Abstract

A general method is provided for constructing Jones’s reflection and transmission matrices of any beam splitter. Derivations are presented for the various known configurations. The method uses Abelès’s matrices and pays special consideration to the different expressions of Jones’s matrices relative to the various beams in an interferometric arrangement. The reversibility of the beam splitter in its action on the amplitude or phase, or both, of an incident light is studied. It is finally suggested that, even for an asymmetric beam splitter configuration, the symmetry of the interferogram can still be preserved by adjusting the thickness of the beam splitter in a prescribed manner.

© 1971 Optical Society of America

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References

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  1. W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge, 1962), Chap. 8, and Appendix 2.
  2. P. Connes, in Annual Review of Astronomy and Astrophysics, L. Goldberg, D. Layzer, J. S. Phillips, Eds. (Annual Reviews, Inc., Palo Alto, 1970), Vol. 8, p. 209.
    [CrossRef]
  3. A. L. Fymat, K. D. Abhyankar, Appl. Opt. 9, 1075 (1970); NASA Technical Brief 70-10405, November1970; Proc. Aspen Int. Conf. on Fourier Spectroscopy, 1970, AFCRL 71-0019, Special Report 114, G. A. Vanasse, A. T. Stair, D. J. Baker, Eds. (1971), p. 377.
    [CrossRef] [PubMed]
  4. Complementary interferograms can also be obtained with the other configurations when using an angle of incidence on the beam splitter larger than 45°. A requirement is always, however, that the separation and the recombination points be different.
  5. F. Abelès, Ann. Phys. (Paris) 5, 596, 706 (1950).
  6. D. Gabor, J. Inst. Elec. Eng. (London) 93, 429 (1946).
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), Chap. 1.

1970 (1)

1950 (1)

F. Abelès, Ann. Phys. (Paris) 5, 596, 706 (1950).

1946 (1)

D. Gabor, J. Inst. Elec. Eng. (London) 93, 429 (1946).

Abelès, F.

F. Abelès, Ann. Phys. (Paris) 5, 596, 706 (1950).

Abhyankar, K. D.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), Chap. 1.

Connes, P.

P. Connes, in Annual Review of Astronomy and Astrophysics, L. Goldberg, D. Layzer, J. S. Phillips, Eds. (Annual Reviews, Inc., Palo Alto, 1970), Vol. 8, p. 209.
[CrossRef]

Fymat, A. L.

Gabor, D.

D. Gabor, J. Inst. Elec. Eng. (London) 93, 429 (1946).

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge, 1962), Chap. 8, and Appendix 2.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), Chap. 1.

Ann. Phys. (Paris) (1)

F. Abelès, Ann. Phys. (Paris) 5, 596, 706 (1950).

Appl. Opt. (1)

J. Inst. Elec. Eng. (London) (1)

D. Gabor, J. Inst. Elec. Eng. (London) 93, 429 (1946).

Other (4)

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), Chap. 1.

W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge, 1962), Chap. 8, and Appendix 2.

P. Connes, in Annual Review of Astronomy and Astrophysics, L. Goldberg, D. Layzer, J. S. Phillips, Eds. (Annual Reviews, Inc., Palo Alto, 1970), Vol. 8, p. 209.
[CrossRef]

Complementary interferograms can also be obtained with the other configurations when using an angle of incidence on the beam splitter larger than 45°. A requirement is always, however, that the separation and the recombination points be different.

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Figures (2)

Fig. 1
Fig. 1

Various beam splitter configurations: (a) unsupported thin film, (b) and (c) thin film in a rectangular and in a cubic sandwich, respectively, (d) same as in (b) with an air gap between the film and the compensator, and (e) double-half deck. A schematic representation of a two-beam interferometer is also given with the definitions of components and beams: → incoming beam from the source, → subbeam in one arm of the instrument with its contributions to the source beam and to the detector beam, ⇒ subbeam in the other arm with its corresponding contributions.

Fig. 2
Fig. 2

Plane wave propagation in a stratified, isotropic medium across a boundary of arbitrary thickness. Case 1 = Fresnel’s case, case 2 = Abelès’s case. (The plane of incidence is the plane of the figure.)

Equations (39)

