Analytical relations for thin-film total internal reflection phase retarders

E. Cojocaru

doi:10.1364/AO.33.007425

Applied Optics
Vol. 33,
Issue 31,
pp. 7425-7430
(1994)
•https://doi.org/10.1364/AO.33.007425

Analytical relations for thin-film total internal reflection phase retarders

E. Cojocaru

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E. Cojocaru, "Analytical relations for thin-film total internal reflection phase retarders," Appl. Opt. 33, 7425-7430 (1994)

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Abstract

Single-layer coatings on total internal reflection boundaries that produce any specific phase retardation are analyzed. One- and two-reflection phase-retarding systems are considered. Simple quadratic equations are obtained for an unknown layer phase thickness at a given angle of incidence and phase retardation. Equations for specific multilayer coatings that are equivalent to single-layer designs are also deduced. Diagrams of solution zones for refractive indices are given comparatively for one- and two-reflection systems.

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Type of Phase Retarder	Phase Shift Δ (k = 0, 1, 2…)	Type of System	P _h
HWR	kπ	(A)	0
		(B)	−y
QWR	(2k + 1)π/2	(A)	∞
		(B)	1/y
Half-QWR	(2k + 1)π/4	(A)	±1
		(B)	(1 ∓ y)/(y ± 1)

Sequence	Double-Layer Coating		Equivalent Single-Layer Design (with Γ₁ = γ)

Refractive index	n ₀/n ₁/n ₂/n _g		N ₀/N ₁/N _g
Effective index (v = p, s)	μ₀ _v /μ₁ _v /μ₂ _v /μ _gv		M₀ _v /M₁ _v /M _gv
	Layer Phase Thicknesses		Refractive Indices			Equivalent Effective Indices

Condition Number	γ₁	γ₂	N ₀	N ₁	N _g	M₀ _v	M₁ _v	M _gv
1	γ	$\frac{π}{2}$	n ₀	n ₁	$\frac{{n_{2}}^{2}}{n_{g}}$	μ₀ _v	μ₁ _v	$\frac{{μ_{2 v}}^{2}}{μ_{g v}}$
2	$\frac{π}{2}$	γ	$\frac{n_{0} {n_{2}}^{2}}{{n_{1}}^{2}}$	n ₂	$\frac{{n_{2}}^{2}}{n_{g}}$	$\frac{μ_{0 v} {μ_{2 v}}^{2}}{{μ_{1 v}}^{2}}$	μ₂ _v	$\frac{{μ_{2 v}}^{2}}{μ_{g v}}$
3	$\frac{π}{2}$	γ	$\frac{n_{0} n_{2}}{n_{1}}$	n ₁	$\frac{n_{1} n_{2}}{n_{g}}$	$\frac{μ_{0 v} μ_{2 v}}{μ_{1 v}}$	μ₁ _v	$\frac{μ_{1 v} μ_{2 v}}{μ_{g v}}$

Sequence	Triple-Layer Coating			Equivalent Single-Layer Design (with Γ₁ = γ)

Refractive index	n ₀/n ₁/n ₂/n _g			N ₀/N ₁/N _g
Effective index (v = p, s)	μ₀ _v /μ₁ _v /μ₂ _v /μ₁ _v /μ _gv			M₀ _v /M₁ _v /M _gv
	Layer Phase Thicknesses			Refractive Indices			Equivalent Effective Indices

