Coherence properties of some isotropic planar sources

William H. Carter

doi:10.1364/JOSAA.1.000716

Journal of the Optical Society of America A
Vol. 1,
Issue 7,
pp. 716-722
(1984)
•https://doi.org/10.1364/JOSAA.1.000716

Coherence properties of some isotropic planar sources

William H. Carter

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William H. Carter, "Coherence properties of some isotropic planar sources," J. Opt. Soc. Am. A 1, 716-722 (1984)

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Abstract

The complex degree of spectral coherence for both the source distribution function and the field amplitude over the source plane are found for isotropic homogeneous planar sources that do not produce evanescent plane waves. This calculation is extended to also include similar sources that are quasi-homogeneous rather than homogeneous. All the calculated coherence functions are expressed in simple mathematical form and are tabulated for easy reference.

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Equations (67)

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$μ_{ρ}^{(0)} (x_{-})$	g(x₋)	J_ω(s)/J_ω(0)	Equations
$3 \frac{j_{1} (k ∣ x_{-} ∣)}{k ∣ x_{-} ∣}$	$3 λ^{2} j_{0} (k ∣ x_{-} ∣)$	$cos θ (Lambertian)$	(32)
$15 \frac{j_{2} (∣ k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{2}}$	$5 λ^{2} \frac{j_{1} (∣ x_{-} ∣)}{k ∣ x_{-} ∣}$	${cos}^{3} θ$	(33)
$105 \frac{j_{3} (k ∣ x_{-} ∣)}{(k ∣ x_{-} ∣)}$	$21 λ^{2} \frac{j_{2} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{2}}$	${cos}^{5} θ$	(34)
⋮	⋮	⋮
$1 \times 3 \times 5 \times \dots \times (2 n + 1) \frac{j_{n} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{n}}$	$\frac{1 \times 3 \times 5 \times \dots \times (2 n - 1)}{(2 n - 1) / (2 n + 1)} λ^{2} \frac{j_{n - 1} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{n - 1}}$	${cos}^{2 n - 1} θ$	(35)
$2 \frac{J_{1} (k ∣ x_{-} ∣)}{k ∣ x_{-} ∣}$	$λ^{2} J_{0} (k ∣ x_{-} ∣)$	$~ 1 (uniform)$	(36)
$8 \frac{J_{2} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{2}}$	$4 λ^{2} \frac{J_{1} (k ∣ x_{-} ∣)}{k ∣ x_{-} ∣}$	${cos}^{2} θ$	(37)
$48 \frac{J_{3} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{3}}$	$12 λ^{2} \frac{J_{2} (k ∣ x_{-} ∣)}{k ∣ x_{-} ∣^{2}}$	${cos}^{4} θ$	(38)
⋮	⋮	⋮
$2^{n + 1} (n + 1)! \frac{J_{n + 1} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{n + 1}}$	$(n + 1) 2^{n} (n - 1)! λ^{2} \frac{J_{n} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{n}}$	${cos}^{2 n} θ$	(39)

μ⁽⁰⁾(x₋)	J_ω(s)/J_ω(0)	Equations
$j_{0} (k ∣ x_{-} ∣)$	$cos θ (Lambertian)$	(59)
$3 \frac{j_{1} (k ∣ x_{-} ∣)}{k ∣ x_{-} ∣}$	${cos}^{3} θ$	(60)
$15 \frac{j_{2} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{2}}$	${cos}^{5} θ$	(61)
⋮	⋮
$1 \times 3 \times 5 \times \dots \times (2 n - 1) \frac{j_{n - 1} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{n - 1}}$	${cos}^{2 n - 1} θ$	(62)
$J_{0} (k ∣ x_{-} ∣)$	$~ 1 (uniform)$	(63)
$2 \frac{J_{1} (k ∣ x_{-} ∣)}{k ∣ x_{-} ∣}$	${cos}^{2} θ$	(64)
$8 \frac{J_{2} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{2}}$	${cos}^{4} θ$	(65)
⋮	⋮
$2^{n} n! \frac{J_{n} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{n}}$	${cos}^{2 n} θ .$	(66)

