Abstract

We introduce the generalized vector Helmholtz-Gauss (gVHzG) beams that constitute a general family of localized beam solutions of the Maxwell equations in the paraxial domain. The propagation of the electromagnetic components through axisymmetric ABCD optical systems is expressed elegantly in a coordinate-free and closed-form expression that is fully characterized by the transformation of two independent complex beam parameters. The transverse mathematical structure of the gVHzG beams is form-invariant under paraxial transformations. Any paraxial beam with the same waist size and transverse spatial frequency can be expressed as a superposition of gVHzG beams with the appropriate weight factors. This formalism can be straightforwardly applied to propagate vector Bessel-Gauss, Mathieu-Gauss, and Parabolic-Gauss beams, among others.

© 2006 Optical Society of America

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  1. M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys, Rev. A 11, 1365-1370 (1975).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  15. R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  26. M. Guizar-Sicairos and J. C. Gutierrez-Vega, "Generalized Helmholtz-Gauss beams and its transformation by paraxial optical systems," Opt. Lett. 31, 2912-2914 (2006).
    [CrossRef] [PubMed]

2006 (4)

K. Volke-Sepulveda and E. Ley-Koo, "General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states," J. Opt. A: Pure Appl. Opt.,  8, 867-877 (2006).
[CrossRef]

C. Lopez-Mariscal, M. A. Bandres, and J. C. Gutierrez-Vega, "Observation of the experimental propagation properties of Helmholtz-Gauss beams," Opt. Eng. 45, 068001 (2006).
[CrossRef]

A. Flores-Perez, J. Hernandez-Hernandez, R. Jauregui, and K. Volke-Sepulveda, "Experimental generation and analysis of first-order TE and TM Bessel modes in free space," Opt. Lett. 31, 1732-1734 (2006).
[CrossRef] [PubMed]

M. Guizar-Sicairos and J. C. Gutierrez-Vega, "Generalized Helmholtz-Gauss beams and its transformation by paraxial optical systems," Opt. Lett. 31, 2912-2914 (2006).
[CrossRef] [PubMed]

2005 (2)

2004 (2)

2003 (2)

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Z. Bouchal, "Nondiffracting optical beams: physical properties, experiments, and applications" Czech. J. Phys. 53,537-578 (2003).
[CrossRef]

2000 (1)

1999 (1)

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D: Appl. Phys. 32, 1455-1461 (1999).
[CrossRef]

1997 (2)

1996 (2)

D. G. Hall, "Vector-beam solutions of Maxwell’s wave equation," Opt. Lett. 21, 9-11 (1996).
[CrossRef] [PubMed]

M. Santarsiero, "Propagation of generalized Bessel-Gauss beams through ABCD optical systems," Opt. Commun. 132, 1-7 (1996).
[CrossRef]

1995 (1)

Z. Bouchal and M. Olivık, "Non-diffractive vector Bessel beams," J. Mod. Opt. 42, 1555-1566 (1995).
[CrossRef]

1994 (1)

1981 (1)

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys, Rev. A 11, 1365-1370 (1975).
[CrossRef]

1970 (1)

1942 (1)

R. D. Spence and C. P. Wells, "The propagation of electromagnetic waves in parabolic pipes," Phys. Rev. 62, 58-62 (1942).
[CrossRef]

1938 (1)

L. J. Chu, "Electromagnetic waves in elliptic hollow pipes of metal," J. Appl. Phys. 9, 583-591 (1938).
[CrossRef]

Bandres, M. A.

Bouchal, Z.

Z. Bouchal, "Nondiffracting optical beams: physical properties, experiments, and applications" Czech. J. Phys. 53,537-578 (2003).
[CrossRef]

Z. Bouchal and M. Olivık, "Non-diffractive vector Bessel beams," J. Mod. Opt. 42, 1555-1566 (1995).
[CrossRef]

Casperson, L. W.

Ch´avez-Cerda, S.

Chu, L. J.

L. J. Chu, "Electromagnetic waves in elliptic hollow pipes of metal," J. Appl. Phys. 9, 583-591 (1938).
[CrossRef]

Clark, G. H.

