Abstract

The spatial shaping of laser beams is a subject of research in modern optics. Recently the introduction of diffractive elements in laser resonators has offered an alternative to external beam-shaping optics by mode shaping within the resonator. We describe the specification of the laser resonator mirrors to obtain by means of internal mode shaping a desired beam outside the resonator. Modal discrimination of the modified resonator and the mirror alignment sensitivity is discussed. Basic features of resonator-originated and external beam shaping are compared.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
  2. H. Aagedal, F. Wyrowski, M. Schmid, “Paraxial beam splitting and shaping,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie-Verlag, Berlin, 1997), pp. 165–188.
  3. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  4. R. Gerchberg, W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  5. F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
    [CrossRef]
  6. D. Peri, “Optical implementation of a phase retrieval algorithm,” Appl. Opt. 26, 1782–1785 (1987).
    [CrossRef] [PubMed]
  7. A. Desfarges, V. Kermene, B. Colombeau, M. Vampouille, C. Froehly, “Wave-front reconstruction with a Fourier hologram in a phase-conjugating mirror oscillator,” Opt. Lett. 20, 1940–1942 (1995).
    [CrossRef] [PubMed]
  8. J. R. Leger, “Diffractive laser resonators,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie-Verlag, Berlin, 1997), pp. 189–215.
  9. J. R. Leger, D. Chen, G. Mowry, “Design and performance of diffractive optics for custom laser resonators,” Appl. Opt. 34, 2498–2509 (1995).
    [CrossRef] [PubMed]
  10. B. Colombeau, M. Vampouille, V. Kermene, A. Desfarges, C. Froehly, “Spatial shaping of coherent waves inside a confocal laser,” Pure Appl. Opt. 3, 757–773 (1994).
    [CrossRef]
  11. C. Pare, P. Belanger, “Custom laser resonators using graded-phase mirrors,” IEEE J. Quantum Electron. 28, 355–362 (1992).
    [CrossRef]
  12. A. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
    [CrossRef]
  13. J. R. Leger, D. Chen, Z. Wang, “Diffractive optical element for mode shaping of a Nd:YAG laser,” Opt. Lett. 19, 108–110 (1994).
    [CrossRef] [PubMed]
  14. D. Chen, Z. Wang, J. R. Leger, “Measurements of the modal properties of a diffractive-optic graded-phase resonator,” Opt. Lett. 20, 663–665 (1995).
    [CrossRef] [PubMed]
  15. E.-B. Kley, B. Schnabel, “E-beam lithography: a suitable technology for fabrication of high-accuracy 2D and 3D surface profiles,” in Microlithography and Metrology in Micromachining, M. T. Postek, ed., Proc. SPIE2640, 71–80 (1995).
    [CrossRef]

1995 (3)

1994 (2)

J. R. Leger, D. Chen, Z. Wang, “Diffractive optical element for mode shaping of a Nd:YAG laser,” Opt. Lett. 19, 108–110 (1994).
[CrossRef] [PubMed]

B. Colombeau, M. Vampouille, V. Kermene, A. Desfarges, C. Froehly, “Spatial shaping of coherent waves inside a confocal laser,” Pure Appl. Opt. 3, 757–773 (1994).
[CrossRef]

1992 (1)

C. Pare, P. Belanger, “Custom laser resonators using graded-phase mirrors,” IEEE J. Quantum Electron. 28, 355–362 (1992).
[CrossRef]

1991 (1)

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

1987 (1)

1972 (1)

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

1961 (1)

A. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Aagedal, H.

H. Aagedal, F. Wyrowski, M. Schmid, “Paraxial beam splitting and shaping,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie-Verlag, Berlin, 1997), pp. 165–188.

Belanger, P.

C. Pare, P. Belanger, “Custom laser resonators using graded-phase mirrors,” IEEE J. Quantum Electron. 28, 355–362 (1992).
[CrossRef]

Bryngdahl, O.

