Abstract

A theoretical analysis of a traveling-wave difference-frequency-generation problem in three-dimensional space was performed. We found that, under a suitable hypothesis of low conversion efficiency in anisotropic nonlinear crystals, the nonlinear interaction gives a diffraction-free propagation condition for particular J0 Bessel beams and modulation instability phenomena for other J0 Bessel beams.

© 2000 Optical Society of America

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References

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  1. H.-A. Bachor, A Guide to Experiments in Quantum Optics (Wiley, New York, 1998).
  2. L. Lugiato and A. Gatti, “Spatial structure of a squeezed vacuum,” Phys. Rev. Lett. 70, 3868–3871 (1993).
    [CrossRef] [PubMed]
  3. A. Gatti and L. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator below threshold,” Phys. Rev. Lett. 52, 1675–1690 (1995).
  4. R. Horák, M. Bertolotti, and C. Sibilia, “A linearized approach for diffraction-free propagation in a planar nonlinear waveguide,” J. Mod. Opt. 41, 1615–1623 (1994).
    [CrossRef]
  5. R. Horák, J. Bajer, M. Bertolotti, and C. Sibilia, “Diffraction-free field in a planar nonlinear waveguide,” Phys. Rev. E 52, 4421–4429 (1995).
    [CrossRef]
  6. P. Di Trapani, G. Valiulis, W. Chinaglia, and A. Andreoni, “Two-dimensional spatial solitary waves from travelling wave parametric amplification of the quantum noise,” Phys. Rev. Lett. 80, 265–268 (1998).
    [CrossRef]
  7. G. J. Milburn and D. F. Walls, Quantum Optics (Springer-Verlag, New York, 1995).
  8. S. M. Barnet and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon, Oxford, UK, 1997).
  9. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  10. T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
    [CrossRef] [PubMed]
  11. P. Di Trapani, A. Berkanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
    [CrossRef]
  12. N. Bloembergen, R. J. Glauber, and N. M. Kroll, Quantum Optics and Electronics (Gordon & Breach, New York, 1965).
  13. G. Assanto and I. Torelli, “Cascading effects in type II second-harmonic generation: applications to all-optical processing,” Opt. Commun. 119, 143–148 (1995).
    [CrossRef]
  14. C. Sibilia, V. Schiavone, M. Bertolotti, R. Horák, and J. Perina, “Nonclassical spatial properties of light propagation in dissipative nonlinear waveguides,” J. Opt. Soc. Am. B 11, 2175–2181 (1994).
    [CrossRef]
  15. S. Trillo and P. Ferro, “Modulational instability in second-harmonic generation,” Opt. Lett. 20, 438–443 (1995).
    [CrossRef] [PubMed]
  16. F. Gori, Elementi di Ottica (Ed. Academica, Rome, 1997).
  17. J. Bajer and R. Horák, “Nondiffractive fields,” Phys. Rev. E 54, 3052–3054 (1996).
    [CrossRef]

1998

P. Di Trapani, G. Valiulis, W. Chinaglia, and A. Andreoni, “Two-dimensional spatial solitary waves from travelling wave parametric amplification of the quantum noise,” Phys. Rev. Lett. 80, 265–268 (1998).
[CrossRef]

P. Di Trapani, A. Berkanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
[CrossRef]

1996

J. Bajer and R. Horák, “Nondiffractive fields,” Phys. Rev. E 54, 3052–3054 (1996).
[CrossRef]

1995

S. Trillo and P. Ferro, “Modulational instability in second-harmonic generation,” Opt. Lett. 20, 438–443 (1995).
[CrossRef] [PubMed]

G. Assanto and I. Torelli, “Cascading effects in type II second-harmonic generation: applications to all-optical processing,” Opt. Commun. 119, 143–148 (1995).
[CrossRef]

A. Gatti and L. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator below threshold,” Phys. Rev. Lett. 52, 1675–1690 (1995).

R. Horák, J. Bajer, M. Bertolotti, and C. Sibilia, “Diffraction-free field in a planar nonlinear waveguide,” Phys. Rev. E 52, 4421–4429 (1995).
[CrossRef]

1994

R. Horák, M. Bertolotti, and C. Sibilia, “A linearized approach for diffraction-free propagation in a planar nonlinear waveguide,” J. Mod. Opt. 41, 1615–1623 (1994).
[CrossRef]

C. Sibilia, V. Schiavone, M. Bertolotti, R. Horák, and J. Perina, “Nonclassical spatial properties of light propagation in dissipative nonlinear waveguides,” J. Opt. Soc. Am. B 11, 2175–2181 (1994).
[CrossRef]

1993

L. Lugiato and A. Gatti, “Spatial structure of a squeezed vacuum,” Phys. Rev. Lett. 70, 3868–3871 (1993).
[CrossRef] [PubMed]

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
[CrossRef] [PubMed]

1987

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

Andreoni, A.

