Young's experiment with electromagnetic spatial coherence wavelets

Roman Castaneda; Juan Carrasquilla; Jorge Garcia-Sucerquia

doi:10.1364/JOSAA.23.002519

Journal of the Optical Society of America A
Vol. 23,
Issue 10,
pp. 2519-2529
(2006)
•https://doi.org/10.1364/JOSAA.23.002519

Young’s experiment with electromagnetic spatial coherence wavelets

Roman Castaneda, Juan Carrasquilla, and Jorge Garcia-Sucerquia

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Roman Castaneda, Juan Carrasquilla, and Jorge Garcia-Sucerquia, "Young's experiment with electromagnetic spatial coherence wavelets," J. Opt. Soc. Am. A 23, 2519-2529 (2006)

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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.

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Right-Side Members of Eq. (9)	$Δ ϑ_{\pm} \to 0$	$Δ ϑ_{+} = Δ ϑ_{-}$	Uniform Fluctuations
$⟨ \cos^{2} ({\bar{ϑ}}_{\pm} + Δ ϑ_{\pm}) ⟩$	$\approx \cos^{2} {\bar{ϑ}}_{\pm}$	$1 ∕ 2$	$1 ∕ 2$
$⟨ \sin^{2} ({\bar{ϑ}}_{\pm} + Δ ϑ_{\pm}) ⟩$	$\approx \sin^{2} {\bar{ϑ}}_{\pm}$	$1 ∕ 2$	$1 ∕ 2$
$⟨ \cos ({\bar{ϑ}}_{+} + Δ ϑ_{+}) \cos ({\bar{ϑ}}_{-} + Δ ϑ_{-}) ⟩$	$\approx \cos {\bar{ϑ}}_{+} \cos {\bar{ϑ}}_{-}$	$\frac{1}{2} \cos ({\bar{ϑ}}_{+} - {\bar{ϑ}}_{-})$	0
$⟨ \sin ({\bar{ϑ}}_{+} + Δ ϑ_{+}) \sin ({\bar{ϑ}}_{+} + Δ ϑ_{-}) ⟩$	$\approx \sin {\bar{ϑ}}_{+} \sin {\bar{ϑ}}_{-}$	$\frac{1}{2} \cos ({\bar{ϑ}}_{+} - {\bar{ϑ}}_{-})$	0
$⟨ \cos ({\bar{ϑ}}_{\pm} + ϑ_{\pm}) \sin ({\bar{ϑ}}_{\mp} + Δ ϑ_{\mp}) ⟩$	$\approx \cos {\bar{ϑ}}_{\pm} \sin {\bar{ϑ}}_{\mp}$	$\pm \frac{1}{2} \sin ({\bar{ϑ}}_{\pm} - {\bar{ϑ}}_{\mp})$	0

