Abstract

A two-order strong interaction matrix formalism is presented for analyzing acoustooptic beam-steering devices operated in the Bragg regime. Matrix solutions are compared with numerical solutions of the coupled differential equations, using ten orders. At the low frequency edge of the Bragg region (Qoverall ≃ 2π) the worst-case discrepancy is ∼1 dB (20%). The accuracy, however, improves rapidly with frequency and transducer length so that, in most cases of practical interest, the matrix method is an acceptable and simple analytic alternative to the numerical solution of the system of coupled differential equations.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Korpel, R. Adler, P. Desmares, T. M. Smith, IEEE J. Quantum Electron. QE-1, 60 (1965).
    [CrossRef]
  2. D. A. Pinnow, IEEE Trans. Sonics Ultrason. SU-18, #4, 209 (Oct.1971).
  3. E. I. Gordon, Proc. IEEE 54, 1391 (1966).
    [CrossRef]
  4. G. A. Coquin, J. P. Griffin, L. K. Anderson, IEEE Trans. Sonics Ultrason. SU-17, 34 (1970).
    [CrossRef]
  5. A. Korpel, in Applied Solid State Science, Vol. 3, R. Wolfe, Ed. (Academic, New York, 1972).
  6. A. Korpel, J. Opt. Soc. Am. 69, 678 (1979).
    [CrossRef]

1979

1971

D. A. Pinnow, IEEE Trans. Sonics Ultrason. SU-18, #4, 209 (Oct.1971).

1970

G. A. Coquin, J. P. Griffin, L. K. Anderson, IEEE Trans. Sonics Ultrason. SU-17, 34 (1970).
[CrossRef]

1966

E. I. Gordon, Proc. IEEE 54, 1391 (1966).
[CrossRef]

1965

A. Korpel, R. Adler, P. Desmares, T. M. Smith, IEEE J. Quantum Electron. QE-1, 60 (1965).
[CrossRef]

Adler, R.

A. Korpel, R. Adler, P. Desmares, T. M. Smith, IEEE J. Quantum Electron. QE-1, 60 (1965).
[CrossRef]

Anderson, L. K.

G. A. Coquin, J. P. Griffin, L. K. Anderson, IEEE Trans. Sonics Ultrason. SU-17, 34 (1970).
[CrossRef]

Coquin, G. A.

G. A. Coquin, J. P. Griffin, L. K. Anderson, IEEE Trans. Sonics Ultrason. SU-17, 34 (1970).
[CrossRef]

Desmares, P.

A. Korpel, R. Adler, P. Desmares, T. M. Smith, IEEE J. Quantum Electron. QE-1, 60 (1965).
[CrossRef]

Gordon, E. I.

E. I. Gordon, Proc. IEEE 54, 1391 (1966).
[CrossRef]

Griffin, J. P.

G. A. Coquin, J. P. Griffin, L. K. Anderson, IEEE Trans. Sonics Ultrason. SU-17, 34 (1970).
[CrossRef]

Korpel, A.

A. Korpel, J. Opt. Soc. Am. 69, 678 (1979).
[CrossRef]

A. Korpel, R. Adler, P. Desmares, T. M. Smith, IEEE J. Quantum Electron. QE-1, 60 (1965).
[CrossRef]

A. Korpel, in Applied Solid State Science, Vol. 3, R. Wolfe, Ed. (Academic, New York, 1972).

Pinnow, D. A.

D. A. Pinnow, IEEE Trans. Sonics Ultrason. SU-18, #4, 209 (Oct.1971).

Smith, T. M.

A. Korpel, R. Adler, P. Desmares, T. M. Smith, IEEE J. Quantum Electron. QE-1, 60 (1965).
[CrossRef]

IEEE J. Quantum Electron.

A. Korpel, R. Adler, P. Desmares, T. M. Smith, IEEE J. Quantum Electron. QE-1, 60 (1965).
[CrossRef]

IEEE Trans. Sonics Ultrason.

D. A. Pinnow, IEEE Trans. Sonics Ultrason. SU-18, #4, 209 (Oct.1971).

G. A. Coquin, J. P. Griffin, L. K. Anderson, IEEE Trans. Sonics Ultrason. SU-17, 34 (1970).
[CrossRef]

J. Opt. Soc. Am.

Proc. IEEE

E. I. Gordon, Proc. IEEE 54, 1391 (1966).
[CrossRef]

Other

A. Korpel, in Applied Solid State Science, Vol. 3, R. Wolfe, Ed. (Academic, New York, 1972).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1
Fig. 1

General configuration of light sound interaction for the −1 order.

