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Dispersion and Absorption of Light

Problem 5.213

A plane monochromatic light wave of intensity \(I_{0}\) falls normally on a plane-parallel plate both of whose surfaces have a reflection coefficient \(\rho .\) Taking into account multiple reflections, find the intensity of the transmitted light if (a) the plate is perfectly transparent, i.e. the absorption is absent; (b) the coefficient of linear absorption is equal to \(x\), and the plate thickness is \(d\).

Reveal Answer
5.213.(a)I=I0(1−ρ)2(1+ρ2+ρ4+⋯ )==I0(1−ρ)2/(1−ρ2);(b)I=I0(1−ρ)2σ(1+σ2ρ2+σ4ρ4+…)==I0σ(1−ρ)2/(1−σ2ρ2), where σ=exp⁡(−xd)\begin{array}{l} 5.213 .(a) I=I_{0}(1-\rho)^{2}\left(1+\rho^{2}+\rho^{4}+\cdots\right)= \\ =I_{0}(1-\rho)^{2} /\left(1-\rho^{2}\right) ;(b) I=I_{0}(1-\rho)^{2} \sigma\left(1+\sigma^{2} \rho^{2}+\sigma^{4} \rho^{4}+\ldots\right)= \\ =I_{0} \sigma(1-\rho)^{2} /\left(1-\sigma^{2} \rho^{2}\right), \text { where } \sigma=\exp (-x d) \end{array}
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