Abstract

We introduce a new representation of coherent laser beams that are usually described in circular cylindrical coordinates. This representation is based on the decomposition of a laser beam of a given azimuthal order into beams exhibiting Cartesian symmetry. These beams, which we call constituent waves, diffract along only one of their transverse dimensions and propagate noncollinearly with the propagation axis. A cylindrically symmetric laser beam is then considered a coherent superposition of constituent waves and is represented by an integral over an angular variable. Such a representation allows for the introduction of the propagation factor M2, defined in terms of one-dimensional root-mean-square (rms) quantities, in the treatment of two-dimensional beams. The representation naturally leads to the definition of a new rms parameter that we call the quality factor Q. It is shown that the quality factor defines in quantitative terms the nondiffracting character of a laser beam. The representation is first applied to characterize Laguerre–Gauss beams in terms of these one-dimensional rms parameters. This analysis reveals an asymptotic link between Laguerre–Gauss beams and one-dimensional Hermite–Gauss beams in the limit of high azimuthal orders. The representation is also applied to Bessel–Gauss beams and demonstrates the geometrical and one-dimensional characters of the Bessel–Gauss beams that propagate in a nondiffracting regime. By using two separate rms parameters, Q and M2, our approach gives an alternative way to describe laser beam propagation that is especially well suited to characterize Bessel-type nondiffracting beams.

© 2005 Optical Society of America

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References

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  1. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  2. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  3. J. E. Durnin, J. J. Miceli, J. H. Eberly, “Experiments with nondiffracting needle beams,” J. Opt. Soc. Am. B 4, P230 (1987).
  4. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  5. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
  6. Y. Li, H. Lee, E. Wolf, “New generalized Bessel–Gaussian beams,” J. Opt. Soc. Am. A 21, 640–646 (2004).
    [CrossRef]
  7. M. A. Porras, R. Borghi, M. Santarsiero, “Relationship between elegant Laguerre–Gauss and Bessel–Gauss beams,” J. Opt. Soc. Am. A 18, 177–184 (2001).
    [CrossRef]
  8. R. M. Herman, T. A. Wiggins, “Bessel-like beams modulated by arbitrary radial functions,” J. Opt. Soc. Am. A 17, 1021–1032 (2000).
    [CrossRef]
  9. S. Chavez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).
  10. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
  11. R. Borghi, M. Santarsiero, “M2 factor of Bessel–Gauss beams,” Opt. Lett. 22, 262–264 (1997).
    [CrossRef] [PubMed]
  12. R. M. Herman, T. A. Wiggins, “Rayleigh range and the M2 factor for Bessel–Gauss beams,” Appl. Opt. 37, 3398–3400 (1998).
    [CrossRef]
  13. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).
  14. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
  15. A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
    [CrossRef]
  16. P.-A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
    [CrossRef] [PubMed]
  17. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, 2000).

2004 (1)

2001 (1)

2000 (1)

1999 (1)

S. Chavez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).

1998 (1)

1997 (1)

1996 (1)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

1991 (2)

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

P.-A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
[CrossRef] [PubMed]

1987 (4)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Experiments with nondiffracting needle beams,” J. Opt. Soc. Am. B 4, P230 (1987).

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

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Bélanger, P.-A.

Borghi, R.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).

Chavez-Cerda, S.

S. Chavez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).

Durnin, J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

Durnin, J. E.

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Experiments with nondiffracting needle beams,” J. Opt. Soc. Am. B 4, P230 (1987).

Eberly, J. H.

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Experiments with nondiffracting needle beams,” J. Opt. Soc. Am. B 4, P230 (1987).

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Gori, F.

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

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, 2000).

Guattari, G.

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

Herman, R. M.

Lee, H.

Li, Y.

Miceli, J. J.

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Experiments with nondiffracting needle beams,” J. Opt. Soc. Am. B 4, P230 (1987).

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Padovani, C.

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

Porras, M. A.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, 2000).

Santarsiero, M.

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Siegman, A. E.

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

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

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

Spagnolo, G. Schirripa

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Wiggins, T. A.

Wolf, E.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

J. Mod. Opt. (2)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

S. Chavez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).

J. Opt. Soc. Am. A (4)

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

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Experiments with nondiffracting needle beams,” J. Opt. Soc. Am. B 4, P230 (1987).

