Abstract

We discuss Young’s experiment with electromagnetic random fields at arbitrary states of coherence and polarization within the framework of the electric spatial coherence wavelets. The use of this approach for the electromagnetic spatial coherence theory allows us to envisage the existence of polarization domains inside the observation plane. We show that it is possible to locally control those polarization domains by means of the correlation properties of the electromagnetic wave. To show the validity of this alternative approach, we derive by means of numerical modeling the classical Fresnel–Arago interference laws.

© 2006 Optical Society of America

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  1. R. Castaneda and J. Garcia-Sucerquia, "Electromagnetic spatial coherence wavelets," J. Opt. Soc. Am. A 23, 81-90 (2006).
    [CrossRef]
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  3. 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]
  4. E. Wolf, "Unified theory of coherence and polarisation of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  5. M. Mujat, A. Dogariu, and E. Wolf, "Law of interference of electromagnetic beams of any state of coherence and polarization and the Fresnel-Arago interference laws," J. Opt. Soc. Am. A 21, 2414-2417 (2004).
    [CrossRef]
  6. J. García, R. Castañeda, F. F. Medina, and G. Matteucci, "Distinguishing between Fraunhofer and Fresnel diffraction by the Young's experiment," Opt. Commun. 200, 15-22 (2001).
    [CrossRef]
  7. M. Born and E. Wolf, Principles of Optics, 6th. ed. (Pergamon, 1993).

2006

2005

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]

2004

2003

E. Wolf, "Unified theory of coherence and polarisation of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

2001

J. García, R. Castañeda, F. F. Medina, and G. Matteucci, "Distinguishing between Fraunhofer and Fresnel diffraction by the Young's experiment," Opt. Commun. 200, 15-22 (2001).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th. ed. (Pergamon, 1993).

Castaneda, R.

Castañeda, R.

J. García, R. Castañeda, F. F. Medina, and G. Matteucci, "Distinguishing between Fraunhofer and Fresnel diffraction by the Young's experiment," Opt. Commun. 200, 15-22 (2001).
[CrossRef]

Dogariu, A.

García, J.

J. García, R. Castañeda, F. F. Medina, and G. Matteucci, "Distinguishing between Fraunhofer and Fresnel diffraction by the Young's experiment," Opt. Commun. 200, 15-22 (2001).
[CrossRef]

Garcia-Sucerquia, J.

Korotkova, O.

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]

Mandel, L.

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

Matteucci, G.

J. García, R. Castañeda, F. F. Medina, and G. Matteucci, "Distinguishing between Fraunhofer and Fresnel diffraction by the Young's experiment," Opt. Commun. 200, 15-22 (2001).
[CrossRef]

Medina, F. F.

J. García, R. Castañeda, F. F. Medina, and G. Matteucci, "Distinguishing between Fraunhofer and Fresnel diffraction by the Young's experiment," Opt. Commun. 200, 15-22 (2001).
[CrossRef]

Mujat, M.

Wolf, E.

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]

M. Mujat, A. Dogariu, and E. Wolf, "Law of interference of electromagnetic beams of any state of coherence and polarization and the Fresnel-Arago interference laws," J. Opt. Soc. Am. A 21, 2414-2417 (2004).
[CrossRef]

E. Wolf, "Unified theory of coherence and polarisation of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

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

M. Born and E. Wolf, Principles of Optics, 6th. ed. (Pergamon, 1993).

J. Opt. Soc. Am. A

Opt. Commun.

J. García, R. Castañeda, F. F. Medina, and G. Matteucci, "Distinguishing between Fraunhofer and Fresnel diffraction by the Young's experiment," Opt. Commun. 200, 15-22 (2001).
[CrossRef]

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]

Phys. Lett. A

E. Wolf, "Unified theory of coherence and polarisation of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

Other

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

M. Born and E. Wolf, Principles of Optics, 6th. ed. (Pergamon, 1993).

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Figures (7)

Fig. 1
Fig. 1

Young’s experiment with electric spatial coherence wavelets. Young’s pinholes are arranged at the A plane. The two orientations of the separation vector R D are shown. Axes 1 and 3 are R A R D 2 r A , along which the wavelets W ( r A + r D 2 , r A r D 2 , R A ± R D 2 ; v ) propagate. Axis 2 is R A r A along which W ± ( r A + r D 2 , r A r D 2 , R A ; v ) propagate. All these wavelets have an influence within the region delimited by the dashed–dotted circle at the observation plane.

