Abstract

We study how Gaussian laser-beam profiles can be modified into a desired form using acousto-optic Bragg diffraction. By exploiting the angular dependence of Bragg diffraction of plane waves by acoustic gratings, we demonstrate that the conversion from a Gaussian-profile beam into either a near-field or a far-field flattop profile is possible.

© 1994 Optical Society of America

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References

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  1. Y. Ohtsuka, A. Tanone, “Acousto-optic intensity modification of a Gaussian laser beam,” Opt. Commun. 39, 70–74 (1981).
    [CrossRef]
  2. Y. Ohtsuka, A. Yasatomo, Y. Imai, “Acousto-optic two-dimensional profile shaping of a Gaussian laser beam,” Appl. Opt. 24, 2813–2819 (1985).
    [CrossRef] [PubMed]
  3. C.-Y. Han, Y. Ishii, K. Murata, “Reshaping collimated laser beams with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
    [CrossRef] [PubMed]
  4. W. B. Veldkamp, C. J. Kastner, “Beam profile shaping for laser radars that use detector arrays,” Appl. Opt. 21, 345–355 (1982).
    [CrossRef] [PubMed]
  5. W. B. Veldkamp, “Laser beam profile shaping with interlaced diffraction gratings,” Appl. Opt. 21, 3209–3212 (1982).
    [CrossRef] [PubMed]
  6. D. Shafer, “Gaussian to flattop intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
    [CrossRef]
  7. L. E. Hargrove, “Effects of ultrasonic waves on Gaussian light beams with diameter comparable to ultrasonic wavelength,” J. Acoust. Soc. Am. 43, 847–851 (1967).
    [CrossRef]
  8. S. R. Jahan, M. A. Karim, “Refracting systems for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21, 27–30 (1989).
    [CrossRef]
  9. M. A. Karim, A. M. Hanafi, F. Hussain, S. Mustafa, Z. Samberid, N. M. Zain, “Realization of a uniform circular source using a two-dimensional binary filter,” Opt. Lett. 10, 470–472 (1985).
    [CrossRef] [PubMed]
  10. B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. 4, 1400–1403 (1965).
    [CrossRef]
  11. R. L. Aagard, “Methods of optimizing the beam shape in a focused coherent optical system,” Appl. Opt. 13, 1633–1638 (1974).
    [CrossRef] [PubMed]
  12. E. Tervonen, A. T. Friberg, J. Turunen, “Acousto-optic conversion of laser beams into flat-top beams,” J. Mod. Opt. 40, 625–635 (1993).
    [CrossRef]
  13. A. Korpel, Acousto-Optics (Dekker, New York, 1988), pp. 95–121.
  14. T.-C. Poon, A. Korpel, “Feynman diagram approach to acousto-optic scattering in the near-Bragg region,” J. Opt. Soc. Am. 71, 1202–1208 (1981).
    [CrossRef]
  15. T.-C. Poon, M. R. Chatterjee, “Transfer function approach to acousto-optic Bragg diffraction of finite optical beams using Fourier integrals,” presented at the 1988 International Union of Radio Science Meeting, Syracuse, N.Y., June 1988.
  16. M. R. Chatterjee, T.-C. Poon, D. N. Sitter, “Transfer function formalism for strong acousto-optic Bragg diffraction of light beams with arbitrary profiles,” Acustica 71, 81–92 (1990).
  17. W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 723–733 (1967).
  18. C. V. Raman, N. S. N. Nath, “The diffraction of light by high frequency sound waves,” Proc. Indian Acad. Sci. Sect. A 2, 406–420 (1935); Proc. Indian Acad. Sci. Sect. A 3, 75–84, 119–125, 459–465 (1936).
  19. P. Phariseau, “On the diffraction of light by progressive supersonic waves,” Proc. Indian Acad. Sci. Sect. A 44, 165–170 (1956).
  20. R. R. Aggarwal, “Diffraction of light by ultrasonic waves (deduction of different theories from the generalized theory of Raman and Nath),” Proc. Indian Acad. Sci. Sect. A 31, 417–426 (1950).
  21. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J 48, 2909–2947 (1969).
  22. P. P. Banerjee, C. Tarn, “A Fourier transform approach to acousto-optic interaction in the presence of propagational diffraction,” Acustica 74181–191 (1991).
  23. S. Min, M. R. Chatterjee, “General integral formalism for acousto-optic and holographic Bragg scattering for arbitrary profiles and orientations,” Acustica 71, 81–92 (1990).
  24. L. N. Magdich, V. Y. Molchanov, “Diffraction of a divergent beam by intense acoustic waves,” Opt. Spectrosc. (USSR) 42, 299–302 (1977).
  25. B. Benlarbi, P. St. J. Russell, L. Solymar, “Bragg diffraction of finite beams by thick gratings: two rival theories,” Appl. Phys. B 28, 63–72 (1982).
    [CrossRef]
  26. M. R. Forshaw, “Diffraction of a narrower laser beam by a thick hologram: experimental results,” Opt. Commun. 12, 279–281 (1974).
    [CrossRef]
  27. R. S. Chu, T. Tamir, “Bragg diffraction of Gaussian beams by periodically modulated media,” J. Opt. Soc. Am. 66, 220–226 (1976).
    [CrossRef]
  28. R. S. Chu, T. Tamir, “Bragg diffraction of Gaussian beams by periodically modulated media for incidence close to a Bragg angle,” J. Opt. Soc. Am. 66, 1438–1440 (1976).
    [CrossRef]
  29. R. S. Chu, J. A. Kong, T. Tamir, “Diffraction of Gaussian beams by a periodically modulated layer,” J. Opt. Soc. Am. 67, 1555–1561 (1977).
    [CrossRef]
  30. M. G. Moharam, T. K. Gaylord, R. Magnusson, “Bragg diffraction of finite beams by thick gratings,” J. Opt. Soc. Am. 70, 300–304 (1980).
    [CrossRef]
  31. B. Benlarbi, P. St. J. Russell, L. Solymar, “Bragg diffraction of Gaussian beams by thick gratings: numerical evaluations by plane-wave decomposition,” Appl. Phys. B 28, 383–390 (1982).
    [CrossRef]
  32. D. G. Hawkins, “Finite beamwidth effects in bulk acousto-optic interactions,” J. Opt. Soc. Am. A 70, 1611 (1980).

