Abstract

The propagation of Helmholtz–Gauss beams in media exhibiting loss or gain is studied. The general expressions for the field propagation, the time-averaged power on propagation, the trajectory of the beam centroid, the beam spreading, the nondiffracting distance, and the far field are derived and discussed. Explicit expressions of these parameters for Bessel–Gauss and cosine-Gauss beams are included. The general expressions can be applied straightforwardly to describe the propagation of Mathieu–Gauss and parabolic-Gauss beams in complex media as well.

© 2006 Optical Society of America

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  1. J. C. Gutiérrez-Vega and M. A. Bandres, "Helmholtz-Gauss waves," J. Opt. Soc. Am. A 22, 289-298 (2005).
    [CrossRef]
  2. A. P. Kiselev, "New structures in paraxial Gaussian beams," Opt. Spectrosc. 96, 479-481 (2004).
    [CrossRef]
  3. M. A. Bandres and J. C. Gutiérrez-Vega, "Vector Helmholtz-Gauss and vector Laplace-Gauss beams," Opt. Lett. 30, 2155-2157 (2005).
    [CrossRef] [PubMed]
  4. C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, "Characterization of Helmholtz-Gauss beams," in Laser Beam Shaping VI, F.Dickey and D.L.Shealy, eds., Proc. SPIE5876, 77-88 (2005).
  5. F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
    [CrossRef]
  6. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, "Alternative formulation of nondiffracting beams: Mathieu beams," Opt. Lett. 25, 1493-1495 (2000).
    [CrossRef]
  7. Z. Bouchal, "Nondiffracting optical beams: physical properties, experiments, and applications," Czech. J. Phys. 53, 537-624 (2003).
    [CrossRef]
  8. A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J. Simonen, P. Pääkkönen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).
  9. D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005).
    [CrossRef]
  10. J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Nondiffracting bulk-acoustic X waves in crystals," Phys. Rev. Lett. 83, 1171-1174 (1999).
    [CrossRef]
  11. S. Velumani, S. K. Narayandass, D. Mangalaraj, and C. P. G. Villaban, "Laser damage studies on hot-wall-deposited cadmium selenide films," J. Mater. Sci. Lett. 16, 1974-1976 (1997).
    [CrossRef]
  12. J. A. Gaspar and P. Halevi, "Linear pulse propagation in an absorbing medium: effect of film thickness," Phys. Rev. B 49, 10742-10744 (1994).
    [CrossRef]
  13. J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, and S. Chávez-Cerda, "Attenuation characteristics in confocal annular elliptic waveguides and resonators," IEEE Trans. Microwave Theory Tech. 50, 1095-1100 (2002).
    [CrossRef]
  14. S. R. Seshadri, "Effect of absorption on the spreading of a laser beam," Opt. Lett. 29, 1179-1181 (2004).
    [CrossRef] [PubMed]
  15. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004).
    [CrossRef] [PubMed]
  16. O. Svelto, Principles of Lasers, 4th ed. (Plenum, 1998).
  17. R. W. D. Nickalls, "A new approach to solving the cubic: Cardan's solution revealed," Math. Gaz. 77, 354-359 (1993).
    [CrossRef]

2005 (3)

2004 (4)

A. P. Kiselev, "New structures in paraxial Gaussian beams," Opt. Spectrosc. 96, 479-481 (2004).
[CrossRef]

A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J. Simonen, P. Pääkkönen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).

S. R. Seshadri, "Effect of absorption on the spreading of a laser beam," Opt. Lett. 29, 1179-1181 (2004).
[CrossRef] [PubMed]

M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004).
[CrossRef] [PubMed]

2003 (1)

Z. Bouchal, "Nondiffracting optical beams: physical properties, experiments, and applications," Czech. J. Phys. 53, 537-624 (2003).
[CrossRef]

2002 (1)

J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, and S. Chávez-Cerda, "Attenuation characteristics in confocal annular elliptic waveguides and resonators," IEEE Trans. Microwave Theory Tech. 50, 1095-1100 (2002).
[CrossRef]

2000 (1)

1999 (1)

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Nondiffracting bulk-acoustic X waves in crystals," Phys. Rev. Lett. 83, 1171-1174 (1999).
[CrossRef]

1997 (1)

S. Velumani, S. K. Narayandass, D. Mangalaraj, and C. P. G. Villaban, "Laser damage studies on hot-wall-deposited cadmium selenide films," J. Mater. Sci. Lett. 16, 1974-1976 (1997).
[CrossRef]

1994 (1)

J. A. Gaspar and P. Halevi, "Linear pulse propagation in an absorbing medium: effect of film thickness," Phys. Rev. B 49, 10742-10744 (1994).
[CrossRef]

1993 (1)

R. W. D. Nickalls, "A new approach to solving the cubic: Cardan's solution revealed," Math. Gaz. 77, 354-359 (1993).
[CrossRef]

1987 (1)

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Bandres, M. A.