Polarization Case		$η (+, -, v)$	$η (+, +, v)$	$η -, -, v$	Visibility
1 Uniform over $[- π, π]$ Statistically independent $ϑ_{\pm}$		$[\begin{matrix} 0.0125 & 0.0011 \\ 0.0011 & 0.0181 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0113 \\ 0.0113 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & - 0.0033 \\ - 0.0033 & 1.0000 \end{matrix}]$	0.0152
2 Uniform over $[- π, π]$ $ϑ_{+} = ϑ_{-}$		$[\begin{matrix} 1.0000 & 0.0068 \\ 0.0068 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0068 \\ 0.0068 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0068 \\ 0.0068 & 1.0000 \end{matrix}]$	1.0000
3 Gaussian ${\bar{ϑ}}_{+} = {\bar{ϑ}}_{-} = \frac{π}{4}$ Statistically independent $ϑ_{\pm}$	$σ = 5 \times 10^{- 4} rad$	$[\begin{matrix} 0.9511 & 0.9506 \\ 0.9506 & 0.9511 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.9040 \\ 0.9040 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.9041 \\ 0.9041 & 1.0000 \end{matrix}]$	0.9510
	$σ = 1.0 rad$	$[\begin{matrix} 0.3728 & 0.3695 \\ 0.3695 & 0.3644 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.1337 \\ 0.1337 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.1297 \\ 0.1297 & 1.0000 \end{matrix}]$	0.3686
4 Gaussian ${\bar{ϑ}}_{+} = {\bar{ϑ}}_{-} = \frac{π}{4}$ $Δ ϑ_{+} = Δ ϑ_{-}$	$σ = 5 \times 10^{- 4} rad$	$[\begin{matrix} 1.0000 & 0.9990 \\ 0.9990 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.9990 \\ 0.9990 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.9990 \\ 0.9990 & 1.0000 \end{matrix}]$	1.0000
	$σ = 1.0 rad$	$[\begin{matrix} 1.0000 & 0.1342 \\ 0.1342 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.1342 \\ 0.1342 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.1342 \\ 0.1342 & 1.0000 \end{matrix}]$	1.0000
5 Gaussian ${\bar{ϑ}}_{+} = \frac{π}{4}$ ${\bar{ϑ}}_{-} = \frac{3 π}{4}$ Statistically independent $ϑ_{\pm}$	$σ = 5 \times 10^{- 4} rad$	$[\begin{matrix} 0.9995 & 0.9995 \\ 0.9995 & 0.9995 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.9990 \\ 0.9990 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & - 0.9990 \\ - 0.9990 & 1.0000 \end{matrix}]$	$2.84 \times 10^{- 4}$
	$σ = 1.0 rad$	$[\begin{matrix} - 0.3332 & 0.3331 \\ 0.3331 & 0.3305 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.1086 \\ 0.1086 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & - 0.1147 \\ - 0.1147 & 1.0000 \end{matrix}]$	0.0054
6 Gaussian ${\bar{ϑ}}_{+} = \frac{π}{4}$ ${\bar{ϑ}}_{-} = \frac{3 π}{4}$ $Δ ϑ_{+} = Δ ϑ_{-}$	$σ = 5 \times 10^{- 4} rad$	$[\begin{matrix} - 0.9990 & 1.0000 \\ 1.0000 & 0.9990 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.9990 \\ 0.9990 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & - 0.9990 \\ - 0.9990 & 1.0000 \end{matrix}]$	$3.88 \times 10^{- 16}$
	$σ = 1.0 rad$	$[\begin{matrix} - 0.1133 & 1.0000 \\ 1.0000 & 0.1133 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.1133 \\ 0.1133 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & - 0.1133 \\ - 0.1133 & 1.0000 \end{matrix}]$	$2.22 \times 10^{- 16}$

Polarization State	$η (+, -, v)$ before Rotation	Visibility before Rotation	Visibility after Rotation
$Δ ϑ_{+} = Δ ϑ_{-}$ with $σ = 5 \times 10^{- 4} rad$	$[\begin{matrix} - 0.9990 & 1.0000 \\ 1.0000 & 0.9990 \end{matrix}]$	$3.88 \times 10^{- 16}$	1.0000
Statistically independent fluctuations with $σ = 5 \times 10^{- 4} rad$	$[\begin{matrix} - 0.9995 & 0.9995 \\ 0.9995 & 0.9995 \end{matrix}]$	$4.01 \times 10^{- 4}$	0.9994
$Δ ϑ_{+} = Δ ϑ_{-}$ with $σ = 1 rad$	$[\begin{matrix} - 0.1386 & 1.0000 \\ 1.0000 & 0.1386 \end{matrix}]$	$3.88 \times 10^{- 16}$	1.0000
Statistically independent fluctuations with $σ = 1 rad$	$[\begin{matrix} - 0.3548 & 0.3739 \\ 0.3739 & 0.3550 \end{matrix}]$	$4.87 \times 10^{- 4}$	0.3738

Right-Side Members of Eq. (9)	$Δ ϑ_{\pm} \to 0$	$Δ ϑ_{+} = Δ ϑ_{-}$	Uniform Fluctuations
$⟨ \cos^{2} ({\bar{ϑ}}_{\pm} + Δ ϑ_{\pm}) ⟩$	$\approx \cos^{2} {\bar{ϑ}}_{\pm}$	$1 ∕ 2$	$1 ∕ 2$
$⟨ \sin^{2} ({\bar{ϑ}}_{\pm} + Δ ϑ_{\pm}) ⟩$	$\approx \sin^{2} {\bar{ϑ}}_{\pm}$	$1 ∕ 2$	$1 ∕ 2$
$⟨ \cos ({\bar{ϑ}}_{+} + Δ ϑ_{+}) \cos ({\bar{ϑ}}_{-} + Δ ϑ_{-}) ⟩$	$\approx \cos {\bar{ϑ}}_{+} \cos {\bar{ϑ}}_{-}$	$\frac{1}{2} \cos ({\bar{ϑ}}_{+} - {\bar{ϑ}}_{-})$	0
$⟨ \sin ({\bar{ϑ}}_{+} + Δ ϑ_{+}) \sin ({\bar{ϑ}}_{+} + Δ ϑ_{-}) ⟩$	$\approx \sin {\bar{ϑ}}_{+} \sin {\bar{ϑ}}_{-}$	$\frac{1}{2} \cos ({\bar{ϑ}}_{+} - {\bar{ϑ}}_{-})$	0
$⟨ \cos ({\bar{ϑ}}_{\pm} + ϑ_{\pm}) \sin ({\bar{ϑ}}_{\mp} + Δ ϑ_{\mp}) ⟩$	$\approx \cos {\bar{ϑ}}_{\pm} \sin {\bar{ϑ}}_{\mp}$	$\pm \frac{1}{2} \sin ({\bar{ϑ}}_{\pm} - {\bar{ϑ}}_{\mp})$	0

