Abstract

In this paper, we propose a method that is capable of describing in exact and analytic form the propagation of nonparaxial scalar and electromagnetic beams. The main features of the method presented here are its mathematical simplicity and the fast convergence in the cases of highly nonparaxial electromagnetic beams, enabling us to obtain high-precision results without the necessity of lengthy numerical simulations or other more complex analytical calculations. The method can be used in electromagnetism (optics, microwaves) as well as in acoustics.

© 2014 Optical Society of America

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  1. A. Carnicer, I. Juvells, D. Maluenda, R. Martínez-Herrero, and P. M. Mejías, “The longitudinal component of paraxial fields,” Eur. J. Phys. 33, 1235–1247 (2012).
    [CrossRef]
  2. P. Varga and P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
    [CrossRef]
  3. A. April, “Nonparaxial TM and TE beams in free space,” Opt. Lett. 33, 1563–1565 (2008).
    [CrossRef]
  4. R. Borghi, A. Ciattoni, and M. Santarsiero, “Exact axial electromagnetic field for vectorial Gaussian and flattened Gaussian boundary distributions,” J. Opt. Soc. Am. A 19, 1207–1211 (2002).
    [CrossRef]
  5. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  6. P. C. Chaumet, “Fully vectorial highly nonparaxial beam close to the waist,” J. Opt. Soc. Am. A 23, 3197–3202 (2006).
    [CrossRef]
  7. A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002).
    [CrossRef]
  8. A. April, “Bessel–Gauss beams as rigorous solutions of the Helmholtz equation,” J. Opt. Soc. Am. A 28, 2100–2107 (2011).
    [CrossRef]
  9. S. Nemoto, “Nonparaxial Gaussian beams,” Appl. Opt. 29, 1940–1946 (1990).
    [CrossRef]
  10. M. Zamboni-Rached and E. Recami, “Sub-luminal wave bullets: exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
    [CrossRef]
  11. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
    [CrossRef]
  12. J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
    [CrossRef]
  13. T. Ito and S. Okazaki, “Pushing the limits of lithography,” Nature 406, 1027–1031 (2000).
    [CrossRef]
  14. Localized Waves: Theory and Applications, H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds. (Wiley, 2008).
  15. Non-Diffracting Waves, H. E. Hernández-Figueroa, E. Recamim, and M. Zamboni-Rached, eds. (Wiley-VCH, 2014).
  16. I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
    [CrossRef]
  17. I. Besieris and A. Shaarawi, “Paraxial localized waves in free space,” Opt. Express 12, 3848–3864 (2004).
    [CrossRef]
  18. M. Zamboni-Rached, E. Recami, and H. Figueroa, “New localized superluminal solutions to the wave equations-with finite total energies and arbitrary frequencies,” Eur. Phys. J. D 21, 217–228 (2002).
    [CrossRef]
  19. M. Zamboni-Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A 79, 013816 (2009).
    [CrossRef]
  20. M. Zamboni-Rached, “Localized waves: structure and applications,” M.Sc. thesis (Physics Department, State University of Campinas, 1999).
  21. M. Zamboni-Rached, “Localized waves in diffractive/dispersive media,” Ph.D. thesis (State University of Campinas, 2004).
  22. I. S. Gradshteyn and I. M. Ryzhik, Integrals, Series and Products, 5th ed. (Academic, 1965).
  23. Classical Electrodynamics, J. D. Jackson, ed., 3rd ed. (Wiley, 1998), pp. 239–242.

2012 (1)

A. Carnicer, I. Juvells, D. Maluenda, R. Martínez-Herrero, and P. M. Mejías, “The longitudinal component of paraxial fields,” Eur. J. Phys. 33, 1235–1247 (2012).
[CrossRef]

2011 (1)

2009 (1)

M. Zamboni-Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A 79, 013816 (2009).
[CrossRef]

2008 (2)

M. Zamboni-Rached and E. Recami, “Sub-luminal wave bullets: exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
[CrossRef]

A. April, “Nonparaxial TM and TE beams in free space,” Opt. Lett. 33, 1563–1565 (2008).
[CrossRef]

2006 (1)

2004 (1)

2002 (4)

M. Zamboni-Rached, E. Recami, and H. Figueroa, “New localized superluminal solutions to the wave equations-with finite total energies and arbitrary frequencies,” Eur. Phys. J. D 21, 217–228 (2002).
[CrossRef]

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002).
[CrossRef]

R. Borghi, A. Ciattoni, and M. Santarsiero, “Exact axial electromagnetic field for vectorial Gaussian and flattened Gaussian boundary distributions,” J. Opt. Soc. Am. A 19, 1207–1211 (2002).
[CrossRef]

2001 (1)

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

2000 (1)

T. Ito and S. Okazaki, “Pushing the limits of lithography,” Nature 406, 1027–1031 (2000).
[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, 108–118 (1998).
[CrossRef]

1990 (1)

1989 (1)

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[CrossRef]

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]

April, A.

Arlt, J.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

Besieris, I.

Besieris, I. M.

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[CrossRef]

Borghi, R.

Carnicer, A.