1993 (1)

E. Tervonen, A. T. Friberg, J. Turunen, “Acousto-optic conversion of laser beams into flat-top beams,” J. Mod. Opt. 40, 625–635 (1993).
[CrossRef]

1991 (1)

P. P. Banerjee, C. Tarn, “A Fourier transform approach to acousto-optic interaction in the presence of propagational diffraction,” Acustica 74181–191 (1991).

1990 (2)

S. Min, M. R. Chatterjee, “General integral formalism for acousto-optic and holographic Bragg scattering for arbitrary profiles and orientations,” Acustica 71, 81–92 (1990).

M. R. Chatterjee, T.-C. Poon, D. N. Sitter, “Transfer function formalism for strong acousto-optic Bragg diffraction of light beams with arbitrary profiles,” Acustica 71, 81–92 (1990).

1989 (1)

S. R. Jahan, M. A. Karim, “Refracting systems for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21, 27–30 (1989).
[CrossRef]

1985 (2)

1983 (1)

1982 (5)

W. B. Veldkamp, C. J. Kastner, “Beam profile shaping for laser radars that use detector arrays,” Appl. Opt. 21, 345–355 (1982).
[CrossRef] [PubMed]

W. B. Veldkamp, “Laser beam profile shaping with interlaced diffraction gratings,” Appl. Opt. 21, 3209–3212 (1982).
[CrossRef] [PubMed]

D. Shafer, “Gaussian to flattop intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
[CrossRef]

B. Benlarbi, P. St. J. Russell, L. Solymar, “Bragg diffraction of finite beams by thick gratings: two rival theories,” Appl. Phys. B 28, 63–72 (1982).
[CrossRef]

