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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 20 — Oct. 2, 2006
  • pp: 8918–8928
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Simulator for general-type beam propagation in turbulent atmosphere

Çağlar Arpali, Canan Yazicioğlu, Halil Tanyer Eyyuboğlu, Serap Altay Arpali, and Yahya Baykal  »View Author Affiliations


Optics Express, Vol. 14, Issue 20, pp. 8918-8928 (2006)
http://dx.doi.org/10.1364/OE.14.008918


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Abstract

A simulator is designed in MATLAB code which gives the propagation characteristics of a general-type beam in turbulent atmosphere. When the required source and medium parameters are entered, the simulator yields the average intensity profile along the propagation axis in a video format. In our simulator, the user can choose the option of a “user defined beam” in which the source and medium parameters are selected as requested by the user by entering numerical values in the relevant menu boxes. Alternatively, the user can proceed with the option of “pre-defined beam” in which the average intensity profiles of beams such as annular, cos-Gaussian, sine-Gaussian, cosh-Gaussian, sinh-Gaussian, their higher-order counterparts and flat-topped can be observed as they propagate in a turbulent atmosphere. Some samples of the simulator output are presented.

© 2006 Optical Society of America

1. Introduction

2. Formulation

A general beam, excluding the multimode composition, emitted from a source co-centric with its own plane will be given as [12

12. Y. Baykal, “Formulation of correlations for general-type beams in atmospheric turbulence,” J. Opt. Soc. Am. A 23, 889–893 (2006). [CrossRef]

]

u(s)==1NAexp(jθ)Hn(axsx+bxt)exp[(0.5kαxsx2+jVxsx)]
×Hm(aysy+bx)exp[(0.5kαysy2+jVytsy)],
(1)

αx=1(kαsx2)+jFx,αy=1(kαsy2)+jFy,
(2)

where ere k=2π/λ is the wave number with λ being the wavelength and j=(-1)0.5.

The average intensity 〈I(p,L)〉, of a general beam on a receiver plane located at L distance away from the source can be found from extended Huygens-Fresnel integral in the following way [13

13. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, (SPIE, Bellingham, Washington, 2005).

]

I(p,L)=(k2πL)2ds12ds22u(s1)u*(s2)
×exp{jk[(ps1)2(ps2)2]2L}exp[ψ(s1,p)+ψ*(s2,p)],
(3)

where the ensemble average term within the integrand is [2

2. 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). [CrossRef]

]

exp[ψ(s1,p)+ψ*(s2,p)]exp[ρ02(s1s2)2],
(4)

with ψ being the fluctuations of the complex amplitude, px and py are the x and y components of the receiver plane vector p, such that p=(px ,py ), ρ 0=(0.545 Cn2k 2 L)-3/5 is the coherence length of a spherical wave propagating in the turbulent medium and Cn2 , the structure constant. We note that Eq.(4) is derived under the quadratic approximation for the Rytov’s phase structure function.

The source field u(s 1), when multiplied by its conjugate u *(s 2), indicated by the sign *, will be

u(s1)u*(s2)=1=1N2=1NA1A2*exp[j(θ1θ2)]
×Hn1(ax1s1x+bx1)exp[(0.5kαx1s1x2+jVx1s1x)]
×Hm1(ay1s1y+by1)exp[(0.5kαy1s1y2+jVy1s1y)]
×Hn2*(ax2s2x+bx2)exp[(0.5kαx2*s2x2jVx2*s2x)]
×Hm2*(ay2s2y+by2)exp[(0.5kαy2*s2y2jVy2*s2y)],
(5)

After substituting Eqs. (4) and (5) into Eq. (3) and solving the resulting integral by the repeated use of Eq. 3.462.2 of Ref. [14

14. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, New York, 2000).

], which is dxxnexp(px2+2qx)=n!exp(q2p)(πp)0.5(qp)nt=0[n2]pt[(4q2)t(n2t)!(t)!],, we obtain the following expression for the average intensity at the receiver plane

I(p,L)=b2ρ04[ρ04(asx1jb)(asx2*+jb)1]1/2ExEySxSy[ρ04(asy1jb)(asy2*+jb)1]1/2,
(6)

where b=k2L,asx1=0.5kαx1+1ρ02,asx2=0.5kαx2+1ρ02, asy1 and asy2 are the y counterparts of asy1 and asy2,

Ex=exp({(Vx1+2bpx)24(asx1jb)+[ρ02(Vx2*+2bpx)(asx1jb)Vx12bpx]24(asx1jb)[ρ04(asx1jb)(asx2*+jb)1]}),
(7)

