Abstract

Perturbation theory for the plane-wave eigenvector equation of crystal optics is developed. General expressions for the refractive-index changes that result from a general homogeneous dielectric perturbation are found for isotropic solids, uniaxial crystals, and biaxial crystals. The expressions may be applied directly to the electro-optic effect for any direction of light propagation, for either state of light polarization, and for any direction of electric-field application. They are also useful for the photoelastic effect when the elastic perturbation is homogeneous.

© 1975 Optical Society of America

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References

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  1. I. P. Kaminow and E. H. Turner, in Handbook of Lasers, edited by R. J. Pressley (Chemical Rubber Co., Cleveland, 1971), p. 447.
  2. B. V. Gisin, Kristallografiya 16, 151 (1971) [Sov. Phys. –Crystallography 16, 118 (1971)]; Kristallografiya 16, 638 (1971) [Sov. Phys. –Crystallography 16, 547 (1971)]; Kvantovaya Elektronika 1, 136 (1971) [Sov. J. Quantum Electron. 1, 686 (1972)].
  3. D. Kalymnios, Electronics Lett. 6, 804 (1970).
    [Crossref]
  4. I. P. Kaminow and E. H. Turner, Appl. Opt. 5, 1612 (1956).
    [Crossref]
  5. M. Lax and D. F. Nelson, Phys. Rev. B 4, 3694 (1971).
    [Crossref]
  6. D. F. Nelson and E. H. Turner, J. Appl. Phys. 39, 3337 (1968).
    [Crossref]
  7. D. F. Nelson, M. Lax, and P. D. Lazay, Proceedings of the IUTAM Symposium on the Photoelastic Effect and Its Applications (Brussels, 1973).
  8. D. F. Nelson and M. Lax, Phys. Rev. Lett. 24, 379 (1970).
    [Crossref]
  9. D. F. Nelson and P. D. Lazay, Phys. Rev. Lett. 17, 1187 (1970).
    [Crossref]
  10. D. F. Nelson and M. Lax, Phys. Rev. B 3, 2778 (1971).
    [Crossref]
  11. J. Chapelle and L. Taurel, C. R. Acad. Sci. (Paris) 240, 743 (1955).
  12. We ignore the special degenerate case when propagation occurs exactly along the optic axes. Such directions are of little interest for use in optical modulators because in biaxial crystals conical refraction occurs.
  13. M. Mason and W. Weaver, The Electromagnetic Field (Dover, New York, 1929), p. 153.

1971 (3)

B. V. Gisin, Kristallografiya 16, 151 (1971) [Sov. Phys. –Crystallography 16, 118 (1971)]; Kristallografiya 16, 638 (1971) [Sov. Phys. –Crystallography 16, 547 (1971)]; Kvantovaya Elektronika 1, 136 (1971) [Sov. J. Quantum Electron. 1, 686 (1972)].

M. Lax and D. F. Nelson, Phys. Rev. B 4, 3694 (1971).
[Crossref]

D. F. Nelson and M. Lax, Phys. Rev. B 3, 2778 (1971).
[Crossref]

1970 (3)

D. F. Nelson and M. Lax, Phys. Rev. Lett. 24, 379 (1970).
[Crossref]

D. F. Nelson and P. D. Lazay, Phys. Rev. Lett. 17, 1187 (1970).
[Crossref]

D. Kalymnios, Electronics Lett. 6, 804 (1970).
[Crossref]

1968 (1)

D. F. Nelson and E. H. Turner, J. Appl. Phys. 39, 3337 (1968).
[Crossref]

1956 (1)

1955 (1)

J. Chapelle and L. Taurel, C. R. Acad. Sci. (Paris) 240, 743 (1955).

Chapelle, J.

J. Chapelle and L. Taurel, C. R. Acad. Sci. (Paris) 240, 743 (1955).

Gisin, B. V.

B. V. Gisin, Kristallografiya 16, 151 (1971) [Sov. Phys. –Crystallography 16, 118 (1971)]; Kristallografiya 16, 638 (1971) [Sov. Phys. –Crystallography 16, 547 (1971)]; Kvantovaya Elektronika 1, 136 (1971) [Sov. J. Quantum Electron. 1, 686 (1972)].

