Generalized parallel-cascaded microring networks for spectral engineering applications

Ashok M. Prabhu; Hai Ling Liew; Vien Van

doi:10.1364/JOSAB.25.001505

Journal of the Optical Society of America B
Vol. 25,
Issue 9,
pp. 1505-1514
(2008)
•https://doi.org/10.1364/JOSAB.25.001505

Generalized parallel-cascaded microring networks for spectral engineering applications

Ashok M. Prabhu, Hai Ling Liew, and Vien Van

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Ashok M. Prabhu, Hai Ling Liew, and Vien Van, "Generalized parallel-cascaded microring networks for spectral engineering applications," J. Opt. Soc. Am. B 25, 1505-1514 (2008)

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Abstract

The transfer characteristics of parallel-cascaded arrays of generalized two-port microring networks with interstage phase shifts are investigated for potential applications in spectral engineering. It is found that these structures can exactly realize optical transfer functions with quadrantal symmetric zeros if the microring networks are even-symmetric. The proposed filter architectures have advantages in that the microring networks have all-positive coupling coefficients, and the array can always be reduced to the canonical form that contains only one interstage π phase-shift element. A method for exactly synthesizing cascaded symmetric microring networks based on the transfer matrix factorization technique is presented, and an example of an eighth-order generalized Chebyshev filter is given to illustrate the application of these microring lattice architectures for realizing very high-order optical transfer functions.

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Array configurations	Stage number	Poles of $G_{k} (s)$	Bus-to-ring energy-coupling coefficients, $μ_{b}$	Ring-to-ring energy-coupling coefficients, $μ_{i, j}$	Inter stage phase shift, $γ_{k}$
{8}	Stage 1	all poles of $Q_{N} (s)$	1.3195	$μ_{12} = μ_{78} = 0.7711$ $μ_{23} = μ_{67} = 0.5577$ $μ_{34} = μ_{56} = 0.3971$ $μ_{18} = 0.0213$ , $μ_{27} = - 0.1321$ $μ_{36} = 0.4700$ , $μ_{45} = - 0.8709$
{6, 2}	Stage 1	$p_{1}, p_{1}^{}$ , $p_{2}, p_{2}^{}$ , $p_{4}, p_{4}^{*}$	1.2340	$μ_{12} = μ_{56} = 0.6634$ , $μ_{23} = μ_{45} = 0.1466$ , $μ_{34} = 0.9748$ , $μ_{25} = 0.3846$ $μ_{16} = 0.1133$
	Stage 2	$p_{3}, p_{3}^{*}$	0.4671	$μ_{12} = 0.9610$	$- 1$
{4, 4}	Stage 1	$p_{1}, p_{1}^{}$ , $p_{3}, p_{3}^{}$	1.0719	$μ_{12} = μ_{34} = 0.3371$ , $μ_{14} = 0.4431$ , $μ_{23} = 0.8396$
	Stage 2	$p_{2}, p_{2}^{}$ , $p_{4}, p_{4}^{}$	0.7695	$μ_{12} = μ_{34} = 0.1126$ , $μ_{14} = 0.7971$ , $μ_{23} = 0.9972$	$- 1$
{4, 2, 2}	Stage 1	$p_{1}, p_{1}^{}$ , $p_{2}, p_{2}^{}$	1.2112	$μ_{12} = μ_{34} = 0.6349$ , $μ_{14} = 0.0788$ , $μ_{23} = 0.3734$
	Stage 2	$p_{4}, p_{4}^{*}$	0.2363	$μ_{12} = 1.0205$	1
	Stage 3	$p_{3}, p_{3}^{*}$	0.4671	$μ_{12} = 0.9610$	$- 1$
{2, 2, 2, 2}	Stage 1	$p_{1}, p_{1}^{*}$	0.9648	$μ_{12} = 0.3217$
	Stage 2	$p_{3}, p_{3}^{*}$	0.4671	$μ_{12} = 0.9610$	1
	Stage 3	$p_{2}, p_{2}^{*}$	0.7323	$μ_{12} = 0.7739$	$- 1$
	Stage 4	$p_{4}, p_{4}^{*}$	0.2363	$μ_{12} = 1.0205$	1

