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Abstract
The optical memory effect is an interesting phenomenon exploited for deep-tissue optical imaging. Besides the widely studied memory effects in the spatial domain to accelerate point scanning speed, the spectral memory effect is also important in multispectral wavefront shaping. Although being theoretically analyzed for decades, quantitative studies of spectral memory effect on a variety of scattering media including biological tissue were rarely reported. In practice, quantifying the range of the spectral memory effect is essential in efficiently shaping broadband light, as it determines the optimum spectral resolution in realizing spatiotemporal focus through scattering media. In this work, we analyze the spectral memory effect based on a diffusion model. An explicit analytical expression involves the illumination wavelength, the diffusion constant, and the sample thickness is derived, which is consistent with the one in the literature. We experimentally quantified the range of spectral correlation for two types of biological tissue, tissue-mimicking phantoms with different concentrations, and diffusers. Specifically, for tissue-mimicking phantoms with calibrated scattering parameters, we show that a correction factor of more than 20 should be inserted, indicating that the range of spectral correlation is much larger than one would expect. This finding is particularly beneficial to multispectral wavefront shaping, as stringent requirements on the spectral resolution could be alleviated by at least one order of magnitude.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Imaging through and inside scattering media has been a long-standing challenge in optics. This challenge originates from the fact that the formation of high-resolution images usually relies on ballistic photons, which decay exponentially with the increased imaging depth. A recent paradigm shows that by exploiting the correlations of few-scattered photons, i.e., the angular memory effect, high-resolution images of objects hidden behind scattering media can be computationally reconstructed [1–3]. The angular memory effect, which was first predicted by Feng et al. in 1988, describes the phenomenon that titling the incident wavefront within a certain angular range results in a corresponding translational shift for the far-field speckle patterns [4]. Besides assisting in computational image recovery, this angular memory effect is directly applicable to increase point scanning speed in microscopy [5,6]. Attracted by this capability, several different types of spatial memory effects, including the rotational memory effect of multimode fibers [7] and the translational memory effect of highly anisotropic scattering media [8], were explored and reported to open up new possibilities in biomedical imaging. Later, numerous efforts have been devoted to understanding the physics of these scattering correlations and extending the operational range [9,10]. Except for studying the spatial memory effect, the desire to overcome optical scattering for broadband light urges investigations on the spectral correlation as well. Such a desire arises from the increasing demand for deep-tissue nonlinear microscopy. By relating the Fourier transformation, it was shown that the temporal speckle width Δf is inversely proportional to the time that photons take to traverse the scattering medium [11]. Thus, Δf is known to have a width of approximately cD/L2, where c, D, and L are the speed of light inside, the diffusion constant of, and the thickness of the scattering medium, respectively [12]. The spectral correlation was also studied for coherent enhanced absorption, which was found to be significantly extended [13].
In the realm of wavefront shaping, the spectral correlation, i.e., the spectral memory effect, has also been widely investigated. For example, the intensity of the achieved focus through scattering media decays as the detuning wavelength increases, following the exact trend of the spectral correlation function [12]. Moreover, an axial spatio-spectral coupling was observed when focusing broadband light through the scattering medium [14]. Such an effect, which was named chromato-axial memory effect, was theoretically modeled and experimentally validated with a large spectral width for a forward-scattering slab [15]. A combined effect of both angular and chromato-axial memory effects was also reported to realize three-dimensional manipulation of broadband light [16]. Quantifying the range of the spectral memory effect is essential in efficiently shaping broadband light. Figure 1(a) illustrates the scattering process for a femtosecond pulse with a broadband spectrum. After scattering, a time-varying speckle pattern with low contrast is formed. Mathematically, this speckle pattern can be decomposed as the superposition of a series of high-contrast speckle patterns that belong to different wavelengths. In general, adjacent wavelengths that exhibiting similar speckle patterns should have large correlations, indicating similar scattering processes experienced. To realize spatiotemporal focusing through scattering media, multispectral wavefront shaping that modulates phase values for different wavelengths was employed, as shown in Fig. 1(b). In practice, the optimum spectral resolution during the modulation process, i.e., $\Delta \mathrm{\lambda } = |{{\mathrm{\lambda }_2}^{\prime} - \textrm{ }{\mathrm{\lambda }_1}^{\prime}} |$, is determined by the range of the spectral memory effect [17,18]. This choice minimizes the redundancy of the modulated information, guaranteeing the efficient usage of the spectrum. As a result, a spatiotemporal focus through the scattering medium can be formed. However, to date, a detailed quantitative analysis on the spectral memory effect for scattering media such as biological tissue was rarely reported experimentally. Since there exists a huge discrepancy between the theoretically predicted value and the experimentally measured data for angular memory effect [19], one would naturally ask whether such a situation also exists for spectral memory effect as well. To answer this question, an experimental analysis of the spectral memory effect is necessary to serve as future guidance for choosing optimum spectral resolution in multispectral wavefront shaping experiments to realize spatiotemporal focusing.
