Abstract

The Green's function model of the acoustooptic modulator proposed by E. I. Gordon et al. is reformulated in Fourier transform space to simplify the mathematics and to underscore the physics. Numerical studies of response to sinusoidal video signals and to square pulse trains indicate that the modulator can be approximated by a linear invariant model with a suitably scaled Gaussian impulse response. An angular scattering window analogy is proposed to explain the characteristics of the modulator.

© 1978 Optical Society of America

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References

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  1. H. V. Hance, J. K. Parks, J. Acoust. Soc. Am. 38, 14 (1965).
    [CrossRef]
  2. A. Korpel, R. Adler, P. Desmares, W. Watson, Appl. Opt. 5, 1667 (1966).
    [CrossRef] [PubMed]
  3. A. H. Rosenthal, IRE Trans. Ultrason. Eng. 8, 1 (1961).
    [CrossRef]
  4. J. Randolph, J. Morrison, “Spatial and Temporal Response of Acousto-Optics Devices,” presented at Electro-Optics 1971, New York Colliseum (September 1971).
  5. L. B. Lambert, IRE Nat. Conv. Rec. 10, Part 6 (1962).
  6. E. I. Gordon, Appl. Opt. 5, 1629 (1966).
    [CrossRef] [PubMed]
  7. D. Maydan, IEEE J. Quantum Electron. QE-6, 15 (1970).
    [CrossRef]
  8. M. G. Cohen, E. I. Gordon, Bell Syst. Tech. J. 44, 693 (1965).
  9. R. W. Dixon, E. I. Gordon, Bell Syst. Tech. J. 46, 367 (1967).
  10. D. M. Henderson, IEEE J. Quantum Electron. QE-8, 184 (1972).
    [CrossRef]
  11. R. A. Adler, IEEE Spectrum 4, 42 (1967).
    [CrossRef]
  12. C. F. Quate, C. D. W. Wilkinson, D. K. Winslow, Proc. IEEE 53, 1604 (1965).
    [CrossRef]
  13. R. W. Damon, W. T. Maloney, D. H. McMahon, in Physical Acoustics, Vol. 7 (Academic, New York, 1970), pp. 277–280
  14. A. Korpel, in Applied Solid State Science, Vol. 3 (Academic, New York, 1972), pp. 100–103.
  15. R. V. Johnson, Appl. Opt. 16, 507 (1977).
    [CrossRef] [PubMed]
  16. C. J. Bouwkamp, Rep. Prog. Phys. 17, 35 (1954).
    [CrossRef]
  17. H. Weyl, Ann. Phys. 60, 481 (1919).
    [CrossRef]
  18. J. A. Lucero, J. A. Duardo, R. V. Johnson, SPIE Proc. 90, 32 (1976).
    [CrossRef]
  19. R. W. Dixon, IEEE J. Quantum Electron. QE-3, 85 (1967).
    [CrossRef]
  20. W. R. Klein, B. D. Cook, IEEE Trans. Sonics Ultrason. SU-14, 123 (1976).
  21. P. F. Panter, Modulation, Noise, and Spectral Analysis (McGraw-Hill, New York, 1965), Chap. 5.
  22. J. Lapierre, D. Phalippou, S. Lowenthal, Appl. Opt. 14, 1949 (1975).

