Abstract

A quantitative measurement of laser-beam quality can be performed by determination of the presence of multiple transverse modes of the laser oscillator and by calculation of their power content. Along this line of argument, we discuss a new approach that, starting from near-field and far-field intensity measurements, can evaluate the complex excitation coefficients of the transverse modes in a laser beam. The exploitation of near-field measurements sharply improves the performances of the technique in those cases in which only far-field measurements are used. The validity of the method is confirmed by several accurate numerical simulations and by some experimental results relative to a multimode Q-switched Nd:YAG laser.

© 1995 Optical Society of America

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References

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  1. D. F. Grevey, K. F. Badawi, J. C. Boquillon, B. Taisne, “Interest of beam quality in materials treatment by high power lasers,” J. Phys. (Paris) 1, 167–170 (1991).
  2. G. Berkovic, E. Shvartsberg, “Phase measurements in surface nonlinear optics: the effect of laser beam quality,” Appl. Phys. B 53, 333–338 (1991).
    [CrossRef]
  3. D. Wright, P. Greve, J. Fleischer, L. Austin, “Laser beam width, divergence and beam propagation factor—an international standardization approach,” Opt. Quantum Electron. 24, S993–S1000 (1992).
    [CrossRef]
  4. A. Cutolo, A. Esposito, T. Isernia, R. Pierri, L. Zeni, “Characterization of the transverse modes in a laser beam: analysis and application to a Q-switched Nd:YAG laser,” Appl. Opt. 31, 2722–2733 (1992).
    [CrossRef] [PubMed]
  5. A. Cutolo, F. Ferreri, T. Isernia, R. Pierri, L. Zeni, “Measurement of the waist and the power distribution across the transverse modes of a laser beam,” Opt. Quantum Electron. 24, S963–S971 (1992).
    [CrossRef]
  6. T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inverse Prob. 11, 183–203 (1995).
    [CrossRef]
  7. T. Isernia, G. Leone, R. Pierri, “Phaseless near field techniques: uniqueness conditions and attainment of the solution,” J. Electromagn. Waves App. 8, 889–908 (1994).
  8. A. Siegman, Lasers (University Science, San Francisco, 1987).
  9. H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef] [PubMed]
  10. P. T. Chen, M. A. Fiddy, “Image reconstruction from power spectral data with use of point zero locations,” J. Opt. Soc. Am. A 11, 2210–2215 (1994).
    [CrossRef]

1995 (1)

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inverse Prob. 11, 183–203 (1995).
[CrossRef]

1994 (2)

T. Isernia, G. Leone, R. Pierri, “Phaseless near field techniques: uniqueness conditions and attainment of the solution,” J. Electromagn. Waves App. 8, 889–908 (1994).

P. T. Chen, M. A. Fiddy, “Image reconstruction from power spectral data with use of point zero locations,” J. Opt. Soc. Am. A 11, 2210–2215 (1994).
[CrossRef]

1992 (3)

D. Wright, P. Greve, J. Fleischer, L. Austin, “Laser beam width, divergence and beam propagation factor—an international standardization approach,” Opt. Quantum Electron. 24, S993–S1000 (1992).
[CrossRef]

A. Cutolo, F. Ferreri, T. Isernia, R. Pierri, L. Zeni, “Measurement of the waist and the power distribution across the transverse modes of a laser beam,” Opt. Quantum Electron. 24, S963–S971 (1992).
[CrossRef]

A. Cutolo, A. Esposito, T. Isernia, R. Pierri, L. Zeni, “Characterization of the transverse modes in a laser beam: analysis and application to a Q-switched Nd:YAG laser,” Appl. Opt. 31, 2722–2733 (1992).
[CrossRef] [PubMed]

1991 (2)

D. F. Grevey, K. F. Badawi, J. C. Boquillon, B. Taisne, “Interest of beam quality in materials treatment by high power lasers,” J. Phys. (Paris) 1, 167–170 (1991).

G. Berkovic, E. Shvartsberg, “Phase measurements in surface nonlinear optics: the effect of laser beam quality,” Appl. Phys. B 53, 333–338 (1991).
[CrossRef]

1966 (1)

Austin, L.

D. Wright, P. Greve, J. Fleischer, L. Austin, “Laser beam width, divergence and beam propagation factor—an international standardization approach,” Opt. Quantum Electron. 24, S993–S1000 (1992).
[CrossRef]

Badawi, K. F.

D. F. Grevey, K. F. Badawi, J. C. Boquillon, B. Taisne, “Interest of beam quality in materials treatment by high power lasers,” J. Phys. (Paris) 1, 167–170 (1991).

Berkovic, G.

G. Berkovic, E. Shvartsberg, “Phase measurements in surface nonlinear optics: the effect of laser beam quality,” Appl. Phys. B 53, 333–338 (1991).
[CrossRef]

Boquillon, J. C.