Bandrés, M. A.

C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, "Characterization of Helmholtz-Gauss beams," in Laser Beam Shaping VI, F.Dickey and D.L.Shealy, eds., Proc. SPIE5876, 77-88 (2005).

Bouchal, Z.

Z. Bouchal, "Nondiffracting optical beams: physical properties, experiments, and applications," Czech. J. Phys. 53, 537-624 (2003).
[CrossRef]

Buchter, S. C.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J. Simonen, P. Pääkkönen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).

Chávez-Cerda, S.

Dholakia, K.

D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

Elfström, H.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J. Simonen, P. Pääkkönen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).

Fagerholm, J.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Nondiffracting bulk-acoustic X waves in crystals," Phys. Rev. Lett. 83, 1171-1174 (1999).
[CrossRef]

Friberg, A. T.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Nondiffracting bulk-acoustic X waves in crystals," Phys. Rev. Lett. 83, 1171-1174 (1999).
[CrossRef]

Gaspar, J. A.

J. A. Gaspar and P. Halevi, "Linear pulse propagation in an absorbing medium: effect of film thickness," Phys. Rev. B 49, 10742-10744 (1994).
[CrossRef]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Gutiérrez-Vega, J. C.

M. A. Bandres and J. C. Gutiérrez-Vega, "Vector Helmholtz-Gauss and vector Laplace-Gauss beams," Opt. Lett. 30, 2155-2157 (2005).
[CrossRef] [PubMed]

J. C. Gutiérrez-Vega and M. A. Bandres, "Helmholtz-Gauss waves," J. Opt. Soc. Am. A 22, 289-298 (2005).
[CrossRef]

M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004).
[CrossRef] [PubMed]

J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, and S. Chávez-Cerda, "Attenuation characteristics in confocal annular elliptic waveguides and resonators," IEEE Trans. Microwave Theory Tech. 50, 1095-1100 (2002).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, "Alternative formulation of nondiffracting beams: Mathieu beams," Opt. Lett. 25, 1493-1495 (2000).
[CrossRef]

C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, "Characterization of Helmholtz-Gauss beams," in Laser Beam Shaping VI, F.Dickey and D.L.Shealy, eds., Proc. SPIE5876, 77-88 (2005).

Hakola, A.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J. Simonen, P. Pääkkönen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).

Halevi, P.

J. A. Gaspar and P. Halevi, "Linear pulse propagation in an absorbing medium: effect of film thickness," Phys. Rev. B 49, 10742-10744 (1994).
[CrossRef]

Iturbe-Castillo, M. D.

Kajava, T.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J. Simonen, P. Pääkkönen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).

Kiselev, A. P.

A. P. Kiselev, "New structures in paraxial Gaussian beams," Opt. Spectrosc. 96, 479-481 (2004).
[CrossRef]

López-Mariscal, C.

C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, "Characterization of Helmholtz-Gauss beams," in Laser Beam Shaping VI, F.Dickey and D.L.Shealy, eds., Proc. SPIE5876, 77-88 (2005).

Mangalaraj, D.

S. Velumani, S. K. Narayandass, D. Mangalaraj, and C. P. G. Villaban, "Laser damage studies on hot-wall-deposited cadmium selenide films," J. Mater. Sci. Lett. 16, 1974-1976 (1997).
[CrossRef]

McGloin, D.

D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

Narayandass, S. K.

S. Velumani, S. K. Narayandass, D. Mangalaraj, and C. P. G. Villaban, "Laser damage studies on hot-wall-deposited cadmium selenide films," J. Mater. Sci. Lett. 16, 1974-1976 (1997).
[CrossRef]

Nickalls, R. W. D.

R. W. D. Nickalls, "A new approach to solving the cubic: Cardan's solution revealed," Math. Gaz. 77, 354-359 (1993).
[CrossRef]

Pääkkönen, P.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J. Simonen, P. Pääkkönen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Rodríguez-Dagnino, R. M.

J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, and S. Chávez-Cerda, "Attenuation characteristics in confocal annular elliptic waveguides and resonators," IEEE Trans. Microwave Theory Tech. 50, 1095-1100 (2002).
[CrossRef]

Salo, J.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Nondiffracting bulk-acoustic X waves in crystals," Phys. Rev. Lett. 83, 1171-1174 (1999).
[CrossRef]

Salomaa, M. M.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Nondiffracting bulk-acoustic X waves in crystals," Phys. Rev. Lett. 83, 1171-1174 (1999).
[CrossRef]

Seshadri, S. R.

