Abstract

We display the results of the numerical simulations of a set of Langevin equations, which describe the dynamics of a degenerate optical parametric oscillator in the Wigner representation. The scan of the threshold region shows the gradual transformation of a quantum image into a classical roll pattern. An experiment on parametric down- conversion in lithium triborate shows strikingly similar results in both the near and the far field, displaying qualitatively the classical features of quantum images.

© 1997 Optical Society of America

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References

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  1. H. Haken , “ Synergetics - An introduction ” ( Springer-Verlag, Berlin 1977 )
  2. G. Nicolis and I. Prigogine , “ Self Organization in Nonequilibrium Systems ” ( Wiley, New York 1977 )
  3. L.A. Lugiato ed., Special Issue on “ Nonlinear Optical Structures, Pattern, Chaos ”, Chaos Soli-tons and Fractals, August/September 1994
  4. A. Gatti and L.A. Lugiato , “ Quantum Images and Critical Fluctuations in the Optical Parametric Oscillator Below Threshold ”, Phys. Rev. A   52 , 1675 ( 1995 )
    [Crossref] [PubMed]
  5. A. Gatti , H. Wiedemann , L.A. Lugiato , I. Marzoli , G.-L. Oppo , and S.M. Barnett , “ Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold ”, Phys. Rev. A , in press (July 1997 )
    [Crossref]
  6. I. Marzoli , A. Gatti , and L.A. Lugiato , “ Spatial quantum signatures in parametric downconver-sion ”, Phys. Rev. Lett.   78 , 2092 ( 1997 )
    [Crossref]
  7. P.D. Drummond , K.J. McNeil , and D.F. Walls , “ Non-equilibrium transitions in sub/second harmonic generation ”, Optica Acta   27 , 321 ( 1980 );“ Non-equilibrium transitions in sub/second harmonic generation. II. Quantum theory ”, Optica Acta   28 , 211 ( 1981 )
    [Crossref]
  8. G.-L. Oppo , M. Brambilla , and L.A. Lugiato , “ Formation and evolution of rolls pattern in optical parametric oscillators ”, Phys. Rev A   49 , 2028 ( 1994 )
    [Crossref] [PubMed]
  9. L. A. Lugiato and A. Gatti , “ Spatial Structure of a Squeezed vacuum ”, Phys. Rev. Lett.   70 , 3868 ( 1993 )
    [Crossref] [PubMed]
  10. E.M. Nagasako , R.W. Boyd , and G.B. Agarwal , “ Vacuum field induced filamentation in laser beam propagation ”, Phys. Rev. A   55 , 1412 ( 1997 )
    [Crossref]
  11. S.X. Dou , D. Josse , and J. Zyss , “ Comparison of collinear and one-beam noncollinear phase matching in optical parametric amplification ”, J. Opt. Soc. Am B   9 , 1312 ( 1992 )
    [Crossref]
  12. P. Di Trapani , A. Andreoni , G. P. Banfi , C. Solcia , R. Danielius , P. Foggi , M. Monguzzi , A. Piskarskas , and C. Sozzi , “ Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation ”, Phys. Rev. A   51 , 3164 ( 1995 )
    [Crossref] [PubMed]

1997 (3)

A. Gatti , H. Wiedemann , L.A. Lugiato , I. Marzoli , G.-L. Oppo , and S.M. Barnett , “ Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold ”, Phys. Rev. A , in press (July 1997 )
[Crossref]

I. Marzoli , A. Gatti , and L.A. Lugiato , “ Spatial quantum signatures in parametric downconver-sion ”, Phys. Rev. Lett.   78 , 2092 ( 1997 )
[Crossref]

E.M. Nagasako , R.W. Boyd , and G.B. Agarwal , “ Vacuum field induced filamentation in laser beam propagation ”, Phys. Rev. A   55 , 1412 ( 1997 )
[Crossref]

1995 (2)

A. Gatti and L.A. Lugiato , “ Quantum Images and Critical Fluctuations in the Optical Parametric Oscillator Below Threshold ”, Phys. Rev. A   52 , 1675 ( 1995 )
[Crossref] [PubMed]

