Abstract

I show that the reflection coefficient of any stratified planar structure can be obtained by using a complex generalization of Einstein's addition theorem for parallel velocities. This result also applies to multiple quantum wells. It provides a new mathematical tool in optics and in quantum theory and may lead to useful algorithms in computing. It may also give a new insight into special relativity. The composition law of velocities, in fact, no longer appears as a specific result of special relativity but rather as the expression, in the particular case of kinematics, of a more general law of physics. The possible use of the composition law of probability amplitude in quantum theory is also presented.

© 1992 Optical Society of America

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References

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  1. L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960).
  2. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  3. J. Lekner, Theory of Reflection (Nijhoff, Norwell, Mass., 1987).
  4. F. Abeles, “Recherche sur la propagation des ondes electromagnétiques sinusoidales dans les milieux stratifiés. Applications aux couches minces,” Ann. Phys. Paris 5, 596–640, 706–782 (1950); “Optics of thin films,” in Advanced Optical Techniques, A. C. S. van Heel, ed. (North-Holland, Amsterdam, 1967), Chap. 5.
  5. J. M. Vigoureux, “Polynomial formulation of reflection and transmission by stratified planar structures,” J. Opt. Soc. Am. A 8, 1697–1701 (1991).
    [CrossRef]
  6. B. L. Van Der Waerden, Modern Algebra (Ungar, New York, 1966), Vol. 1.
  7. S. Lang, Algebra (Addison-Wesley, Reading, Mass., 1965).
  8. J. M. Vigoureux, “The groupoïd of amplitudes of light in stratified planar media and the ‘phase Thomas precession,’” submitted to J. Eur. Opt. Soc.
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).
  10. W. Pauli, Theory of Relativity (Pergamon, New York, 1958), p. 74.

1991 (1)

1950 (1)

F. Abeles, “Recherche sur la propagation des ondes electromagnétiques sinusoidales dans les milieux stratifiés. Applications aux couches minces,” Ann. Phys. Paris 5, 596–640, 706–782 (1950); “Optics of thin films,” in Advanced Optical Techniques, A. C. S. van Heel, ed. (North-Holland, Amsterdam, 1967), Chap. 5.

Abeles, F.

F. Abeles, “Recherche sur la propagation des ondes electromagnétiques sinusoidales dans les milieux stratifiés. Applications aux couches minces,” Ann. Phys. Paris 5, 596–640, 706–782 (1950); “Optics of thin films,” in Advanced Optical Techniques, A. C. S. van Heel, ed. (North-Holland, Amsterdam, 1967), Chap. 5.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Born, M.

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

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960).

Lang, S.

S. Lang, Algebra (Addison-Wesley, Reading, Mass., 1965).

Lekner, J.

J. Lekner, Theory of Reflection (Nijhoff, Norwell, Mass., 1987).

Pauli, W.

W. Pauli, Theory of Relativity (Pergamon, New York, 1958), p. 74.

Van Der Waerden, B. L.

B. L. Van Der Waerden, Modern Algebra (Ungar, New York, 1966), Vol. 1.

Vigoureux, J. M.

J. M. Vigoureux, “Polynomial formulation of reflection and transmission by stratified planar structures,” J. Opt. Soc. Am. A 8, 1697–1701 (1991).
[CrossRef]

J. M. Vigoureux, “The groupoïd of amplitudes of light in stratified planar media and the ‘phase Thomas precession,’” submitted to J. Eur. Opt. Soc.

Wolf, E.

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

Ann. Phys. Paris (1)

F. Abeles, “Recherche sur la propagation des ondes electromagnétiques sinusoidales dans les milieux stratifiés. Applications aux couches minces,” Ann. Phys. Paris 5, 596–640, 706–782 (1950); “Optics of thin films,” in Advanced Optical Techniques, A. C. S. van Heel, ed. (North-Holland, Amsterdam, 1967), Chap. 5.

J. Opt. Soc. Am. A (1)

Other (8)

B. L. Van Der Waerden, Modern Algebra (Ungar, New York, 1966), Vol. 1.

S. Lang, Algebra (Addison-Wesley, Reading, Mass., 1965).

J. M. Vigoureux, “The groupoïd of amplitudes of light in stratified planar media and the ‘phase Thomas precession,’” submitted to J. Eur. Opt. Soc.

