Accurate first-order leaky-wave analysis of antiresonant reflecting optical waveguides

Meng-Huei Sheng; Hung-Wen Chang

doi:10.1364/AO.44.000751

Applied Optics
Vol. 44,
Issue 5,
pp. 751-764
(2005)
•https://doi.org/10.1364/AO.44.000751

Accurate first-order leaky-wave analysis of antiresonant reflecting optical waveguides

Meng-Huei Sheng and Hung-Wen Chang

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Meng-Huei Sheng and Hung-Wen Chang, "Accurate first-order leaky-wave analysis of antiresonant reflecting optical waveguides," Appl. Opt. 44, 751-764 (2005)

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Abstract

We present a closed-form approximation for estimating both the field distribution and complex propagating constant of the antiresonant reflecting optical waveguide (ARROW) based on a first-order leaky-mode analysis. The formula was obtained from a novel coupled-electric–coupled-magnetic matrix method and provides six significant figures of the real part of the propagation constant β of a SiO₂/TiO₂/SiO₂/Si ARROW with an 8-μm core. The accuracy for the quantity of the imaginary part of β is greater than 98.4% for the TE₀ mode and 99.3% for TM₀. The approximate values for field components are 96.1% accurate. In addition, a slight absorption by the substrate will result in modification of the initial improper leaky-mode behavior, which grows exponentially in the substrate, yielding a proper solution.

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Mode	d2 (μm)	CE–CH	Closed Form	Mode Analysis12
TM₀	4	6.9660039 − 3.7659 × 10⁻⁴i	6.9660336 − 3.7934 × 10⁻⁴i	6.9660242 − 3.8574 × 10⁻⁴i
	6	6.9891762 − 5.2174 × 10⁻⁵i	6.9891805 − 5.2389 × 10⁻⁵i	6.9891798 − 5.2780 × 10⁻⁵i
	8	6.9974100 − 1.2682 × 10⁻⁵i	6.9974111 − 1.2714 × 10⁻⁵i	6.9974110 − 1.2769 × 10⁻⁵i
	10	7.0012532 − 4.2137 × 10⁻⁶i	7.0012535 − 4.2209 × 10⁻⁶i	7.0012535 − 4.2328 × 10⁻⁶i
	12	7.0033520 − 1.7087 × 10⁻⁶i	7.0033522 − 1.7108 × 10⁻⁶i	7.0033521 − 1.7143 × 10⁻⁶i
TE₀	4	6.9680773 − 9.7093 × 10⁻⁶i	6.9681093 − 9.5547 × 10⁻⁶i	6.9680796 − 9.7305 × 10⁻⁶i
	6	6.9898083 − 1.3849 × 10⁻⁶i	6.9898126 − 1.3757 × 10⁻⁶i	6.9898086 − 1.3865 × 10⁻⁶i
	8	6.9976806 − 3.4217 × 10⁻⁷i	6.9976816 − 3.4092 × 10⁻⁷i	6.9976806 − 3.4243 × 10⁻⁷i
	10	7.0013929 − 1.1487 × 10⁻⁷i	7.0013933 − 1.1461 × 10⁻⁷i	7.0013930 − 1.1494 × 10⁻⁷i
	12	7.0034334 − 4.6922 × 10⁻⁸i	7.0034335 − 4.6848 × 10⁻⁸i	7.0034334 − 4.6944 × 10⁻⁸i

Mode	d₂ (μm)	β(n₅ = 3.5 − 0.3i)	q₅ (n₅ = 3.5 − 0.3i)	q₅ (n₅ = 3.5)
TM₀	4	6.9660 − 3.7974 × 10⁻⁴i	15.4292 − 1.5895i	15.4154 + 1.7056 × 10⁻i
	6	6.9892 − 5.2361 × 10⁻⁵i	15.4188 − 1.5908i	15.4049 + 2.3673 × 10⁻⁵i
	8	6.9974 − 1.2702 × 10⁻⁵i	15.4151 − 1.5912i	15.4012 + 5.7586 × 10⁻⁶i
	10	7.0013 − 4.2164 × 10⁻⁶i	15.4133 − 1.5913i	15.3994 + 1.9141 × 10⁻⁶i
	12	7.0034 − 1.7089 × 10⁻⁶i	15.4124 − 1.5914i	15.3985 + 7.7640 × 10⁻⁷i
TE₀	4	6.9681 − 9.4451 × 10⁻⁶i	15.4282 − 1.5898i	15.4145 + 4.3356 × 10⁻⁶i
	6	6.9898 − 1.3600 × 10⁻⁶i	15.4185 − 1.5908i	15.4046 + 6.2102 × 10⁻⁷i
	8	6.9977 − 3.3702 × 10⁻⁷i	15.4150 − 1.5912i	15.4011 + 1.5366 × 10⁻⁷i
	10	7.0014 − 1.1330 × 10⁻⁷i	15.4133 − 1.5914i	15.3994 + 5.1625 × 10⁻⁸i
	12	7.0034 − 4.6312 × 10⁻⁸i	15.4124 − 1.5914i	15.3984 + 2.1095 × 10⁻⁸i

