )\delta ({\tau - {\tau_\textrm{S}}} )$ reflected from the target object, time-gated multiple-scattered wave ${\tilde{E}_{\textrm{TM}}}({{{\boldsymbol k}_o},\;{\tau_\textrm{S}};{{\boldsymbol k}_{in}}} )\delta ({\tau - {\tau_\textrm{S}}} )$, and background multiple-scattered wave ${\tilde{E}_{\textrm{BM}}}({{{\boldsymbol k}_o},\;\tau \ne {\tau_\textrm{S}};{{\boldsymbol k}_{in}}} )\;$ from the scattering layer:

Here ${\tau _\textrm{S}} = 2L/c$ is the flight time of single-scattered waves reflected by the target with c the average speed of light in the scattering medium. With the setting of the time-gated detection at $\tau = {\tau _\textrm{S}}$, ${\tilde{E}_{\textrm{BM}}}({{{\boldsymbol k}_o},\tau \ne {\tau_\textrm{S}};{{\boldsymbol k}_{in}}} )$ is removed in the total reflected wave detected at the camera. Since both the incident wave and reflected wave should go through an objective lens, the conditions, $|{{{\boldsymbol k}_{in}}} |\le k\alpha $ and $|{{{\boldsymbol k}_o}} |\le k\alpha $, are to be met. Here $k = 2\pi /\lambda $ is the free-space wavenumber with $\lambda $ the wavelength of the light source in free space, and $\alpha \;$ the numerical aperture of the objective lens. The spectra of single-scattered wave can then be respectively expressed as

In both cases, the spatial bandwidth of the reflected wave is limited by the pupil function $P({{{\boldsymbol k}_\textrm{o}}} )$, whose value is unity when $|{{{\boldsymbol k}_o}} |\le k\alpha $ and zero otherwise. $\tilde{O}$ is the Fourier transform of the object function $O({x,y,z = 0} )$ of the thin target layer located at z = 0. Note that the object of interest itself needs not be thin as the temporal gating and short Rayleigh range set by the high-NA objective lens provide thin optical sectioning within a bulk object. For the multiple-scattered wave, the detected electric field is the superposition of random phasors arising from the numerous scattered waves emanating out of the scattering layer. With a relatively thin scattering medium under a quasi-diffusive regime, the amplitude A follows the Rayleigh probability density function, and the phase $\mathrm{\Phi }$ is uniformly distributed between 0 and 2$\pi $ [25]. The factor $\sqrt {{I_\textrm{S}}} $ is the average amplitude of single-scattered wave for a target with unity reflectance, and $\sqrt {{I_{\textrm{TM}}}} $ the average amplitude of the time-gated multiple-scattered waves having the same flight time as single-scattered waves. $\sqrt {{I_\textrm{S}}} $ is mostly governed by the thickness L of the scattering medium in such a way that the intensity of the single-scattered wave is exponentially attenuated during its round trip to the target, i.e. ${I_\textrm{S}} \propto {e^{ - 2L/{l_\textrm{s}}}}$. Imaging system specifications such as view field, depth of focus, and the width of time gating window affect ${I_{\textrm{TM}}}$ because the collection efficiency of multiple-scattered waves depends on these factors. In general, ${I_{\textrm{TM}}}$ decreases with depth, but much slower rate than ${l_\textrm{s}}.$ From previous experiments [12] and Monte Carlo simulations [13], we found that the time-gated multiple scattering intensity approximately decays exponentially with depth, i.e. ${I_\textrm{S}} \propto {e^{ - 2L/{l_{\textrm{TM}}}}}$. Therefore, the intensity ratio between single- and multiple-scattered waves, $\gamma (L )= {I_\textrm{S}}/{I_{\textrm{TM}}} = exp \left[ { - 2L\left( {\frac{1}{{{l_\textrm{s}}}} - \frac{1}{{{l_{\textrm{TM}}}}}} \right)} \right]$, is a function of imaging depth L. The decay length of $\gamma (L )$ is given by $\frac{1}{{{l_\gamma }}} = \frac{1}{{{l_\textrm{s}}}} - \frac{1}{{{l_{\textrm{TM}}}}}$.

