Abstract

The recent proposal to use Weinger transformation field (WTF) [Opt. Express 17, 4959-4969 (2009)] for describing tightly focused laser beams is investigated here in detail. In order to validate the accuracy of WTF, we derive the numerical field (NF) from the plane wave spectrum method. WTF is compared with NF and Lax series field (LSF). Results show that LSF is accurate close to the beam axis and divergent far from the beam axis, and WTF is always accurate. Moreover, electron dynamics in a tightly focused intense laser beam are simulated by LSF, WTF and NF, respectively. The results obtained by WTF are shown to be accurate.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Strickland, and G. Mourou, "Compression of amplified chirped optical pulses," Opt. Commun. 55(6), 447-449 (1985).
    [CrossRef]
  2. N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, "Accurate description of Gaussian laser beams and electron dynamics," Opt. Commun.  204(1-6), 7-15 (2002).
    [CrossRef]
  3. Y. I. Salamin, and C. H. Keitel, "Electron acceleration by a tightly focused laser beam," Phys. Rev. Lett. 88(9), 095005 (2002).
    [CrossRef]
  4. Y. I. Salamin, G. R. Mocken, and C. H. Keitel, "Electron scattering and acceleration by a tightly focused laser beam," Phys. Rev. ST Accel. Beams 5(10), 101301 (2002).
    [CrossRef]
  5. M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11(4), 1365-1370 (1975).
    [CrossRef]
  6. R. Borghi, and M. Santarsiero, "Summing Lax series for nonparaxial beam propagation," Opt. Lett. 28(10), 774-776 (2003).
    [CrossRef]
  7. J. X. Li, W. P. Zang, and J. G. Tian, "Simulation of Gaussian laser beams and electron dynamics by Weniger transformation method," Opt. Express 17(7), 4959-4969 (2009).
    [CrossRef]
  8. A. Doicu, and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Opt. Commun. 136(1-2), 114-124 (1997).
    [CrossRef]
  9. P. Varga, and P. Török, "The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation," Opt. Commun. 152(1-3), 108-118 (1998).
    [CrossRef]
  10. L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19(3), 1177-1179 (1979).
    [CrossRef]
  11. H. Luo, S. Y. Liu, Z. F. Lin, and C. T. Chan, "Method for accurate description of a radially polarized Gaussian laser beam beyond the paraxial approximation," Opt. Lett. 32(12), 1692-1694 (2007).
    [CrossRef]
  12. Y. I. Salamin, "Fields of a Gaussian beam beyond the paraxial approximation," Appl. Phys. B 86(2), 319-326 (2007).
    [CrossRef]
  13. E. J. Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series," Comput. Phys. Rep. 10(5-6), 189-371 (1989).
    [CrossRef]
  14. B. Richards, and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. A 253(1274), 358-379 (1959).
    [CrossRef]

2009 (1)

2007 (2)

2003 (1)

2002 (2)

Y. I. Salamin, and C. H. Keitel, "Electron acceleration by a tightly focused laser beam," Phys. Rev. Lett. 88(9), 095005 (2002).
[CrossRef]

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, "Electron scattering and acceleration by a tightly focused laser beam," Phys. Rev. ST Accel. Beams 5(10), 101301 (2002).
[CrossRef]

1998 (1)

P. Varga, and P. Török, "The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation," Opt. Commun. 152(1-3), 108-118 (1998).
[CrossRef]

1997 (1)

A. Doicu, and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Opt. Commun. 136(1-2), 114-124 (1997).
[CrossRef]

1989 (1)

E. J. Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series," Comput. Phys. Rep. 10(5-6), 189-371 (1989).
[CrossRef]

1985 (1)

D. Strickland, and G. Mourou, "Compression of amplified chirped optical pulses," Opt. Commun. 55(6), 447-449 (1985).
[CrossRef]

1979 (1)

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19(3), 1177-1179 (1979).
[CrossRef]

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11(4), 1365-1370 (1975).
[CrossRef]

1959 (1)

B. Richards, and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. A 253(1274), 358-379 (1959).
[CrossRef]

Borghi, R.