Mode	d2 (μm)	CE–CH	Closed Form	Mode Analysis12
TM₀	4	6.9660039 − 3.7659 × 10⁻⁴i	6.9660336 − 3.7934 × 10⁻⁴i	6.9660242 − 3.8574 × 10⁻⁴i
	6	6.9891762 − 5.2174 × 10⁻⁵i	6.9891805 − 5.2389 × 10⁻⁵i	6.9891798 − 5.2780 × 10⁻⁵i
	8	6.9974100 − 1.2682 × 10⁻⁵i	6.9974111 − 1.2714 × 10⁻⁵i	6.9974110 − 1.2769 × 10⁻⁵i
	10	7.0012532 − 4.2137 × 10⁻⁶i	7.0012535 − 4.2209 × 10⁻⁶i	7.0012535 − 4.2328 × 10⁻⁶i
	12	7.0033520 − 1.7087 × 10⁻⁶i	7.0033522 − 1.7108 × 10⁻⁶i	7.0033521 − 1.7143 × 10⁻⁶i
TE₀	4	6.9680773 − 9.7093 × 10⁻⁶i	6.9681093 − 9.5547 × 10⁻⁶i	6.9680796 − 9.7305 × 10⁻⁶i
	6	6.9898083 − 1.3849 × 10⁻⁶i	6.9898126 − 1.3757 × 10⁻⁶i	6.9898086 − 1.3865 × 10⁻⁶i
	8	6.9976806 − 3.4217 × 10⁻⁷i	6.9976816 − 3.4092 × 10⁻⁷i	6.9976806 − 3.4243 × 10⁻⁷i
	10	7.0013929 − 1.1487 × 10⁻⁷i	7.0013933 − 1.1461 × 10⁻⁷i	7.0013930 − 1.1494 × 10⁻⁷i
	12	7.0034334 − 4.6922 × 10⁻⁸i	7.0034335 − 4.6848 × 10⁻⁸i	7.0034334 − 4.6944 × 10⁻⁸i

Mode	d₂ (μm)	β(n₅ = 3.5 − 0.3i)	q₅ (n₅ = 3.5 − 0.3i)	q₅ (n₅ = 3.5)
TM₀	4	6.9660 − 3.7974 × 10⁻⁴i	15.4292 − 1.5895i	15.4154 + 1.7056 × 10⁻i
	6	6.9892 − 5.2361 × 10⁻⁵i	15.4188 − 1.5908i	15.4049 + 2.3673 × 10⁻⁵i
	8	6.9974 − 1.2702 × 10⁻⁵i	15.4151 − 1.5912i	15.4012 + 5.7586 × 10⁻⁶i
	10	7.0013 − 4.2164 × 10⁻⁶i	15.4133 − 1.5913i	15.3994 + 1.9141 × 10⁻⁶i
	12	7.0034 − 1.7089 × 10⁻⁶i	15.4124 − 1.5914i	15.3985 + 7.7640 × 10⁻⁷i
TE₀	4	6.9681 − 9.4451 × 10⁻⁶i	15.4282 − 1.5898i	15.4145 + 4.3356 × 10⁻⁶i
	6	6.9898 − 1.3600 × 10⁻⁶i	15.4185 − 1.5908i	15.4046 + 6.2102 × 10⁻⁷i
	8	6.9977 − 3.3702 × 10⁻⁷i	15.4150 − 1.5912i	15.4011 + 1.5366 × 10⁻⁷i
	10	7.0014 − 1.1330 × 10⁻⁷i	15.4133 − 1.5914i	15.3994 + 5.1625 × 10⁻⁸i
	12	7.0034 − 4.6312 × 10⁻⁸i	15.4124 − 1.5914i	15.3984 + 2.1095 × 10⁻⁸i

Abstract

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Applied Optics