Modeling optical memory effects with phase screens

Malchiel Haskel; Adrian Stern

doi:10.1364/OE.26.029231

1. Introduction

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 [1], 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 [2] 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 $U_{i} (r)$ and the output field $U_{o} (r)$ from a scattering layer [Fig. 1(a)] can be expressed as:

U_{o} (r_{o}) = m (r_{o}) U_{i} (r_{i}) = e^{- j ϕ (r_{o})} U_{i} (r_{i})

where the vector

r

denotes the space coordinates, and

ϕ (r_{o})

is the random phase of the RPS

m (r_{o}) = e^{- j ϕ (r_{o})}

. A system block diagram illustrating Eq. (1) is shown in Fig. 1(b).

Fig. 1 (a) The incident and the output wavefronts from a thin diffusive medium layer, (b) system block diagram of the RPS model.

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The RPS model complies well with the so-called “optical memory effect” [4]. 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]).

Fig. 2 The tilt and shift effects, (a) Illustration of the classical optical (tilt) memory effect where a tilt of the input wavefront yields a similar tilt in the output plane - a phenomenon common to disordered media at small angles [4]; (b) shift memory effect, where a shift of the input wavefront yields a shift in the output plane [14] These two effects and their combination are described by the “generalized memory effect model” [15].

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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 $k_{t}$ , i.e. $e^{- j k_{t} r_{i}} U_{i} (r_{i})$ , then the respective output field is tilted as well:

m (r_{o}) (e^{- j k_{t} r_{i}} U_{i} (r_{i})) = (e^{- j ϕ (r_{o})} U_{i} (r_{i})) e^{- j k_{t} r_{i}} = U_{o} (r_{o}) e^{- j k_{t} r_{i}}

2. The generalized memory effect

A new type of correlation between the incident wave and the scattered wave was discovered recently [14], which cannot be described by the system in Fig. 1(b). Judkewitz and his associates have shown in [14] 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 [15], 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 [18].

Let us consider the relation between input and output Wigner distributions, $W_{i} (r_{i}, k_{i})$ and $W_{o} (r_{o}, k_{o})$ , of a general linear system [18–21]:

W_{o} (r_{o}, k_{o}) = \iint_{} K (r_{o}, k_{o}, r_{i}, k_{i}) W_{i} (r_{i}, k_{i}) d r_{i} d k_{i}

where the vector

k

can be considered to be the frequency variable associated with

r

. The Wigner distribution of the field

U (r)

is defined by [18–21]:

W (r, k) = \int U (r+ \frac{1}{2} r') U^{*} (r- \frac{1}{2} r') \exp [- j kr'] d r'

and

K (r_{o}, k_{o}, r_{i}, k_{i})

is the so called “ray-spread function” [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:

{\bar{W}}_{o} (r_{o}, k_{o}) = \iint_{} \bar{K} (r_{o}, k_{o}, r_{i}, k_{i}) W_{i} (r_{i}, k_{i}) d r_{i} d k_{i}

where

\bar{K} (r_{o}, k_{o}, r_{i}, k_{i})

is the ensemble average of

K (r_{o}, k_{o}, r_{i}, k_{i})

. It is straightforward to show (see Appendix A) that the average ray-spread function of a RPS can be approximated by:

\bar{K} (r_{o}, k_{o}, r_{i}, k_{i}) = δ (r_{o} - r_{i}) \int Γ_{m} (| r_{o}^{,} |) e^{- j r_{o}^{,} (k_{o} - k_{i})} d r_{o}^{,}

where

Γ_{m} (| r_{o}^{,} |)

is the phase mask autocorrelation function.

We can see that Eq. (6) depends only on the angular difference $k_{o} - k_{i}$ , 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 $Γ_{m} (| r_{o}^{,} |) = e^{- \frac{σ_{ϕ'}^{2}}{2} {| r_{o}^{,} |}^{2}}$ , where $σ_{ϕ'}^{2}$ is the variance of the derivative of the phase [22]. 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):

\bar{K} (r_{o}, k_{o}, r_{i}, k_{i}) = δ (r_{o} - r_{i}) e^{- \frac{1}{2 σ_{ϕ'}^{2}} {| k_{o} - k_{i} |}^{2}}

In Eq. (7) and in the sequel we ignore constant multiplicative factors. Consequently, the Wigner distribution of the output is:

{\bar{W}}_{o} (r_{o}, k_{o}) = \iint_{} δ (r_{o} - r_{i}) e^{- \frac{1}{2 σ_{ϕ'}^{2}} {| k_{o} - k_{i} |}^{2}} W_{i} (r_{i}, k_{i}) d r_{i} d k_{i} .

