Abstract

A first-order optical system (represented by its 4×4 ABCD matrix) is given in order to obtain a beam that preserves its spatial orientation of the transverse profile under free propagation from a beam with rotating irradiance distribution in free space. Within the formalism of the second-order irradiance moments, this transverse orientation is analyzed in terms of the evolution of the principal axes of the field irradiance distribution. It is shown that the spatial profile of the beam emerging from the proposed optical system does not rotate when light freely propagates. The improvement of the joint near-field and far-field beam spread product at the output of this optical system is also studied.

© 2006 Optical Society of America

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  1. R. Simon, N. Mukunda and E. C. G. Sudarshan, "Partially coherent beams and a generalized ABCD-law," Opt. Commun. 65, 322-328 (1988).
    [CrossRef]
  2. S. Lavi, R. Prochaska and E. Keren, "Generalized beam parameters and transformation law for partially coherent light," Appl. Opt. 27, 3696-3703 (1988).
    [CrossRef] [PubMed]
  3. M. J. Bastiaans, "Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems," Optik 82, 173-181 (1989).
  4. A. E. Siegman, "New developments in laser resonators" in Laser Resonators, Proc. SPIE 1224, 2-14 (1990).
    [CrossRef]
  5. J. Serna, R. Martínez-Herrero and P. M. Mejías, "Parametric characterization of general partially coherent beams propagating through ABCD optical systems," J. Opt. Soc. Am. A 8, 1094-1098 (1991).
    [CrossRef]
  6. H. Weber, "Propagation of higher-order intensity moments in quadratic-index media," Opt. Quantum Electron. 24, 1027-1049 (1992).
    [CrossRef]
  7. ISO 11146, Laser and laser related equipment-Test methods for laser beam widths, divergence angles and beam propagation ratios: ISO 11146-1:2005, Part 1: Stigmatic and simple astigmatic beams; ISO11146-2:2005, Part 2: General astigmatic beams; ISO/TR 11146-3:2004, Part 3: Intrinsic and geometrical laser beam classification, propagation, and details of test method; ISO/TR 11146-3:2004/Cor1:2005 (International Organization for Standardization, Geneva, Switzerland, 2005).
  8. F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero and P. M. Mejías, "Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses," IEEE J. Quantum Electron. 34, 1835-1838 (1998).
    [CrossRef]
  9. P. M. Mejías, R. Martínez-Herrero, G. Piquero and J. M. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarized laser beams," Prog. Quantum Electron. 26, 65-130 (2002), and references therein.
    [CrossRef]
  10. J. A. Arnaud and H. Kogelnik, "Light beams with general astigmatism," Appl. Opt. 8, 1687-1693 (1969).
    [CrossRef] [PubMed]
  11. J. Serna and G. Nemes, "Decoupling of coherent Gaussian beams with general astimatism," Opt. Lett. 18, 1174-1176 (1993).
    [CrossRef]
  12. G. Nemes and A. E. Siegman, "Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics," J. Opt. Soc. Am A 11, 2257-2264 (1994).
    [CrossRef]
  13. G. Nemes, "Synthesis of general astigmatic optical systems, the detwisting procedure and the beam quality factors for general astigmatic laser beams," in Proceedings of the Second Workshop on Laser Beam Characterization, H. Weber, N. Reng, J. Ludtke and P. M. Mejías, eds., Festkorper-Laser Institut Berlin GmbH, Berlin, Germany, 1994, pp. 93-104.
  14. J. Serna, P. M. Mejías and R. Martínez-Herrero, "Rotation of partially coherent beams through free space," Opt. Quantum Electron. 24, 873-880 (1992).
    [CrossRef]
  15. G. Nemes and J. Serna, "Do not use spherical lenses and free spaces to characterize beams: a possible improvement of the ISO/DIS 11146 document," in Proceedings of the Fourth Workshop on Laser Beam and Optics Characterization, A. Giesen and M. Morin, eds. (Verein Deutscher Ingenieure-Technologiezentrum, Düseldorf, Germany, 1997), pp. 29-49.
  16. G. Nemes and J. Serna, "Laser beam characterization with use of second order moments: an overview," in Diode Pumped Solid State Lasers: Applications and Issues, M. W. Dowley, ed., OSA TOPS 17, 200-207 (1998).
  17. J. Serna, F. Encinas and G. Nemes, "Complete spatial characterization of a pulsed doughnut-type beam by use of spherical optics and a cylindrical lens," J. Opt. Soc. Am. A 18, 1726-1733 (2001).
    [CrossRef]
  18. M. J. Bastiaans, "Wigner distribution function and its applications to first-order optics," J. Opt. Soc. Am. 69, 1710-1716 (1979).
    [CrossRef]
  19. A. Walther, "Radiometry and coherence," J. Opt. Soc. Am. 58, 1256-1259 (1968).
    [CrossRef]
  20. A. T. Friberg, E. Tervonen and T. Turunen, "Interpretation and experimental demonstration of twisted Gauss-Schell-mode beams," J. Opt. Soc. Am A 11, 1818-1826 (1994).
    [CrossRef]
  21. A. T. Friberg, C. Gao, B. Eppich and H. Weber, "Generation of partially coherent fields with twist," Proc. SPIE 3110, 317-318 (1997).
    [CrossRef]
  22. R. Simon and K. Bernardo Wolf, "Fractional Fourier transform in two dimensions," J. Opt. Soc. Am. A 17, 2368-2381 (2000).
    [CrossRef]