B. Benlarbi, P. St. J. Russell, L. Solymar, “Bragg diffraction of Gaussian beams by thick gratings: numerical evaluations by plane-wave decomposition,” Appl. Phys. B 28, 383–390 (1982).
[CrossRef]

1981 (2)

Y. Ohtsuka, A. Tanone, “Acousto-optic intensity modification of a Gaussian laser beam,” Opt. Commun. 39, 70–74 (1981).
[CrossRef]

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

1980 (2)

D. G. Hawkins, “Finite beamwidth effects in bulk acousto-optic interactions,” J. Opt. Soc. Am. A 70, 1611 (1980).

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Bragg diffraction of finite beams by thick gratings,” J. Opt. Soc. Am. 70, 300–304 (1980).
[CrossRef]

1977 (2)

R. S. Chu, J. A. Kong, T. Tamir, “Diffraction of Gaussian beams by a periodically modulated layer,” J. Opt. Soc. Am. 67, 1555–1561 (1977).
[CrossRef]

L. N. Magdich, V. Y. Molchanov, “Diffraction of a divergent beam by intense acoustic waves,” Opt. Spectrosc. (USSR) 42, 299–302 (1977).

1976 (2)

1974 (2)

M. R. Forshaw, “Diffraction of a narrower laser beam by a thick hologram: experimental results,” Opt. Commun. 12, 279–281 (1974).
[CrossRef]

R. L. Aagard, “Methods of optimizing the beam shape in a focused coherent optical system,” Appl. Opt. 13, 1633–1638 (1974).
[CrossRef] [PubMed]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J 48, 2909–2947 (1969).

1967 (2)

L. E. Hargrove, “Effects of ultrasonic waves on Gaussian light beams with diameter comparable to ultrasonic wavelength,” J. Acoust. Soc. Am. 43, 847–851 (1967).
[CrossRef]

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 723–733 (1967).

1965 (1)

1956 (1)

P. Phariseau, “On the diffraction of light by progressive supersonic waves,” Proc. Indian Acad. Sci. Sect. A 44, 165–170 (1956).

1950 (1)

R. R. Aggarwal, “Diffraction of light by ultrasonic waves (deduction of different theories from the generalized theory of Raman and Nath),” Proc. Indian Acad. Sci. Sect. A 31, 417–426 (1950).

1935 (1)

C. V. Raman, N. S. N. Nath, “The diffraction of light by high frequency sound waves,” Proc. Indian Acad. Sci. Sect. A 2, 406–420 (1935); Proc. Indian Acad. Sci. Sect. A 3, 75–84, 119–125, 459–465 (1936).

Aagard, R. L.

Aggarwal, R. R.

R. R. Aggarwal, “Diffraction of light by ultrasonic waves (deduction of different theories from the generalized theory of Raman and Nath),” Proc. Indian Acad. Sci. Sect. A 31, 417–426 (1950).

Banerjee, P. P.

P. P. Banerjee, C. Tarn, “A Fourier transform approach to acousto-optic interaction in the presence of propagational diffraction,” Acustica 74181–191 (1991).

Benlarbi, B.

B. Benlarbi, P. St. J. Russell, L. Solymar, “Bragg diffraction of finite beams by thick gratings: two rival theories,” Appl. Phys. B 28, 63–72 (1982).
[CrossRef]

B. Benlarbi, P. St. J. Russell, L. Solymar, “Bragg diffraction of Gaussian beams by thick gratings: numerical evaluations by plane-wave decomposition,” Appl. Phys. B 28, 383–390 (1982).
[CrossRef]

Chatterjee, M. R.

S. Min, M. R. Chatterjee, “General integral formalism for acousto-optic and holographic Bragg scattering for arbitrary profiles and orientations,” Acustica 71, 81–92 (1990).

M. R. Chatterjee, T.-C. Poon, D. N. Sitter, “Transfer function formalism for strong acousto-optic Bragg diffraction of light beams with arbitrary profiles,” Acustica 71, 81–92 (1990).