Opt. Commun. (1)

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

Opt. Lett. (2)

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Other (4)

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).

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

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, 2000).

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

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Figures (11)

Fig. 1
Fig. 1

Typical intensity distribution of a Bessel-type nondiffracting laser beam in (a) the spatial-frequency domain and (b) the spatial domain.

Fig. 2
Fig. 2

(a) Illustration of the decomposition of a laser beam into angular sections in the spatial-frequency domain. (b) Intensity distribution of the constituent wave, in the spatial domain, resulting from the selected angular section in the spatial-frequency domain.

Fig. 3
Fig. 3

Evolution of a constituent wave of a (a) diffracting beam and (b) nondiffracting beam. The solid curves represent the positions x ¯ ( z ) ± 2 σ x ( z ) of the lateral intensity distribution of the constituent wave as a function of z. The dashed lines represent the geometrical projection of the near-field lateral intensity profile given by x ¯ ( z ) ± 2 σ x 0 .

Fig. 4
Fig. 4

Intensity (solid curves) and phase (dashed curves) profiles of the constituent waves of low-order Laguerre–Gauss beams in the spatial and spatial-frequency domains. For each radial order p, the azimuthal order l is indicated to discriminate the curves. The phase profiles are omitted in the spatial-frequency domain; they are flat, and each intensity lobe is in antiphase with its neighbors. These curves were obtained with w 0 = 1 mm and λ = 632.8 nm .

Fig. 5
Fig. 5

(a) Quality factor Q and (b) propagation factor M c 2 of the constituent waves of Laguerre–Gauss beams as a function of the azimuthal order l for different radial order p.

Fig. 6
Fig. 6

Comparison of the exact (solid curves) and the asymptotic (dashed curves) near-field intensity profile of the constituent wave of Laguerre–Gauss modes of radial orders p = 0 , 1, and 2. Exact profiles were obtained with w 0 = 1 mm , λ = 632.8 nm , and l = 100 .

Fig. 7
Fig. 7

Lateral intensity distribution of the constituent wave times 4 π 2 (thin curves) and radial intensity distribution of the resulting two-dimensional Gaussian beam (thick curves) at seven positions along the z axis. For this beam, Q = 1.185 , M c 2 = 1.172 , z R , c = 2.585 z 0 , and Z max = 2.181 z 0 .

Fig. 8
Fig. 8

Intensity (solid curves) and phase (dashed curves) profiles of the constituent waves of low-order Bessel–Gauss beams in the spatial and in the spatial-frequency domains. For each value of β w 0 , the azimuthal order n is indicated to discriminate the curves. The phase profiles in the spatial-frequency domain are omitted, since they are flat. These curves were obtained with w 0 = 1 mm and λ = 632.8 nm .

Fig. 9
Fig. 9

(a) Quality factor Q and (b) propagation factor M c 2 of the constituent waves of a Bessel–Gauss beam as a function of β w 0 and for azimuthal orders n = 0 to 5.

Fig. 10
Fig. 10

Lateral intensity distribution of the constituent wave times 4 π 2 (thin curves) and radial intensity distribution of the resulting two-dimensional Bessel–Gauss beam (thick curves) at seven positions along the z axis. The parameters used are w 0 = 1 mm , β = 8 mm 1 , and λ = 632.8 nm . In such a case, Q = 4.094 , M c 2 1 = 2.596 × 10 6 , z R , c = 1.016 z 0 , and Z max = 0.2481 z 0 .

Fig. 11
Fig. 11

Comparison of the exact (solid curves) on-axis intensity profile of a Bessel–Gauss beam of zero azimuthal order with the geometrical projection (dashed curves) of the near-field lateral intensity distribution of its constituent wave once multiplied by 4 π 2 . These curves were obtained with w 0 = 1 mm and λ = 632.8 nm .

