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Optics Express

  • Editor: Martijn de Sterke
  • Vol. 16, Iss. 20 — Sep. 29, 2008
  • pp: 15834–15846
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Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere

Yangjian Cai, Olga Korotkova, Halil T. Eyyuboğlu, and Yahya Baykal  »View Author Affiliations


Optics Express, Vol. 16, Issue 20, pp. 15834-15846 (2008)
http://dx.doi.org/10.1364/OE.16.015834


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Abstract

Propagation of stochastic electromagnetic beams through paraxial ABCD optical systems operating through turbulent atmosphere is investigated with the help of the ABCD matrices and the generalized Huygens-Fresnel integral. In particular, the analytic formula is derived for the cross-spectral density matrix of an electromagnetic Gaussian Schell-model (EGSM) beam. We applied our analysis for the ABCD system with a single lens located on the propagation path, representing, in a particular case, the unfolded double-pass propagation scenario of active laser radar. Through a number of numerical examples we investigated the effect of local turbulence strength and lens’ parameters on spectral, coherence and polarization properties of the EGSM beam.

© 2008 Optical Society of America

1. Introduction

Over the past several decades, scalar partially coherent beams have found wide applications in optical projection, laser scanning, inertial confinement fusion, free space optical communications, imaging applications and nonlinear optics [1

1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press,1995).

9

9. Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15, 15480–15492 (2007). [CrossRef] [PubMed]

]. Gaussian Schell-model beam (GSM) is a conventional mathematical model for describing a typical scalar partially coherent beam whose spectral density and spectral degree of coherence are Gaussian functions [10

10. E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978). [CrossRef]

11

11. F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1978). [CrossRef]

]. Generation and propagation of a scalar GSM beam in various media and in imaging and non-imaging optical systems are now well understood [12

12. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun 41, 383–387 (1982). [CrossRef]

19

19. Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31, 685–687 (2006) [CrossRef] [PubMed]

].

In the past decades the two important properties of light waves: coherence and polarization were studied separately (cf. [1

1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press,1995).

], [20

20. C. Brosseau, Fundamentals of polarized light-a statistical approach (Wiley, New York, 1998).

]). After the unified theory of coherence and polarization was formulated [21

21. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).

] it became evident that these properties are interrelated. Scalar Gaussian Schell-model (GSM) beams were then extended to electromagnetic domain (called EGSM beams) and studied in details [21

21. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).

38

38. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008). [CrossRef]

].

Propagation characteristics of different types of beams propagating in the turbulent atmosphere are of interest for optical communications, imaging and remote sensing applications [28

28. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004). [CrossRef]

, 33

33. H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52, 1611–1618 (2005). [CrossRef]

, 38

38. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008). [CrossRef]

52

52. H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B 89, 91–97 (2007).

]. In Refs. [28

28. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004). [CrossRef]

], [33

33. H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52, 1611–1618 (2005). [CrossRef]

] and [38

38. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008). [CrossRef]

] various statistical properties of EGSM beams propagating in the atmosphere have been studied. More importantly, it was found that under suitable conditions the EGSM beams may have reduced levels of intensity fluctuations (scintillations) compared to the scalar GSM beams (i.e. fully polarized GSM beam) [38

38. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008). [CrossRef]

], which makes them attractive for free-space optical communications.

In this paper, we analyze various phenomena arising on propagation of an EGSM beam through a paraxial ABCD optical system in a turbulent atmosphere by deriving relating analytic formulas. To illustrate the usefulness of our analytic results we apply them to the case when the ABCD system consists of a single lens which may be located anywhere between the source and the receiving system [see Fig. 1(a)]. Such example may also be used for study of reflection of a beam from a mirror target in a bistatic mode [see Fig. 1(b)], the problem that was studied previously only in the framework of scalar theory [57

57. V. A. Banakh and L. V. Mironov, LIDAR in a turbulent atmosphere (Artech House, Dedham1987).

]. We will pay special attention to spectral properties, and the states of coherence and polarization in such systems.