T.-C. Poon, M. R. Chatterjee, “Transfer function approach to acousto-optic Bragg diffraction of finite optical beams using Fourier integrals,” presented at the 1988 International Union of Radio Science Meeting, Syracuse, N.Y., June 1988.

Chu, R. S.

Cook, B. D.

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 723–733 (1967).

Forshaw, M. R.

M. R. Forshaw, “Diffraction of a narrower laser beam by a thick hologram: experimental results,” Opt. Commun. 12, 279–281 (1974).
[CrossRef]

Friberg, A. T.

E. Tervonen, A. T. Friberg, J. Turunen, “Acousto-optic conversion of laser beams into flat-top beams,” J. Mod. Opt. 40, 625–635 (1993).
[CrossRef]

Frieden, B. R.

Gaylord, T. K.

Han, C.-Y.

Hanafi, A. M.

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, 847–851 (1967).
[CrossRef]

Hawkins, D. G.

D. G. Hawkins, “Finite beamwidth effects in bulk acousto-optic interactions,” J. Opt. Soc. Am. A 70, 1611 (1980).

Hussain, F.

Imai, Y.

Ishii, Y.

Jahan, S. R.

S. R. Jahan, M. A. Karim, “Refracting systems for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21, 27–30 (1989).
[CrossRef]

Karim, M. A.

Kastner, C. J.

Klein, W. R.

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 723–733 (1967).

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J 48, 2909–2947 (1969).

Kong, J. A.

Korpel, A.

Magdich, L. N.

L. N. Magdich, V. Y. Molchanov, “Diffraction of a divergent beam by intense acoustic waves,” Opt. Spectrosc. (USSR) 42, 299–302 (1977).

Magnusson, R.

Min, S.

S. Min, M. R. Chatterjee, “General integral formalism for acousto-optic and holographic Bragg scattering for arbitrary profiles and orientations,” Acustica 71, 81–92 (1990).

Moharam, M. G.

Molchanov, V. Y.

L. N. Magdich, V. Y. Molchanov, “Diffraction of a divergent beam by intense acoustic waves,” Opt. Spectrosc. (USSR) 42, 299–302 (1977).

Murata, K.

Mustafa, S.

Nath, N. S. N.

C. V. Raman, N. S. N. Nath, “The diffraction of light by high frequency sound waves,” Proc. Indian Acad. Sci. Sect. A 2, 406–420 (1935); Proc. Indian Acad. Sci. Sect. A 3, 75–84, 119–125, 459–465 (1936).

Ohtsuka, Y.

Y. Ohtsuka, A. Yasatomo, Y. Imai, “Acousto-optic two-dimensional profile shaping of a Gaussian laser beam,” Appl. Opt. 24, 2813–2819 (1985).
[CrossRef] [PubMed]

Y. Ohtsuka, A. Tanone, “Acousto-optic intensity modification of a Gaussian laser beam,” Opt. Commun. 39, 70–74 (1981).
[CrossRef]

Phariseau, P.

P. Phariseau, “On the diffraction of light by progressive supersonic waves,” Proc. Indian Acad. Sci. Sect. A 44, 165–170 (1956).

Poon, T.-C.

M. R. Chatterjee, T.-C. Poon, D. N. Sitter, “Transfer function formalism for strong acousto-optic Bragg diffraction of light beams with arbitrary profiles,” Acustica 71, 81–92 (1990).

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

T.-C. Poon, M. R. Chatterjee, “Transfer function approach to acousto-optic Bragg diffraction of finite optical beams using Fourier integrals,” presented at the 1988 International Union of Radio Science Meeting, Syracuse, N.Y., June 1988.

Raman, C. V.

C. V. Raman, N. S. N. Nath, “The diffraction of light by high frequency sound waves,” Proc. Indian Acad. Sci. Sect. A 2, 406–420 (1935); Proc. Indian Acad. Sci. Sect. A 3, 75–84, 119–125, 459–465 (1936).

Russell, P. St. J.