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 $\exp (- j k_{t} r_{i}) U_{i} (r_{i})$ yields a frequency shift of the Wigner distribution, which, when substituted in Eq. (8) gives:

{\bar{W}}_{o} (r_{o}, k_{o} - k_{t}) = \iint_{} δ (r_{o} - r_{i}) e^{- \frac{1}{2 σ_{ϕ'}^{2}} {| k_{o} - k_{i} |}^{2}} W_{i} (r_{i}, k_{o} - k_{t}) d r_{i} d k_{i} .

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 [15]. According to the generalized memory model, the ray spread function has the form of:

\bar{K} (\tilde{r}, \tilde{k}) = e^{- \frac{6 l_{t r}}{L} (\frac{{| \tilde{r} |}^{2}}{L^{2}} - \frac{\tilde{k} \cdot \tilde{r}}{k_{0} L} + \frac{{| \tilde{k} |}^{2}}{3 k_{0}^{2}})},

where

k_{0}

is the wave number,

l_{t r}

is the transport mean free path and

L

is the width of the diffusive medium. The expression in Eq. (10) was derived in [15] using a continuous Fokker-Plank scattering model, and was found to be consistent with the experimental results and with the results in [23]. In (10)

\tilde{r} \equiv r_{o} - r_{i} - L k_{i} / k_{0}, \tilde{k} \equiv k_{o} - k_{i}

, which, when substituted in Eq. (10) and in Eq. (5) yields:

{\bar{W}}_{o} (r_{o}, k_{o}) = \iint_{} e^{- \frac{6 l_{t r}}{L} (\frac{{| r_{o} - r_{i} - L k_{i} / k_{0} |}^{2}}{L^{2}} - \frac{(k_{o} - k_{i}) \cdot (r_{o} - r_{i} - L k_{i} / k_{0})}{k_{0} L} + \frac{{| k_{o} - k_{i} |}^{2}}{3 k_{0}^{2}})} W_{i} (r_{i}, k_{i}) d r_{i} d k_{i}

In Eq. (11) we can recognize a dependence on $k_{0} - k_{i}$ similar to that in Eq. (8), but we also see dependence on the coordinate difference $r_{0} - r_{i}$ 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, $m_{1}$ and $m_{2}$ , located between three Linear Canonical Transform (LCT) subsystems specified by respective $M_{1}, M_{2}$ and $M_{3}$ matrices. Each system performs a LCT [24] of the complex field amplitude defined by:

U_{o} (r_{o}) = L (M) U_{i} (r_{i}) = \frac{j}{2} {| B |}^{- 1 / 2} \int e x p [\frac{j}{2} (r_{o}^{t} D B^{- 1} r_{o} - 2 r_{i}^{t} B^{- 1} r_{o} + r_{i}^{t} B^{- 1} A r_{i})] U_{i} (r_{i}) d r_{i}

where

L (M)

denotes the LCT operator with ray matrix

M

as its parameters. The input variables

(r_{i}, k_{i})

and the output variables

(r_{o}, k_{o})

are related by the two dimensional ray transformation matrix

M

relation

[\begin{matrix} r_{o} \\ k_{o} \end{matrix}] = [\begin{matrix} A & B \\ C & D \end{matrix}] [\begin{matrix} r_{i} \\ k_{i} \end{matrix}] = M [\begin{matrix} r_{i} \\ k_{i} \end{matrix}]

and the superscript

^{t}

denotes the transpose of the vector [24]. Equation (12) describes the non-degenerated LCT case for which

| B | \neq 0

, where

| B |

is the determinant of the B matrix. In the general case, M is a 4 × 4 matrix [19]; however, under the assumption of a separable and symmetric system, the matrices

A, B, C, D

reduce to be a scalar (see Appendix B), thus yielding 2 × 2 ray matrix, M, which is more common in the literature.

Fig. 3 The proposed generalized phase screen model.

