Abstract

The effect of a plate of anisotropic material, such as a crystal, on a collimated beam of polarized light may always be represented mathematically as a linear transformation of the components of the electric vector of the light. The effect of a retardation plate, of an anisotropic absorber (plate of tourmaline; Polaroid sheeting), or of a crystal or solution possessing optical activity, may therefore be represented as a matrix which operates on the electric vector of the incident light. Since a plane wave of light is characterized by the phases and amplitudes of the two transverse components of the electric vector, the matrices involved are two-by-two matrices, with matrix elements which are in general complex. A general theory of optical systems containing plates of the type mentioned is developed from this point of view.

© 1941 Optical Society of America

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  1. Registered trade-mark of the Polaroid Corporation. For a description of Polaroid sheeting, see Martin Grabau, J. Opt. Soc. Am. 27, 420 (1937).
    [CrossRef]
  2. In Parts I and II we shall need only the algebraic properties of matrices, as they are presented on pp. 348–352 of E. C. Kemble, Fundamental Principles of Quantum Mechanics (McGraw-Hill Book Company, Inc., New York, 1937). In Part III, we shall find it necessary to use also the transformation properties of matrices; see the reference just given, or V. Rojansky, Introductory Quantum Mechanics (Prentice-Hall, New York, 1938), pp. 285–340.
  3. Max Born, Optik (Springer, Berlin, 1933), p. 23.

1937 (1)

Born, Max

Max Born, Optik (Springer, Berlin, 1933), p. 23.

Grabau, Martin

Kemble, E. C.

In Parts I and II we shall need only the algebraic properties of matrices, as they are presented on pp. 348–352 of E. C. Kemble, Fundamental Principles of Quantum Mechanics (McGraw-Hill Book Company, Inc., New York, 1937). In Part III, we shall find it necessary to use also the transformation properties of matrices; see the reference just given, or V. Rojansky, Introductory Quantum Mechanics (Prentice-Hall, New York, 1938), pp. 285–340.

J. Opt. Soc. Am. (1)

Other (2)

In Parts I and II we shall need only the algebraic properties of matrices, as they are presented on pp. 348–352 of E. C. Kemble, Fundamental Principles of Quantum Mechanics (McGraw-Hill Book Company, Inc., New York, 1937). In Part III, we shall find it necessary to use also the transformation properties of matrices; see the reference just given, or V. Rojansky, Introductory Quantum Mechanics (Prentice-Hall, New York, 1938), pp. 285–340.

Max Born, Optik (Springer, Berlin, 1933), p. 23.

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Equations (28)

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E x = A x exp [ i ( ϵ x + 2 π ν t ) ] , E y = A y exp [ i ( ϵ y + 2 π ν t ) ] ,
E x 1 = E x 0 exp [ - i ( 2 π d / λ ) ( n x - i k x ) ] = N x E x 0 , E y 1 = E y 0 exp [ - i ( 2 π d / λ ) ( n y - i k y ) ] = N y E y 0 .
E x = E x cos ω + E y sin ω , E y = - E x sin ω + E y cos ω .
E x 1 = ( N x cos 2 ω + N y sin 2 ω ) E x 0 + ( N x - N y ) sin ω cos ω E y 0 , E y 1 = ( N x - N y ) sin ω cos ω E x 0 + ( N x sin 2 ω + N y cos 2 ω ) E y 0 .
M ( m 1 m 4 m 3 m 2 ) ,
m 1 = N x cos 2 ω + N y sin 2 ω , m 2 = N x sin 2 ω + N y cos 2 ω , m 3 = m 4 = ( N x - N y ) sin ω cos ω .
ɛ 0 ( E x 0 E y 0 ) ;             ɛ 1 ( E x 1 E y 1 ) .
ɛ 1 = M ɛ 0 .
S ( ω ) ( cos ω - sin ω sin ω cos ω )
N ( N x 0 0 N y ) .
M = S ( ω ) NS ( - ω )
ɛ 1 = S ( ω ) NS ( - ω ) ɛ 0 .
ɛ 1 = S ( ω 1 ) N 1 S ( - ω 1 ) ɛ 0 = M 1 ɛ 0 , ɛ 0 = M 2 ɛ 1 , ·             ·             ·             ·             ·             · ɛ n = M n ɛ n - 1 ,
ɛ n = M n M n - 1 M 2 M 1 ɛ 0 = M ( n ) ɛ 0 ,
M ( n ) M n M n - 1 M 2 M 1 .
M ( n ) = [ S ( ω n ) N n S ( - ω n ) ] [ S ( ω n - 1 ) N n - 1 S ( - ω n - 1 ) ] [ S ( ω 2 ) N 2 S ( - ω 2 ) ] [ S ( ω 1 ) N 1 S ( - ω 1 ) ] .
S ( ω 1 + ω 2 ) S ( ω 1 ) S ( ω 2 ) ,
M ( n ) = S ( ω n ) [ N n S ( ω n , n - 1 ) N 2 S ( ω 2 , 1 ) N 1 S ( ω 1 , n ) ] S ( - ω n ) ,
= S ( ω 1 ) [ S ( ω 1 , n ) N n S ( ω n , n - 1 ) S ( ω 2 , 1 ) N 1 ] S ( - ω 1 ) ,
ω i , j ω j - ω i .
ω i + 1 , i ω i - ω i + 1 + ω ˜ .
ɛ ¯ ( E x E y ) ,
ɛ ¯ final = ɛ ¯ initial M ( n ) ,
I A x 2 + A y 2 = E * x E x + E * y E y = ɛ ¯ * ɛ = ɛ ¯ ɛ *
G ( e i γ 0 0 e - i γ ) .
P ( p 1 0 0 p 2 )             0 p 1 1 0 p 2 1.
tan 2 ψ = tan 2 α cos δ , cos 2 θ = sin 2 α sin δ ,
tan α = A y / A x ,             0 α 1 2 π .