A p-polarised laser beam in air ( ) meets a glass beamsplitter plate ( ) at

PHYSICS · Subjective · Fermi University Physics · Optics & Photonics · Wave propagation and light in media · Wave propagation and light in media

Optics and Photonics, Ch Reflection and transmission at interfaces

Problem

A p-polarised laser beam in air (ni=1.00n_i = 1.00) meets a glass beamsplitter plate (nt=1.50n_t = 1.50) at θi=45.0∘\theta_i = 45.0^\circ. The chapter's Fresnel forms for p are rp=nicos⁡θt−ntcos⁡θinicos⁡θt+ntcos⁡θi,tp=2nicos⁡θinicos⁡θt+ntcos⁡θi,r_p=\frac{n_i\cos\theta_t-n_t\cos\theta_i}{n_i\cos\theta_t+n_t\cos\theta_i}, \qquad t_p=\frac{2n_i\cos\theta_i}{n_i\cos\theta_t+n_t\cos\theta_i}, and the reflectance and transmittance are Rp=rp2\mathcal{R}_p=r_p^2 and Tp=ntcos⁡θtnicos⁡θitp2\mathcal{T}_p = \dfrac{n_t\cos\theta_t}{n_i\cos\theta_i}t_p^2. (a) Write Rp\mathcal{R}_p and Tp\mathcal{T}_p over the common denominator D=(nicos⁡θt+ntcos⁡θi)2D=(n_i\cos\theta_t+n_t\cos\theta_i)^2, expand the two numerators, and show that they add to exactly DD, so that Rp+Tp=1\mathcal{R}_p+\mathcal{T}_p=1 at every angle of incidence for which θt\theta_t is real: any pair of real indices, below any critical angle, where rpr_p and tpt_p are the real amplitude ratios the chapter defined. (b) Find θt\theta_t from Snell's law for this surface and evaluate rpr_p, tpt_p, Rp\mathcal{R}_p and Tp\mathcal{T}_p numerically, confirming the identity to three significant figures.

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