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In Fig. 3, $U_{i} (r_{i})$ , $U' (r')$ , $U'' (r'')$ , $U''' (r''')$ , $U'''' (r'''')$ and $U_{o} (r_{o})$ are the optical fields between the subsystems and the two RPSs: LCT₁, $m_{1}$ , LCT₂, $m_{2}$ , LCT₃, respectively, i.e., $U' = L (M_{1}) U_{i}$ , $U'' = U' \cdot m_{1}$ , $U''' = L (M_{2}) U''$ , $U'''' = U''' \cdot m_{2}$ , $U_{o} = L (M_{3}) U''''$ .

We shall show that with proper choice of the LCT operators, the joint spatial-tilt interaction of the generalized memory effect [15] can be obtained. In our design, the first phase mask $m_{1} (r)$ essentially performs the angular smearing appropriate to the tilt memory effect, and the second phase mask $m_{2} (r)$ 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:

M_{1} = [\begin{matrix} I & \frac{1}{k_{0}} Z \\ 0 & I \end{matrix}], M_{2} = [\begin{matrix} 0 & \frac{f}{k_{0}} I \\ - \frac{k_{0}}{f} I & 0 \end{matrix}], M_{3} = [\begin{matrix} \frac{1}{f} Z & - \frac{f}{k_{0}} I \\ \frac{k_{0}}{f} I & 0 \end{matrix}],

where

I

is the identity matrix, the matrix

0

stands for

0 \cdot I

,

Z = z' \cdot I

with z' being a constant normally associated with the free space propagation distance,

f

is associated with the scaling Fourier transform of optical

2 f

system, and

k_{0}

is the wave number. Indeed, we demonstrate in 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:

\begin{array}{l} \bar{K} (r_{o}, k_{o}, r_{i}, k_{i}) = e^{- \frac{1}{2 σ_{1}^{2}} {| k_{o} - k_{i} |}^{2}} e^{- \frac{1}{2 f^{2} σ_{2}^{2}} {| - k_{0} (r_{o} - r_{i} - \frac{L}{k_{0}} k_{i}) + z (k_{o} - k_{i}) |}^{2}} \\ = e^{- \frac{k_{0}^{2}}{2 σ_{2}^{2} f^{2}} {| r_{o} - r_{i} - \frac{L}{k_{0}} k_{i} |}^{2} + \frac{k_{0} z}{σ_{2}^{2} f^{2}} [k_{o} - k_{i}] \cdot [r_{o} - r_{i} - \frac{L}{k_{0}} k_{i}]} e^{- (\frac{z^{2}}{2 σ_{2}^{2} f^{2}} + \frac{1}{2 σ_{1}^{2}}) {| k_{o} - k_{i} |}^{2}} \end{array}

where

σ_{1}^{2}

and

σ_{2}^{2}

are the variances of the derivatives of the phase terms of

m_{1}

and

m_{2}

, respectively, and

z \equiv L - z'

. Notice that the ray spread function in Eq. (14) involves both spatial and angular smearing of the phase space, which is a central characteristic of the generalized memory effect [15]. Furthermore, with the following choice of parameters:

σ_{1}^{2} = \frac{L k_{0}^{2}}{l_{t r}}, σ_{2}^{2} f^{2} = \frac{L^{3} k_{0}^{2}}{12 l_{t r}}, z = \frac{L}{2},

we obtain the same ray spread function as found in (11).

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 $σ_{1}^{2}$ and $σ_{2}^{2}$ 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 $L$ ) the phase variances decrease as well. However, $σ_{2}^{2}$ is proportional to $L^{3}$ , whereas $σ_{1}^{2}$ is proportional to L; therefore, for thin diffusers, the role of the second mask $m_{2}$ , and hence the translation memory effect is less dominant.

As demonstrated in [15], one can express the generalized tilt/shift field correlation function of the system by applying two forward and two inverse 2D Fourier transforms on $\bar{K}$ in Eq. (14) with the substitution of $σ_{1}^{2}, σ_{2}^{2}$ 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 [15], it is shown that all (tilt, shift, and combined tilt-shift) memory extents are proportional to $\sqrt{l_{t r} / L k_{0}^{2}}$ , which, in our compatible model is determined by $σ_{1}$ .

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 [24], 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.

Fig. 4 A first order optical realization of the generalized RPS model in Fig. 3.