B. Benlarbi, P. St. J. Russell, L. Solymar, “Bragg diffraction of finite beams by thick gratings: two rival theories,” Appl. Phys. B 28, 63–72 (1982).
[CrossRef]

B. Benlarbi, P. St. J. Russell, L. Solymar, “Bragg diffraction of Gaussian beams by thick gratings: numerical evaluations by plane-wave decomposition,” Appl. Phys. B 28, 383–390 (1982).
[CrossRef]

Samberid, Z.

Shafer, D.

D. Shafer, “Gaussian to flattop intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
[CrossRef]

Sitter, D. N.

M. R. Chatterjee, T.-C. Poon, D. N. Sitter, “Transfer function formalism for strong acousto-optic Bragg diffraction of light beams with arbitrary profiles,” Acustica 71, 81–92 (1990).

Solymar, L.

B. Benlarbi, P. St. J. Russell, L. Solymar, “Bragg diffraction of Gaussian beams by thick gratings: numerical evaluations by plane-wave decomposition,” Appl. Phys. B 28, 383–390 (1982).
[CrossRef]

B. Benlarbi, P. St. J. Russell, L. Solymar, “Bragg diffraction of finite beams by thick gratings: two rival theories,” Appl. Phys. B 28, 63–72 (1982).
[CrossRef]

Tamir, T.

Tanone, A.

Y. Ohtsuka, A. Tanone, “Acousto-optic intensity modification of a Gaussian laser beam,” Opt. Commun. 39, 70–74 (1981).
[CrossRef]

Tarn, C.

P. P. Banerjee, C. Tarn, “A Fourier transform approach to acousto-optic interaction in the presence of propagational diffraction,” Acustica 74181–191 (1991).

Tervonen, E.

E. Tervonen, A. T. Friberg, J. Turunen, “Acousto-optic conversion of laser beams into flat-top beams,” J. Mod. Opt. 40, 625–635 (1993).
[CrossRef]

Turunen, J.

E. Tervonen, A. T. Friberg, J. Turunen, “Acousto-optic conversion of laser beams into flat-top beams,” J. Mod. Opt. 40, 625–635 (1993).
[CrossRef]

Veldkamp, W. B.

Yasatomo, A.

Zain, N. M.

Acustica (3)

M. R. Chatterjee, T.-C. Poon, D. N. Sitter, “Transfer function formalism for strong acousto-optic Bragg diffraction of light beams with arbitrary profiles,” Acustica 71, 81–92 (1990).

P. P. Banerjee, C. Tarn, “A Fourier transform approach to acousto-optic interaction in the presence of propagational diffraction,” Acustica 74181–191 (1991).

S. Min, M. R. Chatterjee, “General integral formalism for acousto-optic and holographic Bragg scattering for arbitrary profiles and orientations,” Acustica 71, 81–92 (1990).

Appl. Opt. (6)

Appl. Phys. B (2)

B. Benlarbi, P. St. J. Russell, L. Solymar, “Bragg diffraction of finite beams by thick gratings: two rival theories,” Appl. Phys. B 28, 63–72 (1982).
[CrossRef]

B. Benlarbi, P. St. J. Russell, L. Solymar, “Bragg diffraction of Gaussian beams by thick gratings: numerical evaluations by plane-wave decomposition,” Appl. Phys. B 28, 383–390 (1982).
[CrossRef]

Bell Syst. Tech. J (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J 48, 2909–2947 (1969).

IEEE Trans. Sonics Ultrason. (1)

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 723–733 (1967).