P. Di Trapani , A. Andreoni , G. P. Banfi , C. Solcia , R. Danielius , P. Foggi , M. Monguzzi , A. Piskarskas , and C. Sozzi , “ Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation ”, Phys. Rev. A   51 , 3164 ( 1995 )
[Crossref] [PubMed]

1994 (1)

G.-L. Oppo , M. Brambilla , and L.A. Lugiato , “ Formation and evolution of rolls pattern in optical parametric oscillators ”, Phys. Rev A   49 , 2028 ( 1994 )
[Crossref] [PubMed]

1993 (1)

L. A. Lugiato and A. Gatti , “ Spatial Structure of a Squeezed vacuum ”, Phys. Rev. Lett.   70 , 3868 ( 1993 )
[Crossref] [PubMed]

1992 (1)

S.X. Dou , D. Josse , and J. Zyss , “ Comparison of collinear and one-beam noncollinear phase matching in optical parametric amplification ”, J. Opt. Soc. Am B   9 , 1312 ( 1992 )
[Crossref]

1980 (1)

P.D. Drummond , K.J. McNeil , and D.F. Walls , “ Non-equilibrium transitions in sub/second harmonic generation ”, Optica Acta   27 , 321 ( 1980 );“ Non-equilibrium transitions in sub/second harmonic generation. II. Quantum theory ”, Optica Acta   28 , 211 ( 1981 )
[Crossref]

Agarwal, G.B.

E.M. Nagasako , R.W. Boyd , and G.B. Agarwal , “ Vacuum field induced filamentation in laser beam propagation ”, Phys. Rev. A   55 , 1412 ( 1997 )
[Crossref]

Andreoni, A.

P. Di Trapani , A. Andreoni , G. P. Banfi , C. Solcia , R. Danielius , P. Foggi , M. Monguzzi , A. Piskarskas , and C. Sozzi , “ Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation ”, Phys. Rev. A   51 , 3164 ( 1995 )
[Crossref] [PubMed]

Banfi, G. P.

P. Di Trapani , A. Andreoni , G. P. Banfi , C. Solcia , R. Danielius , P. Foggi , M. Monguzzi , A. Piskarskas , and C. Sozzi , “ Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation ”, Phys. Rev. A   51 , 3164 ( 1995 )
[Crossref] [PubMed]

Barnett, S.M.

A. Gatti , H. Wiedemann , L.A. Lugiato , I. Marzoli , G.-L. Oppo , and S.M. Barnett , “ Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold ”, Phys. Rev. A , in press (July 1997 )
[Crossref]

Boyd, R.W.

E.M. Nagasako , R.W. Boyd , and G.B. Agarwal , “ Vacuum field induced filamentation in laser beam propagation ”, Phys. Rev. A   55 , 1412 ( 1997 )
[Crossref]

Brambilla, M.

G.-L. Oppo , M. Brambilla , and L.A. Lugiato , “ Formation and evolution of rolls pattern in optical parametric oscillators ”, Phys. Rev A   49 , 2028 ( 1994 )
[Crossref] [PubMed]

Danielius, R.

P. Di Trapani , A. Andreoni , G. P. Banfi , C. Solcia , R. Danielius , P. Foggi , M. Monguzzi , A. Piskarskas , and C. Sozzi , “ Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation ”, Phys. Rev. A   51 , 3164 ( 1995 )
[Crossref] [PubMed]

Dou, S.X.

S.X. Dou , D. Josse , and J. Zyss , “ Comparison of collinear and one-beam noncollinear phase matching in optical parametric amplification ”, J. Opt. Soc. Am B   9 , 1312 ( 1992 )
[Crossref]

Drummond, P.D.

P.D. Drummond , K.J. McNeil , and D.F. Walls , “ Non-equilibrium transitions in sub/second harmonic generation ”, Optica Acta   27 , 321 ( 1980 );“ Non-equilibrium transitions in sub/second harmonic generation. II. Quantum theory ”, Optica Acta   28 , 211 ( 1981 )
[Crossref]

Foggi, P.

