Abstract

Nonclassical spatial effects such as squeezing are studied in the transverse domain in a parametric process of difference-frequency interaction inside a nonlinear quadratic bulk, by use of a quantum-perturbative approach. The classical modulation-instability regions correspond to those in which maximum squeezing of the radiation is found. Calculations were performed with all three spatial variables, i.e., in a traveling-wave approximation under the further hypothesis of low conversion efficiency and small quantum fluctuations superimposed upon the strong classical fields.

© 2002 Optical Society of America

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References

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  1. M. D. Levenson, W. H. Richardson, and S. H. Perlmutter, “Stochastic noise in TEM00 laser beam position,” Opt. Lett. 14, 779–781 (1989).
    [CrossRef] [PubMed]
  2. H. A. Bachor, A Guide to Experiments in Quantum Optics (Wiley, New York, 1998).
  3. M. I. Kolobov and I. V. Sokolov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1531–1589 (1999).
    [CrossRef]
  4. M. I. Kolobov and I. V. Sokolov, “Squeezed states of light and quantum noise free optical images,” Phys. Lett. A 140, 101–116 (1989).
    [CrossRef]
  5. L. A. Lugiato, M. Brambilla, and A. Gatti, “Optical pattern formation,” Adv. At., Mol., Opt. Phys. 40, 229–306 (1998).
    [CrossRef]
  6. A. Gatti and L. A. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator below threshold,” Phys. Rev. A 52, 1675–1690 (1995).
    [CrossRef] [PubMed]
  7. L. A. Lugiato and A. Gatti, “Spatial structure of a squeezed vacuum,” Phys. Rev. Lett. 70, 3868–3871 (1993).
    [CrossRef] [PubMed]
  8. S. Severini, C. Sibilia, and M. Bertolotti, “Transverse effects in a parametric downconversion process in three dimensions,” J. Opt. Soc. Am. B 17, 580–585 (2000).
    [CrossRef]
  9. P. Di Trapani, Aberkanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observations of optical vortices and J0 Bessel-like beams in quantum noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
    [CrossRef]
  10. 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]
  11. J. A. Armstrong, N. Bloembergen, J. Ducuing, and S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  12. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  13. A. V. Buryak and Y. S. Kivshar, “Spatial optical solitons governed by quadratic nonlinearity,” Opt. Lett. 19, 1612–1614 (1994).
    [CrossRef] [PubMed]
  14. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, Cambridge, UK, 1995).

2000 (1)

1999 (1)

M. I. Kolobov and I. V. Sokolov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1531–1589 (1999).
[CrossRef]

1998 (2)

L. A. Lugiato, M. Brambilla, and A. Gatti, “Optical pattern formation,” Adv. At., Mol., Opt. Phys. 40, 229–306 (1998).
[CrossRef]

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

1995 (1)

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

1994 (2)

1993 (1)

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

1989 (2)

M. I. Kolobov and I. V. Sokolov, “Squeezed states of light and quantum noise free optical images,” Phys. Lett. A 140, 101–116 (1989).
[CrossRef]

M. D. Levenson, W. H. Richardson, and S. H. Perlmutter, “Stochastic noise in TEM00 laser beam position,” Opt. Lett. 14, 779–781 (1989).
[CrossRef] [PubMed]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Aberkanskis,

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

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bertolotti, M.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Brambilla, M.

L. A. Lugiato, M. Brambilla, and A. Gatti, “Optical pattern formation,” Adv. At., Mol., Opt. Phys. 40, 229–306 (1998).
[CrossRef]

Buryak, A. V.

Chinaglia, W.

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

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Gatti, A.

L. A. Lugiato, M. Brambilla, and A. Gatti, “Optical pattern formation,” Adv. At., Mol., Opt. Phys. 40, 229–306 (1998).
[CrossRef]

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

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

Horák, R.

Kivshar, Y. S.

Kolobov, M. I.

M. I. Kolobov and I. V. Sokolov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1531–1589 (1999).
[CrossRef]

M. I. Kolobov and I. V. Sokolov, “Squeezed states of light and quantum noise free optical images,” Phys. Lett. A 140, 101–116 (1989).
[CrossRef]

Levenson, M. D.

Lugiato, L. A.

L. A. Lugiato, M. Brambilla, and A. Gatti, “Optical pattern formation,” Adv. At., Mol., Opt. Phys. 40, 229–306 (1998).
[CrossRef]

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

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

Minardi, S.

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

Perina, J.

Perlmutter, S. H.

Pershan, S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Richardson, W. H.

Sapone, S.

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

Schiavone, V.

Severini, S.

Sibilia, C.

Sokolov, I. V.

M. I. Kolobov and I. V. Sokolov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1531–1589 (1999).
[CrossRef]

M. I. Kolobov and I. V. Sokolov, “Squeezed states of light and quantum noise free optical images,” Phys. Lett. A 140, 101–116 (1989).
[CrossRef]

Adv. At., Mol., Opt. Phys. (1)

L. A. Lugiato, M. Brambilla, and A. Gatti, “Optical pattern formation,” Adv. At., Mol., Opt. Phys. 40, 229–306 (1998).
[CrossRef]

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

Opt. Lett. (2)

Phys. Lett. A (1)

M. I. Kolobov and I. V. Sokolov, “Squeezed states of light and quantum noise free optical images,” Phys. Lett. A 140, 101–116 (1989).
[CrossRef]

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Phys. Rev. A (1)

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

Phys. Rev. Lett. (2)

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

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

Rev. Mod. Phys. (1)

M. I. Kolobov and I. V. Sokolov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1531–1589 (1999).
[CrossRef]

Other (3)

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

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, Cambridge, UK, 1995).

