Abstract

A method of calculating 2×2 reflection and transmission-coefficient matrices for a multilayer birefringent system, developed for use in a plane stratified magnetoplasma, is applied to an optical system. The diagonal matrix elements refer to the direct reflection and transmission coefficients of the two characteristic or normal modes, and off-diagonal elements refer to inter-mode coupling between them. For waves normally incident on the plane stratified system, in the positive or negative z directions, corresponding reflection and transmission-coefficient matrices, R± and T±, are defined in terms of suitably normalized wave fields. A reciprocity theorem is established wherein it is shown that the matrices R+ and R are symmetric, and the matrix T+ is the transpose of T, provided that the dielectric tensors of the layers composing the system are either (a) all symmetric, or (b) all gyrotropic (magnetoactive), having a common principal axis that is generally the external magnetic-field direction. Optically inactive, isotropic layers may be included in either class, but care must be taken in defining characteristic modes for such layers. Some applications are discussed, including the feasibility of constructing light valves.

© 1971 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964).
  2. G. H. Price, Radio Sci. 68D, 407 (1964).
  3. D. O. Smith, Opt. Acta 12, 13 (1965); Opt. Acta 13, 195 (1966).
    [CrossRef]
  4. S. Teitler and B. W. Henvis, J. Opt. Soc. Am. 60, 830 (1970).
    [CrossRef]
  5. C. Altman and H. Cory, Alta Frequenza 38, (No. Spec.), 180 (1969).
  6. C. Altman and H. Cory, Radio Sci. 4, 459 (1969).
    [CrossRef]
  7. D. L. Caballero, J. Opt. Soc. Am. 37, 176 (1947).
    [CrossRef] [PubMed]
  8. A. Vašíček, Optics of Thin Films (North-Holland, Amsterdam, 1960).
  9. C. Altman and A. Postan, Radio Sci. 6, 483 (1971).
    [CrossRef]
  10. L. D. Landau and Ye. M. Lifshits, Electrodynamics of Continuous Media (Pergamon, New York, 1960).
  11. K. G. Budden, Radio Waves in the Ionosphere (Cambridge U. P., London, 1961).
  12. V. M. Agranovich and V. L. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons (Wiley-Interscience, New York, 1966).
  13. C. Altman and H. Cory, Radio Sci. 4, 449 (1969).
    [CrossRef]
  14. K. Rawer and K. Suchy, in Encyclopedia of Physics, Vol. 49/2, Geophysics III (Springer, Berlin, 1967).
  15. J. A. Ratcliffe, The Magneto-Ionic Theory (Cambridge U. P., London, 1959).
  16. V. L. Ginzburg, The Propagation of Electromagnetic Waves in Plasmas (Pergamon, New York, 1964).
  17. L. I. Mandel’shtam, Lectures on Selected Topics in Optics, in Complete Works, Vol. 5 (Izdat AN SSR, 1950).
  18. Rayleigh, Phil. Trans. 176, 343 (1885); Scientific Papers Vol. II (Dover, New York, 1964), p. 360.
    [CrossRef]
  19. Rayleigh, Nature 64, 577 (1901); Scientific Papers Vol. IV (Dover, New York, 1964), p. 555.
    [CrossRef]

1971 (1)

C. Altman and A. Postan, Radio Sci. 6, 483 (1971).
[CrossRef]

1970 (1)

1969 (3)

C. Altman and H. Cory, Radio Sci. 4, 449 (1969).
[CrossRef]

C. Altman and H. Cory, Alta Frequenza 38, (No. Spec.), 180 (1969).

C. Altman and H. Cory, Radio Sci. 4, 459 (1969).
[CrossRef]

1965 (1)

D. O. Smith, Opt. Acta 12, 13 (1965); Opt. Acta 13, 195 (1966).
[CrossRef]

1964 (1)

G. H. Price, Radio Sci. 68D, 407 (1964).

1947 (1)

1901 (1)

Rayleigh, Nature 64, 577 (1901); Scientific Papers Vol. IV (Dover, New York, 1964), p. 555.
[CrossRef]

1885 (1)

Rayleigh, Phil. Trans. 176, 343 (1885); Scientific Papers Vol. II (Dover, New York, 1964), p. 360.
[CrossRef]

Agranovich, V. M.

V. M. Agranovich and V. L. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons (Wiley-Interscience, New York, 1966).

Altman, C.

