Abstract

The diffraction of light that is due to a converging sound field is interpreted and analyzed in terms of an eikonal ray theory. This theory, in combination with Feynman diagram techniques, indicates that the acousto-optic interaction is physically localized over a width of the sound field given roughly by the geometric mean of the sound’s wavelength and the sound field’s radius of curvature. Analytic results (for two diffracted orders) are in close agreement with 10-order numerical simulations, as long as sound amplitudes are not so strong as to violate assumptions made in the analytic model.

© 1985 Optical Society of America

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References

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  1. A. Korpel, “Acousto-optics,” in, Applied Solid State Science, R. Wolfe, ed. (Academic, New York, 1972), Vol. 3.
  2. A. Korpel, “Eikonal theory of Bragg diffraction imaging,” in Acoustical Holography, A. F. Metherell, L. Larmore, eds. (Plenum, New York, 1970), Vol. 2.
    [CrossRef]
  3. A. Korpel, “Visualization of the cross section of a sound beam by Bragg diffraction of light,” Appl. Phys. Lett. 9, 425–427 (1966).
    [CrossRef]
  4. F. D. Martin, L. Adler, M. A. Breazeale, “Bragg imaging with finite amplitude ultrasonic waves,” J. Appl. Phys. 43, 1480–1487 (1972).
    [CrossRef]
  5. R. S. Chu, T. Tamir, “Diffraction of Gaussian beams by periodically modulated media for incidence close to a Bragg angle,” J. Opt. Soc. Am. 66, 1438–1440 (1976).
    [CrossRef]
  6. R. S. Chu, J. A. Kong, “Diffraction of optical beams with arbitrary profile by periodically modulated layer,” J. Opt. Soc. Am. 70, 1–6 (1980).
    [CrossRef]
  7. A. Korpel, “Two-dimensional plane wave theory of strong ac-ousto-optic interaction in isotropic media,” J. Opt. Soc. Am. 69, 678–683 (1979).
    [CrossRef]
  8. A. Korpel, T. C. Poon, “An explicit formalism for acousto-optic multiple plane wave scattering,” J. Opt. Soc. Am. 70, 817–820 (1980).
    [CrossRef]
  9. T. C. Poon, A. Korpel, “A Feynman diagram approach toward acousto-optic scattering in the near-Bragg region,” J. Opt. Soc. Am. 71, 1202–1208 (1981).
    [CrossRef]
  10. T. Kosaka, T. Saitoh, H. Mada, S. Kobayashi, “Characteristics of an acousto-optic deflection device having a bent ultrasonic wavefront,” Jpn. J. Appl. Phys. 19, 664–666 (1980).
    [CrossRef]
  11. R. Pieper, A. Korpel, “A comparison of phased array Bragg cells operating in the second order,” Appl. Opt. 23, 2921–2934 (1984).
    [CrossRef] [PubMed]
  12. T. C. Poon, A. Korpel, “Second order Bragg diffraction operation of acousto-optic devices,” in Proceedings of the 1981 IEEE Ultrasonics Symposium (Institute of Electrical and Electronics Engineers, New York, 1981), pp. 751–754.
    [CrossRef]
  13. D. H. Achenbach, Wave Propagation in Elastic Solids (North-Holland, Amsterdam, 1973).

1984

1981

1980

1979

1976

1972

F. D. Martin, L. Adler, M. A. Breazeale, “Bragg imaging with finite amplitude ultrasonic waves,” J. Appl. Phys. 43, 1480–1487 (1972).
[CrossRef]

1966

A. Korpel, “Visualization of the cross section of a sound beam by Bragg diffraction of light,” Appl. Phys. Lett. 9, 425–427 (1966).
[CrossRef]

Achenbach, D. H.

D. H. Achenbach, Wave Propagation in Elastic Solids (North-Holland, Amsterdam, 1973).

Adler, L.