Collins, S. A.

Davis, L. W.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Flores-P´erez, A.

Guizar-Sicairos, M.

Guti´errez-Vega, J. C.

Hall, D. G.

Hern´andez-Hern´andez, J.

Iturbe-Castillo, M.D.

J´auregui, R.

L´opez-Mariscal, C.

C. Lopez-Mariscal, M. A. Bandres, and J. C. Gutierrez-Vega, "Observation of the experimental propagation properties of Helmholtz-Gauss beams," Opt. Eng. 45, 068001 (2006).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys, Rev. A 11, 1365-1370 (1975).
[CrossRef]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Ley-Koo, E.

K. Volke-Sepulveda and E. Ley-Koo, "General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states," J. Opt. A: Pure Appl. Opt.,  8, 867-877 (2006).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys, Rev. A 11, 1365-1370 (1975).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys, Rev. A 11, 1365-1370 (1975).
[CrossRef]

Nesterov, A. V.

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D: Appl. Phys. 32, 1455-1461 (1999).
[CrossRef]

Niziev, V. G.

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D: Appl. Phys. 32, 1455-1461 (1999).
[CrossRef]

Olivik, M.

Z. Bouchal and M. Olivık, "Non-diffractive vector Bessel beams," J. Mod. Opt. 42, 1555-1566 (1995).
[CrossRef]

Patsakos, G.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Ruschin, S.

Santarsiero, M.

M. Santarsiero, "Propagation of generalized Bessel-Gauss beams through ABCD optical systems," Opt. Commun. 132, 1-7 (1996).
[CrossRef]

Spence, R. D.

R. D. Spence and C. P. Wells, "The propagation of electromagnetic waves in parabolic pipes," Phys. Rev. 62, 58-62 (1942).
[CrossRef]

Tovar, A. A.

Volke-Sep´ulveda, K.

Volke-Sepulveda, K.

K. Volke-Sepulveda and E. Ley-Koo, "General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states," J. Opt. A: Pure Appl. Opt.,  8, 867-877 (2006).
[CrossRef]

Wells, C. P.

R. D. Spence and C. P. Wells, "The propagation of electromagnetic waves in parabolic pipes," Phys. Rev. 62, 58-62 (1942).
[CrossRef]

Zhan, Q.

Czech. J. Phys. (1)

Z. Bouchal, "Nondiffracting optical beams: physical properties, experiments, and applications" Czech. J. Phys. 53,537-578 (2003).
[CrossRef]

J. Appl. Phys. (1)

L. J. Chu, "Electromagnetic waves in elliptic hollow pipes of metal," J. Appl. Phys. 9, 583-591 (1938).
[CrossRef]

J. Mod. Opt. (1)

Z. Bouchal and M. Olivık, "Non-diffractive vector Bessel beams," J. Mod. Opt. 42, 1555-1566 (1995).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

K. Volke-Sepulveda and E. Ley-Koo, "General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states," J. Opt. A: Pure Appl. Opt.,  8, 867-877 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. D: Appl. Phys. (1)

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D: Appl. Phys. 32, 1455-1461 (1999).
[CrossRef]

Opt. Commun. (1)

M. Santarsiero, "Propagation of generalized Bessel-Gauss beams through ABCD optical systems," Opt. Commun. 132, 1-7 (1996).
[CrossRef]

Opt. Eng. (1)

C. Lopez-Mariscal, M. A. Bandres, and J. C. Gutierrez-Vega, "Observation of the experimental propagation properties of Helmholtz-Gauss beams," Opt. Eng. 45, 068001 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (7)

Phys, Rev. A (1)

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys, Rev. A 11, 1365-1370 (1975).
[CrossRef]

Phys. Rev. (1)

R. D. Spence and C. P. Wells, "The propagation of electromagnetic waves in parabolic pipes," Phys. Rev. 62, 58-62 (1942).
[CrossRef]

Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Other (3)

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

J. A. Stratton, Electromagnetic theory (McGraw-Hill, New York, 1941)

P. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Supplementary Material (8)

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Figures (3)

Fig. 1.
Fig. 1.