2. Theory

Within the validity of the paraxial approximation, propagation of a laser beam through an astigmatic ABCD optical system situated in the turbulent atmosphere can be studied with the help of the following generalized Huygens-Fresnel integral [54

54. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am A 4, 1931 (1987). [CrossRef]

56

56. X. Chu, “Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express 15, 17613–17618 (2007). [CrossRef] [PubMed]

]

E(ρ1,l)=iλ[det(B)]1/2E(r1,0)
×exp[ik2(r1TB1Ar12r1TB1ρ1+ρ1TDB1ρ1)+Ψ(r1,ρ1)]dr1,
(1)

where det stands for the determinant of a matrix, E(r 1,0) and E(ρ1,l) are the electric fields of the laser beam in the source plane (z=0) and the output plane (z=l), respectively. r T 1=(x 1 y 1) and ρ T 1=(ρ 1x ρ 1y) with r 1 and ρ 1 being the position vectors in the source plane and output planes, Ψ(r 1,ρ 1) is the Rytov perturbation being the random part of the complex phase of the beam induced by atmospheric fluctuations, k=2π/λ is the wave number, λ is the wavelength of light. Here we note that A,B,C and Dare the 2×2 sub-matrices of the astigmatic optical system [58, 59], satisfying the following Luneburg relations that describe the symplecticity of an astigmatic optical system [60

60. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, 1964).

]

(B1A)T=B1A,(B1)T=(CDB1A),(DB1)T=DB1.
(2)

Denoting the optical fields at the two arbitrary points r 1,r 2 in the source plane by E(r 1),E(r 2) and the optical fields at the two arbitrary points ρ 1,ρ 2 in the output plane by E(ρ 1),E(ρ 2), respectively, we may write the expressions for the cross-spectral density in the source and output planes as:

W(r1,r20)=E(r1,0)E*(r2,0),W(ρ1,ρ2,l)=E(ρ1,l)E*(ρ2,l).
(3)

Here “<>” denotes ensemble average. Using Eqs. (1) and (3) we find that the cross-spectral density of a scalar partially coherent beam propagating through a general astigmatic optical system is given by the expression

W(ρ1,ρ2,l)=1λ2[det(B)]1/2[det(B*)]1/2W(r1,r2,0)exp[ik2(r1TB1Ar12r1TB1ρ1+ρ1TDB1ρ1)]
×exp[ik2(r2T(B*)1A*r22r2T(B*)1ρ2+ρ2TD*(B*)1ρ2)]exp[Ψ(r1,ρ1)+Ψ*(r2,ρ2)]dr1dr2,
(4)

where “*” denotes the complex conjugate. The expression in the angular brackets in Eq. (4) can be expressed as [5

5. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802 (2002). [CrossRef]

, 42

42. 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]

, 43

43. J. C. Leader, “Atmospheric propagation of partially coherent radiation”, J. Opt. Soc. Am. 68, 175–185 (1978). [CrossRef]

, 54

54. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am A 4, 1931 (1987). [CrossRef]

55

55. H. T. Yura and S. G. Hanson, “Second-order statistics for wave propagation through complex optical systems,” J. Opt. Soc. Am A 6, 564 (1989). [CrossRef]

]

exp[ψ(r1,ρ1)+ψ*(r2,ρ2)]=exp[(r1r2)2ρ02(r1r2)(ρ1ρ2)ρ02(ρ1ρ2)2ρ02],
(5)

ρ0 being the coherence length of a spherical wave propagating in the turbulent medium given by the expression [54

54. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am A 4, 1931 (1987). [CrossRef]

56

56. X. Chu, “Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express 15, 17613–17618 (2007). [CrossRef] [PubMed]

]

ρ0=Det[B]12(1.46k2Cn20lDet[B(z)]56dz)35,
(6)

Here B(z) is the sub-matrix for back-propagpation from output plane to propagation distance z [54

54. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am A 4, 1931 (1987). [CrossRef]

56

56. X. Chu, “Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express 15, 17613–17618 (2007). [CrossRef] [PubMed]

], and C 2 n is the structure constant of turbulent atmosphere. Here, following [42

42. 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]

56

56. X. Chu, “Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express 15, 17613–17618 (2007). [CrossRef] [PubMed]

], we have applied the Kolmogorov turbulence spectrum and a quadratic approximation for wave structure function.

After some arrangement Eq. (4) can be expressed in the tensor form as

W(ρ,l)=k24π2[det(B)]12W(r,0)exp[ik2(rTB1Ar2rTB1ρ+ρTDB1ρ)]
×exp[ik2rTPrik2rTPρik2ρTPρ]dr,
(7)

where d =d r 1 d r 2, T=(r T 1 r T 2), ρ˜ T=(ρ T 1 ρ T 2) and

A=(A0I0IA*),B=(B0I0IB*),C=(C0I0IC*),D=(D0I0ID*),P=2ikρ02(IIII),
(8)

I being a 2×2 unit matrix. In the absence of turbulence (ρ 0→∞, i.e., C 2 n=0), =0, Eq. (7) reduces to the generalized Collins formula for treating propagation of a partially coherent beam through a general astigmatic ABCD optical system in free space [16

16. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27, 216–218 (2002). [CrossRef]

]. Due to its generality, Eq. (7) can be used to investigate the paraxial propagation of any partially coherent beam through a general astigmatic ABCD optical system in a turbulent atmosphere.

