Abstract

A class of partially coherent elliptical sources with twisted Laguerre Gaussian Schell-model (TLGSM) correlation function is proposed, which are capable of producing beams whose intensity profiles may vary substantially. This kind of beam can be viewed as the generalization of the LGSM beams. Properties of the spectral density during propagation in free space and atmospheric turbulence are investigated with varying quantities related to the beam source and the medium. It is shown that the elliptical TLGSM beams evolve in a manner that is much more complex compared to the LGSM beams. In addition, the behaviour of the rotation angle is further analysed by quantitative examples.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (2)

2018 (6)

2017 (1)

2016 (1)

2015 (3)

2014 (3)

2013 (2)

2012 (3)

2011 (1)

2007 (2)

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

1998 (1)

1994 (1)

1993 (1)

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

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

Baykal, Y.

Borghi, R.

Cai, Y.

Chen, Y.

Chen, Z.

Cui, S.

Eyyuboglu, H. T.

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Friberg, A. T.

Gori, F.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ruzhik, “Table of Integrals, Series and Products,” (Academic, 2007).

Guattari, G.

Guo, Q.

Korotkova, O.

Lajunen, H.

Li, Y.

Liang, G.

Liu, L.

Ma, L.

Mandel, L.

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

Mei, Z.

Mukunda, N.

Peng, X.

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

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

Ponomarenko, S. A.

Popov, S.

Pu, J.

Ruzhik, I. M.

I. S. Gradshteyn and I. M. Ruzhik, “Table of Integrals, Series and Products,” (Academic, 2007).

Saastamoinen, T.

Sahin, S.

Santarsiero, M.

Simon, R.

Tervonen, E.

Tong, Z.

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Turunen, J.

Wan, L.

Wang, F.

Wang, H.

Wang, Y.

Wolf, E.

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

Wu, G.

Zhang, H.

Zhang, L.

Zhao, D.

Zhou, Y.

Zhu, S.

J. Opt. Soc. Am. A (5)

Opt. Express (3)

Opt. Lett. (18)

R. Borghi, “Twisting partially coherent light,” Opt. Lett. 43(8), 1627–1630 (2018).
[Crossref]

L. Wan and D. Zhao, “Twisted Gaussian Schell-model array beams,” Opt. Lett. 43(15), 3554–3557 (2018).
[Crossref]

Z. Mei and O. Korotkova, “Twisted EM beams with structured correlations,” Opt. Lett. 43(16), 3905–3908 (2018).
[Crossref]

R. Borghi, F. Gori, G. Guattari, and M. Santarsiero, “Twisted Schell-model beams with axial symmetry,” Opt. Lett. 40(19), 4504–4507 (2015).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref]

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[Crossref]

F. Wang, Y. Cai, H. T. Eyyuboǧlu, and Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012).
[Crossref]

Z. Tong and O. Korotkova, “Beyond the classical Rayleigh limit with twisted light,” Opt. Lett. 37(13), 2595–2597 (2012).
[Crossref]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref]

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref]

S. Cui, Z. Chen, L. Zhang, and J. Pu, “Experimental generation of nonuniformly correlated partially coherent light beams,” Opt. Lett. 38(22), 4821–4824 (2013).
[Crossref]

Y. Cai and S. Zhu, “Orbital angular moment of a partially coherent beam propagating through an astigmatic ABCD optical system with loss or gain,” Opt. Lett. 39(7), 1968–1971 (2014).
[Crossref]

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref]

L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. 39(23), 6656–6659 (2014).
[Crossref]

F. Gori and M. Santarsiero, “Twisted Gaussian Schell-model beams as series of partially coherent modified Bessel-Gauss beams,” Opt. Lett. 40(7), 1587–1590 (2015).
[Crossref]

H. Wang, X. Peng, L. Liu, F. Wang, Y. Cai, and S. A. Ponomarenko, “Generating bona fide twisted Gaussian Schell-model beams,” Opt. Lett. 44(15), 3709–3712 (2019).
[Crossref]

Z. Mei and O. Korotkova, “Random sources for rotating spectral densities,” Opt. Lett. 42(2), 255–258 (2017).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine twisted cross-spectral densities,” Opt. Lett. 43(3), 595–598 (2018).
[Crossref]

Proc. SPIE (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Other (3)

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

I. S. Gradshteyn and I. M. Ruzhik, “Table of Integrals, Series and Products,” (Academic, 2007).

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

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Figures (4)