Physical picture of the decomposition of a gVHzG beam propagating in free space in terms of fundamental vector Gaussian beams whose mean propagation axes lie on the surface of a double cone.

Fig. 3.
Fig. 3.

Propagation of the transverse intensity distribution and electric vector field for generalized vector Bessel-cosine-Gauss, Mathieu-Gauss, and parabolic-Gauss beams. The parameter data for the propagations are included within the text. The movies show the evolution from z=0 to z=4LF . (Movie files: 3.1 MB, 2.5 MB, 3.6 MB, and 2.2 MB)

Fig. 2.
Fig. 2.

Propagation of the transverse intensity distribution and the electric vector field for circularly polarized gVHzG beams constructed with finite superposition of vector Gaussian beams. The parameter data for the propagations are included within the text. The movies show the evolution from z=0 to z=4LF . (Movie files: 2.4 MB, 2.3 MB, 3.3 MB, and 3.3 MB)

Equations (42)

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R = ( r , z ) , r = ( x , y ) = ( r , θ ) ; K = ( k , k z ) , k = ( k x , k y ) = ( k , ϕ ) ,
e 1 ( r 1 ) = exp ( i K r 1 2 2 q 1 ) 1 W ( r 1 ; κ 1 ) , h 1 ( r 1 ) = ( ε κ ) 1 2 z ̂ × e 1 ( r 1 ) ,
W ( r 1 ; κ 1 ) = π π g ( ϕ ) exp [ i κ 1 ( x 1 cos ϕ + y 1 sin ϕ ) ] d ϕ ,
e z , 1 = i K 1 · e 1 = exp ( i K r 1 2 2 q 1 ) ( i κ 1 2 K W + 1 W · r 1 q 1 ) ,
h z , 1 = i K 1 · h 1 = ε μ exp ( i K r 1 2 2 q 1 ) ( z ̂ × 1 W ) · r 1 q 1 .
e 1 ( T E ) ( r 1 ) = exp ( i K r 1 2 2 q 1 ) [ z ̂ × 1 W ] , h 1 ( T E ) ( r 1 ) = ε μ exp ( i K r 1 2 2 q 1 ) 1 W .
e 2 ( r 2 ) = K exp ( i K L 0 ) i 2 π B e 1 ( r 1 ) exp [ i K 2 B ( A r 1 2 2 r 1 · r 2 + D r 2 2 ) ] d 2 r 1 ,
e 2 ( r 2 ) = κ 1 κ 2 exp ( i κ 1 κ 2 B 2 K ) G ( r 2 , q 2 ) 2 W ( r 2 ; κ 2 ) ,
h 2 ( r 2 ) = ( ε μ ) 1 2 z ̂ × e 2 ( r 2 ) ,
G ( r 2 , q 2 ) = exp ( i K L 0 ) A + B q 1 exp ( i K r 2 2 2 q 2 ) ,
q 2 = A q 1 + B C q 1 + D , κ 2 = κ 1 q 1 A q 1 + B .
S j = 1 2 ( ε μ ) 1 2 f j 2 { j W 2 z ̂ + Re [ ( i κ j 2 K W + j W · r j q j ) j W * ] } ,
e ˜ ( k ) = 1 2 π e ( r ) exp ( i k · r ) d 2 r .
e ˜ 1 ( k 1 ) = i exp ( i q 1 κ 1 2 2 K ) exp ( i q 1 k 1 2 2 K ) ˜ 1 W ( k 1 ; q 1 κ 1 K ) ,
