Calculation and optimization of the parameters of a radiating distributed-feedback structure

A simple method is proposed for the calculation – in the second diffraction order – of the reflection, emission, and transmission coefficients of a distributed-feedback waveguide structure. The method is based on a small deviation of the reflection and emission coefficients $R$ and $I$ of a surface wave at $\lambda=\lambda_\infty$ ($\lambda_\infty$ is the wavelength of light corresponding to exact resonance when the grating length is $\ell=\infty$) from the values at $\lambda=\lambda_\ell$ ($\lambda_\ell$ is the wavelength of light corresponding to resonance when the grating has a finite length). The simplicity of the method makes it suitable for the optimization of a distributed-feedback structure in respect of a number of parameters. The method is applicable to thin-film and diffused waveguides of $H$ and $E$ modes.