RCWA-EIS method for interlayer grating coupling

Congshan Wan; Thomas K. Gaylord; Muhannad S. Bakir

doi:10.1364/AO.55.005900

Applied Optics
Vol. 55,
Issue 22,
pp. 5900-5908
(2016)
•https://doi.org/10.1364/AO.55.005900

RCWA-EIS method for interlayer grating coupling

Congshan Wan, Thomas K. Gaylord, and Muhannad S. Bakir

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Congshan Wan, Thomas K. Gaylord, and Muhannad S. Bakir, "RCWA-EIS method for interlayer grating coupling," Appl. Opt. 55, 5900-5908 (2016)

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Abstract

The grating coupling efficiencies for interlayer connection (overlaid chips) were previously calculated using the new rigorous coupled-wave analysis equivalent-index-slab (RCWA-EIS) method. The chip-to-chip coupling efficiencies were determined for rectangular-groove (binary) gratings. In the present work, the search algorithms used in the RCWA-EIS method are optimized giving rise to improved definition of equivalent indices. Further, the versatility of the RCWA-EIS method is demonstrated by extending it to (nonbinary) parallelogramic gratings, sawtooth gratings, and volume gratings. The finite-difference time-domain method is used to verify the results. This demonstrates the flexibility of the RCWA-EIS method in analyzing arbitrary 1D gratings.

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	Initial Values			Optimized Values
Case Number	$θ$ [rad (deg)]	$Λ$ [μm]	$t_{g}$ [μm]	$θ$ [rad (deg)]	$Λ$ [μm]	$t_{g}$ [μm]	$ϕ$ [rad (deg)]	$P C_{c, + 1}$	$D E_{c, + 1}$ at $N = 50$	$β$ $[{μm}^{- 1}]$	$α$ $[{μm}^{- 1}]$
f-pa-1	0.30 (17.19°)	0.55	0.30	0.3005 (17.22°)	0.5373	0.3124	0.9942 (54.96°)	0.8175	0.6354	10.4950	0.0279
f-pa-2	0.25 (14.32°)	0.70	0.30	0.1935 (11.09°)	0.5706	0.2994	0.9665 (55.38°)	0.7330	0.6025	10.2310	0.0302
f-pa-3	0.25 (14.32°)	0.55	0.30	0.2916 (16.71°)	0.5573	0.3323	0.9770 (55.98°)	0.7600	0.6622	10.1087	0.0368
f-pa-4	0.10 (5.73°)	0.52	0.39	0.1956 (11.21°)	0.5589	0.3087	0.9757 (55.90°)	0.7923	0.5828	10.4542	0.0238
b-pa-1	0.10 (5.73°)	0.53	0.34	0.2242 (12.85°)	0.5407	0.3095	0.9911 (56.79°)	0.8038	0.4246	10.7192	0.0139
b-pa-2	0.11 (6.30°)	0.48	0.12	0.1507 (8.63°)	0.4912	0.1575	1.0418 (59.69°)	0.8006	0.3609	12.1829	0.0122

	Initial Values			Optimized Values
Case Number	$θ$ [rad (deg)]	$Λ$ [μm]	$t_{g}$ [μm]	$θ$ [rad (deg)]	$Λ$ [μm]	$t_{g}$ [μm]	$P C_{c, + 1}$	$D E_{c, + 1}$ at $N = 50$	$β$ $[{μm}^{- 1}]$	$α$ $[{μm}^{- 1}]$
f-saw-1	0.10 (5.73°)	0.47	0.37	0.2345 (13.44°)	0.4793	0.3667	0.7931	0.2929	12.1672	0.0096
f-saw-2	0.11 (6.30°)	0.53	0.36	0.1201 (6.88°)	0.5406	0.3712	0.7889	0.3085	11.1369	0.0092
f-saw-3	0.10 (5.73°)	0.52	0.31	0.2057 (11.79°)	0.5434	0.2872	0.8033	0.3370	10.7348	0.0010
f-saw-4	0.13 (7.45°)	0.58	0.25	0.1710 (9.80°)	0.4866	0.3572	0.7922	0.2790	12.2226	0.0089
b-saw-1	0.10 (5.73°)	0.60	0.10	0.2317 (13.28°)	0.5356	0.2079	0.5840	0.2366	10.7997	0.0097
b-saw-2	0.25 (14.32°)	0.50	0.25	0.2376 (13.61°)	0.5088	0.2409	0.4864	0.1940	11.3951	0.0099