Now we apply Eq. (7) to study propagation of an EGSM beam through a general astigmatic ABCD optical system in a turbulent atmosphere. The second-order statistical properties of the EGSM beam can be characterized by the 2×2 cross-spectral density matrix W̃(r 1,r 2,0) specified at any two points with position vectors r 1 and r 2 in the source plane with elements [21

21. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).

25

25. E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).

]

Waβ(r1,r2,0)=AaAβBaβexp[r124σa2r224σβ2(r1r2)22δaβ2],(a=x,y;β=x,y)
(9)

Here Aα, Bαβ=|Bαβ|exp(αβ)=B * αβ, σα and βαδ are independent of position but, in general, depend on the frequency. In Eq. (9) and everywhere else in this paper we have omitted the dependence on the oscillation frequency for conciseness. The nine real parameters Ax, Ay, σx, σy, |Bxy|, ϕxy, δxx, δyy and δxy entering the general model are shown to satisfy several intrinsic constraints and obey some simplifying assumptions (e.g. the phase difference between the x- an y-components of the field is removable, i.e. ϕαβ=0[29

29. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 7, 232–237 (2005). [CrossRef]

,30

30. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005). [CrossRef]

]. The elements of the cross-spectral density matrix in Eq. (9) can alternatively be expressed in the following tensor form [16

16. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27, 216–218 (2002). [CrossRef]

, 27

27. Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A: Pure App. Opt. 5, 453–459 (2003). [CrossRef]

]

Waβ(r)=AαAβBaβexp[ik2rTM0αβ1r],(α=x,y;β=x,y),
(10)

where k=2π/λ is the wave number, λ is the wavelength, T=(r T 1 r T 2), and the 4×4 tensor has the form

M0αβ1=[1ik(12σa2+1δαβ2)Iikδαβ2Iikδαβ2I1ik(12σβ2+1δαβ2)I],
(11)

Substituting from Eq. (10) into Eq. (7), after some vector integration and tensor operations, we obtain (see Appendix A) the following expression for the elements of the cross-spectral density matrix of a EGSM beam after propagating through an astigmatic ABCD optical system in a turbulent atmosphere

Wαβ(ρ,l)=AαAβBαβ[det(A+BM0αβ1+BP)]12exp[ik2ρTM1αβ1ρ]
exp[ik2ρTPρik2ρT(B1T14PT)(M0αβ1+B1A+P)1Pρ],
(12)

where

M1αβ1=(C+DM0αβ1+DP)(A+BM0αβ1+BP)1,
(13)

In the absence of turbulence (when C 2 n=0, and hence, ρ0→∞) =0. Equation (12) then reduces to the propagation formula for an EGSM beam passing through a general astigmatic ABCD optical system in free space [31

31. D. Ge, Y. Cai, and Q. Lin, “Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems,” Chin. Phys. 14, 128–132 (2005). [CrossRef]

], and Eq. (13) reduces to the known tensor ABCD law for a partially coherent beam [16

16. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27, 216–218 (2002). [CrossRef]

, 31

31. D. Ge, Y. Cai, and Q. Lin, “Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems,” Chin. Phys. 14, 128–132 (2005). [CrossRef]

]. Equations (12) and (13) also can be applied to study propagation of an anisotropic EGSM beam whose M -1 0αβ can be expressed as [35

35. H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32, 2215–2217 (2007). [CrossRef] [PubMed]

]

M0αβ1=(12ik(σa2)1+1ik(δαβ2)1ik(δαβ2)1ik(δαβ2)112ik(σβ2)1+1ik(δαβ2)1),
(14)

where σ 2 a σ 2 β and δ 2 αβ all are 2.2 matrices with transpose symmetry [15

15. R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).

, 16

16. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27, 216–218 (2002). [CrossRef]

].

In the absence of an optical system but with presence of atmospheric turbulence, the transformation matrix between the source plane and the output plane is given by

(ABCD)=(IlI0I),
(15)

Eq. (12) reduces to expression

Wαβ(ρ,l)=AαAβBαβ[det(I+BM0αβ1+BP)]12exp[ik2ρT(P+B)ρ]
×exp[ik2ρT(B112P)T(M11+B1+P)1(B112P)ρ],
(16)

with ρ 0=(0 0.545C 2 n k 2 l)-3/5. Equation (16) agrees well with existing propagation formula for a scalar partially coherent GSM beam for atmospheric propagation [48

48. Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157–167 (2007). [CrossRef]

], and it can be applied to study all the second-order statistical properties of isotropic and anisotropic EGSM beams [61

61. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909–16915 (2007). [CrossRef] [PubMed]

].

