Abstract

We compute the eigenmodes of a spatially-broadband optical parametric amplifier with elliptical Gaussian pump and show that the well-amplified eigenmodes can be compactly represented by a low-dimensional subspace of the first few Laguerre- or Hermite-Gaussian (LG or HG, respectively) modes of an appropriate waist size. We also show that the first few eigenmodes are well matched to single LG or HG modes. For sufficiently large pump waists, the optimum waist size of the compact basis is in the vicinity of the geometric average of the pump waist size and the inverse spatial bandwidth of the nonlinear crystal in the parametric amplifier. The use of such compact representation can greatly simplify numerical computation of the spatial eigenmodes of the amplifier and thus lead to improving the experiments on traveling-wave image amplification and spatially-broadband vacuum squeezing.

© 2013 Optical Society of America

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    [CrossRef]
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  3. D. Levandovsky, M. Vasilyev, and P. Kumar, “Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” Pramana56, 281–285 (2001).
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    [CrossRef]
  5. P. L. Voss, K. G. Köprülü, and P. Kumar, “Raman-noise-induced quantum limits for χ(3) nondegenerate phase-sensitive amplification and quadrature squeezing,” J. Opt. Soc. Am. B23(4), 598–610 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  34. E. Lantz, N. Treps, C. Fabre, and E. Brambilla, “Spatial distribution of quantum fluctuations in spontaneous down-conversion in realistic situations: comparison between the stochastic approach and the Green’s function method,” Eur. Phys. J. D29(3), 437–444 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  38. K. G. Köprülü and O. Aytür, “Analysis of Gaussian-beam degenerate optical parametric amplifiers for the generation of quadrature-squeezed states,” Phys. Rev. A60(5), 4122–4134 (1999).
    [CrossRef]
  39. K. G. Köprülü and O. Aytür, “Analysis of the generation of amplitude-squeezed light with Gaussian-beam degenerate optical parametric amplifiers,” J. Opt. Soc. Am. B18(6), 846–854 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  44. D. Levandovsky, M. Vasilyev, and P. Kumar, “Perturbation theory of quantum solitons: continuum evolution and optimum squeezing by spectral filtering,” Opt. Lett.24(1), 43–45 (1999).
    [CrossRef] [PubMed]
  45. D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive nonlinear optical loop mirror,” Opt. Lett.24, 89–91 (1999).
  46. D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive nonlinear optical loop mirror: errata,” Opt. Lett.24, 423 (1999).
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    [CrossRef]
  48. A. Christ, K. Laiho, A. Eckstein, K. N. Cassemiro, and C. Silberhorn, “Probing multimode squeezing with correlation functions,” New J. Phys.13(3), 033027 (2011).
    [CrossRef]
  49. G. Alon, O.-K. Lim, A. Bhagwat, C.-H. Chen, M. Annamalai, M. Vasilyev, and P. Kumar, “Optimization of gain in traveling-wave optical parametric amplifiers by tuning the offset between pump- and signal-waist locations,” Opt. Lett.38(8), 1268–1270 (2013).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2013 (2)

2012 (3)

M. Annamalai and M. Vasilyev, “Phase-sensitive multimode parametric amplification in a parabolic-index waveguide,” IEEE Photonics Technol. Lett.24(21), 1949–1952 (2012).
[CrossRef]

M. Vasilyev and P. Kumar, “Frequency up-conversion of quantum images,” Opt. Express20(6), 6644–6656 (2012).
[CrossRef] [PubMed]

S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun.3, 1026 (2012).
[CrossRef] [PubMed]

2011 (4)

2010 (5)

G. Patera, N. Treps, C. Fabre, and G. J. de Valcárcel, “Quantum theory of synchronously pumped type I optical parametric oscillators: characterization of the squeezed supermodes,” Eur. Phys. J. D56(1), 123–140 (2010).
[CrossRef]

Z. Tong, A. Bogris, C. Lundström, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier,” Opt. Express18(14), 14820–14835 (2010).
[CrossRef] [PubMed]

Z. Dutton, J. H. Shapiro, and S. Guha, “LADAR resolution improvement using receivers enhanced with squeezed-vacuum injection and phase-sensitive amplification,” J. Opt. Soc. Am. B27(6), A63–A72 (2010).
[CrossRef]

B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Multimode nonclassical light generation through the optical-parametric-oscillator threshold,” Phys. Rev. A81(6), 061804 (2010).
[CrossRef]

M. Vasilyev, M. Annamalai, N. Stelmakh, and P. Kumar, “Quantum properties of a spatially-broadband traveling-wave phase-sensitive optical parametric amplifier,” J. Mod. Opt.57(19), 1908–1915 (2010).
[CrossRef]

2009 (6)

P. J. Mosley, A. Christ, A. Eckstein, and C. Silberhorn, “Direct measurement of the spatial-spectral structure of waveguided parametric down-conversion,” Phys. Rev. Lett.103(23), 233901 (2009).
[CrossRef] [PubMed]

M. Vasilyev, N. Stelmakh, and P. Kumar, “Estimation of the spatial bandwidth of an optical parametric amplifier with plane-wave pump,” J. Mod. Opt.56(18-19), 2029–2033 (2009).
[CrossRef]

M. Vasilyev, N. Stelmakh, and P. Kumar, “Phase-sensitive image amplification with elliptical Gaussian pump,” Opt. Express17(14), 11415–11425 (2009).
[CrossRef] [PubMed]

C. Navarrete-Benlloch, G. J. de Valcárcel, and E. Roldán, “Generating highly squeezed hybrid Laguerre-Gauss modes in large-Fresnel-number degenerate optical parametric oscillators,” Phys. Rev. A79(4), 043820 (2009).
[CrossRef]

L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A80(4), 043816 (2009).
[CrossRef]

C. J. McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun.282(4), 583–593 (2009).
[CrossRef]

2008 (2)

E. Lantz and F. Devaux, “Parametric amplification of images: from time gating to noiseless amplification,” J. Sel. Top. Quantum Electron.14(3), 635–647 (2008).
[CrossRef]

L. Lopez, N. Treps, B. Chalopin, C. Fabre, and A. Maître, “Quantum processing of images by continuous wave optical parametric amplification,” Phys. Rev. Lett.100(1), 013604 (2008).
[CrossRef] [PubMed]

2007 (2)

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett.98(8), 083602 (2007).
[CrossRef] [PubMed]

V. Delaubert, M. Lassen, D. R. N. Pulford, H.-A. Bachor, and C. C. Harb, “Spatial mode discrimination using second harmonic generation,” Opt. Express15(9), 5815–5826 (2007).
[CrossRef] [PubMed]

2006 (2)

P. L. Voss, K. G. Köprülü, and P. Kumar, “Raman-noise-induced quantum limits for χ(3) nondegenerate phase-sensitive amplification and quadrature squeezing,” J. Opt. Soc. Am. B23(4), 598–610 (2006).
[CrossRef]

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A73(6), 063819 (2006).
[CrossRef]

2005 (1)

A. Mosset, F. Devaux, and E. Lantz, “Spatially noiseless optical amplification of images,” Phys. Rev. Lett.94(22), 223603 (2005).
[CrossRef] [PubMed]

2004 (2)

E. Brambilla, A. Gatti, M. Bache, and L. A. Lugiato, “Simultaneous near-field and far-field spatial quantum correlations in the high-gain regime of parametric down-conversion,” Phys. Rev. A69(2), 023802 (2004).
[CrossRef]

E. Lantz, N. Treps, C. Fabre, and E. Brambilla, “Spatial distribution of quantum fluctuations in spontaneous down-conversion in realistic situations: comparison between the stochastic approach and the Green’s function method,” Eur. Phys. J. D29(3), 437–444 (2004).
[CrossRef]

2003 (1)

K. Wang, G. Yang, A. Gatti, and L. Lugiato, “Controlling the signal-to-noise ratio in optical parametric image amplification,” J. Opt. B Quantum Semiclassical Opt.5(4), S535–S544 (2003).
[CrossRef]

2001 (2)

D. Levandovsky, M. Vasilyev, and P. Kumar, “Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” Pramana56, 281–285 (2001).

