Abstract

In recent work the laser mode selectivity induced separately by a biprism-like reflector and by an absorbing strip was investigated by numerical analysis. It was shown that each of these elements in an otherwise conventional resonator was suitable to cause the laser to oscillate preferentially on the first odd mode that contains a line singularity, which is a useful dark beam (i.e., a laser beam with a dark central region) for high-resolution metrological applications. We study the combined effect of these two mode-selecting elements and show that the unified analysis leads to much better performance than could be expected from a simple superposition of the performance with each element alone. The results indicate that the mode selectivity can be enhanced by at least a factor of 3 compared with that of laser resonators with biprism-like reflectors alone. Thus a laser equipped with such a combined element will oscillate on a pure first-order mode with high power efficiency. Moreover, calculations show that the resultant dark beam, focused for metrological applications, has a significantly improved shape compared with the dark beam obtained by external modulation of a fundamental Gaussian beam.

© 2006 Optical Society of America

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References

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  1. M. Friedman, T. Kotzer, and J. Shamir, " High-resolution surface reconstruction using optically measured Gabor expansion coefficients," Opt. Eng. 36, 2068- 2073 ( 1997).
    [CrossRef]
  2. M. Friedmann and J. Shamir, " Resolution enhancement by extrapolation of the optically measured spectrum of surface profiles," Appl. Opt. 36, 1747- 1751 ( 1997).
    [CrossRef] [PubMed]
  3. M. Friedmann, E. Paquet, and J. Shamir, " Surface feature reconstruction using scanning beams," CLEO/Europe'96, Technical Digest p. 30, Hamburg, Sept. 8- 13, 1996.
  4. G. Toker, A. Brunfeld, J. Shamir, and B. Spektor, " In-line optical surface roughness determination by laser scanning," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 323- 329 ( 2002).
    [CrossRef]
  5. B. Spektor, G. Toker, J. Shamir, M. Friedman, and A. Brunfeld, " High-resolution surface evaluation using multi-wavelength optical transforms," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 345- 351 ( 2002).
    [CrossRef]
  6. B. Spektor, R. Piestun, and J. Shamir, " Dark beams with a constant notch," Opt. Lett. 21, 456- 458 ( 1996).
    [CrossRef] [PubMed]
  7. R. Piestun, B. Spektor, and J. Shamir, " Wave fields in three dimensions: analysis and synthesis," J. Opt. Soc. Am. A 13, 1837- 1848 ( 1996).
    [CrossRef]
  8. Yu. Ananev, Laser Resonators and the Beam Divergence Problem, Adam Hilger Series on Optics and Optoelectronics (Adam Hilger, 1992).
  9. X. Kol'chenko, A. G. Nikitenko, and Yu. K. Troitskij, " Control of the structure of transverse modes by phase-shifting masks," Sov. J. Quantum Electron. 10, 1013- 1016 ( 1980).
    [CrossRef]
  10. R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, " Transverse mode shaping and selection in laser resonators," in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 325- 386.
    [CrossRef]
  11. Y. Parkhomenko, B. Spektor, and J. Shamir, " Two regions of mode selection in resonators with biprismlike elements," Appl. Opt. 44, 2546- 2552 ( 2005).
    [CrossRef] [PubMed]
  12. Yu. N. Parkhomenko, O. V. Anisimova, and O. N. Galkin, " Lasers with fast electronic synthesis of spectral line shape," in Advances in Laser and Optic Research, W. T. Lavworth, ed. (Nova Science, 2002), Vol. 1, pp. 128- 161.

2005 (1)

2002 (2)

G. Toker, A. Brunfeld, J. Shamir, and B. Spektor, " In-line optical surface roughness determination by laser scanning," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 323- 329 ( 2002).
[CrossRef]

B. Spektor, G. Toker, J. Shamir, M. Friedman, and A. Brunfeld, " High-resolution surface evaluation using multi-wavelength optical transforms," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 345- 351 ( 2002).
[CrossRef]

1997 (2)

M. Friedman, T. Kotzer, and J. Shamir, " High-resolution surface reconstruction using optically measured Gabor expansion coefficients," Opt. Eng. 36, 2068- 2073 ( 1997).
[CrossRef]

M. Friedmann and J. Shamir, " Resolution enhancement by extrapolation of the optically measured spectrum of surface profiles," Appl. Opt. 36, 1747- 1751 ( 1997).
[CrossRef] [PubMed]

1996 (2)

1980 (1)

X. Kol'chenko, A. G. Nikitenko, and Yu. K. Troitskij, " Control of the structure of transverse modes by phase-shifting masks," Sov. J. Quantum Electron. 10, 1013- 1016 ( 1980).
[CrossRef]

Ananev, Yu.