1977

1976

J. A. Lucero, J. A. Duardo, R. V. Johnson, SPIE Proc. 90, 32 (1976).
[CrossRef]

W. R. Klein, B. D. Cook, IEEE Trans. Sonics Ultrason. SU-14, 123 (1976).

1975

J. Lapierre, D. Phalippou, S. Lowenthal, Appl. Opt. 14, 1949 (1975).

1972

D. M. Henderson, IEEE J. Quantum Electron. QE-8, 184 (1972).
[CrossRef]

1970

D. Maydan, IEEE J. Quantum Electron. QE-6, 15 (1970).
[CrossRef]

1967

R. W. Dixon, E. I. Gordon, Bell Syst. Tech. J. 46, 367 (1967).

R. A. Adler, IEEE Spectrum 4, 42 (1967).
[CrossRef]

R. W. Dixon, IEEE J. Quantum Electron. QE-3, 85 (1967).
[CrossRef]

1966

1965

H. V. Hance, J. K. Parks, J. Acoust. Soc. Am. 38, 14 (1965).
[CrossRef]

M. G. Cohen, E. I. Gordon, Bell Syst. Tech. J. 44, 693 (1965).

C. F. Quate, C. D. W. Wilkinson, D. K. Winslow, Proc. IEEE 53, 1604 (1965).
[CrossRef]

1962

L. B. Lambert, IRE Nat. Conv. Rec. 10, Part 6 (1962).

1961

A. H. Rosenthal, IRE Trans. Ultrason. Eng. 8, 1 (1961).
[CrossRef]

1954

C. J. Bouwkamp, Rep. Prog. Phys. 17, 35 (1954).
[CrossRef]

1919

H. Weyl, Ann. Phys. 60, 481 (1919).
[CrossRef]

Adler, R.

Adler, R. A.

R. A. Adler, IEEE Spectrum 4, 42 (1967).
[CrossRef]

Bouwkamp, C. J.

C. J. Bouwkamp, Rep. Prog. Phys. 17, 35 (1954).
[CrossRef]

Cohen, M. G.

M. G. Cohen, E. I. Gordon, Bell Syst. Tech. J. 44, 693 (1965).

Cook, B. D.

W. R. Klein, B. D. Cook, IEEE Trans. Sonics Ultrason. SU-14, 123 (1976).

Damon, R. W.

R. W. Damon, W. T. Maloney, D. H. McMahon, in Physical Acoustics, Vol. 7 (Academic, New York, 1970), pp. 277–280

Desmares, P.

Dixon, R. W.

R. W. Dixon, E. I. Gordon, Bell Syst. Tech. J. 46, 367 (1967).

R. W. Dixon, IEEE J. Quantum Electron. QE-3, 85 (1967).
[CrossRef]

Duardo, J. A.

J. A. Lucero, J. A. Duardo, R. V. Johnson, SPIE Proc. 90, 32 (1976).
[CrossRef]

Gordon, E. I.

R. W. Dixon, E. I. Gordon, Bell Syst. Tech. J. 46, 367 (1967).

E. I. Gordon, Appl. Opt. 5, 1629 (1966).
[CrossRef] [PubMed]

M. G. Cohen, E. I. Gordon, Bell Syst. Tech. J. 44, 693 (1965).

Hance, H. V.

H. V. Hance, J. K. Parks, J. Acoust. Soc. Am. 38, 14 (1965).
[CrossRef]

Henderson, D. M.

D. M. Henderson, IEEE J. Quantum Electron. QE-8, 184 (1972).
[CrossRef]

Johnson, R. V.

R. V. Johnson, Appl. Opt. 16, 507 (1977).
[CrossRef] [PubMed]

J. A. Lucero, J. A. Duardo, R. V. Johnson, SPIE Proc. 90, 32 (1976).
[CrossRef]

Klein, W. R.

W. R. Klein, B. D. Cook, IEEE Trans. Sonics Ultrason. SU-14, 123 (1976).

Korpel, A.

A. Korpel, R. Adler, P. Desmares, W. Watson, Appl. Opt. 5, 1667 (1966).
[CrossRef] [PubMed]

A. Korpel, in Applied Solid State Science, Vol. 3 (Academic, New York, 1972), pp. 100–103.

Lambert, L. B.

L. B. Lambert, IRE Nat. Conv. Rec. 10, Part 6 (1962).

Lapierre, J.

J. Lapierre, D. Phalippou, S. Lowenthal, Appl. Opt. 14, 1949 (1975).

Lowenthal, S.

J. Lapierre, D. Phalippou, S. Lowenthal, Appl. Opt. 14, 1949 (1975).

Lucero, J. A.

J. A. Lucero, J. A. Duardo, R. V. Johnson, SPIE Proc. 90, 32 (1976).
[CrossRef]

Maloney, W. T.

R. W. Damon, W. T. Maloney, D. H. McMahon, in Physical Acoustics, Vol. 7 (Academic, New York, 1970), pp. 277–280

Maydan, D.

D. Maydan, IEEE J. Quantum Electron. QE-6, 15 (1970).
[CrossRef]

McMahon, D. H.

