Abstract

We develop a method for finding the number and shapes of the independently squeezed or amplified modes of a spatially-broadband, travelling-wave, frequency- and polarization-degenerate optical parametric amplifier in the general case of an elliptical Gaussian pump. The obtained results show that for tightly focused pump only one mode is squeezed, and this mode has a Gaussian TEM00 shape. For larger pump spot sizes that support multiple modes, the shapes of the most-amplified modes are close to Hermite- or Laguerre-Gaussian profiles. These results can be used to generate matched local oscillators for detecting high amounts of squeezing and to design parametric image amplifiers that introduce minimal distortion.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields 26(8), 1817–1839 (1982).
    [CrossRef]
  2. 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]
  3. D. Levandovsky, M. Vasilyev, and P. Kumar, “Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” PRAMANA–J. Phys. 56(2-3), 281–285 (2001).
    [CrossRef]
  4. 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]
  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. B 23(4), 598–610 (2006).
    [CrossRef]
  6. 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. Express 18(14), 14820–14835 (2010).
    [CrossRef] [PubMed]
  7. M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A 52(6), 4930–4940 (1995).
    [CrossRef] [PubMed]
  8. M. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71(5), 1539–1589 (1999).
    [CrossRef]
  9. 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]
  10. S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938–1941 (1999) [erratum: Phys. Rev. Lett. 84, 1361–1361 (2000)].
  11. A. Mosset, F. Devaux, and E. Lantz, “Spatially noiseless optical amplification of images,” Phys. Rev. Lett. 94(22), 223603 (2005).
    [CrossRef] [PubMed]
  12. E. Lantz and F. Devaux, “Parametric amplification of images: from time gating to noiseless amplification,” IEEE J. Sel. Top. Quantum Electron. 14(3), 635–647 (2008).
    [CrossRef]
  13. 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]
  14. P. Kumar, V. Grigoryan, and M. Vasilyev, “Noise-free amplification: towards quantum laser radar,” the 14th Coherent Laser Radar Conference, Snowmass, CO, July 2007. http://space.hsv.usra.edu/CLRC/presentations/Kumar.ppt
  15. 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. B 27(6), A63–A72 (2010).
    [CrossRef]
  16. 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.
  17. M. Vasilyev, N. Stelmakh, and P. Kumar, “Phase-sensitive image amplification with elliptical Gaussian pump,” Opt. Express 17(14), 11415–11425 (2009).
    [CrossRef] [PubMed]
  18. 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]
  19. A. La Porta and R. E. Slusher, “Squeezing limits at high parametric gains,” Phys. Rev. A 44(3), 2013–2022 (1991).
    [CrossRef] [PubMed]
  20. 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. B 14(7), 1564–1575 (1997).
    [CrossRef]
  21. 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]
  22. R.-D. Li, S.-K. Choi, C. Kim, and P. Kumar, “Generation of sub-Poissonian pulses of light,” Phys. Rev. A 51(5), R3429–R3432 (1995).
    [CrossRef] [PubMed]
  23. K. G. Köprülü and O. Aytür, “Analysis of Gaussian-beam degenerate optical parametric amplifiers for the generation of quadrature-squeezed states,” Phys. Rev. A 60(5), 4122–4134 (1999).
    [CrossRef]
  24. 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. B 18(6), 846–854 (2001).
    [CrossRef]
  25. 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. B 66(6), 685–699 (1998).
    [CrossRef]
  26. M. Annamalai, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive image amplifier with elliptical Gaussian pump,” in Laser Science, OSA Technical Digest (CD) (Optical Society of America, 2010), paper LTuB5.
  27. 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]
  28. Please note that the definition of deff in our prior work (Refs. 17, 18, and 27) is different from that in the present paper. The prior-work deff denotes the quantity that is more commonly known as the effective χ(2) and equals 2deff in the present paper’s notations. As a result, the nonlinear paraxial wave equation in Refs. 17, 18, and 27 does not have the factor of 2 in front of deff. One fallout of this unfortunate choice of notation in our prior work is that Ref. 17 assumes effective χ(2) = 8.7 pm/V for PPKTP crystal, which is about half of the actual value of that crystal’s nonlinearity, and the resulting pump powers listed in Refs. 17 and 26 are four times larger than those required for the same gain in a real PPKTP crystal. The present paper’s definitions rectify the previous inconsistencies.
  29. E. Lantz and F. Devaux, “Numerical simulation of spatial fluctuations in parametric image amplification,” Eur. Phys. J. D 17(1), 93–98 (2001).
    [CrossRef]
  30. H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13(6), 2226–2243 (1976).
    [CrossRef]
  31. H. P. Yuen, “Multimode two-photon coherent states and unitary representation of the symplectic group,” Nucl. Phys. B 6, 309–313 (1989).
    [CrossRef]
  32. D. Levandovsky, “Quantum noise suppression using optical fibers,” Ph.D. thesis, Northwestern University, 1999.
  33. L. Lopez, S. Gigan, N. Treps, A. Maître, C. Fabre, and A. Gatti, “Multimode squeezing properties of a confocal optical parametric oscillator: Beyond the thin-crystal approximation,” Phys. Rev. A 72(1), 013806 (2005).
    [CrossRef]
  34. 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, OSA Technical Digest (CD) (Optical Society of America, 2011), paper JThB77.
  35. W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: Simultaneous squeezing of multiple modes,” Phys. Rev. A 73(6), 063819 (2006).
    [CrossRef]
  36. J. H. Shapiro and A. Shakeel, “Optimizing homodyne detection of quadrature-noise squeezing by local-oscillator selection,” J. Opt. Soc. Am. B 14(2), 232–249 (1997).
    [CrossRef]
  37. C. J. McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun. 282(4), 583–593 (2009).
    [CrossRef]
  38. 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. D 56(1), 123–140 (2010).
    [CrossRef]
  39. A. Ekert and P. L. Knight, “Entangled quantum systems and the Schmidt decomposition,” Am. J. Phys. 63(5), 415–423 (1995).
    [CrossRef]
  40. C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite hilbert space and entropy control,” Phys. Rev. Lett. 84(23), 5304–5307 (2000).
    [CrossRef] [PubMed]
  41. C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92(12), 127903 (2004).
    [CrossRef] [PubMed]