 

Fig. 1. Types of reflected waves from a target object embedded within a scattering medium. The spatial coordinates of input, sample and output planes are ${{\boldsymbol r}_{in}},{\boldsymbol r}$, and ${{\boldsymbol r}_o}$, respectively, and their associated transverse wavevectors are ${{\boldsymbol k}_{in}},{\boldsymbol k}$, and ${{\boldsymbol k}_o}$. ${\tilde{E}_\textrm{S}}$: single-scattered wave, ${\tilde{E}_{\textrm{TM}}}$: multiple-scattered wave having the same flight time as single-scattered wave, ${\tilde{E}_{BM}}$ : multiple-scattered wave with different flight time from single-scattered wave.

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In the CASS microscopy, ${N_\textrm{m}}$ measurements taken by scanning the angle of illumination are coherently added to make the growth of single-scattering signal proportional to $N_\textrm{m}^2$, while the fluctuation of the multiple-scattering background increases with ${N_\textrm{m}}$. As a consequence, the overall SNR grows in proportion to ${N_\textrm{m}}$, an increase better than that obtained by incoherent imaging by a factor of $\sqrt {{N_\textrm{m}}} $. The maximum number of independent measurements is the same as the number of free modes determined by field of view and the numerical aperture of the imaging system. The overall SNR of the intensity spectrum approximately scales with ${N_\textrm{m}}$, and the target structure can be resolved if ${N_\textrm{m}}$ is large enough to raise SNR above certain threshold. As we shall discuss in the following sections, the SNR varies with the spatial frequency, and its functional shape can be described by the pupil function.

3. Optical transfer function of incoherent imaging

While the optical transfer function of imaging systems has been well investigated for the case of a scattering-free medium [18], there have been little studies on the spatial frequency spectrum of the multiple-scattered waves. We performed a rigorous theoretical analysis on the spatial frequency-dependent intensities of both single- and multiple-scattered waves, and supported the theoretical derivation using numerical simulations. In this section, we provide the analytic analysis of the imaging performance of the incoherent addition of time-gated images. At first, the intensity spectrum of the incoherent imaging is described as

Here $\ast $ stands for the autocorrelation operator. Since the cross term between the single- and multiple-scattered waves tends to be averaged out when assessing the intensity of the images, incoherent imaging spectrum becomes the summation of their individual spectra. For a heuristic purpose, let us consider a periodic structure whose object function is given by $O({x,y} )= \sqrt 2 \cos ({k_x^\textrm{s}x/2} )$. The reflected wave will be ${E_\textrm{S}}({x,y} )= \sqrt {2{I_\textrm{S}}} \cos ({k_x^\textrm{s}x/2} )$, where the amplitude $\sqrt {{I_\textrm{S}}} $ is defined in Eq. (2). Therefore, the intensity spectrum of the single-scattered waves for all ${N_\textrm{m}}$ different angles of illumination uniformly covering the pupil function is given by

Here $N({{{\boldsymbol k}_o}} )= {N_\textrm{m}}P\ast P({{{\boldsymbol k}_o}} )$ is the normalized autocorrelation of the pupil function multiplied by the number of measurements. Therefore, the intensity spectrum of the single-scattered wave at the spatial frequency of ${k_x} = k_x^\textrm{s}$ is given by $\frac{1}{2}N({{k_x} = k_x^\textrm{s},{k_y} = 0} ){I_\textrm{S}}$ [the blue curve in Fig. 2(a) shows $N({{{\boldsymbol k}_o}} )/{N_\textrm{m}}$].

 

Fig. 2. Intensity spectra of single- and multiple-scattered waves for incoherent addition. (a) Blue and red curves are theoretically expected spectra of single and multiple scattering, respectively, derived in Eq. (5) and the main text thereafter. Black curve shows the numerical simulation of the intensity spectrum for a mixture of single- and multiple-scattered waves where γ = 1/50 and N_m = 100. (b) Detailed plots of multiple scattering spectra shown in (a). Red and green curves are the spectra of the average intensity and its standard deviation, respectively, derived from the theory. Black curve and blue dots are, respectively, the numerically simulated intensity spectrum of multiple scattering and its ensemble average.