C. Altman and A. Postan, Radio Sci. 6, 483 (1971).
[CrossRef]

C. Altman and H. Cory, Radio Sci. 4, 449 (1969).
[CrossRef]

C. Altman and H. Cory, Alta Frequenza 38, (No. Spec.), 180 (1969).

C. Altman and H. Cory, Radio Sci. 4, 459 (1969).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Budden, K. G.

K. G. Budden, Radio Waves in the Ionosphere (Cambridge U. P., London, 1961).

Caballero, D. L.

Cory, H.

C. Altman and H. Cory, Radio Sci. 4, 459 (1969).
[CrossRef]

C. Altman and H. Cory, Alta Frequenza 38, (No. Spec.), 180 (1969).

C. Altman and H. Cory, Radio Sci. 4, 449 (1969).
[CrossRef]

Ginzburg, V. L.

V. L. Ginzburg, The Propagation of Electromagnetic Waves in Plasmas (Pergamon, New York, 1964).

V. M. Agranovich and V. L. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons (Wiley-Interscience, New York, 1966).

Henvis, B. W.

Landau, L. D.

L. D. Landau and Ye. M. Lifshits, Electrodynamics of Continuous Media (Pergamon, New York, 1960).

Lifshits, Ye. M.

L. D. Landau and Ye. M. Lifshits, Electrodynamics of Continuous Media (Pergamon, New York, 1960).

Mandel’shtam, L. I.

L. I. Mandel’shtam, Lectures on Selected Topics in Optics, in Complete Works, Vol. 5 (Izdat AN SSR, 1950).

Postan, A.

C. Altman and A. Postan, Radio Sci. 6, 483 (1971).
[CrossRef]

Price, G. H.

G. H. Price, Radio Sci. 68D, 407 (1964).

Ratcliffe, J. A.

J. A. Ratcliffe, The Magneto-Ionic Theory (Cambridge U. P., London, 1959).

Rawer, K.

K. Rawer and K. Suchy, in Encyclopedia of Physics, Vol. 49/2, Geophysics III (Springer, Berlin, 1967).

Rayleigh,

Rayleigh, Nature 64, 577 (1901); Scientific Papers Vol. IV (Dover, New York, 1964), p. 555.
[CrossRef]

Rayleigh, Phil. Trans. 176, 343 (1885); Scientific Papers Vol. II (Dover, New York, 1964), p. 360.
[CrossRef]

Smith, D. O.

D. O. Smith, Opt. Acta 12, 13 (1965); Opt. Acta 13, 195 (1966).
[CrossRef]

Suchy, K.

K. Rawer and K. Suchy, in Encyclopedia of Physics, Vol. 49/2, Geophysics III (Springer, Berlin, 1967).

Teitler, S.

Vašícek, A.

A. Vašíček, Optics of Thin Films (North-Holland, Amsterdam, 1960).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Alta Frequenza (1)

C. Altman and H. Cory, Alta Frequenza 38, (No. Spec.), 180 (1969).

J. Opt. Soc. Am. (2)

Nature (1)

Rayleigh, Nature 64, 577 (1901); Scientific Papers Vol. IV (Dover, New York, 1964), p. 555.
[CrossRef]

Opt. Acta (1)

D. O. Smith, Opt. Acta 12, 13 (1965); Opt. Acta 13, 195 (1966).
[CrossRef]

Phil. Trans. (1)

Rayleigh, Phil. Trans. 176, 343 (1885); Scientific Papers Vol. II (Dover, New York, 1964), p. 360.
[CrossRef]

Radio Sci. (4)

G. H. Price, Radio Sci. 68D, 407 (1964).

C. Altman and H. Cory, Radio Sci. 4, 459 (1969).
[CrossRef]

C. Altman and A. Postan, Radio Sci. 6, 483 (1971).
[CrossRef]

C. Altman and H. Cory, Radio Sci. 4, 449 (1969).
[CrossRef]

Other (9)

K. Rawer and K. Suchy, in Encyclopedia of Physics, Vol. 49/2, Geophysics III (Springer, Berlin, 1967).

J. A. Ratcliffe, The Magneto-Ionic Theory (Cambridge U. P., London, 1959).

V. L. Ginzburg, The Propagation of Electromagnetic Waves in Plasmas (Pergamon, New York, 1964).

L. I. Mandel’shtam, Lectures on Selected Topics in Optics, in Complete Works, Vol. 5 (Izdat AN SSR, 1950).

L. D. Landau and Ye. M. Lifshits, Electrodynamics of Continuous Media (Pergamon, New York, 1960).

K. G. Budden, Radio Waves in the Ionosphere (Cambridge U. P., London, 1961).

V. M. Agranovich and V. L. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons (Wiley-Interscience, New York, 1966).

A. Vašíček, Optics of Thin Films (North-Holland, Amsterdam, 1960).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964).