2010 (4)

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]

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. D 56(1), 123–140 (2010).
[CrossRef]

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. B 27(6), A63–A72 (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. Express 18(14), 14820–14835 (2010).
[CrossRef] [PubMed]

2009 (3)

M. Vasilyev, N. Stelmakh, and P. Kumar, “Phase-sensitive image amplification with elliptical Gaussian pump,” Opt. Express 17(14), 11415–11425 (2009).
[CrossRef] [PubMed]

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

2008 (2)

E. Lantz and F. Devaux, “Parametric amplification of images: from time gating to noiseless amplification,” IEEE 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]

2006 (2)

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: Simultaneous squeezing of multiple modes,” Phys. Rev. A 73(6), 063819 (2006).
[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. B 23(4), 598–610 (2006).
[CrossRef]

2005 (2)

L. Lopez, S. Gigan, N. Treps, A. Maître, C. Fabre, and A. Gatti, “Multimode squeezing properties of a confocal optical parametric oscillator: Beyond the thin-crystal approximation,” Phys. Rev. A 72(1), 013806 (2005).
[CrossRef]

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

2004 (1)

C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92(12), 127903 (2004).
[CrossRef] [PubMed]

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 (3)

D. Levandovsky, M. Vasilyev, and P. Kumar, “Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” PRAMANA–J. Phys. 56(2-3), 281–285 (2001).
[CrossRef]

E. Lantz and F. Devaux, “Numerical simulation of spatial fluctuations in parametric image amplification,” Eur. Phys. J. D 17(1), 93–98 (2001).
[CrossRef]

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. B 18(6), 846–854 (2001).
[CrossRef]

2000 (1)

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite hilbert space and entropy control,” Phys. Rev. Lett. 84(23), 5304–5307 (2000).
[CrossRef] [PubMed]

1999 (4)

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]

K. G. Köprülü and O. Aytür, “Analysis of Gaussian-beam degenerate optical parametric amplifiers for the generation of quadrature-squeezed states,” Phys. Rev. A 60(5), 4122–4134 (1999).
[CrossRef]

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]

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

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. B 66(6), 685–699 (1998).
[CrossRef]

1997 (2)

1995 (3)

A. Ekert and P. L. Knight, “Entangled quantum systems and the Schmidt decomposition,” Am. J. Phys. 63(5), 415–423 (1995).
[CrossRef]

R.-D. Li, S.-K. Choi, C. Kim, and P. Kumar, “Generation of sub-Poissonian pulses of light,” Phys. Rev. A 51(5), R3429–R3432 (1995).
[CrossRef] [PubMed]

M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A 52(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. A 44(3), 2013–2022 (1991).
[CrossRef] [PubMed]

1989 (1)

H. P. Yuen, “Multimode two-photon coherent states and unitary representation of the symplectic group,” Nucl. Phys. B 6, 309–313 (1989).
[CrossRef]

1982 (1)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields 26(8), 1817–1839 (1982).
[CrossRef]

1976 (1)

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13(6), 2226–2243 (1976).
[CrossRef]

Andrekson, P. A.