Fig. 2
Fig. 2

General configuration of phased array transducer and descriptive parameters. Note that i = 1 n L ̅ i = 1 .

Fig. 3
Fig. 3

Stepped phase array configuration and descriptive parameters.

Fig. 4
Fig. 4

Specification of the four-cell standard phased array operating in the −1 order.

Fig. 5
Fig. 5

Symbols for interpretation of the code in the legends on the graphs.

Fig. 6
Fig. 6

Variation of diffracted intensities (0 and −1 orders) with normalized frequency for a single-cell transducer as obtained by matrix and differential equations techniques.

Fig. 7
Fig. 7

Variation of diffracted intensities (−3, −2, −1, 0, and +1 orders) with normalized frequency for a single-cell transducer as obtained by the differential equations technique.

Fig. 8
Fig. 8

Variation of diffracted intensities (0 and −1 orders) with normalized frequency for a two-cell transducer as obtained by matrix and differential equations techniques.

Fig. 9
Fig. 9

Variation of diffracted intensities (−3, −2, −1, 0, and +1 orders) with normalized frequency for a two-cell transducer as obtained by the differential equations technique.

Fig. 10
Fig. 10

Variation of diffracted intensities (0 and −1 orders) with normalized frequency for a three-cell transducer as obtained by matrix and differential equations techniques.

Fig. 11
Fig. 11

Variation of diffracted intensities (0 and −1 orders) with normalized frequency for a four-cell transducer as obtained by matrix and differential equations techniques.

Fig. 12
Fig. 12

Variation of diffracted intensities (0 and −1 orders) with normalized frequency for a four-cell transducer with an offset in the normalized incident angle, ϕ ̅ 0 = 1.10 , as obtained by the matrix and differential equations techniques.

Fig. 13
Fig. 13

Variation of diffracted intensities (0 and −1 orders) with normalized frequency for a four-cell transducer with an offset in the normalized incident angle, ϕ ̅ 0 = 1.15 , as obtained by the matrix and differential equations techniques.

Fig. 14
Fig. 14

Variation of diffracted intensities (0 and −1 orders) with normalized frequency for a four-cell transducer with an offset in the normalized incident angle, ϕ ̅ 0 = 1.20 , as obtained by the matrix and differential equations techniques.

Fig. 15
Fig. 15

Illustration of sliding phase reference (at z) for 0 and −1 orders.

Tables (1)

Tables Icon

Table I Design Parameters for Intensity Plots

Equations (78)

Equations on this page are rendered with MathJax. Learn more.

