Abstract

A complete spatial characterization (in second-order moments) of a doughnut-type beam from a pulsed transversely excited atmospheric CO2 laser is described. It includes the measurement of the orbital angular momentum carried by the beam. The key element in the characterization is the use of a cylindrical lens in addition to the usual spherical optics. Internal features of the beam that would have remained hidden if only spherical optics were employed were revealed by use of the cylindrical lens. The experimental results are compared and agree with a theoretical Laguerre–Gauss mode beam.

© 2001 Optical Society of America

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  1. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  2. S. Lavi, R. Prochaska, E. Keren, “Generalized beam parameters and transformation laws for partially coherent light,” Appl. Opt. 27, 3696–3703 (1988).
    [CrossRef] [PubMed]
  3. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
    [CrossRef]
  4. A. E. Siegman, “New developments in laser resonators,” in Laser Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
    [CrossRef]
  5. J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
    [CrossRef]
  6. J. Serna, J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26, 405–407 (2001).
    [CrossRef]
  7. International Organization for Standardization, Technical Committee/Subcommittee 172/SC9, “Lasers and laser-related equipment—test methods for laser beam parameters—beam widths, divergence angle and beam propagation factor,” (International Organization for Standardization, Geneva, Switzerland, 1999).
  8. G. Nemeş, 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, M. Morin, eds. (Verein Deutscher Ingenieure-Technologiezentrum, Düsseldorf, Germany, 1997), pp. 29–49.
  9. G. Nemeş, J. Serna, “Laser beam characterization with use of second order moments: an overview,” in DPSS Lasers: Applications and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 200–207.
  10. G. Nemeş, 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]
  11. B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity moments,” Opt. Laser Technol. 30, 337–340 (1998).
    [CrossRef]
  12. G. Nemeş, J. Serna, “The ten physical parameters associated with a full general astigmatic beam: a Gauss Schell-model,” in Proceedings of the Fourth Workshop on Laser Beam and Optics Characterization, A. Giesen, M. Morin, eds. (Verein Deutscher Ingenieure-Technologiezentrum, Düsseldorf, Germany, 1997), pp. 92–105.
  13. C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
    [CrossRef]
  14. F. Encinas-Sanz, J. Serna, C. Martı́nez, R. Martı́nez-Herrero, P. M. Mejı́ias, “Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses,” IEEE J. Quantum Electron. 34, 1835–1838 (1998).
    [CrossRef]
  15. C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 17–30 (2000).
    [CrossRef]
  16. P. M. Mejı́as, R. Martı́nez-Herrero, “Time-resolved spatial parametric characterization of pulsed light beams,” Opt. Lett. 20, 660–662 (1995).
    [CrossRef] [PubMed]

2001 (1)

2000 (1)

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 17–30 (2000).
[CrossRef]

1998 (2)

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity moments,” Opt. Laser Technol. 30, 337–340 (1998).
[CrossRef]

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

1997 (1)

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

1995 (1)

1994 (1)

1991 (1)

1988 (2)

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

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

1979 (1)

Bastiaans, M. J.

Encinas-Sanz, F.

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 17–30 (2000).
[CrossRef]

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

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

Eppich, B.

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity moments,” Opt. Laser Technol. 30, 337–340 (1998).
[CrossRef]

Gao, C.

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity moments,” Opt. Laser Technol. 30, 337–340 (1998).
[CrossRef]

Keren, E.

Lavi, S.

Marti´nez, C.

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 17–30 (2000).
[CrossRef]

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

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

Marti´nez-Herrero, R.

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 17–30 (2000).
[CrossRef]

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

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

P. M. Mejı́as, R. Martı́nez-Herrero, “Time-resolved spatial parametric characterization of pulsed light beams,” Opt. Lett. 20, 660–662 (1995).
[CrossRef] [PubMed]

J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

Meji´as, P. M.

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 17–30 (2000).
[CrossRef]

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

P. M. Mejı́as, R. Martı́nez-Herrero, “Time-resolved spatial parametric characterization of pulsed light beams,” Opt. Lett. 20, 660–662 (1995).
[CrossRef] [PubMed]

J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

Meji´ias, P. M.

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

Movilla, J. M.

Mukunda, N.

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

Nemes, G.

G. Nemeş, 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]

G. Nemeş, J. Serna, “Laser beam characterization with use of second order moments: an overview,” in DPSS Lasers: Applications and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 200–207.

G. Nemeş, 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, M. Morin, eds. (Verein Deutscher Ingenieure-Technologiezentrum, Düsseldorf, Germany, 1997), pp. 29–49.

G. Nemeş, J. Serna, “The ten physical parameters associated with a full general astigmatic beam: a Gauss Schell-model,” in Proceedings of the Fourth Workshop on Laser Beam and Optics Characterization, A. Giesen, M. Morin, eds. (Verein Deutscher Ingenieure-Technologiezentrum, Düsseldorf, Germany, 1997), pp. 92–105.

Prochaska, R.

Serna, J.

J. Serna, J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26, 405–407 (2001).
[CrossRef]

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 17–30 (2000).
[CrossRef]

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

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

G. Nemeş, J. Serna, “The ten physical parameters associated with a full general astigmatic beam: a Gauss Schell-model,” in Proceedings of the Fourth Workshop on Laser Beam and Optics Characterization, A. Giesen, M. Morin, eds. (Verein Deutscher Ingenieure-Technologiezentrum, Düsseldorf, Germany, 1997), pp. 92–105.

G. Nemeş, 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, M. Morin, eds. (Verein Deutscher Ingenieure-Technologiezentrum, Düsseldorf, Germany, 1997), pp. 29–49.

G. Nemeş, J. Serna, “Laser beam characterization with use of second order moments: an overview,” in DPSS Lasers: Applications and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 200–207.

