Abstract

A new mathematical formalism, based on a coupled-mode approach, is presented to describe light propagation in birefringent optically active crystals. An arbitrary light wave in the medium is represented as an expansion in terms of linearly polarized eigenmodes; explicit solutions of the equations for the coupled-wave amplitudes show how power is transferred between the principal components in the presence of optical activity. The (generally elliptically polarized) eigenmodes of the optically active crystal are also derived by a diagonalization of the coupled-mode equations. The analysis is extended to include electro-optic crystals, yielding a more general, complex coupling parameter. The predictions of the theory are in excellent agreement with spectral-transmission measurements in AgGaS2 near its isoindex point at 497 nm.

© 1982 Optical Society of America

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References

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  1. G. Szivessy, “Kristaloptik,” in Handbuck der Physik, H. Geiger and K. Sheel, eds. (Springer-Verlag, Berlin, 1928), Vol. 22, pp. 635–904.
  2. M. Born, Optik (Springer-Verlag, Berlin, 1933), pp. 413–420.
  3. E. U. Condon, “Theories of optical rotary power,” Rev. Mod. Phys. 9, 432–457 (1937).
    [Crossref]
  4. G. N. Ramachandran and S. Ramaseshan, “Crystal optics,” in Handbuck der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1961), Vol. 25, No. 1, pp. 76–85, 97–106.
  5. J. P. Mathieu, “Activité optique naturelle,” in Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1961), Vol. 28, pp. 333–431.
  6. J. F. Nye, Physical Properties of Crystals (Clarendon, Oxford, 1976), Chap. 14.
  7. A. Yariv, “Coupled-mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
    [Crossref]
  8. I. P. Kaminow, An Introduction to Electrooptic Devices (Academic, New York, 1974), p. 56.
  9. J. F. Lotspeich, “Iso-index coupled-wave electrooptic filter,” IEEE J. Quantum Electron. QE-15, 904–907 (1979).
    [Crossref]
  10. M. V. Hobden, “Optical activity in a non-enantiomorphous crystal: AgGaS2,” Acta Crystallogr. Sect. A 24, 676–680 (1968).
    [Crossref]
  11. G. D. Boyd, H. Kasper, and J. H. McFee, “Linear and nonlinear optical properties of AgGaS2, CuGaS2, and CuInS2, and theory of the wedge technique for the measurement of nonlinear coefficients,” IEEE J. Quantum Electron. QE-7, 563–573 (1971).
    [Crossref]

1979 (1)

J. F. Lotspeich, “Iso-index coupled-wave electrooptic filter,” IEEE J. Quantum Electron. QE-15, 904–907 (1979).
[Crossref]

1973 (1)

A. Yariv, “Coupled-mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[Crossref]

1971 (1)

G. D. Boyd, H. Kasper, and J. H. McFee, “Linear and nonlinear optical properties of AgGaS2, CuGaS2, and CuInS2, and theory of the wedge technique for the measurement of nonlinear coefficients,” IEEE J. Quantum Electron. QE-7, 563–573 (1971).
[Crossref]

1968 (1)

M. V. Hobden, “Optical activity in a non-enantiomorphous crystal: AgGaS2,” Acta Crystallogr. Sect. A 24, 676–680 (1968).
[Crossref]

1937 (1)

E. U. Condon, “Theories of optical rotary power,” Rev. Mod. Phys. 9, 432–457 (1937).
[Crossref]

Born, M.

M. Born, Optik (Springer-Verlag, Berlin, 1933), pp. 413–420.

Boyd, G. D.

G. D. Boyd, H. Kasper, and J. H. McFee, “Linear and nonlinear optical properties of AgGaS2, CuGaS2, and CuInS2, and theory of the wedge technique for the measurement of nonlinear coefficients,” IEEE J. Quantum Electron. QE-7, 563–573 (1971).
[Crossref]

Condon, E. U.

E. U. Condon, “Theories of optical rotary power,” Rev. Mod. Phys. 9, 432–457 (1937).
[Crossref]

Hobden, M. V.

M. V. Hobden, “Optical activity in a non-enantiomorphous crystal: AgGaS2,” Acta Crystallogr. Sect. A 24, 676–680 (1968).
[Crossref]

Kaminow, I. P.

I. P. Kaminow, An Introduction to Electrooptic Devices (Academic, New York, 1974), p. 56.

Kasper, H.

G. D. Boyd, H. Kasper, and J. H. McFee, “Linear and nonlinear optical properties of AgGaS2, CuGaS2, and CuInS2, and theory of the wedge technique for the measurement of nonlinear coefficients,” IEEE J. Quantum Electron. QE-7, 563–573 (1971).
[Crossref]

Lotspeich, J. F.

J. F. Lotspeich, “Iso-index coupled-wave electrooptic filter,” IEEE J. Quantum Electron. QE-15, 904–907 (1979).
[Crossref]

Mathieu, J. P.