Fig. 1. Illustration of the scattering process for broadband light. (a) After scattering, a femtosecond pulse results in time-varying speckles with low contrast, which is the intensity superposition of high-contrast speckles of various wavelengths. (b) Spatiotemporal focusing by modulating phase values for light with different wavelengths.
2. Diffusion model of the spectral memory effect
We start by analyzing the spectral memory effect with a diffusion model, which approximates the incident light as an isotropic point source located one transport mean free path below the surface [20]. As shown in Fig. 2(a), photons can take various optical paths to reach the target position. Using the diffusion theory, at time t, the laser fluence rate $\Phi $ at a distance r from the source can be estimated as
Fig. 2. Illustration of the physical model based on the diffusion theory. (a) Schematic of different photon paths inside a scattering medium. (b) Normalized photon fluence rate as a function of time delay.
It is worth noting that in previous studies, the exact value of b was rarely discussed for different types of scattering media. Without this correction factor, Eq. (4) is consistent with the expression reported in the literature ($\Delta \mathrm{\lambda }\sim {\mathrm{\lambda }^2}\Delta f/c = {\mathrm{\lambda }^2}D/{L^2}$) [12].
3. Experimental results of the spectral memory effect
Having theoretically analyzed the spectral memory effect based on the diffusion model, we then investigated this property through experiments and tried to quantify correction factor b. A schematic figure of the experimental setup is shown in Fig. 3. A homemade single-frequency fiber laser that supports a tunable wavelength ranging from 1020 nm to 1075 nm was used as the light source. The average output power was measured to be around 140 mW. The frequency of the output light was controlled by setting the voltage of a programmable power supply (Rigol DP832A). A fiber collimator (Thorlabs F220FC-1064) was subsequently used to direct light to the scattering medium. The output beam diameter was about 2 mm right after the collimator but was expanded to about 5 mm at the sample surface. After scattering, the light was collected by a lens (Thorlabs AC254-050-B) and measured with a camera (FLIR GS3-U3-23S6M-C). Both the programmable power supply and the camera were synchronized by a computer. Customized control codes in Matlab were employed to synchronize these devices for automatic wavelength sweeping and speckles capturing. The typical scanning time for a range of 5 nm with a 0.5-nm resolution is about 1 second. Most of the samples we examined are soft, which suffers from stability issues and exhibits considerable decorrelation when becoming thick. To avoid this effect from contaminating our experimental results, we always kept the scanning time to be less than 1/3 of the speckle correlation time of the sample, by varying the scanning range and resolution.