Tables (3)

Tables Icon

Table 1 Quality Factors Q of Laguerre–Gauss Beams of Order ( p , l )

Tables Icon

Table 2 M c 2 Factors of the Constituent Waves of Laguerre–Gauss Beams of Order ( p , l )

Tables Icon

Table 3 Widths 2 σ x 0 w 0 of the Constituent Waves of Laguerre–Gauss Beams of Order ( p , l )

Equations (67)

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U ( x ̃ , y ̃ ) U ( r , θ ) = U ( r ) exp ( + j n θ ) ,
U ̃ ( s x ̃ , s y ̃ ) U ̃ ( s r , ϕ ) = j n U ̃ ( s r ) exp ( + j n ϕ ) ,
U ̃ c ( s x , s y ) H ( s x ) s x j n U ̃ ( s r = s x ) exp ( + j n ϕ ) δ ( s y ) ,
U c ( x , y ) = + + U ̃ c ( s x , s y ) exp [ j 2 π ( s x x + s y y ) ] d s x d s y = j n exp ( + j n ϕ ) U c ( x )
U c ( x ) = + U ̃ c ( s x ) exp ( j 2 π s x x ) d s x ,
U ̃ c ( s x ) H ( s x ) s x U ̃ ( s r = s x ) .
P ̃ ( s x , z ) = exp ( j k z ) exp ( + j π λ z s x 2 ) .
U ̃ c ( s x , z ) = U ̃ c ( s x , 0 ) exp ( j k z ) exp ( + j π λ z s x 2 ) .
U c ( x , z ) = exp ( j k z ) 0 + d s x s x U ̃ c ( s x , 0 ) exp ( + j π λ z s x 2 ) exp ( j 2 π s x x ) .
U c ( x , z ) = exp ( + j k z ) 0 + d s x s x U ̃ c ( s x , 0 ) exp ( j π λ z s x 2 ) exp ( + j 2 π s x x ) .
U ̃ c * ( s x , 0 ) = U ̃ c ( s x , 0 ) .
U c * ( x , z ) = U c ( x , z ) .
U c ( x , z ) 2 = U c ( x , z ) 2 ,
ψ c ( x , z ) = ψ c ( x , z ) .
U c ( x , 0 ) 2 = U c ( x , 0 ) 2 ,
ψ c ( x , 0 ) = ψ c ( x , 0 ) .
k R c ( z ) = + x ψ c ( x , z ) x U c ( x , z ) 2 d x + x 2 U c ( x , z ) 2 d x ,
x = x ̃ cos ϕ + y ̃ sin ϕ ,
y = x ̃ sin ϕ + y ̃ cos ϕ .
U ( x ̃ , y ̃ , z ) = j n 0 2 π d ϕ exp ( + j n ϕ ) U c ( x = x ̃ cos ϕ + y ̃ sin ϕ , z ) .
U ( r , z ) = j n 0 2 π d ϕ exp ( + j n ϕ ) U c ( x = r cos ϕ , z ) .
U ( r , z ) = 0 2 π d ϕ U c ( x = r cos ϕ , z ) .
U ( r = 0 , z ) = 2 π U c ( x = 0 , z ) .
M c 2 = 4 π σ x 0 σ s x ,
z R , c = 4 π σ x 0 2 M c 2 λ .
α k = + α k U c ( α ) 2 d α + U c ( α ) 2 d α ,
σ x ( z ) σ x 0 = [ 1 + ( z z R , c ) 2 ] 1 2 .
Z max 2 σ x 0 tan Θ 2 σ x 0 λ s ¯ x .
Θ = arcsin ( λ s ¯ x ) λ s ¯ x ,
Q z R , c Z max .
Q = s ¯ x 2 σ s x [ 1 ( λ s ¯ x ) 2 ] 1 2 s ¯ x 2 σ s x .
σ x ( ± Z max ) σ x 0 = ( 1 + 1 Q 2 ) 1 2 .
U c ( x , 0 ) U c ( x , 0 ) exp ( j 2 π s ¯ x x ) .
U c ( ± x , z ) U c ( z Θ ± x , 0 ) exp ( j k z ) .
U c ( ± x , z ) U c ( z Θ ± x , 0 ) exp [ j Φ ( z ) ] exp ( j 2 π s ¯ x x ) ,
