耦合模理论-coupled mode theory

Lecture 6: Coupled-mode theory

Mode expansion Single-waveguide mode coupling Multiple-waveguide mode coupling Two-mode coupling Codirectional coupling Contradirectional coupling Phase matching
References: This lecture follows the materials from Photonic Devices, Jia-Ming Liu, Chapter 4.
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Coupled-mode theory


Coupled-mode theory deals with the coupling of spatial modes of different spatial distributions or different polarizations, or both. The normal mode fields spatial dependence in a lossless waveguide at a single frequency can be given as
ˆ ( x, y ) exp(i z ) E (r ) A ( z ) E

ˆ ( x, y ) exp(i z ) H (r ) A ( z ) H

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Single-waveguide mode coupling
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Single-waveguide mode coupling
E i 0 H H iE iP

The fields in the perturbed waveguide are governed by these two equations with P 0. The normal mode fields of the unperturbed waveguide also satisfy these two equations with P = 0.

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Power and orthogonality


The normal modes are orthogonal and can be normalized to have the orthonormality relation. Except for evanescent fields, the energy of the fields in a waveguide flows only in the longitudinal direction. The intensity of a waveguide mode is given by



The orthogonality relation or the orthonormality relation indicate that power cannot be transferred between different modes in a linear, lossless waveguide. For anisotropic or lossy waveguides, the orthogonality conditions for modes of such waveguides have other forms.
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Single-waveguide mode coupling

Using the two Maxwell’s equations, we have
* * * ( E1 H 2 E* H ) i ( E P E P 1 1 2 2 1) 2


If we take (E1, H1) to be those of the perturbed mode fields and (E2, H2) to be the normal mode fields, we have P1 = P and P2 = 0. Substituting these into the above equation and integrating both sides of the resultant equation over the cross section of the waveguide, we have




where the plus sign is for forward-propagating modes while the minus sign is for backward-propagating modes. The electric and magnetic field patterns of a particular mode are represented by the normalized mode field distributions E(x, y) and H(x, y). Here is the Kronecker delta function for discrete modes. For a nonplanar waveguide, = mn and = m’n’, and mm’ nn’ For a planar waveguide, = m, = m’, and mm’
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Power and orthogonality

For TE and TM modes, the power obtained by integrating I(x, y) can be given as 2 2 PTE E dxdy 0
PTM 1 2 H dxdy ( x, y ) 2
ˆ ( x, y ) exp i ( z ) E (r ) A E

ˆ ( x, y ) exp i ( z ) H (r ) A H

where E and H are normalized mode fields satisfying the orthonormality relation, and the summation sums over all discrete indices of the guided modes (and integrates over all continuous indices of the radiation and evanescent modes). In an ideal waveguide where these modes are defined, the normal modes do not couple. Then, the expansion coefficients A are constants that are independent of x, y and z.
E (r ) E ( x, y ) exp i ( z ) H (r ) H ( x, y ) exp i ( z )

The two Maxwell’s equations become
E i 0 H
H iE
The normal modes with fields given by E(r) and H(r) are characteristic solutions of Maxwell’s equations.
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Mode expansion



When there is a spatially dependent perturbation to a waveguide, the modes defined by the unperturbed ideal waveguide are no longer exact normal modes of the perturbed waveguide. They can now be coupled by the perturbation as they propagate along the waveguide. If the fields are still expanded in terms of the normal modes of the unperturbed waveguide, the expansion coefficients are no longer constants of propagation but vary with z as the fields propagate down the waveguide:
ˆ ( E H * E* H ) z ˆ I ( Sion of x and y. The power, P, of the mode is obtained by integrating I(x, y) over the entire transverse cross section of the waveguide.

Consider the coupling between normal modes in a single waveguide that is subject to some perturbation. The spatially dependent perturbation to the waveguide can be represented by a perturbing polarization P(r) at frequency . The following Maxwell’s equations are used
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Power and orthogonality

For TE modes, the orthonormality relation can be given as