propagation of partially coherent beams after a source ... · piers online, vol. 4, no. 1, 2008 84...

PIERS ONLINE, VOL. 4, NO. 1, 2008 84

Propagation of Partially Coherent Beams after a Source PlaneRing Aperture

H. T. Eyyuboglu1, Y. K. Baykal1, and Y. Cai2

1Department of Electronic and Communication Engineering, Cankaya UniversityOgretmenler Cad. 14, Yuzuncuyıl, 06530 Balgat, Ankara, Turkey

2Max-Planck-Research-Group, Institute of Optics, Information and Photonics, University of ErlangenStaudtstr. 7/B2D-91058 Erlangen, Germany

Abstract— The propagation properties of partially coherent beams passing through a sourceplaced ring aperture are examined. The derivation is based on the lowest order general beamformulation, such that our results are applicable to a wide range of beams. In this study, ourfocus is on fundamental Gaussian, cosh-Gaussian, cos-Gaussian, sinh-Gaussian, sine-Gaussianand annular beams. The aperture consists of inner and outer parts, thus the middle hollow partappears in the form of a ring. The propagation environment is turbulent.From the graphical outputs of the beams investigated, it is seen that despite the existence of thecircular ring, during propagation, the beams tend to retain the basic profiles similar to the caseof no aperture, but depending on the inner and outer radius dimensions, the propagated beamsare reduced in intensity levels and become more spread. It is further observed that, when theinner part of the aperture has nonzero radius, ring formations are developed at the outer edgesof the receiver plane intensities.

1. INTRODUCTION

In practical optical communication equipment, an aperture confinement of some type, usually acircular shape, is readily built into the transmission apparatus. Here we also introduce an innerpart so that the eventual configuration is in the form of a ring consisting of an inner and outerradius.

In literature, many sources have investigated the propagation of laser beams with a sourceaperture confinement, usually coupled with the representation of the propagation environment asABCD matrix. In these works, the propagation medium is mostly taken to be free space, ratherthan turbulent atmosphere. A collection of such studies is found in [1–8].

Recently there has also appeared a study concerning the propagation of beams in turbulentatmosphere incorporating a source aperture confinement [9].

Our goal in this paper has been to extend the aperture treatment to the case of selected generalbeams. Hence, this study is aimed at the in-turbulence propagation analysis of beams passingthrough a source aperture configuration. In particular, within the context of general beam, ourgraphical results cover cosh-Gaussian, sine-Gaussian and annular beams.

2. FORMULATION

As shown in [10], the source field of most popular beams contains a summation of two terms. Inradial coordinates, the mutual coherence function of partial coherent version of such beams is givenas

Γs(s1, s2, φ1s, φ2s) = exp{−0.25

σ2s

[s21 + s2

2 − 2s1s2 cos(φ1s − φ2s)]}

2∑

t1=1

2∑

t2=1

Ct1C∗t2 exp{−[kαt1s

21 + jVt1s1(cos φ1s + sinφ1s)]

−[kα∗t2s22 − jV ∗

t2s2(cos φ2s + sinφ2s)]} (1)

s1, s2, φ1s and φ2s refer to the two distinct source coordinates, σs is the partial coherence parameter.k = 2π/λ is the wave number with λ being the wavelength, α = 1/(kα2

0) + j/(2F0) where α0 andF0 respectively indicate radial Gaussian source size and focusing parameter. Via the amplitude

parameters C, displacement parameters V and α together with their t1 and t2 subscripts, it ispossible to configure the general beam so that it will deliver fundamental Gaussian, cosh-Gaussian,cos-Gaussian, sinh-Gaussian, sine-Gaussian and annular beams [11]. We assume that prior topropagation, the source given by Eq. (1) passes through a ring aperture with internal radius of aand outer radius of b as shown in Figure 1.

The average intensity falling on a receiver plane which is L distance apart from the source plane,can be calculated in terms of the Huygens-Fresnel integral as shown below [10]

<Ir(r, φr)> =k2

(2πL)2

b∫

a

2π∫

0

b∫

a

2π∫

0

ds1dφ1sds2dφ2sΓs(s1, s2, φ1s, φ2s)s1s2

× exp{

jk

2L[−2rs1 cos(φr − φ1s) + s2

1 + 2rs2 cos(φr − φ2s)− s22]

}

× exp{−1

ρ20

[s21 + s2

2 − 2s1s2 cos(φ1s − φ2s)]}

(2)

In Eq. (2), the diffraction phenomena is expressed by the first exponential, while the second expo-nential arises due to turbulence. In this second exponential, ρ0 is the coherence length of a sphericalwave propagating in the turbulent medium and under the conditions of Kolmogorov spectrum andquadratic approximation, given by ρ0 = (0.545 C2

nk2L)−3/5, where C2n refractive index structure

constant expressing the turbulence strength. Finally r and φr refer to the radial coordinates onthe receiver plane.

