Coupled-Mode Theory

Integrated Optics Prof. Elias N. Glytsis

School of Electrical & Computer Engineering National Technical University of Athens

17/12/2020

Coupled-Mode Theory

Prof. Elias N. Glytsis, School of ECE, NTUA 2

Waveguide Modes (m-th and n-th):

Guided Modes

Radiation Modes

Orthogonality Conditions

Prof. Elias N. Glytsis, School of ECE, NTUA 3

Perturbation along the Waveguide

Transverse Field Expansion for the Perturbed Waveguide

D. L. Lee, Electromagnetic Prnciples of Integrated Optics, John Wiley & Sons, 1986

Prof. Elias N. Glytsis, School of ECE, NTUA 4

Perturbation along the Waveguide

Neglect Radiation Modes (for problems that light remains mostly guided)

D. L. Lee, Electromagnetic Prnciples of Integrated Optics, John Wiley & Sons, 1986

Prof. Elias N. Glytsis, School of ECE, NTUA 5

Lorentz Reciprocity Theorem

D. L. Lee, Electromagnetic Prnciples of Integrated Optics, John Wiley & Sons, 1986

Prof. Elias N. Glytsis, School of ECE, NTUA 6

Lorentz Reciprocity Theorem

Prof. Elias N. Glytsis, School of ECE, NTUA 7

Coupled-Mode Equations

Usually all modes (guided) are normalized to unit power Pm = 1W Usually perturbation is small (ε/ε’ ≈ 1)

Prof. Elias N. Glytsis, School of ECE, NTUA 8

Application of Coupled-Mode Equations to two-coupled waveguides

Two waveguides (A and B) in proximity The presence of each consists a perturbation to its neighbor

Prof. Elias N. Glytsis, School of ECE, NTUA 9

Application of Coupled-Mode Equations to two-coupled waveguides

Waveguide B consists a perturbation for waveguide A

Assume that both waveguides A and B are single (guided)-mode waveguides.

Prof. Elias N. Glytsis, School of ECE, NTUA 10

Application of Coupled-Mode Equations to two-coupled waveguides

Assume that: (approximate orthogonality condition)

Prof. Elias N. Glytsis, School of ECE, NTUA 11

Application of Coupled-Mode Equations to two-coupled waveguides

Coupling Coefficients

However:

Prof. Elias N. Glytsis, School of ECE, NTUA 12

Application of Coupled-Mode Equations to two-coupled waveguides

Prof. Elias N. Glytsis, School of ECE, NTUA 13

Application of Coupled-Mode Equations to two-coupled planar waveguides (directional coupler)

Prof. Elias N. Glytsis, School of ECE, NTUA 14

Application of Coupled-Mode Equations to two-coupled planar waveguides (directional coupler)

Assume solutions of the form:

Prof. Elias N. Glytsis, School of ECE, NTUA 15

Application of Coupled-Mode Equations to two-coupled planar waveguides (directional coupler)

Assume power transfer from B to A:

Waveguide Powers:

Prof. Elias N. Glytsis, School of ECE, NTUA 16

Application of Coupled-Mode Equations to two-coupled planar waveguides (directional coupler)

Prof. Elias N. Glytsis, School of ECE, NTUA 17

Application of Coupled-Mode Equations to two-coupled planar waveguides (directional coupler)

Prof. Elias N. Glytsis, School of ECE, NTUA 18

Supermode Analysis of Directional Coupler

Prof. Elias N. Glytsis, School of ECE, NTUA 19

Supermode Analysis of Directional Coupler

Assume TE modes:

z = 0

Prof. Elias N. Glytsis, School of ECE, NTUA 20

Supermode Analysis of Directional Coupler

z = 0 z = z0

Prof. Elias N. Glytsis, School of ECE, NTUA 21

Supermode Analysis of Directional Coupler

z = L