f(u)	F(ξ)	Equations
$f (u) = 2 π \int_{0}^{\infty} F (ξ) J_{0} (2 π u ξ) ξ d ξ$	$F (ξ) = 2 π \int_{0}^{\infty} f (u) J_{0} (2 π u ξ) u d u$	(A13)
$2 π 2^{α / 2} Γ (α / 2 + 1) \frac{J_{α / 2 + 1} (2 π u)}{{(2 π u)}^{α / 2 + 1}}$	${[{(1 - ξ^{2})}^{1 / 2}]}^{α} Π (∣ ξ ∣), α > - 2$	(A14)
$2 π 2^{l} l! \frac{J_{l + 1} (2 π u)}{{(2 π u)}^{l + 1}}$	${(1 - ξ^{2})}^{l} Π (∣ ξ ∣), l > - 1$	(A15)
$2 π (1 \times 3 \times 5 \times \dots \times (2 l - 1)) \frac{j_{l} (2 π u)}{{(2 π u)}^{l}}$	${[{(1 - ξ^{2})}^{1 / 2}]}^{2 l - 1} Π (∣ ξ ∣), l \geq 0,$	(A16)

$μ_{ρ}^{(0)} (x_{-})$	g(x₋)	J_ω(s)/J_ω(0)	Equations
$3 \frac{j_{1} (k ∣ x_{-} ∣)}{k ∣ x_{-} ∣}$	$3 λ^{2} j_{0} (k ∣ x_{-} ∣)$	$cos θ (Lambertian)$	(32)
$15 \frac{j_{2} (∣ k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{2}}$	$5 λ^{2} \frac{j_{1} (∣ x_{-} ∣)}{k ∣ x_{-} ∣}$	${cos}^{3} θ$	(33)
$105 \frac{j_{3} (k ∣ x_{-} ∣)}{(k ∣ x_{-} ∣)}$	$21 λ^{2} \frac{j_{2} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{2}}$	${cos}^{5} θ$	(34)
⋮	⋮	⋮
$1 \times 3 \times 5 \times \dots \times (2 n + 1) \frac{j_{n} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{n}}$	$\frac{1 \times 3 \times 5 \times \dots \times (2 n - 1)}{(2 n - 1) / (2 n + 1)} λ^{2} \frac{j_{n - 1} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{n - 1}}$	${cos}^{2 n - 1} θ$	(35)
$2 \frac{J_{1} (k ∣ x_{-} ∣)}{k ∣ x_{-} ∣}$	$λ^{2} J_{0} (k ∣ x_{-} ∣)$	$~ 1 (uniform)$	(36)
$8 \frac{J_{2} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{2}}$	$4 λ^{2} \frac{J_{1} (k ∣ x_{-} ∣)}{k ∣ x_{-} ∣}$	${cos}^{2} θ$	(37)
$48 \frac{J_{3} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{3}}$	$12 λ^{2} \frac{J_{2} (k ∣ x_{-} ∣)}{k ∣ x_{-} ∣^{2}}$	${cos}^{4} θ$	(38)
⋮	⋮	⋮
$2^{n + 1} (n + 1)! \frac{J_{n + 1} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{n + 1}}$	$(n + 1) 2^{n} (n - 1)! λ^{2} \frac{J_{n} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{n}}$	${cos}^{2 n} θ$	(39)

μ⁽⁰⁾(x₋)	J_ω(s)/J_ω(0)	Equations
$j_{0} (k ∣ x_{-} ∣)$	$cos θ (Lambertian)$	(59)
$3 \frac{j_{1} (k ∣ x_{-} ∣)}{k ∣ x_{-} ∣}$	${cos}^{3} θ$	(60)
$15 \frac{j_{2} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{2}}$	${cos}^{5} θ$	(61)
⋮	⋮
$1 \times 3 \times 5 \times \dots \times (2 n - 1) \frac{j_{n - 1} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{n - 1}}$	${cos}^{2 n - 1} θ$	(62)
$J_{0} (k ∣ x_{-} ∣)$	$~ 1 (uniform)$	(63)
$2 \frac{J_{1} (k ∣ x_{-} ∣)}{k ∣ x_{-} ∣}$	${cos}^{2} θ$	(64)
$8 \frac{J_{2} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{2}}$	${cos}^{4} θ$	(65)
⋮	⋮
$2^{n} n! \frac{J_{n} (k ∣ x_{-} ∣)}{{(k ∣ x_{-} ∣)}^{n}}$	${cos}^{2 n} θ .$	(66)

Abstract

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Figures (1)

Tables (3)

Equations (67)

Journal of the Optical Society of America A