F. D. Martin, L. Adler, M. A. Breazeale, “Bragg imaging with finite amplitude ultrasonic waves,” J. Appl. Phys. 43, 1480–1487 (1972).
[CrossRef]

Breazeale, M. A.

F. D. Martin, L. Adler, M. A. Breazeale, “Bragg imaging with finite amplitude ultrasonic waves,” J. Appl. Phys. 43, 1480–1487 (1972).
[CrossRef]

Chu, R. S.

Kobayashi, S.

T. Kosaka, T. Saitoh, H. Mada, S. Kobayashi, “Characteristics of an acousto-optic deflection device having a bent ultrasonic wavefront,” Jpn. J. Appl. Phys. 19, 664–666 (1980).
[CrossRef]

Kong, J. A.

Korpel, A.

R. Pieper, A. Korpel, “A comparison of phased array Bragg cells operating in the second order,” Appl. Opt. 23, 2921–2934 (1984).
[CrossRef] [PubMed]

T. C. Poon, A. Korpel, “A Feynman diagram approach toward acousto-optic scattering in the near-Bragg region,” J. Opt. Soc. Am. 71, 1202–1208 (1981).
[CrossRef]

A. Korpel, T. C. Poon, “An explicit formalism for acousto-optic multiple plane wave scattering,” J. Opt. Soc. Am. 70, 817–820 (1980).
[CrossRef]

A. Korpel, “Two-dimensional plane wave theory of strong ac-ousto-optic interaction in isotropic media,” J. Opt. Soc. Am. 69, 678–683 (1979).
[CrossRef]

A. Korpel, “Visualization of the cross section of a sound beam by Bragg diffraction of light,” Appl. Phys. Lett. 9, 425–427 (1966).
[CrossRef]

T. C. Poon, A. Korpel, “Second order Bragg diffraction operation of acousto-optic devices,” in Proceedings of the 1981 IEEE Ultrasonics Symposium (Institute of Electrical and Electronics Engineers, New York, 1981), pp. 751–754.
[CrossRef]

A. Korpel, “Acousto-optics,” in, Applied Solid State Science, R. Wolfe, ed. (Academic, New York, 1972), Vol. 3.

A. Korpel, “Eikonal theory of Bragg diffraction imaging,” in Acoustical Holography, A. F. Metherell, L. Larmore, eds. (Plenum, New York, 1970), Vol. 2.
[CrossRef]

Kosaka, T.

T. Kosaka, T. Saitoh, H. Mada, S. Kobayashi, “Characteristics of an acousto-optic deflection device having a bent ultrasonic wavefront,” Jpn. J. Appl. Phys. 19, 664–666 (1980).
[CrossRef]

Mada, H.

T. Kosaka, T. Saitoh, H. Mada, S. Kobayashi, “Characteristics of an acousto-optic deflection device having a bent ultrasonic wavefront,” Jpn. J. Appl. Phys. 19, 664–666 (1980).
[CrossRef]

Martin, F. D.

F. D. Martin, L. Adler, M. A. Breazeale, “Bragg imaging with finite amplitude ultrasonic waves,” J. Appl. Phys. 43, 1480–1487 (1972).
[CrossRef]

Pieper, R.

Poon, T. C.

T. C. Poon, A. Korpel, “A Feynman diagram approach toward acousto-optic scattering in the near-Bragg region,” J. Opt. Soc. Am. 71, 1202–1208 (1981).
[CrossRef]

A. Korpel, T. C. Poon, “An explicit formalism for acousto-optic multiple plane wave scattering,” J. Opt. Soc. Am. 70, 817–820 (1980).
[CrossRef]

T. C. Poon, A. Korpel, “Second order Bragg diffraction operation of acousto-optic devices,” in Proceedings of the 1981 IEEE Ultrasonics Symposium (Institute of Electrical and Electronics Engineers, New York, 1981), pp. 751–754.
[CrossRef]

Saitoh, T.

