Abstract

Localized wave solutions of free-space wave equation can be used in numerous applications where the localized transmission of electromagnetic energy is of major importance. However, an optical implementation of localized wave fields has not been accomplished yet, except for an ultrashort version of the Bessel beams or the so called Bessel-X pulses. We propose an approach to constructing realizable optical schemes for generation of localized wave fields. We show that wavelength dispersion of the cone angle of axicons and circular diffraction gratings can be used to generate good approximation to focus wave modes.

© 2000 Optical Society of America

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References

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  1. J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
    [CrossRef]
  2. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  3. P. L. Ovefelt, “Bessel–Gauss pulses,” Phys. Rev. A 44, 3941–3947 (1991).
    [CrossRef]
  4. J. Lu, J. G. Greenleaf, “Nondiffracting X waves—exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
    [CrossRef]
  5. R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3983 (1991).
    [CrossRef] [PubMed]
  6. J. Fagerhorm, A. T. Friberg, J. Huttunen, D. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
    [CrossRef]
  7. P. Saari, H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).
  8. K. Reivelt, P. Saari, “Angular spectrum analysis and synthesis of propagation-invariant femtosecond-domain localized wave fields,” in Ultrafast Processes in Spectroscopy, R. Kaarli, A. Freiberg, P. Saari, eds. (Institute of Physics, University of Tartu, Tartu, Estonia, 1998), pp. 168–175.
  9. H. Sõnajalg, M. Rätsep, P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive media,” Opt. Lett. 22, 310–312 (1997).
    [CrossRef]
  10. A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
    [CrossRef] [PubMed]
  11. P. Saari, K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
    [CrossRef]
  12. A. M. Shaarawi, “Comparison of two localized wave fields generated from dynamic apertures,” J. Opt. Soc. Am. A 14, 1804–1815 (1997).
    [CrossRef]
  13. A. M. Shaarawi, R. W. Ziolkowski, I. M. Besieris, “On the evanescent fields and the causality of the focus wave modes,” J. Math. Phys. 36, 5565–5587 (1995).
    [CrossRef]
  14. Z. Bin, L. Zhu, “Diffraction property of an axicon in oblique illumination,” Appl. Opt. 37, 2563–2568 (1998).
    [CrossRef]
  15. S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986).
  16. S. Chávez-Cerda, G. S. McDonald, G. H. C. New, “Nondiffracting beams: traveling, standing and spiral waves,” Opt. Commun. 123, 225–233 (1996).
    [CrossRef]
  17. In Optics, Opto-Mechanics, Lasers, Instruments (catalog) (Melles Griot, Irvine, Calif., 1995).
  18. E. J. Mayer, J. Möbius, A. Euteneuer, W. W. Rühle, R. Szipöcs, “Ultrabroadband chirped mirrors for femtosecond lasers,” Opt. Lett. 22, 528–530 (1997).
    [CrossRef] [PubMed]

1998 (1)

1997 (5)

1996 (2)

J. Fagerhorm, A. T. Friberg, J. Huttunen, D. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

S. Chávez-Cerda, G. S. McDonald, G. H. C. New, “Nondiffracting beams: traveling, standing and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

1995 (1)

A. M. Shaarawi, R. W. Ziolkowski, I. M. Besieris, “On the evanescent fields and the causality of the focus wave modes,” J. Math. Phys. 36, 5565–5587 (1995).
[CrossRef]

1992 (1)

J. Lu, J. G. Greenleaf, “Nondiffracting X waves—exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[CrossRef]

1991 (2)

R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3983 (1991).
[CrossRef] [PubMed]

P. L. Ovefelt, “Bessel–Gauss pulses,” Phys. Rev. A 44, 3941–3947 (1991).
[CrossRef]

1989 (1)

1987 (1)

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

1983 (1)

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
[CrossRef]

Besieris, I. M.

A. M. Shaarawi, R. W. Ziolkowski, I. M. Besieris, “On the evanescent fields and the causality of the focus wave modes,” J. Math. Phys. 36, 5565–5587 (1995).
[CrossRef]

Bin, Z.

Brittingham, J. N.

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
[CrossRef]

Chávez-Cerda, S.

S. Chávez-Cerda, G. S. McDonald, G. H. C. New, “Nondiffracting beams: traveling, standing and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

Crosignani, B.

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986).

DiPorto, P.

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986).

Durnin, J.

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

Eberly, J. H.

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

Euteneuer, A.

Fagerhorm, J.

J. Fagerhorm, A. T. Friberg, J. Huttunen, D. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Friberg, A. T.

J. Fagerhorm, A. T. Friberg, J. Huttunen, D. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
[CrossRef] [PubMed]

Greenleaf, J. G.

J. Lu, J. G. Greenleaf, “Nondiffracting X waves—exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[CrossRef]

Huttunen, J.

J. Fagerhorm, A. T. Friberg, J. Huttunen, D. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Lu, J.

J. Lu, J. G. Greenleaf, “Nondiffracting X waves—exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[CrossRef]

Mayer, E. J.