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

W. Pauli, Theory of Relativity (Pergamon, New York, 1958), p. 74.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

J. Lekner, Theory of Reflection (Nijhoff, Norwell, Mass., 1987).

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Figures (3)

Fig. 1
Fig. 1

Illustration of notations for the case of n parallel planar surfaces.

Fig. 2
Fig. 2

Reflection of a plane wave in a plane-parallel plate. The overall reflection coefficient can be obtained by taking into account all the ways in which the light can be reflected by each interface that it encounters in going from the source to the detector. The notation is explained in the text.

Fig. 3
Fig. 3

Comparison of (a) the composition law of parallel velocities and (b) the composition law of amplitudes. (a) In special relativity consider three similar observers, K0, K1, K2. Each can measure only velocities relative to his own inertial frame. The only way for K0 to determine the velocity V ( 2 ) of K2 relative to himself (K0) by using his measurement V1 of K1 (relative to himself) and V2 of K2 (relative to K1) is to compose (instead of to add) V1 and V2. (b) An observer K0 located in the first medium n0 can measure the reflection coefficient R01R1 of light on an interface between his own medium n0 and the medium n1. Because he is not in medium n1, he cannot directly measure the reflection coefficient R12 = R2 of light on the interface n1n2. To calculate the complete process of light reflected by the two interfaces by using R1 and R2, he must compose his result R1 with the result of another observer K1 located in the medium n1. The reasoning in (a) and (b) can be generalized to the case of n inertial frames and n interfaces, as is suggested by the dashed lines.

Equations (40)

