During the last decade, optical memory effects have been explored extensively for various applications. In this letter we propose phase screen models to facilitate the analysis and the simulation of wave propagation through optical media that exhibits memory effects. We show that the classical optical memory effect, which implies tilt wave correlations of the input and the scattered fields, can be readily modeled by a single random phase screen. For the recently discovered generalized optical memory effect, which implies the existence of shift wave correlations in addition to the tilt correlation, we propose an appropriate generalized random phase screen model.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Random Phase Screen (RPS) models have been used for nearly seven decades to describe various wave scattering phenomena. Originally, RPS models were introduced to model the fading of radio waves due to fluctuations in ionosphere layers , and later have been found useful to model various wave scattering phenomena, including optical scattering by (large-scale) refractive index fluctuations [2,3]. For example, the RPS model was extensively used for the analysis of speckles in coherent imaging, speckle metrology, atmospheric turbulence and optical scintillations, amongst others (see  and references therein).
The RPS model is extremely useful for the analysis, simulation, and interpretation of optical systems that involve optical scattering due to refractive index fluctuations. Motivated by this, we show in this letter how RPS can be applied to model optical memory effects.
In its simplest form, the RPS model asserts that the relation between the incident field and the output field from a scattering layer [Fig. 1(a)] can be expressed as:Eq. (1) is shown in Fig. 1(b).
The RPS model complies well with the so-called “optical memory effect” . The celebrated “optical memory effect”, also known as the “tilt” memory effect, was discovered based on mesoscopic transport theory [4,5] and it claims that if we slightly tilt the input wavefront to an inhomogeneous random medium, the speckle pattern produced at the output will be simply tilted by the same angle [Fig. 2(a)]. This phenomenon was extensively utilized in novel imaging techniques through disordered media or for imaging with the aid of the disordered media (e.g. [6–13]).
It is evident that the “tilt” memory effect is modeled well by the system in Fig. 1(b); if the input field is tilted along the wave vector , i.e. , then the respective output field is tilted as well:
2. The generalized memory effect
A new type of correlation between the incident wave and the scattered wave was discovered recently , which cannot be described by the system in Fig. 1(b). Judkewitz and his associates have shown in  that, in addition to the tilt-tilt correlation, there also exists a shift-shift correlation effect that links the speckle produced by a beam with the speckle produced by a shifted beam [Fig. 2 (b)]. This spatial translational correlation effect occurs primarily in the case of propagation through anisotropic media and has been found suitable for imaging or scanning inside biological tissue [16,17]. Recently, in , a “generalized optical memory effect” model was proposed that describes the combined shift and tilt memory effect.
In the following we shall generalize the RPS system model shown in Fig. 1(b) to account for the generalized optical memory effect. To this end, we first revisit the RPS and examine its agreement with the shift memory effects through a phase space analysis .
Let us consider the relation between input and output Wigner distributions, and , of a general linear system [18–21]:18–21]:18,20]. Since in this work we are considering wave propagation through random media [Fig. 1(a)], our first concern is the expected value of the output Wigner distribution:Appendix A) that the average ray-spread function of a RPS can be approximated by:
We can see that Eq. (6) depends only on the angular difference, implying that Eq. (6) is angularly shift invariant, which complies with the tilt memory effect. This is further evident if we employ the common Gaussian approximation , where is the variance of the derivative of the phase . This approximation applies for isotropic RPS models with a large phase variance. In such a case the average ray-spread function becomes (see Appendix A):
In Eq. (7) and in the sequel we ignore constant multiplicative factors. Consequently, the Wigner distribution of the output is:
It can be seen in Eq. (8) that a RPS generates a smearing in the phase space that is shift invariant in the k direction. A tilted input field yields a frequency shift of the Wigner distribution, which, when substituted in Eq. (8) gives:
Thus, we can see that the RPS generates a tilt invariant Gaussian spread of the phase space in the k direction. Unfortunately, this pure k-directional smear cannot model the generalized memory effect, which involves joint spatial and angular smearing of the phase space . According to the generalized memory model, the ray spread function has the form of:Eq. (10) was derived in  using a continuous Fokker-Plank scattering model, and was found to be consistent with the experimental results and with the results in . In (10) , which, when substituted in Eq. (10) and in Eq. (5) yields:
In Eq. (11) we can recognize a dependence on similar to that in Eq. (8), but we also see dependence on the coordinate difference which is responsible for a phase space smear in the shift direction. Moreover, Eq. (11) exhibits a cross term implying a combined shift and tilt effect.