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Figures (2)

Fig. 1
Fig. 1

Schematic representation of transfer-matrix elements. Ê1± and Ê2± represent the normalized x components of the electric wave vectors in modes 1 and 2, respectively.

Fig. 2
Fig. 2

(a). Generation of equivalent interface matrices, and (b) schematic representation of the iteration procedure described in Eqs. (12)(15). Waves at normal incidence are drawn obliquely for clarity.

Equations (46)

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r ± = r ˜ ±
t + = t ˜ - ,
R ± = R ˜ ±
T + = T ˜ - ,
ɛ = ( 1 i 2 0 - i 2 1 0 0 0 3 ) .
S = ( 0 / 4 μ 0 ) 1 2 Re ( n ) ( 1 + ρ 2 ) E x 2 ,
ρ = E y / E x = D y / D x = - H x / H y ;
× H = D / t
D = ɛ E ,
E x ± ~ [ n ( 1 - ρ 2 ) ] - 1 2 exp ( i k z n d z ) .
Ê ± = [ n ( 1 + γ ρ 2 ) ] 1 2 E x ± ,             γ = ± 1
ɛ = ɛ R + σ / i ω ,
( Ê 1 , s - , Ê 2 , s - ) = ( Ê 1 , s + , Ê 2 , s + ) ( r 11 s , s + 1 r 12 s , s + 1 r 21 s , s + 1 r 22 s , s + 1 ) = ( Ê 1 , s + , Ê 2 , s + ) r + , ( Ê 1 , s + 1 + , Ê 2 , s + 1 + ) = ( Ê 1 , s + , Ê 2 , s + ) t + ,
( Ê 1 , s - 1 - , Ê 2 , s - 1 - ) = ( Ê 1 , s - 1 + , Ê 2 , s - 1 + ) R s + ( Ê 1 , s + 1 + , Ê 2 , s + 1 + ) = ( Ê 1 , s - 1 + , Ê 2 , s - 1 + ) T s + ,
R s + 1 + = R s + + T s + Δ r + Δ ( I - R s - Δ r + Δ ) - 1 T s -
T s + 1 + = T s + Δ ( I - r + Δ R s - Δ ) - 1 t +
R s + 1 - = r - + t - Δ R s - Δ ( I - r + Δ R s - Δ ) - 1 t +
T s + 1 - = t - Δ ( I - R s - Δ r + Δ ) - 1 T s - ;
Δ = ( exp ( - i n 1 k δ z ) 0 0 exp ( - i n 2 k δ z ) ) ,
D 1 · D 2 = 0 ,
ρ 1 ρ 2 = - 1.
ɛ = Λ ˜ ɛ Λ , Λ = ( cos θ 0 sin θ 0 1 0 - sin θ 0 cos θ ) ,
ρ 2 + c ρ + 1 = 0 ,
ρ 1 ρ 2 = 1.
D 1 , D 2 * = 0 ,
ρ 1 ρ 2 * = - 1.
ρ 1 ρ 2 = - γ ,
Ê 1 ± = ( n 1 ) 1 2 [ ( ρ 2 - ρ 1 ) / ρ 2 ] 1 2 E x 1 ± Ê 2 ± = ( n 2 ) 1 2 [ ( ρ 1 - ρ 2 ) / ρ 1 ] 1 2 E x 2 ±
r 12 s , s + 1 = ± 2 ( n 1 s + 1 - n 2 s + 1 ) ( n 1 s n 2 s ρ 1 s ρ 2 s ) 1 2 × ( δ ρ 1 + δ ρ 2 ) / C
r 21 s , s + 1 = 2 ( n 2 s + 1 - n 1 s + 1 ) ( n 1 s n 2 s ρ 1 s ρ 2 s ) 1 2 × ( δ ρ 1 + δ ρ 2 ) / C
