Abstract

The mathematical expressions for the coherent-mode decomposition of Bessel–Gauss (BG) beams have been derived for the general case, for which the cross correlation of modes is included, and the inverse problem of calculating the mode parameters from the given M2 factor has been solved for BG beams.

© 1999 Optical Society of America

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References

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  1. P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzalez-Urena, eds., Proceedings of the First Workshop on Laser Beam Characterization, Madrid, Spain, June 14–16 1993 (workshop sponsored by Sociedad Española de Optica).
  2. H. Weber, N. Reng, J. Lüdtke, P. M. Mejias, eds., Proceedings of the Second Workshop on Laser Beam Characterization, Berlin, Germany, May 30–June 1, 1994 (workshop sponsored by Festkörper-Laser-Institut Berlin GmbH).
  3. A. E. Siegman, “New development in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
    [CrossRef]
  4. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
    [CrossRef]
  5. K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
    [CrossRef]
  6. K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, 1095–1108 (1992).
    [CrossRef]
  7. K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Measurement of the mode coherence coefficients,” Opt. Quantum Electron. 24, 1119–1127 (1992).
    [CrossRef]
  8. F. Gori, G. Geattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  9. R. Borghi, M. Santarsiero, “M2 factor of Bessel-Gauss beams,” Opt. Lett. 22, 262–264 (1997).
    [CrossRef] [PubMed]
  10. A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).
  11. I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).
  12. H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, p. 187.
  13. A. E. Siegman, Handbook of Laser Beam Propagation and Beam Quality Formulas Using the Spatial-Frequency and Intensity-Moments Analysis (Stanford University, Palo Alto, Calif., 1991).
  14. B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–53 (1993).
    [CrossRef]
  15. C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” Microwaves Opt. Acoust. 2, 105–112 (1978).
    [CrossRef]

1997 (1)

1993 (1)

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–53 (1993).
[CrossRef]

1992 (4)

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

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, 1095–1108 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Measurement of the mode coherence coefficients,” Opt. Quantum Electron. 24, 1119–1127 (1992).
[CrossRef]

1987 (1)

F. Gori, G. Geattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

1978 (1)

C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” Microwaves Opt. Acoust. 2, 105–112 (1978).
[CrossRef]

Borghi, R.

Cai, B.

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–53 (1993).
[CrossRef]

Du, K. M.

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, 1095–1108 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Measurement of the mode coherence coefficients,” Opt. Quantum Electron. 24, 1119–1127 (1992).
[CrossRef]

Erdélyi, A.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Gamo, H.

H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, p. 187.

Geattari, G.

F. Gori, G. Geattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Gori, F.

F. Gori, G. Geattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Gradshtein, I. S.

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Herziger, G.

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Measurement of the mode coherence coefficients,” Opt. Quantum Electron. 24, 1119–1127 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, 1095–1108 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
[CrossRef]

Loosen, P.

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, 1095–1108 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Measurement of the mode coherence coefficients,” Opt. Quantum Electron. 24, 1119–1127 (1992).
[CrossRef]

Lü, B.

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–53 (1993).
[CrossRef]

Magnus, W.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Oberhettinger, F.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Padovani, C.

F. Gori, G. Geattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Rühl, F.

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Measurement of the mode coherence coefficients,” Opt. Quantum Electron. 24, 1119–1127 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, 1095–1108 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Santarsiero, M.

Sheppard, C. J. R.

C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” Microwaves Opt. Acoust. 2, 105–112 (1978).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Handbook of Laser Beam Propagation and Beam Quality Formulas Using the Spatial-Frequency and Intensity-Moments Analysis (Stanford University, Palo Alto, Calif., 1991).

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

Tricomi, F. G.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Weber, H.

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

Wilson, T.

C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” Microwaves Opt. Acoust. 2, 105–112 (1978).
[CrossRef]

Yang, C.

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–53 (1993).
[CrossRef]

Zhang, B.