Upon substituting for Γs(s1, s2, φ1s, φ2s) in Eq. (2) from Eq. (1), the integration in Eq. (2) canbe solved with respect to s1 and φ1s in the manner described in [12–14], leaving the following doubleintegral

<Ir(r, φr)> =0.25k2

πL2

2∑

t1=1

2∑

t2=1

∞∑

t=0

(0.25)t

(t!)2Ct1C

∗t2

(kαt1−

jk

2L+

1ρ20

+0.25σ2

s

)−(t+1) t∑

p1=0

(2t)!!2p1(2t− 2p1)!!

×{[(

kαt1 −jk

2L+

1ρ20

+0.25σ2

s

)a2

]t−p1

exp[−

(kαt1 −

jk

2L+

1ρ20

+0.25σ2

s

)a2

]

−[(

kαt1 −jk

2L+

1ρ20

+0.25σ2

s

)b2

]t−p1

exp[−

(kαt1 −

jk

2L+

1ρ20

+0.25σ2

s

)b2

]}

×b∫

a

2π∫

0

ds2dφ2ss2 exp{−[kα∗t2s22 − jV ∗

t2s2(cos φ2s + sinφ2s)]}

exp{

jk

2L

[2rs2 cos(φr − φ2s)− s2

2

]}

× exp[−s2

2

(1ρ20

+0.25σ2

s

)][−2V 2

t1 −k2r2

L2+

4s22

ρ40

+0.25s2

2

σ4s

− 2Vt1kr

L(cos φr+ sinφr)

−(

4jVt1s2

ρ20

+jVt1s2

σ2s

)(cos φ2s+sin φ2s)−

(4jkrs2

Lρ20

+jkrs2

Lσ2s

)cos(φr−φ2s)+

2s22

ρ20σ

2s

]t

(3)

where (n)!! = 1×3× . . . (n−1) or (n)!! = 2×4× . . . (n−1) depending on whether k is odd or even.As explained in [12, 13], it does not serve much use to continue analytically with Eq. (3), thus it iskept in the present form. Instead the routine mentioned in [12, 13] is employed for relatively rapidevaluation.

3. RESULTS AND DISCUSSIONS

Although, the formulation in Eq. (3) can support a variety of beams, here due to space limitations,we restrict our results to cosh-Gaussian, sine-Gaussian and annular beams. Thus, in this sectionthe intensity plots are offered for cosh-Gaussian, sine-Gaussian and annular beams, while theypropagate in turbulent atmosphere after having passed though the circular aperture of the source

Source Plane Receiver Plane

sx

sy

s

rx

ry

r

Aperture Ring

Pro

pag

atio

n A

xis

rs

a bS

ou

rce

Turbulent Atmosphere

φφ

L

Figure 1: General layout of source and receiver planes and the source ring aperture configuration.

plane. To permit cross-comparisons between the different beam types, for the cosh-Gaussian andsine-Gaussian beams, we use the common settings of, α0 = 1 cm, F0 →∞, V = 200 m−1, σs →∞,C2

n = 10−15 m−2/3. This means that collimated and coherent beams are examined in this study. Forannular beams on the other hand, the above settings are modified in the following way, α01 = 1 cm,α02 = 0.8 cm, V = 0. Here α01 and α02 consecutively refer to primary and the secondary sourcesizes. The individual intensity illustrations are arranged to coincide with propagation lengths ofL = 0, 1, 2, 5 km. The intensity graphs are drawn on the Cartesian transverse planes, where sx,sy and rx, ry will respectively correspond to the Cartesian counterparts of the source and receiverradial coordinates s, φs and r, φr encountered in Eqs. (1) and (3). The vertical axes of the plotsare labelled as IsN or < IrN > meaning that all intensity levels are normalized with respect to thepeak value of the source intensity obtained from Eq. (1) by setting Is(s, φs) = Γs(s, s, φs, φs). It is

