Abstract

In a recent publication [Opt. Lett. 37, 2970 (2012) [CrossRef]  ], a novel class of planar stochastic sources, generating far fields with flat intensity profiles, was introduced. In this paper we examine the behavior of the spectral density and the state of coherence of beamlike fields generated by such sources on propagation in free space and linear isotropic random media. In particular, we find that at sufficiently large distances from the source, the medium destroys the flat intensity profile, even if it remains such for intermediate distances from the source.

© 2012 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  2. A. C. Schell, “The multiple plate antenna,” Ph.D. dissertation (Massachusetts Institute of Technology, 1961).
  3. F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
    [CrossRef]
  4. S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18, 150–156 (2001).
    [CrossRef]
  5. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36, 4104–4106 (2011).
    [CrossRef]
  6. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
    [CrossRef]
  7. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37, 2970–2972 (2012).
    [CrossRef]
  8. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  9. Y. Li, “New expressions for flat-topped beams,” Opt. Commun. 206, 225–234 (2002).
    [CrossRef]
  10. Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A 8, 537–545 (2006).
    [CrossRef]
  11. L. C. Andrews and R. L. Phillips, Laser Beam Propagation in the Turbulent Atmosphere, 2nd ed. (SPIE, 2005).
  12. W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in the degree of coherence of partially coherent electromagnetic beams propagating through turbulent atmosphere,” Opt. Commun. 271, 1–8 (2007).
    [CrossRef]
  13. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).
    [CrossRef]

2012 (1)

2011 (1)

2008 (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).
[CrossRef]

2007 (2)

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in the degree of coherence of partially coherent electromagnetic beams propagating through turbulent atmosphere,” Opt. Commun. 271, 1–8 (2007).
[CrossRef]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
[CrossRef]

2006 (1)

Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A 8, 537–545 (2006).
[CrossRef]

2002 (1)

Y. Li, “New expressions for flat-topped beams,” Opt. Commun. 206, 225–234 (2002).
[CrossRef]

2001 (1)

1994 (1)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

1987 (1)

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation in the Turbulent Atmosphere, 2nd ed. (SPIE, 2005).

Cai, Y.

Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A 8, 537–545 (2006).
[CrossRef]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).
[CrossRef]

Gori, F.

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
[CrossRef]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Korotkova, O.

Lajunen, H.

Li, Y.

Y. Li, “New expressions for flat-topped beams,” Opt. Commun. 206, 225–234 (2002).
[CrossRef]

Liu, L.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in the degree of coherence of partially coherent electromagnetic beams propagating through turbulent atmosphere,” Opt. Commun. 271, 1–8 (2007).
[CrossRef]

Lu, W.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in the degree of coherence of partially coherent electromagnetic beams propagating through turbulent atmosphere,” Opt. Commun. 271, 1–8 (2007).
[CrossRef]

Mandel, L.

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

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation in the Turbulent Atmosphere, 2nd ed. (SPIE, 2005).

Ponomarenko, S. A.

Saastamoinen, T.

Sahin, S.

Santarsiero, M.

Schell, A. C.

A. C. Schell, “The multiple plate antenna,” Ph.D. dissertation (Massachusetts Institute of Technology, 1961).

Sun, J.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in the degree of coherence of partially coherent electromagnetic beams propagating through turbulent atmosphere,” Opt. Commun. 271, 1–8 (2007).
[CrossRef]

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).
[CrossRef]

Wolf, E.

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

Yang, Q.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in the degree of coherence of partially coherent electromagnetic beams propagating through turbulent atmosphere,” Opt. Commun. 271, 1–8 (2007).
[CrossRef]

Zhu, Y.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in the degree of coherence of partially coherent electromagnetic beams propagating through turbulent atmosphere,” Opt. Commun. 271, 1–8 (2007).
[CrossRef]

J. Opt. A (1)

Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A 8, 537–545 (2006).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (4)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

Y. Li, “New expressions for flat-topped beams,” Opt. Commun. 206, 225–234 (2002).
[CrossRef]

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in the degree of coherence of partially coherent electromagnetic beams propagating through turbulent atmosphere,” Opt. Commun. 271, 1–8 (2007).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Opt. Eng. (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008).
[CrossRef]

Opt. Lett. (3)

Other (3)

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

A. C. Schell, “The multiple plate antenna,” Ph.D. dissertation (Massachusetts Institute of Technology, 1961).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation in the Turbulent Atmosphere, 2nd ed. (SPIE, 2005).

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Figures (6)

Fig. 1.
Fig. 1.

Transverse cross section of the spectral density of the MGSM beam propagating in free space versus |ρ|, at several distances from the source plane: (A) 1 m, (B) 100 m, (C) 1000 m, and (D) 10000 m. Several curves correspond to different values of M: M=1 solid curve, M=4 dashed curve, M=10 dotted curve, and M=40 dash-dotted curve. Other beam parameters are λ=632μm, σ=1cm, δ=1mm.

Fig. 2.
Fig. 2.

Spectral degree of coherence of the MGSM beam propagating in free space versus |ρd|, at the same distances from the source, the same values of M, and other source parameters as in Fig. 1.

Fig. 3.
Fig. 3.

Transverse cross section of the spectral density of the MGSM beam propagating in Kolmogorov atmospheric turbulence (α=3.67, C˜n2=1013m2/3, L0=1m; l0=1mm) versus |ρ|, at the same distances from the source, the same values of M, and other source parameters as in Fig. 1.

