Abstract

Acoustooptic 2-D profile shaping of a Gaussian laser beam has been achieved by two plane ultrasonic waves progressing in orthogonal directions. The spot size W of the Gaussian laser beam must be considerable less than the wavelength Λ of the ultrasonic wave at the acoustooptic interaction region. The ultrasonic cell is dealt with as a Raman-Nath 2-D phase grating but serves as a 2-D beam deflector in time for the interaction scheme of interest. The wave front of the Gaussian laser beam must be almost plane in the interaction region. The profile shaping condition is 0.15 ≦ (W/Λ) ≦ 0.30 only when the Raman-Nath parameter dependent on the ultrasonic power has values between υ = 1.0 and 2.0.

© 1985 Optical Society of America

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References

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  1. W. B. Veldkamp, “Laser Beam Profile Shaping with Interlaced Binary Diffraction Gratings,” Appl. Opt. 21, 3209 (1982).
    [CrossRef] [PubMed]
  2. R. L. Aagard, “Optimizing the Beam Shape in a Focused Coherent Optical System Method,” Appl. Opt. 13, 1633 (1974).
    [CrossRef] [PubMed]
  3. W. B. Veldkamp, “Laser Beam Profile Shaping with Binary Gratings,” Opt. Commun. 38, 381 (1981).
    [CrossRef]
  4. W. B. Veldkamp, C. J. Kastner, “Beam Profile Shaping for Laser Radars that Use Detector Arrays,” Appl. Opt. 21, 345 (1982).
    [CrossRef] [PubMed]
  5. W. W. Simmons, G. W. Leppelmeier, B. C. Johnson, “Optical Beam Shaping Devices Using Polarization Effects,” Appl. Opt. 13, 1629 (1974).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  8. D. Shafer, “Gaussian to Flattop Intensity Distributing Lens,” Opt. Laser Technol. 14, 159 (1982).
    [CrossRef]
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    [CrossRef]
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  13. M. A. Brezeale, E. A. Hiedemann, “Investigation of Progressive Waves by Light Refraction,” J. Acoust. Soc. Am. 30, 751 (1958).
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    [CrossRef]

1983 (1)

1982 (3)

1981 (4)

1980 (1)

1974 (2)

1968 (1)

L. E. Hargrove, “Effects of Ultrasonic Waves on Gaussian Light Beams with Diameter Comparable to Ultrasonic Wavelength,” J. Acoust. Soc. Am. 43, 847 (1968).
[CrossRef]

1966 (1)

1958 (1)

M. A. Brezeale, E. A. Hiedemann, “Investigation of Progressive Waves by Light Refraction,” J. Acoust. Soc. Am. 30, 751 (1958).
[CrossRef]

1935 (1)

C. V. Raman, N. S. N. Nath, “The Diffraction of Light by High Frequency Sound Waves: Part I–Part IV,” Proc. Ind. Acad. Sci. Sect. A, 406 (1935; Proc. Ind. Acad. Sci. Sect. A, 2, 413 (1935); Proc. Ind. Acad. Sci. Sect. A, 3, 75 (1936); Proc. Ind. Acad. Sci. Sect. A, 3, 119 (1936).

Aagard, R. L.

Brezeale, M. A.

M. A. Brezeale, E. A. Hiedemann, “Investigation of Progressive Waves by Light Refraction,” J. Acoust. Soc. Am. 30, 751 (1958).
[CrossRef]

Case, S. K.

Han, C.-Y.

Hargrove, L. E.

L. E. Hargrove, “Effects of Ultrasonic Waves on Gaussian Light Beams with Diameter Comparable to Ultrasonic Wavelength,” J. Acoust. Soc. Am. 43, 847 (1968).
[CrossRef]

Haugen, P. R.

Hiedemann, E. A.

M. A. Brezeale, E. A. Hiedemann, “Investigation of Progressive Waves by Light Refraction,” J. Acoust. Soc. Am. 30, 751 (1958).
[CrossRef]

Ishii, Y.

Johnson, B. C.

Kastner, C. J.

Kogelnik, H. W.