Yu. Ananev, Laser Resonators and the Beam Divergence Problem, Adam Hilger Series on Optics and Optoelectronics (Adam Hilger, 1992).

Anisimova, O. V.

Yu. N. Parkhomenko, O. V. Anisimova, and O. N. Galkin, " Lasers with fast electronic synthesis of spectral line shape," in Advances in Laser and Optic Research, W. T. Lavworth, ed. (Nova Science, 2002), Vol. 1, pp. 128- 161.

Brunfeld, A.

G. Toker, A. Brunfeld, J. Shamir, and B. Spektor, " In-line optical surface roughness determination by laser scanning," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 323- 329 ( 2002).
[CrossRef]

B. Spektor, G. Toker, J. Shamir, M. Friedman, and A. Brunfeld, " High-resolution surface evaluation using multi-wavelength optical transforms," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 345- 351 ( 2002).
[CrossRef]

Davidson, N.

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, " Transverse mode shaping and selection in laser resonators," in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 325- 386.
[CrossRef]

Friedman, M.

B. Spektor, G. Toker, J. Shamir, M. Friedman, and A. Brunfeld, " High-resolution surface evaluation using multi-wavelength optical transforms," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 345- 351 ( 2002).
[CrossRef]

M. Friedman, T. Kotzer, and J. Shamir, " High-resolution surface reconstruction using optically measured Gabor expansion coefficients," Opt. Eng. 36, 2068- 2073 ( 1997).
[CrossRef]

Friedmann, M.

M. Friedmann and J. Shamir, " Resolution enhancement by extrapolation of the optically measured spectrum of surface profiles," Appl. Opt. 36, 1747- 1751 ( 1997).
[CrossRef] [PubMed]

M. Friedmann, E. Paquet, and J. Shamir, " Surface feature reconstruction using scanning beams," CLEO/Europe'96, Technical Digest p. 30, Hamburg, Sept. 8- 13, 1996.

Friesem, A. A.

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, " Transverse mode shaping and selection in laser resonators," in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 325- 386.
[CrossRef]

Galkin, O. N.

Yu. N. Parkhomenko, O. V. Anisimova, and O. N. Galkin, " Lasers with fast electronic synthesis of spectral line shape," in Advances in Laser and Optic Research, W. T. Lavworth, ed. (Nova Science, 2002), Vol. 1, pp. 128- 161.

Hasman, E.

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, " Transverse mode shaping and selection in laser resonators," in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 325- 386.
[CrossRef]

Kol'chenko, X.

X. Kol'chenko, A. G. Nikitenko, and Yu. K. Troitskij, " Control of the structure of transverse modes by phase-shifting masks," Sov. J. Quantum Electron. 10, 1013- 1016 ( 1980).
[CrossRef]

Kotzer, T.

M. Friedman, T. Kotzer, and J. Shamir, " High-resolution surface reconstruction using optically measured Gabor expansion coefficients," Opt. Eng. 36, 2068- 2073 ( 1997).
[CrossRef]

Nikitenko, A. G.

X. Kol'chenko, A. G. Nikitenko, and Yu. K. Troitskij, " Control of the structure of transverse modes by phase-shifting masks," Sov. J. Quantum Electron. 10, 1013- 1016 ( 1980).
[CrossRef]

Oron, R.

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, " Transverse mode shaping and selection in laser resonators," in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 325- 386.
[CrossRef]

Paquet, E.

M. Friedmann, E. Paquet, and J. Shamir, " Surface feature reconstruction using scanning beams," CLEO/Europe'96, Technical Digest p. 30, Hamburg, Sept. 8- 13, 1996.