3. Focusing properties of an EGSM beam in a turbulent atmosphere

In this section we study the behavior of spectral density, spectral degree of coherence and the spectral degree of polarization (which we will call for the case under study the focusing properties) of an EGSM beam on propagation in a turbulent atmosphere by applying the formulae derived in section 2.

The propagation geometry is shown in Fig. 1(a). Here the transformation matrix of the total optical system between the source plane and the output plane has the form

(ABCD)=(IfI0I)(I0I(1f)II)(Il1I0II)=(0IfI(1f)I(1l1f)I).
(17)

For 0<zl 1, the transformation matrix for back-propagation from output plane to plane located at distance z from the source is given by

(A(z)B(z)C(z)D(z))=(I(l1z)I0I)(I0I(1f)II)(IfI0II)=((1+zl1f)IfI(1f)I0I).
(18)

For l 1<zl 1+f, the transformation matrix for back-propagation from output plane to plane located at distance z from the source is given by

(A(z)B(z)C(z)D(z))=(I(f+l1z)I0II).
(19)

Substituting the expression for B(z) into Eq. (6), we obtain (after integration)

ρ0=[0.1825Cn2k2(3f+8l1)]35.
(20)
Fig. 1. (a). Focusing geometry, (b). Schematic of laser radar configuration

The spectral density and the degree of polarization of an EGSM beam at point are defined by the expressions

I(ρ1,l)=TrW(ρ1,ρ1,l)
(21)

and

P(ρ1,l)=14DetW(ρ1,ρ1,l)[TrW(ρ1,ρ1,l)]2.
(22)

The spectral degree of coherence of the EGSM beam at a pair of transverse points ρ 1 and ρ 2 is defined by the formula

μ(ρ1,ρ2,l)=TrW(ρ1,ρ2,l)TrW(ρ1,ρ1,l)TrW(ρ2,ρ2,l).
(23)
Fig. 2. Normalized intensity distribution and corresponding cross line (y=0) of an EGSM beam at the geometrical focal plane for three different values of the structure constant of turbulent atmosphere.
Fig. 3. Normalized intensity distribution (cross line, y=0) of an EGSM beam at the geometrical focal plane for different values of the structure constant of turbulent atmosphere and the source correlation coefficients
Fig. 4. Degree of polarization and corresponding cross line (y=0) of an EGSM beam at the geometrical focal plane for three different values of the structure constant of turbulent atmosphere.
Fig. 5. Degree of polarization (cross line, y=0) of an EGSM beam at the geometrical focal plane for different values of the structure constant of turbulent atmosphere and the source correlation coefficients.
Fig. 6. Spectral degree of coherence and corresponding cross line (y1-y2=0) of an EGSM beam at the geometrical focal plane for three different values of the structure constant of turbulent atmosphere.
Fig. 7. Spectral degree of coherence (cross line, y1-y2=0) of an EGSM beam at the geometrical focal lane for different values of the structure constant of turbulent atmosphere and the source correlation coefficients.

On substituting from Eqs. (17) and (20) into Eqs. (12), (13) and (21)(23), we can calculate the statistical properties of an EGSM beams at the geometrical focal plane in a turbulent atmosphere. For all the figures in this paper, the parameters of the source of the beam and of the optical system are chosen to be Ax=Ay=0.707, Bxy=Byx=0.2, σx=σy=1mm, λ=590nm, f=50m and l 1=4.95km. The polarization properties are uniform across the source plane with P(r 1,0)=0.2.

Figures 4 and 5 show the degree of polarization and corresponding cross line (y=0) of an EGSM beam at the geometrical focal plane for different values of the structure constant of turbulent atmosphere and the source correlation coefficients. One finds that the initial uniformly polarized EGSM beam becomes non-uniformly polarized after focusing, and the degree of polarization is of Gaussian profile. It is evident from Fig. 4 that as the strength of atmospheric turbulence increases, the width of the Gaussian profile increases, the value of the on-axis polarization decreases while the value of the off-axis polarization increases gradually. From Figs. 5(a)–5(d), one finds that the shape of the Gaussian profile is affected differently by the refractive index structure parameter C 2 n and by the source correlation coefficients: with increase in C 2 n the distribution becomes shorter and flatter, with increase in source correlations it becomes higher and narrower. The later statement is valid in free space as well.