P. Di Trapani , A. Andreoni , G. P. Banfi , C. Solcia , R. Danielius , P. Foggi , M. Monguzzi , A. Piskarskas , and C. Sozzi , “ Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation ”, Phys. Rev. A   51 , 3164 ( 1995 )
[Crossref] [PubMed]

Gatti, A.

I. Marzoli , A. Gatti , and L.A. Lugiato , “ Spatial quantum signatures in parametric downconver-sion ”, Phys. Rev. Lett.   78 , 2092 ( 1997 )
[Crossref]

A. Gatti , H. Wiedemann , L.A. Lugiato , I. Marzoli , G.-L. Oppo , and S.M. Barnett , “ Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold ”, Phys. Rev. A , in press (July 1997 )
[Crossref]

A. Gatti and L.A. Lugiato , “ Quantum Images and Critical Fluctuations in the Optical Parametric Oscillator Below Threshold ”, Phys. Rev. A   52 , 1675 ( 1995 )
[Crossref] [PubMed]

L. A. Lugiato and A. Gatti , “ Spatial Structure of a Squeezed vacuum ”, Phys. Rev. Lett.   70 , 3868 ( 1993 )
[Crossref] [PubMed]

Haken, H.

H. Haken , “ Synergetics - An introduction ” ( Springer-Verlag, Berlin 1977 )

Josse, D.

S.X. Dou , D. Josse , and J. Zyss , “ Comparison of collinear and one-beam noncollinear phase matching in optical parametric amplification ”, J. Opt. Soc. Am B   9 , 1312 ( 1992 )
[Crossref]

Lugiato, L. A.

L. A. Lugiato and A. Gatti , “ Spatial Structure of a Squeezed vacuum ”, Phys. Rev. Lett.   70 , 3868 ( 1993 )
[Crossref] [PubMed]

Lugiato, L.A.

A. Gatti , H. Wiedemann , L.A. Lugiato , I. Marzoli , G.-L. Oppo , and S.M. Barnett , “ Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold ”, Phys. Rev. A , in press (July 1997 )
[Crossref]

I. Marzoli , A. Gatti , and L.A. Lugiato , “ Spatial quantum signatures in parametric downconver-sion ”, Phys. Rev. Lett.   78 , 2092 ( 1997 )
[Crossref]

A. Gatti and L.A. Lugiato , “ Quantum Images and Critical Fluctuations in the Optical Parametric Oscillator Below Threshold ”, Phys. Rev. A   52 , 1675 ( 1995 )
[Crossref] [PubMed]

G.-L. Oppo , M. Brambilla , and L.A. Lugiato , “ Formation and evolution of rolls pattern in optical parametric oscillators ”, Phys. Rev A   49 , 2028 ( 1994 )
[Crossref] [PubMed]

Marzoli, I.

I. Marzoli , A. Gatti , and L.A. Lugiato , “ Spatial quantum signatures in parametric downconver-sion ”, Phys. Rev. Lett.   78 , 2092 ( 1997 )
[Crossref]

A. Gatti , H. Wiedemann , L.A. Lugiato , I. Marzoli , G.-L. Oppo , and S.M. Barnett , “ Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold ”, Phys. Rev. A , in press (July 1997 )
[Crossref]

McNeil, K.J.

P.D. Drummond , K.J. McNeil , and D.F. Walls , “ Non-equilibrium transitions in sub/second harmonic generation ”, Optica Acta   27 , 321 ( 1980 );“ Non-equilibrium transitions in sub/second harmonic generation. II. Quantum theory ”, Optica Acta   28 , 211 ( 1981 )
[Crossref]

Monguzzi, M.

P. Di Trapani , A. Andreoni , G. P. Banfi , C. Solcia , R. Danielius , P. Foggi , M. Monguzzi , A. Piskarskas , and C. Sozzi , “ Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation ”, Phys. Rev. A   51 , 3164 ( 1995 )
[Crossref] [PubMed]

Nagasako, E.M.

E.M. Nagasako , R.W. Boyd , and G.B. Agarwal , “ Vacuum field induced filamentation in laser beam propagation ”, Phys. Rev. A   55 , 1412 ( 1997 )
[Crossref]

Nicolis, G.