We first examined the spectral memory effect for beef tendon, which can be conveniently acquired and is biology relevant. Figures 4(a) and 4(b) show the photos of the front view and side views of the 8-mm thick beef tendon sample we made. These samples were cut using blades and then sandwiched using two pieces of glass slides. Glass spaces with fixed thickness were employed at two sides of the glass slides to control the thicknesses of the tissue. The incident light is perpendicular to the surface of the glass slides. For visualization purposes, Fig. 4(a) also shows a laser viewing card that was directly illuminated by the incident light, showing an excited region of about 5 mm. Starting from the wavelength of 1060 nm, we continuously swept the output wavelength of the laser and measured the corresponding speckle patterns. Figure 4(c) plots the computed correlation coefficient as a function of the detuning wavelength Δλ for the 8-mm-thick beef tendon. As shown in the figure, the correlation coefficient drops quickly to 1/e for a detuning wavelength of only 0.1 nm. The evolution of speckle patterns is also shown in Figs. 4(d), 4(e), and 4(f), corresponding to the wavelength of 1060.00 nm, 1060.20 nm, and 1060.40 nm, respectively. By scrutinizing the details enclosed by a red circle, the bright spot gradually changes its shape and finally disappears. Theoretically, homothetic dilation of speckles was predicted and it was suggested to compensate for this effect by choosing a virtual image plane located at one-third of the sample thickness [15]. In practice, we neglected this effect and simply positioned the lens whenever a uniformly distributed speckle pattern was observed. Empirically, the relative distance between the rear surface of the sample and the lens was found to be around the focal length of the collecting lens. After repeating the above procedures for beef tendon samples with different thicknesses, we plot Δλ as a function of the tissue thickness L, as shown in Fig. 4(g). Here, the largest thickness was set as 10 mm, due to the consideration of the strength of the transmitted light and the stability issue. A fitting curve that is inversely proportional to the thickness L was also provided (Δλ=8.30/L2), exhibiting good agreement with the experimental data. However, the behaviors of beef tendon samples cannot be well described by Eq. (4), as it exhibits considerable absorption. Recalling the distribution displayed in Fig. 2(b), considerable absorption effectively shrinks the width of the distribution, which extends the correlation range. In any case, we did not retrieve the correction factor based on Eq. (4) but choose to provide the fitting coefficient of experimental data here.
Fig. 4. Characterizing spectral memory effect for beef tendon. (a) and (b) Photos of the front view and side of beef tendon with a thickness of 8 mm, respectively. (c) A typical demonstration of the correlation coefficient as a function of detuning wavelength for 8-mm-thick beef tendon. (d)-(f) Camera measured speckle patterns that correspond to the wavelengths of 1060.00 nm, 1060.20 nm, and 1060.40 nm, respectively, showing the evolution of speckles with wavelengths being continuously tuned. (g) Thickness-dependent spectral memory effect for beef tendon. The error bars were generated using the standard deviations of three independent realizations.
We also examined the spectral memory effect for chicken breast, which is another type of frequently used biological tissue during experiments. Figures 5(a) and 5(b) show the photos of the front view and side views of the 10-mm thick chicken breast sample. Figure 5(c) plots the computed correlation coefficient as a function of the detuning wavelength Δλ for 10-mm-thick chicken breast. The correlation coefficient drops quickly to 1/e for a detuning wavelength of about 0.25 nm. The evolution of speckle patterns is also shown in Figs. 5(d), 5(e), and 5(f), corresponding to the wavelength of 1060.00 nm, 1060.25 nm, and 1060.50 nm, respectively. Similarly, Fig. 5(g) plots Δλ as a function of the tissue thickness L. A fitting curve that is inversely proportional to the thickness L was also provided (Δλ=34.34/L2). Notably, the spectral correlation range of chicken breast is much larger than that of beef tendon. Since chicken breast also exhibits strong absorption, we did not retrieve the correction factor either. In the following experiments, we switched to tissue-mimicking phantoms with calibrated scattering properties and negligible absorption.