U c ( x , z ) + U c ( x , z ) 2 U c ( z Θ , 0 ) exp [ j Φ ( z ) ] cos ( 2 π s ¯ x x ) .
U ( r , z ) = 2 0 r d x [ U c ( x , z ) + U c ( x , z ) ] r 2 x 2 .
0 1 d α cos ( κ α ) 1 α 2 = π 2 J 0 ( κ ) ,
U ( r , z ) 2 π U c ( z Θ , 0 ) exp [ j Φ ( z ) ] J 0 ( 2 π s ¯ x r ) .
U ( r , z ) 2 4 π 2 U c ( z Θ , 0 ) 2 J 0 2 ( 2 π s ¯ x r ) .
U c ( z Θ , 0 ) 2 U ( 0 , z ) 2 4 π 2 .
r 0 2.405 2 π s ¯ x ,
Q 2.405 M c 2 σ x 0 r 0 .
Q 1.446 M c 2 Z max z R , G ,
U ( r , θ ) = A 0 ( 2 r w 0 ) L p l ( 2 r 2 w 0 2 ) exp ( r 2 w 0 2 ) exp ( + j l θ ) ,
U ̃ ( s r , ϕ ) = A 0 π w 0 2 ( 1 ) p ( j 2 π w 0 s r ) l exp ( π 2 w 0 2 s r 2 ) L p l ( 2 π 2 w 0 2 s r 2 ) exp ( + j l ϕ ) .
U ̃ c ( s x ) = H ( s x ) s x A 0 π w 0 2 ( 1 ) p ( 2 π w 0 s x ) l exp ( π 2 w 0 2 s x 2 ) L p l ( 2 π 2 w 0 2 s x 2 ) .
lim l U c ( x , z = 0 ) = A 1 H p ( 2 x w 1 ) exp ( x 2 w 1 2 ) exp ( j 2 π s ¯ x x ) .
U ( r = 0 , z ) 2 = A 0 2 1 + ( z z 0 ) 2
U ̃ c 0 ( s x ) = H ( s x ) s x A 0 π 2 w 0 2 ( 1 ) p exp ( π 2 w 0 2 s x 2 ) L p 0 ( 2 π 2 w 0 2 s x 2 ) .
U ̃ c 0 ( s x , z ) = U ̃ c 0 ( s x ) exp ( j k z ) exp { π 2 w 0 2 [ 1 j ( z z 0 ) ] } .
U c 0 ( x = 0 , z ) = + U ̃ c 0 ( s x , z ) d s x .
U c 0 ( x = 0 , z ) = A 0 2 π exp [ j k z + j 2 p ψ ] 1 j ( z z 0 ) ,
U c 0 ( x = 0 , z ) 2 = 1 4 π 2 A 0 2 1 + ( z z 0 ) 2
U ( r , θ ) = A 0 J n ( β r ) exp ( r 2 w 0 2 ) exp ( + j n θ ) .
U ( s r , ϕ ) = A 0 π w 0 2 j n F n ( π β w 0 2 s r ) exp [ π 2 w 0 2 ( s r β 2 π ) 2 ] exp ( + j n ϕ ) ,
F n ( α ) exp ( α ) I n ( α ) 1 2 π α ( 1 4 n 2 1 8 α + ) .
( β 2 π ) ( 1 π w 0 ) = β w 0 2 .
U ̃ c ( s x ) = H ( s x ) s x A 0 π w 0 2 F n ( π β w 0 2 s x ) exp [ π 2 w 0 2 ( s x β 2 π ) 2 ] .
U ( r , z ) = j n 0 2 π d ϕ exp ( + j n ϕ ) U c ( x = r cos ϕ , z ) ,
U ( r , z ) = j n 0 2 π d ϕ exp ( + j n ϕ ) + d s x H ( s x ) s x U ̃ ( s x , z ) exp ( j 2 π s x r cos ϕ ) .
U ( r , z ) = j n 0 s r d s r U ̃ ( s r , z ) 0 2 π d ϕ exp ( + j n ϕ ) exp ( j 2 π s r r cos ϕ ) .
J n ( z ) = 1 2 π π π exp ( j n α ) exp ( + j z sin α ) d α .
U ( r , z ) = 2 π 0 s r d s r U ̃ ( s r , z ) J n ( 2 π s r r ) .
U ( r , z ) = 0 π d ϕ [ U c ( x = r cos ϕ , z ) + U c ( x = r cos ( π + ϕ ) , z ) ] .
U ( r , z ) = 0 π d ϕ [ U c ( x = r cos ϕ , z ) + U c ( x = r cos ϕ , z ) ] ,
U ( r , z ) = 2 0 r d x U c ( x , z ) + U c ( x , z ) r 2 x 2 .

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