Abstract

We study theoretically and experimentally the interference of light produced by a pair of mutually correlated Schell-model sources. The spatial distributions of the fields produced by the two sources are inverted with respect to each other through their common center in the source plane. When the beams are in phase, a bright spot appears in the center of the spatial distribution of the beam intensity. When the beams have a phase shift ϕ = π, a dark spot appears in the center of the spatial distribution of the beam intensity. Experimental results that illustrate these results are included. Both bright and dark spots diverge more slowly with the increasing distance from the sources than the beam itself.

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References

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  1. Zu-Han Gu, E. R. Mendez, M. Ciftan, T. A. Leskova, and A. A. Maradudin, "Interference of a pair of symmetric Collett-Wolf beams," Opt. Lett. 30, 1605-1607 (2005).
    [CrossRef] [PubMed]
  2. E. Collett and E. Wolf, "Is complete spatial coherence necessary for the generation of highly directional light beams?," Opt. Lett. 2, 27-29 (1978).
    [CrossRef] [PubMed]
  3. E. Wolf and E. Collett, "Partially coherent sources which produce the same far-field intensity distribution as a laser," Opt. Commun. 25, 293-296 (1978).
    [CrossRef]
  4. P. DeSantis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
    [CrossRef]
  5. A. C. Schell, The Multiple Plate Antenna, Ph. D. Dissertation, Massachusetts Institute of Technology, 1961, Section 7.5.
  6. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995), Section 4.3.2.
  7. J. J. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297-300 (1978).
    [CrossRef]
  8. Ref. 6, Section 5.2.1.
  9. R. M. Fitzgerald, T. A. Leskova, and A. A. Maradudin, "Control of coherence of light scattered from a onedimensional randomly rough surface that acts as a Schell-model source," J. Lumin. 125, 147-155 (2007).
    [CrossRef]

2007

R. M. Fitzgerald, T. A. Leskova, and A. A. Maradudin, "Control of coherence of light scattered from a onedimensional randomly rough surface that acts as a Schell-model source," J. Lumin. 125, 147-155 (2007).
[CrossRef]

2005

1979

P. DeSantis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

1978

J. J. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297-300 (1978).
[CrossRef]

E. Collett and E. Wolf, "Is complete spatial coherence necessary for the generation of highly directional light beams?," Opt. Lett. 2, 27-29 (1978).
[CrossRef] [PubMed]

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

Collett, E.

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

E. Collett and E. Wolf, "Is complete spatial coherence necessary for the generation of highly directional light beams?," Opt. Lett. 2, 27-29 (1978).
[CrossRef] [PubMed]

DeSantis, P.

P. DeSantis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

Fitzgerald, R. M.

R. M. Fitzgerald, T. A. Leskova, and A. A. Maradudin, "Control of coherence of light scattered from a onedimensional randomly rough surface that acts as a Schell-model source," J. Lumin. 125, 147-155 (2007).
[CrossRef]

Foley, J. J.

J. J. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297-300 (1978).
[CrossRef]

Gori, F.

P. DeSantis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

Guattari, G.

P. DeSantis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

Leskova, T. A.

R. M. Fitzgerald, T. A. Leskova, and A. A. Maradudin, "Control of coherence of light scattered from a onedimensional randomly rough surface that acts as a Schell-model source," J. Lumin. 125, 147-155 (2007).
[CrossRef]

Maradudin, A. A.

R. M. Fitzgerald, T. A. Leskova, and A. A. Maradudin, "Control of coherence of light scattered from a onedimensional randomly rough surface that acts as a Schell-model source," J. Lumin. 125, 147-155 (2007).
[CrossRef]

Palma, C.

P. DeSantis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

Wolf, E.

E. Collett and E. Wolf, "Is complete spatial coherence necessary for the generation of highly directional light beams?," Opt. Lett. 2, 27-29 (1978).
[CrossRef] [PubMed]

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

Zubairy, M. S.

J. J. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297-300 (1978).
[CrossRef]

Zu-Han Gu,

J. Lumin.

R. M. Fitzgerald, T. A. Leskova, and A. A. Maradudin, "Control of coherence of light scattered from a onedimensional randomly rough surface that acts as a Schell-model source," J. Lumin. 125, 147-155 (2007).
[CrossRef]

Opt. Commun.

J. J. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297-300 (1978).
[CrossRef]

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

P. DeSantis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

Opt. Lett.

Other

Ref. 6, Section 5.2.1.

A. C. Schell, The Multiple Plate Antenna, Ph. D. Dissertation, Massachusetts Institute of Technology, 1961, Section 7.5.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995), Section 4.3.2.

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Figures (8)

Fig. 1.
Fig. 1.

The spatial distribution of the normalized intensity of the beams that are inverted with respect to their common center with (a) and (c) ϕ = 0 and (b) and (d) ϕ = π. The distance from the source plane is x3 = 1 m (a) and (b), and x3 = 100m (c) and (d).

Fig. 2.
Fig. 2.

