Existence and Uniqueness of the Solutions in the SN, DN and CN Waveguide Theories

Abstract

z = snw, dnw and cnw mapping the lines u = const and v = const. onto the z plane respectively:z = snw,−ρ ² A ² cos 2ϕ = −2c ² d ²/k ² s ² (&rho ⁴ + 1/k ²) + &rho ² (c ⁴ d ⁴/k ⁴ s ⁴ − k′⁴/k ⁴),A = c ² + s ² d ²/k ² s ²−ρ ² A ² cos 2ϕ = 2c ² ₁ s ² ₁ k ²/d ² ₁ (ρ ² + 1/k ²) + ρ2(k′⁴ c ⁴ ₁ s ⁴ ₁/d ⁴ − 1),A = s ² ₁ − c ² ₁/d ² ₁ z = dnw,−ρ ² A ² cos 2ϕ = + 2c ² s ²/d ² (ρ ⁴ + k′²) + ρ ² (c ⁴ ks ⁴/d ⁴ − 1),A = s ² − c ²/d ²−ρ ² A ² cos 2ϕ = −2d ² ₁ s ² ₁/c ² ₁ (ρ ⁴ + k′²) + ρ ² (k ⁴ s ⁴ ₁/c ⁴ ₁ − d ⁴ ₁),A = −c ² ₁ + d ² ₁ s ² ₁/c ² ₁ z = cnw,ρ ² A ² cos² 2ϕ + 2(AB + E) cos 2ϕ + B ² − F/ρ ²= [ρ ² A ² cos 2ϕ + 2(AB + E)] cos 2ϕ + B ² − F/ρ ² = 0B ² − F/ρ ² = −2c ² d ²/k ² s ² (ρ ² − k′²/k ²) + ρ ² c ⁴ − d ⁴ s ⁴/k ² s ²,A = −d ² − c ² + d ² c ²/k ² s ² = −d ² s ² − c ² + 2c ² d ²/k ² s ²,AB + E = c ² d ²/k ² s ² (ρ ² − k′²/k ²)+ ρ ² c ⁴/k ⁴ s ⁴ (d ² s ² − c ² + 2k′² d ² s ²),B ² − F/ρ ² = 2k ² s ² ₁/d ² ₁ (ρ ² − k′²/k ²) + ρ ² (c ⁴ ₁ − d ⁴ ₁ s ⁴ ₁/d ⁴ ₁),A = −(d ² ₁ + k′²2/k ² s ² ₁/d ² ₁), (k ² s ² ₁ = d ² ₁ s ² ₁ − c ² ₁ + d ² ₁ c ² ₁)AB + E = −k ² s ² ₁/d ² ₁ (p ⁴ − k′²/k ²) − c ² ₁ ρ ² {1 − k′² s ² ₁/d ⁴ ₁ (2d ² ₁ c ² ₁ + d ² ₁ s ² ₁ − c ² ₁)}when u = const = K/2 and V = const = k′/2, they become Cassinians:ρ ² + 2k′²/k ² cos 2ϕ ρ ² + k′⁴/k ⁴ = k′⁴/k ⁴ + k′²/k ² = k′²/k ⁴ = β ⁴, β = 1/√k′andρ ² − 2cos 2ϕ ρ ² + 1 = k′²/k ² + 1 = 1/k ² = β ⁴, β = 1/√k