Abstract

We theoretically demonstrate beam shaping through adaptive feedback in an acousto-optic device with electrical feedback by using experimentally determined parameters. Cases of positive feedback and negative feedback from undiffracted and diffracted orders are investigated. In addition, we demonstrate the dependence of the final value of the induced grating strength in the acousto-optic cell on the feedback parameters.

© 2005 Optical Society of America

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References

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  1. V. I. Balakshy, “Scanning of images,” Sov. J. Quantum Electron. 6, 965–971 (1979).
  2. J. N. Mait, D. W. Prather, R. A. Athale, “Acousto-optic processing with electronic feedback for morphological filtering,” Appl. Opt. 31, 5688–5699 (1992).
    [CrossRef] [PubMed]
  3. J. Xia, D. Dunn, T.-C. Poon, P. P. Banerjee, “Image edge enhancement by Bragg diffraction,” Opt. Commun. 128, 1–7 (1996).
    [CrossRef]
  4. P. P. Banerjee, D. Cao, T.-C. Poon, “Basic image-processing operations by the use of acousto-optics,” Appl. Opt. 36, 3086–3089 (1997).
    [CrossRef] [PubMed]
  5. D. Cao, P. P. Banerjee, T-C Poon, “Image edge enhancement with two cascaded acousto-optic cells with counterpropagating sound,” Appl. Opt. 37, 3007–3014 (1998).
    [CrossRef]
  6. P. P. Banerjee, D. Cao, T.-C. Poon, “Notch spatial filtering with an acousto-optic modulator,” Appl. Opt. 37, 7532–7537 (1998).
    [CrossRef]
  7. S. Case, “Fourier processing in the object plane,” Opt. Lett. 4, 286–288 (1979).
    [CrossRef] [PubMed]
  8. R. Gonzalez, R. Woods, Digital Image Processing (Addison-Wesley, New York, 1993).
  9. D. Cao, “AO interaction transfer functions and applications to image processing,” Ph.D. thesis (University of Alabama, Huntsville, Ala., 1998).
  10. M. McNeill, T.-C. Poon, “Gaussian-beam shaping by acousto-optic Bragg diffraction,” Appl. Opt. 33, 4508–4515 (1994).
    [CrossRef] [PubMed]
  11. J. Chrostowski, C. Delisle, “Bistable optical switching based on Bragg diffraction,” Opt. Commun. 41, 71–81 (1982).
    [CrossRef]
  12. J. Chrostowski, R. Vallee, C. Delisle, “Self-pulsing and chaos in acousto-optic bistability,” Can. J. Phys. 63, 1143–1148 (1983).
    [CrossRef]
  13. H. Jerominek, C. Delisle, J. Y. D. Pomerleau, R. Tremblay, “Bistability, optical regenerative oscillations and chaos in an integrated acousto-optic device,” Can. J. Phys. 63, 227–233 (1985).
    [CrossRef]
  14. P. P. Banerjee, U. Banerjee, H. Kaplan, “Response of an acousto-optic device with feedback to time-varying inputs,” Appl. Opt. 31, 1842–1852 (1992).
    [CrossRef] [PubMed]
  15. T.-C. Poon, P. P. Banerjee, Contemporary Optical Image Processing with matlab (Elsevier Science, Amsterdam, 2001), pp. 111−118.

