Abstract

A nonaxisymmetric mirror is designed by the same method as a computer-generated hologram for laser beam intensity profile transformation and is fabricated by plasma chemical vaporization machining. We successfully transformed a circular Gaussian beam of a He–Ne laser into a rectangular uniform beam maintaining spatial coherence and using a nonaxisymmetric surface profile mirror. There are ripples in the intensity profile of the transformed rectangular beam. These ripples in the intensity profile result from small ripples on the mirror surface. These results show that we can perform coordinate transformation using these fabricated mirrors, which has so far been possible only by using computer-generated holograms.

© 1997 Optical Society of America

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References

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    [CrossRef] [PubMed]
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1996 (1)

1994 (3)

1993 (1)

Y. Mori, K. Yamamura, K. Yamauchi, K. Yoshii, T. Kataoka, K. Endo, K. Inagaki, H. Kakiuchi, “Plasma CVM (chemical vaporization machining)—an ultra precision machining with high pressure reactive plasma,” Technol. Rep. Osaka Univ. 43, 261–266 (1993).

1992 (4)

1991 (1)

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

1989 (2)

M. T. Eismann, A. M. Tai, J. N. Cederquist, “Iterative design of a holographic beam former,” Appl. Opt. 28, 2641–2650 (1989).
[CrossRef] [PubMed]

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

1983 (1)

1982 (2)

D. H. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol., 159–160 (June1982).

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

1981 (1)

W. Lee, “Method for converting a Gaussian laser beam into a uniform beam,” Opt. Commun. 36, 469–471 (1981).
[CrossRef]

1974 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane figures,” Optik 35, 237–246 (1972).

1961 (1)

Aleksoff, C. C.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Bass, I. L.

Be’langer, P. A.

C. Pa’re, P. A. Be’langer, “Custom laser resonators using graded-phase mirrors,” IEEE J. Quantum Electron. 28, 355–362 (1992).
[CrossRef]

Bennett, H. E.

Bonanno, R. E.

Bryngdahl, O.

Cederquist, J. N.

Chen, D.

Dixit, S. N.

Dorsch, R. G.

Eismann, M. T.

Ellis, K. K.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Endo, K.

Y. Mori, K. Yamamura, K. Yamauchi, K. Yoshii, T. Kataoka, K. Endo, K. Inagaki, H. Kakiuchi, “Plasma CVM (chemical vaporization machining)—an ultra precision machining with high pressure reactive plasma,” Technol. Rep. Osaka Univ. 43, 261–266 (1993).

Fienup, J. R.

Fujii, T.

K. Nemoto, T. Fujii, N. Goto, T. Nayuki, Y. Kanai, “Transformation of the laser beam intensity profile by a deformable mirror,” Opt. Lett. 21, 168–170 (1996).
[CrossRef] [PubMed]

K. Nemoto, T. Fujii, M. Nagano, in “Laser beam forming by aspherical mirror,” in Beam Control, Diagnostics, Standards, and Propagation, L. W. Austin, A. Giesen, D. H. Leslie, H. Weichel, eds., Proc. SPIE2375, 103–108 (1994).

K. Nemoto, T. Fujii, M. Nagano, Y. Kanai, “Laser beam forming by deformable mirror,” in Intense Beams and Applications: Lasers, Ions, and Microwaves, W. E. McDermott, ed., Proc. SPIE2119 (1993).

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane figures,” Optik 35, 237–246 (1972).

Goto, N.

Hackel, R. P.

Hammond, P. R.

Han, C.

Inagaki, K.

Y. Mori, K. Yamamura, K. Yamauchi, K. Yoshii, T. Kataoka, K. Endo, K. Inagaki, H. Kakiuchi, “Plasma CVM (chemical vaporization machining)—an ultra precision machining with high pressure reactive plasma,” Technol. Rep. Osaka Univ. 43, 261–266 (1993).

Ishii, Y.

Jahan, S. R.

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

Kakiuchi, H.

Y. Mori, K. Yamamura, K. Yamauchi, K. Yoshii, T. Kataoka, K. Endo, K. Inagaki, H. Kakiuchi, “Plasma CVM (chemical vaporization machining)—an ultra precision machining with high pressure reactive plasma,” Technol. Rep. Osaka Univ. 43, 261–266 (1993).

Kanai, Y.

