Abstract

Computations of the reflectivity, transmissivity, and efficiency properties for TE, TM, and T45° waves of far ir beam splitters (BS) and of the polarizations induced at both reflection and transmission are described. Effects of variations in the state of polarization, orientation, pointing accuracy, and wavelength of the incident light, as well as variations in refractive index and thickness of the BS, are discussed. These results apply directly to Fourier interferometer-spectrometers. They can be used for optimizing the performance of these instruments. They indicate, in particular, that some advantages may be gained by the use of incident polarized light (angle of polarization smaller than about 45° or negative elliptical polarization) or light of large incidence angle (larger than approximately 60°) or both. A novel method of inversion of experimental results to the end of determining the BS physical parameters is proposed. It makes use of the variations with incident light direction of the BS reflectivity, transmissivity, or efficiency curves.

© 1974 Optical Society of America

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References

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  1. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955).
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).
  3. P. Rouard, Ann. Phys. 7, 291 (1937).
  4. A. W. Crook, J. Opt. Soc. Am. 38, 954 (1948).
    [CrossRef] [PubMed]
  5. A. Vašiček, J. Phys. 11, 342 (1950).
  6. F. Abelès, Ann. Phys. 3, 504 (1948).
  7. A. L. Fymat, Appl. Opt. 10, 2711 (1971).
    [CrossRef] [PubMed]
  8. A. L. Fymat, Appl. Opt. 11, 160 (1972).
    [CrossRef] [PubMed]
  9. A. L. Fymat, Appl. Opt. 10, 2499 (1971).
    [CrossRef] [PubMed]
  10. The difference between Jones’s and Abelès’s matrices may be noted. Abelès’s matrices relate both the reflected and the incident fields to the transmitted field separately for the TE and the TM modes. On the other hand, Jones’s matrices relate the reflected or the transmitted field to the incident field for both modes. The elements of the Jones’s matrices do not involve Fresnel’s coefficients, applicable only to a plane interface, but their generalized expressions to finitely thick films or Abelès’s coefficients.
  11. H. Anders, Thin Films in Optics (Focal Press, London, 1967).
  12. H. A. McLeod, Thin Film Optical Filters (Elsevier, New York, 1969).
  13. E. V. Loewenstein, A. Engelsrath, J. Phys. 28, C2-153 (1967).
  14. In the usual representation of the state of polarization of light in terms of the so-called Stokes intensity parameters, no information is provided on the phase of the light wave. On the other hand, the coherency representations, which make use of the (auto- and cross-) correlations Jij = 〈EiEj*〉, i, j = x, y, 〈…〉 = time-averaged quantity, * = complex conjugate, contain both amplitude and phase information and provide therefore a more appropriate description of polarization for the Fourier spectroscopic method. Among these representations, the polarization coherency matrix (PCM) is the most convenient one because, like the instrumental Jones’s matrix, it is expressed by a two-by-two matrix. The elements of the PCM are the Jij’s.

1972

1971

1967

E. V. Loewenstein, A. Engelsrath, J. Phys. 28, C2-153 (1967).

1950

A. Vašiček, J. Phys. 11, 342 (1950).

1948

1937

P. Rouard, Ann. Phys. 7, 291 (1937).

Abelès, F.

F. Abelès, Ann. Phys. 3, 504 (1948).

Anders, H.

H. Anders, Thin Films in Optics (Focal Press, London, 1967).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

Crook, A. W.

Engelsrath, A.

E. V. Loewenstein, A. Engelsrath, J. Phys. 28, C2-153 (1967).

Fymat, A. L.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955).

Loewenstein, E. V.

E. V. Loewenstein, A. Engelsrath, J. Phys. 28, C2-153 (1967).

McLeod, H. A.

H. A. McLeod, Thin Film Optical Filters (Elsevier, New York, 1969).

Rouard, P.

P. Rouard, Ann. Phys. 7, 291 (1937).

Vašicek, A.

A. Vašiček, J. Phys. 11, 342 (1950).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

Ann. Phys.

P. Rouard, Ann. Phys. 7, 291 (1937).

F. Abelès, Ann. Phys. 3, 504 (1948).

Appl. Opt.