Fig. 2
Fig. 2

Illustration of the simulation principle of Young’s experiment with random electric spatial coherence wavelets.

Fig. 3
Fig. 3

Uniform fluctuations over [ π , π ] with (a) statistically independent ϑ ± and (b) ϑ + = ϑ .

Fig. 4
Fig. 4

Gaussian fluctuations over [ π , π ] with standard deviation σ = 1.0 rad , statistically independent ϑ ± , and (a) ϑ ¯ + = ϑ ¯ = π 4 and (b) ϑ ¯ + = π 4 , ϑ ¯ = 3 π 4 .

Fig. 5
Fig. 5

Gaussian fluctuations over [ π , π ] with standard deviation σ = 1.0 rad , Δ ϑ + = Δ ϑ , and (a) ϑ ¯ + = ϑ ¯ = π 4 and (b) ϑ ¯ + = π 4 , ϑ ¯ = 3 π 4 .

Fig. 6
Fig. 6

Very narrow Gaussian fluctuations over [ π , π ] (standard deviation σ = 5 × 10 4 rad ) with ϑ ¯ + = ϑ ¯ = π 4 for (a) statistically independent ϑ ± and (b) Δ ϑ + = Δ ϑ .

Fig. 7
Fig. 7

Very narrow Gaussian fluctuations over [ π , π ] (standard deviation σ = 5 × 10 4 rad ) with ϑ ¯ + = π 4 , ϑ ¯ = 3 π 4 for statistically independent ϑ ± or Δ ϑ + = Δ ϑ .

Tables (3)

Tables Icon

Table 1 Values of the Quantities That Define the Matrix Elements of the η Tensor under the Following Conditions of Very Narrow Fluctuations at Both Openings ( Δ ϑ ± 0 ) , Equal-Valued Fluctuations ( Δ ϑ + = Δ ϑ ) , and Uniform Fluctuations over the Whole Interval [ π , π ]

Tables Icon

Table 2 Numerical Results of the Simulations of Young’s Experiment with Electric Spatial Coherence Wavelets of Random Electromagnetic Fields a

Tables Icon

Table 3 Numerical Results of the Simulation of Young’s Experiment by Rotating Polarized Modes to Bring Them Parallel a

Equations (24)