G. Nicolis and I. Prigogine , “ Self Organization in Nonequilibrium Systems ” ( Wiley, New York 1977 )

Oppo, G.-L.

A. Gatti , H. Wiedemann , L.A. Lugiato , I. Marzoli , G.-L. Oppo , and S.M. Barnett , “ Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold ”, Phys. Rev. A , in press (July 1997 )
[Crossref]

G.-L. Oppo , M. Brambilla , and L.A. Lugiato , “ Formation and evolution of rolls pattern in optical parametric oscillators ”, Phys. Rev A   49 , 2028 ( 1994 )
[Crossref] [PubMed]

Piskarskas, A.

P. Di Trapani , A. Andreoni , G. P. Banfi , C. Solcia , R. Danielius , P. Foggi , M. Monguzzi , A. Piskarskas , and C. Sozzi , “ Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation ”, Phys. Rev. A   51 , 3164 ( 1995 )
[Crossref] [PubMed]

Prigogine, I.

G. Nicolis and I. Prigogine , “ Self Organization in Nonequilibrium Systems ” ( Wiley, New York 1977 )

Solcia, C.

P. Di Trapani , A. Andreoni , G. P. Banfi , C. Solcia , R. Danielius , P. Foggi , M. Monguzzi , A. Piskarskas , and C. Sozzi , “ Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation ”, Phys. Rev. A   51 , 3164 ( 1995 )
[Crossref] [PubMed]

Sozzi, C.

P. Di Trapani , A. Andreoni , G. P. Banfi , C. Solcia , R. Danielius , P. Foggi , M. Monguzzi , A. Piskarskas , and C. Sozzi , “ Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation ”, Phys. Rev. A   51 , 3164 ( 1995 )
[Crossref] [PubMed]

Trapani, P. Di

P. Di Trapani , A. Andreoni , G. P. Banfi , C. Solcia , R. Danielius , P. Foggi , M. Monguzzi , A. Piskarskas , and C. Sozzi , “ Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation ”, Phys. Rev. A   51 , 3164 ( 1995 )
[Crossref] [PubMed]

Walls, D.F.

P.D. Drummond , K.J. McNeil , and D.F. Walls , “ Non-equilibrium transitions in sub/second harmonic generation ”, Optica Acta   27 , 321 ( 1980 );“ Non-equilibrium transitions in sub/second harmonic generation. II. Quantum theory ”, Optica Acta   28 , 211 ( 1981 )
[Crossref]

Wiedemann, H.

A. Gatti , H. Wiedemann , L.A. Lugiato , I. Marzoli , G.-L. Oppo , and S.M. Barnett , “ Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold ”, Phys. Rev. A , in press (July 1997 )
[Crossref]

Zyss, J.

S.X. Dou , D. Josse , and J. Zyss , “ Comparison of collinear and one-beam noncollinear phase matching in optical parametric amplification ”, J. Opt. Soc. Am B   9 , 1312 ( 1992 )
[Crossref]

J. Opt. Soc. Am B (1)

S.X. Dou , D. Josse , and J. Zyss , “ Comparison of collinear and one-beam noncollinear phase matching in optical parametric amplification ”, J. Opt. Soc. Am B   9 , 1312 ( 1992 )
[Crossref]

Optica Acta (1)

P.D. Drummond , K.J. McNeil , and D.F. Walls , “ Non-equilibrium transitions in sub/second harmonic generation ”, Optica Acta   27 , 321 ( 1980 );“ Non-equilibrium transitions in sub/second harmonic generation. II. Quantum theory ”, Optica Acta   28 , 211 ( 1981 )
[Crossref]

Phys. Rev A (1)

G.-L. Oppo , M. Brambilla , and L.A. Lugiato , “ Formation and evolution of rolls pattern in optical parametric oscillators ”, Phys. Rev A   49 , 2028 ( 1994 )
[Crossref] [PubMed]

Phys. Rev. A (4)

P. Di Trapani , A. Andreoni , G. P. Banfi , C. Solcia , R. Danielius , P. Foggi , M. Monguzzi , A. Piskarskas , and C. Sozzi , “ Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation ”, Phys. Rev. A   51 , 3164 ( 1995 )
[Crossref] [PubMed]