2002 (1)

P. M. Mejías, R. Martínez-Herrero, G. Piquero and J. M. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarized laser beams," Prog. Quantum Electron. 26, 65-130 (2002), and references therein.
[CrossRef]

2001 (1)

2000 (1)

1998 (1)

F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero and P. M. Mejías, "Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses," IEEE J. Quantum Electron. 34, 1835-1838 (1998).
[CrossRef]

1997 (1)

A. T. Friberg, C. Gao, B. Eppich and H. Weber, "Generation of partially coherent fields with twist," Proc. SPIE 3110, 317-318 (1997).
[CrossRef]

1994 (2)

A. T. Friberg, E. Tervonen and T. Turunen, "Interpretation and experimental demonstration of twisted Gauss-Schell-mode beams," J. Opt. Soc. Am A 11, 1818-1826 (1994).
[CrossRef]

G. Nemes and A. E. Siegman, "Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics," J. Opt. Soc. Am A 11, 2257-2264 (1994).
[CrossRef]

1993 (1)

1992 (2)

J. Serna, P. M. Mejías and R. Martínez-Herrero, "Rotation of partially coherent beams through free space," Opt. Quantum Electron. 24, 873-880 (1992).
[CrossRef]

H. Weber, "Propagation of higher-order intensity moments in quadratic-index media," Opt. Quantum Electron. 24, 1027-1049 (1992).
[CrossRef]

1991 (1)

1989 (1)

M. J. Bastiaans, "Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems," Optik 82, 173-181 (1989).

1988 (2)

R. Simon, N. Mukunda and E. C. G. Sudarshan, "Partially coherent beams and a generalized ABCD-law," Opt. Commun. 65, 322-328 (1988).
[CrossRef]

S. Lavi, R. Prochaska and E. Keren, "Generalized beam parameters and transformation law for partially coherent light," Appl. Opt. 27, 3696-3703 (1988).
[CrossRef] [PubMed]

1979 (1)

1969 (1)

1968 (1)

Arnaud, J. A.

Bastiaans, M. J.

M. J. Bastiaans, "Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems," Optik 82, 173-181 (1989).

M. J. Bastiaans, "Wigner distribution function and its applications to first-order optics," J. Opt. Soc. Am. 69, 1710-1716 (1979).
[CrossRef]

Bernardo Wolf, K.

Encinas, F.

Encinas-Sanz, F.

F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero and P. M. Mejías, "Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses," IEEE J. Quantum Electron. 34, 1835-1838 (1998).
[CrossRef]

Eppich, B.

A. T. Friberg, C. Gao, B. Eppich and H. Weber, "Generation of partially coherent fields with twist," Proc. SPIE 3110, 317-318 (1997).
[CrossRef]

Friberg, A. T.

A. T. Friberg, C. Gao, B. Eppich and H. Weber, "Generation of partially coherent fields with twist," Proc. SPIE 3110, 317-318 (1997).
[CrossRef]

A. T. Friberg, E. Tervonen and T. Turunen, "Interpretation and experimental demonstration of twisted Gauss-Schell-mode beams," J. Opt. Soc. Am A 11, 1818-1826 (1994).
[CrossRef]

Gao, C.

A. T. Friberg, C. Gao, B. Eppich and H. Weber, "Generation of partially coherent fields with twist," Proc. SPIE 3110, 317-318 (1997).
[CrossRef]

Keren, E.

Kogelnik, H.

Lavi, S.

Martínez, C.

F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero and P. M. Mejías, "Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses," IEEE J. Quantum Electron. 34, 1835-1838 (1998).
[CrossRef]

Martínez-Herrero, R.

P. M. Mejías, R. Martínez-Herrero, G. Piquero and J. M. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarized laser beams," Prog. Quantum Electron. 26, 65-130 (2002), and references therein.
[CrossRef]

F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero and P. M. Mejías, "Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses," IEEE J. Quantum Electron. 34, 1835-1838 (1998).
[CrossRef]

J. Serna, P. M. Mejías and R. Martínez-Herrero, "Rotation of partially coherent beams through free space," Opt. Quantum Electron. 24, 873-880 (1992).
[CrossRef]

J. Serna, R. Martínez-Herrero and P. M. Mejías, "Parametric characterization of general partially coherent beams propagating through ABCD optical systems," J. Opt. Soc. Am. A 8, 1094-1098 (1991).
[CrossRef]

Mejías, P. M.