Chan, C. T.

Davis, L. W.

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19(3), 1177-1179 (1979).
[CrossRef]

Doicu, A.

A. Doicu, and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Opt. Commun. 136(1-2), 114-124 (1997).
[CrossRef]

Keitel, C. H.

Y. I. Salamin, and C. H. Keitel, "Electron acceleration by a tightly focused laser beam," Phys. Rev. Lett. 88(9), 095005 (2002).
[CrossRef]

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, "Electron scattering and acceleration by a tightly focused laser beam," Phys. Rev. ST Accel. Beams 5(10), 101301 (2002).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11(4), 1365-1370 (1975).
[CrossRef]

Li, J. X.

Lin, Z. F.

Liu, S. Y.

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11(4), 1365-1370 (1975).
[CrossRef]

Luo, H.

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11(4), 1365-1370 (1975).
[CrossRef]

Mocken, G. R.

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, "Electron scattering and acceleration by a tightly focused laser beam," Phys. Rev. ST Accel. Beams 5(10), 101301 (2002).
[CrossRef]

Mourou, G.

D. Strickland, and G. Mourou, "Compression of amplified chirped optical pulses," Opt. Commun. 55(6), 447-449 (1985).
[CrossRef]

Richards, B.

B. Richards, and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. A 253(1274), 358-379 (1959).
[CrossRef]

Salamin, Y. I.

Y. I. Salamin, "Fields of a Gaussian beam beyond the paraxial approximation," Appl. Phys. B 86(2), 319-326 (2007).
[CrossRef]

Y. I. Salamin, and C. H. Keitel, "Electron acceleration by a tightly focused laser beam," Phys. Rev. Lett. 88(9), 095005 (2002).
[CrossRef]

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, "Electron scattering and acceleration by a tightly focused laser beam," Phys. Rev. ST Accel. Beams 5(10), 101301 (2002).
[CrossRef]

Santarsiero, M.

Strickland, D.

D. Strickland, and G. Mourou, "Compression of amplified chirped optical pulses," Opt. Commun. 55(6), 447-449 (1985).
[CrossRef]

Tian, J. G.

Török, P.

P. Varga, and P. Török, "The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation," Opt. Commun. 152(1-3), 108-118 (1998).
[CrossRef]

Varga, P.

P. Varga, and P. Török, "The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation," Opt. Commun. 152(1-3), 108-118 (1998).
[CrossRef]

Weniger, E. J.

E. J. Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series," Comput. Phys. Rep. 10(5-6), 189-371 (1989).
[CrossRef]

Wolf, E.

B. Richards, and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. A 253(1274), 358-379 (1959).
[CrossRef]

Wriedt, T.

A. Doicu, and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Opt. Commun. 136(1-2), 114-124 (1997).
[CrossRef]

Zang, W. P.

Appl. Phys. B (1)

Y. I. Salamin, "Fields of a Gaussian beam beyond the paraxial approximation," Appl. Phys. B 86(2), 319-326 (2007).
[CrossRef]

Comput. Phys. Rep. (1)

E. J. Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series," Comput. Phys. Rep. 10(5-6), 189-371 (1989).
[CrossRef]

Opt. Commun. (3)

A. Doicu, and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Opt. Commun. 136(1-2), 114-124 (1997).
[CrossRef]

P. Varga, and P. Török, "The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation," Opt. Commun. 152(1-3), 108-118 (1998).
[CrossRef]

D. Strickland, and G. Mourou, "Compression of amplified chirped optical pulses," Opt. Commun. 55(6), 447-449 (1985).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (2)

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11(4), 1365-1370 (1975).
[CrossRef]

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19(3), 1177-1179 (1979).
[CrossRef]

Phys. Rev. Lett. (1)