1998

1997

1996

J. Xia, D. Dunn, T.-C. Poon, P. P. Banerjee, “Image edge enhancement by Bragg diffraction,” Opt. Commun. 128, 1–7 (1996).
[CrossRef]

1994

1992

1985

H. Jerominek, C. Delisle, J. Y. D. Pomerleau, R. Tremblay, “Bistability, optical regenerative oscillations and chaos in an integrated acousto-optic device,” Can. J. Phys. 63, 227–233 (1985).
[CrossRef]

1983

J. Chrostowski, R. Vallee, C. Delisle, “Self-pulsing and chaos in acousto-optic bistability,” Can. J. Phys. 63, 1143–1148 (1983).
[CrossRef]

1982

J. Chrostowski, C. Delisle, “Bistable optical switching based on Bragg diffraction,” Opt. Commun. 41, 71–81 (1982).
[CrossRef]

1979

V. I. Balakshy, “Scanning of images,” Sov. J. Quantum Electron. 6, 965–971 (1979).

S. Case, “Fourier processing in the object plane,” Opt. Lett. 4, 286–288 (1979).
[CrossRef] [PubMed]

Athale, R. A.

Balakshy, V. I.

V. I. Balakshy, “Scanning of images,” Sov. J. Quantum Electron. 6, 965–971 (1979).

Banerjee, P. P.

Banerjee, U.

Cao, D.

Case, S.

Chrostowski, J.

J. Chrostowski, R. Vallee, C. Delisle, “Self-pulsing and chaos in acousto-optic bistability,” Can. J. Phys. 63, 1143–1148 (1983).
[CrossRef]

J. Chrostowski, C. Delisle, “Bistable optical switching based on Bragg diffraction,” Opt. Commun. 41, 71–81 (1982).
[CrossRef]

Delisle, C.

H. Jerominek, C. Delisle, J. Y. D. Pomerleau, R. Tremblay, “Bistability, optical regenerative oscillations and chaos in an integrated acousto-optic device,” Can. J. Phys. 63, 227–233 (1985).
[CrossRef]

J. Chrostowski, R. Vallee, C. Delisle, “Self-pulsing and chaos in acousto-optic bistability,” Can. J. Phys. 63, 1143–1148 (1983).
[CrossRef]

J. Chrostowski, C. Delisle, “Bistable optical switching based on Bragg diffraction,” Opt. Commun. 41, 71–81 (1982).
[CrossRef]

Dunn, D.

J. Xia, D. Dunn, T.-C. Poon, P. P. Banerjee, “Image edge enhancement by Bragg diffraction,” Opt. Commun. 128, 1–7 (1996).
[CrossRef]

Gonzalez, R.

R. Gonzalez, R. Woods, Digital Image Processing (Addison-Wesley, New York, 1993).

Jerominek, H.

H. Jerominek, C. Delisle, J. Y. D. Pomerleau, R. Tremblay, “Bistability, optical regenerative oscillations and chaos in an integrated acousto-optic device,” Can. J. Phys. 63, 227–233 (1985).
[CrossRef]

Kaplan, H.

Mait, J. N.

McNeill, M.

Pomerleau, J. Y. D.

H. Jerominek, C. Delisle, J. Y. D. Pomerleau, R. Tremblay, “Bistability, optical regenerative oscillations and chaos in an integrated acousto-optic device,” Can. J. Phys. 63, 227–233 (1985).
[CrossRef]

Poon, T.-C.

Poon, T-C

Prather, D. W.

Tremblay, R.

H. Jerominek, C. Delisle, J. Y. D. Pomerleau, R. Tremblay, “Bistability, optical regenerative oscillations and chaos in an integrated acousto-optic device,” Can. J. Phys. 63, 227–233 (1985).
[CrossRef]

Vallee, R.

J. Chrostowski, R. Vallee, C. Delisle, “Self-pulsing and chaos in acousto-optic bistability,” Can. J. Phys. 63, 1143–1148 (1983).
[CrossRef]

Woods, R.

R. Gonzalez, R. Woods, Digital Image Processing (Addison-Wesley, New York, 1993).

Xia, J.

J. Xia, D. Dunn, T.-C. Poon, P. P. Banerjee, “Image edge enhancement by Bragg diffraction,” Opt. Commun. 128, 1–7 (1996).
[CrossRef]

Appl. Opt.

Can. J. Phys.