P. M. Mejías, R. Martínez-Herrero, G. Piquero and J. M. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarized laser beams," Prog. Quantum Electron. 26, 65-130 (2002), and references therein.
[CrossRef]

F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero and P. M. Mejías, "Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses," IEEE J. Quantum Electron. 34, 1835-1838 (1998).
[CrossRef]

J. Serna, P. M. Mejías and R. Martínez-Herrero, "Rotation of partially coherent beams through free space," Opt. Quantum Electron. 24, 873-880 (1992).
[CrossRef]

J. Serna, R. Martínez-Herrero and P. M. Mejías, "Parametric characterization of general partially coherent beams propagating through ABCD optical systems," J. Opt. Soc. Am. A 8, 1094-1098 (1991).
[CrossRef]

Movilla, J. M.

P. M. Mejías, R. Martínez-Herrero, G. Piquero and J. M. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarized laser beams," Prog. Quantum Electron. 26, 65-130 (2002), and references therein.
[CrossRef]

Mukunda, N.

R. Simon, N. Mukunda and E. C. G. Sudarshan, "Partially coherent beams and a generalized ABCD-law," Opt. Commun. 65, 322-328 (1988).
[CrossRef]

Nemes, G.

Piquero, G.

P. M. Mejías, R. Martínez-Herrero, G. Piquero and J. M. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarized laser beams," Prog. Quantum Electron. 26, 65-130 (2002), and references therein.
[CrossRef]

Prochaska, R.

Serna, J.

Siegman, A. E.

G. Nemes and A. E. Siegman, "Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics," J. Opt. Soc. Am A 11, 2257-2264 (1994).
[CrossRef]

Simon, R.

R. Simon and K. Bernardo Wolf, "Fractional Fourier transform in two dimensions," J. Opt. Soc. Am. A 17, 2368-2381 (2000).
[CrossRef]

R. Simon, N. Mukunda and E. C. G. Sudarshan, "Partially coherent beams and a generalized ABCD-law," Opt. Commun. 65, 322-328 (1988).
[CrossRef]

Sudarshan, E. C. G.

R. Simon, N. Mukunda and E. C. G. Sudarshan, "Partially coherent beams and a generalized ABCD-law," Opt. Commun. 65, 322-328 (1988).
[CrossRef]

Tervonen, E.

A. T. Friberg, E. Tervonen and T. Turunen, "Interpretation and experimental demonstration of twisted Gauss-Schell-mode beams," J. Opt. Soc. Am A 11, 1818-1826 (1994).
[CrossRef]

Turunen, T.

A. T. Friberg, E. Tervonen and T. Turunen, "Interpretation and experimental demonstration of twisted Gauss-Schell-mode beams," J. Opt. Soc. Am A 11, 1818-1826 (1994).
[CrossRef]

Walther, A.

Weber, H.

A. T. Friberg, C. Gao, B. Eppich and H. Weber, "Generation of partially coherent fields with twist," Proc. SPIE 3110, 317-318 (1997).
[CrossRef]

H. Weber, "Propagation of higher-order intensity moments in quadratic-index media," Opt. Quantum Electron. 24, 1027-1049 (1992).
[CrossRef]

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero and P. M. Mejías, "Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses," IEEE J. Quantum Electron. 34, 1835-1838 (1998).
[CrossRef]

J. Opt. Soc. Am A (2)

G. Nemes and A. E. Siegman, "Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics," J. Opt. Soc. Am A 11, 2257-2264 (1994).
[CrossRef]

A. T. Friberg, E. Tervonen and T. Turunen, "Interpretation and experimental demonstration of twisted Gauss-Schell-mode beams," J. Opt. Soc. Am A 11, 1818-1826 (1994).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

R. Simon, N. Mukunda and E. C. G. Sudarshan, "Partially coherent beams and a generalized ABCD-law," Opt. Commun. 65, 322-328 (1988).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (2)

H. Weber, "Propagation of higher-order intensity moments in quadratic-index media," Opt. Quantum Electron. 24, 1027-1049 (1992).
[CrossRef]

J. Serna, P. M. Mejías and R. Martínez-Herrero, "Rotation of partially coherent beams through free space," Opt. Quantum Electron. 24, 873-880 (1992).
[CrossRef]

Optik (1)

M. J. Bastiaans, "Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems," Optik 82, 173-181 (1989).

Proc. SPIE (1)