R. W. Damon, W. T. Maloney, D. H. McMahon, in Physical Acoustics, Vol. 7 (Academic, New York, 1970), pp. 277–280

Morrison, J.

J. Randolph, J. Morrison, “Spatial and Temporal Response of Acousto-Optics Devices,” presented at Electro-Optics 1971, New York Colliseum (September 1971).

Panter, P. F.

P. F. Panter, Modulation, Noise, and Spectral Analysis (McGraw-Hill, New York, 1965), Chap. 5.

Parks, J. K.

H. V. Hance, J. K. Parks, J. Acoust. Soc. Am. 38, 14 (1965).
[CrossRef]

Phalippou, D.

J. Lapierre, D. Phalippou, S. Lowenthal, Appl. Opt. 14, 1949 (1975).

Quate, C. F.

C. F. Quate, C. D. W. Wilkinson, D. K. Winslow, Proc. IEEE 53, 1604 (1965).
[CrossRef]

Randolph, J.

J. Randolph, J. Morrison, “Spatial and Temporal Response of Acousto-Optics Devices,” presented at Electro-Optics 1971, New York Colliseum (September 1971).

Rosenthal, A. H.

A. H. Rosenthal, IRE Trans. Ultrason. Eng. 8, 1 (1961).
[CrossRef]

Watson, W.

Weyl, H.

H. Weyl, Ann. Phys. 60, 481 (1919).
[CrossRef]

Wilkinson, C. D. W.

C. F. Quate, C. D. W. Wilkinson, D. K. Winslow, Proc. IEEE 53, 1604 (1965).
[CrossRef]

Winslow, D. K.

C. F. Quate, C. D. W. Wilkinson, D. K. Winslow, Proc. IEEE 53, 1604 (1965).
[CrossRef]

Ann. Phys.

H. Weyl, Ann. Phys. 60, 481 (1919).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

M. G. Cohen, E. I. Gordon, Bell Syst. Tech. J. 44, 693 (1965).

R. W. Dixon, E. I. Gordon, Bell Syst. Tech. J. 46, 367 (1967).

IEEE J. Quantum Electron.

D. M. Henderson, IEEE J. Quantum Electron. QE-8, 184 (1972).
[CrossRef]

D. Maydan, IEEE J. Quantum Electron. QE-6, 15 (1970).
[CrossRef]

R. W. Dixon, IEEE J. Quantum Electron. QE-3, 85 (1967).
[CrossRef]

IEEE Spectrum

R. A. Adler, IEEE Spectrum 4, 42 (1967).
[CrossRef]

IEEE Trans. Sonics Ultrason.

W. R. Klein, B. D. Cook, IEEE Trans. Sonics Ultrason. SU-14, 123 (1976).

IRE Nat. Conv. Rec.

L. B. Lambert, IRE Nat. Conv. Rec. 10, Part 6 (1962).

IRE Trans. Ultrason. Eng.

A. H. Rosenthal, IRE Trans. Ultrason. Eng. 8, 1 (1961).
[CrossRef]

J. Acoust. Soc. Am.

H. V. Hance, J. K. Parks, J. Acoust. Soc. Am. 38, 14 (1965).
[CrossRef]

Proc. IEEE

C. F. Quate, C. D. W. Wilkinson, D. K. Winslow, Proc. IEEE 53, 1604 (1965).
[CrossRef]

Rep. Prog. Phys.

C. J. Bouwkamp, Rep. Prog. Phys. 17, 35 (1954).
[CrossRef]

SPIE Proc.

J. A. Lucero, J. A. Duardo, R. V. Johnson, SPIE Proc. 90, 32 (1976).
[CrossRef]

Other

P. F. Panter, Modulation, Noise, and Spectral Analysis (McGraw-Hill, New York, 1965), Chap. 5.

R. W. Damon, W. T. Maloney, D. H. McMahon, in Physical Acoustics, Vol. 7 (Academic, New York, 1970), pp. 277–280

A. Korpel, in Applied Solid State Science, Vol. 3 (Academic, New York, 1972), pp. 100–103.

J. Randolph, J. Morrison, “Spatial and Temporal Response of Acousto-Optics Devices,” presented at Electro-Optics 1971, New York Colliseum (September 1971).