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On the other hand, the intensity spectrum of the multiple-scattered waves is proportional to $\sqrt {N({{{\boldsymbol k}_o}} )} $ because the autocorrelation of the Rayleigh density function asymptotically follows the statistics of the summation of $N({{{\boldsymbol k}_o}} )$ different random phasors. Likewise, the standard deviation of the spectrum of the multiple-scattered waves is also proportional to $\sqrt {N({{{\boldsymbol k}_o}} )} $. In fact, the spectra of average intensity and of standard deviation from multiple scattering are approximately given by $2 \times 0.88\sqrt \pi \frac{1}{{{k_0}\alpha \sqrt B }}{I_M}\sqrt {N({{{\boldsymbol k}_o}} )} $[red curves in Fig. 2(a) and 2(b)] and $2 \times 0.44\sqrt \pi \frac{1}{{{k_0}\alpha \sqrt B }}{I_M}\sqrt {N({{{\boldsymbol k}_o}} )} $ [green curve in Fig. 2(b)], respectively, where B is the area of the view field. The multiplication factors arise due to the summation of the band-limited double Rayleigh distribution in the autocorrelation process. We found such factors by fitting the intensity spectrum of multiple-scattered wave obtained by the numerical simulation.

We performed a numerical simulation of single- and multiple-scattered waves for $\gamma = 1/50$, a view field of 65×65 µm², and N_m = 100. First, we generate N_m images of single-scattered wave for each incident angle for an object $O({x,y} )= \sqrt 2 \cos ({k_x^\textrm{s}x/2} )$ with $k_x^\textrm{s} = 0.32k\alpha $ and intensity normalization factor $\sqrt 2 $. The spectrum of single-scattered field ${\tilde{E}_\textrm{S}}$ is then calculated by the Fourier transform after applying the pupil mask $P({{{\boldsymbol k}_o}} )$ to introduce finite bandwidth in the experiment. In the case of multiple-scattered waves, we prepared N_m independent map of electric field spectrum ${\tilde{E}_{\textrm{TM}}}$ shown in Eq. (3) by numerically generating the amplitude A following the Rayleigh probability density function and the phase $\mathrm{\Phi }$ uniformly distributed between 0 and 2π. Individual spectrums are normalized by the average intensity of each spectrum. Finally, we obtain the angular spectrum of the total reflected wave ${\tilde{E}_{\textrm{tot}}} = \left( {\sqrt \gamma {{\tilde{E}}_\textrm{S}} + {{\tilde{E}}_{\textrm{TM}}}} \right)/({\gamma {N_\textrm{m}}/2} )\; $ where $\gamma {N_\textrm{m}}/2$ is normalization factor to compare the spectrum with the blue curve in Fig. 2(a).

The black curve in Fig. 2(a) shows a representative incoherent imaging spectrum of this numerical simulation. The baseline noise spectrum is in excellent agreement with the analytic expectation (red curve), and the single scattering signal given by the periodic object structure, which is the peak of the black curve at $k_x^\textrm{s} = 0.32k\alpha $, corresponds well to the blue curve. We performed numerical simulation only with the multiple scattering noise [Fig. 2(b)] to look into the spatial frequency spectrum of the multiple scattering. The blue dots in Fig. 2(b) show the ensemble averaged intensity spectrum of multiple light scattering over 100 different realizations, which fits well with the analytic expectation (red curve) with only single fit parameter, i.e. multiplication factor. We also confirmed that the standard deviation of the multiple scattering spectrum is well fitted to the expectation (green curve).

4. Optical transfer function of CASS microscopy

In CASS microscopy, spatial input-output correlation provides a way to coherently combine electric fields measured at different ${{\boldsymbol k}_{in}}$. Since the single-scattered waves conserve the in-plane momentum when they get reflected from the target, the momentum difference between reflected and incident waves remains the same as the object spectrum regardless of the incident momentum. Therefore, the addition of reflected waves at the momentum difference space can preferentially accumulate single-scattered waves. This is implemented by adding the measured electric field spectra after subtracting the incidence momentum from each measurement. Mathematically, this is equivalent to adding ${\tilde{E}_{tot}}$ after replacing ${{\boldsymbol k}_o}$ with ${{\boldsymbol k}_o} + {{\boldsymbol k}_{in}}$ in each ${\tilde{E}_{\textrm{tot}}}$, i.e.

It is evident that the subtraction of the incident wavevector causes the amplitudes of single-scattered waves to be added in phase.

