Abstract

We propose the Gaussian content (GC) as an optional quality parameter for the characterization of laser beams. It is defined as the overlap integral of a given field with an optimally defined Gaussian. The definition is especially suited for applications where coherence properties are targeted. Mathematical definitions and basic calculation procedures are given along with results for basic beam profiles. The coherent combination of an array of laser beams and the optimal coupling between a diode laser and a single-mode fiber are elaborated as application examples. The measurement of the GC and its conservation upon propagation are experimentally confirmed.

© 2011 Optical Society of America

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References

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    [CrossRef] [PubMed]
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2009 (3)

P. Zhou, Z. Liu, X. Xu, Z. Chen, and X. Wang, “Beam quality factor for coherently combined fiber laser beams,” Opt. Laser Technol. 41, 268–271 (2009).

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A Pure Appl. Opt. 11, 105705 (2009).
[CrossRef]

P. Zhou, Y. Ma, X. Wang, H. Ma, J. Wang, X. Xu, and Z. Liu, “Coherent beam combination of a hexagonal distributed high-power fiber amplifier array,” Appl. Opt. 48, 6537–6540 (2009).
[CrossRef] [PubMed]

2008 (1)

2005 (3)

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation: Fundamentals, Advanced Concepts and Applications (Springer, 2005).

ISO Standard 11146, “Lasers and laser-related equipment—Test methods for laser beam widths, divergence angles, and beam propagation ratios” (2005).

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
[CrossRef]

1997 (1)

W. D. Herzog, M. S. Unlu, B. B. Goldberg, G. H. Rhodes, and C. Harder, “Beam divergence and waist measurements of laser diodes by near-field scanning optical microscopy,” Appl. Phys. Lett. 70, 688–690 (1997).
[CrossRef]

1995 (1)

1993 (1)

A. E. Siegman, “Defining, measuring, and optimizing laser beam quality,” Proc. SPIE 1868, 2–12 (1993).
[CrossRef]

1992 (2)

H. Weber, “Some historical and technical aspects of beam quality,” Opt. Quantum Electron. 24, S861–S864 (1992).
[CrossRef]

J. Serna, P. M. Mejías, and R. Martínez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, S881–S887 (1992).
[CrossRef]

1991 (1)

1982 (1)

1978 (1)

1975 (2)

D. D. Cook and F. R. Nash, “Gain-induced guiding and astigmatic output beam of GaAs lasers,” J. Appl. Phys. 46, 1660–1662 (1975).
[CrossRef]

W. P. Dumke, “Angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11, 400–402 (1975).
[CrossRef]

Belanger, P. A.

Chen, Z.

P. Zhou, Z. Liu, X. Xu, Z. Chen, and X. Wang, “Beam quality factor for coherently combined fiber laser beams,” Opt. Laser Technol. 41, 268–271 (2009).

Cook, D. D.

D. D. Cook and F. R. Nash, “Gain-induced guiding and astigmatic output beam of GaAs lasers,” J. Appl. Phys. 46, 1660–1662 (1975).
[CrossRef]

Dumke, W. P.

W. P. Dumke, “Angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11, 400–402 (1975).
[CrossRef]

Fan, T. Y.

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
[CrossRef]

Fienup, J. R.

Goldberg, B. B.

W. D. Herzog, M. S. Unlu, B. B. Goldberg, G. H. Rhodes, and C. Harder, “Beam divergence and waist measurements of laser diodes by near-field scanning optical microscopy,” Appl. Phys. Lett. 70, 688–690 (1997).
[CrossRef]

Harder, C.

W. D. Herzog, M. S. Unlu, B. B. Goldberg, G. H. Rhodes, and C. Harder, “Beam divergence and waist measurements of laser diodes by near-field scanning optical microscopy,” Appl. Phys. Lett. 70, 688–690 (1997).
[CrossRef]

G. Hunziker and C. Harder, “Beam quality of InGaAs ridge lasers at high output power,” Appl. Opt. 34, 6118–6122(1995).
[CrossRef] [PubMed]

Herzog, W. D.