T. Kosaka, T. Saitoh, H. Mada, S. Kobayashi, “Characteristics of an acousto-optic deflection device having a bent ultrasonic wavefront,” Jpn. J. Appl. Phys. 19, 664–666 (1980).
[CrossRef]

Tamir, T.

Appl. Opt.

Appl. Phys. Lett.

A. Korpel, “Visualization of the cross section of a sound beam by Bragg diffraction of light,” Appl. Phys. Lett. 9, 425–427 (1966).
[CrossRef]

J. Appl. Phys.

F. D. Martin, L. Adler, M. A. Breazeale, “Bragg imaging with finite amplitude ultrasonic waves,” J. Appl. Phys. 43, 1480–1487 (1972).
[CrossRef]

J. Opt. Soc. Am.

Jpn. J. Appl. Phys.

T. Kosaka, T. Saitoh, H. Mada, S. Kobayashi, “Characteristics of an acousto-optic deflection device having a bent ultrasonic wavefront,” Jpn. J. Appl. Phys. 19, 664–666 (1980).
[CrossRef]

Other

T. C. Poon, A. Korpel, “Second order Bragg diffraction operation of acousto-optic devices,” in Proceedings of the 1981 IEEE Ultrasonics Symposium (Institute of Electrical and Electronics Engineers, New York, 1981), pp. 751–754.
[CrossRef]

D. H. Achenbach, Wave Propagation in Elastic Solids (North-Holland, Amsterdam, 1973).

A. Korpel, “Acousto-optics,” in, Applied Solid State Science, R. Wolfe, ed. (Academic, New York, 1972), Vol. 3.

A. Korpel, “Eikonal theory of Bragg diffraction imaging,” in Acoustical Holography, A. F. Metherell, L. Larmore, eds. (Plenum, New York, 1970), Vol. 2.
[CrossRef]

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Figures (16)

Fig. 1
Fig. 1

Curved sound wave front with incident and diffracted rays of light.

Fig. 2
Fig. 2

Curved sound wave front showing coordinate system.

Fig. 3
Fig. 3

General interaction diagram showing and defining (dashed) Bragg lines.

Fig. 4
Fig. 4

Scattering path representation for Feynman diagram calculations.

Fig. 5
Fig. 5

Geometric interpretation predicting physical location of first-order interaction at point ξs = R(ϕ0ϕB).

Fig. 6
Fig. 6

Generation points for spurious diffracted orders showing qualitative dependence on sound amplitude: (a) low βct, (b) medium βct, (c) high βct, (d) very high βct.

Fig. 7
Fig. 7

Figure defining variable sound-field length q.

Fig. 8
Fig. 8

Intensity plot versus q ¯ for βct = 0.2, Rn = 8.

Fig. 9
Fig. 9

Intensity plot versus q ¯ for βct = 0.7, Rn = 8.

Fig. 10
Fig. 10

Intensity plots versus extended q ¯ for βct = 0.2, Rn = 8: (a) I−3I+1I0, (b) I+2I−1I−2.

Fig. 11
Fig. 11

Intensity plots versus sound amplitude factor (βct) with Rn = 8,Qc = 72π: (a) I−2I−1I0, (b) I−3I+1I+2.

Fig. 12
Fig. 12

Intensity plot versus normalized z coordinate z ¯, with Rn = 8, βct= 3.1, Qc = 72π.

Fig. 13
Fig. 13

First-order diffraction with ϕ0 = −ϕBc.

Fig. 14
Fig. 14

Slow variation of g(z) as required by assumptions made in text.

Fig. 15
Fig. 15

F(z) defined for reference in stationary phase mathematics discussed in the text.

Fig. 16
Fig. 16

Curved wave front with separated interaction regions.