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–53 (1993).
[CrossRef]

Microwaves Opt. Acoust. (1)

C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” Microwaves Opt. Acoust. 2, 105–112 (1978).
[CrossRef]

Opt. Commun. (2)

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–53 (1993).
[CrossRef]

F. Gori, G. Geattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (4)

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

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, 1095–1108 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Measurement of the mode coherence coefficients,” Opt. Quantum Electron. 24, 1119–1127 (1992).
[CrossRef]

Other (7)

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, p. 187.

A. E. Siegman, Handbook of Laser Beam Propagation and Beam Quality Formulas Using the Spatial-Frequency and Intensity-Moments Analysis (Stanford University, Palo Alto, Calif., 1991).

P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzalez-Urena, eds., Proceedings of the First Workshop on Laser Beam Characterization, Madrid, Spain, June 14–16 1993 (workshop sponsored by Sociedad Española de Optica).

H. Weber, N. Reng, J. Lüdtke, P. M. Mejias, eds., Proceedings of the Second Workshop on Laser Beam Characterization, Berlin, Germany, May 30–June 1, 1994 (workshop sponsored by Festkörper-Laser-Institut Berlin GmbH).

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

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Tables (1)

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Table 1 Some Illustrative Numerical Examples for the Coherent-Mode Decomposition of BG Beams

Equations (30)

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E(r, θ)=plcplϕpl(r, θ),
ϕpl(r, θ)=upl exp-α22r2(αr)lLpl(α2r2)exp(ilθ),
E(r, θ)=Jn(βr)exp(-r2/w02)exp(inθ),
cpl=02π0+ϕpl*(r, θ)E(r, θ)rdrdθ.
02π exp[-i(l-n)θ]dθ
=2πl=n0ln,
0+xl exp(-β1x2)Lpl(γx2)Jl(xy)xdx
=2-l-1β1-p-l-1(β1-γ)pyl exp-y24β1
×Lplγy24β1(γ-β1),
cpl=cpn=π1/2α1/w02+α2/2exp-β24(1/w02+α2/2)×p!(p+n)!1/2αβ2(1/w02+α2/2)nα2/2-1/w02α2/2+1/w02p
×Lpnα2β24(1/w02+α2/2)(α2/2-1/w02)(l=n),
cpl=0(ln).
J(r1, θ1, r2, θ2)=cplcpl*ϕ(r1, θ1)ϕ*(r2, θ2)=λppllϕ(r1, θ1)ϕ*(r2, θ2),
λppll=cplcpl*
σr2=1α2Σ(2p+l+1)λppll-Σ[(p+1)(p+l+1)]1/2λp,p+1,llΣλppll.
σk2=α2Σ(2p+l+1)λppll+Σ[(p+1)(p+l+1)]1/2λp,p+1,llΣλppll.
M2=([Σ(2p+l+1)λppll]2-{Σ[(p+l)(p+l+1)]1/2λp,p+1,ll}2)1/2Σλppll.
M2=Σ(2p+l+1)λppllΣλppll.
M2=1+n+νIn+1(ν)In(ν)2-ν21/2,
ν=β2w024.
β2w024=f(M2, n).
λppnn=πα2(1/w02+α2/2)2exp-β22(1/w02+α2/2)p!(p+n)!α2β24(1/w02+α2/2)2nα2w02/2-1α2w02/2+12p×Lpnα2β24(1/w02+α2/2)(α2/2-1/w02)2,
λp,p+1,nn=πα2(1/w02+α2/2)2exp-β22(1/w02+α2/2)p!(p+1)!(p+n)!(p+n+1)!1/2α2β24(1/w02+α2/2)2nα2w02/2-1α2w02/2+12p+1×Lpnα2β24(1/w02+α2/2)(α2/2-1/w02)Lp+1nα2β24(1/w02+α2/2)(α2/2-1/w02),
Cppnn=λppnnΣλppnn,
Cp,p+1,nn=λp,p+1,nnΣλppnn.
λppnn=λ00nn=πw022exp(-ν)(ν/2)nn!,
λp,p+1,nn=0,
M2=n+1,
M2=3,n=2,p, n=0, 2;C0022=100%,
M2=5,n=4,p, n=0, 4;C0044=100%.

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