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Wave optics simulation approach for partial

spatially coherent beams

Xifeng Xiao and David Voelz

Klipsch School of Electrical and Computer Engineering

New Mexico State University, Las Cruces, NM 88003

davvoelz@nmsu.edu

Abstract: A numerical wave optics approach for simulating a partial

spatially coherent beam is presented. The approach involves the application

of a sequence of random phase screens to an initial beam field and the

summation of the intensity results after propagation. The relationship

between the screen parameters and the spatial coherence function for the

beam is developed and the approach is verified by comparing results with

analytic formulations for a Gaussian Schell-model beam. The approach can

be used for modeling applications such as free space optical laser links that

utilize partially coherent beams.

©2006 Optical Society of America

OCIS codes: (030.0030) Coherence and statistical optics; (030.1670) Coherent optical effects;

(060.4510) Optical communications; (060.5060) Phase modulation; (110.4980) Partial

coherence in imaging.

References and links

1.

S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent

atmosphere,” J. Opt. Soc. Am. 69, 1297-1304 (1979).

Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for a weak atmospheric turbulence using a

partially coherent beam,” Radio Sci. 18, 551-556 (1983).

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam:

implication for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794-1802 (2002).

W. Martienssen and E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919-926

(1964).

H. Arsenault and S. Lowenthal, “Partial coherence of an object illuminated with laser light through a

moving diffuser,” Opt. Commun. 1, 451-453 (1970).

E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic

acousto-optic holograms,” J. Opt. Soc. Am. A 9, 796-803 (1992).

D. G. Voelz and K. J. Fitzhenry, “Pseudo-partially coherent beam for free-space laser communication,” in

Free-Space Laser Communications IV, J. C. Ricklin, D. G. Voelz, eds., Proc. SPIE 5550, 218-224 (2004).

X. Xiao and D. G. Voelz, “Wave optics simulation of partially coherent beams,” in Free-Space Laser

Communications V, D. G. Voelz, J. C. Ricklin, eds., Proc. SPIE 5892, 219-227 (2005).

D. G. Voelz, K. A. Bush, and P. S. Idell, “Illumination coherence effects in laser-speckle imaging:

modeling and experimental demonstration,” Appl. Opt. 36, 1781-1788 (1997).

10. J. W. Goodman, Statistical Optics, (John Wiley & Sons, 1985).

11. L. Mandel and E. Wolf, “Radiation from sources of any state of coherence,” in Optical Coherence and

Quantum Optics, (Cambridge University, 1995), pp. 229-337.

12. H. Stark and J. W. Woods, Probability and Random Process with Applications to Signal Processing,

(Prentice Hall, 2002), Chap. 7.

1. Introduction

2.

3.

4.

5.

6.

7.

8.

9.

The partially (spatial) coherent beam (PCB) has been studied extensively over the last four

decades as an approach for improving the performance of various laser system applications

including free-space optical laser communications [1,2,3]. The characteristics of PCBs and

methods for producing such beams have been examined in a number of publications [4,5,6].

#72004 - $15.00 USD

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Received 14 June 2006; revised 25 July 2006; accepted 27 July 2006

7 August 2006 / Vol. 14, No. 16 / OPTICS EXPRESS 6986

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Most studies consider the Gaussian Schell-model (GSM) beam [6], which is an analytically

tractable model where the beam field amplitude distribution and the spatial coherence

function are both Gaussian. However, practical PCBs may not follow the GSM theory and are

better examined in detail through some type of numerical simulation approach such as a wave

optics simulation. An example is the pseudo-partially coherent beam (PPCB) suggested for

use in communications applications [7,8]. In addition, explicit modeling of the optical

systems and propagation scenarios associated with PCB applications may be best done

through wave optics simulations. For these reasons we have developed an approach for

modeling the PCB in a wave optics simulation. The approach can accommodate arbitrary

initial beam amplitude and spatial coherence functions.

In this paper we describe the modeling approach and relate the method to the Schell

theorem that describes the Fraunhofer intensity pattern of a PCB. To validate the simulation

approach, we compare intensity patterns produced by the method with analytical results for a

nominal GSM beam propagation and a GSM beam incident on a two-pinhole aperture.