Y. I. Salamin, and C. H. Keitel, "Electron acceleration by a tightly focused laser beam," Phys. Rev. Lett. 88(9), 095005 (2002).
[CrossRef]

Phys. Rev. ST Accel. Beams (1)

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, "Electron scattering and acceleration by a tightly focused laser beam," Phys. Rev. ST Accel. Beams 5(10), 101301 (2002).
[CrossRef]

Proc. R. Soc. A (1)

B. Richards, and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. A 253(1274), 358-379 (1959).
[CrossRef]

Other (1)

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, "Accurate description of Gaussian laser beams and electron dynamics," Opt. Commun.  204(1-6), 7-15 (2002).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1.

The electric field z component of NF as a function of transverse coordinate x in the line of y=0 at longitudinal coordinate z=zr with a spot size (a) w 0=5λ, (b) w 0=λ, (c) w 0=0.5λ, where w(z)=w 0[1+(z /zr )2]1/2. The initial phase φ 0=0. The different curves represent the fields obtained by the boundary values including different order corrections, (black) ε 0, (red) ε 2, (green dash-dot) ε 4, (blue dash) ε 6.

Fig. 2.
Fig. 2.

The z and x components of electric field as a function of transverse coordinate x in the line of y=0 at longitudinal coordinate z=5zr for a Gaussian beam with a spot size w 0=λ, and initial phase ϕ=0. The black curves represent the components of NF, and the boundary field is accurate to ε 2. The red, green and blue (dash) curves in (a) and (c) is simulated by LSF accurate to ε 3, ε 7 and ε 39 respectively, and those in (b) and (d) is simulated by WTF accurate to ε 3, ε 7 and ε 39 respectively. The insets in (b) and (d) are small parts of (b) and (d). The legend in (a) applies to other figures as well.

Fig. 3.
Fig. 3.

(a) The trajectories. (b) The energy gains of the electron dynamics in a laser beam. The red curves, blue curves and green dash curves represent the electron dynamics obtained by LSF, WTF and NF respectively. The electron is injected toward the beam focus (0, 0, 0). The two black curves represent the beam boundaries. Parameters used in the simulations are injected angle θ=10°, q=10, w 0=5µm, λ=5µm, (z 0, y 0, x 0)=(-5mm, 0, -5tanθmm), initial phase φ 0=0, the initial injection energy γ 0=15, Energy Gain=m 0 c 2(γ-γ 0) and the full interaction time ωt=1.96×106.

Fig. 4.
Fig. 4.

(a) The trajectories. (b) The electron energy gains. The initial injection energy γ 0=200, q=100, and the other parameters are same as those of Fig. 3.

Fig. 5.
Fig. 5.

(a) The trajectories. (b) The electron energy gains. The initial injection energy γ 0=16, q=100, and the other parameters are same as those of Fig. 3.

Equations (53)