W. D. Herzog, M. S. Unlu, B. B. Goldberg, G. H. Rhodes, and C. Harder, “Beam divergence and waist measurements of laser diodes by near-field scanning optical microscopy,” Appl. Phys. Lett. 70, 688–690 (1997).
[CrossRef]

Hodgson, N.

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation: Fundamentals, Advanced Concepts and Applications (Springer, 2005).

Hou, J.

Hunziker, G.

Ji, X.

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A Pure Appl. Opt. 11, 105705 (2009).
[CrossRef]

Jia, X.

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A Pure Appl. Opt. 11, 105705 (2009).
[CrossRef]

Jiang, Z. F.

Liu, M.

Liu, Z.

P. Zhou, Z. Liu, X. Xu, Z. Chen, and X. Wang, “Beam quality factor for coherently combined fiber laser beams,” Opt. Laser Technol. 41, 268–271 (2009).

P. Zhou, Y. Ma, X. Wang, H. Ma, J. Wang, X. Xu, and Z. Liu, “Coherent beam combination of a hexagonal distributed high-power fiber amplifier array,” Appl. Opt. 48, 6537–6540 (2009).
[CrossRef] [PubMed]

Ma, H.

Ma, Y.

Marcuse, D.

Martínez-Herrero, R.

J. Serna, P. M. Mejías, and R. Martínez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, S881–S887 (1992).
[CrossRef]

Mejías, P. M.

J. Serna, P. M. Mejías, and R. Martínez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, S881–S887 (1992).
[CrossRef]

Nash, F. R.

D. D. Cook and F. R. Nash, “Gain-induced guiding and astigmatic output beam of GaAs lasers,” J. Appl. Phys. 46, 1660–1662 (1975).
[CrossRef]

Paré, C.

Rhodes, G. H.

W. D. Herzog, M. S. Unlu, B. B. Goldberg, G. H. Rhodes, and C. Harder, “Beam divergence and waist measurements of laser diodes by near-field scanning optical microscopy,” Appl. Phys. Lett. 70, 688–690 (1997).
[CrossRef]

Serna, J.

J. Serna, P. M. Mejías, and R. Martínez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, S881–S887 (1992).
[CrossRef]

Siegman, A. E.

A. E. Siegman, “Defining, measuring, and optimizing laser beam quality,” Proc. SPIE 1868, 2–12 (1993).
[CrossRef]

Unlu, M. S.

W. D. Herzog, M. S. Unlu, B. B. Goldberg, G. H. Rhodes, and C. Harder, “Beam divergence and waist measurements of laser diodes by near-field scanning optical microscopy,” Appl. Phys. Lett. 70, 688–690 (1997).
[CrossRef]

Wang, J.

Wang, X.

P. Zhou, Y. Ma, X. Wang, H. Ma, J. Wang, X. Xu, and Z. Liu, “Coherent beam combination of a hexagonal distributed high-power fiber amplifier array,” Appl. Opt. 48, 6537–6540 (2009).
[CrossRef] [PubMed]

P. Zhou, Z. Liu, X. Xu, Z. Chen, and X. Wang, “Beam quality factor for coherently combined fiber laser beams,” Opt. Laser Technol. 41, 268–271 (2009).

Weber, H.

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation: Fundamentals, Advanced Concepts and Applications (Springer, 2005).

H. Weber, “Some historical and technical aspects of beam quality,” Opt. Quantum Electron. 24, S861–S864 (1992).
[CrossRef]

Xiao, R.

Xu, X.

P. Zhou, Z. Liu, X. Xu, Z. Chen, and X. Wang, “Beam quality factor for coherently combined fiber laser beams,” Opt. Laser Technol. 41, 268–271 (2009).

P. Zhou, Y. Ma, X. Wang, H. Ma, J. Wang, X. Xu, and Z. Liu, “Coherent beam combination of a hexagonal distributed high-power fiber amplifier array,” Appl. Opt. 48, 6537–6540 (2009).
[CrossRef] [PubMed]

Zhang, T.

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A Pure Appl. Opt. 11, 105705 (2009).
[CrossRef]

Zhou, P.

P. Zhou, Z. Liu, X. Xu, Z. Chen, and X. Wang, “Beam quality factor for coherently combined fiber laser beams,” Opt. Laser Technol. 41, 268–271 (2009).