J. Opt. Soc. Am.

J. Phys.

A. Vašiček, J. Phys. 11, 342 (1950).

E. V. Loewenstein, A. Engelsrath, J. Phys. 28, C2-153 (1967).

Other

In the usual representation of the state of polarization of light in terms of the so-called Stokes intensity parameters, no information is provided on the phase of the light wave. On the other hand, the coherency representations, which make use of the (auto- and cross-) correlations Jij = 〈EiEj*〉, i, j = x, y, 〈…〉 = time-averaged quantity, * = complex conjugate, contain both amplitude and phase information and provide therefore a more appropriate description of polarization for the Fourier spectroscopic method. Among these representations, the polarization coherency matrix (PCM) is the most convenient one because, like the instrumental Jones’s matrix, it is expressed by a two-by-two matrix. The elements of the PCM are the Jij’s.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

The difference between Jones’s and Abelès’s matrices may be noted. Abelès’s matrices relate both the reflected and the incident fields to the transmitted field separately for the TE and the TM modes. On the other hand, Jones’s matrices relate the reflected or the transmitted field to the incident field for both modes. The elements of the Jones’s matrices do not involve Fresnel’s coefficients, applicable only to a plane interface, but their generalized expressions to finitely thick films or Abelès’s coefficients.

H. Anders, Thin Films in Optics (Focal Press, London, 1967).

H. A. McLeod, Thin Film Optical Filters (Elsevier, New York, 1969).

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Figures (5)

Fig. 1
Fig. 1

Wavelength variations of characteristics of a far ir beam splitter of Mylar (angle of incidence of light, θ1 = 45°; thin film thickness, h = 6 μm; refractive index, n = 1.8). (a) Reflectivity and transmissivity for TE, TM, and T45° waves; (b) efficiency for TE and TM waves and total efficiency; (c) degree of polarization of reflected and transmitted light.

Fig. 2
Fig. 2

Refractive index variations of beam splitter properties (n = 1.7, 1.8, 1.9; θ1 = 45°; h = 6 μm) for one period. (a) TE, TM, and total efficiencies; (b) reflected and transmitted polarizations.

Fig. 3
Fig. 3

Thin film thickness variations of beam-splitter properties (h = 6 μm, 12 μm; θ1 = 45°; n = 1.8). (a) Efficiency for TE and TM waves and total efficiency; (b) reflected and transmitted polarizations.

Fig. 4
Fig. 4

Angle of incidence variations of beam-splitter properties [θ1 = 15° (15°) 75°, 45° ±2°; h = 6 μm; n = 1.8] for one period. (a) TE-wave efficiency; (b) TM-wave efficiency (the curve for θ1 = 60° is not drawn because the corresponding efficiency point values are everywhere less than 0.2%; the small horizontal scale indicates the limits of the periods corresponding to the incidence angles indicated); (c) total efficiency; (d) polarization of reflected light; (e) polarization of transmitted light. The hatched areas illustrate the effects of an inaccuracy of ±2° in the incidence angle.

Fig. 5
Fig. 5

Incident polarization variations of the characteristics of a far ir beam splitter of Mylar under the same conditions as in Fig. 1. (a) Reflectivity and Transmissivity for incident light that is natural, completely linearly polarized (ϕ = 30°, 60°), and completely elliptically polarized (ϕ = ±30°); (b) total efficiency; (c) degree of polarization of reflected and transmitted light.

Equations (10)

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R = | r | 2 , R = | r | 2 ,
R 45 = 1 2 ( R + R ) , R = R J x x i + R J y y i J x x i + J y y i = | r | 2 .
T = p l p 1 | t | 2 , T = p l p 1 | t | 2 T 45 = 1 2 ( T + T ) , T = T J x x i + T J y y i J x x i + J y y i = p l p 1 | t | 2
R + T = R + T = R 45 + T 45 = R + T = 1.
P ( r ) = R J x x i R J y y i R J x x i + R J y y i , P ( t ) = T J x x i T J y y i T J x x i + T J y y i ,
η = 4 R T , η = 4 R T , η 45 = 4 R 45 T 45 , η = 4 R T .
R R 45 , T 45 , η = η 45 , P ( r ) = R R R + R , P ( t ) = T T T + T , }
R = ( i A i tan i β ) / ( i A i tan i β ) , i = 0,2,4 ,
S ( n , h ) = x [ R , measured ( x ; n , h ) R , computed ( x ; n , h ) R , measured ( x ; n , h ) ] 2 ,
J = 4 cos θ 1 ν cos θ m ( ν 1 ν + 1 ) 2 ( η O O η ) ,

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