Parkhomenko, Y.

Parkhomenko, Yu. N.

Yu. N. Parkhomenko, O. V. Anisimova, and O. N. Galkin, " Lasers with fast electronic synthesis of spectral line shape," in Advances in Laser and Optic Research, W. T. Lavworth, ed. (Nova Science, 2002), Vol. 1, pp. 128- 161.

Piestun, R.

Shamir, J.

Y. Parkhomenko, B. Spektor, and J. Shamir, " Two regions of mode selection in resonators with biprismlike elements," Appl. Opt. 44, 2546- 2552 ( 2005).
[CrossRef] [PubMed]

B. Spektor, G. Toker, J. Shamir, M. Friedman, and A. Brunfeld, " High-resolution surface evaluation using multi-wavelength optical transforms," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 345- 351 ( 2002).
[CrossRef]

G. Toker, A. Brunfeld, J. Shamir, and B. Spektor, " In-line optical surface roughness determination by laser scanning," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 323- 329 ( 2002).
[CrossRef]

M. Friedman, T. Kotzer, and J. Shamir, " High-resolution surface reconstruction using optically measured Gabor expansion coefficients," Opt. Eng. 36, 2068- 2073 ( 1997).
[CrossRef]

M. Friedmann and J. Shamir, " Resolution enhancement by extrapolation of the optically measured spectrum of surface profiles," Appl. Opt. 36, 1747- 1751 ( 1997).
[CrossRef] [PubMed]

R. Piestun, B. Spektor, and J. Shamir, " Wave fields in three dimensions: analysis and synthesis," J. Opt. Soc. Am. A 13, 1837- 1848 ( 1996).
[CrossRef]

B. Spektor, R. Piestun, and J. Shamir, " Dark beams with a constant notch," Opt. Lett. 21, 456- 458 ( 1996).
[CrossRef] [PubMed]

M. Friedmann, E. Paquet, and J. Shamir, " Surface feature reconstruction using scanning beams," CLEO/Europe'96, Technical Digest p. 30, Hamburg, Sept. 8- 13, 1996.

Spektor, B.

Y. Parkhomenko, B. Spektor, and J. Shamir, " Two regions of mode selection in resonators with biprismlike elements," Appl. Opt. 44, 2546- 2552 ( 2005).
[CrossRef] [PubMed]

G. Toker, A. Brunfeld, J. Shamir, and B. Spektor, " In-line optical surface roughness determination by laser scanning," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 323- 329 ( 2002).
[CrossRef]

B. Spektor, G. Toker, J. Shamir, M. Friedman, and A. Brunfeld, " High-resolution surface evaluation using multi-wavelength optical transforms," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 345- 351 ( 2002).
[CrossRef]

B. Spektor, R. Piestun, and J. Shamir, " Dark beams with a constant notch," Opt. Lett. 21, 456- 458 ( 1996).
[CrossRef] [PubMed]

R. Piestun, B. Spektor, and J. Shamir, " Wave fields in three dimensions: analysis and synthesis," J. Opt. Soc. Am. A 13, 1837- 1848 ( 1996).
[CrossRef]

Toker, G.

B. Spektor, G. Toker, J. Shamir, M. Friedman, and A. Brunfeld, " High-resolution surface evaluation using multi-wavelength optical transforms," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 345- 351 ( 2002).
[CrossRef]

G. Toker, A. Brunfeld, J. Shamir, and B. Spektor, " In-line optical surface roughness determination by laser scanning," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 323- 329 ( 2002).
[CrossRef]

Troitskij, Yu. K.