A. T. Friberg, C. Gao, B. Eppich and H. Weber, "Generation of partially coherent fields with twist," Proc. SPIE 3110, 317-318 (1997).
[CrossRef]

Prog. Quantum Electron. (1)

P. M. Mejías, R. Martínez-Herrero, G. Piquero and J. M. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarized laser beams," Prog. Quantum Electron. 26, 65-130 (2002), and references therein.
[CrossRef]

Other (5)

A. E. Siegman, "New developments in laser resonators" in Laser Resonators, Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

ISO 11146, Laser and laser related equipment-Test methods for laser beam widths, divergence angles and beam propagation ratios: ISO 11146-1:2005, Part 1: Stigmatic and simple astigmatic beams; ISO11146-2:2005, Part 2: General astigmatic beams; ISO/TR 11146-3:2004, Part 3: Intrinsic and geometrical laser beam classification, propagation, and details of test method; ISO/TR 11146-3:2004/Cor1:2005 (International Organization for Standardization, Geneva, Switzerland, 2005).

G. Nemes and J. Serna, "Do not use spherical lenses and free spaces to characterize beams: a possible improvement of the ISO/DIS 11146 document," in Proceedings of the Fourth Workshop on Laser Beam and Optics Characterization, A. Giesen and M. Morin, eds. (Verein Deutscher Ingenieure-Technologiezentrum, Düseldorf, Germany, 1997), pp. 29-49.

G. Nemes and J. Serna, "Laser beam characterization with use of second order moments: an overview," in Diode Pumped Solid State Lasers: Applications and Issues, M. W. Dowley, ed., OSA TOPS 17, 200-207 (1998).

G. Nemes, "Synthesis of general astigmatic optical systems, the detwisting procedure and the beam quality factors for general astigmatic laser beams," in Proceedings of the Second Workshop on Laser Beam Characterization, H. Weber, N. Reng, J. Ludtke and P. M. Mejías, eds., Festkorper-Laser Institut Berlin GmbH, Berlin, Germany, 1994, pp. 93-104.

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Equations (24)

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

h ( r , η , z ) = + W ( r + s 2 , r s 2 ) exp ( i k η s ) d s ,
< x m y n u p v q > 1 I o + x m y n u p v q h r η z d r d η ,
Q = < x 2 + y 2 > < u 2 + v 2 > < xu + yv > 2 .
Q x = < x 2 > < u 2 > < xu > 2 ,
Q y = < y 2 > < v 2 > < yv > 2 .
tan 2 θ = < y 2 > f < v 2 > f < x 2 > f < u 2 > f < uv > f < x 2 + y 2 > f + < xy > f < u 2 + v 2 > f ,
S = 2 2 = a 0 1 a 0 0 b 0 1 b a 0 1 a 0 0 b 0 1 b ,
M = a 0 0 0 0 b 0 0 0 0 1 a 0 0 0 0 1 b .
F = 2 2 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 ,
tan 2 φ ( z ) = 2 < xy > < x 2 > < y 2 > ,
tan 2 φ ( z ) =
= 2 < xy > o + 2 kz ( < xv > o + < yu > o ) + 2 k 2 z 2 < uv > o < x 2 > o < y 2 > o + 2 kz ( < xu > o < yu > o ) + k 2 z 2 ( < u 2 > o < v 2 > o ) ,
< x 2 > o = 1 2 ( a 2 < x 2 > i + < u 2 > i a 2 + 2 < xu > i ) = < x 2 > i < u 2 > i + < xu > i
< y 2 > o = 1 2 ( b 2 < y 2 > i + < v 2 > i b 2 + 2 < yv > i ) = < y 2 > i < v 2 > i + < yv > i
< x 2 > o = < y 2 > o .
tan 2 φ ( z ) = , for any z ,
< x 2 > o + < y 2 > o = 2 < x 2 > i < u 2 > i + ( < xu > i < yu > i ) .
< u 2 > o + < v 2 > o = 2 < x 2 > i < u 2 > i ( < xu > i < yv > i ) ,
Q o = 4 < x 2 > i < u 2 > i ( < xu > i < yv > i ) 2 ,
Q o = 4 < x 2 > i < u 2 > i 4 < xu > i 2 = 4 Q x = 4 Q y .
Q i = 2 < x 2 > i < u 2 > i + < y 2 > i < u 2 > i + < x 2 > i < v 2 > i =
= 2 < x 2 > i < u 2 > i + < x 2 > i < u 2 > i 2 < v 2 > i + < x 2 > i < v 2 > i .
Q o Q i = 2< x 2 > i < u 2 > i < x 2 > i < u 2 > i 2 < v 2 > i < x 2 > i < v 2 > i 4 < xu > i 2 =
= < x 2 > i < v 2 > i ( < u 2 > i < v 2 > i ) 2 4 < xu > i 2 0 ,

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