n3 ||| -d/2 n>n>n neff || d/2 1 n In a general three-layer slab waveguide with...

 

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n3 ||| -d/2 n₂>n₂>n₁ neff || d/2 1 n₁ In a general three-layer slab waveguide with central thickness d, the index step profile is shown as left, and n₂>n²>n₁. The solution is in the form: Um = Aexp[r₂(x + 1)] U₁ = Bcos(k₂x) +Csin(k₂x) Dexp[-V₁ (x - 2) U₁ = (a) Show that the U₂ can also be expressed in a simpler form: (10 pt) UIIB'cos(k₂x + A) (b) Prove that A>0 for fundamental mode when n²>n₁, so the mode intensity peak is shifted toward the cladding layer with the higher refractive index. You might start with comparing the B.C.s at both interfaces (10 pt) n3 ||| -d/2 n₂>n₂>n₁ neff || d/2 1 n₁ In a general three-layer slab waveguide with central thickness d, the index step profile is shown as left, and n₂>n²>n₁. The solution is in the form: Um = Aexp[r₂(x + 1)] U₁ = Bcos(k₂x) +Csin(k₂x) Dexp[-V₁ (x - 2) U₁ = (a) Show that the U₂ can also be expressed in a simpler form: (10 pt) UIIB'cos(k₂x + A) (b) Prove that A>0 for fundamental mode when n²>n₁, so the mode intensity peak is shifted toward the cladding layer with the higher refractive index. You might start with comparing the B.C.s at both interfaces (10 pt)