J. Chrostowski, R. Vallee, C. Delisle, “Self-pulsing and chaos in acousto-optic bistability,” Can. J. Phys. 63, 1143–1148 (1983).
[CrossRef]

H. Jerominek, C. Delisle, J. Y. D. Pomerleau, R. Tremblay, “Bistability, optical regenerative oscillations and chaos in an integrated acousto-optic device,” Can. J. Phys. 63, 227–233 (1985).
[CrossRef]

Opt. Commun.

J. Chrostowski, C. Delisle, “Bistable optical switching based on Bragg diffraction,” Opt. Commun. 41, 71–81 (1982).
[CrossRef]

J. Xia, D. Dunn, T.-C. Poon, P. P. Banerjee, “Image edge enhancement by Bragg diffraction,” Opt. Commun. 128, 1–7 (1996).
[CrossRef]

Opt. Lett.

Sov. J. Quantum Electron.

V. I. Balakshy, “Scanning of images,” Sov. J. Quantum Electron. 6, 965–971 (1979).

Other

R. Gonzalez, R. Woods, Digital Image Processing (Addison-Wesley, New York, 1993).

D. Cao, “AO interaction transfer functions and applications to image processing,” Ph.D. thesis (University of Alabama, Huntsville, Ala., 1998).

T.-C. Poon, P. P. Banerjee, Contemporary Optical Image Processing with matlab (Elsevier Science, Amsterdam, 2001), pp. 111−118.

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Figures (10)

Fig. 1
Fig. 1

Diagram of the sound–light interaction.

Fig. 2
Fig. 2

AO transfer functions as a function of spatial frequency for (a) the 0th order and (b) the −1st order with Q ≅ 21 (corresponding to acoustic frequency of 40 MHz).

Fig. 3
Fig. 3

(a) Spectrum of an input Gaussian profile with a beam waist of 75 µm. (b) Spectrum of a processed profile with α0 = 0.35π at acoustic frequencies 30, 40, 50, and 60 MHz, which correspond to Q = 11, 21, 32, and 46, respectively.

Fig. 4
Fig. 4

Schematic of AO interaction in the presence of feedback, according to Eq. (16).

Fig. 5
Fig. 5

Details of the feedback stage necessary for the beam-shaping scheme. The input is Vdet = RκPdet, with κ as the responsivity of the photodetector and Pdet as the detected power. The output is Vfdbk.

Fig. 6
Fig. 6

Effect of feedback on the AO transfer function: (a) 0th order and (b) −1st order. The sound frequency is taken as 45 MHz, which is equivalent to Q = 26. In this simulation α0 = 0.35π, and the incident power is 10 mW.

Fig. 7
Fig. 7

Beam shapes for 75-µm incident beam of 10 mW, corresponding to the parameters in Fig. 6 for (a) 0th-order feedback and (b) −1st-order feedback. The far-field distance is 1 m.

Fig. 8
Fig. 8

Cross section of the 0th-order beam shapes from Fig. 7 for (a) 0th-order feedback and (b) −1st-order feedback.

Fig. 9
Fig. 9

(a) Δα variation with the feedback amplifier gain factor G for various initial α0 for 0th-order feedback, (b) the corresponding normalized power detected in the undiffracted order, (c) the Δα variation with the feedback amplifier gain factor G for various initial α0 for −1st-order feedback, and (d) the corresponding normalized power detected in the undiffracted order.

Fig. 10
Fig. 10

Experimental results of shaping a 75-µm Gaussian beam as observed by the Spiricon beam profiler at 75 cm away from the exit surface of the AO cell: (a) the three-dimensional plot of the beam shape and (b) the 2-D cross section. This result is for Vrms = 10.67 V. Note that beam shaping is only in the x direction, whereas the y direction remains Gaussian in that AO interaction is only in one dimension.