Fig. 5. Characterizing spectral memory effect for chicken breast. (a) and (b) Photos of the front view and side of chicken breast with a thickness of 10 mm, respectively. (c) A typical demonstration of the correlation coefficient as a function of detuning wavelength for 10-mm-thick chicken breast. (d)-(f) Camera measured speckle patterns that correspond to the wavelengths of 1060.00 nm, 1060.25 nm, and 1060.50 nm, respectively, showing the evolution of speckles with wavelengths being continuously tuned. (g) Thickness-dependent spectral memory effect for chicken breast. The error bars were generated using the standard deviations of three independent realizations.
Figure 6(a) shows a photo of some tissue-mimicking phantoms we made, which look opaque. These phantoms were made by mixing gelatin, water, and intralipid with procedures detailed in Ref. [22]. Compared to the scattering coefficient, the mixture of these three chemical substances exhibits a negligible absorption coefficient around 1060 nm. By controlling the concentrations of intralipid, phantoms with different scattering parameters can be prepared. We start by considering phantoms with an intralipid concentration of 2%. Figure 6(b) plots the correlation coefficient as a function of the detuning wavelength Δλ for a piece of 4-mm-thick tissue-mimicking phantom. Specifically, the correlation coefficient drops to 1/e when the detuning wavelength reaches 0.83 nm. The evolution of speckle patterns is also shown in Figs. 6(c), (d), and (e), corresponding to the wavelength of 1061.25 nm, 1061.50 nm, and 1061.75 nm, respectively. A red circle was also used to highlight how a bright speckle evolves. Spectral memory effect as a function of sample thickness was also investigated, as shown in Fig. 6(f). A fitting curve of Δλ = 13.81/L2 agrees with the experimental data quite well.
Fig. 6. Characterizing spectral memory effect for tissue-mimicking phantoms with an intralipid concentration of 2%. (a) A photo of several tissue-mimicking phantoms with various thicknesses ranging from 4 mm to 10 mm. (b) A typical demonstration of the correlation coefficient as a function of detuning wavelength for 4-mm-thick tissue-mimicking phantom mixed. (c)-(e) Camera measured speckle patterns that correspond to the wavelengths of 1061.25 nm, 1061.50 nm, and 1061.75 nm, respectively, showing the evolution of speckles with wavelengths being continuously tuned. (g) Thickness-dependent spectral memory effect for tissue-mimicking phantoms. The error bars were generated using the standard deviations of three independent realizations.
To determine the reduced scattering coefficient, the method of oblique incidence reflectometry was employed, which is schematically illustrated in Fig. 7(a). Light illuminates the sample with an angle of θ, and the reflected light was measured using the same camera through the same lens. Figure 7(b) shows a typical camera-captured reflected pattern, exhibiting a distorted circle due to the penetrations into the sample. The corresponding intensity distribution along the dashed line is also plotted in Fig. 7(c). A horizontal displacement Δx is determined as the difference between the horizontal positions of the peak value and the centers of the symmetric reflectance profile. Then, the reduced scattering coefficient can be calculated as
where ns and ${\mu _a}$ are the refractive index and absorption coefficient of the phantom, which were set to be 1.33 and 0, respectively. As a result, the reduced scattering coefficient was determined to be 7.24 cm-1. According to Eq. (4), the correction factor b is estimated to be 26.71.Fig. 7. Characterizing reduced scattering coefficients for tissue-mimicking phantoms. (a) Schematics of the experimental setup for oblique incidence reflectometry. (b) A typical camera-captured reflected pattern, exhibiting a distorted circle. (c) The one-dimensional plot of the intensity distribution along the blue dashed line in (b). A horizontal displacement Δx is determined as the difference between the horizontal positions of the peak value (black solid line) and the centers of the symmetric reflectance profile (red dashed line).