Kalymnios, D.

D. Kalymnios, Electronics Lett. 6, 804 (1970).
[Crossref]

Kaminow, I. P.

I. P. Kaminow and E. H. Turner, Appl. Opt. 5, 1612 (1956).
[Crossref]

I. P. Kaminow and E. H. Turner, in Handbook of Lasers, edited by R. J. Pressley (Chemical Rubber Co., Cleveland, 1971), p. 447.

Lax, M.

M. Lax and D. F. Nelson, Phys. Rev. B 4, 3694 (1971).
[Crossref]

D. F. Nelson and M. Lax, Phys. Rev. B 3, 2778 (1971).
[Crossref]

D. F. Nelson and M. Lax, Phys. Rev. Lett. 24, 379 (1970).
[Crossref]

D. F. Nelson, M. Lax, and P. D. Lazay, Proceedings of the IUTAM Symposium on the Photoelastic Effect and Its Applications (Brussels, 1973).

Lazay, P. D.

D. F. Nelson and P. D. Lazay, Phys. Rev. Lett. 17, 1187 (1970).
[Crossref]

D. F. Nelson, M. Lax, and P. D. Lazay, Proceedings of the IUTAM Symposium on the Photoelastic Effect and Its Applications (Brussels, 1973).

Mason, M.

M. Mason and W. Weaver, The Electromagnetic Field (Dover, New York, 1929), p. 153.

Nelson, D. F.

D. F. Nelson and M. Lax, Phys. Rev. B 3, 2778 (1971).
[Crossref]

M. Lax and D. F. Nelson, Phys. Rev. B 4, 3694 (1971).
[Crossref]

D. F. Nelson and P. D. Lazay, Phys. Rev. Lett. 17, 1187 (1970).
[Crossref]

D. F. Nelson and M. Lax, Phys. Rev. Lett. 24, 379 (1970).
[Crossref]

D. F. Nelson and E. H. Turner, J. Appl. Phys. 39, 3337 (1968).
[Crossref]

D. F. Nelson, M. Lax, and P. D. Lazay, Proceedings of the IUTAM Symposium on the Photoelastic Effect and Its Applications (Brussels, 1973).

Taurel, L.

J. Chapelle and L. Taurel, C. R. Acad. Sci. (Paris) 240, 743 (1955).

Turner, E. H.

D. F. Nelson and E. H. Turner, J. Appl. Phys. 39, 3337 (1968).
[Crossref]

I. P. Kaminow and E. H. Turner, Appl. Opt. 5, 1612 (1956).
[Crossref]

I. P. Kaminow and E. H. Turner, in Handbook of Lasers, edited by R. J. Pressley (Chemical Rubber Co., Cleveland, 1971), p. 447.

Weaver, W.

M. Mason and W. Weaver, The Electromagnetic Field (Dover, New York, 1929), p. 153.

Appl. Opt. (1)

C. R. Acad. Sci. (Paris) (1)

J. Chapelle and L. Taurel, C. R. Acad. Sci. (Paris) 240, 743 (1955).

Electronics Lett. (1)

D. Kalymnios, Electronics Lett. 6, 804 (1970).
[Crossref]

J. Appl. Phys. (1)

D. F. Nelson and E. H. Turner, J. Appl. Phys. 39, 3337 (1968).
[Crossref]

Kristallografiya (1)

B. V. Gisin, Kristallografiya 16, 151 (1971) [Sov. Phys. –Crystallography 16, 118 (1971)]; Kristallografiya 16, 638 (1971) [Sov. Phys. –Crystallography 16, 547 (1971)]; Kvantovaya Elektronika 1, 136 (1971) [Sov. J. Quantum Electron. 1, 686 (1972)].

Phys. Rev. B (2)

M. Lax and D. F. Nelson, Phys. Rev. B 4, 3694 (1971).
[Crossref]

D. F. Nelson and M. Lax, Phys. Rev. B 3, 2778 (1971).
[Crossref]

Phys. Rev. Lett. (2)

D. F. Nelson and M. Lax, Phys. Rev. Lett. 24, 379 (1970).
[Crossref]

D. F. Nelson and P. D. Lazay, Phys. Rev. Lett. 17, 1187 (1970).
[Crossref]

Other (4)

D. F. Nelson, M. Lax, and P. D. Lazay, Proceedings of the IUTAM Symposium on the Photoelastic Effect and Its Applications (Brussels, 1973).