K. G. Köprülü and O. Aytür, “Analysis of the generation of amplitude-squeezed light with Gaussian-beam degenerate optical parametric amplifiers,” J. Opt. Soc. Am. B18(6), 846–854 (2001).
[CrossRef]

2000 (1)

S.-K. Choi, M. Vasilyev, and P. Kumar, “Erratum: Noiseless optical amplification of images,” Phys. Rev. Lett.84, 1361 (2000).

1999 (8)

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett.83, 1938–1941 (1999).

W. Imajuku, A. Takada, and Y. Yamabayashi, “Low-noise amplification under the 3 dB noise figure in high-gain phase-sensitive fibre amplifier,” Electron. Lett.35(22), 1954–1955 (1999).
[CrossRef]

D. Levandovsky, M. Vasilyev, and P. Kumar, “Amplitude squeezing of light by means of a phase-sensitive fiber parametric amplifier,” Opt. Lett.24(14), 984–986 (1999).
[CrossRef] [PubMed]

M. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys.71(5), 1539–1589 (1999).
[CrossRef]

K. G. Köprülü and O. Aytür, “Analysis of Gaussian-beam degenerate optical parametric amplifiers for the generation of quadrature-squeezed states,” Phys. Rev. A60(5), 4122–4134 (1999).
[CrossRef]

D. Levandovsky, M. Vasilyev, and P. Kumar, “Perturbation theory of quantum solitons: continuum evolution and optimum squeezing by spectral filtering,” Opt. Lett.24(1), 43–45 (1999).
[CrossRef] [PubMed]

D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive nonlinear optical loop mirror,” Opt. Lett.24, 89–91 (1999).

D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive nonlinear optical loop mirror: errata,” Opt. Lett.24, 423 (1999).

1998 (1)

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B66(6), 685–699 (1998).
[CrossRef]

1997 (1)

1995 (2)

R.-D. Li, S.-K. Choi, C. Kim, and P. Kumar, “Generation of sub-Poissonian pulses of light,” Phys. Rev. A51(5), R3429–R3432 (1995).
[CrossRef] [PubMed]

M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A52(6), 4930–4940 (1995).
[CrossRef] [PubMed]

1994 (1)

C. Kim and P. Kumar, “Quadrature-squeezed light detection using a self-generated matched local oscillator,” Phys. Rev. Lett.73(12), 1605–1608 (1994).
[CrossRef] [PubMed]

1991 (1)

A. La Porta and R. E. Slusher, “Squeezing limits at high parametric gains,” Phys. Rev. A44(3), 2013–2022 (1991).
[CrossRef] [PubMed]

1982 (1)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields26(8), 1817–1839 (1982).
[CrossRef]

1968 (1)

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys.39(8), 3597–3639 (1968).
[CrossRef]

Alon, G.

Andrekson, P. A.

Annamalai, M.

Armstrong, S.

S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun.3, 1026 (2012).
[CrossRef] [PubMed]

Aytür, O.

K. G. Köprülü and O. Aytür, “Analysis of the generation of amplitude-squeezed light with Gaussian-beam degenerate optical parametric amplifiers,” J. Opt. Soc. Am. B18(6), 846–854 (2001).
[CrossRef]

K. G. Köprülü and O. Aytür, “Analysis of Gaussian-beam degenerate optical parametric amplifiers for the generation of quadrature-squeezed states,” Phys. Rev. A60(5), 4122–4134 (1999).
[CrossRef]

Bache, M.

E. Brambilla, A. Gatti, M. Bache, and L. A. Lugiato, “Simultaneous near-field and far-field spatial quantum correlations in the high-gain regime of parametric down-conversion,” Phys. Rev. A69(2), 023802 (2004).
[CrossRef]

Bachor, H.-A.

S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun.3, 1026 (2012).
[CrossRef] [PubMed]

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett.98(8), 083602 (2007).
[CrossRef] [PubMed]

V. Delaubert, M. Lassen, D. R. N. Pulford, H.-A. Bachor, and C. C. Harb, “Spatial mode discrimination using second harmonic generation,” Opt. Express15(9), 5815–5826 (2007).
[CrossRef] [PubMed]

Banaszek, K.

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A73(6), 063819 (2006).
[CrossRef]

Bhagwat, A.

Bhagwat, A. R.

Bogris, A.

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys.39(8), 3597–3639 (1968).
[CrossRef]

Brambilla, E.

E. Brambilla, A. Gatti, M. Bache, and L. A. Lugiato, “Simultaneous near-field and far-field spatial quantum correlations in the high-gain regime of parametric down-conversion,” Phys. Rev. A69(2), 023802 (2004).
[CrossRef]

E. Lantz, N. Treps, C. Fabre, and E. Brambilla, “Spatial distribution of quantum fluctuations in spontaneous down-conversion in realistic situations: comparison between the stochastic approach and the Green’s function method,” Eur. Phys. J. D29(3), 437–444 (2004).
[CrossRef]

Buchhave, P.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett.98(8), 083602 (2007).
[CrossRef] [PubMed]

Cassemiro, K. N.

A. Christ, K. Laiho, A. Eckstein, K. N. Cassemiro, and C. Silberhorn, “Probing multimode squeezing with correlation functions,” New J. Phys.13(3), 033027 (2011).
[CrossRef]

Caves, C. M.

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields26(8), 1817–1839 (1982).
[CrossRef]

Chalopin, B.

B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Direct generation of a multi-transverse mode non-classical state of light,” Opt. Express19(5), 4405–4410 (2011).
[CrossRef] [PubMed]

B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Multimode nonclassical light generation through the optical-parametric-oscillator threshold,” Phys. Rev. A81(6), 061804 (2010).
[CrossRef]

L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A80(4), 043816 (2009).
[CrossRef]

L. Lopez, N. Treps, B. Chalopin, C. Fabre, and A. Maître, “Quantum processing of images by continuous wave optical parametric amplification,” Phys. Rev. Lett.100(1), 013604 (2008).
[CrossRef] [PubMed]

Chen, C.-H.

Choi, S.-K.

S.-K. Choi, M. Vasilyev, and P. Kumar, “Erratum: Noiseless optical amplification of images,” Phys. Rev. Lett.84, 1361 (2000).

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett.83, 1938–1941 (1999).

