## Abstract

The optical memory effect is an interesting phenomenon that has attracted considerable attention in recent decades. Here, we present a new physical picture of the optical memory effect, in which the memory effect and the conventional spatial shift invariance are united. Based on this picture we depict the role of thickness, scattering times, and anisotropy factor and derive equations to calculate the ranges of the angular memory effect (AME) of different scattering components (ballistic light, singly scattered, doubly scattered, etc.), and hence a more accurate equation for the real AME ranges of volumetric turbid media. A conventional random phase mask model is modified according to the new picture. The self-consistency of the simulation model and its agreement with the experiment demonstrate the rationality of the model and the physical picture, which provide powerful tools for more sophisticated studies of the memory-effect-related phenomena and wavefront-sensitive techniques, such as wavefront shaping, optical phase conjugation, and optical trapping in/through scattering media.

© 2019 Chinese Laser Press

## 1. INTRODUCTION

In 1988 Feng *et al.* derived and proposed the angular memory effect (AME), also called the tilt memory effect in a waveguide geometry [1], and later observed the shift-invariant laser speckle through a ground glass disk [2]. In their theory, the AME range is inversely proportional to the thickness of the medium without a consideration of optical parameters of scattering media, such as scattering coefficient and anisotropy factor. They also pointed out that the presence of the memory effect in the multiple-scattering situation is less apparent intuitively, but did not give further explanation. Recently, a method of imaging through the speckle correlation, i.e., speckle autocorrelation imaging [3], has inspired many interesting studies based on the AME [4–13]. However, the field of view is always limited by the small range of the AME, typically $\sim 400$ millidegrees for 1 mm thick chicken breast tissue [13]. Yang *et al.* found that the measured AME ranges of frozen chicken breast tissues were larger than theoretical predictions [14]. Soon afterward, Schott *et al.* [13] found that a large anisotropy factor $g$ could enhance the AME range by more than 1 order of magnitude compared to $g=0$. In 2015, based on the macroscopic characteristics of the scattering transmission matrices of anisotropically scattering media, Judkewitz *et al.* proposed a new type of memory effect, i.e., the shift memory effect from the matrix correlations, and explained its relationship with the AME [15]. Two years later, Osnabrugge *et al.* [16] predicted a more general class of combined shift and tilt correlations in scattering media under the paraxial approximation and demonstrated experimentally with an optical thickness up to 1.76 (multiply scattered light is not dominant) and $g=0.98$. However, the equations relating the AME range $\mathrm{\Delta}\theta $ to the thickness $d$ are still not far from the original $\mathrm{\Delta}\theta \approx \lambda /\pi d$. How the thickness, scattering times (the number of scattering events a photon experienced, or equivalently a product of the scattering coefficient and the photon path length), and anisotropy factor come into play remains unclear.

In this paper, we present a new physical picture of the memory effect that can unify the memory effect and shift invariance. The roles of the thickness, scattering times, anisotropy factor, and so on in the memory effect are interpreted in a more physical and intuitive manner. The AME range of each scattering component is analyzed and a new equation for estimating the AME range is derived. Experimental results and simulations based on a modified multiple-phase-mask model are presented to validate the physical picture.

## 2. PRINCIPLE

If the diameter of a laser beam illuminating an aperture is smaller than the aperture, the beam propagates as if no aperture exists. The tilt in the incident beam at an angle $\theta $ leads to the same amount of tilt in the output beam [see Fig. 1(a)]. Under the paraxial condition, the distortion in the beam spot on the observation plane is trivial, and the aperture can be considered as a shift-invariant system. Once we introduce a random phase distribution to the aperture—also called a random phase mask—we see speckles on the observation plane instead of a bright spot. Tilting the incident laser beam causes the entire speckle pattern to shift, as shown in Fig. 1(b), which is referred to as the memory effect—more specifically, the AME. While a single spot is considered in the shift invariance [Fig. 1(a)], the entire speckle pattern is used as the reference for speckle correlation in the memory effect [Fig. 1(b)].