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W ( r A + r D 2 , r A r D 2 , r A ; v ) = W ( r A + r D 2 , r A r D 2 , R A + R D 2 ; v ) + W ( r A + r D 2 , r A r D 2 , R A R D 2 ; v ) + W + ( r A + r D 2 , r A r D 2 , R A ; v ) + W ( r A + r D 2 , r A r D 2 , R A ; v ) ,
W ( r A + r D 2 , r A r D 2 , R A ± R D 2 ; v ) = ( λ z ) 2 E ( r A , v ) η ( r A , r A , v ) E ( r A , v ) × δ ( r A R A R D 2 ) e i ( k z ) r D r A
W ± ( r A + r D 2 , r A r D 2 , R A ; v ) = ( λ z ) 2 δ ( r A R A ) e ± i ( k z ) ( r A r A ) R D e i ( k z ) r D r A × E ( r A ± R D 2 , v ) η ( r A ± R D 2 , r A R D 2 , v ) × E ( r A R D 2 , v )
E ( r A ± r D 2 , v ) = [ E x ( r A ± r D 2 , v ) 2 1 2 0 0 E y ( r A ± r D 2 , v ) 2 1 2 ] ,
P ( r A , r A ; v ) = 1 4 det [ S ( r A , r A ; v ) ] tr 2 [ S ( r A , r A ; v ) ] ,
S ind ( ± ) ( r A ; v ) = tr [ E ( R A ± R D 2 , v ) η ( R A ± R D 2 , R A ± R D 2 , v ) E ( R A ± R D 2 , v ) ]
S pair ( ± ) ( r A ; v ) = tr [ E ( R A ± R D 2 , v ) η ( R A ± R D 2 , R A R D 2 , v ) E ( R A R D 2 , v ) ] e ± i ( k z ) ( R A r A ) R D
η ( + , , v ) = [ cos ϑ + cos ϑ cos 2 ϑ + 1 2 cos 2 ϑ 1 2 cos ϑ + sin ϑ cos 2 ϑ + 1 2 sin 2 ϑ 1 2 sin ϑ + cos ϑ sin 2 ϑ + 1 2 cos 2 ϑ 1 2 sin ϑ + sin ϑ sin 2 ϑ + 1 2 sin 2 ϑ 1 2 ]
E ( ± , v ) = 1 2 E 0 ( v ) 2 1 2 [ 1 0 0 1 ] , η ( ± , ± ) = [ 1 0 0 1 ] , η ( ± , ) = [ 0 0 0 0 ] ( for ϑ + ϑ ) , η ( ± , ) = [ 1 0 0 1 ] ( for ϑ + = ϑ )
W ( r A + r D 2 , r A r D 2 , r A ; v ) = ( λ z ) 2 E 0 ( v ) 2 cos ( k 2 z R D r D ) [ 1 0 0 1 ] δ ( r A R A ) e ( i k ) z r A r D
W ( r A + r D 2 , r A r D 2 , r A ; v ) = ( λ z ) 2 E 0 ( v ) 2 δ ( r A R A ) e i ( k z ) r A r D × [ cos ( k 2 z R D r D ) + cos [ k z ( r A r A ) R D ] 0 0 cos ( k 2 z R D r D ) + cos [ k z ( r A r A ) R D ] ]
W ( r A + r D 2 , r A r D 2 , r A ; v ) = ( λ z ) 2 E 0 ( v ) 2 × [ cos ( k 2 z r D R D ) 0 0 cos ( k 2 z r D R D ) ] × δ ( r A R A ) e i ( k z ) r A r D
W ( r A + r D 2 , r A r D 2 , r A ; v ) = ( λ z ) 2 E 0 ( v ) 2 δ ( r A R A ) e i ( k z ) r A r D [ cos ( k 2 z r D R D ) + cos ( ϑ ¯ + ϑ ¯ ) cos [ k z ( r A r A ) R D ] i sin ( ϑ ¯ + ϑ ¯ ) sin [ k z ( r A r A ) R D ] i sin ( ϑ ¯ + ϑ ¯ ) sin [ k z ( r A r A ) R D ] cos ( k 2 z r D R D ) + cos ( ϑ ¯ + ϑ ¯ ) cos [ k z ( r A r A ) R D ] ] ,
P ( r A , R A ; v ) = sin ( ϑ ¯ + ϑ ¯ ) sin [ k z ( R A r A ) R D ] 1 + cos ( ϑ ¯ + ϑ ¯ ) cos [ k z ( R A r A ) R D ] .
W 1 + 2 ( r A + r D 2 , r A r D 2 , r A ; v ) = ( λ z ) 2 E 0 ( v ) 2 δ ( r A R A ) e i ( k z ) r A r D { cos [ k 2 z R D r D ] + cos [ k z ( r A r A ) R D ] } [ 1 0 0 1 ] .