I. P. Kaminow and E. H. Turner, in Handbook of Lasers, edited by R. J. Pressley (Chemical Rubber Co., Cleveland, 1971), p. 447.

We ignore the special degenerate case when propagation occurs exactly along the optic axes. Such directions are of little interest for use in optical modulators because in biaxial crystals conical refraction occurs.

M. Mason and W. Weaver, The Electromagnetic Field (Dover, New York, 1929), p. 153.

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Figures (1)

FIG. 1
FIG. 1

Uniform normal traction (or pressure) of − n p is applied to the top surface of an infinite slab of a piezoelectric crystal whose bottom surface rests on a stationary plane. The paralellograms represent a volume element of the slab in the undeformed (A), strained-without-rotation (B), and deformed (C) configurations. In configuration B, the strains S13 = S31 are indicated.

Equations (110)

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( κ - 1 + Δ κ - 1 ) · l = ( 1 / n 2 ) l ,
( Δ κ - 1 ) i j = r i j k S E k ,
( Δ κ - 1 ) i j = r i j k T E k ,
r i j k T = r i j k S + p i j m n d k m n .
( Δ κ - 1 ) i j = ( p i j k l + δ i [ k κ l ] j - 1 + δ j [ k κ l ] i - 1 - r i j m S n m n n e n k l 0 n · κ · n ) u k , l .
e n k l s k l i j d n i j ,
( Δ κ - 1 ) i j = [ p i j k l s ¯ k l m n + 2 n k κ k ( i - 1 s ¯ j ) l m n n l - 2 n ( i κ j ) k - 1 s ¯ k l m n n l - r i j k S n k n l d l m n 0 n p κ p q L n q ] t m n ,
s ¯ k l m n s k l m n - n r d r k l n s d s i j 0 n · κ L · n ,
κ p q L κ p q + e p r s d q r s / 0 .
× ( × E ) + 1 c 2 κ · 2 E t 2 = 0 ,
E = 0 α ( s , ω ) e i ( k 0 α s · z - ω t ) ,
( 1 - s s ) · 0 α = 1 ( n 0 α ) 2 κ · 0 α ,
n 0 α = n 0 α ( s , ω ) k 0 α c / ω .
D 0 α κ · 0 α .
( 1 - s s ) · κ - 1 · D 0 α = 1 ( n 0 α ) 2 D 0 α .
D 0 α · 0 β = δ α β .
n 0 ( 1 ) = κ 1 / 2 n 0 ,             0 ( 1 ) = e n 0 ,             D 0 ( 1 ) = n 0 e ,
n 0 ( 2 ) = κ 1 / 2 n 0 ,             0 ( 2 ) = s × e n 0 ,             D 0 ( 2 ) = n 0 ( s × e ) ,
n 0 ( l ) = ,             0 ( l ) = s n 0 ,             D 0 ( l ) = n 0 s ,
e · s = 0.