Leppelmeier, G. W.

Li, T.

Løkberg, O. J.

Murata, K.

Nath, N. S. N.

C. V. Raman, N. S. N. Nath, “The Diffraction of Light by High Frequency Sound Waves: Part I–Part IV,” Proc. Ind. Acad. Sci. Sect. A, 406 (1935; Proc. Ind. Acad. Sci. Sect. A, 2, 413 (1935); Proc. Ind. Acad. Sci. Sect. A, 3, 75 (1936); Proc. Ind. Acad. Sci. Sect. A, 3, 119 (1936).

Ohtsuka, Y.

Y. Ohtsuka, A. Tanone, “Acoustooptic Intensity Modification of a Gaussian Laser Beam,” Opt. Commun. 39, 70 (1981).
[CrossRef]

Raman, C. V.

C. V. Raman, N. S. N. Nath, “The Diffraction of Light by High Frequency Sound Waves: Part I–Part IV,” Proc. Ind. Acad. Sci. Sect. A, 406 (1935; Proc. Ind. Acad. Sci. Sect. A, 2, 413 (1935); Proc. Ind. Acad. Sci. Sect. A, 3, 75 (1936); Proc. Ind. Acad. Sci. Sect. A, 3, 119 (1936).

Rhodes, P. W.

Scott, P. W.

Shafer, D.

D. Shafer, “Gaussian to Flattop Intensity Distributing Lens,” Opt. Laser Technol. 14, 159 (1982).
[CrossRef]

Shealy, D. L.

Simmons, W. W.

Southwell, W. H.

Tanone, A.

Y. Ohtsuka, A. Tanone, “Acoustooptic Intensity Modification of a Gaussian Laser Beam,” Opt. Commun. 39, 70 (1981).
[CrossRef]

Veldkamp, W. B.

Appl. Opt. (9)

J. Acoust. Soc. Am. (2)

L. E. Hargrove, “Effects of Ultrasonic Waves on Gaussian Light Beams with Diameter Comparable to Ultrasonic Wavelength,” J. Acoust. Soc. Am. 43, 847 (1968).
[CrossRef]

M. A. Brezeale, E. A. Hiedemann, “Investigation of Progressive Waves by Light Refraction,” J. Acoust. Soc. Am. 30, 751 (1958).
[CrossRef]

Opt. Commun. (2)

W. B. Veldkamp, “Laser Beam Profile Shaping with Binary Gratings,” Opt. Commun. 38, 381 (1981).
[CrossRef]

Y. Ohtsuka, A. Tanone, “Acoustooptic Intensity Modification of a Gaussian Laser Beam,” Opt. Commun. 39, 70 (1981).
[CrossRef]

Opt. Laser Technol. (1)

D. Shafer, “Gaussian to Flattop Intensity Distributing Lens,” Opt. Laser Technol. 14, 159 (1982).
[CrossRef]

Proc. Ind. Acad. Sci. Sect. A (1)

C. V. Raman, N. S. N. Nath, “The Diffraction of Light by High Frequency Sound Waves: Part I–Part IV,” Proc. Ind. Acad. Sci. Sect. A, 406 (1935; Proc. Ind. Acad. Sci. Sect. A, 2, 413 (1935); Proc. Ind. Acad. Sci. Sect. A, 3, 75 (1936); Proc. Ind. Acad. Sci. Sect. A, 3, 119 (1936).

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Figures (5)

Fig. 1
Fig. 1

Optical system for beam profile shaping.

Fig. 2
Fig. 2

Three-dimensional display of time-averaged intensity profile for (W/Λ) = 0.5.

Fig. 3
Fig. 3

Three-dimensional display of time-averaged intensity profile for (W/Λ) = 0.3.

Fig. 4
Fig. 4

Longitudinal cross-sectional time-averaged intensity profile. The four profiles from top to bottom in each figure are plotted, respectively with υ = 0, 1.0, 1.5, and 2.0.

Fig. 5
Fig. 5

Comparison of measured and computed profiles. The upper pair denotes the result in the 45° direction from the x axis. The right and left traces show those obtained with experiments and computations, respectively.