3. Generalized phase screen model
3.1 The model
To account for the generalized memory effect we propose the model shown in Fig. 3. The model consists of two RPSs, and , located between three Linear Canonical Transform (LCT) subsystems specified by respective and matrices. Each system performs a LCT  of the complex field amplitude defined by:24]. Equation (12) describes the non-degenerated LCT case for which, where is the determinant of the B matrix. In the general case, M is a 4 × 4 matrix ; however, under the assumption of a separable and symmetric system, the matrices reduce to be a scalar (see Appendix B), thus yielding 2 × 2 ray matrix, M, which is more common in the literature.
In Fig. 3, ,, , , and are the optical fields between the subsystems and the two RPSs: LCT1, , LCT2, , LCT3, respectively, i.e., , ,,,.
We shall show that with proper choice of the LCT operators, the joint spatial-tilt interaction of the generalized memory effect  can be obtained. In our design, the first phase mask essentially performs the angular smearing appropriate to the tilt memory effect, and the second phase mask is responsible for the phase space smear in the spatial direction, which accounts for the shift memory effect. The LCT subsystems in the proposed model [Fig. 3] are responsible for balancing the joint tilt and shift effects. By choosing the first LCT system to describe a forward free space propagation, the second one to describe a Fourier transform, and the last one to describe an inverse Fourier transform followed by free space propagation, we get the respective matrices:Appendix B that with this choice, the Wigner distribution of the output is related to that of the input by Eq. [REMOVED GOTOBUTTON FIELD] with the ray spread function:Eq. (14) involves both spatial and angular smearing of the phase space, which is a central characteristic of the generalized memory effect . Furthermore, with the following choice of parameters:
Now let us examine our generalized phase screen model [Fig. 3] with the particular choice of parameters in accordance with Eq. (13) and Eq. (15). We see that the phase derivatives variances and are proportional to L, implying that the ray spread is stronger for wide diffusers, as expected. On the other hand, by narrowing the diffuser (i.e., by decreasing) the phase variances decrease as well. However, is proportional to , whereas is proportional to L; therefore, for thin diffusers, the role of the second mask , and hence the translation memory effect is less dominant.
As demonstrated in , one can express the generalized tilt/shift field correlation function of the system by applying two forward and two inverse 2D Fourier transforms on in Eq. (14) with the substitution of from Eq. (15), from which it is possible to evaluate the correlation lengths in the spatial and tilt domains. From such a calculation, as in , it is shown that all (tilt, shift, and combined tilt-shift) memory extents are proportional to, which, in our compatible model is determined by.
3.2 Optical implementation
Each of the LCT subsystems in Fig. 3 can be naturally considered as representing a first-order optical system. Considering a classical interpretation of the matrices in Eq. (13) as ray transform ABCD matrices , the generalized screen model in Fig. 3 has a first order optical realization as shown in Fig. 4. The system in Fig. 4 can be used to simulate numerically or optically the generalized memory effect. Moreover, by realizing the two RPSs with spatial light modulators it is possible to optically simulate a dynamical (time-changing) generalized memory effect.
Let us take a closer look at the system in Fig. 4. It can be seen that for the set-up reduces to a 4f system. Therefore, in such a case, the two masks placed in the input and Fourier planes of the 4f system, respectively, perform orthogonal operations in the Wigner space. Moreover, since each RPS generates a unidirectional spread in the Wigner space [Eq. (8)] the two masks generate a separable and independent smear in the r-k space. This implies independent shift and tilt effects. Now, considering the relations Eq. (15), we see that a small L yields small values of and , which, in turn, yield , resulting in , as expected.
For the more interesting case of a finite L, the input field to the 4f system and its output field both undergo a defocusing by. Thus, the system ceases to be a linear shift invariant system, making it possible to simulate the coupling between the tilt and the shift effects by rendering the effect of the two RSPs not independent. This is also corroborated by the dependence of all the parameters in Eq. (15) on L and .
