Abstract

We present a real-time method to determine the beam propagation ratio M 2 of laser beams. The all-optical measurement of modal amplitudes yields M 2 parameters conform to the ISO standard method. The experimental technique is simple and fast, which allows to investigate laser beams under conditions inaccessible to other methods.

©2011 Optical Society of America

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References

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  1. A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, M. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.
  2. International Organization for Standardization, ISO 11146-1/2/3 Test methods for laser beam widths, divergence angles and beam propagation ratios – Part 1: Stigmatic and simple astigmatic beams / Part 2: General astigmatic beams / Part 3: Intrinsic and geometrical laser beam classification, propagation and details of test methods (ISO, Geneva, 2005).
    [PubMed]
  3. B. Schäfer and K. Mann, “Determination of beam parameters and coherence properties of laser radiation by use of an extended Hartmann-Shack wave-front sensor,” Appl. Opt. 41, 2809–2817 (2002).
    [Crossref] [PubMed]
  4. B. Eppich, G. Mann, and H. Weber, “Measurement of the four-dimensional Wigner distribution of paraxial light sources,” in Optical Design and Engineering II, L. Mazuray and R. Wartmann, eds., Proc. SPIE 5962, 59622D (2005).
  5. R. W. Lambert, R. Cortés-Martínez, A. J. Waddie, J. D. Shephard, M. R. Taghizadeh, A. H. Greenaway, and D. P. Hand, “Compact optical system for pulse-to-pulse laser beam quality measurement and applications in laser machining,” Appl. Opt. 43, 5037–5046 (2004).
    [Crossref] [PubMed]
  6. A. E. Siegman, Lasers (University Science Books, 1986).
  7. S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
    [Crossref]
  8. E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
    [Crossref]
  9. A. Cutolo, T. Isernia, I. Izzo, R. Pierri, and L. Zeni, “Transverse mode analysis of a laser beam by near-and far-field intensity measurements,” Appl. Opt. 34, 7974–7978 (1995).
    [Crossref] [PubMed]
  10. M. Santarsiero, F. Gori, R. Borghi, and G. Guattari, “Evaluation of the modal structure of light beams composed of incoherent mixtures of Hermite-Gaussian modes,” Appl. Opt. 38, 5272–5281 (1999).
    [Crossref]
  11. X. Xue, H. Wei, and A. G. Kirk, “Intensity-based modal decomposition of optical beams in terms of Hermite-Gaussian functions,” J. Opt. Soc. Am. A 17, 1086–1091 (2000).
    [Crossref]
  12. N. Andermahr, T. Theeg, and C. Fallnich, “Novel approach for polarization-sensitive measurements of transverse modes in few-mode optical fibers,” Appl. Phys. B 91, 353–357 (2008).
    [Crossref]
  13. V. A. Soifer and M. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, 1994).
  14. M. Duparré, B. Lüdge, and S. Schröter, “On-line characterization of Nd:YAG laser beams by means of modal decomposition using diffractive optical correlation filters,” in Optical Design and Engineering II, L. Mazuray and R. Wartmann, eds., Proc. SPIE 5962, 59622G (2005).
  15. T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express 17, 9347–9356 (2009).
    [Crossref] [PubMed]
  16. D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distributionof multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010).
    [Crossref] [PubMed]
  17. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [Crossref] [PubMed]
  18. H. Laabs and B. Ozygus, “Excitation of Hermite-Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
    [Crossref]
  19. G. Szegö, Orthogonal Polynomials (Amer. Math. Soc., 1975).

2010 (1)

2009 (1)

2008 (1)

N. Andermahr, T. Theeg, and C. Fallnich, “Novel approach for polarization-sensitive measurements of transverse modes in few-mode optical fibers,” Appl. Phys. B 91, 353–357 (2008).
[Crossref]

2004 (1)

2002 (1)

2000 (1)

1999 (1)

1998 (1)

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
[Crossref]

1996 (1)

H. Laabs and B. Ozygus, “Excitation of Hermite-Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
[Crossref]

1995 (1)

1989 (1)

E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[Crossref]

1966 (1)

Andermahr, N.