K. Nemoto, T. Fujii, N. Goto, T. Nayuki, Y. Kanai, “Transformation of the laser beam intensity profile by a deformable mirror,” Opt. Lett. 21, 168–170 (1996).
[CrossRef] [PubMed]

K. Nemoto, T. Fujii, M. Nagano, Y. Kanai, “Laser beam forming by deformable mirror,” in Intense Beams and Applications: Lasers, Ions, and Microwaves, W. E. McDermott, ed., Proc. SPIE2119 (1993).

Karim, N. M.

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

Kataoka, T.

Y. Mori, K. Yamamura, K. Yamauchi, K. Yoshii, T. Kataoka, K. Endo, K. Inagaki, H. Kakiuchi, “Plasma CVM (chemical vaporization machining)—an ultra precision machining with high pressure reactive plasma,” Technol. Rep. Osaka Univ. 43, 261–266 (1993).

Lawson, J. K.

Lee, W.

W. Lee, “Method for converting a Gaussian laser beam into a uniform beam,” Opt. Commun. 36, 469–471 (1981).
[CrossRef]

Leger, J. R.

Lohmann, A. W.

Malyak, P. H.

Manes, K. R.

Mori, Y.

Y. Mori, K. Yamamura, K. Yamauchi, K. Yoshii, T. Kataoka, K. Endo, K. Inagaki, H. Kakiuchi, “Plasma CVM (chemical vaporization machining)—an ultra precision machining with high pressure reactive plasma,” Technol. Rep. Osaka Univ. 43, 261–266 (1993).

Murata, K.

Nagano, M.

K. Nemoto, T. Fujii, M. Nagano, in “Laser beam forming by aspherical mirror,” in Beam Control, Diagnostics, Standards, and Propagation, L. W. Austin, A. Giesen, D. H. Leslie, H. Weichel, eds., Proc. SPIE2375, 103–108 (1994).

K. Nemoto, T. Fujii, M. Nagano, Y. Kanai, “Laser beam forming by deformable mirror,” in Intense Beams and Applications: Lasers, Ions, and Microwaves, W. E. McDermott, ed., Proc. SPIE2119 (1993).

Nayuki, T.

Neagle, B. D.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Nemoto, K.

K. Nemoto, T. Fujii, N. Goto, T. Nayuki, Y. Kanai, “Transformation of the laser beam intensity profile by a deformable mirror,” Opt. Lett. 21, 168–170 (1996).
[CrossRef] [PubMed]

K. Nemoto, T. Fujii, M. Nagano, Y. Kanai, “Laser beam forming by deformable mirror,” in Intense Beams and Applications: Lasers, Ions, and Microwaves, W. E. McDermott, ed., Proc. SPIE2119 (1993).

K. Nemoto, T. Fujii, M. Nagano, in “Laser beam forming by aspherical mirror,” in Beam Control, Diagnostics, Standards, and Propagation, L. W. Austin, A. Giesen, D. H. Leslie, H. Weichel, eds., Proc. SPIE2375, 103–108 (1994).

Pa’re, C.

C. Pa’re, P. A. Be’langer, “Custom laser resonators using graded-phase mirrors,” IEEE J. Quantum Electron. 28, 355–362 (1992).
[CrossRef]

Porteus, J. O.

Powell, H. T.

Roberts, N. C.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane figures,” Optik 35, 237–246 (1972).

Shafer, D. H.

D. H. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol., 159–160 (June1982).

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986, pp. 805–810.

Sinzinger, S.

Tai, A. M.

Wang, Z.

Yamamura, K.

Y. Mori, K. Yamamura, K. Yamauchi, K. Yoshii, T. Kataoka, K. Endo, K. Inagaki, H. Kakiuchi, “Plasma CVM (chemical vaporization machining)—an ultra precision machining with high pressure reactive plasma,” Technol. Rep. Osaka Univ. 43, 261–266 (1993).

Yamauchi, K.

Y. Mori, K. Yamamura, K. Yamauchi, K. Yoshii, T. Kataoka, K. Endo, K. Inagaki, H. Kakiuchi, “Plasma CVM (chemical vaporization machining)—an ultra precision machining with high pressure reactive plasma,” Technol. Rep. Osaka Univ. 43, 261–266 (1993).

Yoshii, K.