A brief insight into the operation of the system in Fig. 4 as the generator of a joint tilt-shift memory effect can be obtained by following the methodology used in , where the influence of each individual mask on the maskless system is investigated. Ignoring the masks in Fig. 4, the 4f system is just a relay system. Consequently, the output field is just the result of free space propagation of by a distance L, as expected for propagation with negligible scattering along the slab.
To include the influence of m1, this maskless output field needs to be convolved with a scaled Fourier transform of m1 and multiplied by a scaled quadratic phase function before and after the convolution , where the scaling is proportional to the free space propagation of L/2. This result needs to be convolved again with a scaled Fourier transform of m2  to obtain the overall output of the system. The precise evaluation of the output field using the method in  is beyond the scope of our work here; we only wish to point out here that the convolution with the scaled Fourier transform of m1 accounts for the tilt effect and the convolution with the scaled Fourier transform of m2 accounts for the shift effect. The quadratic phase function multiplication prior to the first convolution causes a cross dependency between the tilt and shift effects (represented by the cross term involving in Eq. (11) and the respective term in Eq. (14)).
Please note that our model, being based on LCT is valid only in the paraxial regime. Similarly the models in  and , are valid in the regime where the light is propagating mainly in the forward direction.
In summary, the ubiquitous RPS model was generalized to apply to the tilt and tilt-shift memory effects. To account for the classical memory effect a conventional model with a single RPS can be used. For the generalized memory effect, which involves tilt-shift correlations, two RPSs are required. The two RPSs are sandwiched between three LCT operators that can be considered to represent simple first order optical systems, such as a 4f system and free space propagation. We derived a mathematical model of the proposed RPS system. This mathematical model is valid through its equivalence to that obtained in previous publications. The physical processing is still open for modeling.
In a similar way that the classical RPS model is useful for the analysis and simulation of scattering through thin diffusers, the generalized phase screen models allow analysis of a series of scattering events in thick diffusing media. In addition to offering a practical analytical tool, the RPS models offer simple ways to build numerical or optical simulators of the memory effects. These simulators can be easily adapted for temporal non-stationary conditions by implementing dynamic RPSs.
Appendix A Wigner distribution analysis of the Random Phase Screen Model
In this appendix we derive the Wigner distrubution of the field at the output of the RPS system shown in Fig. 1(a). The input–output field realtion in Eq. (1) of the RPS model [Fig. 1(b)] can be written as the linear integal:
with the system function :
together with Eq. (17) yields the ray spread function of the RSP:
The expected value of this RSP ray spread function in Eq. (19) is given by:
where denotes the expectraion operator and is the autocorrelation of the stationary RPS . For a stationary Gaussian random pahse-only tramsitance function the autocorrelation function is :
where is the phase derivative's variance . Substituting Eq. (22) in Eq. (20) yields:
In Eq. (23) and in the sequel, we ignore constant multiplicative factors. Consequently, from Eq. (5), the Wigner distribution of the output is:
Appendix B Wigner distribution analysis of the Generalized Phase Screen Model
For simplicity we shall assume that the LCTs in our model [Fig. 3] are separable and symmetric in Cartesian coordinates (Ch.6 in ). This assumption allows to express the 4×4 ray transform matrix , which defines the LCT in Eq. (12), in a simpler form:
and the ray-spread function describes an affine transform :
where the four scalars A, B, C and D (different from the 2×2 sub matrices A, B, C and D) are the elements of the inverse matrix :
In the following we shall derive the ray-spread function of each of the five components in Fig. 3 and then we shall cascade them to obtaining the overall ray-spread function. The ray-spread functions of the first, third and fifth subsystems (the LCTs) can be expressed in terms of Eq. (25) and Eq. (27), and the ray-spread functions of the second and third components (the RPSs) can be expressed in terms of Eq. (23). The ray spread functions of the first LCT sub system in Fig. 3 is
the ray spread functions of the first mask is
the ray spread functions of the second LCT sub system is:
the ray spread functions of the second mask is
and finally the ray spread functions of the third LCT system is
The ray spread function of any two cascaded linear systems with respective ray spread functions and is given by :
It is easy to verify that by substituting the values of the matrices of Eq. (29) into Eq. (39) and choosing the parameters in Eq. (15) we obtain the same ray spread function within Eq. (11), which was obtained in  by employing a continuous Fokker-Plank scattering model.
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