S.-K. Choi, R.-D. Li, C. Kim, and P. Kumar, “Traveling-wave optical parametric amplifier: investigation of its phase-sensitive and phase-insensitive gain response,” J. Opt. Soc. Am. B14(7), 1564–1575 (1997).
[CrossRef]

R.-D. Li, S.-K. Choi, C. Kim, and P. Kumar, “Generation of sub-Poissonian pulses of light,” Phys. Rev. A51(5), R3429–R3432 (1995).
[CrossRef] [PubMed]

Christ, A.

A. Christ, K. Laiho, A. Eckstein, K. N. Cassemiro, and C. Silberhorn, “Probing multimode squeezing with correlation functions,” New J. Phys.13(3), 033027 (2011).
[CrossRef]

P. J. Mosley, A. Christ, A. Eckstein, and C. Silberhorn, “Direct measurement of the spatial-spectral structure of waveguided parametric down-conversion,” Phys. Rev. Lett.103(23), 233901 (2009).
[CrossRef] [PubMed]

Cohadon, P. F.

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B66(6), 685–699 (1998).
[CrossRef]

de Valcárcel, G. J.

G. Patera, N. Treps, C. Fabre, and G. J. de Valcárcel, “Quantum theory of synchronously pumped type I optical parametric oscillators: characterization of the squeezed supermodes,” Eur. Phys. J. D56(1), 123–140 (2010).
[CrossRef]

C. Navarrete-Benlloch, G. J. de Valcárcel, and E. Roldán, “Generating highly squeezed hybrid Laguerre-Gauss modes in large-Fresnel-number degenerate optical parametric oscillators,” Phys. Rev. A79(4), 043820 (2009).
[CrossRef]

Delaubert, V.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett.98(8), 083602 (2007).
[CrossRef] [PubMed]

V. Delaubert, M. Lassen, D. R. N. Pulford, H.-A. Bachor, and C. C. Harb, “Spatial mode discrimination using second harmonic generation,” Opt. Express15(9), 5815–5826 (2007).
[CrossRef] [PubMed]

Devaux, F.

E. Lantz and F. Devaux, “Parametric amplification of images: from time gating to noiseless amplification,” J. Sel. Top. Quantum Electron.14(3), 635–647 (2008).
[CrossRef]

A. Mosset, F. Devaux, and E. Lantz, “Spatially noiseless optical amplification of images,” Phys. Rev. Lett.94(22), 223603 (2005).
[CrossRef] [PubMed]

Dutton, Z.

Eckstein, A.

A. Christ, K. Laiho, A. Eckstein, K. N. Cassemiro, and C. Silberhorn, “Probing multimode squeezing with correlation functions,” New J. Phys.13(3), 033027 (2011).
[CrossRef]

P. J. Mosley, A. Christ, A. Eckstein, and C. Silberhorn, “Direct measurement of the spatial-spectral structure of waveguided parametric down-conversion,” Phys. Rev. Lett.103(23), 233901 (2009).
[CrossRef] [PubMed]

Fabre, C.

B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Direct generation of a multi-transverse mode non-classical state of light,” Opt. Express19(5), 4405–4410 (2011).
[CrossRef] [PubMed]

B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Multimode nonclassical light generation through the optical-parametric-oscillator threshold,” Phys. Rev. A81(6), 061804 (2010).
[CrossRef]

G. Patera, N. Treps, C. Fabre, and G. J. de Valcárcel, “Quantum theory of synchronously pumped type I optical parametric oscillators: characterization of the squeezed supermodes,” Eur. Phys. J. D56(1), 123–140 (2010).
[CrossRef]

L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A80(4), 043816 (2009).
[CrossRef]

L. Lopez, N. Treps, B. Chalopin, C. Fabre, and A. Maître, “Quantum processing of images by continuous wave optical parametric amplification,” Phys. Rev. Lett.100(1), 013604 (2008).
[CrossRef] [PubMed]

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett.98(8), 083602 (2007).
[CrossRef] [PubMed]

E. Lantz, N. Treps, C. Fabre, and E. Brambilla, “Spatial distribution of quantum fluctuations in spontaneous down-conversion in realistic situations: comparison between the stochastic approach and the Green’s function method,” Eur. Phys. J. D29(3), 437–444 (2004).
[CrossRef]

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B66(6), 685–699 (1998).
[CrossRef]

Gatti, A.

E. Brambilla, A. Gatti, M. Bache, and L. A. Lugiato, “Simultaneous near-field and far-field spatial quantum correlations in the high-gain regime of parametric down-conversion,” Phys. Rev. A69(2), 023802 (2004).
[CrossRef]

K. Wang, G. Yang, A. Gatti, and L. Lugiato, “Controlling the signal-to-noise ratio in optical parametric image amplification,” J. Opt. B Quantum Semiclassical Opt.5(4), S535–S544 (2003).
[CrossRef]

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B66(6), 685–699 (1998).
[CrossRef]

Guha, S.

Hage, B.

S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun.3, 1026 (2012).
[CrossRef] [PubMed]

Harb, C. C.

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett.98(8), 083602 (2007).
[CrossRef] [PubMed]

V. Delaubert, M. Lassen, D. R. N. Pulford, H.-A. Bachor, and C. C. Harb, “Spatial mode discrimination using second harmonic generation,” Opt. Express15(9), 5815–5826 (2007).
[CrossRef] [PubMed]

Imajuku, W.

W. Imajuku, A. Takada, and Y. Yamabayashi, “Low-noise amplification under the 3 dB noise figure in high-gain phase-sensitive fibre amplifier,” Electron. Lett.35(22), 1954–1955 (1999).
[CrossRef]

Janousek, J.

S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun.3, 1026 (2012).
[CrossRef] [PubMed]

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett.98(8), 083602 (2007).
[CrossRef] [PubMed]

Karlsson, M.

Kim, C.

S.-K. Choi, R.-D. Li, C. Kim, and P. Kumar, “Traveling-wave optical parametric amplifier: investigation of its phase-sensitive and phase-insensitive gain response,” J. Opt. Soc. Am. B14(7), 1564–1575 (1997).
[CrossRef]

R.-D. Li, S.-K. Choi, C. Kim, and P. Kumar, “Generation of sub-Poissonian pulses of light,” Phys. Rev. A51(5), R3429–R3432 (1995).
[CrossRef] [PubMed]

C. Kim and P. Kumar, “Quadrature-squeezed light detection using a self-generated matched local oscillator,” Phys. Rev. Lett.73(12), 1605–1608 (1994).
[CrossRef] [PubMed]

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys.39(8), 3597–3639 (1968).
[CrossRef]

Kolobov, M.

M. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys.71(5), 1539–1589 (1999).
[CrossRef]

Kolobov, M. I.

M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A52(6), 4930–4940 (1995).
[CrossRef] [PubMed]

Köprülü, K. G.

Kumar, P.

