Abstract

A beam width measure based on Rényi entropy was introduced by Luis [Opt. Lett 31, 3644 (2006)] . That one-dimensional analysis was limited to beam profiles with rectangular symmetry. In this Letter, we derive a general Rényi beam width measure that accounts for the diffraction properties of beams with profiles of arbitrary symmetry. We also show that the square of this measure has a quadratic dependence as a function of the propagation coordinate, so that it can be applied to propagation through arbitrary ABCD paraxial systems. The Rényi beam propagation factor, here introduced, is discussed in examples where the M2 factor seems to have a limited effectiveness in describing the beam spreading.

© 2008 Optical Society of America

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References

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  1. A. E. Siegman, G. Nemes, and J. Serna, in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.
  2. P. Vaveliuk, B. Ruiz, and A. Lencina, Opt. Lett. 32, 927 (2007).
    [CrossRef] [PubMed]
  3. A. Luis, Opt. Lett. 31, 3644 (2006).
    [CrossRef] [PubMed]
  4. A. Rényi, Probability Theory (North-Holland, 1970).
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  6. A. E. Siegman, IEEE J. Quantum Electron. 27, 1146 (1991).
    [CrossRef]
  7. A. E. Siegman, Lasers (University Science Books, 1986).

2007 (1)

2006 (1)

1991 (1)

A. E. Siegman, IEEE J. Quantum Electron. 27, 1146 (1991).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Lencina, A.

Luis, A.

Nemes, G.

A. E. Siegman, G. Nemes, and J. Serna, in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.

Rényi, A.

A. Rényi, Probability Theory (North-Holland, 1970).

Ruiz, B.

Serna, J.

A. E. Siegman, G. Nemes, and J. Serna, in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.

Siegman, A. E.

A. E. Siegman, IEEE J. Quantum Electron. 27, 1146 (1991).
[CrossRef]

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

A. E. Siegman, G. Nemes, and J. Serna, in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.

Vaveliuk, P.

IEEE J. Quantum Electron. (1)

A. E. Siegman, IEEE J. Quantum Electron. 27, 1146 (1991).
[CrossRef]

Opt. Lett. (2)

Other (4)

A. Rényi, Probability Theory (North-Holland, 1970).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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

A. E. Siegman, G. Nemes, and J. Serna, in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.

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Figures (2)

Fig. 1
Fig. 1

Diffracted exponential (circular) and Gaussian profiles (in units of w 0 2 ) versus k x in units of 1 w 0 with w 0 being the Gaussian beam waist. The inset was placed in log scale to enhance the long tail of the diffracted exponential beam.

Fig. 2
Fig. 2

Diffracted top-hat and Gaussian profiles versus k x . The inset in log scale enhances the secondary peaks of the diffracted circular top-hat beam.

Equations (11)

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Δ x = d x d y x 2 U ( x , y ) 2 d x d y U ( x , y ) 2 ,
Δ k x = d k x d k y k x 2 U ̃ ( k x , k y ) 2 d k x d k y U ̃ ( k x , k y ) 2 ,
Δ x 1 D = d x x 2 U x ( x ) 2 d x U x ( x ) 2 ,
Δ k x 1 D = d k x k x 2 U ̃ k x ( k x ) 2 d k x U ̃ k x ( k x ) 2 .
A x 1 D = ( U x ( x ) 2 d x ) 2 4 π U x ( x ) 4 d x ,
A k x 1 D = ( U ̃ k x ( k x ) 2 d k x ) 2 4 π U ̃ k x ( k x ) 4 d k x .
A x = U ( x , y ) 2 U ( 0 , y ) 2 d x d y U ( x , 0 ) 2 d x 4 π U ( 0 , 0 ) 2 U ( x , y ) 4 d x d y ,
A k x = U ̃ ( k x , k y ) 2 U ̃ ( 0 , k y ) 2 d k x d k y U ̃ ( k x , 0 ) 2 d k x 4 π U ̃ ( 0 , 0 ) 2 U ̃ ( k x , k y ) 4 d k x d k y ,
A x 1 D ( z ) = ( z Φ ( ν ) 2 d ν ) 2 z Φ ( ν ) 4 d ν ,
A x ( z ) = z 3 Φ ( υ , ν ) 2 Φ ( 0 , ν ) 2 d υ d ν Φ ( υ , 0 ) 2 d υ z 2 Φ ( 0 , 0 ) 2 Φ ( υ , ν ) 4 d υ d ν ,
R x 2 = 2 A x A k x , R y 2 = 2 A y A k y ,

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