Figures 6 and 7 show the spectral degree of coherence and corresponding cross lines (y1-y2=0) of an EGSM beam versus the spatial difference vectors x1-x2 and y1-y2 at the geometrical focal plane for different values of the structure constant of turbulent atmosphere and the source correlation coefficients. One can see from these figures that the spectral degree of coherence is of Gaussian profile. The width of the Gaussian profile decreases as the value of the structure constant increases, which means the atmospheric turbulence degrades the coherence of the EGSM beam. Similar phenomenon is known for laser (coherent) Gaussian beams [62

62. T. L. Ho, “Coherence degradation of Gaussian beams in a turbulent atmosphere,” J. Opt. Soc. Am 60, 667–673(1970). [CrossRef]

]. One also finds from Fig. 7 that the initial source correlation coefficients have obvious influence on the spectral degree of coherence of the focused EGSM beam in free space, while in turbulent atmosphere their influence is small although the width of the Gaussian distribution increases gradually as the value of the structure constant increases. This is caused by the fact the influence of atmospheric turbulence on spectral degree of coherence largely surpasses the influence of the initial source correlation coefficients at sufficiently large propagation distances from the source.

4. Summary

We have derived laws for the cross-spectral density matrix of an EGSM beam propagating through a paraxial ABCD optical system in the turbulent atmosphere based on the generalized Huygens-Fresnel integral with the help of a tensor method. In particular, we have obtained the closed-form propagation formula for the cross-spectral density matrix of an EGSM beam. The statistical properties of an EGSM beam focused by a thin lens in the turbulent atmosphere have been studied as a numerical example. We have found that the beam spot of an EGSM beam with higher values of the initial source correlation coefficients can be focused more tightly, the initial uniformly polarized EGSM beam will become non-uniformly polarized after focusing, the atmospheric turbulence will degrade the coherence of the EGSM beam, and the EGSM beam with lower values of the source correlation coefficients is less affected by the atmospheric turbulence. The focusing properties of an EGSM beam can be closely controlled by the structure constant of the turbulent atmosphere and the statistical properties of the EGSM beam.

Our results might find uses in optimization of bistatic LIDAR systems.

Appendix A. Derivation of propagation Eq. (12)

Substituting Eq. (10) into Eq. (7) and after some arrangement, we obtain

W(ρ˜,l)=k2AαAβBαβ4π2[det(B˜)]12exp[ik2ρ˜T(D˜B˜1+P˜)ρ˜]
×exp[ik2r˜T(M0αβ1+B˜1A˜+P˜)r˜]exp[ikr˜T(B˜112P˜)ρ˜]dr˜
=k2AαAβBαβ4π2[det(B˜)]12exp[ik2ρ˜T(D˜B˜1+P˜)ρ˜]
×exp[ik2ρ˜T(B˜112P˜)T(M0αβ1+B˜1A˜+P˜)1(B˜112P˜)ρ˜]
×exp[ik2(M0αβ1+B˜1A˜+P˜)12r˜(M0αβ1+B˜1A˜+P˜)12(B˜112P˜)ρ˜]dr˜,(A1)
(A1)

Then after applying the integral formula

exp(ax2)dx=π/a,
(A2)

Eq. (A1) reduces (after vector integration) to the expression

W(ρ˜,l)=AαAβBαβ[det(B˜)]12[det(M0αβ1+B˜1A˜+P˜)]12exp[ik2ρ˜T(D˜B˜1+P˜)ρ˜]
exp[ik2ρ˜T(B˜112P˜)T(M0αβ1+B˜1A˜+P˜)1(B˜112P˜)ρ˜],
(A3)

By applying the following operations

[det(B˜)]12[det(M0αβ1+B˜1A˜+P˜)]12=[det(A˜+B˜M0αβ1+B˜P˜)]12,
(A4)
D˜B˜1B˜1T(M0αβ1+B˜1A˜+P˜)1B˜1=D˜B˜1B˜1(A˜+B˜M0αβ1+B˜P˜)1
=[D˜B˜1(A˜+B˜M0αβ1+B˜P˜)B˜1](A˜+B˜M0αβ1+B˜P˜)1
=[D˜B˜1A˜+D˜M0αβ1+D˜P˜B˜1](A˜+B˜M0αβ1+B˜P˜)1
=(C˜+D˜M0αβ1+D˜P˜)(A˜+B˜M0αβ1+B˜P˜)1,
(A5)

and by setting

M1αβ1=(C˜+D˜M0αβ1+D˜P˜)(A˜+B˜M0αβ1+B˜P˜)1,
(A6)

then Eq. (A3) reduces to Eq. (12) in the text. In(A5) and (A6) we have used the Luneburg relations (Eq. (2)) and the relations B̃1T=B̃1 and P̃T=P̃.

