Abstract

The propagation of a novel class of paraxial spatially partially coherent beams exhibiting Bessel-type correlations is studied in free space and in paraxial optical systems. We show that, under certain conditions, such beams can have functionally identical forms of the absolute value of the complex degree of spatial coherence not only at the source plane and in the far zone, but also at all finite propagation distances. Under these conditions the degree of spatial coherence properties of the field is a shape-invariant quantity, but the spatial intensity distribution is only approximately shape-invariant. The main properties of this class of model beams are demonstrated experimentally by passing a coherent Gaussian beam through a rotating wedge and measuring the coherence of the ensuing beams with a Young-type interferometer realized with a digital micromirror device.

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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2017 (2)

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Progr. Opt. 62, 157–223 (2017).
[Crossref]

M. Koivurova, A. Halder, H. Partanen, and J. Turunen, “Bessel-correlated supercontinuum fields,” Opt. Express 25, 23974–23988 (2017).
[Crossref] [PubMed]

2014 (2)

2010 (2)

2009 (1)

J. Turunen and A. T. Friberg, ”Propagation-invariant optical fields,” Progr. Opt. 54, 1–88 (2009).

2008 (2)

2006 (1)

1999 (1)

R. Borghi, “Superposition scheme for J0-correlated partially coherent sources,” IEEE J. Quantum Electron. 35, 849–856 (1999).
[Crossref]

1997 (1)

C. Palma and G. Cincotti, “Imaging of J0-correlated Bessel-Gauss beams,” IEEE J. Quantum Electron. 33, 1032–1040 (1997).
[Crossref]

1994 (1)

1993 (1)

1992 (1)

G. Cincotti, F. Gori, and M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A: Math. Gen. 25(20), L1191-L1194 (1992).
[Crossref]

1988 (1)

1987 (2)

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

J. M. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

1983 (2)

F. Gori, “Mode propagation of the fields generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[Crossref]

F. Gori and G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[Crossref]

1982 (1)

1980 (2)

F. Gori, “Directionality and spatial coherence,” Opt. Acta 27, 1025–1034 (1980).
[Crossref]

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

1978 (1)

F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional light beams,” Opt. Commun. 27, 185–188 (1978).
[Crossref]

1970 (1)

Borghi, R.

Cai, Y.

Chen, Y.

Cincotti, G.

C. Palma and G. Cincotti, “Imaging of J0-correlated Bessel-Gauss beams,” IEEE J. Quantum Electron. 33, 1032–1040 (1997).
[Crossref]

G. Cincotti, F. Gori, and M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A: Math. Gen. 25(20), L1191-L1194 (1992).
[Crossref]

Collins, S. A.

Durnin, J. M.

J. M. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Eberly, J. H.

J. M. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Friberg, A. T.

Gbur, G.

G. Gbur and T. Visser, “The structure of partially coherent fields,” Progr. Opt. 55, 285-341 (2010).
[Crossref]

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and C. F. Li, “Partially correlated thin annular sources: The scalar case,” J. Opt. Soc. Am. A 25, 2826–2832 (2008).
[Crossref]

F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33, 1857–1859 (2008).
[Crossref] [PubMed]

G. Cincotti, F. Gori, and M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A: Math. Gen. 25(20), L1191-L1194 (1992).
[Crossref]

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

F. Gori, “Mode propagation of the fields generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[Crossref]

F. Gori and G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[Crossref]

F. Gori, “Directionality and spatial coherence,” Opt. Acta 27, 1025–1034 (1980).
[Crossref]

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional light beams,” Opt. Commun. 27, 185–188 (1978).
[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]

F. Gori and G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[Crossref]

Halder, A.

Koivurova, M.

Li, C. F.

Liu, L.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Progr. Opt. 62, 157–223 (2017).
[Crossref]

Liu, X.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Progr. Opt. 62, 157–223 (2017).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Miceli, J. J.

J. M. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Mukunda, N.

Padovani, C.

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

Palma, C.