	Initial Values			Optimized Values
Case Number	$θ$ [rad (deg)]	$Λ$ [μm]	$ϕ$ [rad (deg)]	$θ$ [rad (deg)]	$Λ$ [μm]	$ϕ$ [rad (deg)]	$P C_{c, + 1}$	$D E_{c, + 1}$ at $N = 50$	$β$ $[{μm}^{- 1}]$	$α$ $[{μm}^{- 1}]$
vol-1	0.30 (17.19°)	0.70	1.20 (68.75°)	0.1041 (5.96°)	0.7019	0.9255 (53.03°)	0.5968	0.3010	6.7298	0.0098
vol-2	0.20 (11.46°)	0.80	1.10 (63.03°)	0.2107 (12.07°)	0.7864	1.1041 (63.26°)	0.6022	0.2810	6.2883	0.0080
vol-3	0.30 (17.19°)	0.70	1.10 (63.03°)	0.2933 (16.80°)	0.6974	0.9630 (55.18°)	0.6274	0.2684	6.2235	0.0079
vol-4	0.20 (11.46°)	0.80	1.00 (57.30°)	0.1847 (10.58°)	0.8318	1.0985 (62.94°)	0.7415	0.2722	5.9822	0.0055

	Binary Case 1		Binary Case 2
	Method 1	Method 2	Method 1	Method 2
Quantity	Iteration 1	Iteration 1	Iteration 1	Iteration 1	Iteration 2	Iteration 3	Iteration 4
$n_{g, eq 1}$	$0.9876 - 4.8328 j$	$- 0.9890 + 4.7668 j$	$3.2720 - 5.1527 j$	$3.1778 - 4.6698 j$	$3.3593 - 4.7174 j$	$3.1524 - 3.9359 j$	$4.1782 - 8.1183 j$
$n_{g, eq 2}$	$6.4085 + 1.5522 j$	$6.3905 + 1.5414 j$	$6.8213 + 2.0172 j$	$6.7104 + 2.0089 j$	$6.9422 + 2.1722 j$	$6.4213 + 2.0822 j$	$7.2202 + 1.6512 j$
$n_{g, eq 3}$	$5.0465 - 3.7908 j$	$5.0146 - 3.8093 j$	$6.6727 - 4.1771 j$	$6.5670 - 4.2384 j$	$4.4964 - 3.3923 j$	$3.1862 - 0.5344 j$	$5.1783 + 1.3997 j$
$n_{g, eq 4}$	$2.8623 + 1.2968 j$	$2.9104 + 1.3127 j$	$3.1786 + 1.6805 j$	$- 3.4811 - 1.8417 j$	$5.2289 - 2.5631 j$	$5.6797 - 3.2270 j$	$4.0158 - 2.0805 j$
$n_{g, eq 5}$	–	–	–	–	$- 3.6444 - 2.7911 j$	$2.7077 - 0.8281 j$	$6.9473 - 2.9823 j$
$n_{g, eq 6}$	–	–	–	–	–	$- 3.6547 - 2.6036 j$	$1.1492 - 0.5535 j$
$n_{g, eq 7}$	–	–	–	–	–	–	$- 3.2478 - 2.2459 j$
func. val.	$1 \times 10^{- 9}$	$1 \times 10^{- 4}$	$1 \times 10^{- 13}$	$2.4 \times 10^{- 3}$	$1.6 \times 10^{- 2}$	$5.8 \times 10^{- 2}$	$4 \times 10^{- 4}$
$α$ ( ${μm}^{- 1}$ )	0.0375	0.0352	0.0445	0.0428	1.6284	0.617	0.0101
$P C_{c, + 1}$	0.4914	0.4913	0.5011	–	–	–	0.5010

	Initial Values			Optimized Values
Case Number	$θ$ [rad (deg)]	$Λ$ [μm]	$t_{g}$ [μm]	$θ$ [rad (deg)]	$Λ$ [μm]	$t_{g}$ [μm]	$ϕ$ [rad (deg)]	$P C_{c, + 1}$	$D E_{c, + 1}$ at $N = 50$	$β$ $[{μm}^{- 1}]$	$α$ $[{μm}^{- 1}]$
f-pa-1	0.30 (17.19°)	0.55	0.30	0.3005 (17.22°)	0.5373	0.3124	0.9942 (54.96°)	0.8175	0.6354	10.4950	0.0279
f-pa-2	0.25 (14.32°)	0.70	0.30	0.1935 (11.09°)	0.5706	0.2994	0.9665 (55.38°)	0.7330	0.6025	10.2310	0.0302
f-pa-3	0.25 (14.32°)	0.55	0.30	0.2916 (16.71°)	0.5573	0.3323	0.9770 (55.98°)	0.7600	0.6622	10.1087	0.0368
f-pa-4	0.10 (5.73°)	0.52	0.39	0.1956 (11.21°)	0.5589	0.3087	0.9757 (55.90°)	0.7923	0.5828	10.4542	0.0238
b-pa-1	0.10 (5.73°)	0.53	0.34	0.2242 (12.85°)	0.5407	0.3095	0.9911 (56.79°)	0.8038	0.4246	10.7192	0.0139
b-pa-2	0.11 (6.30°)	0.48	0.12	0.1507 (8.63°)	0.4912	0.1575	1.0418 (59.69°)	0.8006	0.3609	12.1829	0.0122

Abstract

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Equations (17)

Applied Optics