-2

0

2 -2

0

20

0.5

1

s y

axisin

cm

Source intensity plot for a cosh-Gaussian beam

sx axis in cm

IsN

-50

5 -5

0

50

0.02

0.04

0.06

0.08

r y

axisin

cm

Receiver intensity plot for a cosh-Gaussian beam at L = 1 km

rx axis in cm



-100

10-10

0

100

0.01

0.02

r y

axisin

cm


rx axis in cm



-40-20

020

40 -40

-20

0

20

400

1

2

3

x 10-3

r y

axisin

cm


rx axis in cm



a = 0 , b = 2 cm

Plot 1 Plot 2

Plot 3 Plot 4

Figure 2: Progress of an cosh-Gaussian beam along propagation axis with aperture ring dimensions of a = 0,b = 2 cm.

worth pointing out that in order to obtain a problem-free (or reliable) intensity profile, as many as26 terms had to be included in the most inner summation appearing in the first line of Eq. (3).

Figure 2 shows the progress of a cosh-Gaussian beam along the propagation axis with theradius of interior ring, a = 0, that of the outer ring, b = 2 cm. Judging by the source intensityplot (Plot 1), it is seen that an outer aperture radius of b = 2 cm allows almost all the sourceintensity to be emitted from the source plane. In this manner, when Figures 2 and 3 are compared,where the latter shows the propagation of the same cosh-Gaussian beam with the outer radiusof the aperture extending to infinity, hardly any noticeable difference is found. A closer jointexamination of Figures 2 and 3 will reveal that, the finite aperture (i.e., Figure 2) neverthelessimposes more spreading on the beam, whilst also lowering its peak amplitudes. This is evidentfrom the comparison of Plots 2, 3 and 4 of Figures 2 and 3.

-2

0

2 -2

0

20

0.5

1

s y

axisin

cm


sx axis in cm

IsN

-50

5 -5

0

50

0.05

0.1

r y

axisin

cm


rx axis in cm



-100

10 -10

0

100

0.01

0.02

0.03

r y

axisin

cm


rx axis in cm



-40-20

020

40 -40

-20

0

20

400

2

4

6

x 10-3

r y

axisin

cm


rx axis in cm



Plot 1 Plot 2

Plot 3 Plot 4

a = 0 , b =∞

Figure 3: Progress of a cosh-Gaussian beam along propagation axis with aperture ring dimensions of a = 0,b = ∞.

In Figure 4, we introduce a finite interior part, that is a = 1 cm, for the source aperture of thecosh-Gaussian beam. From Figure 4, it is seen that the existence of the nonzero inner part givesrise to the emergence of bigger side lobes, formation of outer rings and more beam spreading. It isinteresting to note from Figures 2 and 4 that, the presence of an aperture at the source plane doesnot hinder the beam evolution phases. That is, the evolutions discussed and demonstrated in [15]for the unapetured propagation of cosh-Gaussian beam are valid here as well. Consequently evenwith an aperture confinement, as observed from Figures 2 and 4, the cosh-Gaussian beam will stilltransform into a cos-Gaussian beam upon propagation.

In Figure 5, the progress of the sine-Gaussian beam is displayed for and aperture configurationof a = 0, b = 1 cm. Figure 5 proves that, this particular sine-Gaussian beam does not seem to beaffected by the presence of the aperture, such that it conveniently transforms into a sinh-Gaussianbeam after propagation. Considering the outer aperture size of 1 cm in relation foot-print of sourcebeam from Plot 1 of Figure 5, it easy to see that an aperture opening of 1 cm is able to transmitthe whole source beam intensity. Figure 6 shows the sine-Gaussian beam propagation for the caseof a = 1 cm, b = 2 cm. According to Figure 6, sine-Gaussian beam will again yield a sinh-Gaussianbeam after propagation, but due to the existence of the inner part of the aperture, outer rings willbe formed as well.

-2

0

2 -2

0

20

0.5

1

s y

axisin

cm


sx axis in cm

IsN

-10

0

10 -10

0

100

0.02

0.04

r y

axisin

cm


rx axis in cm



-20

0

20 -20

0

200

0.005

0.01

r y

axisin

cm


rx axis in cm



-50

0

50 -50

0

500

0.5

1

1.5

x 10-3

r y

axisin

cm


rx axis in cm



a = 1 cm , b = 2 cm

Plot 1 Plot 2

Plot 3 Plot 4

Figure 4: Progress of a cosh-Gaussian beam along propagation axis with aperture ring dimensions of a = 1 cm,b = 2 cm.