For both systems, the light fields on the observation planes can be written as

When inserting another aperture/mask into the optical path of System 2 to obtain Systems 3 and 4, as shown in Fig. 2, the field on the observation plane becomes

For System 3, it is obviously a shift-invariant system; its shift-invariant range is the same as a system consisting of two unmasked apertures. For System 4, since Mask 2 has internal structures, a shift in the input wavefront on Mask 2 will result in a different output, breaking the shift invariance. For better understanding, we take the Fourier transform of Eq. (4) and obtain

If Mask 1 is just an unmasked aperture, will the memory effect remain? If ${A}_{1}({f}_{x},{f}_{y})=\delta ({f}_{x},{f}_{y})$, Eq. (5) can be simplified as

The intensity correlation function of two speckle patterns can be expressed as

where $\u27e8\cdots \u27e9$ denotes the ensemble average and $I(x,y)=U(x,y){U}^{*}(x,y)$. For a practical random phase Mask 2—typically a ground glass disk—its fluctuation can be modelled by a complex circular Gaussian random process with zero mean [18]. Then, the field correlation can be expressed asIf we define $\mathrm{\Delta}{x}_{2}={x}_{2}-{x}_{2}^{\prime}$ and $\mathrm{\Delta}{y}_{2}={y}_{2}-{y}_{2}^{\prime}$, Eq. (10) becomes

Here, we ask ourselves the following questions. How do we connect the phase mask model to a real scattering medium with some thickness? If the space between Planes 1 and 2 is filled with scattering medium, can we concentrate all the scattering effect onto Plane 2 and replace it with an equivalent phase mask?

We know that each occurrence of scattering has a phase function corresponding to an angular distribution of the intensity of the scattered light. A commonly used approximation is the Henyey–Greenstein phase function [20]:

Now, we can calculate the overall angular distribution of the scattered light intensity as

After performing a transformation from ${f}_{x}-{f}_{y}$ coordinates to $\theta -\phi $ coordinates, where ${f}_{x}=\mathrm{sin}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\phi /\lambda $ and ${f}_{y}=\mathrm{sin}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\phi /\lambda $ are the spatial frequencies along the $x$ and $y$ axes, respectively, Eq. (13) becomesIf $d$ is equal to the mean free path (MFP), i.e., ${\mu}_{s}d=1$, Eq. (19) becomes

For other thicknesses, we can always fit the $\sqrt{2}(n-1)\sigma /\kappa $ for different scattering components and calculate the corresponding AME ranges. As scattering times increase, the corresponding AME range decreases. The entire AME range is a weighted sum of all scattering components.

## 3. SIMULATION AND EXPERIMENTAL RESULTS

From the above derivations, it is clear that the DC component $\delta ({f}_{x},{f}_{y})$, i.e., the ballistic light, has an important contribution to the memory effect. On the basis of the physical picture, we design a new multiple-phase-mask model, which models a thick scattering medium as multiple phase masks evenly separated (optional) in sequence. The product of the interval between adjacent masks and the number of phase masks equals the medium thickness. In the conventional multiple-phase-mask model [13], several successive evenly spaced phase masks are applied, and light scattering is assumed to occur only at the masks. A single mask represents a single scattering event; therefore, the spatial frequency distribution of the mask is determined by only the phase function, and the distance between two adjacent masks is set to be the MFP. However, the memory effect of the conventional model is much smaller than that of a real scattering medium. Now, we know that the main problem of this model is the exclusion of ballistic light. By using Eq. (19) to generate a phase mask (see Appendix B for the workflow), we integrate more ballistic light into the simulation (see Appendix C for a comparison of the new and conventional phase mask models), and the simulated memory effect curve agrees well with the experimental results, as shown in Fig. 3. The angular distributions of scattered light intensity calculated using Eq. (17) are compared with the standard Monte Carlo (MC) simulation, in which scattered photons are classified according to their directions instead of positions. The results agree well, which means that Eq. (17) is an accurate description of the angular distribution of scattered light. The simulation in Fig. 3(c) shows that less scattered light has a larger AME range. A direct way to expand the AME range is to select less-scattered photons.

In the experiment, the turbid sample is made of silicon microspheres, porcine gelatin, and distilled water with a scattering coefficient of $10\text{\hspace{0.17em}}{\mathrm{mm}}^{-1}$ and a thickness of 1 mm. The microspheres have an average diameter of 2.5 μm and a refractive index of 1.45, and the refractive index of gelatin is 1.33. According to the Mie theory, the anisotropy factor $g=0.98$. The AME curve was measured by scanning a time-reversed focus (see Appendix D for a schematic figure of the principle) [8,17,21].

## 4. DISCUSSION AND CONCLUSION

The relative shift between the wavefront and the scattering medium is the reason for the break of the AME. When the correlation lengths of the wavefront and equivalent phase mask are smaller, the AME range is smaller. The translation memory effect, i.e., the shift memory effect, is also influenced by the relative shift between the wavefront and the cross section in the medium. Essentially, they are the same. With the aid of the new picture, we can explain the phenomena of different AME ranges for turbid media with the same physical thickness or optical thickness. For a thick turbid medium, the AME range of multiply scattered light is small; the main contribution to a measurable AME range is the ballistic and singly scattered components. To expand the AME range, we can select more ballistic and singly scattered light in detection. Compared to the conventional multiple phase mask models, the new one is closer to simulate coherent wavefront propagation in scattering media.