Simonen, J.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J. Simonen, P. Pääkkönen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).

Svelto, O.

O. Svelto, Principles of Lasers, 4th ed. (Plenum, 1998).

Turunen, J.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J. Simonen, P. Pääkkönen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).

Velumani, S.

S. Velumani, S. K. Narayandass, D. Mangalaraj, and C. P. G. Villaban, "Laser damage studies on hot-wall-deposited cadmium selenide films," J. Mater. Sci. Lett. 16, 1974-1976 (1997).
[CrossRef]

Villaban, C. P. G.

S. Velumani, S. K. Narayandass, D. Mangalaraj, and C. P. G. Villaban, "Laser damage studies on hot-wall-deposited cadmium selenide films," J. Mater. Sci. Lett. 16, 1974-1976 (1997).
[CrossRef]

Contemp. Phys. (1)

D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

Czech. J. Phys. (1)

Z. Bouchal, "Nondiffracting optical beams: physical properties, experiments, and applications," Czech. J. Phys. 53, 537-624 (2003).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, and S. Chávez-Cerda, "Attenuation characteristics in confocal annular elliptic waveguides and resonators," IEEE Trans. Microwave Theory Tech. 50, 1095-1100 (2002).
[CrossRef]

J. Mater. Sci. Lett. (1)

S. Velumani, S. K. Narayandass, D. Mangalaraj, and C. P. G. Villaban, "Laser damage studies on hot-wall-deposited cadmium selenide films," J. Mater. Sci. Lett. 16, 1974-1976 (1997).
[CrossRef]

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

Math. Gaz. (1)

R. W. D. Nickalls, "A new approach to solving the cubic: Cardan's solution revealed," Math. Gaz. 77, 354-359 (1993).
[CrossRef]

Opt. Commun. (2)

A. Hakola, S. C. Buchter, T. Kajava, H. Elfström, J. Simonen, P. Pääkkönen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Opt. Lett. (4)

Opt. Spectrosc. (1)

A. P. Kiselev, "New structures in paraxial Gaussian beams," Opt. Spectrosc. 96, 479-481 (2004).
[CrossRef]

Phys. Rev. B (1)

J. A. Gaspar and P. Halevi, "Linear pulse propagation in an absorbing medium: effect of film thickness," Phys. Rev. B 49, 10742-10744 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Nondiffracting bulk-acoustic X waves in crystals," Phys. Rev. Lett. 83, 1171-1174 (1999).
[CrossRef]

Other (2)

O. Svelto, Principles of Lasers, 4th ed. (Plenum, 1998).

C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, "Characterization of Helmholtz-Gauss beams," in Laser Beam Shaping VI, F.Dickey and D.L.Shealy, eds., Proc. SPIE5876, 77-88 (2005).

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Figures (4)

Fig. 1
Fig. 1

Plot of P ¯ ( z ) f ( z ) 1 for the tilted plane-wave-Gaussian beam, the cosine-Gauss beam, and the zeroth- and first-order Bessel–Gauss beams.

Fig. 2
Fig. 2

Behavior of the tilted plane-wave-Gaussian beam components of the HzG beams in (a) medium with purely real refraction index, (b) absorbing medium, and (c) gain medium.

Fig. 3
Fig. 3

Constant amplitude contours of the plane-wave-Gaussian beam in absorptive media; dashed curve depicts the curve given by Eq. (19).

Fig. 4
Fig. 4

Propagation along the plane ( x , z ) of a cosine-Gauss beam with parameters λ = 632.8 nm , k 0 = 2 π λ , k t = 0.001 k 0 , within a lossy medium with refraction index n = 1 + i 0.01 . For visualization purposes the field amplitude is normalized at each plane z.

Equations (50)

Equations on this page are rendered with MathJax. Learn more.

