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In this problem we will consider a few different dielectric slab waveguides. For this problem, all the dielectrics are nonmagnetic. (a) Consider a dielectric slab waveguide. This waveguide is designed to guide mil- limeter waves, and the dielectric core has a relative permittivity of 4 and sits in a cladding region composed of air. Find the thickness of this guide such that only the TE₁ or TM₁ mode can propagate at frequencies up to 300 GHz. (b) Let's consider a different dielectric slab waveguide. This waveguide has a thickness of 2 cm and is composed of a core region with a relative permittivity of 7 embedded in a cladding region with a relative permittivity of 3. Find the cutoff frequency asso- ciated with the TE2 mode. (c) Consider a dielectric slab waveguide with a thickness of 4 µm. The waveguide guides light with a free-space wavelength of 1 µm. The core has a refractive index of 1.6, and the cladding has a refractive index of 1.5. How many modes can propagate without loss in this waveguide?