D. F. Grevey, K. F. Badawi, J. C. Boquillon, B. Taisne, “Interest of beam quality in materials treatment by high power lasers,” J. Phys. (Paris) 1, 167–170 (1991).

Chen, P. T.

Cutolo, A.

A. Cutolo, A. Esposito, T. Isernia, R. Pierri, L. Zeni, “Characterization of the transverse modes in a laser beam: analysis and application to a Q-switched Nd:YAG laser,” Appl. Opt. 31, 2722–2733 (1992).
[CrossRef] [PubMed]

A. Cutolo, F. Ferreri, T. Isernia, R. Pierri, L. Zeni, “Measurement of the waist and the power distribution across the transverse modes of a laser beam,” Opt. Quantum Electron. 24, S963–S971 (1992).
[CrossRef]

Esposito, A.

Ferreri, F.

A. Cutolo, F. Ferreri, T. Isernia, R. Pierri, L. Zeni, “Measurement of the waist and the power distribution across the transverse modes of a laser beam,” Opt. Quantum Electron. 24, S963–S971 (1992).
[CrossRef]

Fiddy, M. A.

Fleischer, J.

D. Wright, P. Greve, J. Fleischer, L. Austin, “Laser beam width, divergence and beam propagation factor—an international standardization approach,” Opt. Quantum Electron. 24, S993–S1000 (1992).
[CrossRef]

Greve, P.

D. Wright, P. Greve, J. Fleischer, L. Austin, “Laser beam width, divergence and beam propagation factor—an international standardization approach,” Opt. Quantum Electron. 24, S993–S1000 (1992).
[CrossRef]

Grevey, D. F.

D. F. Grevey, K. F. Badawi, J. C. Boquillon, B. Taisne, “Interest of beam quality in materials treatment by high power lasers,” J. Phys. (Paris) 1, 167–170 (1991).

Isernia, T.

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inverse Prob. 11, 183–203 (1995).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, “Phaseless near field techniques: uniqueness conditions and attainment of the solution,” J. Electromagn. Waves App. 8, 889–908 (1994).

A. Cutolo, A. Esposito, T. Isernia, R. Pierri, L. Zeni, “Characterization of the transverse modes in a laser beam: analysis and application to a Q-switched Nd:YAG laser,” Appl. Opt. 31, 2722–2733 (1992).
[CrossRef] [PubMed]

A. Cutolo, F. Ferreri, T. Isernia, R. Pierri, L. Zeni, “Measurement of the waist and the power distribution across the transverse modes of a laser beam,” Opt. Quantum Electron. 24, S963–S971 (1992).
[CrossRef]

Kogelnik, H.

Leone, G.

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inverse Prob. 11, 183–203 (1995).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, “Phaseless near field techniques: uniqueness conditions and attainment of the solution,” J. Electromagn. Waves App. 8, 889–908 (1994).

Li, T.

Pierri, R.

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inverse Prob. 11, 183–203 (1995).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, “Phaseless near field techniques: uniqueness conditions and attainment of the solution,” J. Electromagn. Waves App. 8, 889–908 (1994).

A. Cutolo, A. Esposito, T. Isernia, R. Pierri, L. Zeni, “Characterization of the transverse modes in a laser beam: analysis and application to a Q-switched Nd:YAG laser,” Appl. Opt. 31, 2722–2733 (1992).
[CrossRef] [PubMed]

A. Cutolo, F. Ferreri, T. Isernia, R. Pierri, L. Zeni, “Measurement of the waist and the power distribution across the transverse modes of a laser beam,” Opt. Quantum Electron. 24, S963–S971 (1992).
[CrossRef]

Shvartsberg, E.

G. Berkovic, E. Shvartsberg, “Phase measurements in surface nonlinear optics: the effect of laser beam quality,” Appl. Phys. B 53, 333–338 (1991).
[CrossRef]

Siegman, A.

A. Siegman, Lasers (University Science, San Francisco, 1987).

Taisne, B.

D. F. Grevey, K. F. Badawi, J. C. Boquillon, B. Taisne, “Interest of beam quality in materials treatment by high power lasers,” J. Phys. (Paris) 1, 167–170 (1991).

Wright, D.

D. Wright, P. Greve, J. Fleischer, L. Austin, “Laser beam width, divergence and beam propagation factor—an international standardization approach,” Opt. Quantum Electron. 24, S993–S1000 (1992).
[CrossRef]

Zeni, L.