The intensity spectrum of CASS microscopy is given by the autocorrelation of the synthesized electric field:

For single-scattered waves, the signal amplitude is added coherently such that the field spectrum scales with $N({{{\boldsymbol k}_o}} )$, and its intensity spectrum is the autocorrelation of this field spectrum. Therefore, the single scattering spectrum for the same periodic object considered in the incoherent imaging analysis is given as

Here $({N\ast N} )({{{\boldsymbol k}_o}} )= N_\textrm{m}^2\;({P\ast P} )\ast ({P\ast P} )({{{\boldsymbol k}_o}} )$, indicating that the single scattering intensity is proportional to $N_\textrm{m}^2$. In addition, the bandwidth expands up to $4{k_0}\alpha $, meaning that the spatial resolution of CASS microscopy can be better than that of the incoherent imaging by a factor of two. Blue curve in Fig. 3(a) shows $({N\ast N} )({{{\boldsymbol k}_o}} )/{N_\textrm{m}}$ which is the single scattering spectrum for the CASS microscopy depending on $k_x^\textrm{s}$. The intensity reflectance of the object considered in this derivation is ${|{O({x,y} )} |^2} = 2{\cos ^2}({k_x^\textrm{s}x/2} )$, whose spectrum is given by $\{{\delta ({{k_x} - k_x^\textrm{s},{k_y}} )+ \delta ({{k_x} + k_x^\textrm{s},{k_y}} )+ 2\delta ({{k_x},{k_y}} )} \}/2$. Therefore, the single scattering contribution to the optical transfer function of CASS imaging is obtained as $({N\ast N} )({{k_x},{k_y}} ){I_\textrm{S}}$.

 

Fig. 3. Intensity spectra of CASS microscopy. (a) Blue and red curves are theoretical spectra of single and multiple scattering, respectively, derived in Eq. (8) and the main text thereafter. Black curve shows the spectrum of numerical simulation under the same condition as Fig. 2. (b) Detailed plots of multiple scattering spectra shown in (a). Red and green curves are the spectra of the average intensity and its standard deviation, respectively, derived from the theory. Black curve and blue dots are, respectively, the numerically simulated intensity spectrum of multiple scattering and its ensemble average.

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In the case of multiple scattering [Fig. 3(b)], the number of phasors added for a given ${{\boldsymbol k}_o}$ is determined by the $N({{{\boldsymbol k}_o}} )$ in the electric field. Since the intensity spectrum is the autocorrelation of the electric field, the average intensity spectrum is proportional to $\sqrt {({N\ast N} )({{{\boldsymbol k}_o}} )} $ and its standard deviation follows the same trend. In fact, the spectra of average intensity and its standard deviation are given by $4 \times 0.3\sqrt \pi \frac{1}{{{k_0}\alpha \sqrt B }}{I_\textrm{M}}\sqrt {({N\ast N} )({{{\boldsymbol k}_o}} )} $ [red curve in Fig. 3(b)] and $4 \times 0.15\sqrt \pi \frac{1}{{{k_0}\alpha \sqrt B }}{I_\textrm{M}}\sqrt {({N\ast N} )({{{\boldsymbol k}_o}} )} $ [green curve in Fig. 3(b)], where once again the additional factors arise due to the autocorrelation of the band-limited random phasors. Considering the contribution of multiple scattering noise to each intensity spectrum, the effective optical transfer function of the CASS imaging in the presence of multiple scattering is given as $OT{F_\textrm{E}}({{{\boldsymbol k}_o}} )= ({N\ast N} )({{{\boldsymbol k}_o}} ){I_\textrm{S}} + 1.2\sqrt \pi \frac{1}{{{k_0}\alpha \sqrt B }}{I_\textrm{M}}\sqrt {({N\ast N} )({{{\boldsymbol k}_o}} )} $.

We simulated the multiple-scattered waves in the same way as the incoherent imaging case and plotted the spectrum of CASS microscopy in the presence of both single- and multiple-scattered waves [black curve in Fig. 3(a)]. The peak intensity is close to the single scattering signal predicted by the analytic theory. The discrepancy is mainly due to the cross term between single- and multiple-scattered waves. The intensity spectrum of the multiple-scattered waves is plotted in Fig. 3(b) (black curve) and its ensemble average is shown as blue dots. The results are in excellent agreement with the analytic prediction (red curve) with the multiplication factor.

