Analysis of Holographic Thin Film Grating Coupler

K. Ogawa; W. S. C. Chang

doi:10.1364/AO.12.002167

Applied Optics
Vol. 12,
Issue 9,
pp. 2167-2171
(1973)
•https://doi.org/10.1364/AO.12.002167

Analysis of Holographic Thin Film Grating Coupler

K. Ogawa and W. S. C. Chang

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K. Ogawa and W. S. C. Chang, "Analysis of Holographic Thin Film Grating Coupler," Appl. Opt. 12, 2167-2171 (1973)

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Abstract

The holographic grating coupler was used experimentally by H. Kogelnik and T. P. Sosnowski to obtain high coupling efficiency in thin film waveguides. This paper presents a perturbation analysis of that grating coupler. The calculated results agree very well with the experimental data. The analysis indicates that the maximum efficiency will depend on the effective length of the evanescent tail of the guided wave mode in the gelatin. The longer the evanescent tail is, the higher will be the excitation efficiency; the maximum possible efficiency is, of course, 81%. The maximum efficiency does not depend significantly on the refractive index of the grating, or the slant angle of the grooves, or the thickness of the gelatin (provided that the thickness is larger than the length of the evanescent tail).

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(i) f_a(x′)

	f_a(x′)
x′ ≥ t	exp[jσ(x′ − t)] + R(θ) exp[−jσ(x′ − t)]
t > x′ > 0	$\frac{2 σ h (cos h x^{'} + j \frac{ρ}{h} sin h x^{'})}{(σ + ρ) h cos h t + j (h^{2} + ρ σ) sin h t}$
0 ≥ x′	$\frac{2 σ h exp (j ρ x^{'})}{(σ + ρ) h cos h t + j (h^{2} + ρ σ) sin h t}$
	$R (θ) = \frac{(σ - ρ) h cos h t + j (σ ρ - h^{2}) sin h t}{(σ + ρ) h cos h t + j (h^{2} + ρ σ) sin h t}$
	$σ = n_{g} k cos θ h = k {({n_{f}}^{2} - {n_{g}}^{2} {sin}^{2} θ)}^{\frac{1}{2}}$
	$ρ = k {({n_{s}}^{2} - {n_{g}}^{2} {sin}^{2} θ)}^{\frac{1}{2}}$

(ii)f_s(x′)

	f_s(x′)
x′ ≥ t	$\frac{2 ρ h exp [- j σ (x^{'} - t)]}{h (σ + ρ) cos h t + j (h^{2} + σ ρ) sin h t}$
t > x′ > 0	$\frac{2 ρ h {cos h (t - x^{'}) + j \frac{σ}{h} sin h (t - x^{'})}}{h (σ + ρ) cos h t + j (h^{2} + σ ρ) sin h t}$
0 ≥ x′	exp(−jρx′) + R′(θ) exp(jρx′)
	$R^{'} (θ) = \frac{(ρ - σ) h cos h t + j (ρ σ - h^{2}) sin h t}{(σ + ρ) h cos h t + j (h^{2} + ρ σ) sin h t}$
	$ρ = n_{s} k cos θ h = k {({n_{f}}^{2} - {n_{s}}^{2} {sin}^{2} θ)}^{\frac{1}{2}}$
	$σ = k {({n_{g}}^{2} - {n_{s}}^{2} {sin}^{2} θ)}^{\frac{1}{2}}$

(iii)f_ss(x′)

	f_ss(x′)
x′ ≥ t	$\frac{2 ρ h exp [- P (x^{'} - t)]}{(h ρ cos h t + P ρ sin h t) - j (h P cos h t - h^{2} sin h t)}$
t > x′ > 0	$\frac{2 ρ h [cos h (t - x^{'}) + (P / h) sin h (t - x^{'})]}{(h ρ cos h t + ρ P sin h t) - j (h P cos h t - h^{2} sin h t)}$
0 ≥ x′	exp(−jρx′) + R′(θ) exp(jρx′)
	$R^{'} (θ) = \frac{(ρ h cos h t + ρ P sin h t) + j (h P cos h t - h^{2} sin h t)}{(h ρ cos h t + ρ P sin h t) - j (h P cos h t - h^{2} sin h t)}$
	$ρ = n_{s} k cos θ h = k {({n_{f}}^{2} - {n_{s}}^{2} {sin}^{2} θ)}^{\frac{1}{2}}$
	$P = k {({n_{s}}^{2} {sin}^{2} θ - {n_{g}}^{2})}^{\frac{1}{2}}$

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Applied Optics