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Scattering Matrix Math

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Scattering Matrix Method

Multi-layered structure:

Electric Field:

Transmission Matrix:

Inverse Transmission Matrix:

Propagation Matrix:

Exponent:

Complex Refractive Index:

Optical Admittance:

 for TE polarization

 for TM polarization

Admittance of free space

Transfer matrix across first boundary from the incident medium:

Propagation matrix across first material layer:

Transfer matrix across second boundary:

Propagation matrix across second material layer:

Transfer matrix across last boundary into exit medium:

Combined transfer and propagation matrix from incident to exit mediums:

Generalized transfer matrix for multiple layers:

Simplified transfer matrix for multiple layers:

Transfer matrix cell representation for reflection, transmission coefficients:

Reflection Coefficient

Exit medium assumed to be infinite in extent thus the backward propagating wave is zero

Thus:

Another way.

Again setting backward propagating wave to zero:

Yields:

Simplifies to:

Simplifying further yields:

Where Y is the effective index of the layered structure

Transmission Coefficient

Again, exit medium assumed to be infinite in extent thus the backward propagating wave is zero

Thus