Annamalai, M.

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]

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. B 18(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. A 60(5), 4122–4134 (1999).
[CrossRef]

Banaszek, K.

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: Simultaneous squeezing of multiple modes,” Phys. Rev. A 73(6), 063819 (2006).
[CrossRef]

Bogris, A.

Caves, C. M.

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields 26(8), 1817–1839 (1982).
[CrossRef]

Chalopin, B.

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]

Choi, S.-K.

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. B 66(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. D 56(1), 123–140 (2010).
[CrossRef]

Devaux, F.

E. Lantz and F. Devaux, “Parametric amplification of images: from time gating to noiseless amplification,” IEEE 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 and F. Devaux, “Numerical simulation of spatial fluctuations in parametric image amplification,” Eur. Phys. J. D 17(1), 93–98 (2001).
[CrossRef]

Dutton, Z.

Eberly, J. H.

C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92(12), 127903 (2004).
[CrossRef] [PubMed]

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite hilbert space and entropy control,” Phys. Rev. Lett. 84(23), 5304–5307 (2000).
[CrossRef] [PubMed]

Ekert, A.

A. Ekert and P. L. Knight, “Entangled quantum systems and the Schmidt decomposition,” Am. J. Phys. 63(5), 415–423 (1995).
[CrossRef]

Fabre, C.

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. D 56(1), 123–140 (2010).
[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]

L. Lopez, S. Gigan, N. Treps, A. Maître, C. Fabre, and A. Gatti, “Multimode squeezing properties of a confocal optical parametric oscillator: Beyond the thin-crystal approximation,” Phys. Rev. A 72(1), 013806 (2005).
[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. B 66(6), 685–699 (1998).
[CrossRef]

Gatti, A.

L. Lopez, S. Gigan, N. Treps, A. Maître, C. Fabre, and A. Gatti, “Multimode squeezing properties of a confocal optical parametric oscillator: Beyond the thin-crystal approximation,” Phys. Rev. A 72(1), 013806 (2005).
[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. B 66(6), 685–699 (1998).
[CrossRef]

Gigan, S.

L. Lopez, S. Gigan, N. Treps, A. Maître, C. Fabre, and A. Gatti, “Multimode squeezing properties of a confocal optical parametric oscillator: Beyond the thin-crystal approximation,” Phys. Rev. A 72(1), 013806 (2005).
[CrossRef]

Guha, S.

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]

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. B 14(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. A 51(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]

Knight, P. L.

A. Ekert and P. L. Knight, “Entangled quantum systems and the Schmidt decomposition,” Am. J. Phys. 63(5), 415–423 (1995).
[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. A 52(6), 4930–4940 (1995).
[CrossRef] [PubMed]

Köprülü, K. G.

Kumar, P.

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. Express 17(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. B 23(4), 598–610 (2006).
[CrossRef]

D. Levandovsky, M. Vasilyev, and P. Kumar, “Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” PRAMANA–J. Phys. 56(2-3), 281–285 (2001).
[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]

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. B 14(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. A 51(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. A 44(3), 2013–2022 (1991).
[CrossRef] [PubMed]

Lantz, E.

E. Lantz and F. Devaux, “Parametric amplification of images: from time gating to noiseless amplification,” IEEE 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 and F. Devaux, “Numerical simulation of spatial fluctuations in parametric image amplification,” Eur. Phys. J. D 17(1), 93–98 (2001).
[CrossRef]

Law, C. K.

C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92(12), 127903 (2004).
[CrossRef] [PubMed]

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite hilbert space and entropy control,” Phys. Rev. Lett. 84(23), 5304–5307 (2000).
[CrossRef] [PubMed]

Levandovsky, D.

D. Levandovsky, M. Vasilyev, and P. Kumar, “Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” PRAMANA–J. Phys. 56(2-3), 281–285 (2001).
[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]

Li, R.-D.

Lopez, L.

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]

L. Lopez, S. Gigan, N. Treps, A. Maître, C. Fabre, and A. Gatti, “Multimode squeezing properties of a confocal optical parametric oscillator: Beyond the thin-crystal approximation,” Phys. Rev. A 72(1), 013806 (2005).
[CrossRef]

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. B 66(6), 685–699 (1998).
[CrossRef]

Lugiato, L. A.