n 0 ( o ) = ( κ 11 ) 1 / 2 ,             0 ( o ) = s × c N ( o ) ,             D 0 ( o ) = ( n 0 ( o ) ) 2 s × c N ( o ) ,
N ( o ) n 0 ( o ) [ 1 - ( s · c ) 2 ] 1 / 2
n 0 ( e ) = ( κ 11 κ 33 κ 11 [ 1 - ( s · c ) 2 ] + κ 33 ( s · c ) 2 ) 1 / 2 , 0 ( e ) = κ - 1 · [ s × ( s × c ) ] N ( e ) ,             D 0 ( e ) = s × ( s × c ) N ( e ) ,
N ( e ) 1 n 0 ( e ) [ 1 - ( s · c ) 2 ] 1 / 2 .
n 0 ( l ) = ,             0 ( l ) = s ( s · κ · s ) 1 / 2 ,             D 0 ( l ) = κ · s ( s · κ · s ) 1 / 2 .
n 0 ( ± ) = ( κ 11 κ 33 κ 11 sin 2 1 2 ( θ 1 ± θ 2 ) + κ 33 cos 2 1 2 ( θ 1 ± θ 2 ) ) 1 / 2 , 0 i ( ± ) = s i [ ( n 0 ( ± ) ) 2 - κ i i ] N ( ± ) ,             D 0 i ( ± ) = κ i i s i [ ( n 0 ( ± ) ) 2 - κ i i ] N ( ± ) , N ( ± ) [ i = 1 3 S i 2 κ i i [ ( n 0 ( ± ) ) 2 - κ i i ] 2 ] 1 / 2 ,
a 1 = [ sin β , 0 , cos β ] ,             a 2 = [ - sin β , 0 , cos β ] .
tan β = [ κ 11 - 1 - κ 22 - 1 κ 22 - 1 - κ 33 - 1 ] 1 / 2 .
n 0 ( l ) = ,             0 ( l ) = s ( s · κ · s ) 1 / 2 ,             D 0 ( l ) = κ · s ( s · κ · s ) 1 / 2 .
( κ + Δ κ ) - 1 = κ - 1 + Δ κ - 1
Δ κ - 1 = - κ - 1 · Δ κ · κ - 1 .
( 1 - s s ) · ( κ - 1 + Δ κ - 1 ) · D α = 1 ( n α ) 2 D α .
α = ( κ + Δ κ ) - 1 · D α = ( κ - 1 + Δ κ - 1 ) · D α
α · D β = δ α β .
Γ ( 1 - s s ) · κ - 1 ,
Δ Γ ( 1 - s s ) · Δ κ - 1 ,
γ α 1 / ( n α ) 2 .
f α = n = 0 λ n f n α ,
n λ n ( Γ + λ Δ Γ ) · D n α = n , m λ m + n γ m α D n α .
Γ · D 1 α + Δ Γ · D 0 α = γ 0 α D 1 α + γ 1 α D 0 α .
D 1 α = β = 1 3 a β α D 0 β .
0 β · Δ Γ · D 0 α = a β α ( γ 0 α - γ 0 β ) + γ 1 α δ α β .
γ 1 α = 0 α · Δ Γ · D 0 α = 0 α · ( 1 - s s ) · Δ κ - 1 · D 0 α = γ 0 α D 0 α · Δ κ - 1 · D 0 α
a β α = 0 β · Δ Γ · D 0 α γ 0 α - γ 0 β = γ 0 β D 0 β · Δ κ - 1 · D 0 α γ 0 α - γ 0 β
m , n λ m + n D m α · ( κ - 1 + λ Δ κ - 1 ) · D n α = 1 ,
D 0 α · Δ κ - 1 · D 0 α + 2 D 1 α · κ - 1 · D 0 α = 0.
a α α = - 1 2 D 0 α · Δ κ - 1 · D 0 α
D α = ( 1 - 1 2 D 0 α · Δ κ - 1 · D 0 α ) D 0 α + γ = 1 , γ α 3 D 0 γ · Δ κ - 1 · D 0 α ( n 0 γ / n 0 α ) 2 - 1 D 0 γ .
1 ( n α ) 2 = 1 ( n 0 α ) 2 + D 0 α · Δ κ - 1 · D 0 α ( n 0 α ) 2
Δ n α n α - n 0 α = - 1 2 n 0 α D 0 α · Δ κ - 1 · D 0 α ,