5. Comparison between incoherent imaging and CASS microscopy

In sections 3 and 4, we analytically derived the optical transfer functions of incoherent addition and CASS microscopy, respectively, in the presence of multiple-scattered waves and confirmed their validity using numerical simulations. From these derivations, we are now ready to compare the SNR depending on the spatial frequency spectrum of the target object. We define the SNR by the ratio between single scattering intensity and the standard deviation of the multiple scattering noise at each spatial frequency. Following this definition, the SNR of incoherent imaging is given as

 

Fig. 4. (a) A comparison of SNR between incoherent imaging (blue curve) and CASS imaging (red curve) as a function of spatial frequency. $\gamma = 1/50$, a view field of B=65×65 µm², and N_m = 100. The threshold SNR, SNR_th, required for the resolving of the structures was considered 10 (black line). (b) Maximum resolvable spatial frequency as a function of ${I_{\textrm{TM}}}/{I_\textrm{S}}$ for incoherent imaging and CASS imaging.

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The resolving power is determined by the maximum spatial frequency ${k_{\textrm{max}}}$ at which the SNR is above a certain threshold value, SNR_th. Theoretically, any signal greater than SNR_th=1 is resolvable, but here we chose SNR_th = 10 as a safe criterion (black curve). In Fig. 4(b), we plotted achievable spatial resolution ` in unit of diffraction-limited spatial resolution ${\mathrm{\Delta }_0}$ with respect to the imaging depth $\frac{L}{{{l_\gamma }}} = \frac{1}{2}\textrm{ln}\frac{{{I_{\textrm{TM}}}}}{{{I_\textrm{S}}}} ={-} \frac{1}{2}\textrm{ln}\; \gamma $. With the increase of depth, ${\mathrm{\Delta }_{\textrm{res}}}$ of incoherent imaging increases steeply first, while that of CASS microscopy stays small until it experiences a steep increase at a much thicker depth.

6. Experimental results

For experimental demonstration of the spatial frequency-dependent SNR for both CASS microscopy and conventional incoherent imaging, we implemented a reflectance-mode CASS microscopy system with Mach-Zehnder configuration (Fig. 5) [16]. A supercontinuum laser (center wavelength: 675 nm, band width: 15 nm, coherence length: 18$\; \mu $m) was used as a low coherence light source. The output beam from the laser was sent to a galvanometer scanning mirror (GM) for scanning the angle of illumination, and then divided into sample and reference beams by a polarizing beam splitter (PBS). The GM was scanned with 4500 different incident angles uniformly covering the full numerical aperture of the objective lens (OL, 60x, 1.0 NA). For each incidence angle, the waves reflected from the sample were captured by the same OL and relayed to the camera. The reference beam transmitted through the PBS was reflected from the reference mirror (RM), which scanned reference pathlength for selecting the temporal gating. It was then sent to the diffraction grating (DG), and its first order diffraction was selected by the iris diaphragm (ID) and combined with the sample beam to create an interferogram. Using the Hilbert transform, we obtained both amplitude and phase maps of the reflected waves. For the test sample, we used a target similar to Siemens Star target for investigating various spatial frequency components. On the top of the target object, we placed a scattering layer composed of randomly dispersed 1 µm-diameter ZnO particles in Polydimethylsiloxane (PDMS). The scattering mean free path and the thickness of the scattering layer were l_s = 48.5 µm and 4.5 l_s, respectively, at a wavelength of 675 nm.

 

Fig. 5. Experimental schematic of the reflection-mode CASS microscopy system. PBS: polarized beam splitter, OL: objective lens (Nikon, NA:1, x60), GM: galvanometer mirror, DG: diffraction grating, ID: the iris diaphragm, WP: wave plate. RM: reference mirror. The blue ray shows the path of the reflected wave from a point on the object plane to the camera.