C. Palma and G. Cincotti, “Imaging of J0-correlated Bessel-Gauss beams,” IEEE J. Quantum Electron. 33, 1032–1040 (1997).
[Crossref]

F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional light beams,” Opt. Commun. 27, 185–188 (1978).
[Crossref]

Partanen, H.

Ramírez-Sánchez, V.

Santarsiero, M.

Simon, R.

Starikov, A.

Tervo, J.

Tervonen, E.

Turunen, J.

Vahimaa, P.

Visser, T.

G. Gbur and T. Visser, “The structure of partially coherent fields,” Progr. Opt. 55, 285-341 (2010).
[Crossref]

Wang, F.

Wolf, E.

Yu, J.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Progr. Opt. 62, 157–223 (2017).
[Crossref]

IEEE J. Quantum Electron. (2)

C. Palma and G. Cincotti, “Imaging of J0-correlated Bessel-Gauss beams,” IEEE J. Quantum Electron. 33, 1032–1040 (1997).
[Crossref]

R. Borghi, “Superposition scheme for J0-correlated partially coherent sources,” IEEE J. Quantum Electron. 35, 849–856 (1999).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

J. Phys. A: Math. Gen. (1)

G. Cincotti, F. Gori, and M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A: Math. Gen. 25(20), L1191-L1194 (1992).
[Crossref]

Opt. Acta (1)

F. Gori, “Directionality and spatial coherence,” Opt. Acta 27, 1025–1034 (1980).
[Crossref]

Opt. Commun. (5)

F. Gori, “Mode propagation of the fields generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[Crossref]

F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional light beams,” Opt. Commun. 27, 185–188 (1978).
[Crossref]

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

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

F. Gori and G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

J. M. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Progr. Opt. (3)

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Progr. Opt. 62, 157–223 (2017).
[Crossref]

G. Gbur and T. Visser, “The structure of partially coherent fields,” Progr. Opt. 55, 285-341 (2010).
[Crossref]

J. Turunen and A. T. Friberg, ”Propagation-invariant optical fields,” Progr. Opt. 54, 1–88 (2009).

Other (1)

L. Mandel and E. Wolf, Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 Geometry. A Gaussian beam with 1/e2 waist radius w0 is incident on a rotating prism with wedge angle α, central thickness d, and refractive index n. In the paraxial domain (small values of α) the output beam at z = 0 is a spatially displaced [by R (1−1/n)] and angularly deflected [by βα(n – 1)] Gaussian beam.
Fig. 2
Fig. 2 Evolution of scaled coordinates for the complex degree of coherence of self-Fourier-transforming Bessel-correlated fields (blue solid line) and the incident Gaussian beam (red dashed line).
Fig. 3
Fig. 3 Absolute values of the complex degree of spatial coherence of self-Fourier-transforming Bessel-correlated fields for various values of the ratio r = R/w0. (a) r = 0.5. (b) r = 0.75. (c) r = 1. (d) r = 2.
Fig. 4
Fig. 4 Intensity profiles of self-Fourier-transforming Bessel-correlated fields. (a) Profiles at z = 0 for various values of r = R/w0. Black: r = 0.5. Red: r = 0.75. Green: r = 1. Blue: r = 2. (b) Profiles with r = 0.5 at distances z = 0 (black), z = zR (red), z = 2zR (green), z = 4zR (blue), and z = 8zR (purple). For clarity the profiles have been normalized to their maximum values.
Fig. 5
Fig. 5 Experimental setup: A is a single-mode HeNe laser source, BF a beam forming system, W the rotating wedge, L an imaging lens, DMD a digital micro-mirror device, and P is a plane at a (variable) distance z from the exit plane of the wedge.
Fig. 6
Fig. 6 Simulated (top row) and measured (bottom row) absolute value of the complex degree of spatial coherence in the (x1, x2) co-ordinate system when Eq. (22) is nearly satisfied: (a,e) z = 0 mm. (b,f) z = 2 mm. (c,g) z = 4 mm. (d,h) z = 6 mm. w0 ≈ 40 µm.
Fig. 7
Fig. 7 Intensity cross-sections at y = 0 µm when (22) is nearly satisfied. Blue: z = 0 mm. Black: z = 2 mm. Red: z = 4 mm. Pink: z = 6 mm. Green: z = 8 mm. Maroon: z = 10 mm. w0 ≈ 40 µm.
Fig. 8
Fig. 8 Same as Fig. 6, but under experimental conditions where Eq. (22) is not satisfied: (a) z = 0 mm. (b) z = 2 mm. (c) z = 4 mm. (d) z = 6 mm. w0 ≈ 100 µm.
Fig. 9
Fig. 9 Intensity cross-sections when (22) is not satisfied. Blue: z = 0 mm. Black: z = 4 mm. Red: z = 8 mm. Pink: z = 10 mm. w0 ≈ 100 µm.