2. Simulation approach

The simulation approach is outlined as follows: a spatially random phase screen is applied to a

deterministic beam field, the result is propagated and the intensity is formed. The process is

repeated many times with different realizations of the phase screen and the resulting intensity

patterns are averaged to produce the PCB intensity. Analytically, we can consider the

sequence of screens to approximate a phase function that is spatially random and time-varying

and the received intensity as a time-averaged quantity. The method is analogous to an

approach for modeling the effects of temporal coherence involving a spectral intensity

average [9].

Consider the field described by

pUtqpU

,();,(

0

=

′

() t

;

qptq

A

,)

, (1)

where U0(p, q) is the deterministic field of an initial beam defined at spatial position (p, q) and

tA is a transmittance function given as

(

jtqptA

exp;,

ξ=

where ξ (p, q; t) is a random phase introduced at time t. After propagation, the time-averaged

intensity of the Fraunhofer pattern is

)()()

tqp

;, , (2)

,)(

2

λ

exp);,(

)(

1

z

),(

2

2

t

dpdqqypx

z

jtqpUyxI

∫∫

⎟

⎠

⎞

⎜

⎝

⎛

+

′

=

π

λ

(3)

where z is the propagation distance, λ is the wavelength and the angular brackets with

subscript t denote a time average. Equation (3) is representative of the process implemented

in our simulation approach. Expanding the squared-modulus quantity and interchanging the

averaging and integration operations we have

1

),(

2211

′′

=

tqpUtqpU

z

λ

()

.)()(

2

λ

exp);,(*);,(

)(

22112121

2 ∫∫∫∫

⎥⎦

⎤

⎢⎣

⎡

−+−

dqdpdq dpyqqxpp

z

jyxI

t

π

(4)

, 2

We introduce the following average and difference notation

, 2/ )(

21

qq

+=

,

21

ppp

−=Δ

qq

=Δ

the first increment, the correlation of the transmittance function

/ )

2

(

1

ppp

+=

q

21

q

−

. Assuming the random phase is stationary in

() t

;

qpqpR

,,,

2211

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Received 14 June 2006; revised 25 July 2006; accepted 27 July 2006

7 August 2006 / Vol. 14, No. 16 / OPTICS EXPRESS 6987

Page 3

=

a result, R(p1, q1, p2, q2; t) = R(Δp, Δq; t) and Eq. (4) can be expressed as [10]

(

∫∫

qp

z

λ

)(

()()()()

tqpjtqpj

;, exp;, exp

2211

ξξ

, depends only on the coordinate differences Δp, Δq. As

)()

ΔΔ

⎥⎦

⎤

⎢⎣

⎡

Δ+ΔΔΔΔΔ=

qpddqy px

z

jtqpRyxI

t

λ

π

2

exp);,(,

1

),(

0

2

U

, (5)

where U0 is the autocorrelation function of the deterministic beam U0

∫∫

⎜

⎝

⎟

⎠

⎞

⎜

⎝

⎛

Δ

−

Δ

−

⎟

⎠

⎞⎛

Δ

+

Δ

+=ΔΔ

qdpd

q

q

p

pU

q

q

p

pUqp

2

,

2

*

2

,

2

),(

000

U

. (6)

Assuming the time-average is equivalent to an ensemble average, in Eq. (5), we could

write

),();,(

qpRtqpR

t

. Equation (5) is a statement of Schell’s theorem [10],

which describes the Fraunhofer intensity profile for a partial spatially coherent beam with a

spatial coherence function corresponding in this case to

approach is consistent with Schell’s theorem.

In the numerical simulation, the time varying transmittance function is approximated by a

sequence of independent, random phase screens

integer i∈ [1,K]. For each realization of the transmittance function, the propagation of the

deterministic field-transmittance product

(

U

0

standard numerical propagation technique such as an FFT-based approach. A K-average of

the resulting intensity frames creates the simulated intensity profile of the PCB.