In summary, we present a new physical picture for the optical memory effect, whose validity is not limited by the thickness of the medium. In this picture, the memory effect and shift invariance are united with the intensity correlation. We depict the role of thickness, scattering times, and anisotropy factor, analyze the AME ranges of different scattering components, and derive a new equation to estimate the AME range, which is affected by the characteristics of the random phase masks and the distance. In other words, the AME range of a thick turbid medium is not determined just by its thickness but also by the scattering coefficient and anisotropy factor. Based on the new physical picture, we modify the traditional random phase mask model to a more physical one. The new picture and more flexible new model provide powerful tools for further study of phenomena based on the memory effect, and wavefront sensitive techniques, such as wavefront shaping [22–24], optical phase conjugation [25–27], and optical trapping [28] in/through scattering media. It will impact biomedical imaging, seeing through clouds, photodynamic therapy, and light manipulation in/through turbid media.

## APPENDIX A: DERIVATION OF THE SPECKLE CORRELATION AND AME RANGE

Substituting Eq. (4) into Eq. (9), we obtain

For a common phase mask such as a ground glass disk, $\u27e8{A}_{2}({x}_{2},{y}_{2}){A}_{2}^{*}({x}_{2}^{\prime},{y}_{2}^{\prime})\u27e9=\mathrm{exp}\{-\frac{{[2\pi (n-1)]}^{2}{\sigma}^{2}}{{\lambda}^{2}{\kappa}^{2}}[{({x}_{2}-{x}_{2}^{\prime})}^{2}+{({y}_{2}-{y}_{2}^{\prime})}^{2}]\}$, where $\sigma $ and $\kappa $ are the standard deviation of the height and the transverse correlation length, respectively, $n$ is its refractive index [30–32], and the radius of Aperture 2 is $r$. Then, the counterpart of Eq. (A13) is

## APPENDIX B: WORKFLOW FOR GENERATING A PHASE MASK

By performing a transformation from $\theta $−$\phi $ coordinates to ${f}_{x}$−${f}_{y}$ coordinates for Eq. (19), we obtain the spatial power spectral density of the phase mask $S({f}_{x},{f}_{y})={|M({f}_{x},{f}_{y})|}^{2}$. Then, the $x\u2013y$ domain mask $M(x,y)$ can be obtained by performing a Fourier transform of $M({f}_{x},{f}_{y})$. We know that the $x\u2013y$ domain presentation of a phase mask must be phase only; in other words, the absolute value of a phase mask must be uniform in the $x\u2013y$ space, which is the demand of the time-reversal symmetry of a pure scattering medium. However, the Fourier transform of $M({f}_{x},{f}_{y})$ does not guarantee a phase-only mask in the $x\u2013y$ domain. Fortunately, the total angular distribution function $P(\theta )$ limits only the amplitude of $M({f}_{x},{f}_{y})$, because $P(\theta )$ represents only the intensity distribution of the scattered light. Thus, $M({f}_{x},{f}_{y})$ can be assigned any phase distribution, which allows us to find a special phase distribution that leads to a phase-only mask $M(x,y)$. This is a typical problem that can be solved by the Gerchberg–Saxton (G-S) algorithm [33] with the detailed algorithm workflow shown in Fig. 4.

## APPENDIX C: COMPARISON OF THE NEW AND CONVENTIONAL PHASE MASK MODELS

By considering more ballistic light, the new multiple-random-phase-mask model has a larger AME range than the conventional one, as shown in Fig. 5. In the new model, the distance between two adjacent masks is not necessarily the MFP of the scattering medium; instead, it can be flexibly adjusted according to computational resources and the requirements for precision. In the simulation, we set the distance to be equal to or half of the MFP and use angular spectrum diffraction theory to calculate the free propagation of light between two adjacent masks. The consistency between the blue and orange curves in Figs. 5(f) and 5(g) verifies the reliability and self-consistency of the new phase mask model.

## APPENDIX D: SCHEMATIC FIGURE OF THE PRINCIPLE OF SCANNING A TIME-REVERSED FOCUS

The schematic of the principle of scanning a time-reversed focus is shown in Fig. 6.

## Funding

National Key Research and Development Program of China Stem Cell and Translational Research (2016YFC0100602).

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