J. Acoust. Soc. Am. (1)

L. E. Hargrove, “Effects of ultrasonic waves on Gaussian light beams with diameter comparable to ultrasonic wavelength,” J. Acoust. Soc. Am. 43, 847–851 (1967).
[CrossRef]

J. Mod. Opt. (1)

E. Tervonen, A. T. Friberg, J. Turunen, “Acousto-optic conversion of laser beams into flat-top beams,” J. Mod. Opt. 40, 625–635 (1993).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (1)

D. G. Hawkins, “Finite beamwidth effects in bulk acousto-optic interactions,” J. Opt. Soc. Am. A 70, 1611 (1980).

Opt. Commun. (2)

M. R. Forshaw, “Diffraction of a narrower laser beam by a thick hologram: experimental results,” Opt. Commun. 12, 279–281 (1974).
[CrossRef]

Y. Ohtsuka, A. Tanone, “Acousto-optic intensity modification of a Gaussian laser beam,” Opt. Commun. 39, 70–74 (1981).
[CrossRef]

Opt. Laser Technol. (2)

S. R. Jahan, M. A. Karim, “Refracting systems for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21, 27–30 (1989).
[CrossRef]

D. Shafer, “Gaussian to flattop intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
[CrossRef]

Opt. Lett. (1)

Opt. Spectrosc. (USSR) (1)

L. N. Magdich, V. Y. Molchanov, “Diffraction of a divergent beam by intense acoustic waves,” Opt. Spectrosc. (USSR) 42, 299–302 (1977).

Proc. Indian Acad. Sci. Sect. A (3)

C. V. Raman, N. S. N. Nath, “The diffraction of light by high frequency sound waves,” Proc. Indian Acad. Sci. Sect. A 2, 406–420 (1935); Proc. Indian Acad. Sci. Sect. A 3, 75–84, 119–125, 459–465 (1936).

P. Phariseau, “On the diffraction of light by progressive supersonic waves,” Proc. Indian Acad. Sci. Sect. A 44, 165–170 (1956).

R. R. Aggarwal, “Diffraction of light by ultrasonic waves (deduction of different theories from the generalized theory of Raman and Nath),” Proc. Indian Acad. Sci. Sect. A 31, 417–426 (1950).

Other (2)

T.-C. Poon, M. R. Chatterjee, “Transfer function approach to acousto-optic Bragg diffraction of finite optical beams using Fourier integrals,” presented at the 1988 International Union of Radio Science Meeting, Syracuse, N.Y., June 1988.

A. Korpel, Acousto-Optics (Dekker, New York, 1988), pp. 95–121.

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Figures (10)

Fig. 1
Fig. 1

Sound–light interaction configuration.

Fig. 2
Fig. 2

Diffraction geometry for upshifted Bragg operation.

Fig. 3
Fig. 3

Numerical plots of |H1(δ)| as a function of the peak phase delay, α, for Q = 20: (a) solution involving two diffracted orders, (b) solution involving ten diffracted orders.

Fig. 4
Fig. 4

Numerical plots of |H1(δ)| as a function of α for Q = 40: (a) solution involving two diffracted orders, (b) solution involving ten diffracted orders.

Fig. 5
Fig. 5

Numerical plots of |H1(δ)| as a function of α for (a) Q = 80, (b) Q = 160.

Fig. 6
Fig. 6

Beam distortion of input Gaussian beam profile and its disappearance: Distortion occurs at α = π but disappears at the stronger sound pressure, α = 3π. σ1/Λ = 4.5, Q = 160.

Fig. 7
Fig. 7

Design curves showing the 1/e point of the input Gaussian spectrum coincide with the first zeros of Hreq(δ) and H1(δ). Q = 242, σ1/Λ = 10, α = π.

Fig. 8
Fig. 8

Near-field flattop for (a) σ1/Λ = 10, Q = 242, and (b) σ1/Λ = 4, Q = 97, illustrating that a thinner Gaussian input beam requires a smaller Q value to achieve a flattop output. In both (a) and (b), α = π.

Fig. 9
Fig. 9

More uniform flattop outputs obtained by the fine tuning of Q or α for the case shown in Fig. 8(a).

Fig. 10
Fig. 10

Far-field flattop generation for an output beam profile obtained by a depression in the transfer function to offset the Gaussian peak. σ1/Λ = 10, Q = 225, α = 7.4.