-2

0

2 -2

0

20

0.5

1

s y

axisin

cm

Source intensity plot for a sine-Gaussian beam

sx axis in cm

IsN

-10

0

10 -10

0

100

2

4

6

x 10-3

r y

axisin

cm

Receiver intensity plot for a sine-Gaussian beam at L = 1 km

rx axis in cm



-20

0

20 -20

0

200

0.5

1

1.5

x 10-3

r y

axisin

cm


rx axis in cm



-50

0

50 -50

0

500

1

2

3

x 10-4

r y

axisin

cm


rx axis in cm



a = 0 , b = 1 cm

Plot 1 Plot 2

Plot 3 Plot 4

Figure 5: Progress of a sine-Gaussian beam along propagation axis with aperture ring dimensions of a = 0,b = 1 cm.

-2

0

2 -2

0

20

0.5

1

s y

axisin

cm

Source intensity plot for a sine-Gaussian beam

sx axis in cm

IsN

-10

0

10 -10

0

100

0.5

1

1.5

x 10-3

r y

axisin

cm


rx axis in cm



-20

0

20 -20

0

200

1

2

3

x 10-4

r y

axisin

cm


rx axis in cm



-50

0

50 -50

0

500

2

4

6

x 10-5

r y

axisin

cm


rx axis in cm



a = 1 , b = 2 cm

Plot 1 Plot 2

Plot 3 Plot 4

Figure 6: Progress of a sine-Gaussian beam along propagation axis with aperture ring dimensions of a = 1 cm,b = 2 cm.

-2-1

01

2 -2

0

20

0.5

1

s y

axisin

cm

Source intensity plot for a annular-Gaussian beam

sx axis in cm

IsN

-10-5

05

10 -10

0

100

0.02

0.04

0.06

r y

axisin

cm

Receiver intensity plot for a annular-Gaussian beam at L = 1 km

rx axis in cm



-20-10

010

20 -20

0

200

0.005

0.01

0.015

r y

axisin

cm


rx axis in cm



-40-20

020

40 -40-20

0

20400

1

2

x 10-3

r y

axisin

cm


rx axis in cm



a = 1 cm , b = 2 cm

Plot 1 Plot 2

Plot 3 Plot 4

Figure 7: Progress of an annular beam along propagation axis with aperture ring dimensions of a = 1 cm,b = 2 cm.


Finally we provide the illustration of annular beam. To this end, Figure 7 shows that, when asource aperture confinement of a = 1 cm, b = 2 cm is used, annular beam will generate an outerring of a complete circle in addition to the central peak. This behaviour is similar to unaperturedsituations [16], except that the outer ring persists for greater distances.

4. CONCLUSION

We have analyzed the effect of a ring aperture placed on the source plane of the general beampropagating in turbulent atmosphere. From the illustrations presented, it is observed that theaperture does not have a major effect on the beam profile provided that its outer radius is in therange of beam foot-print and has no interior part. When the ring aperture contains an inner parthowever, it causes somewhat excessive spreading and the formation of outer rings in the receivedintensity profiles.

ACKNOWLEDGMENT

Y. Cai gratefully acknowledges support from the Alexander von Humboldt Foundation.

REFERENCES

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3. Ji, J. and B. Lu, “Matrix formulation of axis-symmetric laser beams through paraxial ABCDsystem containing hard-edged apertures: A comparative study,” Optik, Vol. 116, No. 5, 219–225, 2005.

4. Zhang, B., Y. Zhang, and J. Li, “Spectral intensity characteristics of partially coherent flat-tened multi-Gaussian beams with an axial shadow passing though paraxial ABCD opticalsystems,” Opt. Commun., Vol. 279, No. 1, 7–12, 2007.

5. Du, X. and D. Zhao, “Propagation of decentered elliptical Gaussian beams in apertured andnonsymmetrical optical systems,” J. Opt. Soc. Am., Vol. 23, No. 3, 625–631, 2006.

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9. Chu, X., Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circularaperture in turbulent atmosphere,” Appl. Phys. B, Vol. 87, No. 3, 547–552, 2007.

10. Eyyuboglu, H. T. and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun., Vol. 278, No. 1, 17–22, 2007.

11. Arpali, C., C. Yazıcıoglu, H. T. Eyyuboglu, S. A. Arpali, and Y. Baykal, “Simulator for general-type beam propagation in turbulent atmosphere,” Opt. Exp., Vol. 14, No. 20, 8918–8928, 2006.

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propagation of partially coherent beams after a source ... · piers online, vol. 4, no. 1, 2008 84...

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