Equations (40)

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W ( x 1 , y 1 , x 2 , y 2 , z ) = 1 2 π 0 2 π e ( x 1 , y 1 , z ; ϕ ) e ( x 2 , y 2 , z ; ϕ ) d ϕ ,
e ( x , y , 0 ; ϕ ) = e 0 exp [ ( x R cos ϕ ) 2 + ( y R sin ϕ ) 2 w 0 2 ] exp [ i k 0 sin β ( x cos ϕ + y sin ϕ R ) ] ,
W ( x 1 , y 1 , x 2 , y 2 , 0 ) = S 0 exp ( x 1 2 + x 2 2 + y 1 2 + y 2 2 + 2 R 2 w 0 2 ) I 0 [ a ( x 1 , y 1 , x 2 , y 2 , 0 ) ] ,
a ( x 1 , y 1 , x 2 , y 2 , 0 ) = 4 R w 0 2 [ ( x ¯ i z R sin β R Δ x 2 ) 2 + ( y ¯ i z R sin β R Δ y 2 ) 2 ] 1 / 2 ,
S ( x , y , 0 ) = W ( x , x , y , y , 0 ) = S 0 exp [ 2 ( x 2 + y 2 + R 2 ) w 0 2 ] I 0 [ 4 R w 0 2 x 2 + y 2 ]
μ ( x 1 , x 2 , y 1 , y 2 , 0 ) = W ( x 1 , x 2 , y 1 , y 2 , 0 ) S ( x 1 , y 1 , 0 ) S ( x 2 , y 2 , 0 ) = I 0 [ a ( x 1 , y 1 , x 2 , y 2 , 0 ) ] I 0 [ a ( x 1 , y 1 , x 1 , y 1 , 0 ) ] I 0 [ a ( x 2 , y 2 , x 2 , y 2 , 0 ) ] .
e ( x , y , z ; ϕ ) = k 0 i 2 π z exp ( i k 0 z ) exp [ i k 0 2 z ( x 2 + y 2 ) ] × e ( x , y , 0 ; ϕ ) exp [ i k 0 2 z ( x 2 + y 2 ) ] exp [ i k 0 z ( x x + y y ) ] d x d y
e ( x , y , z ; ϕ ) = e 0 1 + i z / z R exp [ i k 0 z ( 1 1 2 sin 2 β 1 + i z / z R ) ] exp [ ( x R cos ϕ ) 2 + ( y R sin ϕ ) 2 w 0 2 ( 1 + i z / z R ) ] × exp [ i k 0 sin β 1 + i z / z R ( x cos ϕ + y sin ϕ R ) ] .
W ( x 1 , y 1 , x 2 , y 2 , z ) = | e 0 | 2 1 + ( z / z R ) 2 × exp [ k 0 z R sin 2 β ( z / z R ) 2 1 + ( z / z R ) 2 ] exp [ x 1 2 + y 1 2 + R 2 w 0 2 ( 1 i z / z R ) ] exp [ x 2 2 + y 2 2 + R 2 w 0 2 ( 1 + i z / z R ) ] × 1 2 π 0 2 π exp { [ c ( z ) x ¯ i d ( z ) Δ x / 2 ] cos ϕ + [ c ( z ) y ¯ i d ( z ) Δ y / 2 ] sin ϕ } d ϕ ,
c ( z ) = 4 w 0 2 R + z sin β 1 + ( z / z R ) 2 = 4 R w 2 ( z ) ( 1 + z R sin β ) ,
d ( z ) = 2 k 0 R z / z R 2 + sin β 1 + ( z / z R ) 2 = 4 R w 2 ( z ) ( z z R + z R R sin β ) ,
w ( z ) = w 0 [ 1 + ( z / z R ) 2 ] 1 / 2 .