Equations (26)

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d E ˜ n d ξ = j α 2 exp { j 2 Q ξ [ ϕ inc ϕ B + ( 2 n 1 ) ] } E ˜ n 1 j α 2 exp { j 2 Q ξ [ ϕ inc ϕ B + ( 2 n + 1 ) ] } E ˜ n + 1 ,
d E ˜ 0 d ξ = j α 2 exp ( j Q ξδ / 2 ) E ˜ 1 ,
d E ˜ 1 d ξ = j α 2 exp ( j Q ξδ / 2 ) E ˜ 0 .
E ˜ 0 ( ξ ) = E ˜ inc exp ( j δ Q ξ / 4 ) { cos [ ( δ Q / 4 ) 2 + ( α / 2 ) 2 ] 1 / 2 ξ + j δ Q 4 sin [ ( δ Q / 4 ) 2 + ( α / 2 ) 2 ] 1 / 2 ξ [ ( δ Q / 4 ) 2 + ( α / 2 ) 2 ] 1 / 2 } ,
E ˜ 1 ( ξ ) = E ˜ inc exp ( j δ Q ξ / 4 ) × { j α 2 sin [ ( δ Q / 4 ) 2 + ( α / 2 ) 2 ] 1 / 2 ξ [ ( δ Q / 4 ) 2 + ( α / 2 ) 2 ] 1 / 2 } .
E ˜ 0 = E ˜ inc cos ( α 2 ξ ) ,
E ˜ 1 = j E ˜ inc sin ( α 2 ξ ) .
H 0 ( δ ) = E ˜ 0 ( ξ ) | ξ = 1 E ˜ inc ,
H 1 ( δ ) = E ˜ 1 ( ξ ) | ξ = 1 E ˜ inc ,
E ˜ out ( δ ) = E ˜ in ( δ ) H ( δ ) .
E ( r ) = F 1 [ E ˜ ( ϕ ) ] = E ˜ ( ϕ ) exp ( j 2 π λ ϕ r ) d ( ϕ λ ) ,
E ˜ ( ϕ ) = F [ E ( r ) ] = E ( r ) exp ( j 2 π λ ϕ r ) d r ,
E 0 ( r ) = E ˜ in ( δ ) H 0 ( δ ) exp ( j 2 π δ 2 Λ r ) d ( δ 2 Λ ) ,
E 1 ( r ) = E ˜ in ( δ ) H 1 ( δ ) exp ( j 2 π δ 2 Λ r ) d ( δ 2 Λ ) ,
E in ( r ) = E inc σ 1 ( 2 π ) 1 / 2 exp ( r 2 / 2 σ 1 2 ) ,
E ˜ in ( δ ) = E inc exp [ 1 2 ( πσ 1 Λ ) 2 δ 2 ] .
E 1 ( r ) = E inc exp [ 1 2 ( πσ 1 Λ ) 2 δ 2 ] { ( j α 2 ) exp ( j δ Q / 4 ) × sinc [ ( δ Q / 4 ) 2 + ( α / 2 ) 2 ] 1 / 2 } × exp ( j 2 π δ 2 Λ r ) d ( δ 2 Λ ) ,
F 1 [ E ˜ in ( δ ) H req ( δ ) ] = rect ( r σ 0 ) ,
H req ( δ ) = sinc ( σ 0 δ 2 Λ ) { exp [ 1 2 ( π σ 1 Λ δ ) 2 ] } ,
H 1 ( δ ) = sinc [ ( δ Q / 4 ) 2 + ( α / 2 ) 2 ] 1 / 2 ,
δ 1 / e = 2 Λ πσ 1 ,
δ req = 2 Λ σ 0 ,
δ 1 = 4 Q [ ( n π ) 2 ( α / 2 ) 2 ] 1 / 2 .
Q = 4 π 2 σ 1 Λ [ ( n π ) 2 ( α / 2 ) 2 ] 1 / 2 ,
α = 4 [ ( n π ) 2 1 8 ( Q Λ πσ 1 ) 2 ] 1 / 2 .
E ˜ in ( δ ) H req ( δ ) = rect ( δ σ 0 ) .

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