G. Alon, O.-K. Lim, A. Bhagwat, C.-H. Chen, M. Annamalai, M. Vasilyev, and P. Kumar, “Optimization of gain in traveling-wave optical parametric amplifiers by tuning the offset between pump- and signal-waist locations,” Opt. Lett.38(8), 1268–1270 (2013).
[CrossRef] [PubMed]

A. R. Bhagwat, G. Alon, O.-K. Lim, C.-H. Chen, M. Annamalai, M. Vasilyev, and P. Kumar, “Fundamental eigenmode of traveling-wave phase-sensitive optical parametric amplifier: experimental generation and verification,” Opt. Lett.38(15), 2858–2860 (2013).
[CrossRef] [PubMed]

M. Vasilyev and P. Kumar, “Frequency up-conversion of quantum images,” Opt. Express20(6), 6644–6656 (2012).
[CrossRef] [PubMed]

M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps,” Opt. Express19(27), 26710–26724 (2011).
[CrossRef] [PubMed]

M. Vasilyev, M. Annamalai, N. Stelmakh, and P. Kumar, “Quantum properties of a spatially-broadband traveling-wave phase-sensitive optical parametric amplifier,” J. Mod. Opt.57(19), 1908–1915 (2010).
[CrossRef]

M. Vasilyev, N. Stelmakh, and P. Kumar, “Phase-sensitive image amplification with elliptical Gaussian pump,” Opt. Express17(14), 11415–11425 (2009).
[CrossRef] [PubMed]

M. Vasilyev, N. Stelmakh, and P. Kumar, “Estimation of the spatial bandwidth of an optical parametric amplifier with plane-wave pump,” J. Mod. Opt.56(18-19), 2029–2033 (2009).
[CrossRef]

P. L. Voss, K. G. Köprülü, and P. Kumar, “Raman-noise-induced quantum limits for χ(3) nondegenerate phase-sensitive amplification and quadrature squeezing,” J. Opt. Soc. Am. B23(4), 598–610 (2006).
[CrossRef]

D. Levandovsky, M. Vasilyev, and P. Kumar, “Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” Pramana56, 281–285 (2001).

S.-K. Choi, M. Vasilyev, and P. Kumar, “Erratum: Noiseless optical amplification of images,” Phys. Rev. Lett.84, 1361 (2000).

D. Levandovsky, M. Vasilyev, and P. Kumar, “Amplitude squeezing of light by means of a phase-sensitive fiber parametric amplifier,” Opt. Lett.24(14), 984–986 (1999).
[CrossRef] [PubMed]

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett.83, 1938–1941 (1999).

D. Levandovsky, M. Vasilyev, and P. Kumar, “Perturbation theory of quantum solitons: continuum evolution and optimum squeezing by spectral filtering,” Opt. Lett.24(1), 43–45 (1999).
[CrossRef] [PubMed]

D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive nonlinear optical loop mirror,” Opt. Lett.24, 89–91 (1999).

D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive nonlinear optical loop mirror: errata,” Opt. Lett.24, 423 (1999).

S.-K. Choi, R.-D. Li, C. Kim, and P. Kumar, “Traveling-wave optical parametric amplifier: investigation of its phase-sensitive and phase-insensitive gain response,” J. Opt. Soc. Am. B14(7), 1564–1575 (1997).
[CrossRef]

R.-D. Li, S.-K. Choi, C. Kim, and P. Kumar, “Generation of sub-Poissonian pulses of light,” Phys. Rev. A51(5), R3429–R3432 (1995).
[CrossRef] [PubMed]

C. Kim and P. Kumar, “Quadrature-squeezed light detection using a self-generated matched local oscillator,” Phys. Rev. Lett.73(12), 1605–1608 (1994).
[CrossRef] [PubMed]

La Porta, A.

A. La Porta and R. E. Slusher, “Squeezing limits at high parametric gains,” Phys. Rev. A44(3), 2013–2022 (1991).
[CrossRef] [PubMed]

Laiho, K.

A. Christ, K. Laiho, A. Eckstein, K. N. Cassemiro, and C. Silberhorn, “Probing multimode squeezing with correlation functions,” New J. Phys.13(3), 033027 (2011).
[CrossRef]

Lam, P. K.

S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun.3, 1026 (2012).
[CrossRef] [PubMed]

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett.98(8), 083602 (2007).
[CrossRef] [PubMed]

Lantz, E.

E. Lantz and F. Devaux, “Parametric amplification of images: from time gating to noiseless amplification,” J. Sel. Top. Quantum Electron.14(3), 635–647 (2008).
[CrossRef]

A. Mosset, F. Devaux, and E. Lantz, “Spatially noiseless optical amplification of images,” Phys. Rev. Lett.94(22), 223603 (2005).
[CrossRef] [PubMed]

E. Lantz, N. Treps, C. Fabre, and E. Brambilla, “Spatial distribution of quantum fluctuations in spontaneous down-conversion in realistic situations: comparison between the stochastic approach and the Green’s function method,” Eur. Phys. J. D29(3), 437–444 (2004).
[CrossRef]

Lassen, M.

V. Delaubert, M. Lassen, D. R. N. Pulford, H.-A. Bachor, and C. C. Harb, “Spatial mode discrimination using second harmonic generation,” Opt. Express15(9), 5815–5826 (2007).
[CrossRef] [PubMed]

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett.98(8), 083602 (2007).
[CrossRef] [PubMed]

Levandovsky, D.

Li, R.-D.

Lim, O.-K.

Lopez, L.

L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A80(4), 043816 (2009).
[CrossRef]

L. Lopez, N. Treps, B. Chalopin, C. Fabre, and A. Maître, “Quantum processing of images by continuous wave optical parametric amplification,” Phys. Rev. Lett.100(1), 013604 (2008).
[CrossRef] [PubMed]

Lugiato, L.

K. Wang, G. Yang, A. Gatti, and L. Lugiato, “Controlling the signal-to-noise ratio in optical parametric image amplification,” J. Opt. B Quantum Semiclassical Opt.5(4), S535–S544 (2003).
[CrossRef]

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B66(6), 685–699 (1998).
[CrossRef]

Lugiato, L. A.

E. Brambilla, A. Gatti, M. Bache, and L. A. Lugiato, “Simultaneous near-field and far-field spatial quantum correlations in the high-gain regime of parametric down-conversion,” Phys. Rev. A69(2), 023802 (2004).
[CrossRef]

M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A52(6), 4930–4940 (1995).
[CrossRef] [PubMed]

Lundström, C.

Lvovsky, A. I.

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A73(6), 063819 (2006).
[CrossRef]

Maître, A.

L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A80(4), 043816 (2009).
[CrossRef]

L. Lopez, N. Treps, B. Chalopin, C. Fabre, and A. Maître, “Quantum processing of images by continuous wave optical parametric amplification,” Phys. Rev. Lett.100(1), 013604 (2008).
[CrossRef] [PubMed]

Marte, M. A. M.

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B66(6), 685–699 (1998).
[CrossRef]

McKinstrie, C. J.

Morizur, J.-F.

S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun.3, 1026 (2012).
[CrossRef] [PubMed]

Mosley, P. J.

P. J. Mosley, A. Christ, A. Eckstein, and C. Silberhorn, “Direct measurement of the spatial-spectral structure of waveguided parametric down-conversion,” Phys. Rev. Lett.103(23), 233901 (2009).
[CrossRef] [PubMed]

Mosset, A.

A. Mosset, F. Devaux, and E. Lantz, “Spatially noiseless optical amplification of images,” Phys. Rev. Lett.94(22), 223603 (2005).
[CrossRef] [PubMed]

Navarrete-Benlloch, C.