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$U_{i}$

Let us take a closer look at the system in Fig. 4. It can be seen that for $L \to 0$ 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 $σ_{1}^{2}$ and $σ_{2}^{2}$ , which, in turn, yield $m_{1} \to 1, m_{2} \to 1$ , resulting in $U_{o} \approx U_{i}$ , 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 $L / 2$ . 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 $l_{t r}$ .

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 [25], 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 $U_{o}$ is just the result of free space propagation of $U_{i}$ 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 [25], 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 [25] to obtain the overall output of the system. The precise evaluation of the output field using the method in [25] 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 $(k_{o} - k_{i}) \cdot (r_{o} - r_{i} - L k_{i} / k_{0})$ 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 [14] and [15], are valid in the regime where the light is propagating mainly in the forward direction.

4. Conclusions

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:

U_{o} (r_{o}) = \int h_{r r} (r_{o}, r_{i}) U_{i} (r_{i}) d r_{i}

with the system function [20]:

h_{r r} (r_{o}, r_{i}) = e^{- j ϕ (r_{o})} δ (r_{o} - r_{i})

Using the input output relation [18], [20], [24]:

K (r_{o}, k_{o}, r_{i}, k_{i}) = \int \int h_{r r} (r_{o} + \frac{1}{2} r_{o}^{,} - r_{i} - \frac{1}{2} r_{i}^{,}) h_{r r}^{*} (r_{o} - \frac{1}{2} r_{o}^{,} - r_{i} + \frac{1}{2} r_{i}^{,}) e^{- j k_{o} r_{o}^{,} + j k_{i} r_{i}^{,}} d r_{o}^{,} d r_{i}^{,}

together with Eq. (17) yields the ray spread function of the RSP:

\begin{array}{l} K (r_{o}, k_{o}, r_{i}, k_{i}) = \\ \int \int e^{- j ϕ (r_{o} + \frac{1}{2} r_{o}^{,})} e^{+ j ϕ (r_{o} - \frac{1}{2} r_{o}^{,})} δ (r_{o} + \frac{1}{2} r_{o}^{,} - r_{i} - \frac{1}{2} r_{i}^{,}) δ (r_{o} - \frac{1}{2} r_{o}^{,} - r_{i} + \frac{1}{2} r_{i}^{,}) e^{- j k_{o} r_{o}^{,} + j k_{i} r_{i}^{,}} d r_{o}^{,} d r_{i}^{,} = \\ = \int e^{- j ϕ (r_{o} + \frac{1}{2} r_{o}^{,})} e^{+ j ϕ (r_{o} - \frac{1}{2} r_{o}^{,})} e^{- j k_{o} r_{o}^{,}} d r_{o}^{,} \int δ (r_{o} + \frac{1}{2} r_{o}^{,} - r_{i} - \frac{1}{2} r_{i}^{,}) δ (r_{o} - \frac{1}{2} r_{o}^{,} - r_{i} + \frac{1}{2} r_{i}^{,}) e^{+ j k_{i} r_{i}^{,}} d r_{i}^{,} = \\ = \int e^{- j ϕ (r_{o} + \frac{1}{2} r_{o}^{,})} e^{- j ϕ (r_{o} - \frac{1}{2} r_{o}^{,})} e^{- j k_{o} r_{o}^{,}} δ (2 r_{o} - 2 r_{i}) e^{j k_{i} (2 r_{o} + r_{o}^{,} - 2 r_{i})} d r_{o}^{,} = \\ = δ (r_{o} - r_{i}) e^{j 2 k_{i} (r_{o} - r_{i})} \int e^{- j ϕ (r_{o} + \frac{1}{2} r_{o}^{,})} e^{+ j ϕ (r_{o} - \frac{1}{2} r_{o}^{,})} e^{- j k_{o} r_{o}^{,}} e^{j k_{i} r_{o}^{,}} d r_{o}^{,} = \\ = δ (r_{o} - r_{i}) \int e^{- j ϕ (r_{o} + \frac{1}{2} r_{o}^{,})} e^{+ j ϕ (r_{o} - \frac{1}{2} r_{o}^{,})} e^{- j r_{o}^{,} (k_{o} - k_{i})} d r_{o}^{,} \end{array}