A. Cutolo, A. Esposito, T. Isernia, R. Pierri, L. Zeni, “Characterization of the transverse modes in a laser beam: analysis and application to a Q-switched Nd:YAG laser,” Appl. Opt. 31, 2722–2733 (1992).
[CrossRef] [PubMed]

A. Cutolo, F. Ferreri, T. Isernia, R. Pierri, L. Zeni, “Measurement of the waist and the power distribution across the transverse modes of a laser beam,” Opt. Quantum Electron. 24, S963–S971 (1992).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

G. Berkovic, E. Shvartsberg, “Phase measurements in surface nonlinear optics: the effect of laser beam quality,” Appl. Phys. B 53, 333–338 (1991).
[CrossRef]

Inverse Prob. (1)

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inverse Prob. 11, 183–203 (1995).
[CrossRef]

J. Electromagn. Waves App. (1)

T. Isernia, G. Leone, R. Pierri, “Phaseless near field techniques: uniqueness conditions and attainment of the solution,” J. Electromagn. Waves App. 8, 889–908 (1994).

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

J. Phys. (Paris) (1)

D. F. Grevey, K. F. Badawi, J. C. Boquillon, B. Taisne, “Interest of beam quality in materials treatment by high power lasers,” J. Phys. (Paris) 1, 167–170 (1991).

Opt. Quantum Electron. (2)

D. Wright, P. Greve, J. Fleischer, L. Austin, “Laser beam width, divergence and beam propagation factor—an international standardization approach,” Opt. Quantum Electron. 24, S993–S1000 (1992).
[CrossRef]

A. Cutolo, F. Ferreri, T. Isernia, R. Pierri, L. Zeni, “Measurement of the waist and the power distribution across the transverse modes of a laser beam,” Opt. Quantum Electron. 24, S963–S971 (1992).
[CrossRef]

Other (1)

A. Siegman, Lasers (University Science, San Francisco, 1987).

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Figures (2)

Fig. 1
Fig. 1

Schematic of the measurement setup.

Fig. 2
Fig. 2

Experimental results: (a) the actual laser-beam profile measured by the camera; (b) the laser-beam profile relative to the coefficients reported in Table 4.

Tables (4)

Tables Icon

Table 1 Numerical Simulation with ∊ M = 5%, Waist = 0.0726 mm

Tables Icon

Table 2 Numerical Simulation with ∊ M = 20%, Waist = 0.0726 mm

Tables Icon

Table 3 Numerical Simulation with Eight-Bit Quantization, Waist = 0.0726 mm

Tables Icon

Table 4 Experimental Results with Retrieved Waist = 0.10 mm

Equations (15)

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u n m ( x , y , z ) = [ 2 n + m - 1 π n ! m ! / w ( z ) ] × H n [ 2 1 / 2 x / w ( z ) ] H m [ 2 1 / 2 y / w ( z ) ] × exp ( j { ( n + m + 1 ) arctan ( z / z R ) - ( x 2 + y 2 ) / [ 2 R ( z ) ] - k z } - ( x 2 + y 2 ) / w 2 ( z ) ) ,
w ( z ) = w 0 [ 1 + ( z / z R ) 2 ] 1 / 2 , R ( z ) = z [ 1 + ( z / z R ) 2 ] ,
ω n m = π c / d { ( q + 1 ) + 1 / π ( m + n + 1 ) × arccos [ ( 1 - d / R 1 ) 1 / 2 ( 1 - d / R 2 ) 1 / 2 ] } ,
E ( x , y , z , t ) = n , m c n m exp ( i ω n m t ) u n m ( x , y , z ) .
I ˜ ( x , y , z , t ) = E ( x , y , z , t ) 2 = n , m c n m exp [ i ( ω n m - ω 00 ) t ] u n m ( x , y , z ) 2 ,
I ˜ ( x , y , z , t ) = n , m ρ n m F n m u n m ( x , y , z ) 2 ,
F n m = exp [ i ( ω n m - ω 00 ) t + i ϕ n m ] .
I ( x , y , z , t ) = n m ρ n m 2 u n m ( x , y , z ) 2 + i , j n j , m i ρ n m ρ i j u n m u i j * F n m F i j * T M ,
I ( x , y , z , t ) = n m ρ n m 2 u n m ( x , y , z ) 2 + i , j n j , m i ρ n m ρ i j u n m u i j * F n m F i j * ,
I ( x , y , z , t ) = n m ρ n m 2 u n m ( x , y , z ) 2 .
I ( x , y ) = E ( x , y ) E * ( x , y ) ,
Ψ = Ψ 1 near field + Ψ 1 far field ,
Ψ 1 = p W p [ n m ρ n m 2 u n m ( x p , y p , z p ) 2 + i , j n j , m i ρ n m ρ i j u n m u i j * F n m F i j * T M - M p 2 ] 2 ,
Ψ = Ψ 1 near field ; α = 1 + Ψ 1 far field ; α = β 2 ,
Ψ 1 = p W p ( n m ρ n m 2 u n m ( x p , y p , z p ) 2 + i , j n j , m i ρ n m ρ i j u n m u i j * F n m F i j * T M - α M p 2 ) 2 .

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