t 11 s , s + 1 = 2 ( n 1 s n 1 s + 1 ) 1 2 ( n 2 s + n 2 s + 1 ) × ( ρ 2 s - ρ 1 s ) 1 2 ( ρ 2 s + 1 - ρ 1 s + 1 ) 1 2 × [ ( ρ 2 s ρ 2 s + 1 ) 1 2 - ( ρ 1 s ρ 1 s + 1 ) 1 2 ] / C
t 12 s , s + 1 = 2 ( n 1 s n 2 s + 1 ) 1 2 ( n 1 s + 1 + n 2 s ) × ( ρ 1 s - ρ 2 s ) 1 2 ( ρ 2 s + 1 - ρ 1 s + 1 ) 1 2 × [ ( ρ 2 s ρ 1 s + 1 ) 1 2 - ( ρ 1 s ρ 2 s + 1 ) 1 2 ] / C ;
Ê 1 , f + = Ê 1 , i + T 11 +             and             Ê 2 , f + = Ê 1 , i + T 12 + = 0 ,
T 12 + = 0.
Ê 1 , i - = Ê 1 , f - T 11 - = - Ê 1 , f + T 11 - = - Ê 1 , f + T 11 + Ê 2 , i - = Ê 1 , f - T 12 - = - Ê 1 , f + T 21 +
( Ê 1 , i - , Ê 2 , i - ) = - ( Ê 1 , i + , Ê 2 , i + ) T + T - = - ( Ê 1 , i + , Ê 2 , i + ) T + T ˜ + ,
T 11 + = a e i ϕ ,             T 22 + = a e i ( ϕ + 2 δ ) ,
T 11 + T 11 - = ( T 11 + ) 2 = a 2 e 2 i ϕ T 22 + T 22 - = a 2 e 2 i ( ϕ + 2 δ ) ,
E s = E s , 1 + E s , 2 = ( E s , 1 + + E s , 1 - ) + ( E s , 2 + + E s , 2 - ) E s + 1 = E s + 1 , 1 + E s + 1 , 2 .
E y + = ρ E x + E y - = ρ E x - z 0 H y + = n E x + z 0 H y - = - n E x - z 0 H x - = - ρ n E x + z 0 H x - = ρ n E x - ,
( E 1 , s - , E 2 , s - ) = ( E 1 , s + , E 2 , s + ) r ( x ) s , s + 1 ( E 1 , s + 1 + , E 2 , s + 1 + ) = ( E 1 , s + , E 2 , s + ) t ( x ) s , s + 1 .
C ( r ( x ) 11 s , s + 1 r ( x ) 12 s , s + 1 t ( x ) 11 s , s + 1 r ( x ) 12 s , s + 1 ) = ( - 1 - ρ 1 s n 1 s n 1 s ρ 1 s ) ,
C = ( 1 1 - 1 - 1 ρ 1 s ρ 2 s - ρ 1 s + 1 - ρ 2 s + 1 n 1 s n 2 s n 1 s + 1 n 2 s + 1 n 1 s ρ 1 s n 2 s ρ 2 s n 1 s + 1 ρ 1 s + 1 n 2 s + 1 ρ 2 s + 1 ) .
r ( x ) 12 s , s + 1 = 2 n 1 s ( n 2 s + 1 - n 1 s + 1 ) ( ρ 1 s + 1 - ρ 1 s ) × ( ρ 2 s + 1 - ρ 1 s ) / C t ( x ) 11 s , s + 1 = 2 n 1 s ( n 2 s - n 2 s + 1 ) ( ρ 2 s + 1 - ρ 1 s ) × ( ρ 2 s - ρ 1 s ) / C t ( x ) 12 s , s + 1 = 2 n 1 s ( n 1 s + 1 - n 2 s ) ( ρ 1 s - ρ 2 s ) ( ρ 1 s + 1 - ρ 1 s ) / C .
t 12 s , s + 1 = ( n 2 s + 1 n 1 s ) 1 2 ( ρ 1 s + 1 - ρ 2 s + 1 ρ 1 s + 1 ) 1 2 × ( ρ 2 s ρ 2 s - ρ 1 s ) 1 2 t ( x ) 12 s , s + 1 ,
ρ 1 s ρ 2 s = ρ 1 s + 1 ρ 2 s + 1 ( = ± 1 )