The expected value of this RSP ray spread function in Eq. (19) is given by:

\begin{array}{l} \bar{K} (r_{o}, k_{o}, r_{i}, k_{i}) ≜ E [K (r_{o}, k_{o}, r_{i}, k_{i})] = E [δ (r_{o} - r_{i}) \int e^{- j ϕ (r_{o} + \frac{1}{2} r_{o}^{,})} e^{+ j ϕ (r_{o} - \frac{1}{2} r_{o}^{,})} e^{- j r_{o}^{,} (k_{o} - k_{i})} d r_{o}^{,}] = \\ δ (r_{o} - r_{i}) \int E [e^{- j ϕ (r_{o} + \frac{1}{2} r_{o}^{,})} e^{+ j ϕ (r_{o} - \frac{1}{2} r_{o}^{,})}] e^{- j r_{o}^{,} (k_{o} - k_{i})} d r_{o}^{,} = δ (r_{o} - r_{i}) \int Γ_{m} (| r_{o}^{,} |) e^{- j r_{o}^{,} (k_{o} - k_{i})} d r_{o}^{,} \end{array}

where $E [\cdot]$ denotes the expectraion operator and $Γ_{m} (| r_{o}^{,} |)$ is the autocorrelation of the stationary RPS $m (r_{o}) = e^{- j ϕ (r_{o})}$ . For a stationary Gaussian random pahse-only tramsitance function the autocorrelation function is [22]:

Γ_{m} (| r_{o}^{,} |) = e^{- \frac{1}{2} D_{ϕ}^{} (r_{o}^{,})}

where $D_{ϕ}^{} (r_{o}^{,})$ is the structure functions [22]. In the common case that the phase variance is sufficently large, then Eq. (21) is approximately Gaussian [22,26]:

Γ_{m} (| r_{o}^{,} |) = e^{- \frac{σ_{ϕ'}^{2}}{2} {| r_{o}^{,} |}^{2}}

where $σ_{ϕ'}^{2}$ is the phase derivative's variance [22]. Substituting Eq. (22) in Eq. (20) yields:

\bar{K} (r_{o}, k_{o}, r_{i}, k_{i}) = δ (r_{o} - r_{i}) \int e^{- \frac{σ_{ϕ'}^{2}}{2} {| r_{o}^{,} |}^{2}} e^{- j r_{o}^{,} (k_{o} - k_{i})} d r_{o}^{,} = δ (r_{o} - r_{i}) e^{- \frac{{| (k_{o} - k_{i}) |}^{2}}{2 σ_{ϕ'}^{2}}} .

In Eq. (23) and in the sequel, we ignore constant multiplicative factors. Consequently, from Eq. (5), the Wigner distribution of the output is:

\begin{array}{l} {\bar{W}}_{o} (r_{o}, k_{o}) = \iint_{} \bar{K} (r_{o}, k_{o}, r_{i}, k_{i}) W_{i} (r_{i}, k_{i}) d r_{i} d k_{i} \\ = \iint_{} δ (r_{o} - r_{i}) e^{- \frac{{| k_{o} - k_{i} |}^{2}}{2 σ_{ϕ'}^{2}}} W_{i} (r_{i}, k_{i}) d r_{i} d k_{i} . \end{array}

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 [24]). This assumption allows to express the 4×4 ray transform matrix $Μ$ , which defines the LCT in Eq. (12), in a simpler form:

[\begin{matrix} r_{o} \\ k_{o} \end{matrix}] = [\begin{matrix} a I & b I \\ c I & d I \end{matrix}] [\begin{matrix} r_{i} \\ k_{i} \end{matrix}] = M [\begin{matrix} r_{i} \\ k_{i} \end{matrix}]

where the scalars a, b, c and d replace the respective 2×2 sub matrices A, B, C and D. Consequently, the LCT in Eq. (12) can be written as ([27], Ch.6 in [24]):

U_{o} (r_{o}) = L (M) U_{i} (r_{i}) = \frac{j}{2 b} \int e x p [\frac{j}{2 b} (r_{o}^{2} d - 2 r_{o} r_{i} + r_{i}^{2} a)] U_{i} (r_{i}) d r_{i}

and the ray-spread function describes an affine transform [19]:

K (r_{o}, k_{o}, r_{i}, k_{i}) = δ (r_{i} - A r_{o} - B k_{o}) δ (k_{i} - C r_{o} - D k_{o}),

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 [20]:

[\begin{matrix} A I & B I \\ C I & D I \end{matrix}] = M^{- 1} = {[\begin{matrix} a I & b I \\ c I & d I \end{matrix}]}^{- 1}