M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A 52(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. A 73(6), 063819 (2006).
[CrossRef]

Maître, A.

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]

L. Lopez, S. Gigan, N. Treps, A. Maître, C. Fabre, and A. Gatti, “Multimode squeezing properties of a confocal optical parametric oscillator: Beyond the thin-crystal approximation,” Phys. Rev. A 72(1), 013806 (2005).
[CrossRef]

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. B 66(6), 685–699 (1998).
[CrossRef]

McKinstrie, C. J.

Mosset, A.

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

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. D 56(1), 123–140 (2010).
[CrossRef]

Radzewicz, C.

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: Simultaneous squeezing of multiple modes,” Phys. Rev. A 73(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. B 66(6), 685–699 (1998).
[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. B 66(6), 685–699 (1998).
[CrossRef]

Shakeel, A.

Shapiro, J. H.

Slusher, R. E.

A. La Porta and R. E. Slusher, “Squeezing limits at high parametric gains,” Phys. Rev. A 44(3), 2013–2022 (1991).
[CrossRef] [PubMed]

Stelmakh, N.

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. Express 17(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]

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.

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. D 56(1), 123–140 (2010).
[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]

L. Lopez, S. Gigan, N. Treps, A. Maître, C. Fabre, and A. Gatti, “Multimode squeezing properties of a confocal optical parametric oscillator: Beyond the thin-crystal approximation,” Phys. Rev. A 72(1), 013806 (2005).
[CrossRef]

Vasilyev, M.

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. Express 18(14), 14820–14835 (2010).
[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. Express 17(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]

D. Levandovsky, M. Vasilyev, and P. Kumar, “Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” PRAMANA–J. Phys. 56(2-3), 281–285 (2001).
[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]

Voss, P. L.

Walmsley, I. A.

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite hilbert space and entropy control,” Phys. Rev. Lett. 84(23), 5304–5307 (2000).
[CrossRef] [PubMed]

Wang, K.

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]

Wasilewski, W.

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: Simultaneous squeezing of multiple modes,” Phys. Rev. A 73(6), 063819 (2006).
[CrossRef]

Yamabayashi, Y.

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]

Yang, G.

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]

Yuen, H. P.

H. P. Yuen, “Multimode two-photon coherent states and unitary representation of the symplectic group,” Nucl. Phys. B 6, 309–313 (1989).
[CrossRef]

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13(6), 2226–2243 (1976).
[CrossRef]

Am. J. Phys. (1)

A. Ekert and P. L. Knight, “Entangled quantum systems and the Schmidt decomposition,” Am. J. Phys. 63(5), 415–423 (1995).
[CrossRef]

Appl. Phys. B (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. B 66(6), 685–699 (1998).
[CrossRef]

Electron. Lett. (1)

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)

E. Lantz and F. Devaux, “Numerical simulation of spatial fluctuations in parametric image amplification,” Eur. Phys. J. D 17(1), 93–98 (2001).
[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. D 56(1), 123–140 (2010).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

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

J. Mod. Opt. (2)

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 (5)

Nucl. Phys. B (1)

H. P. Yuen, “Multimode two-photon coherent states and unitary representation of the symplectic group,” Nucl. Phys. B 6, 309–313 (1989).
[CrossRef]

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 (2)

Opt. Lett. (1)

Phys. Rev. A (7)

A. La Porta and R. E. Slusher, “Squeezing limits at high parametric gains,” Phys. Rev. A 44(3), 2013–2022 (1991).
[CrossRef] [PubMed]

L. Lopez, S. Gigan, N. Treps, A. Maître, C. Fabre, and A. Gatti, “Multimode squeezing properties of a confocal optical parametric oscillator: Beyond the thin-crystal approximation,” Phys. Rev. A 72(1), 013806 (2005).
[CrossRef]

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: Simultaneous squeezing of multiple modes,” Phys. Rev. A 73(6), 063819 (2006).
[CrossRef]

R.-D. Li, S.-K. Choi, C. Kim, and P. Kumar, “Generation of sub-Poissonian pulses of light,” Phys. Rev. A 51(5), R3429–R3432 (1995).
[CrossRef] [PubMed]

K. G. Köprülü and O. Aytür, “Analysis of Gaussian-beam degenerate optical parametric amplifiers for the generation of quadrature-squeezed states,” Phys. Rev. A 60(5), 4122–4134 (1999).
[CrossRef]