J. P. Mathieu, “Activité optique naturelle,” in Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1961), Vol. 28, pp. 333–431.

McFee, J. H.

G. D. Boyd, H. Kasper, and J. H. McFee, “Linear and nonlinear optical properties of AgGaS2, CuGaS2, and CuInS2, and theory of the wedge technique for the measurement of nonlinear coefficients,” IEEE J. Quantum Electron. QE-7, 563–573 (1971).
[Crossref]

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Clarendon, Oxford, 1976), Chap. 14.

Ramachandran, G. N.

G. N. Ramachandran and S. Ramaseshan, “Crystal optics,” in Handbuck der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1961), Vol. 25, No. 1, pp. 76–85, 97–106.

Ramaseshan, S.

G. N. Ramachandran and S. Ramaseshan, “Crystal optics,” in Handbuck der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1961), Vol. 25, No. 1, pp. 76–85, 97–106.

Szivessy, G.

G. Szivessy, “Kristaloptik,” in Handbuck der Physik, H. Geiger and K. Sheel, eds. (Springer-Verlag, Berlin, 1928), Vol. 22, pp. 635–904.

Yariv, A.

A. Yariv, “Coupled-mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[Crossref]

Acta Crystallogr. Sect. A (1)

M. V. Hobden, “Optical activity in a non-enantiomorphous crystal: AgGaS2,” Acta Crystallogr. Sect. A 24, 676–680 (1968).
[Crossref]

IEEE J. Quantum Electron. (3)

G. D. Boyd, H. Kasper, and J. H. McFee, “Linear and nonlinear optical properties of AgGaS2, CuGaS2, and CuInS2, and theory of the wedge technique for the measurement of nonlinear coefficients,” IEEE J. Quantum Electron. QE-7, 563–573 (1971).
[Crossref]

J. F. Lotspeich, “Iso-index coupled-wave electrooptic filter,” IEEE J. Quantum Electron. QE-15, 904–907 (1979).
[Crossref]

A. Yariv, “Coupled-mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[Crossref]

Rev. Mod. Phys. (1)

E. U. Condon, “Theories of optical rotary power,” Rev. Mod. Phys. 9, 432–457 (1937).
[Crossref]

Other (6)

G. N. Ramachandran and S. Ramaseshan, “Crystal optics,” in Handbuck der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1961), Vol. 25, No. 1, pp. 76–85, 97–106.

J. P. Mathieu, “Activité optique naturelle,” in Handbuch der Physik, S. Flügge, ed. (Springer-Verlag, Berlin, 1961), Vol. 28, pp. 333–431.

J. F. Nye, Physical Properties of Crystals (Clarendon, Oxford, 1976), Chap. 14.

I. P. Kaminow, An Introduction to Electrooptic Devices (Academic, New York, 1974), p. 56.

G. Szivessy, “Kristaloptik,” in Handbuck der Physik, H. Geiger and K. Sheel, eds. (Springer-Verlag, Berlin, 1928), Vol. 22, pp. 635–904.

M. Born, Optik (Springer-Verlag, Berlin, 1933), pp. 413–420.

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Figures (6)

Fig. 1
Fig. 1

Coordinate axis systems used to define crystal orientation and wave-propagation vector in optically active uniaxial crystal.

Fig. 2
Fig. 2

Experimental arrangement of AgGaS2 crystal elements for measurement of spectral-transmittance characteristics.

Fig. 3
Fig. 3

Sample spectrophotometer traces of transmittance of a 1.48-mm sample of (110) AgGaS2 between crossed polarizers for collimated light at selected external angles of incidence in the (110) plane. (Configuration of Fig. 2).

Fig. 4
Fig. 4

Additional spectrophotometer traces as in Fig. 3, including corresponding computer simulations based on analytical model.

Fig. 5
Fig. 5

Two-element antiparallel configuration for illustrating cancellation of optical coupling through optical activity in AgGaS2 at the isoindex wavelength.

Fig. 6
Fig. 6

Examples of spectrophotometer traces for the two-element antiparallel configuration of Fig. 5, together with corresponding computer simulations. Angles shown are external angles of incidence.

Equations (34)