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R 2 = R 1 R 2 = R 1 + R 2 1 + R ¯ 1 R 2 ,
R n R ( 1 , 2 , , n ) = m 0 S 2 m + 1 n m 0 S 2 m n ¯ .
S 0 n = 1 , S 1 n = i = 1 n R i = R 1 + R 2 + + R n , S 2 n = 1 i < j n R i R ¯ j = R 1 R ¯ 2 + R 1 R ¯ 3 + + R 1 R ¯ n + R 2 R ¯ 3 + , S 3 n = 1 i < j < k n R i R ¯ j R k , S p n = 1 i < j < < w n R i R ¯ j R k R ¯ 1 R w 0 ( p terms in each factor ) , S n n = R 1 R ¯ 2 R 3 R ¯ 4 R n 0 .
R j R j , j + 1 = r j , j + 1 exp [ - 2 i ( β 1 + β 2 + + β j ) ] ,
R ¯ j R ¯ j , j + 1 = r j , j + 1 exp [ + 2 i ( β 1 + β 2 + + β j ) ]
β j = 2 π λ n j cos θ j ( z j - z j - 1 ) = q j d j ,
S 1 3 = R 1 + R 2 + R 3 , S 2 3 = R 1 R ¯ 2 + R 1 R ¯ 3 + R 2 R ¯ 3 , S 3 3 = R 1 R ¯ 2 R 3 .
R 2 = R 1 + R 2 1 + R ¯ 1 R 2 = r 12 + r 23 exp ( - i 2 β 2 ) 1 + r 12 r 23 exp ( - i 2 β 2 ) exp ( - i 2 β 1 ) ,
R 3 = R 1 + R 2 + R 3 + R 1 R ¯ 2 R 3 1 + R ¯ 1 R 2 + R ¯ 1 R 3 + R ¯ 2 R 3 ,
R 3 = [ r 12 + r 23 exp ( - 2 i β 2 ) ] + [ r 12 r 23 + exp ( - 2 i β 2 ) ] r 34 exp ( - 2 i β 3 ) [ 1 + r 12 r 23 exp ( - 2 i β 2 ) ] + [ r 23 + r 12 exp ( - 2 i β 2 ) ] r 34 exp ( - 2 i β 3 ) exp ( - 2 i β 1 ) .
R 4 = R 1 + R 2 + R 3 + R 4 + R 1 R ¯ 2 R 3 + R 1 R ¯ 2 R 4 + R 1 R ¯ 3 R 4 + R 2 R ¯ 3 R 4 1 + R ¯ 1 R 2 + R ¯ 1 R 3 + R ¯ 1 R 4 + R ¯ 2 R 3 + R ¯ 2 R 4 + R ¯ 3 R 4 + R ¯ 1 R 2 R ¯ 3 R 4 .
R 2 = R 1 R 2 = R 1 + R 2 1 + R ¯ 1 R 2 A 2
V 2 = V 1 V 2 = V 1 + V 2 1 + V 1 V 2 c 2 .
V 3 = V 1 + V 2 + V 3 + V 1 V 2 V 3 1 + V 1 V 2 + V 1 V 3 + V 2 V 3 ,
V n = m 0 P 2 m + 1 n m 0 P 2 m n ,
R ( 1 , 2 , , n ) R n = R 1 [ R 2 ( R n ) ] ,
R ( 1 , 2 , , n ) = R 1 R ( 2 , 3 , , n ) = R 1 + R ( 2 , 3 , , n ) 1 + R ¯ 1 R ( 2 , 3 , , n ) .
R ( i ) R i .
R ( 1 , 2 , 3 ) = R 1 ( R 2 R 3 ) R 1 + R ( 2 , 3 ) 1 + R ¯ 1 R ( 2 , 3 ) ,
R ( 2 , 3 ) = R 2 + R 3 1 + R ¯ 2 R 3 ,
R ( 1 , 2 , 3 ) = R 1 + R 2 + R 3 + R 1 R ¯ 2 R 3 1 + R ¯ 1 R 2 + R ¯ 1 R 3 + R ¯ 2 R 3 ,
R ( 1 , 2 , 3 , 4 ) = R 1 + R ( 2 , 3 , 4 ) 1 + R ¯ 1 R ( 2 , 3 , 4 ) ,
R 1 R ( 2 , 3 , , n ) = R 1 + R ( 2 , 3 , , n ) 1 + R ¯ 1 R ( 2 , 3 , , n )
R 1 + ( m 0 S 2 m + 1 n - 1 / m 0 S 2 m n - 1 ¯ ) 1 + R ¯ 1 ( m 0 S 2 m + 1 n - 1 / m 0 S 2 m n - 1 ¯ ) = R 1 m 0 S 2 m n - 1 ¯ + m 0 S 2 m + 1 n - 1 m 0 S 2 m n - 1 ¯ + R ¯ 1 m 0 S 2 m n - 1 ,
1 i < j < < w n R i R ¯ j R k R w 0             ( p terms in each factor ) = 2 i < j < < w n R i R ¯ j R k R w 0             ( p terms in each factor ) + R 1 ( 2 i < j < < w n R i R ¯ j R w 0 ¯ )             ( p - 1 terms in each factor ) ,
S p n = S p n - 1 + R 1 S p - 1 n - 1 ¯ .
R 1 m 0 S 2 m n - 1 ¯ + m 0 S 2 m + 1 n - 1 = m 0 ( S 2 m + 1 n - 1 + R 1 S 2 m n - 1 ¯ ) = m 0 S 2 m + 1 n .
m 0 S 2 m n - 1 ¯ + R ¯ 1 m 0 S 2 m + 1 n - 1 = m 0 S 2 m n - 1 ¯ + R ¯ 1 m 0 S 2 m - 1 n - 1 = m 0 S 2 m n ¯ .
D : ( R 1 , R 2 ) R 2 = R 1 R 2 = R 1 + R 2 1 + R ¯ 1 R 2
R 1 , t 1 t 1 R 2 , - t 1 t 1 R ¯ 1 R 2 2 , ( - 1 ) 2 t 1 t 1 R ¯ 1 2 R 3 2 , , ( - 1 ) n t 1 t 1 R ¯ 1 n R 2 n + 1 .
R = R 1 + R 2 ,
R = R 1 R 2 = R 1 + R 2 1 + R ¯ 1 R 2 .
ψ = φ 1 + φ 2 1 + φ ¯ 1 φ 2
V 2 = V 1 V 2 = V 1 + V 2 1 + V 1 V 2 ,
V n = V 1 ( V 2 { V 3 [ ( V n - 1 V n ) ] } ) .
V n = m 0 P 2 m + 1 n m 0 P 2 m n
V 1 = V 1 = P 1 1 P 0 1 m 0 P 2 m + 1 1 m 0 P 2 m 1 .
V 2 = V 1 + V 2 1 + V 1 V 2 = S 1 2 S 0 2 + S 2 2 m 0 P 2 m + 1 2 m 0 P 2 m 2
V 2 [ V 3 ( V n ) ] = m 0 P 2 m + 1 n m 0 P 2 m n
V n = V 1 + ( m 0 P 2 m + 1 n / m 0 P 2 m n ) 1 + V 1 ( m 0 P 2 m + 1 n / m 0 P 2 m n ) .

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