We further investigated the spectral memory effect for tissue-mimicking phantoms with intralipid concentrations of 3% and 4%. In general, the higher the concentration, the larger the scattering strength. Figures 8(a) and 8(b) illustrate Δλ as a function of thickness for intralipid concentrations of 3% and 4%, respectively. Fitting curves that are inversely proportional to the square of thickness were also generated and plotted. The fitting parameter is numerically found to decrease with the increased intralipid concentration. For these phantoms, the reduced scattering coefficients were measured to be 11.75 cm-1 (3%) and 21.16 cm-1 (4%). Thus, the correction factor b is estimated as 24.95 (3%) and 21.31 (4%). Notably, b obtained for tissue-mimicking phantoms with different intralipid concentrations are close, showing that the actual correlation range of this type of scattering medium is more than one order of magnitude larger than one would generally think. This observation also indicates that a much coarse spectral resolution can be chosen for multispectral wavefront shaping.
Fig. 8. Characterizing spectral memory effect for tissue-mimicking phantoms with intralipid concentrations of (a) 3% and (b) 4%. The error bars were generated using the standard deviations of three independent realizations.
As a final demonstration, we examined the spectral memory effect for diffusers (Thorlabs DG10-120), which are commonly used as scattering media in a variety of wavefront shaping experiments. As shown in Fig. 9(a), for two stacked diffusers, it takes about 2.88 nm for the correlation to drop to 1/e. This observation agrees with the conventional cognition that diffusers are generally not that scattering. Notably, Fig. 9(b) illustrates the evolution of Δλ as a function of the number of the stacked diffusers adopted. Although diffusers are difficult to be modeled with specific scattering coefficient and thickness, this observation is consistent with the trend that Δλ decreases inversely proportional to the square of optical thickness, as predicted in Eq. (4).
Fig. 9. (a) A typical correlation coefficient as a function of detuning wavelength for two stacked diffusers. (b) Characterizing the thickness-dependent spectral memory effect for diffusers. The error bars were generated using the standard deviations of three independent realizations.
4. Discussion and conclusion
We characterized the range of spectral correlation for different types of scattering media, including both biological samples and tissue-mimicking phantoms. Table 1 summarizes the results obtained from different types of scattering media. For biological tissue, only the fitting coefficients are provided, due to the considerable absorption. For tissue-mimicking phantoms, the correction factors b obtained experimentally are much larger than 1, showing that the actual correlation ranges are larger than we generally think. This finding is similar to the situation reported in a recent study, where the experimentally quantified range of angular memory is much wider than their corresponding theoretical values [19]. In Ref. [19], it was conjectured that the near-diagonal structure of the transmission matrix of a highly anisotropic medium in k-space [8] also contributed to the enhanced angular memory effect. We expect that a similar effect also exists for the spectral memory effect of forward-scattering media as well. Recently, a large spectral width of the chromate-axial memory effect has been reported through a forward scattering slab [15]. Thus, Eq. (4) may not fully describe the spectral memory effect. In other words, other scattering parameters, such as anisotropy g and scattering coefficient μs, may explicitly exist in the analytical expression. Future works should focus on physical modeling and rigorous finite-difference-time-domain simulations to solve this issue. Moreover, our experimental results also confirm that spectral correlation is inversely proportional to the thickness square, which is in agreement with Eq. (4) and many theoretical models predicted in the literature [11,12].
Table 1. Experimental results for spectral memory effect of scattering media.
In conclusion, we described the spectral memory effect based on a diffusion model and experimentally quantified this effect for several types of scattering media. Experimental results show that a correction factor that differs among different types of scattering media should be inserted, indicating that the actual correlation range should be much larger, i.e., more than 20 times larger, than one would originally expect from the theory. This finding is particularly beneficial to multispectral wavefront shaping by alleviating stringent requirements on the choice of spectral resolution. Before a quantitatively correct formula is derived, it is always suggested to experimentally quantify the correlation range to determine the optimum spectral resolution for the target scattering medium in advance.
Funding
National Key Research and Development Program of China (2018YFB1802300); National Natural Science Foundation of China (12004446); Fundamental and Applied Basic Research Project of Guangzhou (202102020603); Shenzhen-Hong Kong Cooperation Zone for Technology and Innovation (HZQB-KCZYB-2020082).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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