Siegman, A. E.

Simon, R.

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

Weber, H.

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity moments,” Opt. Laser Technol. 30, 337–340 (1998).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

F. Encinas-Sanz, J. Serna, C. Martı́nez, R. Martı́nez-Herrero, P. M. Mejı́ias, “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. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

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

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

Opt. Laser Technol. (1)

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity moments,” Opt. Laser Technol. 30, 337–340 (1998).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 17–30 (2000).
[CrossRef]

Other (5)

International Organization for Standardization, Technical Committee/Subcommittee 172/SC9, “Lasers and laser-related equipment—test methods for laser beam parameters—beam widths, divergence angle and beam propagation factor,” (International Organization for Standardization, Geneva, Switzerland, 1999).

G. Nemeş, 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, M. Morin, eds. (Verein Deutscher Ingenieure-Technologiezentrum, Düsseldorf, Germany, 1997), pp. 29–49.

G. Nemeş, J. Serna, “Laser beam characterization with use of second order moments: an overview,” in DPSS Lasers: Applications and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 200–207.

G. Nemeş, J. Serna, “The ten physical parameters associated with a full general astigmatic beam: a Gauss Schell-model,” in Proceedings of the Fourth Workshop on Laser Beam and Optics Characterization, A. Giesen, M. Morin, eds. (Verein Deutscher Ingenieure-Technologiezentrum, Düsseldorf, Germany, 1997), pp. 92–105.

A. E. Siegman, “New developments in laser resonators,” in Laser Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

The cylindrical lens (cyl) is used to decouple the xv+yu term.

Fig. 2
Fig. 2

Typical beam profiles obtained with the 8-mm diaphragm (a) with no cylindrical lens and (b) after the cylindrical lens, both taken at z=650 mm. (c) Experimental values of x2, xy, and y2 obtained before and after the cylindrical lens with the 8-mm diaphragm. Before the lens the beam is rotationally symmetric, and after the lens the beam is a simple astigmatic beam aligned along the lens axes. Note that the rotation angle is negligible.  

Fig. 3
Fig. 3

Typical beam profiles obtained with the 11-mm diaphragm (a) with no cylindrical lens and (b) after the cylindrical lens, both taken at z=725 mm. (c) Experimental values of x2, xy, and y2 obtained before and after the cylindrical lens with the 11-mm diaphragm. Only pulses that rotate clockwise were considered after the lens. Before the lens the beam is rotationally symmetric, but after the lens the beam profiles rotate as shown.

Fig. 4
Fig. 4

Beam profiles taken at z=775 mm after the cylindrical lens: (a) clockwise rotated (75% of the pulses), (b) aligned elliptical (flattened) profile (20% of the pulses), (c) counterclockwise rotated (5% of the pulses).

Fig. 5
Fig. 5

Beam irradiance profiles at z=920 mm (after the cylindrical lens): (a) an actual experimental profile, (b) a pure (theoretical) LG0-1 beam.

Equations (48)

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P=WMMtU=x2xyxuxvxyy2yuyvxuyuu2uvxvyvuvv2.
Meff4=4k2 (detP)1/21,
t=2k2T=2k2Tr(WU-M2)1.
a=t-Meff4.
aM=12 (Meff4-1)2.
0aaM.
J¯zL=Ic (xv-yu),
lz=J¯zLωI=k(xv-yu)=2km.
x2z=x20+2xu0z+u2z2,
xyz=xy0+(xv0+yu0)z+uvz2,
 
y2z=y20+2yv0z+v2z2.
uvL=uv-xvfx.
m=12(xv-yu)
=(uv-uvL)fx-12(xv+yu).
x2=(5.08±0.02)-[(12.04±0.04)×10-3]z+[(7.69±0.03)×10-6]z2,
xy=(51±5) × 10-3-[(90±20)×10-6]z+[(47±9)×10-9]z2,
y2=(5.37±0.01)-[(12.64±0.04)×10-3]z+[(7.97±0.02)×10-6]z2.
x2L=(23.3±0.1)-[(67.8±0.3)×10-3]z+[(49.5±0.2)×10-6]z2,
xyL=(1.8±0.3)-[(5.7±0.7)×10-3]z+[(4.5±0.5)×10-6]z2,
y2L=(4.51±0.07)-[(10.2±0.2)×10-3]z+[(6.2±0.1)×10-6]z2,
WzL=(730±10) × 10-3 (14.6±0.7) × 10-3(14.6±0.7) × 10-3(770±10) × 10-3,
 
MzL=-(1.66±0.02) × 10-3 -(19±2)× 10-6+m-(19±2) × 10-6-m-(1.80×0.03)× 10-3,
 
UzL=(7.69±0.03) × 10-6 (47±9) × 10-9(47±9) × 10-9(7.97±0.02)× 10-6,
 
WLzL=(760±40) × 10-3 (20±20) × 10-3(20±20) × 10-3(770±10) × 10-3,
 
MLzL=-(5.83±0.2) × 10-3 -(300±100)× 10-6+mL-(300±100) × 10-6-mL-(1.53±0.03)× 10-3,
ULzL=(49.5±0.2) × 10-6 (4.5±0.5) × 10-6(4.5±0.5) × 10-6(6.2±0.1) × 10-6,
 
m=-(810±90) × 10-6mm.
Meff4=4.03±0.01,
t=4.03±0.01,
a=(2.0±0.2) × 10-3,
aM=4.58±0.03
lz=0.
Meff4=3.1±0.2,
t=5.0±0.2,
a=1.8±0.4,
aM=2.2±0.4,
lz=-(1.0±0.1).
Meff4=3,
t=5,
a=2,
aM=2,
lz=-.

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