d ψ 0 dz = jaS ψ 1 exp ( jKz Δ ϕ ) , d ψ 1 dz = jaS * ψ 0 exp ( + jKz Δ ϕ ) ,
ψ 0 ( z ) = exp [ jKz ( ϕ 0 ϕ B ) / 2 ] A 0 ( z ) , ψ 1 ( z ) = exp [ jKz ( ϕ 0 ϕ B ) / 2 ] A 1 ( z ) .
d A 0 dz = + jK Δ ϕ 2 A 0 jaS A 1 , d A 1 dz = jaS * A 0 jK Δ ϕ 2 A 1 .
d A 0 , i d z i = jK Δ ϕ 2 A 0 , i ja S i A 1 , i , d A 1 , i d z i = ja S i * A 0 , i jK Δ ϕ 2 A 1 , i .
A 0 , i ( z i = 0 ) = A 0 ( i 1 ) , A 1 , i ( z i = 0 ) = A 1 ( i 1 ) ,
d A 0 , i dz ( z i = 0 ) = jK Δ ϕ 2 A 0 ( i 1 ) ja S i A 1 ( i 1 ) , d A 1 , i dz ( z i = 0 ) = ja S i * A 0 ( i 1 ) j K Δ ϕ 2 A 1 ( i 1 ) .
| jr jK Δ ϕ 2 ja S i ja S i * jr + jK Δ ϕ 2 | = 0 .
r = ± ( K Δ ϕ 2 ) 2 + a 2 | S i | 2 ± γ i .
A 0 , i = C 1 cos ( γ i z i ) + C 2 sin ( γ i z i ) , A 1 , i = C 3 cos ( γ i z i ) + C 4 sin ( γ i z i ) .
A 0 ( i 1 ) = C 1 , A 1 ( i 1 ) = C 3 , jK Δ ϕ 2 A 0 ( i 1 ) ja S i A 1 ( i 1 ) = C 2 γ i , jA S i * A 0 ( i 1 ) jK Δ ϕ 2 A 1 ( i 1 ) = C 4 γ i ,
A 0 , i = [ cos ( γ i z i ) + j K Δ ϕ 2 γ i sin ( γ i z i ) ] A 0 ( i 1 ) j a S i γ i sin ( γ i z i ) A 1 ( i 1 ) , A 1 , i = j a S i * 2 γ i sin ( γ i z i ) A 0 ( i 1 ) + [ cos ( γ i z i ) j K Δ ϕ 2 γ i sin ( γ i z i ) ] A 1 ( i 1 ) .
( A 0 ( i ) A 1 ( i ) ) = Γ i ( A 0 ( i 1 ) A 1 ( i 1 ) ) ,
Γ i = [ cos ( γ i L i ) + j K Δ ϕ 2 γ i sin ( γ i L i ) ja S i γ i sin ( γ i L i ) ja S i * γ i sin ( γ i L i ) cos ( γ i L i ) j K Δ ϕ 2 γ i sin ( γ i L i ) ] .
A 0 ( 1 ) = A 0 ( 0 ) cos ( a | S | L i ) , A 1 ( 1 ) = j A 0 ( 0 ) sin ( a | S | L i ) ,
Γ i = [ exp ( j γ i L i ) 0 0 exp ( j γ i L i ) ] ,
L i ̅ = L i / L .
( Γ i ) 11 = cos ( γ i L L i ̅ ) + j K Δ ϕ 2 γ i sin ( γ i L L i ̅ ) ,
( Γ i ) 12 = j a S i γ i sin ( γ i L L i ̅ ) ,
( Γ i ) 21 = j a S i * γ i sin ( γ i L L i ̅ ) ,
( Γ i ) 22 = cos ( γ i L L i ̅ ) j K Δ ϕ 2 γ i sin ( γ i L L i ̅ ) .
K = 2 k ϕ B ,
K c = 2 k ϕ B c .
f ̅ = f / f c ,
K = f ̅ K c ,
ϕ B = f ̅ ϕ B c .
Q = f ̅ 2 Q c ,
Q c = K c 2 L / k = 2 K c L ϕ B c .
Q i = f ̅ 2 Q i , c = f ̅ 2 L i ̅ Q c .
ϕ 0 ̅ = ϕ 0 / ϕ B c .
α i a | S i | L ,
β i γ i L .
K Δ ϕ L 2 = KL ϕ B ( ϕ 0 ϕ B 1 ) / 2 = 1 2 [ f ̅ 2 2 Q c ( ϕ ̅ 0 / f ̅ 1 ) ] ,
β i = γ i L = 2 + α i 2 .
S i = | S i | exp ( j ϕ i ) ,
( Γ i ) 11 = cos ( L i ̅ 2 + α i 2 ) + j 2 + α i 2 sin ( L i ̅ 2 + α i 2 ) ,
( Γ i ) 12 = j α i 2 + α i 2 exp ( j θ i ) sin ( L i ̅ 2 + α i 2 ) ,
( Γ i ) 21 = j α i 2 + α i 2 exp ( j θ i ) sin ( L i ̅ 2 + α i 2 ) ,
( Γ i ) 22 = cos ( L i ̅ 2 + α i 2 ) j 2 + α i 2 sin ( L i ̅ 2 + α i 2 ) .
L i ̅
α i a | S i | L = π / 2 .
b i = l = 1 i h l ,
θ i = K b i + θ i E .
θ i = ( K c b i ) f ̅ + θ i E .