Condition Number	γ₁	γ₂	γ₃	N ₀	N ₁	N _g	M₀ _v	M₁ _v	M _gv
1	γ	$\frac{π}{2}$	$\frac{π}{2}$	n ₀	n ₁	$\frac{{n_{2}}^{2} n_{g}}{{n_{1}}^{2}}$	μ₀ _v	μ₁ _v	$\frac{{μ_{2 v}}^{2} μ_{g v}}{{μ_{1 v}}^{2}}$
2	γ	π	$\frac{π}{2}$	n ₀	n ₂	$\frac{{n_{1}}^{2}}{n_{g}}$	μ₀ _v	μ₁ _v	$\frac{{μ_{1 v}}^{2}}{μ_{g v}}$
3	γ	$\frac{π}{2}$	π	n ₀	n ₁	$\frac{{n_{2}}^{2}}{n_{g}}$	μ₀ _v	μ₁ _v	$\frac{{μ_{2 v}}^{2}}{μ_{g v}}$
4	$\frac{π}{2}$	γ	$\frac{π}{2}$	n ₀	$\frac{{n_{1}}^{2}}{n_{2}}$	n _g	μ₀ _v	$\frac{{μ_{1 v}}^{2}}{μ_{2 v}}$	μ _gv
5	γ	π	$\frac{π}{2}$	n ₀	n ₂	$\frac{{n_{1}}^{2}}{n_{g}}$	μ₀ _v	μ₂ _v	$\frac{{μ_{1 v}}^{2}}{μ_{g v}}$
6	$\frac{π}{2}$	γ	π	$\frac{n_{0} {n_{2}}^{2}}{{n_{1}}^{2}}$	n ₂	$\frac{{n_{2}}^{2}}{n_{g}}$	$\frac{μ_{0 v} {μ_{2 v}}^{2}}{{μ_{1 v}}^{2}}$	μ₂ _v	$\frac{{μ_{2 v}}^{2}}{μ_{g v}}$
7	$\frac{π}{2}$	$\frac{π}{2}$	γ	$\frac{n_{0} {n_{2}}^{2}}{{n_{1}}^{2}}$	n ₁	n _g	$\frac{μ_{0 v} {μ_{2 v}}^{2}}{{μ_{1 v}}^{2}}$	μ₁ _v	μ _gv
8	π	$\frac{π}{2}$	γ	$\frac{n_{0} {n_{1}}^{2}}{{n_{2}}^{2}}$	n ₁	$\frac{{n_{1}}^{2}}{n_{g}}$	$\frac{μ_{0 v} {μ_{1 v}}^{2}}{{μ_{2 v}}^{2}}$	μ₁ _v	$\frac{{μ_{1 v}}^{2}}{μ_{g v}}$
9	$\frac{π}{2}$	π	γ	n ₀	n ₁	$\frac{{n_{1}}^{2}}{n_{g}}$	μ₀ _v	μ₁ _v	$\frac{{μ_{1 v}}^{2}}{μ_{g v}}$

Type of Phase Retarder	Phase Shift Δ (k = 0, 1, 2…)	Type of System	P _h
HWR	kπ	(A)	0
		(B)	−y
QWR	(2k + 1)π/2	(A)	∞
		(B)	1/y
Half-QWR	(2k + 1)π/4	(A)	±1
		(B)	(1 ∓ y)/(y ± 1)

Sequence	Double-Layer Coating		Equivalent Single-Layer Design (with Γ₁ = γ)

Refractive index	n ₀/n ₁/n ₂/n _g		N ₀/N ₁/N _g
Effective index (v = p, s)	μ₀ _v /μ₁ _v /μ₂ _v /μ _gv		M₀ _v /M₁ _v /M _gv
	Layer Phase Thicknesses		Refractive Indices			Equivalent Effective Indices

Condition Number	γ₁	γ₂	N ₀	N ₁	N _g	M₀ _v	M₁ _v	M _gv
1	γ	$\frac{π}{2}$	n ₀	n ₁	$\frac{{n_{2}}^{2}}{n_{g}}$	μ₀ _v	μ₁ _v	$\frac{{μ_{2 v}}^{2}}{μ_{g v}}$
2	$\frac{π}{2}$	γ	$\frac{n_{0} {n_{2}}^{2}}{{n_{1}}^{2}}$	n ₂	$\frac{{n_{2}}^{2}}{n_{g}}$	$\frac{μ_{0 v} {μ_{2 v}}^{2}}{{μ_{1 v}}^{2}}$	μ₂ _v	$\frac{{μ_{2 v}}^{2}}{μ_{g v}}$
3	$\frac{π}{2}$	γ	$\frac{n_{0} n_{2}}{n_{1}}$	n ₁	$\frac{n_{1} n_{2}}{n_{g}}$	$\frac{μ_{0 v} μ_{2 v}}{μ_{1 v}}$	μ₁ _v	$\frac{μ_{1 v} μ_{2 v}}{μ_{g v}}$

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Applied Optics