n = n r + i n i .
E 0 ( r t ) = exp ( r 2 w 0 2 ) W ( r t ; k t ) ,
W ( r t ; k t ) = π π A ( θ ) exp [ i k t ( x cos θ + y sin θ ) ] d θ ,
[ x x + y y + ( 2 i n k 0 ) z ] Ψ ( r ) = 0 .
E ( r ) = exp ( i k t 2 z 2 n k 0 ζ ) [ exp ( i n k 0 z ) ζ exp ( r 2 ζ w 0 2 ) ] W ( x ζ , y ζ ; k t ) ,
ζ ( z ) = 1 + i z n L 0
= ( 1 + n i z n 2 L 0 ) + i ( n r z n 2 L 0 )
ζ r + i ζ i
S z = Re ( E × H * ) 2 z ̂ = ϵ 0 c n r 2 E 2 ,
P ( z ) = n r ϵ 0 c exp ( 2 k 0 z n i ) 2 ζ 2 exp [ k t 2 z k 0 n 2 ζ 2 ( n i + z L 0 ) ] × exp ( 2 ζ r ζ 2 w 0 2 r 2 ) W ( x ζ , y ζ ; k t ) 2 d A .
ζ r = 1 + n i z n 2 L 0 > 0
z c r = n 2 L 0 n i .
P ¯ ( z ) = P ( z ) P ( 0 ) ,
E P W ( r ) = exp ( i k t 2 z 2 n k 0 ζ ) [ exp ( i n k 0 z ) ζ exp ( r 2 ζ w 0 2 ) ] exp ( i k t x ζ ) .
P ¯ P W ( z ) = f ( z ) exp [ 2 γ 2 ζ i 2 ζ r ζ 2 ] ,
f ( z ) 1 ζ r exp [ k t 2 z k 0 n 2 ζ 2 ( n i + z L 0 ) ] exp ( 2 k 0 z n i ) .
P ¯ G B ( z ) = 1 ζ r exp ( 2 k 0 z n i ) ,
P ¯ C G ( z ) = f ( z ) [ 1 + exp ( 2 γ 2 ζ r ) ] [ 1 + exp ( 2 γ 2 ) ] exp ( 2 γ 2 [ 1 ζ r ζ 2 ] ) .
P ¯ B G ( z ) = f ( z ) exp ( γ 2 [ ζ r 2 ζ i 2 ζ 2 ζ r 1 ] ) I m ( γ 2 ζ r ) I m ( γ 2 ) ,
x = 1 P ( z ) A x S z d A ,
x = k t n r k 0 n 2 z 1 + z ( n i n 2 L 0 1 ) ,
x z = r C = k t w 0 2 2 n r n i .
d E P W ( x , z ) d x x = x max = 0 ,
w 2 ( z ) = 4 ( x 2 x 2 )
= w 0 2 ζ 2 ζ r , ζ r > 0 ,
z w n i L 0 2 ,
w 2 ( z w ) w 0 2 ( 1 n i 2 4 n 2 ) .
z ND = n r γ 2 1 L 0 .
k t 2 n r 2 k 0 2 n 4 z 2 ( 1 + n i z n 2 L 0 ) 2 = w 0 2 ζ 2 1 + n i z n 2 L 0 .
z 3 + α 2 z 2 + α 1 z + α 0 = 0 ,
α 2 L 0 n i ( n 2 + 2 n i 2 γ 2 n r 2 ) ,
α 1 3 n 2 L 0 2 ,
α 0 n 4 L 0 3 n i .
Δ = 4 α 1 3 α 1 2 α 2 2 + 4 α 0 α 2 3 18 α 0 α 1 α 2 + 27 α 0 2 .
Δ 4 α 0 α 2 3 = n 4 L 0 6 n i 4 ( n 2 + 2 n i 2 γ 2 n r 2 ) 3 ,
n 2 + 2 n i 2 γ 2 n r 2 > 10 1 ,
sign ( Δ ) = sign ( n 2 + 2 n i 2 γ 2 n r 2 ) .
γ 2 > n 2 + 2 n i 2 n r 2 .
γ 2 > 1 ,
z = p 3 u u α 2 3 ,
u = exp ( i π m 3 ) ( q 2 + q 2 4 + p 3 27 ) 1 3 , m = 0 , 1 , 2 ,
p = α 1 α 2 2 3 ,
q = α 0 + 2 α 2 3 9 α 2 α 1 27 .
z ND = w 0 k 0 n 2 n r k t .
z ND = w 0 k 0 n r k t ,
z ND = α 0 α 2 = n 2 n r γ 2 ( 2 n i 2 + n 2 ) n r 2 L 0
z ND = α 0 α 2 = n 2 n r γ 2 1 3 n i 2 n r 2 L 0
ζ ( z ) i z n L 0 .
W ( x ζ , y ζ ; k t ) = π π A ( θ ) exp [ i k t ζ ( x cos θ + y sin θ ) ] d θ .
E x = L 0 n i z exp ( γ 2 ) exp ( i k 0 n z ) exp ( i k 0 n r r 2 2 z ) exp ( L 0 γ 2 n r 2 z n i ) × π π A ( θ ) exp { n i k 0 2 z [ ( x r C cos θ ) 2 + ( y r C sin θ ) 2 ] } d θ .

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