	Bus-to-ring field coupling coefficients, $κ_{b}$	Ring-to-ring field coupling coefficients, $κ_{i, j}$
Stage 1	0.4126	$κ_{12} = κ_{34} = 0.0499$ , $κ_{14} = 0.0656$ , $κ_{23} = 0.1244$
Stage 2	0.2962	$κ_{12} = κ_{34} = 0.0167$ , $κ_{14} = 0.1181$ , $κ_{23} = 0.1477$

Array configurations	Stage number	Poles of $G_{k} (s)$	Bus-to-ring energy-coupling coefficients, $μ_{b}$	Ring-to-ring energy-coupling coefficients, $μ_{i, j}$	Inter stage phase shift, $γ_{k}$
{8}	Stage 1	all poles of $Q_{N} (s)$	1.3195	$μ_{12} = μ_{78} = 0.7711$ $μ_{23} = μ_{67} = 0.5577$ $μ_{34} = μ_{56} = 0.3971$ $μ_{18} = 0.0213$ , $μ_{27} = - 0.1321$ $μ_{36} = 0.4700$ , $μ_{45} = - 0.8709$
{6, 2}	Stage 1	$p_{1}, p_{1}^{}$ , $p_{2}, p_{2}^{}$ , $p_{4}, p_{4}^{*}$	1.2340	$μ_{12} = μ_{56} = 0.6634$ , $μ_{23} = μ_{45} = 0.1466$ , $μ_{34} = 0.9748$ , $μ_{25} = 0.3846$ $μ_{16} = 0.1133$
	Stage 2	$p_{3}, p_{3}^{*}$	0.4671	$μ_{12} = 0.9610$	$- 1$
{4, 4}	Stage 1	$p_{1}, p_{1}^{}$ , $p_{3}, p_{3}^{}$	1.0719	$μ_{12} = μ_{34} = 0.3371$ , $μ_{14} = 0.4431$ , $μ_{23} = 0.8396$
	Stage 2	$p_{2}, p_{2}^{}$ , $p_{4}, p_{4}^{}$	0.7695	$μ_{12} = μ_{34} = 0.1126$ , $μ_{14} = 0.7971$ , $μ_{23} = 0.9972$	$- 1$
{4, 2, 2}	Stage 1	$p_{1}, p_{1}^{}$ , $p_{2}, p_{2}^{}$	1.2112	$μ_{12} = μ_{34} = 0.6349$ , $μ_{14} = 0.0788$ , $μ_{23} = 0.3734$
	Stage 2	$p_{4}, p_{4}^{*}$	0.2363	$μ_{12} = 1.0205$	1
	Stage 3	$p_{3}, p_{3}^{*}$	0.4671	$μ_{12} = 0.9610$	$- 1$
{2, 2, 2, 2}	Stage 1	$p_{1}, p_{1}^{*}$	0.9648	$μ_{12} = 0.3217$
	Stage 2	$p_{3}, p_{3}^{*}$	0.4671	$μ_{12} = 0.9610$	1
	Stage 3	$p_{2}, p_{2}^{*}$	0.7323	$μ_{12} = 0.7739$	$- 1$
	Stage 4	$p_{4}, p_{4}^{*}$	0.2363	$μ_{12} = 1.0205$	1

	Bus-to-ring field coupling coefficients, $κ_{b}$	Ring-to-ring field coupling coefficients, $κ_{i, j}$
Stage 1	0.4126	$κ_{12} = κ_{34} = 0.0499$ , $κ_{14} = 0.0656$ , $κ_{23} = 0.1244$
Stage 2	0.2962	$κ_{12} = κ_{34} = 0.0167$ , $κ_{14} = 0.1181$ , $κ_{23} = 0.1477$

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Equations (39)

Journal of the Optical Society of America B