We note that the transmittance function in Eq. (2) could also be an absorbing screen that

affects the magnitude of the deterministic beam. With the appropriate function, this type of

screen can also be used to simulate a partially coherent beam. However, the random phase

screen has the advantage of preserving the total power of the deterministic beam.

ΔΔ=ΔΔ

),(

qpR

ΔΔ

. Thus our modeling

()()

()

ii

A

qpjqpt

, exp,

ξ=

where i is an

)()

qptqp

A

,,

⋅

can be accomplished using a

3. Phase screen for Gaussian Schell-model beam

The key to applying the simulation technique to a PCB of interest is defining the phase screen

transmittance function since the transmittance autocorrelation function is equivalent to the

beam’s coherence function. For illustration purposes we discuss the GSM beam, which has a

Gaussian coherence function [11]. We begin by modeling the random phase function as

(

tqprtqp

;,,,

=ξ

)()() q

,

pf

⊗

, (7)

where ⊗ indicates a spatial convolution, r(q, p; t) is a spatially uncorrelated or “delta

correlated” random signal with standard deviation σ r and f (p, q) is a Gaussian function that

acts as a filter with normalized area and standard deviation σ f

⎟

⎠

⎟

⎞

⎜

⎝

⎜

⎛

+

−

⋅

=

2

f

22

2

f

2

exp

2

1

),(

qp

qpf

σσπ

. (8)

The parameter σ r is related to the amplitude variation of the screen, something like optical

path length variation, and σ f is a transverse spatial correlation length parameter. Both

parameters have units of length. The convolution of Eq. (7) results in a spatially correlated

random function. Following the approach by Goodman [10], we obtain the transmittance

correlation function

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7 August 2006 / Vol. 14, No. 16 / OPTICS EXPRESS 6988

Page 4

()()

⎟

⎠

⎞

⎜

⎝

⎛−=−=ΔΔ

Δ

2

2211

2

1

exp);,( exp);,(exp),(

ξ

σξ

j

ξ

j

t

tqptqpqpR

, (9)

where );,();,(

1122

tqptqp

ξξξ−=Δ

and note that a multiplicative phase term

()

t

j

ξ

Δ

exp

has been dropped from the right side of Eq. (9) because

ttt

tqptqp

);,();,(

1122

ξξξ−=Δ

= 0.

2

ξ

σΔ

is the variance of ξΔ and

(

ξ

)

() )

q

,( ) 0 , 0 (

ξ

2);,();,(

2

1122

2

ptqptqp

t

ΔΔΓ−Γ=−=

Δξξ

ξσ

, (10)

where

be [12]

() ⋅Γξ

indicates the spatial autocorrelation function of the phase ξ and can be taken to

[] )

q

,(),(),(),(

pfqpfqpRqp

rr

Δ−Δ−⊗ΔΔ⊗ΔΔ=ΔΔΓ

∗

ξ

. (11)

Again, we write

) tp

;

, which is found to be

t

rrrr

tqpRqpR

);,(),(

ΔΔ=ΔΔ

as the autocorrelation function of

(

qr

,

),(),(

2

r

qpqpR

rr

ΔΔ=ΔΔδσ

, (12)

where (

) qp ΔΔ ,

δ

is the delta function. The second part of the right side of Eq. (11) is derived as

⎟

⎠

⎟

⎞

⎜

⎝

⎜

⎛

Δ

2

f

+

σ

Δ

−

⋅

=Δ−Δ−⊗ΔΔ

∗

2

2

f

4

)(

exp

4

1

),(),(

qp

qpfqpf

σπ

. (13)

Substituting Eqs. (12) and (13) in Eq. (11), we find

⎟

⎠

⎟

⎞

⎜

⎝

⎜

⎛

Δ

2

f

+

σ

Δ

−

⋅

=ΔΔΓ

22

2

f

2

r

σ

4

exp

4

),(

qp

qp

π

σ

ξ

. (14)