Acknowledgments

Y. Cai gratefully acknowledges the supports from the Alexander von Humboldt Foundation and the Ministry of Science and Technology of China (grant no. 2006CB921403 & 2006AA06A204). O. Korotkova’s research is funded by the AFOSR (grant FA 95500810102).

References and links

1.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press,1995).

2.

D. Kermisch, “Partially coherent image processing by laser scanning,” J. Opt. Soc. Am 65, 887–891 (1975). [CrossRef]

3.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984). [CrossRef]

4.

A. Belendez, L. Carretero, and A. Fimia, “The use of partially coherent light to reduce the efficiency of silve-halide noise gratings,” Opt. Commun. 98, 236–240 (1993). [CrossRef]

5.

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802 (2002). [CrossRef]

6.

Y. Cai and S. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29, 2716–2718 (2004). [CrossRef] [PubMed]

7.

Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).

8.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93, 068103 (2004). [CrossRef] [PubMed]

9.

Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15, 15480–15492 (2007). [CrossRef] [PubMed]

10.

E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978). [CrossRef]

11.

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1978). [CrossRef]

12.

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun 41, 383–387 (1982). [CrossRef]

13.

Q. S. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988). [CrossRef]

14.

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). [CrossRef]

15.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).

16.

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27, 216–218 (2002). [CrossRef]

17.

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phy. Lett. 86, 021112 (2005). [CrossRef]

18.

F. Wang, Y. Cai, and S. He, “Experimental observation of coincidence fractional Fourier transform with a partially coherent beam,” Opt. Express 14, 6999–7004 (2006). [CrossRef] [PubMed]

19.

Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31, 685–687 (2006) [CrossRef] [PubMed]

20.

C. Brosseau, Fundamentals of polarized light-a statistical approach (Wiley, New York, 1998).

21.

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).

22.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998). [CrossRef]

23.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 3, 1–9 (2001). [CrossRef]

24.

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source”, Opt. Lett. 29, 1173–1175 (2004). [CrossRef] [PubMed]

25.

E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).

26.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9–16 (2002). [CrossRef]

27.

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A: Pure App. Opt. 5, 453–459 (2003). [CrossRef]

28.

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004). [CrossRef]

29.

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 7, 232–237 (2005). [CrossRef]

30.

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005). [CrossRef]

31.

D. Ge, Y. Cai, and Q. Lin, “Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems,” Chin. Phys. 14, 128–132 (2005). [CrossRef]

32.

O Korotkova and E Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation”, Opt. Commun. 246, 35–43 (2005). [CrossRef]

33.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52, 1611–1618 (2005). [CrossRef]

34.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15, 353–364 (2005). [CrossRef]

35.

H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32, 2215–2217 (2007). [CrossRef] [PubMed]

36.

W. Gao and O. Korotkova, “Changes in the state of polarization of a random electromagnetic beam propagating through tissue”, Opt. Comm. 270, 474–478 (2007). [CrossRef]

37.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25, 1016–1021 (2008). [CrossRef]

38.

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008). [CrossRef]

39.

A. Ishimaru, Wave propagation and scattering in random media, (Academic Press, New York,1978) Vol. 2.

40.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Washington, 2001). [CrossRef]

41.

V. A. Banakh and V. L. Mironov, “Phase approximation of the Huygens - Kirchhoff method in problems of space-limited optical-beam propagation in turbulent atmosphere,” Opt. Lett. 4, 259–261 (1979). [CrossRef] [PubMed]

42.

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]

43.

J. C. Leader, “Atmospheric propagation of partially coherent radiation”, J. Opt. Soc. Am. 68, 175–185 (1978). [CrossRef]

44.

T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003). [CrossRef]

45.

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

46.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006). [CrossRef]

47.

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

48.

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157–167 (2007). [CrossRef]

49.

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32, 2405–2407 (2007). [CrossRef] [PubMed]

50.

R. J. Noriega-Manez and J. C. Gutierrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15, 16328–16341 (2007). [CrossRef] [PubMed]

51.

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891–2901 (2007). [CrossRef]

52.

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B 89, 91–97 (2007).

53.

A. Gerrard and J. M. Burch, Introduction to matrix methods in optics (John Wiley and Sons, 1975).

54.

H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am A 4, 1931 (1987). [CrossRef]

55.

H. T. Yura and S. G. Hanson, “Second-order statistics for wave propagation through complex optical systems,” J. Opt. Soc. Am A 6, 564 (1989). [CrossRef]

56.

X. Chu, “Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express 15, 17613–17618 (2007). [CrossRef] [PubMed]

57.