C. Navarrete-Benlloch, G. J. de Valcárcel, and E. Roldán, “Generating highly squeezed hybrid Laguerre-Gauss modes in large-Fresnel-number degenerate optical parametric oscillators,” Phys. Rev. A79(4), 043820 (2009).
[CrossRef]

Patera, G.

G. Patera, N. Treps, C. Fabre, and G. J. de Valcárcel, “Quantum theory of synchronously pumped type I optical parametric oscillators: characterization of the squeezed supermodes,” Eur. Phys. J. D56(1), 123–140 (2010).
[CrossRef]

Pulford, D. R. N.

Radzewicz, C.

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A73(6), 063819 (2006).
[CrossRef]

Ritsch, H.

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B66(6), 685–699 (1998).
[CrossRef]

Rivière de la Souchère, A.

L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A80(4), 043816 (2009).
[CrossRef]

Roldán, E.

C. Navarrete-Benlloch, G. J. de Valcárcel, and E. Roldán, “Generating highly squeezed hybrid Laguerre-Gauss modes in large-Fresnel-number degenerate optical parametric oscillators,” Phys. Rev. A79(4), 043820 (2009).
[CrossRef]

Scazza, F.

B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Direct generation of a multi-transverse mode non-classical state of light,” Opt. Express19(5), 4405–4410 (2011).
[CrossRef] [PubMed]

B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Multimode nonclassical light generation through the optical-parametric-oscillator threshold,” Phys. Rev. A81(6), 061804 (2010).
[CrossRef]

Schwob, C.

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B66(6), 685–699 (1998).
[CrossRef]

Shapiro, J. H.

Silberhorn, C.

A. Christ, K. Laiho, A. Eckstein, K. N. Cassemiro, and C. Silberhorn, “Probing multimode squeezing with correlation functions,” New J. Phys.13(3), 033027 (2011).
[CrossRef]

P. J. Mosley, A. Christ, A. Eckstein, and C. Silberhorn, “Direct measurement of the spatial-spectral structure of waveguided parametric down-conversion,” Phys. Rev. Lett.103(23), 233901 (2009).
[CrossRef] [PubMed]

Slusher, R. E.

A. La Porta and R. E. Slusher, “Squeezing limits at high parametric gains,” Phys. Rev. A44(3), 2013–2022 (1991).
[CrossRef] [PubMed]

Stelmakh, N.

M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps,” Opt. Express19(27), 26710–26724 (2011).
[CrossRef] [PubMed]

M. Vasilyev, M. Annamalai, N. Stelmakh, and P. Kumar, “Quantum properties of a spatially-broadband traveling-wave phase-sensitive optical parametric amplifier,” J. Mod. Opt.57(19), 1908–1915 (2010).
[CrossRef]

M. Vasilyev, N. Stelmakh, and P. Kumar, “Estimation of the spatial bandwidth of an optical parametric amplifier with plane-wave pump,” J. Mod. Opt.56(18-19), 2029–2033 (2009).
[CrossRef]

M. Vasilyev, N. Stelmakh, and P. Kumar, “Phase-sensitive image amplification with elliptical Gaussian pump,” Opt. Express17(14), 11415–11425 (2009).
[CrossRef] [PubMed]

Takada, A.

W. Imajuku, A. Takada, and Y. Yamabayashi, “Low-noise amplification under the 3 dB noise figure in high-gain phase-sensitive fibre amplifier,” Electron. Lett.35(22), 1954–1955 (1999).
[CrossRef]

Tong, Z.

Treps, N.

S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun.3, 1026 (2012).
[CrossRef] [PubMed]

B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Direct generation of a multi-transverse mode non-classical state of light,” Opt. Express19(5), 4405–4410 (2011).
[CrossRef] [PubMed]

B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Multimode nonclassical light generation through the optical-parametric-oscillator threshold,” Phys. Rev. A81(6), 061804 (2010).
[CrossRef]

G. Patera, N. Treps, C. Fabre, and G. J. de Valcárcel, “Quantum theory of synchronously pumped type I optical parametric oscillators: characterization of the squeezed supermodes,” Eur. Phys. J. D56(1), 123–140 (2010).
[CrossRef]

L. Lopez, B. Chalopin, A. Rivière de la Souchère, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A80(4), 043816 (2009).
[CrossRef]

L. Lopez, N. Treps, B. Chalopin, C. Fabre, and A. Maître, “Quantum processing of images by continuous wave optical parametric amplification,” Phys. Rev. Lett.100(1), 013604 (2008).
[CrossRef] [PubMed]

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett.98(8), 083602 (2007).
[CrossRef] [PubMed]

E. Lantz, N. Treps, C. Fabre, and E. Brambilla, “Spatial distribution of quantum fluctuations in spontaneous down-conversion in realistic situations: comparison between the stochastic approach and the Green’s function method,” Eur. Phys. J. D29(3), 437–444 (2004).
[CrossRef]

Vasilyev, M.

A. R. Bhagwat, G. Alon, O.-K. Lim, C.-H. Chen, M. Annamalai, M. Vasilyev, and P. Kumar, “Fundamental eigenmode of traveling-wave phase-sensitive optical parametric amplifier: experimental generation and verification,” Opt. Lett.38(15), 2858–2860 (2013).
[CrossRef] [PubMed]

G. Alon, O.-K. Lim, A. Bhagwat, C.-H. Chen, M. Annamalai, M. Vasilyev, and P. Kumar, “Optimization of gain in traveling-wave optical parametric amplifiers by tuning the offset between pump- and signal-waist locations,” Opt. Lett.38(8), 1268–1270 (2013).
[CrossRef] [PubMed]

M. Annamalai and M. Vasilyev, “Phase-sensitive multimode parametric amplification in a parabolic-index waveguide,” IEEE Photonics Technol. Lett.24(21), 1949–1952 (2012).
[CrossRef]

M. Vasilyev and P. Kumar, “Frequency up-conversion of quantum images,” Opt. Express20(6), 6644–6656 (2012).
[CrossRef] [PubMed]

M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive parametric image amplifiers with circular and elliptical Gaussian pumps,” Opt. Express19(27), 26710–26724 (2011).
[CrossRef] [PubMed]

Z. Tong, C. Lundström, M. Karlsson, M. Vasilyev, and P. A. Andrekson, “Noise performance of a frequency nondegenerate phase-sensitive amplifier with unequalized inputs,” Opt. Lett.36(5), 722–724 (2011).
[CrossRef] [PubMed]

Z. Tong, A. Bogris, C. Lundström, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier,” Opt. Express18(14), 14820–14835 (2010).
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M. Vasilyev, M. Annamalai, N. Stelmakh, and P. Kumar, “Quantum properties of a spatially-broadband traveling-wave phase-sensitive optical parametric amplifier,” J. Mod. Opt.57(19), 1908–1915 (2010).
[CrossRef]

M. Vasilyev, N. Stelmakh, and P. Kumar, “Estimation of the spatial bandwidth of an optical parametric amplifier with plane-wave pump,” J. Mod. Opt.56(18-19), 2029–2033 (2009).
[CrossRef]

M. Vasilyev, N. Stelmakh, and P. Kumar, “Phase-sensitive image amplification with elliptical Gaussian pump,” Opt. Express17(14), 11415–11425 (2009).
[CrossRef] [PubMed]

D. Levandovsky, M. Vasilyev, and P. Kumar, “Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” Pramana56, 281–285 (2001).