e ˜ 2 ( k 2 ) = κ 1 K κ 2 q 2 exp ( i q 2 κ 2 2 + κ 1 κ 2 B 2 K ) G ˜ ( k 2 , q 2 ) ˜ 2 W ( k 2 ; B κ 2 2 K κ 1 ) ,
G ˜ ( k 2 , q 2 ) = ( i q 2 K ) exp ( i K L 0 ) A + B q 1 exp ( i q 2 k 2 2 2 K )
E = α E ( T M ) + β E ( T E ) ,
W = 1 2 ( W x i W y ) u ̂ + + 1 2 ( W x + i W y ) u ̂ .
e j ± ( r j ) = 2 f j ( W x j i W y j ) u ̂ ± ,
q 2 = q 1 + L , κ 2 = κ 1 q 1 q 1 + L ,
e 2 ( r 2 ) = π π g ( ϕ ) g 2 ( r 2 ; ϕ ) d ϕ ,
g 2 ( r 2 ; ϕ ) = i exp ( i κ 1 2 2 K L ζ ) exp ( i K L ) ζ exp ( i K r 2 2 2 q 1 ζ ) exp ( i κ 1 · r 2 ζ ) κ 1 ,
r gen ( z ) = q 1 2 κ 1 𝒥 K q 1 𝒥 + κ 1 𝓡 q 1 𝒥 + κ 1 𝒥 q 1 𝓡 K q 1 𝒥 ( z z 1 ) .
z vertex = z 1 q 1 2 κ 1 𝒥 κ 1 𝓡 q 1 𝒥 + κ 1 𝒥 q 1 𝓡 .
[ A B C D ] = [ cos ( L a ) a sin ( L a ) sin ( L a ) a cos ( L a ) ] .
q 2 = a q 1 cos ( L a ) + a sin ( L a ) q 1 sin ( L a ) + a cos ( L a ) , κ 2 = κ 1 q 1 q 1 cos ( L a ) + a sin ( L a ) ,
W ( r 1 ) = n = 1 N A n exp [ i κ 1 r 1 cos ( θ 1 ϕ n ) ] ,
x = 1 f ( cosh 2 ξ cos 2 η ) ( sinh ξ cos η ξ cosh ξ sin η η ) ,
y = 1 f ( cosh 2 ξ cos 2 η ) ( cosh ξ sin η ξ sinh ξ cos η η ) ,
x = 1 u 2 + v 2 ( u u v v ) ,
y = 1 u 2 + v 2 ( v u u v ) ,
e 2 ( r 2 ) = K exp ( i K L 0 ) i 2 π B d 2 r 1 exp ( i K r 1 2 2 q 1 ) exp [ i K 2 B ( A r 1 2 2 r 1 · r 2 + D r 2 2 ) ]
× 1 [ π π d ϕ g ( ϕ ) exp [ i κ 1 ( x 1 cos ϕ + y 1 sin ϕ ) ] ] ,
e x = K exp ( i K L 0 ) i 2 π B d 2 r 1 exp ( i K r 1 2 2 q 1 ) exp [ i K 2 B ( A r 1 2 2 r 1 · r 2 + D r 2 2 ) ]
× i κ 1 π π d ϕ g ( ϕ ) cos ϕ exp [ i κ 1 ( x 1 cos ϕ + y 1 sin ϕ ) ] ,
e x = K κ 1 exp ( i K L 0 ) 2 π B exp ( i K D r 2 2 2 B ) π π d ϕ g ( ϕ ) cos ϕ
× d 2 r 1 exp [ i K r 1 2 2 ( 1 q 1 + A B ) ] exp [ i K B r 1 · r 2 + i κ 1 ( x 1 cos ϕ + y 1 sin ϕ ) ] ,
π a 2 exp { 1 4 a 2 [ ( κ 1 cos ϕ K B x 2 ) 2 + ( κ 1 sin ϕ K B y 2 ) 2 ] } ,
e x = exp ( i K L 0 ) exp ( κ 1 2 4 a 2 ) exp ( i K D r 2 2 2 B ) exp ( K 2 r 2 2 4 a 2 B 2 )
× π π [ K κ 1 2 a 2 B cos ϕ d ϕ ] g ( ϕ ) exp [ K κ 1 2 a 2 B ( x 2 cos ϕ + y 2 sin ϕ ) ] .
e x = κ 1 κ 2 exp ( i κ 1 κ 2 B 2 K ) G ( r 2 , q 2 ) x 2 W ( r 2 ; κ 2 )
e 2 ( r 2 ) = κ 1 κ 2 exp ( i κ 1 κ 2 B 2 K ) G ( r 2 , q 2 ) 2 W ( r 2 ; k 2 )

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