M. I. Kolobov and L. A. Lugiato, “Noiseless amplification of optical images,” Phys. Rev. A 52(6), 4930–4940 (1995).
[CrossRef] [PubMed]

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13(6), 2226–2243 (1976).
[CrossRef]

Phys. Rev. D Part. Fields (1)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields 26(8), 1817–1839 (1982).
[CrossRef]

Phys. Rev. Lett. (5)

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]

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

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite hilbert space and entropy control,” Phys. Rev. Lett. 84(23), 5304–5307 (2000).
[CrossRef] [PubMed]

C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92(12), 127903 (2004).
[CrossRef] [PubMed]

PRAMANA–J. Phys. (1)

D. Levandovsky, M. Vasilyev, and P. Kumar, “Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” PRAMANA–J. Phys. 56(2-3), 281–285 (2001).
[CrossRef]

Rev. Mod. Phys. (1)

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

Other (7)

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938–1941 (1999) [erratum: Phys. Rev. Lett. 84, 1361–1361 (2000)].

P. Kumar, V. Grigoryan, and M. Vasilyev, “Noise-free amplification: towards quantum laser radar,” the 14th Coherent Laser Radar Conference, Snowmass, CO, July 2007. http://space.hsv.usra.edu/CLRC/presentations/Kumar.ppt

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, N. Stelmakh, M. Vasilyev, and P. Kumar, “Spatial modes of phase-sensitive image amplifier with elliptical Gaussian pump,” in Laser Science, OSA Technical Digest (CD) (Optical Society of America, 2010), paper LTuB5.

Please note that the definition of deff in our prior work (Refs. 17, 18, and 27) is different from that in the present paper. The prior-work deff denotes the quantity that is more commonly known as the effective χ(2) and equals 2deff in the present paper’s notations. As a result, the nonlinear paraxial wave equation in Refs. 17, 18, and 27 does not have the factor of 2 in front of deff. One fallout of this unfortunate choice of notation in our prior work is that Ref. 17 assumes effective χ(2) = 8.7 pm/V for PPKTP crystal, which is about half of the actual value of that crystal’s nonlinearity, and the resulting pump powers listed in Refs. 17 and 26 are four times larger than those required for the same gain in a real PPKTP crystal. The present paper’s definitions rectify the previous inconsistencies.

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, OSA Technical Digest (CD) (Optical Society of America, 2011), paper JThB77.

D. Levandovsky, “Quantum noise suppression using optical fibers,” Ph.D. thesis, Northwestern University, 1999.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Magnitude (absolute value) of the coefficients of the coupling matrix Bmm(0) either as a function of (m + m′) in the direction of main diagonal m = m′ (red) or as a function of (mm′) in the direction orthogonal to the main diagonal for m + m ′ = 20 (blue), 50 (green), 80 (purple), and 100 (black). Filled circles correspond to the exact Eq. (8) and solid curves to the approximate Eq. (10).

Fig. 2
Fig. 2

Eigenvalue (gain and squeezing) spectra for the first 16 (a) and the first 101 (b) PSA modes. The vertical scale is linear in (a) and logarithmic in (b). The legend indicates the pump spot sizes. The horizontal dashed black line in (a) 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.

Fig. 3
Fig. 3

(a) Eigenvalue spectra for the second data set, where the pump powers are chosen to provide the same peak intensity as that in the 100 × 100 μm2 case. (b) Blue circles: pump power needed for the PSA gain of ~15 for mode #0 in Fig. 2(a), as a function of the pump spot size a0px × a0py. Red triangles: PSA gain of mode #0 for the four spot sizes in (a). Blue dashed line: linear power dependence (guide for the reader’s eyes).

Fig. 4
Fig. 4

xy-profiles of the few most prominent eigenmodes of the PSA for five different pump spot sizes. The distance scale is shown below the profile of mode #0 for the 200 × 200 μm2 case. The inset graph shows the number of well-amplified PSA modes [those with gains above the –3-dB line in Fig. 2(a)] versus the pump power.

Fig. 5
Fig. 5

xy-profiles of the few most prominent eigenmodes of the PSA for the two largest elliptic pump spot sizes considered (800 × 50 and 400 × 100 μm2). The distance scale is shown above the profile of mode #0 for the 800 × 50 μm2 case and is the same as that in Fig. 4.

Fig. 9
Fig. 9

HG representation (i.e., |Amn|2) of mode #14 for the four largest pump spot sizes studied. Pump power for each spot size is chosen such that the gain for mode #0 is ~15. The corresponding xy-profiles are shown in Fig. 4 (for the left two graphs) and in Fig. 5 (for the right two graphs).