D 0 a γ = 1 m S γ a D 0 γ .
D 1 a = b = 1 m a b a D 0 b .
Γ · b a b a D 0 b + Δ Γ · D 0 a = γ 0 a b a b a D 0 b + γ 1 a D 0 a .
Γ · D 0 a = γ 0 a D 0 a ,
0 β · D 0 a = S β a
b a b a S β b γ 0 b + 0 β · Δ Γ · D 0 a = γ 0 a b a b a S β b + γ 1 a S β a .
γ = 1 m S γ a [ γ 0 β D 0 β · Δ κ - 1 · D 0 γ - γ 1 a δ β γ ] = 0
det [ γ 0 β D 0 β · Δ κ - 1 · D 0 γ - γ 1 a δ β γ ] = 0 ,
D 0 a · 0 a = 1.
a a b + a b a = - D 0 a · Δ κ - 1 · D 0 b .
a a b = a b a = - 1 2 D 0 a · Δ κ - 1 · D 0 b .
S 1 a ( Δ 11 - γ 1 a ) + S 2 a Δ 12 = 0 ,
S 1 a Δ 12 + S 2 a ( Δ 22 - γ 1 a ) = 0 ,
Δ 11 1 ( n 0 ) 2 D 0 1 · Δ κ - 1 · D 0 1 = e · Δ κ - 1 · e ,
Δ 12 1 ( n 0 ) 2 D 0 1 · Δ κ - 1 · D 0 2 = e · Δ κ - 1 · ( s × e ) ,
Δ 22 1 ( n 0 ) 2 D 0 2 · Δ κ - 1 · D 0 2 = ( s × e ) · Δ κ - 1 · ( s × e ) .
γ 1 ( ± ) = 1 2 { Δ 11 + Δ 22 ± [ ( Δ 11 - Δ 22 ) 2 + 4 Δ 12 2 ] 1 / 2 } ,
S 1 ± = - Δ 12 S 2 ± Δ 11 - γ 1 ±
0 ( ± ) · D 0 ( ± ) = 1 ,
S 2 ± = [ Δ 12 2 ( Δ 11 - γ 1 ± ) 2 ] - 1 / 2 .
D 0 ( ± ) = [ 1 + Δ 12 2 ( Δ 11 - γ 1 ± ) 2 ] - 1 / 2 [ - Δ 12 Δ 11 - γ 1 ± D 0 1 + D 0 2 ] .
γ 1 ( ± ) = 1 ( n 0 ) 2 D 0 ( ± ) · Δ κ - 1 · D 0 ( ± ) ,
Δ n ( ± ) = - 1 4 ( n 0 ) 3 { Δ 11 + Δ 22 ± [ ( Δ 11 - Δ 22 ) 2 + 4 Δ 12 2 ] 1 / 2 }
0 ( ± ) · Δ Γ · D 0 ( ) = 0 ,
0 ( 1 ) ( ) · Δ Γ · D 0 ( 2 ) ( ) = 0 ( 2 ) ( ) · Δ Γ · D 0 ( 1 ) ( ) = 0 ,
· Δ κ - 1 · ( s × ) = 0 ,
Δ n + = - 1 2 ( n 0 ) 3 Δ - - = - 1 2 ( n 0 ) 3 · Δ κ - 1 · ,
Δ n - = - 1 2 ( n 0 ) 3 Δ + + = - 1 2 ( n 0 ) 3 ( s × ) · Δ κ - 1 · ( s × ) ,
D 0 ( + ) = n 0 ,
D ( - ) = n 0 ( s × ) .
Δ n ( o ) = - ( n 0 ( o ) ) 5 2 ( N ( o ) ) 2 ( s × c ) · Δ κ - 1 · ( s × c ) = - ( n 0 ( o ) ) 3 2 [ s 2 2 ( Δ κ - 1 ) 11 + s 1 2 ( Δ κ - 1 ) 22 - 2 s 1 s 2 ( Δ κ - 1 ) 12 s 1 2 + s 2 2 ]
Δ n ( e ) = - n 0 ( e ) 2 ( N ( e ) ) 2 [ s × ( s × c ) ] · Δ κ - 1 · [ s × ( s × c ) ] = - ( n 0 ( e ) ) 3 2 1 s 1 2 + s 2 2 [ s 1 2 s 3 2 ( Δ κ - 1 ) 11 + s 2 2 s 3 2 ( Δ κ - 1 ) 22 + ( s 1 2 + s 2 2 ) 2 ( Δ κ - 1 ) 33 + 2 s 1 s 2 s 3 2 ( Δ κ - 1 ) 12 - 2 s 2 s 3 ( s 1 2 + s 2 2 ) ( Δ κ - 1 ) 23 - 2 s 1 s 3 ( s 1 2 + s 2 2 ) ( Δ κ - 1 ) 13 ]
s = [ sin θ cos φ , sin θ sin φ , cos θ ] .