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For matching the temporal pulse fronts of the reference and the sample beam, the incident angles of both beams are scanned synchronously. As a result, we acquired the input wavefront corrected output field maps, ${\tilde{E}_\textrm{S}}({\mathrm{\Delta }{\boldsymbol k},{\tau_\textrm{S}};{{\boldsymbol k}_{in}}} )$, for a set of different input angles at a fixed flight time (${\tau _\textrm{S}}$). First, we processed the obtained ${\tilde{E}_\textrm{S}}({\mathrm{\Delta }{\boldsymbol k},{\tau_\textrm{S}};{{\boldsymbol k}_{in}}} )$ by incoherent addition (Fig. 6(a)] and the CASS microscopy [Fig. 6(b)] for comparing their SNR for $\gamma $ = 0.21. In the case of incoherent addition, the periodic pattern between the two white dashed circles in Fig. 6(b) was hardly resolved while it was clearly resolved in the case of CASS microscopy. This confirms that the SNR of the CASS microscopy is higher than that of incoherent addition. Second, we compared the SNR of the CASS images with different values of $\gamma $. For finely tuning $\gamma $, we obtained images while varying the illumination area. As the illumination area expands, only the intensity of the multiple-scattered waves increases, whereas the intensity of the single-scattered waves is unchanged. Figures 6(b) and 6(c) show the CASS images obtained with the illumination area of 27.5 × 27.5 µm² and 110 × 110 µm², respectively, and the measured $\gamma $ was reduced from 0.21 to 0.074. The periodic pattern between the two white dashed circles was hardly resolved when $\gamma $=0.074 while it was clearly resolved in the case of $\gamma $=0.21. This shows that the achievable spatial resolution decreases with the reduction of $\gamma $.

 

Fig. 6. (a) Incoherent image and (b) CASS image for the case of $\gamma $ = 0.21. (c) CASS image when $\gamma $ = 0.074. Scale bar, 3 µm. (d) The SNR of incoherent addition (red) and that of CASS microscopy (blue) depending on spatial frequency for $\gamma $ = 0.21. (e) The SNR of CASS microscopy with $\gamma $=0.074 (red) and $\gamma $=0.21 (blue) depending on spatial frequency.

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For quantitative analysis, we obtained the SNR for circles with different radii in the Siemens Star-like target. The grating patterns in the circles with different radii correspond to different spatial frequencies (${k_\textrm{R}}$). One-dimensional profile was extracted from each radial profile, and the signal was determined by the peak intensity in the Fourier transform of the obtained profile. Note that we extracted the one-dimensional profiles of different radii to have the same length for normalization purpose. Therefore, the peaks of the different spectrums appear at the same value of wavevector in this analysis. A circle without a grating pattern in the target was used for determining the noise level. We obtained the spectrum of this circle’s radial profile and measured the standard deviation around the same wavevector of the periodic structures. Figure 6(d) compares the obtained SNR for incoherent image in Fig. 6(a) (red dots) and CASS image in Fig. 6(b) (blue dots) depending on spatial frequency. In case of incoherent addition, the SNR was lower than the cut-off SNR around the spatial frequency ($0.8 < {k_R}/k\alpha < 1$) corresponding to the periodic pattern in the second smallest circle, whereas the SNR of CASS microscopy was higher than the cut-off SNR. Here, the maximum achievable spatial frequencies were approximately 0.6 $\textrm{k}\alpha $ and 1.0 $\textrm{k}\alpha $ for incoherent addition and CASS microscopy, respectively, when cut-off SNR is 10. Figure 6(e) compares the SNR of CASS microscopy for different $\gamma $. For $\gamma $=0.074 (red dots), the SNR was lower than the cut-off SNR around the spatial frequency of $0.8 < {k_R}/k\alpha < 1$. On the contrary, the SNR for $\gamma $=0.21 was higher than the cut-off SNR in the same spatial frequency range. The achievable maximum spatial frequencies were 0.6 $\textrm{k}\alpha $ and 1.0 $k\alpha $ for $\gamma $=0.074 and $\gamma $=0.21, respectively. These experimental results confirmed that the spatial resolving power is gradually lost as the higher spatial frequency components are lost first with the increase of multiple scattering noise.

7. Summary

In this study, we investigated the effect of multiple scattering noise to the optical transfer function of CASS microscopy. With the theoretical derivation and numerical simulations, we obtained signal to noise ratio at each spatial frequency for a given single-to-multiple scattering intensity ratio set by the target depth. This analysis showed that the SNR of CASS microscopy is higher than that of incoherent addition over the entire spectral range. By finding the maximum spatial frequency at which SNR is larger than certain threshold value, we obtained the quantitative assessment of the achievable spatial resolution for a given target depth. This analysis made it clear that the spatial resolving power is lowered with the increase of target depth due to the decrease of SNR with respect to the increase of spatial frequency. We performed an experiment with a resolution target having various spatial frequency components and confirmed the spatial frequency-dependent signal to noise ratio for both CASS microscopy and conventional incoherent imaging. Our study provides a theoretical framework through which the spatial resolution attainable by the different imaging systems can be determined for the given initial condition of the relative intensity ratio between single- and multiple-scattered waves.