Using Eq. (25) and Eq. (28), we may rewrite the respective inverse matrices of the LCT system in Eq. (13) as

$M_{1}^{- 1} = [\begin{matrix} A_{1} I & B_{1} I \\ C_{1} I & D_{1} I \end{matrix}] = [\begin{matrix} 1 I & - \frac{L - z}{k_{0}} I \\ 0 I & 1 I \end{matrix}], M_{2}^{- 1} = [\begin{matrix} A_{2} I & B_{2} I \\ C_{2} I & D_{2} I \end{matrix}] = [\begin{matrix} 0 I & - \frac{f}{k_{0}} I \\ \frac{k_{0}}{f} I & 0 I \end{matrix}], M_{3}^{- 1} = [\begin{matrix} A_{3} I & B_{3} I \\ C_{3} I & D_{3} I \end{matrix}] = [\begin{matrix} 0 I & \frac{f}{k_{0}} I \\ - \frac{k_{0}}{f} I & \frac{z}{f} I \end{matrix}]$

(29)

where we have defined

z' \equiv L - z

in Eq. [REMOVED GOTOBUTTON FIELD], for future convenience.

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

K_{1} (r', k', r_{i}, k_{i}) = δ (r_{i} - A_{1} r' - B_{1} k') δ (k_{i} - C_{1} r' - D_{1} k')

the ray spread functions of the first mask is

{\bar{K}}_{2} (r'', k'', r', k') = δ (r'' - r') e^{- \frac{1}{2 σ_{1}^{2}} {| k'' - k' |}^{2}}

the ray spread functions of the second LCT sub system is:

K_{3} (r''', k''', r'', k'') = δ (r'' - A_{2} r''' - B_{2} k''') δ (k'' - C_{2} r''' - D_{2} k''')

the ray spread functions of the second mask is

{\bar{K}}_{4} (r'''', k'''', r''', k''') = δ (r'''' - r''') e^{- \frac{1}{2 σ_{2}^{2}} {| k'''' - k''' |}^{2}}

and finally the ray spread functions of the third LCT system is

K_{5} (r_{o}, k_{o}, r'''', k'''') = δ (r'''' - A_{3} r_{o} - B_{3} k_{o}) δ (k'''' - C_{3} r_{o} - D_{3} k_{o})

The ray spread function of any two cascaded linear systems with respective ray spread functions $K_{α} (r', k', r_{i}, k_{i})$ and $K_{β} (r_{o}, k_{0}, r', k')$ is given by [19]:

K_{α β} (r_{o}, k_{0}, r_{i}, k_{i}) = \int \int K_{α} (r', k', r_{i}, k_{i}) K_{β} (r_{o}, k_{0}, r', k') d r' d k'

By using Eq. (35) with Eq. (30) and Eq. (31) in it, we may find the ray spread function of the sub system from the input plane up to the first mask in Fig. 3:

\begin{array}{l} {\bar{K}}_{12} (r'', k'', r_{i}, k_{i}) = \\ = \int \int δ (r_{i} - A_{1} r' - B_{1} k') δ (k_{i} - C_{1} r' - D_{1} k') \sqrt{2 π} σ_{1} δ (r'' - r') e^{- \frac{σ_{1}^{2}}{2} {| k'' - k' |}^{2}} d r' d k' \\ \overset{}{=} \int e^{- \frac{1}{2 σ_{1}^{2}} {| k'' - k' |}^{2}} δ (r_{i} - A_{1} r'' - B_{1} k') δ (k_{i} - C_{1} r'' - D_{1} k') d k' \\ = e^{- \frac{1}{2 σ_{1}^{2}} {| k'' - \frac{1}{D_{1}} k_{i} + \frac{C_{1}}{D_{1}} r'' |}^{2}} δ (r_{i} - r'' (A_{1} - \frac{C_{1}}{D_{1}}) - \frac{B_{1}}{D_{1}} k_{i}) . \end{array}

