## Abstract

The condition of minimum deviation (MD) by a transparent optically isotropic prism is re-derived, and expressions for the intensity transmittances ${T}_{p}\left(\theta \right)$ and ${T}_{s}\left(\theta \right)$ of an uncoated prism of refractive index *n* and prism angle *α* for incident *p*- and *s*-polarized light and their derivatives with respect to the internal angle of refraction *θ* are obtained. When the MD condition $\left(\theta =\alpha /2\right)$ is satisfied, ${T}_{s}$ is maximum and ${T}_{p}$ is maximum or minimum. The transmission ellipsometric parameters ${\psi}_{t},{\Delta}_{t}$ of a symmetrically coated prism are also shown to be locally stationary with respect to *θ* at $\theta =\alpha /2$. The constraint on $\left(n,\alpha \right)$ for maximally flat transmittance (MFT) of *p*-polarized light at and near the MD condition is determined. The transmittance ${T}_{p}$ of prisms represented by points that lie below the locus $\left(n,\alpha \right)$ of MFT exhibits oscillation as a function of *θ*. No similar behavior is found for the *s* polarization. Magnitudes and angular positions of the maxima and minima of the oscillatory ${T}_{p}$-versus-*θ* curves are also calculated as functions of *α* for a ZnS prism of refractive index $n=2.35$ in the visible.

© 2010 Optical Society of America

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### Equations (22)

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(1)
$$\gamma \left(\theta \right)={\text{sin}}^{-1}\left(n\text{\hspace{0.17em} sin \hspace{0.17em}}\theta \right)-\theta .$$
(2)
$${\gamma}_{t}\left(\theta \right)=\gamma \left(\theta \right)+\gamma \left(\alpha -\theta \right).$$
(3)
$${\gamma}_{t}^{\prime}\left(\theta \right)=d{\gamma}_{t}/d\theta ={\gamma}^{\prime}\left(\theta \right)-{\gamma}^{\prime}\left(\alpha -\theta \right).$$
(4)
$${\gamma}_{t\text{\hspace{0.17em} min}}=2\left\{{\text{sin}}^{-1}\left[n\text{\hspace{0.17em} sin}\left(\alpha /2\right)\right]-\left(\alpha /2\right)\right\}.$$
(5)
$$n=\frac{\text{sin}\left[\left({\gamma}_{t\text{\hspace{0.17em} min}}+\alpha \right)/2\right]}{\text{sin}\left(\alpha /2\right)}.$$
(6)
$${\tau}_{\nu}=1-{R}_{\nu},\text{\hspace{0.5em} \hspace{0.5em}}\nu =p,s,$$
(7)
$${\tau}_{p}\left(\theta \right)=\frac{4n\text{\hspace{0.17em} cos \hspace{0.17em}}\theta {\left(1-{n}^{2}\text{\hspace{0.17em}}{\text{sin}}^{2}\text{\hspace{0.17em}}\theta \right)}^{1/2}}{\left({n}^{2}+1\right)-\left({n}^{4}+1\right){\text{sin}}^{2}\text{\hspace{0.17em}}\theta +2n\text{\hspace{0.17em} cos \hspace{0.17em}}\theta {\left(1-{n}^{2}\text{\hspace{0.17em}}{\text{sin}}^{2}\text{\hspace{0.17em}}\theta \right)}^{1/2}},$$
(8)
$${\tau}_{s}\left(\theta \right)=\frac{4n\text{\hspace{0.17em} cos \hspace{0.17em}}\theta {\left(1-{n}^{2}\text{\hspace{0.17em}}{\text{sin}}^{2}\text{\hspace{0.17em}}\theta \right)}^{1/2}}{1+{n}^{2}\text{\hspace{0.17em} cos \hspace{0.17em}}2\theta +2n\text{\hspace{0.17em} cos \hspace{0.17em}}\theta {\left(1-{n}^{2}\text{\hspace{0.17em}}{\text{sin}}^{2}\text{\hspace{0.17em}}\theta \right)}^{1/2}},$$
(9)
$${T}_{\nu}={\tau}_{\nu}\left(\theta \right){\tau}_{\nu}\left(\alpha -\theta \right),\text{\hspace{0.5em} \hspace{0.5em}}\nu =p,s.$$
(10)
$${T}_{\nu}^{\prime}\left(\theta \right)={\tau}_{\nu}^{\prime}\left(\theta \right){\tau}_{\nu}\left(\alpha -\theta \right)-{\tau}_{\nu}\left(\theta \right){\tau}_{\nu}^{\prime}\left(\alpha -\theta \right),$$
(11)
$${T}_{\nu}^{\u2033}\left(\theta \right)={\tau}_{\nu}^{\u2033}\left(\theta \right){\tau}_{\nu}\left(\alpha -\theta \right)-2{\tau}_{\nu}^{\prime}\left(\theta \right){\tau}_{\nu}^{\prime}\left(\alpha -\theta \right)+{\tau}_{\nu}\left(\theta \right){\tau}_{\nu}^{\u2033}\left(\alpha -\theta \right),$$
(12)
$${T}_{\nu}^{\u2034}\left(\theta \right)={\tau}_{\nu}^{\u2034}\left(\theta \right){\tau}_{\nu}\left(\alpha -\theta \right)-3{\tau}_{\nu}^{\u2033}\left(\theta \right){\tau}_{\nu}^{\prime}\left(\alpha -\theta \right)+3{\tau}_{\nu}^{\prime}\left(\theta \right){\tau}_{\nu}^{\u2033}\left(\alpha -\theta \right)-{\tau}_{\nu}\left(\theta \right){\tau}_{\nu}^{\u2034}\left(\alpha -\theta \right).$$
(13)
$${T}_{\nu}^{\prime}\left(\alpha /2\right)=0,\text{\hspace{0.5em} \hspace{0.5em}}\nu =p,s.$$
(14)
$${T}_{\nu}^{\u2033}\left(\alpha /2\right)=2{\tau}_{\nu}\left(\alpha /2\right){\tau}_{\nu}^{\u2033}\left(\alpha /2\right)-2{\left[{\tau}_{\nu}^{\prime}\left(\alpha /2\right)\right]}^{2}.$$
(15)
$${T}_{\nu}^{\u2033}\left(\alpha /2\right)=0,$$
(16)
$${\tau}_{\nu}\left(\alpha /2\right){\tau}_{\nu}^{\u2033}\left(\alpha /2\right)-{\left[{\tau}_{\nu}^{\prime}\left(\alpha /2\right)\right]}^{2}=0.$$
(17)
$${\psi}_{t}={\text{tan}}^{-1}{\left({T}_{p}/{T}_{s}\right)}^{1/2},$$
(18)
$${\Delta}_{t}\left(\theta \right)={\Delta}_{1t}\left(\theta \right)+{\Delta}_{1t}\left(\alpha -\theta \right),$$
(19)
$${\Delta}_{t}^{\prime}\left(\alpha /2\right)=0.$$
(20)
$$\alpha =2{\theta}_{Bi}=2\text{\hspace{0.17em}}{\text{tan}}^{-1}\left(1/n\right).$$
(21)
$$\alpha =2{\theta}_{c}={\text{sin}}^{-1}\left(1/n\right).$$
(22)
$${T}_{p\text{\hspace{0.17em} min}}\left(0\right)=16{n}^{2}/{\left(n+1\right)}^{4}.$$