S.-K. Choi, M. Vasilyev, and P. Kumar, “Erratum: Noiseless optical amplification of images,” Phys. Rev. Lett.84, 1361 (2000).

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett.83, 1938–1941 (1999).

D. Levandovsky, M. Vasilyev, and P. Kumar, “Amplitude squeezing of light by means of a phase-sensitive fiber parametric amplifier,” Opt. Lett.24(14), 984–986 (1999).
[CrossRef] [PubMed]

D. Levandovsky, M. Vasilyev, and P. Kumar, “Perturbation theory of quantum solitons: continuum evolution and optimum squeezing by spectral filtering,” Opt. Lett.24(1), 43–45 (1999).
[CrossRef] [PubMed]

D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive nonlinear optical loop mirror,” Opt. Lett.24, 89–91 (1999).

D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive nonlinear optical loop mirror: errata,” Opt. Lett.24, 423 (1999).

Voss, P. L.

Wagner, K.

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Wang, K.

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Appl. Phys. B (1)

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W. Imajuku, A. Takada, and Y. Yamabayashi, “Low-noise amplification under the 3 dB noise figure in high-gain phase-sensitive fibre amplifier,” Electron. Lett.35(22), 1954–1955 (1999).
[CrossRef]

Eur. Phys. J. D (2)

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[CrossRef]

G. Patera, N. Treps, C. Fabre, and G. J. de Valcárcel, “Quantum theory of synchronously pumped type I optical parametric oscillators: characterization of the squeezed supermodes,” Eur. Phys. J. D56(1), 123–140 (2010).
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M. Vasilyev, M. Annamalai, N. Stelmakh, and P. Kumar, “Quantum properties of a spatially-broadband traveling-wave phase-sensitive optical parametric amplifier,” J. Mod. Opt.57(19), 1908–1915 (2010).
[CrossRef]

M. Vasilyev, N. Stelmakh, and P. Kumar, “Estimation of the spatial bandwidth of an optical parametric amplifier with plane-wave pump,” J. Mod. Opt.56(18-19), 2029–2033 (2009).
[CrossRef]

J. Opt. B Quantum Semiclassical Opt. (1)

K. Wang, G. Yang, A. Gatti, and L. Lugiato, “Controlling the signal-to-noise ratio in optical parametric image amplification,” J. Opt. B Quantum Semiclassical Opt.5(4), S535–S544 (2003).
[CrossRef]

J. Opt. Soc. Am. B (4)

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E. Lantz and F. Devaux, “Parametric amplification of images: from time gating to noiseless amplification,” J. Sel. Top. Quantum Electron.14(3), 635–647 (2008).
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Nat. Commun. (1)

S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun.3, 1026 (2012).
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New J. Phys. (1)

A. Christ, K. Laiho, A. Eckstein, K. N. Cassemiro, and C. Silberhorn, “Probing multimode squeezing with correlation functions,” New J. Phys.13(3), 033027 (2011).
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Opt. Commun. (1)

C. J. McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun.282(4), 583–593 (2009).
[CrossRef]

Opt. Express (6)

Opt. Lett. (7)

Z. Tong, C. Lundström, M. Karlsson, M. Vasilyev, and P. A. Andrekson, “Noise performance of a frequency nondegenerate phase-sensitive amplifier with unequalized inputs,” Opt. Lett.36(5), 722–724 (2011).
[CrossRef] [PubMed]

D. Levandovsky, M. Vasilyev, and P. Kumar, “Amplitude squeezing of light by means of a phase-sensitive fiber parametric amplifier,” Opt. Lett.24(14), 984–986 (1999).
[CrossRef] [PubMed]

A. R. Bhagwat, G. Alon, O.-K. Lim, C.-H. Chen, M. Annamalai, M. Vasilyev, and P. Kumar, “Fundamental eigenmode of traveling-wave phase-sensitive optical parametric amplifier: experimental generation and verification,” Opt. Lett.38(15), 2858–2860 (2013).
[CrossRef] [PubMed]

G. Alon, O.-K. Lim, A. Bhagwat, C.-H. Chen, M. Annamalai, M. Vasilyev, and P. Kumar, “Optimization of gain in traveling-wave optical parametric amplifiers by tuning the offset between pump- and signal-waist locations,” Opt. Lett.38(8), 1268–1270 (2013).
[CrossRef] [PubMed]

D. Levandovsky, M. Vasilyev, and P. Kumar, “Perturbation theory of quantum solitons: continuum evolution and optimum squeezing by spectral filtering,” Opt. Lett.24(1), 43–45 (1999).
[CrossRef] [PubMed]

D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive nonlinear optical loop mirror,” Opt. Lett.24, 89–91 (1999).

D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive nonlinear optical loop mirror: errata,” Opt. Lett.24, 423 (1999).

Phys. Rev. A (9)

E. Brambilla, A. Gatti, M. Bache, and L. A. Lugiato, “Simultaneous near-field and far-field spatial quantum correlations in the high-gain regime of parametric down-conversion,” Phys. Rev. A69(2), 023802 (2004).
[CrossRef]

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A73(6), 063819 (2006).
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[CrossRef]

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[CrossRef]

B. Chalopin, F. Scazza, C. Fabre, and N. Treps, “Multimode nonclassical light generation through the optical-parametric-oscillator threshold,” Phys. Rev. A81(6), 061804 (2010).
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Phys. Rev. Lett. (7)

M. Lassen, V. Delaubert, J. Janousek, K. Wagner, H.-A. Bachor, P. K. Lam, N. Treps, P. Buchhave, C. Fabre, and C. C. Harb, “Tools for multimode quantum information: modulation, detection, and spatial quantum correlations,” Phys. Rev. Lett.98(8), 083602 (2007).
[CrossRef] [PubMed]

L. Lopez, N. Treps, B. Chalopin, C. Fabre, and A. Maître, “Quantum processing of images by continuous wave optical parametric amplification,” Phys. Rev. Lett.100(1), 013604 (2008).
[CrossRef] [PubMed]

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett.83, 1938–1941 (1999).

S.-K. Choi, M. Vasilyev, and P. Kumar, “Erratum: Noiseless optical amplification of images,” Phys. Rev. Lett.84, 1361 (2000).

A. Mosset, F. Devaux, and E. Lantz, “Spatially noiseless optical amplification of images,” Phys. Rev. Lett.94(22), 223603 (2005).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

Pramana (1)

D. Levandovsky, M. Vasilyev, and P. Kumar, “Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” Pramana56, 281–285 (2001).

Rev. Mod. Phys. (1)

M. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys.71(5), 1539–1589 (1999).
[CrossRef]

Other (5)

P. Kumar, V. Grigoryan, and M. Vasilyev, “Noise-free amplification: towards quantum laser radar,” in 14th Coherent Laser Radar Conference, Snowmass, CO, July 2007.

O.-K. Lim, G. Alon, Z. Dutton, S. Guha, M. Vasilyev, and P. Kumar, “Optical resolution enhancement with phase-sensitive preamplification,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (CD) (Optical Society of America, 2010), paper CTuPP7.

M. Annamalai, M. Vasilyev, and P. Kumar, “Impact of phase-sensitive-amplifier's mode structure on amplified image quality,” Conference on Lasers and Electro-Optics 2012, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CF1B.7.