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

2A1c22At2=0 .
2Ψ2ikΨz=0 .
2Ψ4iΨζ+ε22Ψζ2=0 ,
ψ=Σn=0ε2nψ2n.
2Ψ04iΨ0ζ=0 ,
2Ψ2n+24iΨ2n+2ζ+2Ψ2nζ2=0. n0
Ψ0=fefρ2,f=i(ζ+i),ρ2=ξ2+υ2.
exp[ik(z2+r2)12]=exp [ikzi(zrz)ρ2] Σn=0 ε2n a2n (ρ,zrz).
ψ2n=(C2n(f)+a2n(ρ,f))ψ0.
ψ2n=D2nψ0,D2n=a2n(ρ,f)+(n+1)fD2n24,
ψ2=(f2f3ρ44)ψ0,
ψ4=(3f283f4ρ416f5ρ68+f6p832)ψ0,
Ψ6=(3f383f5ρ416f6ρ683f7ρ864+f8ρ1032f9ρ12384)ψ0,
Ex=iEψ0Σn=0mε2nEnx(f,ρ,ξ),
Ey=iEψ0ξυΣn=0mε2nEny(f,ρ),
Ez=Eψ0ξΣn=0mε2n+1Enz(f,ρ),
Bx=0 ,
By=iEψ0Σn=0mε2nBny(f,ρ),
Bz=E ψ0 Σn=0m ε2n+1 Bnz (f,ρ),
Ex=iEψ0[1+ε2(f2ξ2f3ρ44)+ε4(f28f3ρ24+f4(ρ2ξ2ρ416)+f5(ρ4ξ24ρ68)
+f6ρ832)+ε6(3f3163f4ρ2169f5ρ432+15f6ρ4ξ216ρ12f9384+f7 (3ρ6ξ28ρ816)
+f8(ρ8ξ232+ρ1032)],
Ey=iEψ0ξυ[ε2f2+ε4(f4ρ2f5ρ44)+ε6(15f6ρ4163f7ρ68+f8ρ832)] ,
Ez=E ψ0 ξ [εf+ε3(f22+f3ρ2f4ρ44)+ε5(3f383f4ρ28+17f5ρ4163f6ρ68+f7ρ832)
+ε7 (3f483f5ρ283f6ρ416+33f7ρ63229f8ρ864+f9ρ1016f10ρ12384)],
By=iEψ0[1+ε2(f2ρ22f3ρ44)+ε4(f28+f3ρ24+5f4ρ416f5ρ64+f6ρ832)
+ε6(3f316+3f4ρ216+9f5ρ432+5f6ρ6327f7ρ832+3f8ρ1064f9ρ12384)] ,
Bz=E ψ0 υ [εf+ε3(f22+f3ρ22f4ρ44)+ε5(3f38+3f4ρ28+3f5ρ416f6ρ64+f7ρ832)
+ε7(3f48+3f5ρ28+3f6ρ416+f7ρ61613f8ρ864+3f9ρ1064f10ρ12384)].
δm=Σn=0mmn(1+n)m1Snan+1Σn=0mmn(1+n)m11an+1,
(b)j=b(b+1)(b+2)(b+j1),mn=m!(mn)!n!.
Ez=E ψ0 ξ Σn=0mmn(1+n)m1Snezε2(n+1)+1En+1zΣn=0mmn(1+n)m11ε2(n+1)+1En+1z ,
E=iE02πk2g(kx,ky)F(kx,ky)exp[i(kxx+kyy+kzz)]dkxdky,
B=iE02πkm(kx,ky)F(kx,ky)exp[i(kxx+kyy+kzz)]dkxdky,
F(kx,ky)=12πψ(x,y,0)exp[i(kxx+kyy)]dxdy,
g(kx,ky)=(k2kx2)ikxkyjkxkzk,m(kx,ky)=kzjkyk.
ψ(x,y,0)=Σn=0mε2nΨ2n(x,y,0).(m0)
kx=κcosθs,ky=κsinθs,x=rcosθp,y=r sin θp ,
E=iE02πk202πg(κ)P(κ)exp[iκrcos(θsθp)]κdκdθs,
B=iE02πk02πm(κ)P(κ)exp[iκrcos(θsθp)]κdκdθs,
P(κ)=F (κ)exp[iz(k2κ2)12].
Ex=iE02k2[I0(e)+cos(2θp)I2(e)],
Ey=iE02k2sin(2θp)I2(e),
Ez=E0k2cos(2θp)I1(e),
Bx=0 ,
By=iE0kI0(m),
Bz=E0ksinθpI0(m),
I0(e)=0(2k2κκ3)P(κ)J0(rκ)dκ,
I1(e)=0κ2(k2κ2)12P(κ)J1(rκ)dκ,
I2(e)=0κ3P(κ)J2(rκ)dκ,
I0(m)=0κ(k2κ2)12P(κ)J0(rκ)dκ,
I1(m)=0κ2P(κ)J1(rκ)dκ,
dpdt=e(E+β×B),dχdt=ecβ.E,

Metrics