P. Di Trapani, G. Valiulis, W. Chinaglia, and A. Andreoni, “Two-dimensional spatial solitary waves from travelling wave parametric amplification of the quantum noise,” Phys. Rev. Lett. 80, 265–268 (1998).
[CrossRef]

Assanto, G.

G. Assanto and I. Torelli, “Cascading effects in type II second-harmonic generation: applications to all-optical processing,” Opt. Commun. 119, 143–148 (1995).
[CrossRef]

Bajer, J.

J. Bajer and R. Horák, “Nondiffractive fields,” Phys. Rev. E 54, 3052–3054 (1996).
[CrossRef]

R. Horák, J. Bajer, M. Bertolotti, and C. Sibilia, “Diffraction-free field in a planar nonlinear waveguide,” Phys. Rev. E 52, 4421–4429 (1995).
[CrossRef]

Berkanskis, A.

P. Di Trapani, A. Berkanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
[CrossRef]

Bertolotti, M.

R. Horák, J. Bajer, M. Bertolotti, and C. Sibilia, “Diffraction-free field in a planar nonlinear waveguide,” Phys. Rev. E 52, 4421–4429 (1995).
[CrossRef]

R. Horák, M. Bertolotti, and C. Sibilia, “A linearized approach for diffraction-free propagation in a planar nonlinear waveguide,” J. Mod. Opt. 41, 1615–1623 (1994).
[CrossRef]

C. Sibilia, V. Schiavone, M. Bertolotti, R. Horák, and J. Perina, “Nonclassical spatial properties of light propagation in dissipative nonlinear waveguides,” J. Opt. Soc. Am. B 11, 2175–2181 (1994).
[CrossRef]

Chinaglia, W.

P. Di Trapani, G. Valiulis, W. Chinaglia, and A. Andreoni, “Two-dimensional spatial solitary waves from travelling wave parametric amplification of the quantum noise,” Phys. Rev. Lett. 80, 265–268 (1998).
[CrossRef]

P. Di Trapani, A. Berkanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
[CrossRef]

Di Trapani, P.

P. Di Trapani, G. Valiulis, W. Chinaglia, and A. Andreoni, “Two-dimensional spatial solitary waves from travelling wave parametric amplification of the quantum noise,” Phys. Rev. Lett. 80, 265–268 (1998).
[CrossRef]

Durnin, J.

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

Eberly, J. H.

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

Ferro, P.

Gatti, A.

A. Gatti and L. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator below threshold,” Phys. Rev. Lett. 52, 1675–1690 (1995).

L. Lugiato and A. Gatti, “Spatial structure of a squeezed vacuum,” Phys. Rev. Lett. 70, 3868–3871 (1993).
[CrossRef] [PubMed]

Herminghaus, S.

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
[CrossRef] [PubMed]

Horák, R.

J. Bajer and R. Horák, “Nondiffractive fields,” Phys. Rev. E 54, 3052–3054 (1996).
[CrossRef]

R. Horák, J. Bajer, M. Bertolotti, and C. Sibilia, “Diffraction-free field in a planar nonlinear waveguide,” Phys. Rev. E 52, 4421–4429 (1995).
[CrossRef]

R. Horák, M. Bertolotti, and C. Sibilia, “A linearized approach for diffraction-free propagation in a planar nonlinear waveguide,” J. Mod. Opt. 41, 1615–1623 (1994).
[CrossRef]

C. Sibilia, V. Schiavone, M. Bertolotti, R. Horák, and J. Perina, “Nonclassical spatial properties of light propagation in dissipative nonlinear waveguides,” J. Opt. Soc. Am. B 11, 2175–2181 (1994).
[CrossRef]

Lugiato, L.

A. Gatti and L. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator below threshold,” Phys. Rev. Lett. 52, 1675–1690 (1995).

L. Lugiato and A. Gatti, “Spatial structure of a squeezed vacuum,” Phys. Rev. Lett. 70, 3868–3871 (1993).
[CrossRef] [PubMed]

Miceli Jr., J. J.

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

Minardi, S.

P. Di Trapani, A. Berkanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
[CrossRef]

Perina, J.

Sapone, S.

P. Di Trapani, A. Berkanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
[CrossRef]

Schiavone, V.

Sibilia, C.

