Volume holographic grating couplers: rigorous analysis by use of the finite-difference frequency-domain method

Shun-Der Wu; Elias N. Glytsis

doi:10.1364/AO.43.001009

Applied Optics
Vol. 43,
Issue 5,
pp. 1009-1023
(2004)
•https://doi.org/10.1364/AO.43.001009

Volume holographic grating couplers: rigorous analysis by use of the finite-difference frequency-domain method

Shun-Der Wu and Elias N. Glytsis

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Shun-Der Wu and Elias N. Glytsis, "Volume holographic grating couplers: rigorous analysis by use of the finite-difference frequency-domain method," Appl. Opt. 43, 1009-1023 (2004)

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Abstract

Two configurations of volume holographic grating couplers are rigorously analyzed by means of the finite-difference frequency-domain method (FDFD) for both TE and TM polarizations and for 0- and 45-deg output coupling. The two configurations depend on the position of the grating coupler, which can be placed either in the film or in the cover waveguide region. The FDFD results are compared with those obtained by the rigorous coupled-wave analysis in conjunction with the leaky-mode approach (RCWA–LM). Because the FDFD method is a rigorous solution of the Maxwell equations, it simulates the VHGC configuration and takes into account the waveguide-coupler discontinuity effects as well as the multimode excitation and interference effects, all of which are neglected by the traditional RCWA–LM.

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Coupling Angle θ_c (deg)	TE Polarization		TM Polarization
Coupling Angle θ_c (deg)	Λ (nm)	ϕ (deg)	Λ (nm)	ϕ (deg)
VHGCs in the Waveguide Film Region
0	473.7	-45.28	473.9	-45.30
45	651.5	-59.52	651.7	-59.56
VHGCs in the Waveguide Cover Region
0	470.6	-44.91	470.9	-44.94
45	647.3	-58.91	647.7	-58.96

Outcoupling Angle, θ_c = 0°
Numerical Method	TE Polarization (Λ = 473.7 nm, ϕ = -45.28°)
Numerical Method	L _g (μm)	CE_u (%)	CE_d (%)	CE_t (%)	CE_r (%)	BR (%)	α_l (1/μm)
FDFD	100	46.72	2.33	50.92	0.03	95.25	0.34 × 10^-2
	250	76.38	3.96	19.64	0.02	95.07	0.33 × 10^-2
RCWA–LM	100	45.60	0.02	54.38	0.00	99.96	0.31 × 10^-2
	250	78.16	0.03	21.81	0.00	99.96	0.31 × 10^-2
	TM Polarization (Λ = 473.9 nm, ϕ =-45.30°)
FDFD	100	0.13	0.03	99.84	0.00	81.25	0.80 × 10^-5
	250	0.29	0.05	99.66	0.00	85.29	0.68 × 10^-5
RCWA–LM	100	0.09	0.00	99.91	0.00	97.67	0.44 × 10^-5
	250	0.22	0.00	99.78	0.00	97.67	0.44 × 10^-5
Outcoupling Angle, θ_c = 45°
	TE Polarization (Λ = 651.5 nm, ϕ =-59.52°)
FDFD	100	48.68	4.07	47.25	0.00	92.28	0.38 × 10^-2
	250	77.20	6.48	16.32	0.00	92.26	0.36 × 10^-2
RCWA–LM	100	45.74	3.84	50.42	0.00	92.26	0.34 × 10^-2
	250	75.61	6.34	18.05	0.00	92.26	0.34 × 10^-2
	TM Polarization (Λ = 651.7 nm, ϕ =-59.56°)
FDFD	100	13.25	0.32	86.43	0.00	97.64	0.73 × 10^-3
	250	29.79	0.68	69.53	0.00	97.77	0.73 × 10^-3
RCWA–LM	100	13.55	0.26	86.19	0.00	98.14	0.74 × 10^-3
	250	30.45	0.58	68.97	0.00	98.14	0.74 × 10^-3