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Figures (27)

Fig. 1
Fig. 1

Block diagram of the acoustooptic modulator.

Fig. 2
Fig. 2

Definition of the Thin-grating model.

Fig. 3
Fig. 3

The diffraction orders of the scattered light beam.

Fig. 4
Fig. 4

Stroboscopic schlieren photographs of the sound field. Shown in these photographs are end-on views of two representative glass modulators, with the transducer on the right. These photographs were taken by the techniques described in Ref. 22.

Fig. 5
Fig. 5

Temporal frequency spectrum of the sound field.

Fig. 6
Fig. 6

Temporal frequency spectrum of the scattered light field.

Fig. 7
Fig. 7

Bragg regime modulator. When the sound field is thick, the incident light energy is selectively coupled into only one of the several diffraction orders.

Fig. 8
Fig. 8

Light throughput efficiency. Fast temporal response can be achieved by focusing the incident light beam, but the coupling efficiency degrades. The A parameter, defined in the text, is a measure of the degree of focusing.

Fig. 9
Fig. 9

Overlap of the deflected and undeflected light beams. Incomplete separation of the diffraction orders degrades the extinction of the modulated light beam. The B parameter, defined in the text, is a measure of the separation of the diffraction orders.

Fig. 10
Fig. 10

The static contrast (extinction) ratio.

Fig. 11
Fig. 11

The modulator's frequency response kernel (A = ¼π sequence). The modulator's temporal response to an arbitrary video signal (in the limit of low scattering efficiency) is completely specified by a 2-D frequency response kernel, shown here in isometric projection.

Fig. 12
Fig. 12

The modulator's frequency response kernel (A = ½π sequence). This kernel is similar to an MTF. However, two independent frequency variables are required to characterize the harmonic coupling which is introduced by the square law nonlinearity in the optical detector.

Fig. 13
Fig. 13

The modulator's frequency response kernel (orthographic projection). Orientation 1 refers to scattering into the +1 diffraction order; orientation 2 refers to scattering into the −1 diffraction order.

Fig. 14
Fig. 14

Modulator response to a sinusoidal video signal with dc offset (A = ¼π sequence).

Fig. 15
Fig. 15

Modulator response to a sinusoidal video signal with dc offset (A = ½π sequence).

Fig. 16
Fig. 16

Response time adjustment factor. The thin grating model (A = 0) of the acoustooptic coupling can yield reasonably accurate temporal response predictions if the true incident light beam radius w is replaced by an effectively larger radius. The larger radius accounts for the acoustic transit time contribution of the finite sound field depth L. The required rescaling is shown in this figure. Res caling is assumed in the A = 0 curves in Figs. 14, 15, 17, and 18.

Fig. 17
Fig. 17

Modulator response to a square pulse video train (A = ¼π sequence).

Fig. 18
Fig. 18

Modulator response to a square pulse video train (A = ½π sequence).

Fig. 19
Fig. 19

The Cohen-Gordon technique for mapping the angular scattering windows.

Fig. 20
Fig. 20

The scattering windows for a Bragg modulator.

Fig. 21
Fig. 21

The scatting windows for a Raman-Nath modulator.

Fig. 22
Fig. 22

Scattering in a Bragg modulator. The far-field diffraction profile of the scattered light is defined by the product of the incident light diffraction profile with the modulator's scattering window.

Fig. 23
Fig. 23

Focusing the incident light beam. When the incident light beam is well collimated (top sequence in this illustration), its far-field diffraction pattern (shaded Gaussian) is narrow compared with the angular passband defined by the modulator's angular scattering window (top center sketch in this figure). As a result, the scattered light profile is negligibly distorted with respect to the incident light profile (top right illustration). When the incident light is focused hard to minimize the modulator response time (bottom sequence), the spread of the corresponding diffraction pattern is increased (shaded Gaussian, bottom sequence). The angular passband of the modulator filters out a significant fraction of the incident light power (bottom center illustration) and distorts the scattered light profile (solid line, bottom right illustration) compared with the incident light profile (dashed line).

Fig. 24
Fig. 24

Splitting the quiescent window by video modulation.

Fig. 25
Fig. 25

Splitting the scattered light beam profile by video modulation. A sinusoidal video signal is assumed in Figs. 2126.