Theoretical plot of the evolution along the x3 axis of the cross section x2 = 0 of the normalized intensity of the beams that are inverted with respect to their common center with ϕ = 0 (a) and ϕ = π (b).

Fig. 3.
Fig. 3.

The spatial distribution of the normalized intensity of the beams that have the spectral degree of coherence of the Lorentzian form and are inverted with respect to their common center with (a) and (c) ϕ = 0 and (b) and (d) ϕ = π. The distance from the source plane is x3 = 1 m (a) and (b), and x3 = 100m (c) and (d).

Fig. 4.
Fig. 4.

The intensity (a) and the normalized intensity (b) of the beam as a function of x1 at x2 = 0 at a distance from the source plane x3 = 500 m. The spectral degree of coherence of in the source plane has a Lorentzian form (black curve) or a Gaussian form (red curve).

Fig. 5.
Fig. 5.

Plots of the normalized intensity of the beams that have the spectral degree of coherence of the form g(0)(u) = sinc(√3u1/σg)sinc(√3u2/σg), and are inverted with respect to their common center with ϕ = 0 (a) and (c) and ϕ = π (b) and (d). The distance from the source plane is x3 = 1 m (a) and (b), and x3 = 100m (c) and (d).

Fig. 6.
Fig. 6.

Plots of the normalized intensity of the beams whose spectral density in the source plane is a constant within a circular domain and their spectral degree of coherence has a Gaussian form. The beams fields are inverted with respect to their common center with ϕ = 0 (a) and (c) and ϕ = π (b) and (d). The distance from the source plane is x3 = 1 m (a) and (b), and x3 = 1Km (c) and (d).

Fig. 7.
Fig. 7.

Schematic diagram of the experimental arrangement used.

Fig. 8.
Fig. 8.

Experimental gray-level images of the normalized intensity of the symmetric beams for (a) constructive interference and (b) destructive interference. The horizontal lines in the images show the positions where the intensity scans in the lower graphs were taken.

Equations (52)