W ( r A + r D 2 , r A r D 2 , r A ; v ) = ( λ z ) 2 E 0 ( v ) 2 δ ( r A R A ) e i ( k z ) r A r D [ cos 2 ϑ ¯ e i ( k 2 z ) R D r D + cos 2 ϑ ¯ + e i ( k 2 z ) R D r D + 2 cos ϑ ¯ + cos ϑ ¯ cos [ k z ( r A r A ) R D ] sin ϑ ¯ cos ϑ ¯ e i ( k 2 z ) R D r D + sin ϑ ¯ + cos ϑ ¯ + e i ( k 2 z ) R D r D + sin ϑ ¯ + cos ϑ ¯ e i ( k z ) ( r A r A ) R D + sin ϑ ¯ cos ϑ ¯ + e i ( k z ) ( r A r A ) R D sin ϑ ¯ cos ϑ ¯ e i ( k 2 z ) R D r D + sin ϑ ¯ + cos ϑ ¯ + e i ( k 2 z ) R D r D + cos ϑ ¯ + sin ϑ ¯ e i ( k z ) ( r A r A ) R D + cos ϑ ¯ sin ϑ ¯ + e i ( k z ) ( r A r A ) R D × sin 2 ϑ ¯ e i ( k 2 z ) R D r D + sin 2 ϑ ¯ + e i ( k 2 z ) R D r D + 2 sin ϑ ¯ + sin ϑ ¯ cos [ k z ( r A r A ) R D ] ] .
W ̃ ( r A + r D 2 , r A r D 2 , r A ; v ) = W ̃ ( r A + r D 2 , r A r D 2 , R A + R D 2 ; v ) + W ( r A + r D 2 , r A r D 2 , R A R D 2 ; v ) + W ̃ + ( r A + r D 2 , r A r D 2 , R A ; v ) + W ̃ ( r A + r D 2 , r A r D 2 , R A ; v ) ,
W ( r A + r D 2 , r A r D 2 , R A R D 2 ; v ) = ( λ z ) 2 E ( r A , v ) η ( r A , r A , v ) E ( r A , v ) × δ ( r A R A + R D 2 ) e i k z r D r A ,
W ̃ ( r A + r D 2 , r A r D 2 , R A + R D 2 ; v ) = ( λ z ) 2 δ ( r A R A R D 2 ) e i ( k z ) r D r A R ( γ ) E ( r A , v ) η ( r A , r A , v ) E ( r A , v ) R t ( γ ) ,
W ̃ + ( r A + r D 2 , r A r D 2 , R A ; v ) = ( λ z ) 2 δ ( r A R A ) e i ( k z ( ( r A r A ) R D e i ( k z ( r D r A R ( γ ) E ( r A + R D 2 , v ) η ( r A + R D 2 , r A R D 2 , v ) E ( r A R D 2 , v ) ,
W ̃ ( r A + r D 2 , r A r D 2 , R A ; v ) = ( λ z ) 2 δ ( r A R A ) e i ( k z ) ( r A r A ) R D e i ( k z ) r D r A E ( r A R D 2 , v ) η ( r A R D 2 , r A + R D 2 , v ) E ( r A + R D 2 , v ) R t ( γ ) ,
W ̃ ( r A + r D 2 , r A r D 2 , r A ; v ) = ( λ z ) 2 E 0 ( v ) 2 cos [ k 2 z R D r D ] [ 1 0 0 1 ] δ ( r A R A ) e i ( k z ) r A r D
W ̃ ( r A + r D 2 , r A r D 2 , r A ; v ) = ( λ z ) 2 E 0 ( v ) 2 δ ( r A R A ) e i ( k z ) r A r D × [ cos [ k 2 z R D r D ] + cos [ k z ( r A r A ) R D ] 0 0 cos [ k 2 z R D r D ] + cos [ k z ( r A r A ) R D ] ]
W ̃ ( r A + r D 2 , r A r D 2 , r A ; v ) = ( λ z ) 2 E 0 ( v ) 2 δ ( r A R A ) e i ( k z ) r A r D [ cos 2 ϑ ¯ e i ( k 2 z ) R D r D + sin 2 ϑ ¯ + e i ( k 2 z ) R D r D + 2 sin ϑ ¯ + cos ϑ ¯ cos [ k z ( r A r A ) R D ] sin ϑ ¯ cos ϑ ¯ e i ( k 2 z ) R D r D sin ϑ ¯ + cos ϑ ¯ + e i ( k 2 z ) R D r D cos ϑ ¯ + cos ϑ ¯ e i ( k z ) ( r A r A ) R D + sin ϑ ¯ + sin ϑ ¯ e i ( k z ) ( r A r A ) R D sin ϑ ¯ cos ϑ ¯ e i ( k 2 z ) R D r D sin ϑ ¯ + cos ϑ ¯ + e i ( k 2 z ) R D r D + sin ϑ ¯ + sin ϑ ¯ e i ( k z ) ( r A r A ) R D cos ϑ ¯ + cos ϑ ¯ e i ( k z ) ( r A r A ) R D cos 2 ϑ ¯ + e i ( 2 z ) R D r D + sin 2 ϑ ¯ e i ( k 2 z ) R D r D 2 cos ϑ ¯ + sin ϑ ¯ cos [ k z ( r A r A ) R D ] ] .

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