X. Kol'chenko, A. G. Nikitenko, and Yu. K. Troitskij, " Control of the structure of transverse modes by phase-shifting masks," Sov. J. Quantum Electron. 10, 1013- 1016 ( 1980).
[CrossRef]

Appl. Opt. (2)

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

Opt. Eng. (1)

M. Friedman, T. Kotzer, and J. Shamir, " High-resolution surface reconstruction using optically measured Gabor expansion coefficients," Opt. Eng. 36, 2068- 2073 ( 1997).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (2)

G. Toker, A. Brunfeld, J. Shamir, and B. Spektor, " In-line optical surface roughness determination by laser scanning," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 323- 329 ( 2002).
[CrossRef]

B. Spektor, G. Toker, J. Shamir, M. Friedman, and A. Brunfeld, " High-resolution surface evaluation using multi-wavelength optical transforms," in Interferometry XI: Techniques and Analysis, K. Creath and J. Schmit, eds., Proc. SPIE 4777, 345- 351 ( 2002).
[CrossRef]

Sov. J. Quantum Electron. (1)

X. Kol'chenko, A. G. Nikitenko, and Yu. K. Troitskij, " Control of the structure of transverse modes by phase-shifting masks," Sov. J. Quantum Electron. 10, 1013- 1016 ( 1980).
[CrossRef]

Other (4)

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, " Transverse mode shaping and selection in laser resonators," in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 325- 386.
[CrossRef]

Yu. N. Parkhomenko, O. V. Anisimova, and O. N. Galkin, " Lasers with fast electronic synthesis of spectral line shape," in Advances in Laser and Optic Research, W. T. Lavworth, ed. (Nova Science, 2002), Vol. 1, pp. 128- 161.

M. Friedmann, E. Paquet, and J. Shamir, " Surface feature reconstruction using scanning beams," CLEO/Europe'96, Technical Digest p. 30, Hamburg, Sept. 8- 13, 1996.

Yu. Ananev, Laser Resonators and the Beam Divergence Problem, Adam Hilger Series on Optics and Optoelectronics (Adam Hilger, 1992).

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of the modified laser cavity with flat-roof mirror M1, spherical mirror M2, and an amplitude mask.

Fig. 2
Fig. 2

Dependence of |Λ|2 on α for three modes (TEM00, solid curve; TEM10, dashed curve; TEM20, dotted–dashed curve) for g = 0.8 and for several aperture parameters of the mirrors: (a) b 1 = 3.05, b 2 = 3.8; (b) b 1 = 3.2, b 2 = 3.8; (c) b 1 = 3.2, b 2 = 3.2. For all cases the width of the absorbing strip was chosen as w = w0 .

Fig. 3
Fig. 3

Dependence of S = | Λ 1 | 2 | Λ 0 | 2 on α in the cavity with a strip (solid curve) and without a mask (dashed curve), with g = 0.8, b 1 = 3.2, and b 2 = 3.2.

Fig. 4
Fig. 4

Dependence of S = |Λ1|2 − |Λ0|2 on α in the resonator with a strip, g = 0.8, b 1 = 3.2 and various apertures of M 2: b 2 = 3.8 (C 1), b = 3.5 (C 2), b = 3.2 (C 3), b = 2.9 (C 4).

Fig. 5
Fig. 5

Dependence of S = |Λ1|2 − |Λ0|2 on strip width w for three values of α, all with g = 0.8, b 1 = 3.2, and b 2 = 3.2. Width w is normalized to the selected width, w 0, that corresponds to Fig. 2(c).

Fig. 6
Fig. 6

Normalized intensity distributions (a) at the output of a resonator with g = 0.8 and b 1 = 3.2 for the TEM10 mode (solid curves) with α = −0.6 and b 2 = 3.2, and for the TEM00 mode (dashed curves) with α = 0 and b 2 = 3.5 and (b) at the focal plane of a lens when the TEM00 mode is transmitted by a mask with a π phase shift along the central line.

Fig. 7
Fig. 7

Dependence of S on α, with normalized displacement δ as a parameter: (a) small displacements, (b) large displacements.

Equations (4)

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

γ u 2 ( x 2 ) = a 1 a 1 P 1 ( x 1 ) P 2 ( x 2 ) G ( x 2 , x 1 ) u 1 ( x 1 ) d x 1 ,
γ u 1 ( x 1 ) = a 2 a 2 P 1 ( x 1 ) P 2 ( x 2 ) G ( x 2 , x 1 ) u 2 ( x 2 ) d x 2 ,
S ( b 1 , b 2 , g , w , α ) = | Λ 1 | 2 | Λ j | 2 ,
S ( b 1 , b 2 , g , w 0 , 0 ) = 0.

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