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W ( 0 ) x 0 x 0 = U ( x , 0 ω ) U * x 0 ω ,
W ( 0 ) x 0 x 0 = [ S ( 0 ) ( x ) ] 1 2 g ( 0 ) ( x x ) [ S ( 0 ) ( x ) ] 1 2 .
g ( 0 ) ( x x ) = g ( 0 ) ( x x ) * ,
0 g ( 0 ) ( x x ) 1
g ( 0 ) ( 0 ) = 1 .
U x x 3 ω = ( ω 2 πic x 3 ) exp [ i ω c x 3 ] d 2 x exp [ i ω 2 c x 3 ( x x ) 2 ] U x 0 ω .
U ( x , x 3 ω ) = U 1 x x 3 ω + U 2 x x 3 ω exp ( ) ,
U 2 x 1 x 2 x 3 | ω = U 1 x 1 x 2 x 3 | ω .
I ( x , x 3 ω ) = U ( x , x 3 ω ) 2
= ( ω 2 πc x 3 ) 2 d 2 x d 2 x exp [ i ω 2 c x 3 ( x x ) 2 ]
× exp [ i ω 2 c x 3 ( x x ) 2 ] U x 0 ω U * x 0 ω
= ( ω 2 πc x 3 ) 2 d 2 x d 2 x exp [ i ω 2 c x 3 ( x x ) 2 ]
× exp [ i ω 2 c x 3 ( x x ) 2 ] { 2 W ( 0 ) x 0 x 0
+ U 1 x 0 ω U 2 * x 0 ω exp ( ) + U 1 * x 0 ω U 2 x 0 ω exp ( ) } .
U 1 ( x , 0 ω ) U 2 * x 0 ω = U 1 x 1 x 2 0 ω U 1 * x 1 x 2 0 ω
= W ( 0 ) ( x , 0 x , 0 )
U 1 * ( x , 0 ω ) U 2 x 0 ω = U 2 * x 1 x 2 0 ω U 2 x 1 x 2 0 ω
= W ( 0 ) ( x , 0 x , 0 ) .
I ( x , x 3 ω ) = I + x x 3 ω + cos ϕ I ( x , x 3 ω ) ,
I ± x x 3 ω = 2 ( ω 2 πc x 3 ) 2 d 2 x d 2 x exp [ i ω 2 c x 3 ( x x ) 2 ]
× exp [ i ω 2 c x 3 ( x x ) 2 ] W ( 0 ) ( x , 0 | ± x " , 0 ) .
I ± ( x , x 3 ω ) = 2 ( ω 2 πc x 3 ) 2 d 2 x d 2 x g ( 0 ) ( x x ) [ S ( 0 ) ( x ) ] 1 2
× [ S ( 0 ) ( ± x " ) ] 1 2 exp [ i ω 2 c x 3 ( x 2 x 2 ) ] exp [ i ω c x 3 ( x x ) · x ] .
I + x x 3 ω = 2 ( ω 2 πc x 3 ) 2 d 2 u g ( 0 ) ( u ) exp [ i ω 2 c x 3 u 2 ] exp [ i ω c x 3 x · u ]
× d 2 x exp { i ω c x 3 [ u · x ] } [ S ( 0 ) ( | x + u | ) ] 1 2 [ S ( 0 ) ( x ) ] 1 2
I x x 3 ω = 2 ( ω 2 πc x 3 ) 2 d 2 u g ( 0 ) ( u ) exp [ i ω 2 c x 3 u 2 ] exp [ i ω c x 3 x · u ]
× d 2 x exp { i ω c x 3 [ ( u 2 x ) · x ] } [ S ( 0 ) ( | x + u | ) ] 1 2 [ S ( 0 ) ( x ) ] 1 2 .
S ( 0 ) ( x ) = exp ( x 2 / 2 σ s 2 ) .
I + x x 3 ω = 1 π ( ω 2 σ s 2 c 2 x 3 2 ) d 2 u g ( 0 ) ( u ) exp [ i ω c x 3 x · u ]
× exp { u 2 [ 1 8 σ s 2 + ω 2 σ s 2 2 c 2 x 3 2 ] } ,
I x x 3 ω = 1 π ( ω 2 σ s 2 c 2 x 3 2 ) exp [ 2 ω 2 σ s 2 c 2 x 3 2 x 2 ] d 2 u g ( 0 ) ( u ) exp [ 2 ω 2 σ s 2 c 2 x 3 2 u · x ]
× exp { u 2 [ 1 8 σ s 2 + ω 2 σ s 2 2 c 2 x 3 2 ] } .
I + ( x , x 3 ω ) = 1 π ( ω 2 σ s 2 c 2 x 3 2 ) d 2 u g ( 0 ) ( u ) exp ( i ω c x 3 x · u ) ,
I ( x , x 3 ω ) = 1 π ( ω 2 σ s 2 c 2 x 3 2 ) exp { 2 ( ω σ s c x 3 ) 2 x 2 } d 2 u g ( 0 ) ( u ) .
Δ i = 1 2 ( ω / c ) σ s .
I x x 3 ω = 2 ( 1 + cos ϕ ) 1 + ( c x 3 2 ω σ s 2 ) 2 exp { x 2 2 σ s 2 [ 1 + ( c x 3 2 ω σ s 2 ) 2 ] } .
I x x 3 ω = A c π ( ω σ s c x 3 ) 2 { 1 + cos ϕ exp [ 2 ( ω σ s c x 3 ) 2 x 2 ] } ,
A c = d 2 u g ( 0 ) ( u )
g ( 0 ) ( u ) = 1 u 2 2 σ g 2 + o ( u 2 ) .
I x x 3 ω = 4 σ s 2 σ eff 2 ( x 3 ) exp ( x 2 σ eff 2 ( x 3 ) ) { 1 + cos ϕ exp [ 4 σ s 2 σ g 2 x 2 σ eff 2 ( x 3 ) ] } ,
σ eff 2 ( x 3 ) = 2 σ s 2 + c 2 x 3 2 ω 2 ( 1 2 σ s 2 + 2 σ g 2 ) .
σ eff ( x 3 ) = 2 ( ω / c ) σ g x 3 = Δ b x 3 ,
I x x 3 ω = ( 2 ω c σ s σ g x 3 ) 2 exp { ( x / x 3 ) 2 Δ b 2 } [ 1 + cos ϕ exp ] { ( x / x 3 ) 2 Δ i 2 } ,
I + x x 3 ω = 1 π ( ω 2 σ s 2 c 2 x 3 2 ) 0 d u u g ( 0 ) ( u ) exp [ ( ω 2 σ s 2 2 c 2 x 3 2 + 1 8 σ s 2 ) u 2 ]
× π π d ϕ u exp [ i ω c x 3 x u cos ( ϕ ϕ u ) ]
= 2 ( ω 2 σ s 2 c 2 x 3 2 ) 0 d u u g ( 0 ) ( u ) J 0 ( ω c x 3 x u ) exp [ ( ω 2 σ s 2 2 c 2 x 3 2 + 1 8 σ s 2 ) u 2 ] ,
I x x 3 ω = 1 π ( ω 2 σ s 2 c 2 x 3 2 ) exp ( 2 ω 2 σ s 2 c 2 x 3 2 x 2 ) 0 d u u g ( 0 ) ( u )
× exp [ ( ω 2 σ s 2 2 c 2 x 3 2 + 1 8 σ 3 2 ) u 2 ] π π d ϕ u exp [ 2 ω 2 σ s 2 c 2 x 3 2 x u cos ( ϕ ϕ u ) ]
= 2 ( ω 2 σ s 2 c 2 x 3 2 ) exp ( 2 ω 2 σ s 2 c 2 x 3 2 x 2 ) 0 d u u g ( 0 ) ( u ) I 0 ( 2 ( ω σ s c x 3 ) 2 x u )
× exp [ ( ω 2 σ s 2 2 c 2 x 3 2 + 1 8 σ s 2 ) u 2 ] ,
S ( 0 ) ( x ) = { 1 , x < σ s 0 , x > σ s ,
g ( 0 ) ( x ) = exp ( x 2 / 2 σ g 2 ) .

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