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Figures (4)

Fig. 1
Fig. 1

Difference-frequency generation process. Typically no input field at the new frequency ω2 is applied to the input of the crystal (χ(2) nonlinear media).

Fig. 2
Fig. 2

Photon-energy description of the three fields involved in the interaction. Creation of one photon at ω1 must be accompanied by the creation of one photon at ω2 and annihilation of one photon at ω3. Dashed lines represent virtual energetic states.

Fig. 3
Fig. 3

Optimized squeezing-spectrum distributions log[Smax] for field 1. Adimensional longitudinal variable Z (linked to the physical one, z, by Z=zΔk, where Δk is the phase mismatch), inside the crystal, assumes values (a) 0+, (b) 0.01, (c) 0.1, (d) 0.2, and (e) 0.3.

Fig. 4
Fig. 4

θopt for field 1 versus the adimensional longitudinal Z variable and amplitude ratios b, defined as b2=|Q10|/|Q30|.

Equations (30)

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i Z-1+2X2+2Y2Q1=-Q3Q2*,
i Z-1+k1k22X2+2Y2Q2=-k1k2Q3Q1*,
i Z-1+k1k32X2+2Y2Q3=-k1k3Q1Q2,
Qi(X, Y, Z)=Ai(x, y, z)exp(-iΔk·z) 8πχωjωk2c2k1Δk2,
X=x2k1Δk,Y=y2k1Δk,Z=zΔk.
Qˆj(X, Y, Z)=Qj(Z)+qˆj(X, Y, Z),j=1,2,3,
i Z-1+2X2+2Y2Q1=-k1k2Q3*Q3Q1,
Q2k1k2Q1*Q3,
i Z-1+k1k3 2X2+2Y2Q3=-k1k3k2Q1*Q1Q2.
i Z-1+X2+Y2qˆ1
=-k1k2[|Q3|2qˆ1+Q1(Q3*qˆ3+Q3qˆ3+)],
i Z-1+k1k3 X2+Y2qˆ3
=-k12k1k3[|Q1|2qˆ3+Q3(Q1*qˆ1+Q1qˆ1+)],
qj(X, Y, Z)=exp[iφj(Z)]2π dσdη×exp[-i(σX+ηY)]cj(σ, η, Z),
i Z c(σ, η, Z)=M(σ, η)·c(σ, η, Z),
c(σ, η, Z)=T(σ, η)·c(σ, η, 0),
qˆj(X, Y, Z)=exp[iφj(Z)]2π dσdη×[tj,1(σ, η, Z)cˆ1(σ, η, 0)+tj,2(σ, η, Z)cˆ1+(-σ,-η, 0)+tj,3(σ, η, Z)cˆ3(σ, η, 0)+tj,4(σ, η, Z)cˆ3+(-σ,-η, 0)]×exp[-i(σX+ηY)],
[cˆ3(σ, η, 0),cˆ3+(-σ,-η, 0)]
=k1k3[cˆ1(σ, η, 0),cˆ1(-σ,-η, 0)],
[cˆ1(σ, η, 0),cˆ1+(-σ,-η, 0)]
=δ(σ+σ)δ(η+η),
Qˆ(θ)=12[Qˆ exp(iθ)+Qˆ+ exp(-iθ)].
(ΔQˆ(θ))2=(Qˆ(θ)-Qˆ(θ))2=14[qˆqˆ++qˆ+qˆ+2 Re{exp(2iθ)qˆqˆ}].
(ΔQˆ(θ))2=dσdηS(θ),
dσdηS(θ)(σ, η, Z)
=[ΔQˆ(θ)(X, Y, Z)]2=14[qˆj(X, Y, Z)qˆj+(X, Y, Z)+qˆj+(X, Y, Z)qˆj(X, Y, Z)+2 Re{exp(2iθ)qˆj(X, Y, Z)qˆj(X, Y, Z)}].
0|cˆj(σ, η, 0)cˆj+(-σ,-η, 0|)0
=0|[cˆj(σ, η, 0)cˆj+(-σ,-η, 0)]+cˆj+(-σ,-η, 0)cˆj(σ, η, 0)|0=0|[cˆj(σ, η, 0)cˆj+(-σ,-η, 0)]0,
Sj(θ)(σ, η, Z)=k=14|tj,k(σ, η, Z)|2+14π Re{exp[2i(φj(Z)+θ)]×[tj,1(σ, η, Z)tj,2(σ, η, Z)+tj,3(σ, η, Z)tj,4(σ, η, Z)]}.
Sjmax(σ, η, Z)=k=14|tj,k|2-2|tj,1(σ, η, Z)tj,2(σ, η, Z)+tj,3(σ, η, Z)tj,4(σ, η, Z)|,j=1,3.

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