Tables (4)

Tables Icon

Table 1 Symbols Used in This Papera

Tables Icon

Table 2 Parameters Used in Numerical Tests (1–3)

Tables Icon

Table 3 Comparison of First-Order Intensities Taken from Numerical and Analytic Methods

Tables Icon

Table 4 Phase Coefficients

Equations (81)

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S ( t ) = Re [ S exp ( j ω t ) ] .
π / 2 π / 2 | S p | 2 R d ϕ = constant ,
S p = S 0 [ ( R 0 / R ) ] 1 / 2 exp [ j K ( R R 0 ) ] ,
R = [ ( R 0 x ) 2 + ξ 2 ] 1 / 2 = ( R 0 x ) { 1 + [ ξ / ( R 0 x ) ] 2 } 1 / 2 .
x constant ( within the region of interaction )
x R 0 .
| ξ / ( R 0 ) | 1 .
R ( R 0 x ) [ 1 + 0.5 ξ 2 / ( R 0 x ) 2 ] .
R ( R 0 x ) + 0.5 ξ 2 ( 1 + x / R 0 ) / R 0 .
R R 0 [ 1 x / R 0 + ξ 2 / ( 2 R 0 2 ) + x ξ 2 / ( 2 R 0 3 ) ] .
R R 0 [ magnitude evaluation of Eq . ( 1 ) ] .
R R 0 [ 1 x / R 0 + ξ 2 / ( 2 R 0 2 ) ] [ Phase evaluation of Eq . ( 1 ) ] .
S p S 0 exp [ j K x + j K ξ 2 / ( 2 R ) ] ,
E n ( z ) = E inc δ on j a n 1 z S n 1 + ( z ) E n 1 d z j a n + 1 z S n + 1 ( z ) E n + 1 d z , δ on = 1 for n = 0 = 0 for n 0 ,
ϕ n = ϕ 0 + 2 n ϕ B ,
ϕ B 0.5 K / k = π f / ( k V s ) .
S n + = S [ z , z tan ( ϕ n + ϕ B ) | S n + | exp ( j ψ n + )
S n = S * [ z , z tan ( ϕ n ϕ B ) ] | S n | exp ( j ψ n )
a n = k n 0 2 p / ( 4 cos ϕ n ) ,
a n = k n 0 2 p / 4 a , all n ,
ψ n 1 + = K z ϕ B [ ϕ 0 / ϕ B + ( 2 n 1 ) ] + K ξ 2 / ( 2 R ) ,
ψ n + 1 = K z ϕ B [ ϕ 0 / ϕ B + ( 2 n + 1 ) ] K ξ 2 / ( 2 R ) ,
ξ = z L / 2.
| S n | | S n + | | S |
E 0 ( z ) = E inc j a | S | z e j ψ E 1 d z ,
E 1 ( z ) = j a | S | z e j ψ E 0 d z ,
ψ = K z ϕ B ( ϕ 0 / ϕ B 1 ) K ξ 2 / ( 2 R ) .
E inc ( j a | S | ) 3 0 L exp [ j ψ ( z 2 ) ] 0 z 2 exp [ j ψ ( z 1 ) ] × 0 z 1 exp [ j ψ ( z ) ] d z d z 1 d z 2 .
2 E inc ( j a | S | Λ R / 2 ) 3 exp { j [ ψ ( z s ) π / 4 ] } , ξ s = R ( ϕ 0 ϕ B ) ,
E 1 = E inc [ 2 j b / ( 1 + b 2 ) ] exp [ j ψ ( z s ) π / 4 ] ,
b = ( a | S | Λ R / 2 ) .
E 0 = E inc ( 1 b 2 ) / ( 1 + b 2 ) .
| E 1 | 2 + | E 0 | 2 = ( E inc ) 2 .
| E 1 | a | S | E inc Λ R , b 1 .
L I = Λ R .
β c t = 4 b / π = a | S | Λ c R / ( π / 2 ) .