Fig. 4.
Fig. 4.

Spectral degree of coherence of the MGSM beam propagating in Kolmogorov atmospheric turbulence (α=3.67, C˜n2=1013m2/3, L0=1m; l0=1mm) versus |ρd|, at the same distances from the source, the same values of M, and other source parameters as in Fig. 1.

Fig. 5.
Fig. 5.

Effect of C˜n2 on the intensity distribution: (A) C˜n2=1014m2/3, (B) C˜n2=1012m2/3 at z=1000m. Other atmospheric parameters are the same as in Fig. 4.

Fig. 6.
Fig. 6.

Effect of α on the intensity distribution (A) α=3.1, (B) α=3.9 at z=1000m. Other atmospheric parameters are the same as in Fig. 4.

Equations (22)

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W(0)(ρ1,ρ2;ω)=S(0)(ρ1;ω)S(0)(ρ2;ω)μ(0)(ρ2ρ1;ω).
S(0)(ρ;ω)=exp[ρ22σ(ω)2],
μ(0)(ρ2ρ1;ω)=1C0m=1M(Mm)(1)m1mexp[|ρ2ρ1|22mδ(ω)2],
W(0)(ρ1,ρ2;ω)=exp[ρ12+ρ224σ2]1C0m=1M(Mm)(1)m1mexp[|ρ2ρ1|22mδ2].
14σ2+1δ22π2λ2,
W(ρ1,ρ2,z;ω)=k2(2πz)2W(0)(ρ1,ρ2;ω)×exp[ik(ρ1ρ1)2(ρ2ρ2)22z]dρ1dρ2,
W(ρ1,ρ2,z;ω)=1Δ2(z)exp[(ρ1+ρ2)28σ2Δ2(z)]×exp[(ρ1ρ2)22ϱ2Δ2(z)]exp[ik(ρ22ρ12)2R(z)],
ϱ2=(14σ2+1δ2)1,Δ2(z)=1+z2k2σ2ϱ2,R(z)=z(1+k2σ2ϱ2z2).
δm=mδ.
W(ρ1,ρ2,z;ω)=1C0m=1M(Mm)(1)m1m1Δm2(z)exp[(ρ1+ρ2)28σ2Δm2(z)]×exp[(ρ1ρ2)22ϱm2Δm2(z)]exp[ik(ρ22ρ12)2Rm(z)],
ϱm2=(14σ2+1δm2)1,Δm2(z)=1+z2k2σ2ϱm2,Rm(z)=z(1+k2σ2ϱm2z2).
S(ρ,z;ω)=W(ρ,ρ,z;ω)=1C0m=1M(Mm)(1)m1mΔm2(z)exp[ρ22σ2Δm2(z)].
μ(ρd,z;ω)=W(ρ1,ρ2,z;ω)S(0,z;ω)=m=1M(Mm)(1)m1m1Δm2(z)exp[ρd22ϱm2Δm2(z)]m=1M(Mm)(1)m1m1Δm2(z).
W(ρ1,ρ2,z;ω)=k2(2πz)2W(0)(ρ1,ρ2;ω)×exp[ik(ρ1ρ1)2(ρ2ρ2)22z]×exp[ψ*(ρ1,ρ1,z;ω)+ψ(ρ2,ρ2,z;ω)]Rdρ1dρ2,
exp[ψ*(ρ1,ρ1,z;ω)+ψ(ρ2,ρ2,z;ω)]Rexp{4π4z3λ2[(ρ1ρ2)2+(ρ1ρ2)(ρ1ρ2)+(ρ1ρ2)20κ3Φn(κ)dκ]},
W(ρ1,ρ2,z;ω)=1Δ2(z)exp[(ρ1+ρ2)28σ2Δ2(z)]×exp[(ρ1ρ2)2(12ϱ2Δ2(z)+T(1+σ2)T2z22k2σ2Δ2(z))]exp[ik(ρ22ρ12)2R(z)],
Δ2(z)=1+z2k2σ2ϱ2+2Tz2k2σ2,R(z)=k2σ2Δ2(z)zk2σ2Δ2(z)+Tz2k2σ2,T=13π2k2z0κ3Φn(κ)dκ.
W(ρ1,ρ2,z;ω)=1C0m=1M(Mm)(1)m1m×1Δm2(z)exp[(ρ1+ρ2)28σ2Δm2(z)]×exp[(ρ1ρ2)2(12ϱm2Δm2(z)+T(1+σ2)T2z22k2σ2Δm2(z))]exp[ik(ρ22ρ12)2Rm(z)],
Δm2(z)=1+z2k2σ2ϱm2+2Tz2k2σ2,Rm(z)=k2σ2Δm2(z)zk2σ2Δm2(z)+Tz2k2σ2.
Φn(κ)=A(α)C˜n2exp[κ2/κm2](κ2+κ02)α/2,0κ<,
κ0=2πL0,κm=c(α)l0,c(α)=[2π3Γ(5α2)A(α)]1α5,A(α)=14π2Γ(α1)cos(απ2).
I=0κ3Φn(κ)dκ=A(α)2(α2)C˜n2κm2αβexp(κ02κm2)Γ(2α2,κ02κm2)2κ04α,

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