R. Horák, J. Bajer, M. Bertolotti, and C. Sibilia, “Diffraction-free field in a planar nonlinear waveguide,” Phys. Rev. E 52, 4421–4429 (1995).
[CrossRef]

R. Horák, M. Bertolotti, and C. Sibilia, “A linearized approach for diffraction-free propagation in a planar nonlinear waveguide,” J. Mod. Opt. 41, 1615–1623 (1994).
[CrossRef]

C. Sibilia, V. Schiavone, M. Bertolotti, R. Horák, and J. Perina, “Nonclassical spatial properties of light propagation in dissipative nonlinear waveguides,” J. Opt. Soc. Am. B 11, 2175–2181 (1994).
[CrossRef]

Torelli, I.

G. Assanto and I. Torelli, “Cascading effects in type II second-harmonic generation: applications to all-optical processing,” Opt. Commun. 119, 143–148 (1995).
[CrossRef]

Trapani, P. Di

P. Di Trapani, A. Berkanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
[CrossRef]

Trillo, S.

Valiulis, G.

P. Di Trapani, G. Valiulis, W. Chinaglia, and A. Andreoni, “Two-dimensional spatial solitary waves from travelling wave parametric amplification of the quantum noise,” Phys. Rev. Lett. 80, 265–268 (1998).
[CrossRef]

Wulle, T.

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
[CrossRef] [PubMed]

J. Mod. Opt.

R. Horák, M. Bertolotti, and C. Sibilia, “A linearized approach for diffraction-free propagation in a planar nonlinear waveguide,” J. Mod. Opt. 41, 1615–1623 (1994).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

G. Assanto and I. Torelli, “Cascading effects in type II second-harmonic generation: applications to all-optical processing,” Opt. Commun. 119, 143–148 (1995).
[CrossRef]

Opt. Lett.

Phys. Rev. E

J. Bajer and R. Horák, “Nondiffractive fields,” Phys. Rev. E 54, 3052–3054 (1996).
[CrossRef]

R. Horák, J. Bajer, M. Bertolotti, and C. Sibilia, “Diffraction-free field in a planar nonlinear waveguide,” Phys. Rev. E 52, 4421–4429 (1995).
[CrossRef]

Phys. Rev. Lett.

P. Di Trapani, G. Valiulis, W. Chinaglia, and A. Andreoni, “Two-dimensional spatial solitary waves from travelling wave parametric amplification of the quantum noise,” Phys. Rev. Lett. 80, 265–268 (1998).
[CrossRef]

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

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
[CrossRef] [PubMed]

P. Di Trapani, A. Berkanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
[CrossRef]

L. Lugiato and A. Gatti, “Spatial structure of a squeezed vacuum,” Phys. Rev. Lett. 70, 3868–3871 (1993).
[CrossRef] [PubMed]

A. Gatti and L. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator below threshold,” Phys. Rev. Lett. 52, 1675–1690 (1995).

Other

H.-A. Bachor, A Guide to Experiments in Quantum Optics (Wiley, New York, 1998).

F. Gori, Elementi di Ottica (Ed. Academica, Rome, 1997).

N. Bloembergen, R. J. Glauber, and N. M. Kroll, Quantum Optics and Electronics (Gordon & Breach, New York, 1965).

G. J. Milburn and D. F. Walls, Quantum Optics (Springer-Verlag, New York, 1995).

S. M. Barnet and P. M. Radmore, Methods in Theoretical Quantum Optics (Clarendon, Oxford, UK, 1997).

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Figures (3)

Fig. 1
Fig. 1

Modulation instability versus transverse spatial frequencies σ and η (corresponding to the nondimensional transverse spatial variables X and Y in the Fourier domain). (a) g(σ, η) for all four octants of interest, (b)–(d) three particular octants. All graphs are given in arbitrary units.

Fig. 2
Fig. 2

Wave vector distribution for the small spatial perturbations qj in Z=0 (at the input of the crystal). Beams so realized are exactly J0 Bessel beams whose (half) angular aperture is θ=arcsin |ktj/kj|.

Fig. 3
Fig. 3

Field amplitudes for three fields (a)–(c) (three J0-Bessel beams) during propagation inside a nonlinear quadratic crystal (such as LBO) for Z=0 (first row), Z=0.03 (second row), and Z=0.05 (third row), given in arbitrary units. On the xy axes are nondimensional spatial chords in arbitrary units, and on the z axis is the field amplitude that we consider (we focus on the transverse field). During propagation only the field in (b), which has a right-angular aperture for which there is diffraction-free propagation, remains unaltered; the fields in (a) and (c) are attenuated. A stationary homogeneous isotropic linear medium is assumed for Z<0 and a nonlinear anisotropic quadratic medium for Z0.

