Prism Coupler - Theory of The Prism Coupler

Theory of The Prism Coupler

A prism coupler may be explained in terms of the reciprocity theorem. The reciprocity theorem permits the relative power coupled into the thin film by an incident beam to be computed from the solution to a reciprocal problem. In the reciprocal problem, a waveguide mode in the film (travelling to the left in the first figure) is incident on the prism coupler. Barring significant scattering at the prism interface, the waveguide mode in the reciprocal problem retains its form as a mode and propagates under the prism, losing power as it propagates due to radiation into the prism. The power in the prism emerges as a collimated beam at an angle determined by the propagation constant of the waveguide mode and the refractive index of the prism. Radiation into the prism occurs because the evanescent tail of the waveguide mode touches the bottom of the prism. The waveguide mode tunnels through the tunneling layer.

Efficient coupling of light into the film occurs when the incident beam (arriving from the left shown in the first figure), evaluated at the bottom face of the prism, has the same shape as the radiated beam in the reciprocal problem. When the power in both the incident beam and the reciprocal waveguide mode is normalized, the fractional coupling amplitude is expressed as an integral over the product of the incident wave and the radiated reciprocal field. The integral is a surface integral taken over the bottom face of the prism. From such an integral we deduce three key features:

1. To couple in a significant fraction of the incident power, the incident beam must arrive at the angle that renders it phase matched to the waveguide mode.
2. The transverse behavior of the waveguide mode launched in the film (transverse to the direction of propagation) will be essentially that of the incident beam.
3. If the thickness of the tunneling layer is adjusted appropriately, it is possible, in principle, to couple nearly all the light in the beam into the waveguide film.

Suppressing the transverse part of the representation for the fields, and taking x as direction to the left in Fig. 1, the waveguide mode in the reciprocal problem takes the monotonically decreasing form

where α(x) is the attenuation rate and is the propagation constant of the waveguide mode. The associated transverse field at the bottom of the prism takes the form

with A a normalization constant. The transverse field of the incident beam will have the form

where f(x) is a normalized Gaussian, or other beam form, and β_in is the longitudinal component of the propagation constant of the incident beam. When β_in = β_w, integration of

yields the coupling amplitude. Adjusting α(x) permits the coupling to approach unity, barring significant geometry dependent diffractive effects.