n = 4 , Q c = 2 π n = 8 π , α i = a | S i | L = π / 2 , θ 1 E = 0 , θ 2 E = π , θ 3 E = 0 , θ 4 E = π ,
b 1 K c = 0 , b 2 K c = π , b 3 K c = 2 π , b 4 K c = 3 π ,
( A 0 ( 0 ) A 1 ( 0 ) ) = A 0 ( 0 ) ( 1 0 ) .
( A 0 ( n ) A 1 ( n ) ) = Γ n Γ n 1 Γ 2 Γ 1 ( 1 0 ) .
( A 0 ( i ) A 1 ( i ) ) = Γ i ( A 0 ( i 1 ) A 1 ( i 1 ) ) = ( ( Γ i ) 11 A 0 ( i 1 ) + ( Γ i ) 12 A 1 ( i 1 ) ( Γ i ) 21 A 0 ( i 1 ) + ( Γ i ) 22 A 1 ( i 1 ) ) .
( A 0 ( n ) A 1 ( n ) )
I 0 = A 0 ( n ) ( A 0 ( n ) ) * , I 1 = A 1 ( n ) ( A 0 ( n ) ) * .
θ i E
ψ 0 ( z ) = exp [ jKz ( ϕ 0 ϕ B ) / 2 ] A 0 ( z ) , ψ 1 ( z ) = exp [ jKz ( ϕ 0 ϕ 0 ) / 2 ] A 1 ( z ) .
ψ 0 ( z ) = ψ 0 ( 0 ) exp [ jkz cos ( ϕ B + Δ ϕ ) ] ,
ψ 1 ( z ) = ψ 1 ( 0 ) exp [ jkz cos ( ϕ B Δ ϕ ) ] .
cos ( ϕ B + Δ ϕ ) = cos ϕ B cos Δ ϕ sin ϕ B sin Δ ϕ ,
k cos ( ϕ B + Δ ϕ ) k cos ϕ B cos Δ ϕ ( K / 2 ) Δ ϕ ,
k sin ϕ B k ϕ B = K / 2 .
k cos ( ϕ B Δ ϕ ) k cos ϕ B cos Δ ϕ + ( K / 2 ) Δ ϕ .
ψ 0 ( z ) = ψ 0 ( 0 ) exp [ jzk cos ϕ B cos ( Δ ϕ ) + jKz Δ ϕ / 2 ] , ψ 1 ( z ) = ψ 1 ( 0 ) exp [ jkz cos ϕ B cos ( Δ ϕ ) jKz Δ ϕ / 2 ] .
ψ 0 ( 0 ) = ψ 0 ( z ) exp [ jKz ( ϕ 0 ϕ B ) / 2 ] , ψ 1 ( 0 ) = ψ 1 ( z ) exp [ + jKz ( ϕ 0 ϕ B ) / 2 ] .
L = i = 1 n L i .
( A 0 ( 2 ) A 1 ( 2 ) ) = Γ 2 ( A 0 ( 1 ) A 1 ( 1 ) ) = Γ 2 Γ 1 ( A 0 ( 0 ) A 1 ( 0 ) ) ,
Γ 1 = [ j K γ Δ ϕ 2 sin γ L 1 + cos γ L 1 ja S γ sin γ L 1 ja S γ sin γ L 1 j K γ Δ ϕ 2 sin γ L 1 + cos γ L 1 ] ,
Γ 2 = [ j K Γ Δ ϕ 2 sin γ L 2 + cos γ L 2 ja S γ sin γ L 2 ja S γ sin γ L 2 j K γ Δ ϕ 2 sin γ L 2 + cos γ L 2 ] ,
( A 0 ( 0 ) A 1 ( 0 ) ) .
Γ = Γ 2 Γ 1 .
( Γ ) 11 = ( j K γ Δ ϕ 2 sin γ L 2 + cos γ L 2 ) ( j K γ Δ ϕ 2 sin γ L 1 + cos γ L 1 ) + ( ja S γ ) 2 sin γ L 1 sin γ L 2 ,
( Γ ) 12 = ( j K γ Δ ϕ 2 sin γ L 2 + cos γ L 2 ) ( ja S γ sin γ L 1 ) + ( j K γ Δ ϕ 2 sin γ L 1 + cos γ L 1 ) ( ja S γ sin γ L 2 ) ,
( Γ ) 21 = ( j K γ Δ ϕ 2 sin γ L 1 + cos γ L 1 ) ( ja S γ sin γ L 2 ) + ( j K γ Δ ϕ 2 sin γ L 2 + cos γ L 2 ) ( ja S γ sin γ L 1 ) ,
( Γ ) 22 = ( j K γ Δ ϕ 2 sin γ L 1 + cos γ L 1 ) ( j K γ Δ ϕ 2 sin γ L 2 + cos γ L 2 ) + ( ja S γ ) 2 sin γ L 2 sin γ L 1 .
L = L 1 + L 2 ,
γ 2 = ( K Δ ϕ / 2 ) 2 + a 2 S 2 ,
cos ( x ± y ) = cos x cos y sin x sin y , sin ( x ± y ) = sin x cos y ± cos x sin y .
( Γ ) 11 = K 2 ( Δ ϕ 2 ) 2 sin γ L 1 sin γ L 2 + ( ja S ) 2 sin γ L 1 sin γ L 2 γ 2 + cos γ L 1 cos γ L 2 + j K 2 Δ ϕ 2 ( cos γ L 1 sin γ L 2 + sin γ L 2 cos γ L 1 ) .
( Γ ) 11 = j K γ Δ ϕ 2 sin γ L + cos γ L .
( Γ ) 12 = ja S γ sin γ L ,
( Γ ) 21 = ja S γ sin γ L ,
( Γ ) 22 = j K γ Δ ϕ γ sin γ L + cos γ L .

Metrics