Abstract

Besides the Pockels effect, crystals of the dihydrogen phosphate type (symmetry class 4¯2 m) also exhibit an indirect electro-optic effect or piezo-optic effect. This effect is examined both experimentally and theoretically, and expressions for the spatial distribution, amplitude, and frequency response of modulation are derived for the case of a (zxt)0°-cut square plate. The frequency response is found to depend upon the light-beam diameter. The amplitude of the piezo-optic resonances is found to be nearly independent of frequency.

© 1965 Optical Society of America

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References

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  1. E. P. Tawil, Compt. Rend. 183, 1099 (1926).
  2. B. Zwicker and P. Scherrer, Helv. Phys. Acta 17, 346 (1944).
  3. B. H. Billings, J. Opt. Soc. Am. 39, 797, 802 (1949).
    [Crossref]
  4. D. Gainon, H. Jaffe, and T. Sliker, J. Appl. Phys. 35, 1166 (1964).
    [Crossref]
  5. R. O’B. Carpenter, J. Opt. Soc. Am. 40, 225 (1950).
    [Crossref]
  6. H. Ekstein, Phys. Rev. 66, 108 (1944).
    [Crossref]
  7. N. W. McLachlan, Theory of Vibrations (Dover Publications, Inc., New York, 1951), p. 29.
  8. H. Jaffe and J. F. Stephany, Sci. Am. 207, 156 (July1962).
    [Crossref]
  9. H. B. Huntington, “The Elastic Constants of Crystals” in Solid State Physics (Academic Press Inc., New York, 1958), Vol. 7.
    [Crossref]

1964 (1)

D. Gainon, H. Jaffe, and T. Sliker, J. Appl. Phys. 35, 1166 (1964).
[Crossref]

1962 (1)

H. Jaffe and J. F. Stephany, Sci. Am. 207, 156 (July1962).
[Crossref]

1950 (1)

1949 (1)

1944 (2)

H. Ekstein, Phys. Rev. 66, 108 (1944).
[Crossref]

B. Zwicker and P. Scherrer, Helv. Phys. Acta 17, 346 (1944).

1926 (1)

E. P. Tawil, Compt. Rend. 183, 1099 (1926).

Billings, B. H.

Carpenter, R. O’B.

Ekstein, H.

H. Ekstein, Phys. Rev. 66, 108 (1944).
[Crossref]

Gainon, D.

D. Gainon, H. Jaffe, and T. Sliker, J. Appl. Phys. 35, 1166 (1964).
[Crossref]

Huntington, H. B.

H. B. Huntington, “The Elastic Constants of Crystals” in Solid State Physics (Academic Press Inc., New York, 1958), Vol. 7.
[Crossref]

Jaffe, H.

D. Gainon, H. Jaffe, and T. Sliker, J. Appl. Phys. 35, 1166 (1964).
[Crossref]

H. Jaffe and J. F. Stephany, Sci. Am. 207, 156 (July1962).
[Crossref]

McLachlan, N. W.

N. W. McLachlan, Theory of Vibrations (Dover Publications, Inc., New York, 1951), p. 29.

Scherrer, P.

B. Zwicker and P. Scherrer, Helv. Phys. Acta 17, 346 (1944).

Sliker, T.

D. Gainon, H. Jaffe, and T. Sliker, J. Appl. Phys. 35, 1166 (1964).
[Crossref]

Stephany, J. F.

H. Jaffe and J. F. Stephany, Sci. Am. 207, 156 (July1962).
[Crossref]

Tawil, E. P.

E. P. Tawil, Compt. Rend. 183, 1099 (1926).

Zwicker, B.

B. Zwicker and P. Scherrer, Helv. Phys. Acta 17, 346 (1944).

Compt. Rend. (1)

E. P. Tawil, Compt. Rend. 183, 1099 (1926).

Helv. Phys. Acta (1)

B. Zwicker and P. Scherrer, Helv. Phys. Acta 17, 346 (1944).

J. Appl. Phys. (1)

D. Gainon, H. Jaffe, and T. Sliker, J. Appl. Phys. 35, 1166 (1964).
[Crossref]

J. Opt. Soc. Am. (2)

Phys. Rev. (1)

H. Ekstein, Phys. Rev. 66, 108 (1944).
[Crossref]

Sci. Am. (1)

H. Jaffe and J. F. Stephany, Sci. Am. 207, 156 (July1962).
[Crossref]

Other (2)

H. B. Huntington, “The Elastic Constants of Crystals” in Solid State Physics (Academic Press Inc., New York, 1958), Vol. 7.
[Crossref]

N. W. McLachlan, Theory of Vibrations (Dover Publications, Inc., New York, 1951), p. 29.

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Figures (8)

Fig. 1
Fig. 1

Theoretical frequency response for a light beam of small diameter, at the center of the crystal.

Fig. 2
Fig. 2

Theoretical frequency response for a light beam that covers the entire crystal.

Fig. 3
Fig. 3

Method for observing shear waves.

Fig. 4
Fig. 4

Degree of modulation vs applied voltage for an arbitrary point on the crystal.

Fig. 5
Fig. 5

Contour lines of local minimum retardation for j = 3 and m = 5.