Equations (26)

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2 E t 2 υ 2 2 E = ( 0 ) 2 E t 2 ,
E ( x , z ; t ) = m Re { Ψ m ( x , y , z ) × exp [ j ( ω m t k m x x k m z z ) ] } â y .
ω m = ω 0 + m Ω ,
ϕ B K 2 k = λ 0 2 Λ n 0 , ϕ m ϕ inc + m K / k ,
ɛ = ɛ 0 C Re [ S e ( x , z ) exp j ( Ω t K x ) ] ,
2 Ψ m x 2 2 j ( k m x Ψ m x + k m z Ψ m z ) + ( k 2 C 2 ) × { A Ψ ( m 1 ) exp [ j ( k ( m 1 ) z k m z ) z ] + A * Ψ ( m + 1 ) exp [ j ( k ( m + 1 ) z k m z ) z ] } + 2 Ψ m y 2 + 2 Ψ m z 2 = 0 .
| 2 Ψ m z 2 | | k m z Ψ m z | .
2 Ψ m x 2 2 j ( k m x Ψ m x + k m z Ψ m z ) + ( k 2 C 2 ) × { A Ψ ( m 1 ) exp [ j ( k ( m 1 ) z k m z ) z ] + A * Ψ ( m + 1 ) exp [ j ( k ( m + 1 ) z k m z ) z ] } + 2 Ψ m y 2 0 .
Ψ ̂ m z = j ( k x 2 + 2 k x k m x ) 2 k m z Ψ ̂ m + j k y 2 2 k m z Ψ ̂ m j D Ψ ̂ m + 1 j E Ψ ̂ m 1 ,
D = ( k C A * / 4 ) exp { j k [ cos ( ϕ m + 1 ) cos ( ϕ m ) ] z } ,
E = ( k C A / 4 ) exp { j k [ cos ( ϕ m 1 ) cos ( ϕ m ) ] z } .
Ψ ̂ m ( k x , k y ; z ) = J x y [ Ψ m ( x , y , z ) ] = Ψ m ( x , y , z ) exp ( j k x x + j k y y ) d x d y .
Ψ ̂ 0 z = j ( k x 2 + 2 k x k 0 x ) 2 k 0 z Ψ ̂ 0 + j k y 2 2 k 0 z Ψ ̂ 0 j E Ψ ̂ 1 ,
Ψ ̂ 1 z = j ( k x 2 + 2 k x k 1 x ) 2 k 1 z Ψ ̂ 1 + j k y 2 2 k 1 z Ψ ̂ 1 j D Ψ ̂ 0 ,
H 0 ( k x , z = L ) = exp { j [ ( k x 2 + k y 2 ) L 2 k ] } × ( cos { [ ( k x k 0 x L k ) 2 + ( α 2 ) 2 ] 1 / 2 } + ( j k x k 0 x L k ) sin { [ ( k x k 0 x L k ) 2 + ( α 2 ) 2 ] 1 / 2 } [ ( k x k 0 x L k ) 2 + ( α 2 ) 2 ] 1 / 2 ) ,
H 1 ( k x , z = L ) = exp { j [ ( k x 2 + k y 2 ) L 2 k ] } × ( j α 2 ) sin { [ ( k x k 0 x L k ) 2 + ( α 2 ) 2 ] 1 / 2 } [ ( k x k 0 x L k ) 2 + ( α 2 ) 2 ] 1 / 2 ,
α = α 0 ± Δ α .
V det = κ R P det ,
V fdbk = [ ( κ P det ) R G ] = [ ( κ P det ) R ( R 2 R 1 ) ] .
α = K ( V rms ) 2 = K ( V gen + V fdbk ) 2 ,
α = α 0 + Δ α = K V gen 2 + 2 K V gen V fdbk + K V fdbk 2
α 0 = K V gen 2 ,
Δ α = 2 K V gen V fdbk + K V fdbk 2 .
I 0 = I inc cos 2 ( α / 2 ) ,
I 1 = I inc sin 2 ( α / 2 ) ,
α = K V rms 2 ,

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