Consequently, the transmittance autocorrelation function is

⎪⎭

⎪

⎬

⎫

⎪⎩

⎪

⎨

⎧

⎥

⎦

⎥

⎤

⎢

⎣

⎢

⎡

⎟

⎠

⎟

⎞

⎜

⎝

⎜

⎛

Δ

2

f

+

σ

Δ

−−

⋅

−=ΔΔ

22

2

f

2

r

σ

4

exp1

4

exp),(

qp

qpR

π

σ

. (15)

We note if

)4/(

2

f

2

r

σ⋅πσ

>> 1, then Eq. (15) can be approximated as

()

⎥

⎦

⎥

⎤

⎢

⎣

⎢

⎡

Δ+Δ−

≈ΔΔ

2

g

22

2

exp ),(

qp

qpR

σ

, (16)

where the variance of this Gaussian function is

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7 August 2006 / Vol. 14, No. 16 / OPTICS EXPRESS 6989

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2

r

4

f

2

g

8

σ

σ

⋅

π

σ

=

. (17)

Thus, Eq. (16) provides a Gaussian correlation function as required for the GSM beam.

For implementation in a wave optics simulation, the GSM transmittance phase screen can

be created by convolving two arrays defined by the discrete functions

[][] 2/, 2/,2/ , 2/;

2

)(

2

f

)(

exp

2

1

⋅

σ

),(

22

2

f

MMmNNn

mmnn

mnf

−∈−∈

⎟

⎠

⎟

⎞

⎜

⎝

⎜

⎛

Δ⋅+

σ

Δ⋅

−=

π

,

(18)

()

2 / 1

),(

),(

mn

mn rand

Δ

mnr

Δ⋅

=

, (19)

where n and m are integers, N and M are the dimensions of the array (number of samples) in

the horizontal and vertical directions, and, Δn = Ln/N and Δm = Lm/M represent the physical

spacings between the samples where Ln and Lm are the physical lengths of the horizontal and

vertical sides of the array. rand(n,m) denotes a random number function with variance σr

that provides a value for each (n, m) position. The division by (ΔnΔm)1/2 in Eq. (19) is

required for conversion to the unit-less sample domain. The convolution is easily

accomplished in the computer using the Fourier convolution theorem by Fourier transforming

the functions in Eqs. (18) and (19), multiplying the results point-wise, inverse transforming

and finally multiplying by Δn⋅Δm to correctly approximate the convolution integral.

Note there is some flexibility in the choice of σ r and σ f when modeling the GSM beam.

A range of combinations can yield a specific coherence size parameter [Eq. (17)] and satisfy

the approximation required in going from Eq. (15) to Eq. (16). We found in practice the

choice of a relatively larger value of σ f tends to produce a smoother intensity pattern with

fewer number of frames. However, aliasing can become a problem when σ f is chosen too

large where the filter function extends significantly beyond the effective boundaries of the

numerical array.

4. Simulation examples

To verify the wave optics approach, we compared simulation results with analytic results for a

GSM beam. The analytic expressions for the GSM beam are presented here for reference.

For a Gaussian transmitted beam field with initial peak value of 1 and radius w0 (1/e field

value), after propagating a distance z, the optical intensity (irradiance) can be expressed as [11]

⎡

−

Δ

0

)(

wz

with

()() ⎥

⎥

⎦

⎤

⎢

⎣

⎢

Δ⋅

+

=

2

2

22

2

)(

) ( 2

exp

1

),,(

z

yx

zyxI

, (20)

2 / 1

⎤

2

g

2

0

2

2

0

1

2

⋅

1)(

⎥

⎦

⎥

⎢

⎣

⎢

⎡

⎟

⎠

⎟

⎞

⎜

⎝

⎜

⎛

+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+=Δ

w

σ

wk

z

z

, (21)

where k = 2π/λ is the wave number, λ is the nominal wavelength, and σ g

the Gaussian coherence function, which corresponds to the parameter defined in Eq. (17).

Figure 1 shows a comparison of simulation and analytic intensity patterns for two sets of

parameters. The array size was 256×256 points and a uniform random number generator was

2 is the variance of

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7 August 2006 / Vol. 14, No. 16 / OPTICS EXPRESS 6990