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What is the physical reason behind why a wave in the cladding of a waveguide cannot phase match to a guided wave?

I read that the most common solutions to coupling waves to a guided wave is to use prism coupling or a grating. But I have trouble understanding why we can't just couple directly to a guided wave with an incident wave that has the same k.

Here is a diagram of using prism coupling to couple to a guided wave (blue). (Source: Photonic Microsystems: Micro and Nanotechnology Applied to Optical Devices by O. Solgaard)

From below diagram, the text says that a wave in the cladding ($n_c$ layer) cannot couple into the guided wave since it cannot be phase-matched. So a prism or grating can be used instead. I don't understand this point.

enter image description here

photonica
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1 Answers1

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Note: The goal is not to match the phase (wavenumber) of the exciting wave to the phase (wavenumber) of the cladding but rather match it to the phase (propagating factor) of the propagating mode that depends on the cladding/core/substrate.

If you denote by $\kappa_0$ the free-space wavenumber of the plane wave incident upon the prism, inside the prism it will be $n_p \kappa_0$ and along the coupling surface $\kappa_z=n_p \kappa_0 \sin \theta$ in the $\hat z$ direction along the fiber, that is the plane wave propagating in the prism and along the axial direction of the fiber it is periodic with $\lambda_z=\frac{2\pi}{\kappa_z}$.

To couple this wave into the fiber and efficiently have it excite a mode whose propagation factor is $\beta$, one should make sure that $\beta = \kappa_z$, otherwise the fiber mode when it is excited by one of the peaks of the wave in the prism will not have its next peak excited by the next peak of the same plane wave in phase. This is phase matching to the mode.

hyportnex
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