A. Carnicer, I. Juvells, D. Maluenda, R. Martínez-Herrero, and P. M. Mejías, “The longitudinal component of paraxial fields,” Eur. J. Phys. 33, 1235–1247 (2012).
[CrossRef]

Chaumet, P. C.

Ciattoni, A.

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002).
[CrossRef]

R. Borghi, A. Ciattoni, and M. Santarsiero, “Exact axial electromagnetic field for vectorial Gaussian and flattened Gaussian boundary distributions,” J. Opt. Soc. Am. A 19, 1207–1211 (2002).
[CrossRef]

Crosignani, B.

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002).
[CrossRef]

Curtis, J. E.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

Dholakia, K.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

Figueroa, H.

M. Zamboni-Rached, E. Recami, and H. Figueroa, “New localized superluminal solutions to the wave equations-with finite total energies and arbitrary frequencies,” Eur. Phys. J. D 21, 217–228 (2002).
[CrossRef]

Garces-Chavez, V.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Integrals, Series and Products, 5th ed. (Academic, 1965).

Grier, D. G.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

Ito, T.

T. Ito and S. Okazaki, “Pushing the limits of lithography,” Nature 406, 1027–1031 (2000).
[CrossRef]

Juvells, I.

A. Carnicer, I. Juvells, D. Maluenda, R. Martínez-Herrero, and P. M. Mejías, “The longitudinal component of paraxial fields,” Eur. J. Phys. 33, 1235–1247 (2012).
[CrossRef]

Koss, B. A.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[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]

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]

Maluenda, D.

A. Carnicer, I. Juvells, D. Maluenda, R. Martínez-Herrero, and P. M. Mejías, “The longitudinal component of paraxial fields,” Eur. J. Phys. 33, 1235–1247 (2012).
[CrossRef]

Martínez-Herrero, R.

A. Carnicer, I. Juvells, D. Maluenda, R. Martínez-Herrero, and P. M. Mejías, “The longitudinal component of paraxial fields,” Eur. J. Phys. 33, 1235–1247 (2012).
[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]

Mejías, P. M.

A. Carnicer, I. Juvells, D. Maluenda, R. Martínez-Herrero, and P. M. Mejías, “The longitudinal component of paraxial fields,” Eur. J. Phys. 33, 1235–1247 (2012).
[CrossRef]

Nemoto, S.

Okazaki, S.

T. Ito and S. Okazaki, “Pushing the limits of lithography,” Nature 406, 1027–1031 (2000).
[CrossRef]

Porto, P. D.

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002).
[CrossRef]

Recami, E.

M. Zamboni-Rached and E. Recami, “Sub-luminal wave bullets: exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
[CrossRef]

M. Zamboni-Rached, E. Recami, and H. Figueroa, “New localized superluminal solutions to the wave equations-with finite total energies and arbitrary frequencies,” Eur. Phys. J. D 21, 217–228 (2002).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Integrals, Series and Products, 5th ed. (Academic, 1965).

Santarsiero, M.

Shaarawi, A.

Shaarawi, A. M.

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[CrossRef]

Sibbett, W.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

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, 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, 108–118 (1998).
[CrossRef]

Zamboni-Rached, M.

M. Zamboni-Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A 79, 013816 (2009).
[CrossRef]

M. Zamboni-Rached and E. Recami, “Sub-luminal wave bullets: exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
[CrossRef]

M. Zamboni-Rached, E. Recami, and H. Figueroa, “New localized superluminal solutions to the wave equations-with finite total energies and arbitrary frequencies,” Eur. Phys. J. D 21, 217–228 (2002).
[CrossRef]

M. Zamboni-Rached, “Localized waves: structure and applications,” M.Sc. thesis (Physics Department, State University of Campinas, 1999).

M. Zamboni-Rached, “Localized waves in diffractive/dispersive media,” Ph.D. thesis (State University of Campinas, 2004).

Ziolkowski, R. W.

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[CrossRef]

Appl. Opt. (1)

Eur. J. Phys. (1)

A. Carnicer, I. Juvells, D. Maluenda, R. Martínez-Herrero, and P. M. Mejías, “The longitudinal component of paraxial fields,” Eur. J. Phys. 33, 1235–1247 (2012).
[CrossRef]

Eur. Phys. J. D (1)

M. Zamboni-Rached, E. Recami, and H. Figueroa, “New localized superluminal solutions to the wave equations-with finite total energies and arbitrary frequencies,” Eur. Phys. J. D 21, 217–228 (2002).
[CrossRef]

J. Math. Phys. (1)

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[CrossRef]

J. Opt. Soc. Am. A (3)

Nature (1)

T. Ito and S. Okazaki, “Pushing the limits of lithography,” Nature 406, 1027–1031 (2000).
[CrossRef]

Opt. Commun. (4)

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002).
[CrossRef]

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

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (3)

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

M. Zamboni-Rached and E. Recami, “Sub-luminal wave bullets: exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
[CrossRef]

M. Zamboni-Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A 79, 013816 (2009).
[CrossRef]

Other (6)

M. Zamboni-Rached, “Localized waves: structure and applications,” M.Sc. thesis (Physics Department, State University of Campinas, 1999).