a | S | Λ R = π / 2 ,
| E 1 | ( a | S | Λ c R ) E 0 , β c t 1 ,
χ d = 90 ° ϕ B c ,
| E + 1 | ( a | S | Λ c R ) E 0 , β c t 1 .
χ b = 90 ° + 2 ϕ B c ,
χ a = 90 ° + 3 ϕ B c ,
R n R λ 2 / Λ c 3 .
q ¯ = q / Λ c R .
z ¯ = z / L
Q c = L K c 2 / k
f = f c ( fixed ) , β c t ( fixed ) , R n ( fixed ) , q ¯ i ( independent variable ) .
2 × ( 2 R ϕ B c + 0.5 Λ c R ) / Λ c R = 2 R n + 1.
R n = 8 ,
Q c = 72 π = 2 π L λ / Λ c 2 ,
f c = 1 ,
0 < β c t < 5 ,
β c t 3.1 ( second - order direct maximum ) ,
ξ 2 = 4 Λ c 2 / λ [ point b in Fig . 6 ( c ) ] .
L = 36 Λ c 2 / λ
L exp ( j ψ 1 + ) z 1 exp ( j ψ 0 + ) d z d z 1 ,
ψ 1 + = K z ( ϕ B c + 3 ϕ B ) + K ξ 2 / ( 2 R ) , ψ 0 + = K z ( ϕ B c + ϕ B ) + K ξ 2 / ( 2 R ) , ξ = z L / 2 .
ξ + 1 = R ( ϕ B c ϕ B ) ,
ξ + 1 ( f = f c ) = 0
ξ + 2 = R ( ϕ B c 3 ϕ B ) ,
ξ + 2 ( f = f c ) = + 2 R ϕ B c ,
I 1 j a | S | z exp ( j ψ 0 ) E 0 d z ,
I + 1 j a | S | z exp ( j ψ 0 + ) E 0 d z ,
ψ 0 = K z ( ϕ B c ϕ B ) K ξ 2 / ( 2 R ) , ψ 0 + = K z ( ϕ B c + ϕ B ) + K ξ 2 / ( 2 R ) .
ξ 1 = R ( ϕ B c ϕ B ) , ξ + 1 = R ( ϕ B c + ϕ B ) .
I ( z ) = z g ( z ) exp [ ± j ψ ( z ) ] d z ,
ψ ( z ) = K L [ ( z / L ) ϕ B ( ϕ 0 / ϕ B 1 ) ξ 2 / ( 2 R L ) ] .
d ψ / d z | z s = 0
z s = L / 2 + R ( ϕ 0 ϕ B ) , ξ s = R ( ϕ 0 ϕ B ) .
ψ ( z ) ψ ( z s ) + ψ | z s ( z z s ) 2 / 2 .
I ( z ) = exp [ ± j ψ ( z s ) ] g ( z s ) z exp [ ± ψ | z s ( z z s ) 2 / 2 ] d z ,
exp ( ± j a y 2 / 2 ) d y = ( 1 ± j ) ( x / a ) for a > 0 = ( 1 j ) ( x / | a | ) for a < 0
I ( z ) F ( z ) g ( z s ) exp [ ± j ψ ( z s ) ] ( 1 ± j ) × [ π / ψ ( z s ) ] 0.5 when ψ ( z s ) > 0 F ( z ) g ( z s ) exp [ ± j ψ ( z s ) ] ( 1 j ) × [ π / | ψ ( z s ) | ] 0.5 when ψ ( z s ) < 0 ,
F ( z ) = 1 when z > z s , F ( z ) = 0.5 when z = z s , F ( z ) = 0 when z < z s .
I ( z ) F ( z ) g ( z s ) exp { ± j [ ψ ( z s ) π / 4 ] } Λ R ,
L exp [ j ψ ( z 1 ) ] z 1 exp [ j ψ ( z ) ] d z d z 1 .
exp [ j ( ψ ( z s ) π / 4 ) ] Λ R L exp [ j ψ ( z 1 ) ] F ( z 1 ) d z 1 .
( Λ R ) 2 / 2 provided that z s < L .
R ϕ B > R Λ
R ϕ B c > R Λ c or R > Λ / ϕ B c 2 .
R n > 4 .

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