Fig. 1.
Fig. 1. Distribution of the spectral density of the circular TLGSM beam at the source plane with parameters of $\sigma =1.3\textrm{cm}$, $a=0.3$, $g=2\times 10^{-5}\textrm{m}^2$ (upper row), $g=9\times 10^{-6}\textrm{m}^2$ (lower row), and $m=0$ (first column), $m=1$ (second column), $m=2$ (third column).
Fig. 2.
Fig. 2. Changes in the spectral density generated by the circular TLGSM beam on propagation in free space (top two rows) and atmospheric turbulence (third row). The related quantities are: $\sigma =1.3\textrm{cm}$, $m=2$, $\tilde {C}_n^2=5\times 10^{-13}\textrm{m}^{2/3}$, $a=0.03$ for Figs. 2(a)–2(c), and $a=0.3$ for Figs. 2(d)–2(i), $g=4\times 10^{-5}\textrm{m}^2$ (left column), $g=2\times 10^{-5}\textrm{m}^2$ (middle column), and $g=9\times 10^{-6}\textrm{m}^2$ (right column).
Fig. 3.
Fig. 3. Evolution of the spectral density of the elliptical TLGSM beam on propagation in free space (top three rows) and atmospheric turbulence (last row). The parameters are set as: $\sigma _x=1.3\textrm{cm}$, $\sigma _y=0.8\textrm{cm}$, $m=2$, $g=9\times 10^{-6}\textrm{m}^2$, $\tilde {C}_n^2=5\times 10^{-13}\textrm{m}^{2/3}$, $a=0.1$ [Figs. 3(a)–3(d)], and $a=0.3$ [Figs. 3(e)–3(h)], the propagation distance $z=0$ (first row), $z=300\textrm{m}$ (second row), and $z=600\textrm{m}$ (last two rows)
Fig. 4.
Fig. 4. Rotation angles of the intensity ellipse of the TLGSM beam on propagation with other parameter $\sigma _x=1.3\textrm{cm}$, $\sigma _y=0.8\textrm{cm}$, $m=1$, $a=0.3$ and $g=2\times 10^{-5}\textrm{m}^2$.

Equations (28)