V. A. Banakh and L. V. Mironov, LIDAR in a turbulent atmosphere (Artech House, Dedham1987).

58.

J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” in Progress in Optics, Vol. XI, E. Wolf, ed. (North-Holland, 1973), pp. 247–304.

59.

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik 85, 67 (1990).

60.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, 1964).

61.

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909–16915 (2007). [CrossRef] [PubMed]

62.

T. L. Ho, “Coherence degradation of Gaussian beams in a turbulent atmosphere,” J. Opt. Soc. Am 60, 667–673(1970). [CrossRef]

OCIS Codes
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(030.1640) Coherence and statistical optics : Coherence
(260.5430) Physical optics : Polarization

ToC Category:
Atmospheric and oceanic optics

History
Original Manuscript: May 29, 2008
Revised Manuscript: August 30, 2008
Manuscript Accepted: September 12, 2008
Published: September 22, 2008

Citation
Yangjian Cai, Olga Korotkova, Halil T. Eyyuboglu, and Yahya Baykal, "Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere," Opt. Express 16, 15834-15846 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15834


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References

  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  2. D. Kermisch, "Partially coherent image processing by laser scanning," J. Opt. Soc. Am 65, 887-891 (1975). [CrossRef]
  3. Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression," Phys. Rev. Lett. 53, 1057-1060 (1984). [CrossRef]
  4. A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silve-halide noise gratings," Opt. Commun. 98, 236-240 (1993). [CrossRef]
  5. J. C. Ricklin and F. M. Davidson, "Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication," J. Opt. Soc. Am. A 19, 1794-1802 (2002). [CrossRef]
  6. Y. Cai and S. Zhu, "Ghost interference with partially coherent radiation," Opt. Lett. 29, 2716-2718 (2004). [CrossRef] [PubMed]
  7. Y. Cai and S. Zhu, "Ghost imaging with incoherent and partially coherent light radiation," Phys. Rev. E 71, 056607 (2005).
  8. T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004). [CrossRef] [PubMed]
  9. Y. Cai and U. Peschel, "Second-harmonic generation by an astigmatic partially coherent beam," Opt. Express 15, 15480-15492 (2007). [CrossRef] [PubMed]
  10. E. Wolf and E. Collett, "Partially coherent sources which produce same far-field intensity distribution as a laser," Opt. Commun. 25, 293-296 (1978). [CrossRef]
  11. F. Gori, "Collet-Wolf sources and multimode lasers," Opt. Commun. 34, 301-305 (1978). [CrossRef]
  12. A. T. Friberg and R. J. Sudol, "Propagation parameters of Gaussian Schell-model beams," Opt. Commun 41, 383-387 (1982). [CrossRef]
  13. Q. S. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988). [CrossRef]
  14. 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). [CrossRef]
  15. R. Simon, E. C. G. Sudarshan, and N. Mukunda, "Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants," Phys. Rev. A 31, 2419-2434 (1985).
  16. Q. Lin, Y. Cai, "Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams," Opt. Lett. 27, 216-218 (2002). [CrossRef]
  17. Y. Cai, Q. Lin, and S. Zhu, "Coincidence fractional Fourier transform with entangled photon pairs and incoherent light," Appl. Phy. Lett. 86, 021112 (2005). [CrossRef]
  18. F. Wang, Y. Cai, and S. He, "Experimental observation of coincidence fractional Fourier transform with a partially coherent beam," Opt. Express 14, 6999-7004 (2006). [CrossRef] [PubMed]
  19. Y. Cai and L. Hu, "Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system," Opt. Lett. 31, 685-687 (2006) [CrossRef] [PubMed]
  20. C. Brosseau, Fundamentals of polarized light-a statistical approach (Wiley, New York, 1998).
  21. E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
  22. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi and G. Guattari, "Beam coherence-polarization matrix," Pure Appl. Opt. 7, 941-951 (1998). [CrossRef]
  23. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A: Pure Appl. Opt. 3, 1-9 (2001). [CrossRef]
  24. O. Korotkova, M. Salem, and E. Wolf, "Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source," Opt. Lett. 29, 1173-1175 (2004). [CrossRef] [PubMed]
  25. E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).
  26. G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi and A. Mondello, "Synthesis of partially polarized Gaussian Schell-model sources," Opt. Commun. 208, 9-16 (2002). [CrossRef]
  27. Y. Cai, D. Ge, and Q. Lin, "Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams," J. Opt. A: Pure App. Opt. 5, 453-459 (2003). [CrossRef]
  28. O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004). [CrossRef]