W ( x 1 , y 1 , x 2 , y 2 , z ) = S 0 w 0 2 w 2 ( z ) exp [ x 1 2 + x 2 2 + y 1 2 + y 2 2 + 2 ( R z sin β ) 2 w 2 ( z ) ] × I 0 [ a ( x 1 , y 1 , x 2 , y 2 , z ) ] exp [ i k 0 2 R ( z ) ( x 2 2 x 1 2 + y 2 2 y 1 2 ) ] ,
a ( x 1 , y 1 , x 2 , y 2 , z ) = { [ c ( z ) x ¯ i d ( z ) Δ x / 2 ] 2 + [ c ( z ) y ¯ i d ( z ) Δ y / 2 ] 2 } 1 / 2 = 4 R w 2 ( z ) { [ ( 1 + z R sin β ) x ¯ i ( z z R + z R R sin β ) Δ x 2 ] 2 + [ ( 1 + z R sin β ) y ¯ i ( z z R + z R R sin β ) Δ y 2 ] 2 } 1 / 2
R ( z ) = z + z R 2 / z .
S ( x , y , z ) = S 0 w 0 2 w 2 ( z ) exp { 2 w 2 ( z ) [ x 2 + y 2 + ( R z sin β ) 2 ] } I 0 [ 4 R w 2 ( z ) ( 1 + z R sin β ) x 2 + y 2 ] .
μ ( x 1 , x 2 , y 1 , y 2 , z ) = I 0 [ a ( x 1 , y 1 , x 2 , y 2 , z ) ] I 0 [ a ( x 1 , y 1 , x 1 , y 1 , z ) ] I 0 [ a ( x 2 , y 2 , x 2 , y 2 , z ) ] exp [ i k 0 2 R ( z ) ( x 2 2 x 1 2 + y 2 2 y 1 2 ) ] .
W ( ) ( x 1 , y 1 , x 2 , y 2 , z ) = S 0 ( z R z ) 2 exp [ ( z R z ) 2 x 1 2 + x 2 2 + y 1 2 + y 2 2 + 2 ( R z sin β ) 2 w 0 2 ] × I 0 [ a ( ) ( x 1 , y 1 , x 2 , y 2 , z ) ] exp [ i k 0 2 z ( x 2 2 x 1 2 + y 2 2 y 1 2 ) ]
a ( ) ( x 1 , y 1 , x 2 , y 2 , z ) = 2 R k 0 z [ ( z R R sin β x ¯ i Δ x 2 ) 2 + ( z R R sin β y ¯ i Δ y 2 ) 2 ] 1 / 2 .
S ( ) ( x , y , z ) = S 0 ( z R z ) 2 exp [ 2 ( z R z ) 2 x 2 + y 2 + z 2 sin 2 β w 0 2 ] I 0 [ 2 k 0 sin β z R z x 2 + y 2 ]
μ ( ) ( x 1 , y 1 , x 2 , y 2 , z ) = I 0 [ a ( ) ( x 1 , y 1 , x 2 , y 2 , z ) ] I 0 [ a ( ) ( x 1 , y 1 , x 1 , y 1 , z ) ] I 0 [ a ( ) ( x 2 , y 2 , x 2 , y 2 , z ) ] exp [ i k 0 2 z ( x 2 2 x 1 2 + y 2 2 y 1 2 ) ] .
z R sin β = R
a ( x 1 , y 1 , x 2 , y 2 , 0 ) = 4 R w 0 2 [ ( x ¯ i Δ x / 2 ) 2 + ( y ¯ i Δ y / 2 ) 2 ] 1 / 2 .
a ( ) ( x 1 , y 1 , x 2 , y 2 , z ) = 2 R k 0 z [ ( x ¯ i Δ x / 2 ) 2 + ( y ¯ i Δ y / 2 ) 2 ] 1 / 2 .
a ( ) ( x 1 , y 1 , x 2 , y 2 , z ) = 4 R w 0 2 { [ x ¯ ( z ) i Δ x ( x ) / 2 ] 2 + [ y ¯ ( z ) i Δ y ( z ) / 2 ] 2 } 1 / 2 .