Polarization Case		$η (+, -, v)$	$η (+, +, v)$	$η -, -, v$	Visibility
1 Uniform over $[- π, π]$ Statistically independent $ϑ_{\pm}$		$[\begin{matrix} 0.0125 & 0.0011 \\ 0.0011 & 0.0181 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0113 \\ 0.0113 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & - 0.0033 \\ - 0.0033 & 1.0000 \end{matrix}]$	0.0152
2 Uniform over $[- π, π]$ $ϑ_{+} = ϑ_{-}$		$[\begin{matrix} 1.0000 & 0.0068 \\ 0.0068 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0068 \\ 0.0068 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0068 \\ 0.0068 & 1.0000 \end{matrix}]$	1.0000
3 Gaussian ${\bar{ϑ}}_{+} = {\bar{ϑ}}_{-} = \frac{π}{4}$ Statistically independent $ϑ_{\pm}$	$σ = 5 \times 10^{- 4} rad$	$[\begin{matrix} 0.9511 & 0.9506 \\ 0.9506 & 0.9511 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.9040 \\ 0.9040 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.9041 \\ 0.9041 & 1.0000 \end{matrix}]$	0.9510
	$σ = 1.0 rad$	$[\begin{matrix} 0.3728 & 0.3695 \\ 0.3695 & 0.3644 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.1337 \\ 0.1337 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.1297 \\ 0.1297 & 1.0000 \end{matrix}]$	0.3686
4 Gaussian ${\bar{ϑ}}_{+} = {\bar{ϑ}}_{-} = \frac{π}{4}$ $Δ ϑ_{+} = Δ ϑ_{-}$	$σ = 5 \times 10^{- 4} rad$	$[\begin{matrix} 1.0000 & 0.9990 \\ 0.9990 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.9990 \\ 0.9990 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.9990 \\ 0.9990 & 1.0000 \end{matrix}]$	1.0000
	$σ = 1.0 rad$	$[\begin{matrix} 1.0000 & 0.1342 \\ 0.1342 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.1342 \\ 0.1342 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.1342 \\ 0.1342 & 1.0000 \end{matrix}]$	1.0000
5 Gaussian ${\bar{ϑ}}_{+} = \frac{π}{4}$ ${\bar{ϑ}}_{-} = \frac{3 π}{4}$ Statistically independent $ϑ_{\pm}$	$σ = 5 \times 10^{- 4} rad$	$[\begin{matrix} 0.9995 & 0.9995 \\ 0.9995 & 0.9995 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.9990 \\ 0.9990 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & - 0.9990 \\ - 0.9990 & 1.0000 \end{matrix}]$	$2.84 \times 10^{- 4}$
	$σ = 1.0 rad$	$[\begin{matrix} - 0.3332 & 0.3331 \\ 0.3331 & 0.3305 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.1086 \\ 0.1086 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & - 0.1147 \\ - 0.1147 & 1.0000 \end{matrix}]$	0.0054
6 Gaussian ${\bar{ϑ}}_{+} = \frac{π}{4}$ ${\bar{ϑ}}_{-} = \frac{3 π}{4}$ $Δ ϑ_{+} = Δ ϑ_{-}$	$σ = 5 \times 10^{- 4} rad$	$[\begin{matrix} - 0.9990 & 1.0000 \\ 1.0000 & 0.9990 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.9990 \\ 0.9990 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & - 0.9990 \\ - 0.9990 & 1.0000 \end{matrix}]$	$3.88 \times 10^{- 16}$
	$σ = 1.0 rad$	$[\begin{matrix} - 0.1133 & 1.0000 \\ 1.0000 & 0.1133 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.1133 \\ 0.1133 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & - 0.1133 \\ - 0.1133 & 1.0000 \end{matrix}]$	$2.22 \times 10^{- 16}$

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Journal of the Optical Society of America A