  29. T. Shirai, O. Korotkova, and E. Wolf, "A method of generating electromagnetic Gaussian Schell-model beams," J. Opt. A: Pure Appl. Opt. 7, 232-237 (2005). [CrossRef]
  30. H. Roychowdhury and O. Korotkova, "Realizability conditions for electromagnetic Gaussian Schell-model sources," Opt. Commun. 249, 379-385 (2005). [CrossRef]
  31. D. Ge, Y. Cai and Q. Lin, "Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems," Chin. Phys. 14, 128-132 (2005). [CrossRef]
  32. O Korotkova, E Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation", Opt. Commun. 246, 35-43 (2005). [CrossRef]
  33. H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005). [CrossRef]
  34. O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005). [CrossRef]
  35. H. Wang, X. Wang, A. Zeng, and K. Yang, "Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation," Opt. Lett. 32, 2215-2217 (2007). [CrossRef] [PubMed]
  36. W. Gao and O. Korotkova, "Changes in the state of polarization of a random electromagnetic beam propagating through tissue," Opt. Comm. 270, 474-478 (2007). [CrossRef]
  37. F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, "Realizability condition for electromagnetic Schell-model sources," J. Opt. Soc. Am. A 25, 1016-1021 (2008). [CrossRef]
  38. O. Korotkova, "Scintillation index of a stochastic electromagnetic beam propagating in random media," Opt. Commun. 281, 2342-2348 (2008). [CrossRef]
  39. A. Ishimaru, Wave propagation and scattering in random media, (Academic Press, New York, 1978) Vol. 2.
  40. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Washington, 2001). [CrossRef]
  41. V. A. Banakh and V. L. Mironov, "Phase approximation of the Huygens - Kirchhoff method in problems of space-limited optical-beam propagation in turbulent atmosphere," Opt. Lett. 4, 259-261 (1979). [CrossRef] [PubMed]
  42. 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]
  43. J. C. Leader, "Atmospheric propagation of partially coherent radiation", J. Opt. Soc. Am. 68, 175-185 (1978). [CrossRef]
  44. T. Shirai, A. Dogariu, and E. Wolf, "Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. A 20, 1094-1102 (2003). [CrossRef]
  45. H. T. Eyyuboglu and Y. Baykal, "Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere," Opt. Express 12, 4659-4674 (2004). [CrossRef] [PubMed]
  46. Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 041117 (2006). [CrossRef]
  47. Y. Cai and S. He, "Propagation of various dark hollow beams in a turbulent atmosphere," Opt. Express 14, 1353-1367 (2006). [CrossRef] [PubMed]
  48. Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, "Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere," Opt. Commun. 278, 157-167 (2007). [CrossRef]
  49. Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, "Scintillation index of elliptical Gaussian beam in turbulent atmosphere," Opt. Lett. 32, 2405-2407 (2007). [CrossRef] [PubMed]
  50. R. J. Noriega-Manez and J. C. Gutierrez-Vega, "Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere," Opt. Express 15, 16328-16341 (2007). [CrossRef] [PubMed]
  51. H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Complex degree of coherence for partially coherent general beams in atmospheric turbulence," J. Opt. Soc. Am. A 24, 2891-2901 (2007). [CrossRef]
  52. H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Degree of polarization for partially coherent general beams in turbulent atmosphere," Appl. Phys. B 89, 91-97 (2007).
  53. A. Gerrard and J. M. Burch, Introduction to matrix methods in optics (John Wiley and Sons, 1975).
  54. H. T. Yura and S. G. Hanson, "Optical beam wave propagation through complex optical systems," J. Opt. Soc. Am A 4, 1931 (1987). [CrossRef]
  55. H. T. Yura and S. G. Hanson, "Second-order statistics for wave propagation through complex optical systems," J. Opt. Soc. Am A 6, 564 (1989). [CrossRef]
  56. X. Chu, "Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere," Opt. Express 15, 17613-17618 (2007). [CrossRef] [PubMed]
  57. V. A. Banakh and L. V. Mironov, LIDAR in a turbulent atmosphere (Artech House, Dedham 1987).
  58. J. A. Arnaud, "Hamiltonian theory of beam mode propagation," in Progress in Optics, Vol. XI, E. Wolf, ed. (North-Holland, 1973), pp. 247-304.
  59. Q. Lin, S. Wang, J. Alda, E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik 85, 67 (1990).
  60. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, 1964).
  61. X. Du, D. Zhao, and O. Korotkova, "Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere," Opt. Express 15, 16909-16915 (2007). [CrossRef] [PubMed]
  62. T. L. Ho, "Coherence degradation of Gaussian beams in a turbulent atmosphere," J. Opt. Soc. Am 60, 667-673(1970). [CrossRef]

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