Equations (20)

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L max 2 = n 0 Λ 4 G 2 / 4 υ λ 2 ,
L max / Λ 1
n ( x 0 , y 0 ; t ) = n 0 + Δ n 1 sin ( Ω 1 t K 1 x 0 ) + Δ n 2 sin ( Ω 2 t K 2 y 0 + δ ) ,
T ( x 0 , y 0 ; t ) = exp [ i k L n ( x 0 , y 0 ; t ) ] .
E ( x 0 , y 0 ; t ) = W 0 / W ( z 0 ) exp [ ( x 0 2 + y 0 2 ) ( 1 / W 2 ( z 0 ) + i k / 2 R ) + i ( ω t k z 0 + ϕ ) ] ,
W 2 ( z 0 ) = W 0 2 [ 1 + ( λ z 0 / π W 0 2 ) 2 ] , R ( z 0 ) = z 0 [ 1 + ( π W 0 2 / λ z 0 ) 2 ] , ϕ = tan 1 ( λ z 0 / π W 0 2 ) ,
U ( x , y , z ; t ) = C + + E ( x 0 , y 0 , z 0 ; t ) T ( x 0 , y 0 ; t ) × exp [ i ( k / z ) ( x x 0 + y y 0 ) ] d x 0 d y 0 ,
exp ( i υ sin θ ) = m = J m ( υ ) exp ( i m θ )
T ( x 0 , y 0 ; t ) = m = n = J m ( υ 1 ) J n ( υ 2 ) exp { i [ ( m Ω 1 + n Ω 2 ) t + m δ + k n 0 L m K 1 x 0 n K 2 y 0 ] } ,
α = 1 / W 2 ( z 0 ) + i k / 2 R .
U ( x , y , z ; t ) = C m = n = J m ( υ 1 ) J n ( υ 2 ) × exp { i [ ( m Ω 1 + n Ω 2 ) t + m δ ] } × + + exp { x 0 2 α + i x 0 [ ( k x / z ) + m K 1 ] y 0 2 α + i y 0 [ ( k y / z ) + n K 2 ] } d x 0 d y 0 ,
| C | 2 = W 0 2 / [ λ z W ( z 0 ) ] 2 .
+ exp { x 0 2 α + i x 0 [ ( k x / z ) + m K ] } d x 0 = π / α exp { ( π 2 / α ) [ ( x / λ z ) + ( m / Λ ) ] 2 }
U ( x , y , z ; t ) = C m = n = J m ( υ 1 ) J n ( υ 2 ) ( π / α ) × exp ( i [ ( m Ω 1 + n Ω 2 ) t + m δ ] ( π 2 / α ) { [ ( x / λ z ) + ( m / Λ 1 ) ] 2 + [ ( y / λ z ) + ( n / Λ 2 ) ] 2 } ) .
1 / α = W 2 [ 1 ( i π W 2 / λ R ) ] 1 + ( π W 2 / λ R ) 2 ,
1 + ( π W 2 / λ R ) 2 = ( W / W 0 ) 2 .
1 / α = W 0 2 [ 1 ( i π W 2 / λ R ) ] .
I ( x , y , z ) = | U ( x , y , z ; t ) | 2 / | U ( 0 , 0 , z ; t ) | 2 υ 1 = υ 2 = 0 = m = n = J m 2 ( υ 1 ) J m 2 ( υ 2 ) × exp { 2 π 2 ( W 0 / Λ 1 ) 2 [ ( Λ 1 x / λ z ) + m ] 2 2 π 2 ( W 0 / Λ 2 ) 2 [ ( Λ 2 y / λ z ) + n ] 2 } ,
exp { i [ ( m h ) Ω 1 + ( n q ) Ω 2 ] t } = { 0 when m h and n q , 1 when m = h and n = q .
I ( x , y , z ) = m = n = J m 2 ( υ ) J n 2 ( υ ) exp ( 2 π 2 ( W 0 / Λ ) 2 × { [ ( Λ x / λ z ) + m ] 2 + [ ( Λ y / λ z ) + n ] 2 } ) .

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