McDonald, G. S.

S. Chávez-Cerda, G. S. McDonald, G. H. C. New, “Nondiffracting beams: traveling, standing and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

Miceli, J. J.

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

Möbius, J.

Morgan, D.

J. Fagerhorm, A. T. Friberg, J. Huttunen, D. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

New, G. H. C.

S. Chávez-Cerda, G. S. McDonald, G. H. C. New, “Nondiffracting beams: traveling, standing and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

Ovefelt, P. L.

P. L. Ovefelt, “Bessel–Gauss pulses,” Phys. Rev. A 44, 3941–3947 (1991).
[CrossRef]

Rätsep, M.

Reivelt, K.

P. Saari, K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[CrossRef]

K. Reivelt, P. Saari, “Angular spectrum analysis and synthesis of propagation-invariant femtosecond-domain localized wave fields,” in Ultrafast Processes in Spectroscopy, R. Kaarli, A. Freiberg, P. Saari, eds. (Institute of Physics, University of Tartu, Tartu, Estonia, 1998), pp. 168–175.

Rühle, W. W.

Saari, P.

H. Sõnajalg, M. Rätsep, P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive media,” Opt. Lett. 22, 310–312 (1997).
[CrossRef]

P. Saari, K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[CrossRef]

P. Saari, H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).

K. Reivelt, P. Saari, “Angular spectrum analysis and synthesis of propagation-invariant femtosecond-domain localized wave fields,” in Ultrafast Processes in Spectroscopy, R. Kaarli, A. Freiberg, P. Saari, eds. (Institute of Physics, University of Tartu, Tartu, Estonia, 1998), pp. 168–175.

Salomaa, M. M.

J. Fagerhorm, A. T. Friberg, J. Huttunen, D. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Shaarawi, A. M.

A. M. Shaarawi, “Comparison of two localized wave fields generated from dynamic apertures,” J. Opt. Soc. Am. A 14, 1804–1815 (1997).
[CrossRef]

A. M. Shaarawi, R. W. Ziolkowski, I. M. Besieris, “On the evanescent fields and the causality of the focus wave modes,” J. Math. Phys. 36, 5565–5587 (1995).
[CrossRef]

Solimeno, S.

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986).

Sõnajalg, H.

Szipöcs, R.

Turunen, J.

Vasara, A.

Zhu, L.

Ziolkowski, R. W.

A. M. Shaarawi, R. W. Ziolkowski, I. M. Besieris, “On the evanescent fields and the causality of the focus wave modes,” J. Math. Phys. 36, 5565–5587 (1995).
[CrossRef]

R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3983 (1991).
[CrossRef] [PubMed]

Appl. Opt. (1)

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

J. Lu, J. G. Greenleaf, “Nondiffracting X waves—exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[CrossRef]

J. Appl. Phys. (1)

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
[CrossRef]

J. Math. Phys. (1)

A. M. Shaarawi, R. W. Ziolkowski, I. M. Besieris, “On the evanescent fields and the causality of the focus wave modes,” J. Math. Phys. 36, 5565–5587 (1995).
[CrossRef]

J. Opt. Soc. Am. A (2)

Laser Phys. (1)

P. Saari, H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).

Opt. Commun. (1)

S. Chávez-Cerda, G. S. McDonald, G. H. C. New, “Nondiffracting beams: traveling, standing and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (2)

P. L. Ovefelt, “Bessel–Gauss pulses,” Phys. Rev. A 44, 3941–3947 (1991).
[CrossRef]

R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3983 (1991).
[CrossRef] [PubMed]

Phys. Rev. E (1)

J. Fagerhorm, A. T. Friberg, J. Huttunen, D. Morgan, M. M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Phys. Rev. Lett. (2)

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

P. Saari, K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[CrossRef]

Other (3)

In Optics, Opto-Mechanics, Lasers, Instruments (catalog) (Melles Griot, Irvine, Calif., 1995).

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986).

K. Reivelt, P. Saari, “Angular spectrum analysis and synthesis of propagation-invariant femtosecond-domain localized wave fields,” in Ultrafast Processes in Spectroscopy, R. Kaarli, A. Freiberg, P. Saari, eds. (Institute of Physics, University of Tartu, Tartu, Estonia, 1998), pp. 168–175.

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Figures (5)

Fig. 1
Fig. 1

Angular spectrum representation of an optically realizable FWM: (a) support of the angular spectrum, (b) frequency spectrum of the source pulse, (c) spatial amplitude distribution. The parameters are chosen as follows: k1=8.4×106 (λ∼750 nm), k2=1.05×107 (λ∼600 nm), k3=1.4×107 (λ∼450 nm), duration of source pulse is <3 fs, cone angle of the Bessel beam with central wavelength (600 nm) is θFWM(k2)0.22°, constant 2β=80 [see Eqs. (4) and (7)]. The striped region outlines the backward-propagating components (kz<0) within the support of the angular spectrum.