N. Andermahr, T. Theeg, and C. Fallnich, “Novel approach for polarization-sensitive measurements of transverse modes in few-mode optical fibers,” Appl. Phys. B 91, 353–357 (2008).
[Crossref]

Borchardt, J.

Borghi, R.

Cortés-Martínez, R.

Cutolo, A.

Duparré, M.

Eppich, B.

B. Eppich, G. Mann, and H. Weber, “Measurement of the four-dimensional Wigner distribution of paraxial light sources,” in Optical Design and Engineering II, L. Mazuray and R. Wartmann, eds., Proc. SPIE 5962, 59622D (2005).

Fallnich, C.

N. Andermahr, T. Theeg, and C. Fallnich, “Novel approach for polarization-sensitive measurements of transverse modes in few-mode optical fibers,” Appl. Phys. B 91, 353–357 (2008).
[Crossref]

Flamm, D.

Friberg, A.

E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[Crossref]

Golub, M.

V. A. Soifer and M. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, 1994).

Gori, F.

Greenaway, A. H.

Guattari, G.

Hand, D. P.

Isernia, T.

Izzo, I.

Kaiser, T.

Kirk, A. G.

Kogelnik, H.

Laabs, H.

H. Laabs and B. Ozygus, “Excitation of Hermite-Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
[Crossref]

Lambert, R. W.

Li, T.

Lüdge, B.

M. Duparré, B. Lüdge, and S. Schröter, “On-line characterization of Nd:YAG laser beams by means of modal decomposition using diffractive optical correlation filters,” in Optical Design and Engineering II, L. Mazuray and R. Wartmann, eds., Proc. SPIE 5962, 59622G (2005).

Mann, G.

B. Eppich, G. Mann, and H. Weber, “Measurement of the four-dimensional Wigner distribution of paraxial light sources,” in Optical Design and Engineering II, L. Mazuray and R. Wartmann, eds., Proc. SPIE 5962, 59622D (2005).

Mann, K.

Ozygus, B.

H. Laabs and B. Ozygus, “Excitation of Hermite-Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
[Crossref]

Pierri, R.

Saghafi, S.

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
[Crossref]

Santarsiero, M.

Schäfer, B.

Schmidt, O. A.

Schröter, S.

Schulze, C.

Shephard, J. D.

Sheppard, C. J. R.

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
[Crossref]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, M. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.

Soifer, V. A.

V. A. Soifer and M. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, 1994).

Szegö, G.

G. Szegö, Orthogonal Polynomials (Amer. Math. Soc., 1975).

Taghizadeh, M. R.

Tervonen, E.

E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[Crossref]

Theeg, T.

N. Andermahr, T. Theeg, and C. Fallnich, “Novel approach for polarization-sensitive measurements of transverse modes in few-mode optical fibers,” Appl. Phys. B 91, 353–357 (2008).
[Crossref]

Turunen, J.

E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[Crossref]

Waddie, A. J.

Weber, H.

B. Eppich, G. Mann, and H. Weber, “Measurement of the four-dimensional Wigner distribution of paraxial light sources,” in Optical Design and Engineering II, L. Mazuray and R. Wartmann, eds., Proc. SPIE 5962, 59622D (2005).

Wei, H.

Xue, X.

Zeni, L.

Appl. Opt. (5)

Appl. Phys. B (2)

N. Andermahr, T. Theeg, and C. Fallnich, “Novel approach for polarization-sensitive measurements of transverse modes in few-mode optical fibers,” Appl. Phys. B 91, 353–357 (2008).
[Crossref]

E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[Crossref]

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

Opt. Commun. (1)

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
[Crossref]

Opt. Express (1)

Opt. Laser Technol. (1)

H. Laabs and B. Ozygus, “Excitation of Hermite-Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
[Crossref]

Opt. Lett. (1)

Other (7)

G. Szegö, Orthogonal Polynomials (Amer. Math. Soc., 1975).

V. A. Soifer and M. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, 1994).

M. Duparré, B. Lüdge, and S. Schröter, “On-line characterization of Nd:YAG laser beams by means of modal decomposition using diffractive optical correlation filters,” in Optical Design and Engineering II, L. Mazuray and R. Wartmann, eds., Proc. SPIE 5962, 59622G (2005).

