Classical Optics / Lagrange multipliers

1. The problem statement, all variables and given/known data



A ray of light enters a glass block of refractive index n and thickness d with angle of incidence θ₁ . Part of the ray refracts at some angle θ₂ such that Snell's law is obeyed, and the rest undergoes specular reflection. The refracted ray reflects off the bottom of the block and then refracts back out. How should θ₁ be chosen to maximize the perpendicular distance x between the ray which reflects off the surface, and the ray which reflects off the bottom of the block and refracts back into the atmosphere?



This problem seems a natural fit for Lagrange multipliers, but I am open to other approaches.



2. Relevant equations



If we let x(θ₁,θ₂) be the objective function and sin(θ₁) = n sin(θ₂) the constraint, then we get



∇f(θ₁,θ₂,λ) = 0, where f(θ₁,θ₂,λ) = x(θ₁,θ₂) - λ(sin(θ₁) - n sin(θ₂))



3. The attempt at a solution



I derived x to be equal to d tan(θ₂) / sin(θ₁), giving a final answer of θ₁ = arsin(sqrt(n²-1)), but this does not seem physically reasonable. Can anyone spot where I went wrong?