## Abstract

Polarization interference filters are described which can have a pass band ranging from a fraction of an angstrom to several hundred angstroms in width. The pass band can be shifted to any desired region of the spectrum. These tunable filters are based on the fixed filters discussed by Lyot and Evans. The transmission band is formed by the superposition of the polarized channel spectra, produced by x-cut plates of quartz or other birefringent media placed between parallel polarizers. The birefringent plates have thicknesses in the ratio 1:2:4 etc. The tuning is accomplished by changing the retardation of successive elements so that transmission maxima in the various channel spectra coincide at the desired wave-length. The retardation change can be made mechanically, for example, by stretching supplemental plastic sheets in series with the filter elements, or can be made electrically by using Kerr cells or crystals with high electro-optic coefficients, such as ammonium dihydrogen phosphate. The additional retardation never has to exceed a full wave at the wave-length of peak transmission. The measured transmission of an experimental filter is shown. The electrical tuning method is particularly adapted to cathode-ray oscillograph presentation of spectra. The filter also has possible application in color reproduction and colorimetry. With a pass band of a half-angstrom line of sight motion of solar prominences could be determined by the use of the Doppler shift of the prominence radiation.

© 1947 Optical Society of America

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### Equations (34)

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(1)
$$T={({A}^{\prime}cos\theta cos2\theta cos4\theta \cdots cos{2}^{N-1}\theta )}^{2},$$
(2)
$$cosn\theta =({e}^{ni\theta}+{e}^{-ni\theta})/2.$$
(3)
$$cosn\theta =({e}^{ni\theta}/2)(1+{e}^{-2ni\theta}).$$
(4)
$${T}^{\frac{1}{2}}={A}^{\prime}\frac{{e}^{i\theta}}{2}(1+{e}^{-2i\theta})\frac{{e}^{2i\theta}}{2}(1+{e}^{-4i\theta})\frac{{e}^{4i\theta}}{2}(1+{e}^{-8i\theta})\cdots \times \frac{exp[{2}^{N-1}i\theta ]}{2}(1+exp[-{2}^{N}i\theta ]).$$
(5)
$${T}^{\frac{1}{2}}=({A}^{\prime}/{2}^{N}){e}^{i\theta \text{\u2211}_{N=1}^{N}{2}^{N-1}}\times [1+{e}^{-2i\theta}+{e}^{-4i\theta}+{e}^{-6i\theta}+\cdots {e}^{-i\theta \text{\u2211}_{N=1}^{N}{2}^{N}}].$$
(6)
$$\begin{array}{cc}\text{\u2211}_{N=1}^{N}{2}^{N-1}={2}^{N}-1;& \text{\u2211}_{N=1}^{N}{2}^{N}={2}^{N+1}-2,\end{array}$$
(7)
$${T}^{\frac{1}{2}}=\frac{{A}^{\prime}}{{2}^{N}}(exp[i\theta ({2}^{N}-1)])\frac{1-exp[-2i\theta ({2}^{N})]}{1-exp[-2i\theta ]};$$
(8)
$${T}^{\frac{1}{2}}=\frac{{A}^{\prime}}{{2}^{N}}\frac{exp[{2}^{N}i\theta ]-exp[-{2}^{N}i\theta ]}{2i}\cdot \frac{2i}{{e}^{i\theta}-{e}^{-i\theta}},$$
(9)
$$T={[{A}^{\prime}sin{2}^{N}\theta /{2}^{N}sin\theta ]}^{2}.$$
(10)
$$T=A({sin}^{2}\alpha /{\alpha}^{2})\cdot ({sin}^{2}n\phi /{sin}^{2}\phi ),$$
(11)
$$\begin{array}{c}\alpha =(\pi /\mathrm{\lambda})dsin\theta ,\\ \phi =(\pi /\mathrm{\lambda})(d+b)sin\theta .\end{array}$$
(12)
$${\theta}_{Nx}={2}^{N-1}{\theta}_{1x}=x\pi +{2}^{N-1}p\pi .$$
(13)
$${\theta}_{Nn}=({n}_{N}+{2}^{N-1}p)\pi ,$$
(14)
$${{\theta}^{\prime}}_{Nx}=a{\theta}_{Nx}.$$
(15)
$$a=({2}^{N-1}p+{n}_{N})/({2}^{N-1}p+x).$$
(16)
$$a=[{2}^{N-1}p+({2}^{N-K}){n}_{K}]/[{2}^{N-1}p+x],$$
(17)
$${a}_{i}={a}_{K}={a}_{K+1}\cdots ={a}_{i}.$$
(18)
$${T}^{\prime}={A}^{2}{[cos{a}_{1}\theta cos2{a}_{2}\theta cos4{a}_{3}\theta \cdots ]}^{2}.$$
(19)
$${T}^{\prime}=\text{\u220f}_{i=1}^{p}{(sin{2}^{{n}_{i}}{a}_{i}\theta /{2}^{{n}_{i}}sin{a}_{i}\theta )}^{2},$$
(20)
$$\text{\u2211}{n}_{i}=N.$$
(22)
$$\begin{array}{ccc}p=2,& {n}_{K}=0,& {a}_{1}=32/33.\end{array}$$
(23)
$${T}_{1}={[cosa\theta cos2a\theta cos4a\theta cos8a\theta cos16\theta ]}^{2}.$$
(24)
$${T}_{3}={[cos{a}_{1}\theta cos2{a}_{1}\theta cos4{a}_{1}\theta cos8{a}_{2}\theta cos16\theta ]}^{2}.$$
(25)
$${T}_{12}={[cos(8/11)\theta cos(20/11)\theta cos4\theta cos8\theta cos16\theta ]}^{2}.$$
(26)
$${T}_{12}={[cos(12/11)\theta (cos24/11)\theta cos4\theta cos8\theta cos16\theta ]}^{2}.$$
(27)
$$T=A{cos}^{2}a\theta .$$
(28)
$$T=A{cos}^{2}(a\theta +\delta ),$$
(29)
$$\mathrm{\Delta}{\delta}_{K}=2({\theta}_{K0}-{{\theta}^{\prime}}_{K0}),$$
(30)
$$\mathrm{\Delta}{\delta}_{K}={2}^{K}({a}_{K}-1){\theta}_{1,0}.$$
(31)
$$\mathrm{\Delta}{\delta}_{K}=({2}^{N}{n}_{K}-{2}^{K}x/{2}^{N-1}p+x){\theta}_{1,0}.$$
(32)
$$x={2}^{N-1}p({\mathrm{\lambda}}_{0}-\mathrm{\lambda})/\mathrm{\lambda},$$
(33)
$$\mathrm{\Delta}{\delta}_{K}=\frac{\mathrm{\lambda}({n}_{K}+{2}^{K-1}p)-{2}^{K-1}p{\mathrm{\lambda}}_{0}}{p{\mathrm{\lambda}}_{0}}\cdot 2\pi ,$$
(34)
$$({2}^{K}p{\mathrm{\lambda}}_{0}/2{n}_{K}+1+{2}^{K}p)<\mathrm{\lambda}<({2}^{K}p{\mathrm{\lambda}}_{0}/2{n}_{K}-1+{2}^{K}p).$$