Ey is attained by changing all x subscripts to y in Ex ,

Sx=1=1N2=1Nx1=0[n12]nx1=0n12x1kx1=0[nx12]nx11=0nx12kx1x2=0[n22]nx2=0n22x2kx2=0[(nx11+nx2)2]Tx1Tx2A1A2*exp[j(θ1θ2)]
×(j)nx1+nx22kx12kx2(1)x1+x22n1+n2x1x2nx1nx2
×(n12x1)(n22x2)(n12x1nx1)(nx12kx1nx11)(n22x2nx2)(nx1)!(nx12kx1)!(kx1)!
×(nx11+nx2)!(nx11+nx22kx2)!(kx2)!(ax1)nx1(ax2*)nx2(bx1)n12x1nx1(bx2*)n22z2nx2(ρ02)nx2
×[ρ04(asx1jb)(asx2*+jb)1]nx11nx2+x2(asx1jb)nx1+x1+x2(Vx1+2bpx)nx12kxnx11
×[ρ02(Vx2*+2bpx)(asx1jb)+Vx1+2bpx]nx11+nx22kx2,
(8)

Sy is constructed by changing all x subscripts to y in Sx given by Eq. (8), Tℓx 1=1×3×....(2xi -1) for x 1≠0 where i=1,2, all appearances in the form of (B1B2) represent binomial coefficients, hence (B1B2)=B1![(B1B2)!B2!], ! is the factorial notation, the square brackets, [], placed in the upper limits of some summations mean that the integral part of the expression within the square brackets is to be taken.

3. Results and discussion

We have designed a simulator that will generate the average receiver intensity of a general beam, when the necessary source and medium parameters, namely N, A , θ , αsxℓ , αsyℓ , Fxℓ , Fyℓ , n , m , Vxℓ , Vyℓ , λ,Cn2 ,L are supplied to Eq. (6). Presently, the remaining parameters are internally set as αxℓ =1/αsxℓ , ayℓ =1/αsyℓ , bxℓ =byℓ =0.

Our simulator offers the choice of “user defined beam” and “pre-defined beam”. We emphasize that Eq. (6) together with Eqs. (7), (8) along with the associated definitions given in the text unify the expressions for propagation of a wide variety of basic beam types into a single expression facilitating numerical investigation. This unification is realized by designing a simulator that implements the propagation of general-type beams as provided in Eq. (6). A selection of such beams, also named as “pre-defined” types when running the simulator, are tabulated below with the relevant parameter usage.

Table 1. Parameter Usage for Pre-Defined Beam Types

table-icon
View This Table

For all beams, there exists x-y symmetry. In this context, n, m are positive integers, αs , F and V are positive numeric values conforming to the following equalities for cos-Gaussian, cosh-Gaussian, sine-Gaussian, sinh-Gaussian, Hermite-cosh-Gaussian, Hermite-sine-Gaussian, Hermite-sinh-Gaussian, Hermite-cos-Gaussian beams, n=n 1=n 2, m=m 1=m 2, αs =αsx 1=αsx 2=αsy 1=αsy 2, F=Fx 1=Fx 2=Fy 1=Fy 2, V=Vx 1=Vx 2=Vy 1=Vy 2. For higherorder annular beam αs 1>αs 2.

Fig. 1. User interface for (a) user defined beam and (b) pre-defined beam (annular case).

Below, sample outputs from the simulator are provided. Figure 2 illustrates three pictures of a general beam during the program run captured at L=1 km with the settings given in the Fig. Here the upper left picture is the intensity plot of the beam viewed perpendicular to the propagation axis, where more brightness means higher intensity spots. The second picture to the right in the upper row refers to the contour plot of the beam. Finally the right one of the second row is the 3D picture of the beam. Figure 3 displays the same for a pre-defined Hermite-cosh-Gaussian beam at L=2 km, where all the settings are again given in the Fig. In Figs. 2 and 3, 〈IrN〉 means that for the receiver intensity, the following normalization is applied

IrN=I(p,L)Max[u(s)u*(s)],
(9)

where the denominator of the right hand side corresponds to the maximum value of the source plane intensity. We note that Fig. 3 matches exactly the second intensity profile picture in Fig. 6 of Ref. [7

7. H. T. Eyyuboğlu and Y. Baykal, “Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere,” J. Opt. Soc. Am. A 22, 2709–2718 (2005). [CrossRef]

].

Fig. 2. Different views of a general beam with given parameters along the propagation axis at L=1 km.
Fig. 3. Different views of an Hermite-cosh-Gaussian beam along the propagation axis at L=2 km.
Fig. 4. (2.0 MB) Progress of a general beam belonging to Fig. 2 along the propagation axis at (a) L=0 km, (b) 1 km, (c) 3 km and (d) 5 km.
Fig. 5. (1.81 MB) Progress of an Hermite-sine-Gaussian beam along the propagation axis at (a) L=0 km, (b) 1 km, (c) 3 km and (d) 5 km.
Fig. 6. (1.82 MB) Progress of a higher-order annular beam along the propagation axis at (a) L=0 km, (b) 1 km, (c) 3 km and (d) 5 km.
Fig. 7. (1.27 MB) Progress of a flat-topped beam along the propagation axis at (a) L=0 km, (b) 1 km, (c) 6 km and (d) 10 km.