M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive image amplifier with higher-order Gaussian pump and phase mismatch,” in IEEE Photonics Conference, Arlington, VA, October 9–13, 2011, paper TuO4.
[CrossRef]

M. Annamalai, M. Vasilyev, N. Stelmakh, and P. Kumar, “Compact representation of spatial modes of phase-sensitive image amplifier,” in Conference on Lasers and Electro-Optics 2011, OSA Technical Digest (CD) (Optical Society of America, 2011), paper JThB77.

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Figures (12)

Fig. 1
Fig. 1

(a) Eigenvalue (gain and squeezing factor) spectra for the first 16 PSA modes. The vertical scale is linear. The legend indicates the pump waist dimensions a0px × a0py in μm2. The horizontal dashed black line marks the –3-dB level from the gain of the fundamental eigenmode #0. The pump powers are chosen in each case to achieve approximately the same gain of ~15 for mode #0. (b) Dependence of the signal’s compact-basis waist size (obtained by best match to eigenmode #0) on the corresponding pump waist size plotted on a double logarithmic scale. “Circular” cases represent 25 × 25, 100 × 100, and 200 × 200 μm2 pump waist sizes (here, the horizontal scale corresponds to a0p), whereas “100×Y” cases represent 100 × 25, 100 × 50, 100 × 100, 100 × 200, and 100 × 400 μm2 pump waist sizes (here, the horizontal scale corresponds to a0py). Blue symbols correspond to pump powers that ensure a gain of ~15 for the eigenmode #0. Red symbols correspond to a pump power of ~4.1 kW (which produces a gain of ~15 for the eigenmode #0 in the case of 200 × 200 μm2 pump waist size). The three dotted lines indicate slopes proportional to a0p, a 0p 1/2 , and a 0p 1/3 .

Fig. 2
Fig. 2

Overlap integral |A00|2 of the eigenmode #0 of the PSA (400 × 100 μm2 pump waist size and pump power P0 = 4.3 kW) with HG00 modes of various waist sizes a0sx and a0sy. The best overlap of 99.3% occurs for a HG00 signal beam of waist size 83 × 48 μm2, which is then taken to be the waist size for the compact basis HGc0.

Fig. 3
Fig. 3

xy and HG representations of the most amplified eigenmode #0 for the PSAs with pump waist sizes (left to right) of 100 × 100, 100 × 50, 100 × 25, and 25 × 25 μm2 and pump powers P0 corresponding to gains of ~15 for the eigenmode #0. Row (a): the xy intensity profiles on the same scale. Row (b): the HG representations |Amn|2 in our original basis (signal waist is 21/2 times larger than the pump waist). Overlap |A00|2 with the HG00 mode is (left to right): 35.4%, 49.9%, 60.9%, and 96.6%. Row (c): the HG representations |Amn|2 in the compact basis HGc0 with a0sx × a0sy sizes of (left to right) 49.6 × 49.6, 51.5 × 40.1, 51.5 × 31.8, and 32.2 × 32.2 μm2, which are optimized for the best overlap of mode HG00 with the eigenmode #0, yielding |A00|2 of 98.9%, 98.4%, 98.4%, and 97.7%, respectively.

Fig. 4
Fig. 4

xy, HG, and LG representations of the eigenmodes #0 to #6 of the PSA with 100 × 100 μm2 pump waist size and pump power P0 = 1.25 kW (gain of mode #0 is 15.3). Note that the eigenmodes #2, #4, and #7 are degenerate with the eigenmodes #1, #3, and #6, respectively. Column 1: xy profiles. Column 2: HG representations |Amn|2 in our original basis (signal waist is 21/2 times larger than the pump waist). Also shown are HG representations |Amn|2 (column 3) and LG representations |Ap|l||2 (column 4) in the compact bases HGc0 and LGc0, respectively, with a0sx = a0sy = a0s = 49.6 μm, which are optimized for best overlap (98.9%) with the eigenmode #0.

Fig. 5
Fig. 5

xy, HG, and LG representations of the eigenmodes #8 to #14 of the PSA with 100 × 100 μm2 pump waist size and pump power P0 = 1.25 kW (gain of mode #0 is 15.3). Note that the eigenmodes #9, #11, and #13 are degenerate with the eigenmodes #8, #10, and #12, respectively. Columns are the same as those in Fig. 4.

Fig. 6
Fig. 6

xy, HG, and LG representations of the eigenmodes #0 to #6 of the PSA with 200 × 200 μm2 pump waist size and pump power P0 = 4.06 kW (gain of mode #0 is 15.1). Note that the eigenmodes #2, #4, and #7 are degenerate with the eigenmodes #1, #3, and #6, respectively. Column 1: xy profiles. Column 2: HG representations |Amn|2 in our original basis (signal waist is 21/2 times larger than the pump waist). Column 3: LG representations |Ap|l||2 in the compact basis LGc0 with a0s = 61.2 μm, optimized for best overlap (99.4%) with the eigenmode #0. Column 4: LG representations |Ap|l||2 in the compact basis LGc5 with a0s = 69.8 μm, optimized for best overlap (96.5%) with the eigenmode #5.

Fig. 7
Fig. 7

xy, HG, and LG representations of the eigenmodes #8 to #14 of the PSA with 200 × 200 μm2 pump waist size and pump power P0 = 4.06 kW (gain of mode #0 is 15.1). Note that the eigenmodes #9, #11, and #13 are degenerate with the eigenmodes #8, #10, and #12, respectively. Columns are the same as those in Fig. 6.

Fig. 8
Fig. 8

xy and HG representations of some of the main eigenmodes of the PSA with 800 × 50 μm2 pump waist size and pump power P0 = 5.4 kW (gain of mode #0 is 15.7). Column 1: xy profiles. Column 2: HG representations |Amn|2 in our original basis (signal waist is 21/2 times larger than the pump waist). Column 3: HG representations |Amn|2 in the compact basis HGc0 with a0sx = 113.1 μm, a0sy = 38.4 μm, optimized for best overlap (99.0%) with the eigenmode #0. Column 4: HG representations |Amn|2 in the compact basis HGc4 with a0sx = 128.2 μm, a0sy = 41.1 μm, optimized for best overlap (98.3%) with the eigenmode #4.

Fig. 9
Fig. 9

Radial field profiles of the output eigenmodes #0 to #14 of the PSA with 200 × 200 μm2 pump waist size and pump power P0 = 4.06 kW (gain of mode #0 is 15.1), compared to the radial profiles fpl(ρ,0,0,a0s,ks) of the dominant LGpl modes of the corresponding compact representation in basis LGc5 with a0s = 69.8 μm, optimized for best overlap with eigenmode #5. The legend lists the eigenmode number, dominant LGpl mode, and their overlap |Ap|l||2, represented by the tallest bar on the corresponding graph in column 4 of Fig. 6 or Fig. 7. Solid red line – real part of the eigenmode profile, solid blue line – imaginary part of the eigenmode profile, dashed black line – dominant LGpl mode profile fpl(ρ,0,0,a0s,ks) at z = 0. To compare the eigenmode with fpl(ρ,0,0,a0s,ks) of Eq. (11), we have normalized the eigenmode by its most dominant mode coefficient Apl, and linearly back-propagated the eigenmode to the center of the crystal z = 0 (this situation corresponds to the output eigenmode profile observable by projecting the center of the crystal onto the image plane of a 1:1 telescope).