M. Zamboni-Rached, “Localized waves in diffractive/dispersive media,” Ph.D. thesis (State University of Campinas, 2004).

I. S. Gradshteyn and I. M. Ryzhik, Integrals, Series and Products, 5th ed. (Academic, 1965).

Classical Electrodynamics, J. D. Jackson, ed., 3rd ed. (Wiley, 1998), pp. 239–242.

Localized Waves: Theory and Applications, H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds. (Wiley, 2008).

Non-Diffracting Waves, H. E. Hernández-Figueroa, E. Recamim, and M. Zamboni-Rached, eds. (Wiley-VCH, 2014).

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Figures (4)

Fig. 1.
Fig. 1.

Nonparaxial scalar beam. (a) Nonparaxial Gaussian spectrum, Eq. (9), represented by the continuous line and its representation through the Fourier series, Eq. (6), with N=2, 4, 6 and 10. (b) 3D intensity pattern of the resulting nonparaxial beam given by Eq. (8) with N=10 and its orthogonal projection.

Fig. 2.
Fig. 2.

Radially polarized nonparaxial beam. (a) Rectangular nonparaxial spectrum given by Eq. (28) and its representation through Eq. (6) with N=80. (b) and (c) 3D intensity patterns for the transverse, Eρ, and longitudinal, Ez, components of the electric field with their orthogonal projections on the plane z=0. (d) Vectorial diagram of Eρ on the plane z=0.

Fig. 3.
Fig. 3.

(a) 3D intensity pattern of the electric field of an azimuthally polarized nonparaxial beam obtained from Eq. (32) and its orthogonal projection. (b) Beam intensity pattern on the plane z=0. (c) Zoom with a vectorial diagram of the electric field on z=0.

Fig. 4.
Fig. 4.

(a) 3D intensity pattern of the longitudinal electric field component, Ez, on the plane z=0 and its orthogonal projection. (b) Same for the transverse component Ey. (c) Vectorial diagram of Ey on the plane z=0.

Equations (42)

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ψ(ρ,z,t)=aexp[iωt]ω/cω/cJ0(ρω2/c2kz2)exp[ikzz]S(kz)dkz,
ψ(ρ,z,t)=aexp[iωt]ω/cω/cJ0(ρω2/c2kz2)exp[ikzz]12cωdkz.
ψ(ρ,z,t)=aexp[iωt]sinc[ω2c2ρ2+ω2c2z2],
ψ(ρ,z,t)=a2cωexp[iωt]ω/cω/cJ0(ρω2c2kz2)exp[i(z+2πnK)kz]dkz.
ψ(ρ,z,t)=aexp[iωt]sinc[ω2c2ρ2+(zωc+πn)2].
S(kz)=n=Rnexp[i2πnKkz],
Rn=1Kω/cω/cS(kz)exp[i2πKnkz]dkz,
ψ(ρ,z,t)=aKexp[iωt]n=Rnsinc[ω2c2ρ2+(zωc+πn)2].
S(kz)={bπexp[b2(kzk¯z)2],0kzω/c0,otherwise,
Rnexp[i2πKnk¯z]exp[π2n2K2b2],
B=×A,
E=ic2ω(·A)+iωA.
A(ρ,z,t)=z^Az(ρ,z,t)=z^A(ρ,z)exp(iωt),
V(ρ,z,t)=ic2ω[zAz(ρ,z,t)].
Bρ=0,
Bϕ=ρAz(ρ,z,t),
Bz=0,
Eρ=ic2ω2ρzAz(ρ,z,t),
Eϕ=0,
Ez=ic2ω22zAz(ρ,z,t)+iωAz.
Eρ=aKi(ω2c2)exp[iωt]n=Rn(cnπ+ωz)ρ(3sinhh53coshh4sinhh3),
Eϕ=0,
Ez=aKiωexp[iωt]n=Rn[coshh2+sinhh3ρ2ω2c2(3sinhh53coshh4sinhh3)],
Bρ=0,
Bϕ=aKω2c2exp[iωt]n=Rnρ(sinhh3coshh2),
Bz=0,
h=ω2c2ρ2+(zωc+πn)2.
S(kz)={1kzmaxkzmin=1Δkz,forkzminkzkzmax0,otherwise,
Rn=i2πnΔkz[exp(i2πKnkzmax)exp(i2πKnkzmin)].
{EcBB1cE.
Eρ=0,
Eϕ=aKω2cexp[iωt]n=Rnρ(sinhh3coshh2),
Ez=0,
Bρ=aKi(ω2c3)exp[iωt]n=Rn(cnπ+ωz)ρ(3sinhh53coshh4sinhh3),
Bϕ=0,
Bz=aKiωcexp[iωt]n=Rn[coshh2+sinhh3ρ2ω2c2(3sinhh53coshh4sinhh3)],
Ez=yEydz.
Ey=zψ,
Ez=yψ.
Ex=0,
Ey=aKexp[iωt](ωc)n=Rn(nπ+ωcz)(cos[h]h2sin[h]h3),
Ez=aKexp[iωt](ωc)n=Rnωcρsinϕ(sin[h]h3cos[h]h2),

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