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Chen, D.

Colombeau, B.

A. Desfarges, V. Kermene, B. Colombeau, M. Vampouille, C. Froehly, “Wave-front reconstruction with a Fourier hologram in a phase-conjugating mirror oscillator,” Opt. Lett. 20, 1940–1942 (1995).
[CrossRef] [PubMed]

B. Colombeau, M. Vampouille, V. Kermene, A. Desfarges, C. Froehly, “Spatial shaping of coherent waves inside a confocal laser,” Pure Appl. Opt. 3, 757–773 (1994).
[CrossRef]

Desfarges, A.

A. Desfarges, V. Kermene, B. Colombeau, M. Vampouille, C. Froehly, “Wave-front reconstruction with a Fourier hologram in a phase-conjugating mirror oscillator,” Opt. Lett. 20, 1940–1942 (1995).
[CrossRef] [PubMed]

B. Colombeau, M. Vampouille, V. Kermene, A. Desfarges, C. Froehly, “Spatial shaping of coherent waves inside a confocal laser,” Pure Appl. Opt. 3, 757–773 (1994).
[CrossRef]

Fox, A.

A. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Froehly, C.

A. Desfarges, V. Kermene, B. Colombeau, M. Vampouille, C. Froehly, “Wave-front reconstruction with a Fourier hologram in a phase-conjugating mirror oscillator,” Opt. Lett. 20, 1940–1942 (1995).
[CrossRef] [PubMed]

B. Colombeau, M. Vampouille, V. Kermene, A. Desfarges, C. Froehly, “Spatial shaping of coherent waves inside a confocal laser,” Pure Appl. Opt. 3, 757–773 (1994).
[CrossRef]

Gerchberg, R.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Kermene, V.

A. Desfarges, V. Kermene, B. Colombeau, M. Vampouille, C. Froehly, “Wave-front reconstruction with a Fourier hologram in a phase-conjugating mirror oscillator,” Opt. Lett. 20, 1940–1942 (1995).
[CrossRef] [PubMed]

B. Colombeau, M. Vampouille, V. Kermene, A. Desfarges, C. Froehly, “Spatial shaping of coherent waves inside a confocal laser,” Pure Appl. Opt. 3, 757–773 (1994).
[CrossRef]

Kley, E.-B.

E.-B. Kley, B. Schnabel, “E-beam lithography: a suitable technology for fabrication of high-accuracy 2D and 3D surface profiles,” in Microlithography and Metrology in Micromachining, M. T. Postek, ed., Proc. SPIE2640, 71–80 (1995).
[CrossRef]

Leger, J. R.

Li, T.

A. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Mowry, G.

Pare, C.

C. Pare, P. Belanger, “Custom laser resonators using graded-phase mirrors,” IEEE J. Quantum Electron. 28, 355–362 (1992).
[CrossRef]

Peri, D.

Saxton, W.

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

Schmid, M.

H. Aagedal, F. Wyrowski, M. Schmid, “Paraxial beam splitting and shaping,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie-Verlag, Berlin, 1997), pp. 165–188.

Schnabel, B.

E.-B. Kley, B. Schnabel, “E-beam lithography: a suitable technology for fabrication of high-accuracy 2D and 3D surface profiles,” in Microlithography and Metrology in Micromachining, M. T. Postek, ed., Proc. SPIE2640, 71–80 (1995).
[CrossRef]

Siegman, A.

A. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

Vampouille, M.

A. Desfarges, V. Kermene, B. Colombeau, M. Vampouille, C. Froehly, “Wave-front reconstruction with a Fourier hologram in a phase-conjugating mirror oscillator,” Opt. Lett. 20, 1940–1942 (1995).
[CrossRef] [PubMed]

B. Colombeau, M. Vampouille, V. Kermene, A. Desfarges, C. Froehly, “Spatial shaping of coherent waves inside a confocal laser,” Pure Appl. Opt. 3, 757–773 (1994).
[CrossRef]

Wang, Z.