A. E. Siegman, Lasers (University Science Books, 1986).

B. Eppich, G. Mann, and H. Weber, “Measurement of the four-dimensional Wigner distribution of paraxial light sources,” in Optical Design and Engineering II, L. Mazuray and R. Wartmann, eds., Proc. SPIE 5962, 59622D (2005).

A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, M. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.

International Organization for Standardization, ISO 11146-1/2/3 Test methods for laser beam widths, divergence angles and beam propagation ratios – Part 1: Stigmatic and simple astigmatic beams / Part 2: General astigmatic beams / Part 3: Intrinsic and geometrical laser beam classification, propagation and details of test methods (ISO, Geneva, 2005).
[PubMed]

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Figures (4)
Fig. 1
Fig. 1 Scheme of experimental setup. Upper branch: M 2 determination using a CGH. Lower branch: M 2 determination by a caustic measurement conform to the ISO standard.

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Fig. 2
Fig. 2 Modal spectrum of an arbitrarily chosen mode mixture (left) with corresponding measured and reconstructed near field intensity distributions (right).

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Fig. 3
Fig. 3 Comparison of the M x 2 and M y 2 factors determined by the CGH technique and the ISO standard method, respectively.

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Fig. 4
Fig. 4 M x 2 and M y 2 on a time scale of 30 s taken with a rate of 4 Hz. To vary the beam quality the laser resonator was tuned continuously. The insets depict the waist intensities at selected points in time.

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Equations (11)

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HG mn ( x ,   y ) = 1 w 0 2 2 m + n π m ! n ! H m ( 2 w 0 y ) H n ( 2 w 0 y ) exp ( − x 2 + y 2 w 0 2 ) ,
U ( x ,   y ) = ∑ m = 0 ∞ ∑ n = 0 ∞ c mn HG mn ( x ,   y ) ,                       c mn = ∬ − ∞ ∞ HG mn * U   d x   d y ,
M x ,   y 2 = π λ d x ,   y θ x ,   y 4 ,
d x = 8 [ σ x 2 + σ y 2 + γ ( σ x 2 − σ y 2 ) 2 + 4 ( σ x y 2 ) 2 ] 1 / 2 ,                   γ = σ x 2 − σ y 2 | σ x 2 − σ y 2 | .
M x , y 2 = d x , y 2 4 w 0 2 .
I = | U | 2 = ∑ p = 0 ∞ I p           with           I p = ∑ k , m ρ k , p − k ρ m , p − m HG k , p − k HG m , p − m ,
σ x 2 = ∬ − ∞ ∞ x 2 I ( x ,   y ) d x   d y = w 0 2 4 ∑ m , n ρ mn 2 ( 2 m + 1 ) ,
σ x y 2 = ∬ − ∞ ∞ x y I ( x ,   y ) d x   d y = w 0 2 2 ∑ m , n ρ m + 1 ,   n ρ m ,   n + 1 m + 1 n + 1 ,
∬ − ∞ ∞ x 2 HG k l HG mn d x   d y = w 0 2 4 δ ln [ ( 2 m + 1 ) δ km + max ( k ,   m ) max ( k , m ) − 1 δ | k − m | , 2 ] ,
∬ − ∞ ∞ x y HG k l HG mn d x   d y = w 0 2 4 max ( k ,   m ) δ | k − m | , 1 max ( l ,   n ) δ | l − n | , 1 ,
M x 2 = ∑ m , n ρ mn 2 ( m + n + 1 ) + γ [ ( ∑ m , n ρ mn 2 ( m − n ) ) 2 + 4 ( ∑ m , n ρ m + 1 , n ρ m , n + 1 m + 1 n + 1 ) 2 ] 1 / 2 .

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