Similarly, by using Eq. (35) with Eq. (36) and Eq. (32) we find the ray spread function of the system from the input plane up to the first LCT system:

\begin{array}{l} {\bar{K}}_{13} (r''', k''', r_{i}, k_{i}) = \int \int \begin{array}{l} e^{- \frac{1}{2 σ_{1}^{2}} {| k'' - \frac{1}{D_{1}} k_{i} + \frac{C_{1}}{D_{1}} r'' |}^{2}} δ (r_{i} - r'' (A_{1} - \frac{C_{1}}{D_{1}}) - \frac{B_{1}}{D_{1}} k_{i}) \times \\ δ (r'' - A_{2} r''' - B_{2} k''') δ (k'' - C_{2} r''' - D_{2} k''') d r'' d k'' \end{array} = \\ \overset{}{=} \int e^{- \frac{1}{2 σ_{1}^{2}} {| (C_{2} r''' + D_{2} k''') - \frac{1}{D_{1}} k_{i} + \frac{C_{1}}{D_{1}} r'' |}^{2}} δ (r_{i} - r'' (A_{1} - \frac{C_{1}}{D_{1}}) - \frac{B_{1}}{D_{1}} k_{i}) δ (r'' - A_{2} r''' - B_{2} k''') d r'' \\ = e^{- \frac{1}{2 σ_{1}^{2}} {| r''' (C_{2} + \frac{C_{1}}{D_{1}} A_{2}) + k''' (\frac{C_{1}}{D_{1}} B_{2} + D_{2}) - \frac{1}{D_{1}} k_{i} |}^{2}} δ (r_{i} - \frac{B_{1}}{D_{1}} k_{i} - (A_{1} - \frac{C_{1}}{D_{1}}) (A_{2} r''' + B_{2} k''')) . \end{array}

The ray spread function relating the input plane with the plane after the second phase mask is found by substituting using Eq. (35) with Eq. (37) and Eq. (33):

\begin{array}{l} {\bar{K}}_{14} (r'''', k'''', r_{i}, k_{i}) \approx \\ \int \int \begin{array}{l}  \end{array} e^{- \frac{1}{2 σ_{1}^{2}} {| r''' (C_{2} + \frac{C_{1}}{D_{1}} A_{2}) + k''' (\frac{C_{1}}{D_{1}} B_{2} + D_{2}) - \frac{1}{D_{1}} k_{i} |}^{2}} δ (r_{i} - \frac{B_{1}}{D_{1}} k_{i} - (A_{1} - \frac{C_{1}}{D_{1}}) (A_{2} r''' + B_{2} k''')) \times \\ \sqrt{2 π} σ_{2} δ (r'''' - r''') e^{- \frac{σ_{2}^{2}}{2} {| k'''' - k''' |}^{2}} d r''' d k''' \\ \overset{}{=} \int \begin{array}{l}  \end{array} e^{- \frac{1}{2 σ_{1}^{2}} {| r'''' (C_{2} + \frac{C_{1}}{D_{1}} A_{2}) + k''' (\frac{C_{1}}{D_{1}} B_{2} + D_{2}) - \frac{1}{D_{1}} k_{i} |}^{2}} e^{- \frac{1}{2 σ_{2}^{2}} {| k'''' - k''' |}^{2}} \times \\ δ (r_{i} - \frac{B_{1}}{D_{1}} k_{i} - (A_{1} - \frac{C_{1}}{D_{1}}) (A_{2} r'''' + B_{2} k''')) d k''' \\ \propto \int \begin{array}{l}  \end{array} e^{- \frac{1}{2 σ_{1}^{2}} {| r'''' (C_{2} + \frac{C_{1}}{D_{1}} A_{2}) + k''' (\frac{C_{1}}{D_{1}} B_{2} + D_{2}) - \frac{1}{D_{1}} k_{i} |}^{2}} e^{- \frac{1}{2 σ_{2}^{2}} {| k'''' - k''' |}^{2}} \times \\ δ (\frac{1}{(A_{1} - C_{1} / D_{1}) B_{2}} r_{i} - \frac{B_{1}}{(A_{1} - C_{1} / D_{1}) B_{2} D_{1}} k_{i} - \frac{A_{2}}{B_{2}} r'''' - k''') d k''' \\ = e^{- \frac{1}{2 σ_{1}^{2}} {| - \frac{1}{B_{2}} r'''' + \frac{(C_{1} B_{2} + D_{1} D_{2})}{(A_{1} - C_{1} / D_{1}) B_{2} D_{1}} r_{i} - (\frac{B_{1} (C_{1} B_{2} + D_{1} D_{2})}{(A_{1} - C_{1} / D_{1}) B_{2} D_{1}} + 1) \frac{k_{i}}{D_{1}} |}^{2}} e^{- \frac{1}{2 σ_{2}^{2}} {| k'''' - (- \frac{A_{2}}{B_{2}} r'''' + \frac{1}{(A_{1} - C_{1} / D_{1}) B_{2}} r_{i} - \frac{B_{1}}{(A_{1} - C_{1} / D_{1}) B_{2} D_{1}} k_{i}) |}^{2}} . \end{array}

