Transmission and dispersion of coupled double-ring resonators

Xin Liu; Mei Kong; He Feng

doi:10.1364/JOSAB.29.000068

Journal of the Optical Society of America B
Vol. 29,
Issue 1,
pp. 68-74
(2012)
•https://doi.org/10.1364/JOSAB.29.000068

Transmission and dispersion of coupled double-ring resonators

Xin Liu, Mei Kong, and He Feng

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Xin Liu, Mei Kong, and He Feng, "Transmission and dispersion of coupled double-ring resonators," J. Opt. Soc. Am. B 29, 68-74 (2012)

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Abstract

Based on the symmetry between the transfer characteristics of two coupled double-ring resonators (CDRRs) with symmetrical loss and gain distributions, the transmission and dispersion characteristics of CDRRs with gain in both rings and gain in one ring and loss in the other are analyzed systematically. It is shown that besides coupled-resonator-induced transparency and absorption as in total lossy structures, transmission spectra of inverse transparency and absorption can also exist when gain is introduced. The lasing threshold, classification of transmission response, and group velocity control of different loss and gain structures at resonance and split modes are provided.

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Condition	Transmittance Difference	Transmission Response
$t_{1} < t_{1}^{(m s)}$ , $t_{1}^{(cr)}$	$Δ τ > 0$	CRIT
$t_{1} < t_{1}^{(m s)}$ , $t_{1}^{(cr)}$	$Δ τ < 0$	ICRIT
$t_{1}^{(m s)} > t_{1} > t_{1}^{(cr)}$	$Δ τ < 0$	CRIA
$t_{1}^{(m s)} > t_{1} > t_{1}^{(cr)}$	$Δ τ > 0$	ICRIA

Gain or Loss	Condition	Transmission Response	Group Velocity Control
Gain or Loss	Condition	Transmission Response	$ϕ = 0$	$ϕ = ϕ_{sp}$
$a_{1} < t_{1} < 1$ , $a_{2} > 1$	$a_{2} \| τ_{1, 0} \| < a_{1} a_{2} < t_{2} < 1$	CRIT	Slow	Fast
	$a_{2} \| τ_{1, 0} \| < t_{2} < a_{1} a_{2} < 1$	CRIT	Slow	Slow
		CRIA	Slow	Slow
	$t_{2} < a_{2} \| τ_{1, 0} \| < a_{1} a_{2} < 1$	CRIA	Fast	Slow
	$a_{2} \| τ_{1, 0} \| < t_{2} < 1 < a_{1} a_{2}$	ICRIT	Slow	Slow
	$t_{2} < a_{2} \| τ_{1, 0} \| < 1 < a_{1} a_{2}$	ICRIT	Fast	Slow
	$t_{2} < 1 < a_{2} \| τ_{1, 0} \| < a_{1} a_{2}$	ICRIT	Fast	Slow
$t_{1} < a_{1} < 1$ , $a_{2} > 1$	$a_{1} a_{2} < t_{2} < 1$	CRIT	Slow	Fast
	$t_{2} < a_{1} a_{2} < 1$	${CRIT}^{1}$ ^a	Slow	Slow
		${CRIA}^{2}$	Slow	Slow
	$t_{2} < 1 < a_{1} a_{2}$	${CRIA}^{3}$	Slow	Slow
$t_{1} < a_{1}^{- 1} < 1$ , $a_{2} < 1$	$a_{1} a_{2} < t_{2} < 1$	CRIT	Slow	Fast
	$t_{2} < (a_{1} a_{2})^{- 1} < 1$	${ICRIT}^{1}$	Slow	Slow
		${ICRIT}^{2}$	Slow	Slow
	$t_{2} < 1 < (a_{1} a_{2})^{- 1}$	${CRIT}^{3}$	Slow	Slow

Condition	Transmittance Difference	Transmission Response
$t_{1} < t_{1}^{(m s)}$ , $t_{1}^{(cr)}$	$Δ τ > 0$	CRIT
$t_{1} < t_{1}^{(m s)}$ , $t_{1}^{(cr)}$	$Δ τ < 0$	ICRIT
$t_{1}^{(m s)} > t_{1} > t_{1}^{(cr)}$	$Δ τ < 0$	CRIA
$t_{1}^{(m s)} > t_{1} > t_{1}^{(cr)}$	$Δ τ > 0$	ICRIA

Gain or Loss	Condition	Transmission Response	Group Velocity Control
Gain or Loss	Condition	Transmission Response	$ϕ = 0$	$ϕ = ϕ_{sp}$
$a_{1} < t_{1} < 1$ , $a_{2} > 1$	$a_{2} \| τ_{1, 0} \| < a_{1} a_{2} < t_{2} < 1$	CRIT	Slow	Fast
	$a_{2} \| τ_{1, 0} \| < t_{2} < a_{1} a_{2} < 1$	CRIT	Slow	Slow
		CRIA	Slow	Slow
	$t_{2} < a_{2} \| τ_{1, 0} \| < a_{1} a_{2} < 1$	CRIA	Fast	Slow
	$a_{2} \| τ_{1, 0} \| < t_{2} < 1 < a_{1} a_{2}$	ICRIT	Slow	Slow
	$t_{2} < a_{2} \| τ_{1, 0} \| < 1 < a_{1} a_{2}$	ICRIT	Fast	Slow
	$t_{2} < 1 < a_{2} \| τ_{1, 0} \| < a_{1} a_{2}$	ICRIT	Fast	Slow
$t_{1} < a_{1} < 1$ , $a_{2} > 1$	$a_{1} a_{2} < t_{2} < 1$	CRIT	Slow	Fast
	$t_{2} < a_{1} a_{2} < 1$	${CRIT}^{1}$ ^a	Slow	Slow
		${CRIA}^{2}$	Slow	Slow
	$t_{2} < 1 < a_{1} a_{2}$	${CRIA}^{3}$	Slow	Slow
$t_{1} < a_{1}^{- 1} < 1$ , $a_{2} < 1$	$a_{1} a_{2} < t_{2} < 1$	CRIT	Slow	Fast
	$t_{2} < (a_{1} a_{2})^{- 1} < 1$	${ICRIT}^{1}$	Slow	Slow
		${ICRIT}^{2}$	Slow	Slow
	$t_{2} < 1 < (a_{1} a_{2})^{- 1}$	${CRIT}^{3}$	Slow	Slow

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Journal of the Optical Society of America B