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k η = ω c n η
k ξ = ω c n ξ ,
n η - 2 = n e - 2 cos 2 θ + n 0 - 2 sin 2 θ , n ξ = n 0 .
| P ξ P η | = ɛ 0 | n ξ 2 - 1 i a - i a n η 2 - 1 | | E ξ E η | .
E ( ξ , η , ζ ) = ξ ^ E ξ ( ζ ) exp [ i ( ω t - k ξ ζ ) ] + η ^ E η ( ζ ) exp [ i ( ω t - k η ζ ) ] + c . c .
× × E = = - μ 0 2 t 2 ( ɛ 0 E + P ) .
d E ξ d ζ = k 0 2 a 2 k ξ E η exp [ - i ( k ξ - k η ) ζ ] , d E η d ζ = - k 0 2 a 2 k η E ξ exp [ i ( k ξ - k η ) ζ ] ,             k 0 = ω ( μ 0 ɛ 0 ) 1 / 2 ,
| d 2 E ξ , η d ζ 2 | k 2 ξ , η E ξ , η .
E η E η n η ,             E ξ E ξ n ξ ,
d E ξ d ζ = k ξ η E η exp [ - i ( k ξ - k η ) ζ ] , d E η d ζ = - κ ξ η E ξ exp [ i ( k ξ - k η ) ζ ] ,
κ ξ η = k 0 a 2 ( n ξ n η ) 1 / 2 .
E 1 = | E ξ ( ζ ) E η ( ζ ) | 1 = | E ξ ( ζ ) e - i k ξ ζ E η ( ζ ) e - i k η ζ | 1 = | i κ * δ + ( κ 2 + δ 2 ) 1 / 2 1 | exp { - i [ k - ( κ 2 + δ 2 ) 1 / 2 ] ζ } , E 2 = | E ξ ( ζ ) E η ( ζ ) | 2 = | E ξ ( ζ ) e - i k ξ ζ E η ( ζ ) e - i k η ζ | 2 = | i κ * δ + ( κ 2 + δ 2 ) 1 / 2 1 | exp { - i [ k + ( κ 2 + δ 2 ) 1 / 2 ] ζ } ,
k = ½ ( k η + k ξ ) , κ 2 = κ ξ η 2 ,             δ = ½ ( k η - k ξ ) .
E 1 = | i 1 | exp [ - i ( k - κ ) ζ ] , E 2 = | - i 1 | exp [ - i ( k + κ ) ζ ] ,
E 1 = | 0 1 | e - i k ξ ζ , E 2 = | 1 0 | e - i k η ζ ,
E ξ ( ζ ) = E ξ ( ζ ) 1 + E ξ ( ζ ) 2 .
E ξ = E 0 κ ξ η e - i δ ζ ( κ 2 + δ 2 ) 1 / 2 sin [ ( κ 2 + δ 2 ) 1 / 2 ] ζ , E η = E 0 e i δ ζ { cos [ ( κ 2 + δ 2 ) 1 / 2 ] ζ - i δ ( κ 2 + δ 2 ) 1 / 2 sin [ ( κ 2 + δ 2 ) 1 / 2 ] ζ } .
Δ ɛ ξ η = - ɛ 0 n ξ 2 n η 2 r ξ η k E k ,
| P ξ P η | = ɛ 0 | n ξ 2 - 1 i a - n ξ 2 n η 2 r ξ η k E k - i a - n ξ 2 n η 2 r ξ η k E k n η 2 - 1 | | E ξ E η | ,
κ ξ η κ + i Γ ,             κ η ξ - κ + i Γ ,
κ = k 0 a 2 ( n ξ n η ) 1 / 2 .
Γ = ½ k 0 ( n ξ n η ) 3 / 2 r ξ η k E k .
E ξ ( ζ ) = E 0 ( κ + i Γ ) e - i δ ζ ( κ 2 + Γ 2 + δ 2 ) 1 / 2 sin [ ( κ 2 + Γ 2 + δ 2 ) 1 / 2 ] ζ , E η ( ζ ) = E o e i δ ζ { cos [ ( κ 2 + Γ 2 + δ 2 ) 1 / 2 ] ζ - i δ sin [ ( κ 2 + Γ 2 + δ 2 ) 1 / 2 ] ζ ( κ 2 + Γ 2 + δ 2 ) 1 / 2 } ,
κ = i j ρ i j l i l j ,
ρ 11 = - ρ 22 = ρ ,
κ = ρ cos 2 θ sin 2 ψ ,
δ = π λ ( n e - n 0 ) cos 3 θ cos ψ = π λ C ( λ 0 - λ ) cos 3 θ cos ψ ,
n η - n ξ = ( n e - n 0 ) cos 2 θ ,
| E ξ ( ζ ) E η ( ζ ) | = | G 11 G 12 G 21 G 22 | | E ξ ( 0 ) E η ( 0 ) |
G 11 = cos S ζ + i ( δ S ) sin S ζ , G 12 = ( κ + i Γ S ) sin S ζ , G 21 = ( - κ + i Γ S ) sin S ζ , G 22 = cos S ζ - i ( δ S ) sin S ζ ,
S ( κ 2 + Γ 2 + δ 2 ) 1 / 2 ,
τ = E ξ ( 2 L ) 2 = G 12 ( 2 L ) 2 = κ 2 sin 2 [ 2 L ( κ 2 + δ 2 ) 1 / 2 ] κ 2 + δ 2 .
λ 0 = 0.497 μ m , ρ = 9.1 × 10 - 3 rad / μ m ( = 522° mm - 1 ) , L = 740 μ m , C = 1.87 μ m - 1 .
G = G ( - κ , L ) · G ( κ , L )