A. Gatti and L.A. Lugiato , “ Quantum Images and Critical Fluctuations in the Optical Parametric Oscillator Below Threshold ”, Phys. Rev. A   52 , 1675 ( 1995 )
[Crossref] [PubMed]

A. Gatti , H. Wiedemann , L.A. Lugiato , I. Marzoli , G.-L. Oppo , and S.M. Barnett , “ Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold ”, Phys. Rev. A , in press (July 1997 )
[Crossref]

E.M. Nagasako , R.W. Boyd , and G.B. Agarwal , “ Vacuum field induced filamentation in laser beam propagation ”, Phys. Rev. A   55 , 1412 ( 1997 )
[Crossref]

Phys. Rev. Lett. (2)

I. Marzoli , A. Gatti , and L.A. Lugiato , “ Spatial quantum signatures in parametric downconver-sion ”, Phys. Rev. Lett.   78 , 2092 ( 1997 )
[Crossref]

L. A. Lugiato and A. Gatti , “ Spatial Structure of a Squeezed vacuum ”, Phys. Rev. Lett.   70 , 3868 ( 1993 )
[Crossref] [PubMed]

Other (3)

H. Haken , “ Synergetics - An introduction ” ( Springer-Verlag, Berlin 1977 )

G. Nicolis and I. Prigogine , “ Self Organization in Nonequilibrium Systems ” ( Wiley, New York 1977 )

L.A. Lugiato ed., Special Issue on “ Nonlinear Optical Structures, Pattern, Chaos ”, Chaos Soli-tons and Fractals, August/September 1994

Supplementary Material (1)

» Media 1: MOV (459 KB)     

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Figures (5)

Fig. 1
Fig. 1

Scheme of the OPO cavity. The mirror M1 has a high reflectivity, M2 is completely reflecting. E in is the input field of frequency 2ωs .

Fig. 2
Fig. 2

Roll pattern. a)Near field distribution of the most amplified quadrature; b) far field intensity.

Fig. 3
Fig. 3

Click in the space to start the Quickmovie of the scan of the threshold region of the OPO obtained by varying the input field amplitude as shown in the bottom part of the animation. Parameters are γ10 = 1, Δ0 = 1, Δ1 = -1, nth = 200. Threshold value of the input field is E = 2. [Media 1]

Fig. 4
Fig. 4

Schematic lay out of the experimental set up. Parametric emission is accomplished by means of single pump pulses travelling in the LBO crystal. Signal and idler waves are preferentially generated along two cones, whose apertures fulfill the phase matching requirement. Far-field and near-field intensity distributions of the signal are detected by means of CCD camera and suitable objective optics.

Fig. 5
Fig. 5

a) Single-shot far-field intensity profile of the signal field. λs = 0.8μm, Δλ S = 80nm Crystal oriented for θ = 87 o , Φ = 11 .6 o b) Corresponding near-field intensity distribution.

Equations (10)

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

t A 0 ( x , t ) = γ 0 [ ( 1 + i Δ 0 i γ 1 2 γ 0 2 ) A 0 ( x , t ) + E A 1 2 ( x , t )
+ 2 γ 0 n th ξ 0 ( x , t ) ] ,
t A 1 ( x , t ) = γ 1 [ ( 1 + i Δ 1 i 2 ) A 1 ( x , t ) + A 0 ( x , t ) A 1 * ( x , t )
+ 1 γ 0 n th ξ 1 ( x , t ) ] ,
Δ 0 = ω 0 2 ω s γ 0 Δ 1 = ω 1 ω s γ 1 ,
2 = 2 x 2 + 2 y 2
l d = ( c 2 2 ω s γ 1 ) 1 / 2
ξ 1 * ( x , y ) ξ 1 ( x , t′ ) = ξ 0 * ( x , y ) ξ 0 ( x , t′ ) = 1 2 δ ( x x′ ) δ ( t t′ ) ,
n th = γ 1 2 g 2 l d 2 ,
E th = 1 + Δ 0 2

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