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W ( 0 ) ( ρ 1 , ρ 2 , ω ) = E ( ρ 1 , ω ) E ( ρ 2 , ω ) ,
W ( 0 ) ( ρ 1 , ρ 2 ) = p ( v ) H 0 ( ρ 1 , v ) H 0 ( ρ 2 , v ) d 2 v ,
p ( v ) = g m + 1 π m ! ( v x 2 + v y 2 ) m exp [ g ( v x 2 + v y 2 ) ] ,
H 0 ( ρ , v ) = τ ( ρ ) exp { [ ( a y + i x ) v x ( a x i y ) v y ] } ,
W ( 0 ) ( ρ 1 , ρ 2 ) = τ ( ρ 1 ) τ ( ρ 2 ) n = 0 m C m n F n ( ξ 1 ) F m n ( ξ 2 ) ,
F n ( ξ ) = x 2 n exp ( g x 2 i ξ x ) d x .
0 x 2 n exp ( β 2 x 2 ) cos ( a x ) d x = ( 1 ) n π ( 2 β ) 2 n + 1 exp ( a 2 4 β 2 ) H 2 n ( a 2 β ) ,
H 2 n ( x ) = ( 1 ) n 2 2 n n ! L n 1 / 2 ( x 2 ) ,
F n ( ξ 1 ) = π n ! g n + 1 / 2 exp ( ξ 1 2 4 g ) L n 1 / 2 ( ξ 1 2 4 g ) .
n = 0 m L n α ( x ) L m n β ( y ) = L m α + β + 1 ( x + y ) .
W ( 0 ) ( ρ 1 , ρ 2 ) = τ ( ρ 1 ) τ ( ρ 2 ) exp ( ξ 1 2 + ξ 2 2 4 g ) L m ( ξ 1 2 + ξ 2 2 4 g ) ,
ξ 1 = y 2 y 1 + i a ( x 1 + x 2 ) , ξ 2 = x 2 x 1 i a ( y 1 + y 2 ) .
τ ( ρ ) = exp ( x 2 σ x 2 y 2 σ y 2 ) ,
W ( 0 ) ( ρ 1 , ρ 2 ) = exp ( x 1 2 + x 2 2 4 w x 2 ) exp ( y 1 2 + y 2 2 4 w y 2 ) exp [ ( ρ 1 ρ 2 ) 2 2 δ 2 ] × L m [ a 2 2 g ( ρ 1 2 + ρ 2 2 ) + ( ρ 1 ρ 2 ) 2 2 δ 2 i u ( x 2 y 1 x 1 y 2 ) ] exp [ i u ( x 1 y 2 x 2 y 1 ) ] .
1 4 w x 2 = 1 σ x 2 a 2 2 g , 1 4 w y 2 = 1 σ y 2 a 2 2 g , 1 2 δ 2 = 1 + a 2 4 g , u = a / g .
S ( ρ , z ) = W ( 0 ) ( ρ 1 , ρ 2 , ) G ( ρ ρ 1 ) G ( ρ ρ 2 ) × exp [ Ψ ( ρ , ρ 1 , z ) + Ψ ( ρ , ρ 2 , z ) ] m d 2 ρ 1 d 2 ρ 2 ,
G ( ρ ρ ) = i k 2 π z exp ( i k | ρ ρ | 2 2 z ) ,
exp [ Ψ ( ρ , ρ 1 , z ) + Ψ ( ρ , ρ 2 , z ) ] m exp [ π 2 k 2 z 3 ( ρ 1 ρ 2 ) 2 0 κ 3 Φ n ( κ ) ] ,
Φ n ( κ ) = A ( α ) C ~ n 2 exp [ ( κ 2 / κ m 2 ) ] / ( κ 2 + κ 0 2 ) α / 2 , 0 κ , 3 < α < 4 ,
A ( α ) = 1 4 π 2 Γ ( α 1 ) cos ( α π / 2 ) ,
c ( α ) = [ 2 π Γ ( 5 α / 2 ) A ( α ) 3 ] 1 / ( α 5 ) .
F = 0 κ 3 Φ n ( κ ) d κ = A ( α ) 2 ( α 2 ) C ~ n 2 [ κ m 2 α β exp ( κ 0 2 / κ m 2 ) Γ ( 2 α 2 , κ 0 2 κ m 2 ) 2 κ 0 4 α ] ,
S ( ρ , z ) = k 2 4 π 2 z 2 p ( v ) | H ( ρ , v , z ) | 2 d 2 v ,
| H ( ρ , v , z ) | 2 = | H 0 ( ρ , v ) G ( ρ ρ , z ) | 2 exp [ π 2 k 2 z F 3 ( ρ 1 ρ 2 ) 2 ] d 2 ρ 1 d 2 ρ 2 .
| H ( ρ , v , z ) | 2 = π 2 p x 1 p x 2 p y 1 p y 2 exp ( a 1 v x 2 a 2 v x b 1 v y 2 b 2 v y + d v x v y ) × exp { k 2 4 z 2 { [ 1 p x 1 + 1 p x 2 ( 1 T p x 1 ) 2 ] x 2 + [ 1 p y 1 + 1 p y 2 ( 1 T p y 1 ) 2 ] y 2 } } ,
T = π 2 k 2 z F / 3 , p i 1 = 1 σ i 2 + i k 2 z + T , p i 2 = 1 σ i 2 i k 2 z + T T 2 p i 1 ( i = x , y ) ,
a 1 = 1 4 [ 1 p x 1 + ( p x 1 T ) 2 p x 2 p x 1 2 a 2 p y 1 a 2 ( p y 1 + T ) 2 p y 2 p y 1 2 ] , a 2 = k 2 z [ ( 1 p x 1 + ( p x 1 T ) 2 p x 2 p x 1 2 ) x + i a ( 1 p y 1 + T 2 p y 1 2 p y 2 p y 1 2 ) y ] , b 1 = 1 4 [ 1 p y 1 + ( p y 1 T ) 2 p y 2 p y 1 2 a 2 p x 1 a 2 ( p x 1 + T ) 2 p x 2 p x 1 2 ] , b 2 = k 2 z [ ( 1 p y 1 + ( p y 1 T ) 2 p y 2 p y 1 2 ) y i a ( 1 p x 1 + T 2 p x 1 2 p x 2 p x 1 2 ) x ] , d = i a 2 ( 1 p x 1 1 p y 1 + T 2 p x 1 2 p x 2 p x 1 2 + p y 1 2 T 2 p y 2 p y 1 2 ) .
S ( ρ , z ) = k 2 4 z 2 p 1 p 2 ( g G ) m + 1 L m ( k 2 4 z 2 4 z 2 / σ 4 + a 2 k 2 4 G p 1 2 p 2 2 ρ 2 ) × exp { k 2 4 z 2 [ 1 p 1 + 1 p 2 ( 1 T p 1 ) 2 4 z 2 / σ 4 + a 2 k 2 4 G p 1 2 p 2 2 ] ρ 2 } ,

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