Fig. 26
Fig. 26

The resulting evolution of the scattered light's far-field diffraction profile.

Fig. 27
Fig. 27

Carrier frequency response of an acoustooptic deflector.

Equations (49)

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G out ( θ out , f out ) = G in ( θ in , f in ) G a ( θ a , f a ) ,
f out = f in + f a ,
k out sin ( θ out ) = k in sin ( θ in ) + k a cos ( θ a ) ,
k out cos ( θ out ) = k in cos ( θ in ) k a sin ( θ a ) ,
k out = [ ( 2 π ) / c ] n out f out ,
k in = [ ( 2 π ) / c ] n in f in ,
k a = [ ( 2 π ) / υ ] f a ,
G in ( θ in , f in ) = in ( θ in ) δ ( f in c λ ) + in * ( θ in ) δ ( f in + c λ ) ,
in ( θ in ) = exp [ ( θ in Θ in ) 2 / θ o 2 ] ,
θ o = λ / ( π w ) ,
S ( x = 0 , z , t ) = W ( z ) V ( t ) cos ( 2 π f c t ) ,
ν ( f υ ) = dtV ( t ) exp ( 2 π i f υ t ) ,
a ( θ α , f a ) = dzW ( z ) exp ( 2 π i z υ f a sin θ a ) ,
G a ( θ a , f a ) = ½ a ( θ a , f a ) [ ν ( f a f c ) + ν ( f a + f c ) ] .
ν ( f υ ) = ν * ( f υ ) ,
G out ( θ out , f out ) = G out * ( θ out , f out ) .
sin θ B = λ f a / ( 2 n υ ) .
f a = ± f c + f υ .
θ B = ± θ c + θ υ ,
θ c = λ f c / ( 2 n υ ) ,
θ υ = λ f υ / ( 2 n υ ) ,
G out ( θ out , f out ) = G + 1 ( θ out , f out ) + G 1 ( θ out , f out ) ,
G ± 1 ( θ out , f out ) = ½ ν ( f υ ) a ( θ a , f a ) in ( θ in )
f out = c λ ± f c + f υ ,
f a = ± f c + f υ ,
θ in = θ out 2 θ c 2 θ υ ,
θ a = θ out θ c θ υ .
a ( θ out , f υ ) = + dzW ( z ) exp ( i arg ) ,
arg = 2 π z υ ( ± f c + f υ ) ( θ out θ c θ υ ) .
W ( z ) δ ( z ) implies a ( θ out , f υ ) 1 ( thin grating model ) .
P ( t ) = d θ out | out ( θ out , t ) | 2 .
P ( t ) = d t 1 d t 2 R ( t t 1 , t t 2 ) V ( t 1 ) V * ( t 2 ) ,
P ( t ) = d f 1 d f 2 ρ ( f 1 , f 2 ) ν ( f 1 ) ν * ( f 2 ) exp [ 2 π i ( f 1 f 2 ) t ] ,
ρ ( f 1 , f 2 ) = d θ out in ( arg 1 ) a ( arg 2 ) in * ( arg 3 ) a * ( arg 4 ) ,
arg 1 = θ out 2 θ c 2 θ 1 ,
arg 2 = θ out θ c θ 1 ,
arg 3 = θ out 2 θ c 2 θ 2 ,
arg 4 = θ out θ c θ 2 ,
θ 1 , 2 = λ f 1 , 2 / ( 2 n υ ) ,
R ( t 1 , t 2 ) = R * ( t 2 , t 1 ) ,
ρ ( f 1 , f 2 ) = ρ * ( f 2 , f 1 ) ,
R ( t 1 , t 2 ) = R ( t 1 , t 2 ) ,
ρ ( f 1 , f 2 ) = ρ ( f 1 , f 2 ) ,
W ( z ) = rect ( z / L ) = { 1 for | z | ½ L 0 otherwise .
A = L θ c / w ,
B = υ / ( f c w ) .
N = A / B .
V ( t ) = 1 + M cos ( 2 π f υ t ) ,
P video ( t ) = V ( t ) 2 = 1 + ½ M 2 + 2 M cos ( 2 π f υ t ) + ½ M 2 cos ( 4 π f υ t ) .

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