Wyrowski, F.

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

H. Aagedal, F. Wyrowski, M. Schmid, “Paraxial beam splitting and shaping,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie-Verlag, Berlin, 1997), pp. 165–188.

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

A. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

IEEE J. Quantum Electron. (1)

C. Pare, P. Belanger, “Custom laser resonators using graded-phase mirrors,” IEEE J. Quantum Electron. 28, 355–362 (1992).
[CrossRef]

Opt. Lett. (3)

Optik (1)

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

Pure Appl. Opt. (1)

B. Colombeau, M. Vampouille, V. Kermene, A. Desfarges, C. Froehly, “Spatial shaping of coherent waves inside a confocal laser,” Pure Appl. Opt. 3, 757–773 (1994).
[CrossRef]

Rep. Prog. Phys. (1)

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Other (5)

A. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

H. Aagedal, F. Wyrowski, M. Schmid, “Paraxial beam splitting and shaping,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie-Verlag, Berlin, 1997), pp. 165–188.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. R. Leger, “Diffractive laser resonators,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie-Verlag, Berlin, 1997), pp. 189–215.

E.-B. Kley, B. Schnabel, “E-beam lithography: a suitable technology for fabrication of high-accuracy 2D and 3D surface profiles,” in Microlithography and Metrology in Micromachining, M. T. Postek, ed., Proc. SPIE2640, 71–80 (1995).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Sketch of the experimental setup for generation of the complex beam.

Fig. 2
Fig. 2

Amplitude |U b (x, y, z f )| of the desired far-field beam pattern.

Fig. 3
Fig. 3

(a) Phase of the diffractive element transmission function (here and in the following figures the phase values are represented as gray levels), (b) amplitude of its simulated far field for illumination with a Gaussian beam.

Fig. 4
Fig. 4

(a) Amplitude and (b) phase of the desired beam U b (x, y, z o ) in the output plane of the laser resonator.

Fig. 5
Fig. 5

Resonator and its behavior when the effect of the outcoupling mirror is considered in the mirror design. Height profiles of (a) the outcoupling mirror and (b) the backresonator mirror; (c) simulated amplitude and (d) simulated phase of the fundamental mode U +(x, y, z o ); (e) far-field amplitude and far-field (f) phase of the outcoupled laser beam U b (x, y, z f ).

Fig. 6
Fig. 6

Calculated round-trip losses for the fundamental and the second-order modes as functions of the mirror aperture size.

Fig. 7
Fig. 7

Sketch of the experimental setup for checking the alignment sensitivity of two diffractive elements.

Fig. 8
Fig. 8

(a) Shape of the object for the alignment check, (b) phase of the diffractive element for phase conjugation (central part), and (c) intensity measured in the image plane.

Equations (16)

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

Ubx, y, zo=-1Ubx, y, zf,
Ubx, y, zo=Tex, yUox, y, zo.
Ubx, y, zo=Uox, y, zo,
Ro/bx, y=exp-i2kho/bx, y
Tox, y=expikhox, yn-1
U-x, y, zo=Rox, yU+x, y, zo,
Uox, y, zo=Tox, yU+x, y, zo.
|U+x, y, zo|=|U-x, y, zo|=|Uox, y, zo|
Rox, y=exp-i2φ+x, y, zo,
Rbx, y=exp-i2φ-x, y, zb.
ho/bx, y=φ±x, y, zo/bk.
U+x, y, zo=Ubx, y, zo.
Uox, y, zo=expikhox, yn-1U+x, y, zo.
|Uox, y, zo|expiφox, y, zo=expin-1φ+x, y, zo×|U+x, y, zo|expiφ+x, y, zo=|U+x, y, zo×|expinφ+x, y, zo.
φ+x, y, zo=φox, y, zon.
U-x, y, zb=LU-x, y, zo,

Metrics