Fig. 2
Fig. 2

Qualitative description of a Bessel beam as a superposition of diverging (dotted lines) and converging (solid lines) conical waves in the plane z=0: (a) free-space evolution of the conical components, (b) diffraction on a circular opaque mask, (c) refraction on an axicon.  

Fig. 3
Fig. 3

Circular diffraction grating on the surface of an axicon: a composite optical element that can be used for the generation of FWM’s.

Fig. 4
Fig. 4

Optical setup for the generation of FWM’s. A plane-wave pulse is incident on an annular slit (AS). A Bessel-X pulse with cone angle θ0 behind a Fourier lens (L) is incident on the composite optical element (AG).

Fig. 5
Fig. 5

Numerical evaluation of the wave field generated by means of the optical setup depicted in Fig. 4: (a) support of the angular spectrum of the generated wave field (solid curve) and of the FWM (dashed curve), (b) exact form of the deviation δθ(k)=θFWM(k)-θG(k), (c) the spatial amplitude distribution of the generated wave field at distances z=5 cm and z=60 cm behind the element AG.

Equations (42)

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Ψ(ρ, z, t)=0dk0πdθA(k, θ)J0(kρ sin θ)×exp[ik(z cos θ-ct)].
ΨB(ρ, z, t)=J0(kρ sin θ)exp[ik(z cos θ-ct)],
vg=cdkdkz=1cddk[k cos θFWM(k)]-1=cγ,
cos θFWM(k)=γ(k-2β)k.
A(k, θ)=A˜(k)δcos θ-γ(k-2β)k
ΨFWM(ρ, z, t)=0dk0πdθA˜(k)δcos θ-γ(k-2β)k×J0(kρ sin θ)exp[ik(z cos θ-ct)],
ΨFWM(ρ, z, t)=exp-i2βzc0dkA˜(k)×J0kρ1-γ(k-2β)k21/2×exp[ik(γz-ct)].
A=(k, θ)=A˜(k)δ {sin θ-tan α[1-n(k)]},
Ψ(ρ, z, t)=0dkA˜(k)J0(kρ{tan α[1-n(k)]})×exp{ik[z cos(arcsin{tan α[1-n(k)]})-ct]},
Ψ(ρ, z, t)=0dkA˜(k)J0kρ2πkd×expikz cosarcsin2πkd-ct.
sin θ(k)=sin[arctan(D/f)]sin θ0,
Ψ(ρ, z, t)=0dkA˜(k)J0(kρ sin θ0)×exp[ik(z cos θ0-ct)].
J0(ξ)  122πξexpiξ-π4+exp-iξ-π4.
ΨB(ρ, 0, t)122πkρ sin θ1/2expikρ sin θ-iπ4+exp-ikρ sin θ+iπ4exp(-ikct)
sin θG(k)=2πkd+n(k)×sin-α+arcsin1n(k)sin(θ0+α).
θG(k)=2πkd+α[1-n(k)]+θ0.
12[2/(πkρ sin θ0]1/2exp(ikρ sin θ0).
exp(ikρ{2π/kd+tan α[1-n(k)]}).
Ψ(ρ, z)=1iλzexpikz+ρ22z×0Ddρf(ρ)exp[ikμ(ρ)],
f(ρ)=ρ122πkρsin θ01/22πJ0kρρz
μ(ρ)=ρ22z-ρ2πkd+tan α[1-n(k)]+sin θ0.
ρc=z2πkd+tan α[1-n(k)]+sin θ0z sin θSP(k)
1iλzexpikz+ρ22z0Ddρf(ρ)exp[ikμ(ρ)]
1iλzexpikz+ρ22zf(ρc)exp[ikμ(ρc)][kμ(2)(ρc)]1/2,
Ψ(ρ, z)1iλzexpikz+ρ22z1k(1z) z sin θSP(k)
× 122πkz sin θSP(k)sin θ01/2
× 2πJ0kρz sin θSP(k)z
× expik[z sin θSP(k)]22z-z sin2 θSP(k)
J0(kρ sin θSP(k))expikρ22z
× exp[ikz cos θSP(k)],
1-sin 2θ/2(1-sin2 θ)1/2=cos θ
Ψ(ρ, z, t)J0(kρ sin θSP(k))exp[ikz cos θSP(k)].
arccosγ(k-2β)k=2πkd+α[1-n(k)]+θ0.
arccosγ(km-2β)km=2πkmd+α[1-n(km)]+θ0,
m=1, 2, 3.
k1=8.4×106(λ750nm),
k2=1.05×107(λ600nm),
k3=1.4×107(λ450nm).
θ0=4.9683×10-3 rad,
α=1.3866×10-2 rad,
d=3.7509×10-4 m.
Ψ(ρ, z, t)=0dkA˜(k)J0(kρ sin θG(k))×exp{ik[z cos θG(k)-ct]}.

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