4. Conclusion

A simulator is designed to provide the progress of the average received intensity profile of a general-type beam as it propagates in a turbulent horizontal link. This progress is shown on the screen at regular intervals of propagation length during the running of the program and the entire intensity plots are recorded in video format. Through the use of this simulator, after entering the appropriate source and medium parameters, the user is able to observe the average received intensity profiles in the above described formats in the form of “user defined beam” and “pre-defined beam” such as annular, cos-Gaussian, sine-Gaussian, cosh-Gaussian, sinh-Gaussian, their higher-order counterparts and flat-topped beams when propagating in turbulent atmosphere. As limiting case solutions, if the structure constant is entered as zero, our simulator also yields the intensity profiles of general-type beams in video format along the free space path.

References and links

1.

Z. I. Feizulin and Y. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron 10, 33–35 (1967). [CrossRef]

2.

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). [CrossRef]

3.

R. L. Phillips and L. C. Andrews, “Spot size and divergence for Laguerre Gaussian beams of any order,” Appl. Opt. 22, 643–644 (1983). [CrossRef] [PubMed]

4.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002). [CrossRef]

5.

H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. 44, 976–983 (2005). [CrossRef] [PubMed]

6.

H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Express 12, 4659–4674 (2004). [CrossRef] [PubMed]

7.

H. T. Eyyuboğlu and Y. Baykal, “Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere,” J. Opt. Soc. Am. A 22, 2709–2718 (2005). [CrossRef]

8.

Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam in a turbulent atmosphere,” Opt. Lett. 31, 568–570 (2006). [CrossRef] [PubMed]

9.

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14, 1353–1367 (2006). [CrossRef] [PubMed]

10.

H. T. Eyyuboğlu, Ç. Arpali, and Y. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14, 4196–4207 (2006). [CrossRef] [PubMed]

11.

H. T. Eyyuboğlu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. 26425–34 (2006). [CrossRef]

12.

Y. Baykal, “Formulation of correlations for general-type beams in atmospheric turbulence,” J. Opt. Soc. Am. A 23, 889–893 (2006). [CrossRef]

13.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, (SPIE, Bellingham, Washington, 2005).

14.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, New York, 2000).

OCIS Codes
(010.1290) Atmospheric and oceanic optics : Atmospheric optics
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(010.3310) Atmospheric and oceanic optics : Laser beam transmission

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: August 8, 2006
Revised Manuscript: September 12, 2006
Manuscript Accepted: September 19, 2006
Published: October 2, 2006

Citation
Çaglar Arpali, Canan Yazicioglu, Halil T. Eyyuboglu, Serap A. Arpali, and Yahya Baykal, "Simulator for general-type beam propagation in turbulent atmosphere," Opt. Express 14, 8918-8928 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-8918


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References

  1. Z. I. Feizulin and Y. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron 10, 33-35 (1967). [CrossRef]
  2. 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). [CrossRef]
  3. R. L. Phillips and L. C. Andrews, "Spot size and divergence for Laguerre Gaussian beams of any order," Appl. Opt. 22, 643-644 (1983). [CrossRef] [PubMed]
  4. C. Y. Young, Y. V. Gilchrest, and B. R. Macon, "Turbulence induced beam spreading of higher order mode optical waves," Opt. Eng. 41, 1097-1103 (2002). [CrossRef]
  5. H. T. Eyyuboğlu and Y. Baykal, "Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere," Appl. Opt. 44, 976-983 (2005). [CrossRef] [PubMed]
  6. H. T. Eyyuboğlu and Y. Baykal, "Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere," Opt. Express 12, 4659-4674 (2004). [CrossRef] [PubMed]
  7. H. T. Eyyuboğlu and Y. Baykal, "Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere," J. Opt. Soc. Am. A 22, 2709-2718 (2005). [CrossRef]
  8. Y. Cai and S. He, "Average intensity and spreading of an elliptical Gaussian beam in a turbulent atmosphere," Opt. Lett. 31, 568-570 (2006). [CrossRef] [PubMed]
  9. Y. Cai and S. He, "Propagation of various dark hollow beams in a turbulent atmosphere," Opt. Express 14, 1353-1367 (2006). [CrossRef] [PubMed]
  10. H. T. Eyyuboğlu, Ç. Arpali, and Y. Baykal, "Flat topped beams and their characteristics in turbulent media," Opt. Express 14, 4196-4207 (2006). [CrossRef] [PubMed]
  11. H. T. Eyyuboğlu, S. Altay, and Y. Baykal, "Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence," Opt. Commun. 26425-34 (2006). [CrossRef]
  12. Y. Baykal, "Formulation of correlations for general-type beams in atmospheric turbulence," J. Opt. Soc. Am. A 23, 889-893 (2006). [CrossRef]
  13. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, (SPIE, Bellingham, Washington, 2005).
  14. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, New York, 2000).

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