Equations (47)

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i z-1+2x2+2y2Q1=-Q3Q2*,
i z-1+k1k22x2+2y2Q2=-k1k2Q3Q1*,
i z-1+k1k32x2+2y2Q3=-k1k3Q1Q2,
Q1(X, Y, Z)=A1(x, y, z)exp(-iΔk Z) 8πχω2ω32c2k1Δk,
Q2(X, Y, Z)=A2(x, y, z)exp(-iΔk Z) 8πχω1ω32c2k1Δk,
Q3(X, Y, Z)=A3(x, y, z)exp(-iΔk Z) 8πχω1ω22c2k1Δk.
X=x2k1Δk;Y=y2k1Δk;Z=zΔk.
i Z-1+2X2+2Y2Q1=k1k2|Q3|2Q1,
i Z-1+k1k32X2+2Y2Q3=-k12k2k3|Q1|2Q3.
Qj(X, Y, Z)=Qj(Z)+qj(X, Y, Z),
i Z-1+2X2+2Y2q1
=-k1k2[|Q3(Z)|2q1+Q1(Z)×(Q3*(Z)q3+Q3(Z)q3*)],
i Z-1+k1k32X2+2Y2q3
=-k1k2k3[|Q1(Z)|2q3+Q3(Z)×(Q1*(Z)q1+Q1(Z)q1*)].
qj(X, Y, Z)=exp[iφj(Z)]2πdσdη×exp[-i(σX+ηY)]cj(σ, η, Z),
j=1, 2, 3,
i Zc(σ, η, Z)=M(σ, η)·c(σ, η, Z),
c(σ, η, Z)=c1(σ, η, Z)c1*(σ, η, Z)c3(σ, η, Z)c3*(σ, η, Z)
M(σ, η)=a10a2a20-a1-a2-a2a4a4a30-a4-a40-a3, a1=(σ2+η2),a2=-k1k2|Q10Q30|,a3=k1k3(σ2+η2),a4=-k12k2k3|Q10Q30|.
c(σ, η, Z)=T(σ, η, Z)·c(σ, η, 0),
c(σ, η, Z)Z.
g(σ, η)=12(Im{λ}+|Im{λ}|)
ρ0=k1k2|Q10Q30|
Im{λ}<0.
ti,j(σ, η, Z=0)=δi,j,
ti,j(σ, η, Z)=ti,j(-σ, η, Z)=ti,j(σ, -η, Z)=ti,j(-σ, -η, Z).
σ=Ξ cos(α),η=Ξ sin(α).
t˜i,j(Ξ, α, Z)=t˜i,j(Ξ, Z),
t˜i,j(Ξ, α, Z)=ti,j(Ξ cos(α),Ξ sin(α), Z).
σ=Ξ cos(α),X=ρ cos(ϕ),
η=Ξ sin(α),Y=ρ sin(ϕ),
q˜j(ρ, ϕ, Z)=12π002πc˜j(Ξ, α, z)×exp[-iΞρ cos(α-ϕ)]ΞdΞdα.
c˜j (Ξ, α, 0)=c˜j (Ξ, 0)=2πq0ktjδ(Ξ+ktj),j=1, 3,
q˜j(ρ, ϕ, 0)=q002π exp[-iktj ρ cos(α-ϕ)]dα,
qj(X, Y, Z)=12πk=14--tj,k(σ, η, Z)ck(σ, η, 0)×exp[-i(σX+ηY)]dσdη,
q˜j(ρ, ϕ, Z)=12πk=14002πtj,k(Ξ, Z)c˜k(Ξ, 0)×exp[-iΞρ cos(α-ϕ)]ΞdΞdα.
|Qj(X, Y, Z)|2=|Qj0|2+C2J02(k0X2+Y2)+2|Qj0|CJ0(k0X2+Y2),
i Z-1+k1k2|Q30|2+2X2+2Y2q1=0,
X=ρ cos(ϕ),
Y=ρ sin(ϕ).
i Z-1+k1k2|Q30|2+1ρρρ ρq1=0,
i Z-1+k1k2|Q30|2-4π2Ξ2c1(Ξ, Z)=0.
c1(Ξ, Z)=f(Z)δ(Ξ-kt),
i Z-1+k1k2|Q30|2-4π2kt2 f (Z)=0.
Zf(Z)=0
k1k2|Q30|2-1-4π2kt2=0,
kt=12πk1k2|Q30|2-11/2.

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