Fig. 6
Fig. 6

Photographs of piezo-optic resonances: (a) m = 1, j = 6; (b) m = 5, j = 3; (c) m = 13, j about one; (d) m = 29, j is small; (e) non-shear mode; (f) non-shear mode.

Fig. 7
Fig. 7

Relative amplitude of resonances vs theoretical value and experimental values of k(m). k(m) = wm/Mw; ΦWQ(w,m)/Q.

Fig. 8
Fig. 8

Piezo-optic resonances at about 4 Mc.

Equations (44)

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n x = n o + 1 2 n o 3 r 63 E z , n y = n o - 1 2 n o 3 r 63 E z , n z = n e
r 63 = r 63 + p 66 d 36 ,
r 63 ( ω , x , y ) = r 63 + p 66 d 36 i = 0 a 2 i + 1 Q ( ω , 2 i + 1 ) ϕ ( x , y , 2 i + 1 ) .
where             ϕ ( x , y , m ) = 1 2 [ sin ( π m x / a ) + sin ( π m y / a ) ] , m = 1 , 3 , 5 , ,
Q ( ω , m ) = { [ 1 - ( ω / ω m ) 2 ] 2 + 4 k 2 ω 2 / ω m 4 } - 1 2 ,
i = 0 a 2 i + 1 Q ( 0 , 2 i + 1 ) ϕ ( x , y , 2 i + 1 ) = 1 ,
4 π i = 0 1 2 i + 1 sin ( 2 i + 1 ) π x / a = + 1             for             0 < x < a .
a m = 4 / m π .
ω m = m ω k ( m ) ,
k ( m ) = [ 1 + 8 / π 2 m 2 - 16 / π 4 m 4 ] 1 2 ,
r 63 ( ω m , x , y ) = r 63 + 2 · p 66 d 36 ω k ( m ) ϕ ( x , y , m ) / π k ,
I / I 0 = 1 2 ( 1 - cos 2 π Γ ) .
Γ = n 0 3 r 63 ( ω , x , y ) V 0 sin ω t / λ ,
V λ / 2 p = λ / 2 n 0 3 r 63
V λ / 2 d = λ / 2 n 0 3 p 66 d 36
V λ / 2 * ( ω , m ) = V λ / 2 d / a m Q ( ω m , m )
I I 0 = 1 2 { 1 - cos [ π ( 1 V λ / 2 p + ϕ ( x , y , m ) V λ / 2 * ) V 0 sin ω t ] } .
cos ( k sin ω t ) = J 0 ( k ) + 2 m = 1 J 2 m ( k ) cos 2 m ω t ,
Ī I 0 = 1 2 [ 1 - J 0 ( π V 0 V λ / 2 p + π V 0 ϕ ( x , y , m ) V λ / 2 * ) ] .
I 2 m I 0 = J 2 m ( π V 0 V λ / 2 p + π V 0 ϕ ( x , y , m ) V λ / 2 * ) cos 2 m ω t
[ 1 / V λ / 2 p + ϕ ( x , y , m ) / V λ / 2 * ] - 1 ,
1 / V λ / 2 0 = 1 / V λ / 2 p + 1 / V λ / 2 * ,
Ī max / I 0 = 1 2 [ I - J 0 ( π V 0 / V λ / 2 0 ) ] .
Δ n = n 0 3 r 63 E z
Δ n max = λ V 0 / 2 L V λ / 2 0 ,
f ( ω , x , y ) = α + ( 1 - α ) m odd a m Q ( ω , m ) ϕ ( x , y , m ) ,
α = ( 1 + p 66 d 63 / r 63 ) - 1 .
f ( ω ) = α + ( 1 - α ) m odd [ 8 Q ( ω , m ) / m 2 π 2 ] .
Δ ω / ω m = - ( 1 / 4 Q 2 + 1 / 32 Q 4 + )
Δ ω = ω - p ω .
f ( Δ ω ) = α + ( 1 - α ) [ 8 π 2 m p 1 m 2 - p 2 + 8 π 2 m p 2 p ( m 2 - p ) 2 Δ ω ω 0 + 8 p 2 π 2 Q ( Δ ω ) ] ,
8 π 2 m = p 1 m 2 - p 2 = 2 π 2 p 2
8 π 2 m p 1 ( m 2 - p 2 ) 2 = 1 2 π 2 ( π 2 3 p 2 - 3 p 4 ) ,
Q ( Δ ω ) = Q p - 2 Q p 3 ( Δ ω / p ω ) 2 + ,
Q p = p ω / 2 k .
Δ ω / p ω = ( 1 / 4 Q 3 ) ( 1 3 - 3 / π 2 p 2 ) .
J 1 [ ( sin π m x a - 1 + sin π m y a - 1 ) π V 0 / 2 V * λ / 2 ] = 0.
V 0 / V λ / 2 * = π - 1 J 1 , 2 j - 1 ( 0 ) ,
1 2 ( sin π m x a - 1 + sin π m y a - 1 ) = J 1 , 2 i - 1 ( 0 ) / J 1 , 2 j - 1 ( 0 ) ,
Q = 2 ω / π k ,
a m Q ( ω m , m ) / Q = k ( m ) .
V x y = L ω / π
V x y = [ s 66 E ρ ] - 1 2 ,
s 66 E = 16.2 × 10 - 11 m 2 / N .