Finally, using the same procedure with Eq. (38) and Eq. (34), we find the ray spread function of the overall system from the input plane up to the third LCT system:

\begin{array}{l} {\bar{K}}_{15} (r_{0}, k_{0}, r_{i}, k_{i}) \approx \\ \iint \begin{array}{l} e^{- \frac{1}{2 σ_{1}^{2}} {| - \frac{1}{B_{2}} r'''' + \frac{(C_{1} B_{2} + D_{1} D_{2})}{(A_{1} - C_{1} / D_{1}) B_{2} D_{1}} r_{i} - (\frac{B_{1} (C_{1} B_{2} + D_{1} D_{2})}{(A_{1} - C_{1} / D_{1}) B_{2} D_{1}} + 1) \frac{k_{i}}{D_{1}} |}^{2}} e^{- \frac{1}{2 σ_{2}^{2}} {| k'''' + \frac{A_{2}}{B_{2}} r'''' - \frac{1}{(A_{1} - C_{1} / D_{1}) B_{2}} r_{i} + \frac{B_{1}}{(A_{1} - C_{1} / D_{1}) B_{2} D_{1}} k_{i} |}^{2} \times} \\ δ (r'''' - A_{3} r_{o} - B_{3} k_{o}) δ (k'''' - C_{3} r_{o} - D_{3} k_{o}) d r'''' d k'''' \end{array} \\ \overset{}{=} \int \begin{array}{l} e^{- \frac{1}{2 σ_{1}^{2}} {| - \frac{1}{B_{2}} r'''' + \frac{(C_{1} B_{2} + D_{1} D_{2})}{(A_{1} D_{1} - C_{1}) B_{2}} r_{i} - (\frac{B_{1} (C_{1} B_{2} + D_{1} D_{2})}{(A_{1} D_{1} - C_{1}) B_{2}} + 1) \frac{k_{i}}{D_{1}} |}^{2}} e^{- \frac{1}{2 σ_{2}^{2}} {| (C_{3} x_{o} + D_{3} ω_{o}) + \frac{A_{2}}{B_{2}} r'''' - \frac{D_{1}}{(A_{1} D_{1} - C_{1}) B_{2}} r_{i} + \frac{B_{1}}{(A_{1} D_{1} - C_{1}) B_{2}} k_{i} |}^{2}} \times \\ δ (r'''' - A_{3} r_{o} - B_{3} k_{o}) d r'''' \end{array} \\ = e^{- \frac{1}{2 σ_{1}^{2}} {| - \frac{A_{3}}{B_{2}} r_{o} - \frac{B_{3}}{B_{2}} k_{o} + \frac{(C_{1} B_{2} + D_{1} D_{2})}{(A_{1} D_{1} - C_{1}) B_{2}} r_{i} - (\frac{B_{1} (C_{1} B_{2} + D_{1} D_{2})}{(A_{1} D_{1} - C_{1}) B_{2}} + 1) \frac{k_{i}}{D_{1}} |}^{2}} e^{- \frac{1}{2 σ_{2}^{2}} {| (C_{3} + \frac{A_{2} A_{3}}{B_{2}}) r_{o} + (D_{3} + \frac{A_{2} B_{3}}{B_{2}}) k_{o} - \frac{D_{1}}{(A_{1} D_{1} - C_{1}) B_{2}} r_{i} + \frac{B_{1}}{(A_{1} D_{1} - C_{1}) B_{2}} k_{i} |}^{2}} \end{array}

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 [15] by employing a continuous Fokker-Plank scattering model.

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Modeling optical memory effects with phase screens

Abstract

1. Introduction

2. The generalized memory effect

3. Generalized phase screen model

3.1 The model

3.2 Optical implementation

4. Conclusions

Appendix A Wigner distribution analysis of the Random Phase Screen Model

Appendix B Wigner distribution analysis of the Generalized Phase Screen Model

References

Cited By

Figures (4)

Equations (38)

Optics Express