Δ n ( o ) = - 1 2 ( n 0 ( o ) ) 3 [ ( Δ κ - 1 ) 11 sin 2 φ + ( Δ κ - 1 ) 22 cos 2 φ - ( Δ κ - 1 ) 12 sin 2 φ ]
Δ n ( e ) = - 1 2 ( n 0 ( e ) ) 3 [ ( Δ κ - 1 ) 11 cos 2 θ cos 2 φ + ( Δ κ - 1 ) 22 cos 2 θ sin 2 φ + ( Δ κ - 1 ) 33 sin 2 θ + ( Δ κ - 1 ) 12 cos 2 θ sin 2 φ - ( Δ κ - 1 ) 23 sin 2 θ sin φ - ( Δ κ - 1 ) 13 sin 2 θ cos φ ] .
D 0 ( o ) = ( n 0 ( o ) ) 2 0 ( o ) = n 0 ( o ) [ sin φ , - cos φ , 0 ] ,
D 0 ( e ) = ( n 0 ( o ) ) 2 0 ( e ) = n 0 ( o ) [ cos φ , sin φ , 0 ] .
D 0 ( o ) · Δ κ - 1 · D 0 ( e ) = 0.
tan 2 φ = 2 ( Δ κ - 1 ) 12 ( Δ κ - 1 ) 11 - ( Δ κ - 1 ) 22 .
cos 2 φ = 1 2 [ 1 + 1 ( 1 + tan 2 2 φ ) 1 / 2 ] ,
sin 2 φ = 1 2 [ 1 - 1 ( 1 + tan 2 2 φ ) 1 / 2 ] ,
Δ n ( e o ) = 1 4 ( n 0 ( o ) ) 3 { ( Δ κ - 1 ) 11 + ( Δ κ - 1 ) 22 ± [ ( ( Δ κ - 1 ) 11 - ( Δ κ - 1 ) 22 ) 2 + 4 ( ( Δ κ - 1 ) 12 ) 2 ] 1 / 2 } ,
Δ n ( ± ) = - n 0 ( ± ) 2 ( N ( ± ) ) 2 i , j = 1 3 s i κ i i ( Δ κ - 1 ) i j s j κ j j [ ( n 0 ( ± ) ) 2 - κ i i ] [ ( n 0 ( ± ) ) 2 - κ j j ]
( Δ κ - 1 ) i j = ( p i j k l + δ i [ k κ l ] j - 1 + δ j [ k κ l ] i - 1 ) u k , l + γ i j k S E k .
t = - n n p ,
t i j = c i j k l S k l - e k i j E k ,
c i j k l s k l m n = s i j k l c k l m n = 1 2 ( δ i m δ j n + δ i n δ j m ) .
E = - 1 0 n n · P ,
P i = 0 ( κ i j - δ i j ) E j + e i j k S j k ,
E k = - n k n l d l i j t i j 0 n p κ p q L n q .
S m n = s ¯ m n i j t i j ,
u m , n = S m n + Ω m n .
u 1 , 3 = 2 S 13 ,             u 2 , 3 = 2 S 23 ,
u 3 , 1 = 0 ,             u 3 , 2 = 0
u i , j = S i j
n l u [ l , n ] n m - n l u [ l , m ] n n + 2 n l S l [ n n m ] = 0.
u [ l , m ] = α l m p n p + β l m p t p + γ l m p ( n × t ) p ,
n l u [ l , n ] n m - n l u [ l , m ] n n = 2 β ( n × t ) [ m n n ] + 2 γ [ n × ( n × t ) ] [ m n n ] = 2 β ( n × t ) [ m n n ] + 2 γ n [ m t n ] = β m n p t p + γ m n p ( n × t ) p .
Ω m n = n l u [ l , n ] n m - n l u [ l , m ] n n = 2 n l S l [ m n n ] . = n l ( s ¯ l m i j n n - s ¯ l n i j n m ) t i j ,
( Δ κ - 1 ) i j = [ p i j k l s ¯ k l m n + 2 n k κ k - 1 ( i s ¯ j ) l m n n l - 2 n ( i κ j ) - 1 k s ¯ k l m n n l - r i j k S n k n l d l m n 0 n p κ p q L n q ] t m n .