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E ( σ , t ) = ( E x ( σ , t ) E y ( σ , t ) )
E j ( σ , t ) = A j ( σ , t ) exp ( - i τ , ) ,
E ( i ) = ( A - A ) exp ( - i τ i ) , E ( r ) = ( R - R ) exp ( - i τ r ) , E ( t ) = ( T - T ) exp ( - i τ t ) ,
TE wave : [ A + R p 1 ( A - R ) ] = M ( 1 p l ) T ,
TM wave : cos θ 1 q 1 [ A + R q 1 ( A - R ) ] = cos θ l q l M ( 1 q l ) T .
p = ν cos θ ,             q = ν - 1 cos θ ,
M ( h = 0 ) M ( h = 0 ) 1.
M ( h ) = [ cos β - ( i / p ) sin β - i p sin β cos β ] , M ( h ) = [ cos β - ( i / q ) sin β - i q sin β cos β ] ,
M N ( h N ) = j M j ( h j ) ,             M N ( h N ) = j M j ( h j ) ;
r = R A = a - b a + b , r = R A = c - d c + d , t = T A = 2 p 1 a + b , t = T A = 2 q l cos θ 1 / cos θ l c + d , }
a = ( m 11 + m 12 p l ) p 1 , c = ( m 11 + m 12 q l ) q 1 , b = ( m 21 + m 22 p 1 ) , d = ( m 21 + m 22 q l ) . }
a = [ cos β - i ( p l / p ) sin β ] p 1 ,             c = [ cos β - i ( q l / q ) sin β ] q 1 , b = - i p sin β + p l cos β ,             d = i q sin β + q l cos β . }
R = ( r 0 0 r ) ,             T = ( t 0 0 t ) .
R = ( r 0 0 r ) ,             T = ( t 0 0 t ) ,
τ = 2 p 1 p 1 + p 1 ,             τ = 2 q l cos θ 1 / cos θ l q 1 + q l .
T F = ( τ 0 0 r ) ,
T F = ( τ 0 0 τ ) .
K = n K n ,             n = 1 , 2 , , N .
Incident beam :             S ( r ) = T F , 1 R T F , 1 ( mirror 1 ) , S ( t ) = T F , l T T F , 1 ( mirror 2 )
Beam 1 :             S ( r ) = T F , 1 R T F , 1 ( source ) , S ( t ) = T F , l T T F , 1 ( detector ) ,
Beam 2 :             S ( r ) = T F , l R T F , l ( detector ) , S ( t ) = T F , 1 T T F , l ( source ) .
Incident beam :             S ( r ) = T F , 1 R T F , 1 ( mirror 1 ) , S ( t ) = T F , l T F , l T T F , 1 ( mirror 2 ) ,
Beam 1 :             S ( r ) = T F , 1 R T F , 1 ( source ) , S ( t ) = TT F , 1 ( detector ) ,
Beam 2 :             S ( r ) = T F , 1 T T F , l T F , l ( detector ) , S ( t ) = R T F , l T F , l ( source ) .
Incident beam :             S ( r ) = R ( 1 ) ( mirror 1 ) , S ( t ) = T F T ( 1 ) ( mirror 2 ) ,
Beam 1 :             S ( r ) = T F R ( 2 ) T F ( source ) , S ( t ) = T ( 2 ) T F ( detector ) ,
Beam 2 :             S ( r ) = R ( 2 ) ( detector ) , S ( t ) = T F T ( 2 ) ( source ) ,
r = r , r = r , t = t , t = t .
( p l - p 1 ) sin 2 β { [ ( 2 p + p 1 + p l p ) ( p l - p 1 ) 2 p + ( 2 p - p 1 + p l p ) ( p l + p 1 ) 2 p ] × cos 2 β + 4 ( p 2 - p 1 p l p 2 ) sin 2 β } = 0.
( p 1 + p l ) 2 ( p l 2 - p 1 2 ) cos 2 β + [ p l 2 ( p + p l p ) 2 - p 1 2 ( p + p 1 p ) ] sin 2 β = 0 ,
tan 2 β = ( p 1 + p l ) 2 ( p l 2 - p 1 2 ) / [ p 1 2 ( p + p 1 p ) 2 + p l 2 ( p + p l p ) 2 ]
( q 1 + q l ) 2 ( q 1 2 - q l 2 ) cos 2 β + [ q 1 2 ( q + q l q ) 2 - q l 2 ( q + q 1 q ) ] sin 2 β = 0
tan 2 β = ( q 1 + q l ) 2 ( q 1 2 - q l 2 ) / [ q 1 2 ( q + q 1 q ) 2 - q 1 2 ( q + q l q ) ] .
tan ( arg r ) = 2 R e ( a ) J m ( b ) - J m ( a ) R e ( b ) [ R e ( a ) ] 2 + [ J m ( a ) ] 2 - [ R e ( b ) ] 2 - [ J m ( b ) ] 2 ,
tan ( arg t ) = - J m ( a ) + J m ( b ) R e ( a ) + R e ( b ) ,
tan ( arg r ) = 2 R e ( c ) J m ( d ) - J m ( c ) R e ( d ) [ R e ( c ) ] 2 + [ J m ( c ) ] 2 - [ R e ( d ) ] 2 - [ J m ( d ) ] 2 ,
tan ( arg t ) = - J m ( c ) + J m ( d ) R e ( c ) + R e ( d ) .
tan ( arg r ) = p p 1 ( p l 2 - p 2 ) sin 2 β p 2 ( p 1 2 - p l 2 ) cos 2 β + ( p l 2 p 1 2 - p 4 ) sin 2 β ,
tan ( arg t ) = p l p 1 + p 2 p ( p 1 - p l ) tan β ,

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