Fig. 6
Fig. 6

Mode #0 in the HG representation (i.e., |Amn|2 for mode #0) for various pump spot sizes. The gain for the PSA mode #0 is ~15 in all cases. The corresponding xy-profiles are shown in Fig. 4 (for all graphs in the top row and the first graph in the bottom row of the present Figure) and in Fig. 5 (for the last two graphs in the bottom row of the present Figure).

Fig. 7
Fig. 7

HG representation (i.e., |Amn|2) of the few most prominent eigenmodes of the PSA for 100 × 100 μm2 pump spot size, corresponding to the xy-profiles shown in Fig. 4. Pump power P0 = 1.25 kW and the gain of mode #0 is 15.3. Note that mode #1 is degenerate with mode #2, mode #3 with #4, mode #6 with #7, mode #8 with #9, mode #10 with #11, and mode #12 with #13.

Fig. 8
Fig. 8

HG representation (i.e., |Amn|2) of the few most prominent eigenmodes of the PSA for 200 × 200 μm2 pump spot size, corresponding to the xy-profiles shown in Fig. 4. Pump power P0 = 4.06 kW and the gain of mode #0 is 15.1. Note that mode #1 is degenerate with mode #2, mode #3 with #4, mode #6 with #7, mode #8 with #9, mode #10 with #11, and mode #12 with #13.

Equations (31)

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

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 0x p , k p ) g 0 (y,z, a 0y p , k p ), E s ( ρ ,z)= m,n A mn (z) 2 ε 0 n s c g m (x,z, 2 a 0x p , k s ) g n (y,z, 2 a 0y p , 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 ].
A mn (z) z =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 0x p a 0y p ,
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 ),
B m m (ξ)= e i(m+ m +1/2) tan 1 ξ 1+ ξ 2 4 B m m (0),
| B m m (0)| e (m m ) 2 4(m+ m ) π(m+ m ) ,
E p ( ρ ,z)= P 0 2 ε 0 n p c e i θ p f 00 (r,φ,z, a 0 p , k p ), E s ( ρ ,z)= p,l A pl (z) 2 ε 0 n s c f pl (r,φ,z, 2 a 0 p , k s ),
f pl (r,φ,z, a 0 ,k)= p! (p+|l|)!π a 2 (z) ( r a(z) ) |l| L p |l| ( r 2 a 2 (z) ) e ilφ e iθ(z) e r 2 2 a 2 (z) e ik r 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 (r,φ,z, a 0 ,k) f p l * (r,φ,z, a 0 ,k)rdr dφ= δ p p δ l l .
f 00 (r,φ,z, a 0 ,k)= g 0 (x,z, a 0 ,k)× g 0 (y,z, a 0 ,k).
A pl (z) z =i e i θ p κ e iΔkz p Λ p p |l| (z/ z R ) A p ,l * (z),
κ= 2 ω s 2 d eff 2 ε 0 n s 2 n p c 3 P 0 π ( a 0 p ) 2 ,
Λ p p |l| (ξ)= e 2i(p+ p +|l|+1/2) tan 1 ξ 1+ ξ 2 (p+ p +|l|)! 2 p+ p +|l|+1 p! p !(p+|l|)!( p +|l|)! ,
Λ p p |l| (ξ)= e 2i(p+ p +|l|+1/2) tan 1 ξ 1+ ξ 2 Λ p p |l| (0),
Λ p p |l| (0) e (p p ) 2 + l 2 2(p+ p +|l|) 2π(p+ p +|l|) ,
A mn (z)=[ X mn (z) Y mn (z) ],
A mn (L/2)= G mn m n (L/2,L/2) A m n (L/2),
λ(L/2)= A mn T G mn m n G m n m n T A m n A mn T A mn .
G mn m n G m n m n T A m n =λ(L/2) A mn ,
A mn λ =[ X mn Y mn ], A mn 1/λ =[ Y mn X mn ].
(2 m max +1/2) tan 1 ( L/2 z R ) m max L z R =π,
m max =π z R L = π k p ( a 0 p ) 2 L .
m max x n max y = π 2 z Rx z Ry L 2 = ( π k p L ) 2 ( a 0x p a 0y p ) 2 .
Number of PSA modes= π k p L a 0x p a 0y p =π z Rx z Ry L ,
(2 m max +1/2) Δz z R 2 m max Δz z R π 150 ,
Δz π z R 300 m max = L 300 .

Metrics