Y. Mori, K. Yamamura, K. Yamauchi, K. Yoshii, T. Kataoka, K. Endo, K. Inagaki, H. Kakiuchi, “Plasma CVM (chemical vaporization machining)—an ultra precision machining with high pressure reactive plasma,” Technol. Rep. Osaka Univ. 43, 261–266 (1993).

Appl. Opt. (7)

IEEE J. Quantum Electron. (1)

C. Pa’re, P. A. Be’langer, “Custom laser resonators using graded-phase mirrors,” IEEE J. Quantum Electron. 28, 355–362 (1992).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

W. Lee, “Method for converting a Gaussian laser beam into a uniform beam,” Opt. Commun. 36, 469–471 (1981).
[CrossRef]

Opt. Eng. (1)

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Opt. Laser Technol. (2)

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

D. H. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol., 159–160 (June1982).

Opt. Lett. (3)

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane figures,” Optik 35, 237–246 (1972).

Technol. Rep. Osaka Univ. (1)

Y. Mori, K. Yamamura, K. Yamauchi, K. Yoshii, T. Kataoka, K. Endo, K. Inagaki, H. Kakiuchi, “Plasma CVM (chemical vaporization machining)—an ultra precision machining with high pressure reactive plasma,” Technol. Rep. Osaka Univ. 43, 261–266 (1993).

Other (3)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986, pp. 805–810.

K. Nemoto, T. Fujii, M. Nagano, in “Laser beam forming by aspherical mirror,” in Beam Control, Diagnostics, Standards, and Propagation, L. W. Austin, A. Giesen, D. H. Leslie, H. Weichel, eds., Proc. SPIE2375, 103–108 (1994).

K. Nemoto, T. Fujii, M. Nagano, Y. Kanai, “Laser beam forming by deformable mirror,” in Intense Beams and Applications: Lasers, Ions, and Microwaves, W. E. McDermott, ed., Proc. SPIE2119 (1993).

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Figures (12)

Fig. 1
Fig. 1

Surface profile of a mirror for intensity transformation from Gaussian beam to rectangular beam. The design parameters are as follows: z = 1.4 m, λ = 0.6328 µm, u 0 = r 0 = 6 mm.

Fig. 2
Fig. 2

Surface profile of a fabricated nonaxisymmetric mirror.

Fig. 3
Fig. 3

Error of the mirror surface that was fabricated when we scanned an electrode along only one axis. The main error is a spherical one resulting from the mirror bending owing to reflection coating a single side.

Fig. 4
Fig. 4

Experimental setup. The screen is in continual rotation during observation.

Fig. 5
Fig. 5

Beam profile (a) without and (b) with transformation.

Fig. 6
Fig. 6

Interference fringe pattern of transformed beam (a) without and (b) with transformation.

Fig. 7
Fig. 7

Beam profile after Fresnel propagation (N eq = 5.3 from the image location).

Fig. 8
Fig. 8

Ripples on the fabricated surface of the mirror in Fig. 2. The length of the measured area in the center of the mirror is 2 mm. The power component of the mirror surface is subtracted from the real surface profile.

Fig. 9
Fig. 9

Calculated beam profile that was transformed when we used a fabricated mirror (a) without small ripples and (b) with small ripples on the mirror surface. The height of the ripples is 10 nm in this simulation.

Fig. 10
Fig. 10

Error of the mirror surface that was fabricated when we scanned by changing the direction alternately between two axes. Mirror bending has no influence because of the 3-mm mirror thickness. (a) Bird’s-eye view of the error. The error has a dc component of approximately 0.1 µm. (b) Contour profile of the same error as in (a). The contour interval is 10 nm.

Fig. 11
Fig. 11

Beam profile after transformation when we used the fabricated mirror with the fabrication error shown in Fig. 10.

Fig. 12
Fig. 12

Microscopic roughness of the fabricated mirror surface. The length of the measured line is approximately 7 µm.

Equations (6)

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

au, v=jk/2πzexp-jkz×- ax, yexp-jϕx, yexpjkx-u2+y-v2/2zdxdy,
S x, y/x=u -x/2z,
S x, y/y=v -y/2z,
v x, y/x=u x, y/y.
S0t=2r0erf2r0/u0t erf2t/u0+u02πexp-2t2u02 -1-t2, Sx, y=-S0x+S0y/4z,
Sx, y=S0x, y+A sin2πx/L,

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