Outcoupling Angle, θ_c = 0°
Numerical Method	TE Polarization (Λ = 470.6 nm, ϕ = -44.91°)
Numerical Method	L _g (μm)	CE_u (%)	CE_d (%)	CE_t (%)	CE_r (%)	BR (%)	α_l (1/μm)
FDFD	100	41.37	1.19	57.37	0.70	97.20	0.28 × 10^-2
	250	72.88	3.03	23.71	0.38	96.01	0.29 × 10^-2
RCWA–LM	100	42.34	0.09	57.57	0.00	99.78	0.28 × 10^-2
	250	74.68	0.17	25.15	0.00	99.78	0.28 × 10^-2
	TM Polarization (Λ = 470.9 nm, ϕ =-44.94°)
FDFD	100	0.35	0.02	99.62	0.01	94.59	0.19 × 10^-4
	250	0.86	0.05	99.08	0.01	94.51	0.19 × 10^-4
RCWA–LM	100	0.33	0.02	99.65	0.00	95.45	0.18 × 10^-4
	250	0.84	0.04	99.12	0.00	95.45	0.18 × 10^-4
Outcoupling Angle, θ_c = 45°
	TE Polarization (Λ = 647.3 nm, ϕ =-58.91°)
FDFD	100	42.35	4.08	53.57	0.00	91.21	0.31 × 10^-2
	250	73.42	7.29	19.29	0.00	90.99	0.33 × 10^-2
RCWA–LM	100	42.94	4.11	52.95	0.00	91.27	0.32 × 10^-2
	250	72.65	6.95	20.40	0.00	91.27	0.32 × 10^-2
	TM Polarization (Λ = 647.7 nm, ϕ =-58.96°)
FDFD	100	14.53	0.30	85.17	0.00	97.98	0.80 × 10^-3
	250	33.69	0.70	65.61	0.00	97.96	0.84 × 10^-3
RCWA–LM	100	17.27	0.22	82.51	0.00	98.72	0.96 × 10^-3
	250	37.68	0.49	61.83	0.00	98.72	0.96 × 10^-3

Coupling Angle θ_c (deg)	TE Polarization		TM Polarization
Coupling Angle θ_c (deg)	Λ (nm)	ϕ (deg)	Λ (nm)	ϕ (deg)
VHGCs in the Waveguide Film Region
0	473.7	-45.28	473.9	-45.30
45	651.5	-59.52	651.7	-59.56
VHGCs in the Waveguide Cover Region
0	470.6	-44.91	470.9	-44.94
45	647.3	-58.91	647.7	-58.96

Outcoupling Angle, θ_c = 0°
Numerical Method	TE Polarization (Λ = 473.7 nm, ϕ = -45.28°)
Numerical Method	L _g (μm)	CE_u (%)	CE_d (%)	CE_t (%)	CE_r (%)	BR (%)	α_l (1/μm)
FDFD	100	46.72	2.33	50.92	0.03	95.25	0.34 × 10^-2
	250	76.38	3.96	19.64	0.02	95.07	0.33 × 10^-2
RCWA–LM	100	45.60	0.02	54.38	0.00	99.96	0.31 × 10^-2
	250	78.16	0.03	21.81	0.00	99.96	0.31 × 10^-2
	TM Polarization (Λ = 473.9 nm, ϕ =-45.30°)
FDFD	100	0.13	0.03	99.84	0.00	81.25	0.80 × 10^-5
	250	0.29	0.05	99.66	0.00	85.29	0.68 × 10^-5
RCWA–LM	100	0.09	0.00	99.91	0.00	97.67	0.44 × 10^-5
	250	0.22	0.00	99.78	0.00	97.67	0.44 × 10^-5
Outcoupling Angle, θ_c = 45°
	TE Polarization (Λ = 651.5 nm, ϕ =-59.52°)
FDFD	100	48.68	4.07	47.25	0.00	92.28	0.38 × 10^-2
	250	77.20	6.48	16.32	0.00	92.26	0.36 × 10^-2
RCWA–LM	100	45.74	3.84	50.42	0.00	92.26	0.34 × 10^-2
	250	75.61	6.34	18.05	0.00	92.26	0.34 × 10^-2
	TM Polarization (Λ = 651.7 nm, ϕ =-59.56°)
FDFD	100	13.25	0.32	86.43	0.00	97.64	0.73 × 10^-3
	250	29.79	0.68	69.53	0.00	97.77	0.73 × 10^-3
RCWA–LM	100	13.55	0.26	86.19	0.00	98.14	0.74 × 10^-3
	250	30.45	0.58	68.97	0.00	98.14	0.74 × 10^-3

Abstract

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Tables (3)

Equations (9)

Applied Optics