P. Zhou, Y. Ma, X. Wang, H. Ma, J. Wang, X. Xu, and Z. Liu, “Coherent beam combination of a hexagonal distributed high-power fiber amplifier array,” Appl. Opt. 48, 6537–6540 (2009).
[CrossRef] [PubMed]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

W. D. Herzog, M. S. Unlu, B. B. Goldberg, G. H. Rhodes, and C. Harder, “Beam divergence and waist measurements of laser diodes by near-field scanning optical microscopy,” Appl. Phys. Lett. 70, 688–690 (1997).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. P. Dumke, “Angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11, 400–402 (1975).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
[CrossRef]

J. Appl. Phys. (1)

D. D. Cook and F. R. Nash, “Gain-induced guiding and astigmatic output beam of GaAs lasers,” J. Appl. Phys. 46, 1660–1662 (1975).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A Pure Appl. Opt. 11, 105705 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Express (1)

Opt. Laser Technol. (1)

P. Zhou, Z. Liu, X. Xu, Z. Chen, and X. Wang, “Beam quality factor for coherently combined fiber laser beams,” Opt. Laser Technol. 41, 268–271 (2009).

Opt. Lett. (1)

Opt. Quantum Electron. (2)

J. Serna, P. M. Mejías, and R. Martínez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, S881–S887 (1992).
[CrossRef]

H. Weber, “Some historical and technical aspects of beam quality,” Opt. Quantum Electron. 24, S861–S864 (1992).
[CrossRef]

Proc. SPIE (1)

A. E. Siegman, “Defining, measuring, and optimizing laser beam quality,” Proc. SPIE 1868, 2–12 (1993).
[CrossRef]

Other (2)

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation: Fundamentals, Advanced Concepts and Applications (Springer, 2005).

ISO Standard 11146, “Lasers and laser-related equipment—Test methods for laser beam widths, divergence angles, and beam propagation ratios” (2005).

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Figures (5)

Fig. 1
Fig. 1

GC and 1 / M 2 parameters for super- Gaussian beams as a function of their order n. The M 2 parameter was inverted in order to fit the same graph.

Fig. 2
Fig. 2

Input field consisting of seven beams of width 2 w g and period Λ. Dotted line, BFG.

Fig. 3
Fig. 3

GC as a function of beams in the array, for different values of the normalized period Λ / w g . Dashed lines for N = 20 (green and red online), local maxima of the overlap integral from which the GC is chosen.

Fig. 4
Fig. 4

Parameters of the optimized Gaussian and GC computed independently at more than 100 planes in the vicinity of the focal plane as a function of z for a low-quality beam. (a) Gaussian width w ( z ) , (b) spherical phase radius of curvature R ( z ) , (c) GC parameter. Red lines, standard Gaussian fit. The inset shows a sampled profile (intensity and phase) near the waist of the beam.

Fig. 5
Fig. 5

Similar to Fig. 4, but measuring a higher quality beam.

Tables (1)

Tables Icon

Table 1 Gaussian Content, W opt and M 2 Values of Simple Beam Profiles

Equations (6)

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U ( x , y ) = n , m A n . m w GH n , m w ( x , y ) , A n , m w = [ U ( x , y ) · ( GH n , m w ( x , y ) ) * ] ,
GC Max w , R | [ 2 π ] 1 / 4 1 w 1 / 2 [ U ( x ) ] * exp ( i k x 2 2 ( 1 R i 2 k w 2 ) ) d x | 2 .
U ( x ) · exp [ ( x / w opt ) 2 ( 2 ( x / w opt ) 2 1 / 2 ] d x = 0 ,
GC [ 2 π ] 1 / 2 1 w opt | U ( x ) exp ( ( x 2 w opt 2 ) ) d x | 2 .
GC = 2 w opt w g w opt 2 + w g 2 exp ( 2 d 2 w opt 2 + w g 2 ) w opt = [ 2 d 2 + ( 4 d 4 + w g 4 ) 1 / 2 ] 1 / 2 .
U ( x , y ) = exp ( | x | / ε ) exp ( y 2 2 σ 2 + i y x 2 2 R ) .

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