Question 17.1: What is the fraction of externally incident unpolarized radi...
What is the fraction of externally incident unpolarized radiation that is transmitted through a glass window in air? The window is 0.75 cm thick, radiation is incident at θ_i = 50°, n_{glass} = 1.53, and κ_{glass} = 0.1 cm^{–1}.
To find the path length through the glass, evaluate χ = sin^{–1}(sin θ_i/n) = sin^{–1}(sin 50°/1.53) = 30°. The path length is S = 0.75/cos χ = 0.866 cm. The transmittance is τ = exp(–κS) = exp(–0.1 ×0.866) = 0.917. The surface reflectivities for the two components of polarization are
\rho _{\parallel }=\frac{\tan ^2(\theta _i-\chi )}{\tan ^2(\theta _i+\chi )}=0.00412 ; \rho _{\bot }=\frac{\sin ^2(\theta _i-\chi )}{\sin ^2(\theta _i+\chi )}=0.1206
Then the overall transmittance for each component is
T _{\parallel }=\tau \frac{1-\rho _{\parallel }}{1+\rho _{\parallel }}\frac{1-\rho ^2_{\parallel }}{1-\rho ^2_{\parallel }\tau ^2} =0.917\frac{0.9959}{1.00411}\frac{1-0.00002}{1-0.00001}=0.9095
T _{\bot }=\tau \frac{1-\rho _{\bot }}{1+\rho _{\bot }}\frac{1-\rho ^2_{\bot }}{1-\rho ^2_{\bot }\tau ^2} =0.917\frac{0.8794}{1.1206}\frac{1-0.0145}{1-0.0122}=0.7179
For unpolarized incident radiation, one-half the energy is in each component. Hence T = (T_{||} + T_⊥)/2= 0.814.