S ( ) ( x , y , z ) = S 0 ( z R z ) 2 exp { 2 [ x 2 ( z ) + y 2 ( z ) + R 2 ] w 0 2 } I 0 [ 4 R w 0 2 x 2 ( z ) + y 2 ( z ) ] .
a ( x 1 , y 1 , x 2 , y 2 , z ) = 4 R w 2 ( z ) ( 1 + z z R ) [ ( x ¯ i Δ x / 2 ) 2 + ( y ¯ i Δ y / 2 ) 2 ] 1 / 2 .
S ( x , y , z ) = S 0 exp { 2 w 2 ( z ) [ x 2 + y 2 + R 2 ( 1 z z R ) 2 ] } I 0 [ 4 R w 2 ( z ) ( 1 + z z R ) x 2 + y 2 ] .
x ( z ) = x 1 + z / z R 1 + ( z / z R ) 2 , y ( z ) = y 1 + z / z R 1 + ( z / z R ) 2 ,
a ( x 1 , y 1 , x 2 , x 2 , z ) = 4 R w 0 2 { [ x ¯ ( z ) i Δ x ( z ) / 2 ] 2 + [ y ¯ ( z ) i Δ y ( z ) / 2 ] 2 } 1 / 2
S ( x , y , z ) = S 0 w 0 2 w 2 ( z ) exp ( 2 w 0 2 1 + ( z / z R ) 2 ( 1 + z / z R ) 2 { x 2 ( z ) + y 2 ( z ) + R 2 [ 1 ( z / z R ) 2 1 + ( z / z R ) 2 ] 2 } ) × I 0 [ 4 R w 0 2 x 2 ( z ) + y 2 ( z ) ] ,
x ( z ) = x [ 1 + ( z / z R ) 2 ] 1 , y ( z ) = y [ 1 + ( z / z R ) 2 ] 1
e ( x , y , L ; ϕ ) = k 0 i 2 π B exp ( i k 0 L ) exp [ i k 0 D 2 B ( x 2 + y 2 ) ] × e ( x , y , 0 ; ϕ ) exp [ i k 0 A 2 B ( x 2 + y 2 ) ] exp [ i k 0 B ( x x + y y ) ] d x d y ,
e ( x , y , L ; ϕ ) = e 0 A + i B / z R exp ( i k 0 L ) exp ( i k 0 2 B sin 2 β A + i B / z R ) exp ( R 2 w 0 2 A A + i B / z R ) × exp ( i k 0 R sin β A A + i B / z R ) exp [ i k 0 2 C + i D / z R A + i B / z R ( x 2 + y 2 ) ] × exp [ i k 0 sin β i R / z R A + i B / z R ( x cos ϕ + y sin ϕ ) ] ,
w L = w 0 [ A 2 + ( B / z R ) 2 ] 1 / 2
R L = A 2 + ( B / z R ) 2 A C + B D / z R 2 .
W ( x 1 , y 1 , x 2 , y 2 , z ) = S 0 w 0 2 w L 2 exp [ x 1 2 + x 2 2 + y 1 2 + y 2 2 + 2 ( A R B sin β ) 2 w L 2 ] × I 0 [ a ( x 1 , y 1 , x 2 , y 2 , L ) ] exp [ i k 0 2 R L ( x 1 2 x 2 2 + y 1 2 y 2 2 ) ] ,
a ( x 1 , y 1 , x 2 , y 2 , L ) = [ ( c L x ¯ i d L Δ x / 2 ) 2 + ( c L y ¯ i d L Δ y / 2 ) 2 ] 1 / 2 ,
c L = 4 R w L 2 ( 1 + B R sin β ) ,
d L = 4 R w L 2 ( B z R + A z R R sin β ) .

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