Fig. 10
Fig. 10

Horizontal field profiles at y = 0 of the output eigenmodes #0 to #14 of the PSA with 800 × 50 μm2 pump waist size and pump power P0 = 5.4 kW (gain of mode #0 is 15.7), compared to the horizontal profiles fm(x,0,a0sx,ks) × f0(0,0,a0sy,ks) of the dominant HGm0 modes of the corresponding compact representation in basis HGc4 with a0sx = 128.2 μm and a0sy = 41.1 μm, optimized for best overlap with eigenmode #4. The legend lists the eigenmode number, dominant HGm0 mode, and their overlap |Am0|2, represented by the tallest bar on the corresponding graph in column 4 of Fig. 8. Solid red line – real part of the eigenmode profile, solid blue line – imaginary part of the eigenmode profile, dashed black line – dominant HGm0 mode profile at z = 0. To compare the eigenmode with fm(x,0,a0sx,ks) × f0(0,0,a0sy,ks) of Eq. (3), we have normalized the eigenmode by its most dominant mode coefficient Am0, and linearly back-propagated the eigenmode to the center of the crystal z = 0 (this situation corresponds to the output eigenmode profile observable by projecting the center of the crystal onto the image plane of a 1:1 telescope).

Fig. 11
Fig. 11

(a) Overlaps of the various eigenmodes from Figs. 610 with the respective most dominant HG or LG mode in the eigenmode’s compact representation. (b) Magnified version of (a), but with added overlap curves (magenta) for the compact LGc bases with optimized z-offsets that are equal to 3.5 mm for basis LGc0 matched to the eigenmode #0 and to 3 mm for basis LGc5 matched to the eigenmode #5.

Fig. 12
Fig. 12

(a) Right vertical scale: Phases of the most dominant modes HGn0, HG(n+2)0, and HG(n+4)0 in the compact HG representation (basis HGc4 optimized for the eigenmode #4) of the nth output eigenmode of a PSA with 800 × 50 μm2 pump waist size and pump power P0 = 5.4 kW. (b) Right vertical scale: Phases of the most dominant modes LGp|l| in the compact LG representation (basis LGc5 optimized for the eigenmode #5 with the basis’ waist location offset from the pump waist location by + 3 mm in the z-direction) of the output eigenmodes of the PSA with 200 × 200 μm2 pump waist size and pump power P0 = 4.06 kW. For the eigenmodes with non-zero |l|, in (b) we plot the average phase of LGpl and LGp(–l) (their difference phase determines azimuthal rotation of the eigenmode). Left vertical scales on both (a) and (b): the corresponding eigenvalue (gain) spectrum [also shown in Fig. 1(a)], which crosses the –3-dB level from the gain of the eigenmode #0 between the eigenmodes #9 and #10 in (a) and between the eigenmodes #5 and #6 in (b). Pump phase θp = –π/2.

Equations (22)

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E s ( ρ ,z) z = i 2 k s ρ 2 E s ( ρ ,z)+ i2 ω s d eff n s c E p ( ρ ,z) E s * ( ρ ,z)exp( iΔkz ),
E p ( ρ ,z)= P 0 2 ε 0 n p c e i θ p g 0 (x,z, a 0px , k p ) g 0 (y,z, a 0py , k p ), E s ( ρ ,z)= m,n A mn (z) 2 ε 0 n s c g m (x,z, 2 a 0px , k s ) g n (y,z, 2 a 0py , k s ),
g m (β,z, a 0 ,k)= H m [β/a(z)] 2 m m! π 1/2 a(z) e iθ(z) e β 2 2 a 2 (z) e ik β 2 2R(z)
g m (β,z, a 0 ,k) g m * (β,z, a 0 ,k) dβ= δ m m ,
z R =k a 0 2 , a(z)= a 0 1+ (z/ z R ) 2 , θ(z)=( m+1/2 ) tan 1 (z/ z R ), R(z)=z[1+ ( z R /z) 2 ].
d A mn (z) dz =i e i θ p κ e iΔkz m , n B m m (z/ z Rx ) B n n (z/ z Ry ) A m n * (z),
κ= 2 ω s 2 d eff 2 ε 0 n s 2 n p c 3 P 0 π a 0px a 0py ,
B m m (ξ)={ e i(m+ m +1/2) tan 1 ξ 1+ ξ 2 4 (1) m m 2 (m+ m 1)!! 2 m+ m +1 m! m ! for even (m+ m ), 0 for odd (m+ m ).
E s ( ρ ,z)= m,n A mn (z) 2 ε 0 n s c g m (x,z, a 0sx , k s ) g n (y,z, a 0sy , k s ),
E p ( ρ ,z)= P 0 2 ε 0 n p c e i θ p f 00 (ρ,φ,z, a 0p , k p ), E s ( ρ ,z)= p,l A pl (z) 2 ε 0 n s c f pl (ρ,φ,z, a 0s , k s ),
f pl (ρ,φ,z, a 0 ,k)= p! (p+|l|)!π a 2 (z) ( ρ a(z) ) |l| L p |l| ( ρ 2 a 2 (z) ) e ilφ e iθ(z) e ρ 2 2 a 2 (z) e ik ρ 2 2R(z) ,
z R =k a 0 2 , a(z)= a 0 1+ (z/ z R ) 2 , θ(z)=( 2p+|l|+1 ) tan 1 (z/ z R ), R(z)=z[1+ ( z R /z) 2 ],
0 2π 0 f pl (ρ,φ,z, a 0 ,k) f p l * (ρ,φ,z, a 0 ,k)ρdρ dφ= δ p p δ l l .
f 00 (ρ,φ,z, a 0 ,k)= g 0 (x,z, a 0 ,k)× g 0 (y,z, a 0 ,k).
( a 0sx j , a 0sy j )= argmax a 0sx , a 0sy (0,) | g m * (x,L/2, a 0sx , k s ) g n * (y,L/2, a 0sy , k s ) E s j ( ρ ,L/2)dxdy | 2
a 0s j = argmax a 0s (0,) | 0 2π 0 f pl * (ρ,φ,L/2, a 0s , k s ) E s j ( ρ ,L/2)ρdρ dφ | 2 .
D m m (z, a 0sx , a 0px )= { e i[ (m+ m +1) tan 1 ( z z Rsx ) 1 2 tan 1 ( z z Rpx ) ] 1+ (z/ z Rpx ) 2 4 2 m+ m πm! m ! k=0 (m+ m )/2 c 2k Γ( 1+2k 2 ) ξ x (z) 1+2k 2 for even (m+ m ), 0 for odd (m+ m ),
ξ x (z)= δ x (z)i γ x (z), δ x (z)= a sx 2 (z) 2 a px 2 (z) +1, γ x (z)= k p a sx 2 (z) 2 ( 1 R px (z) 1 R sx (z) ) ,
N x = a 0px q c = π k p a 0px 2 L = π z Rpx L .
a 0sx N x = a 0px ,.
a 0sx N x = 1 q c = L π k p .
a 0sx = a 0px q c = a 0px L π z Rpx 4 ,

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