OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 2 — Jan. 28, 2013
  • pp: 1374–1394

Schmidt decompositions of parametric processes I: Basic theory and simple examples

C. J. McKinstrie and M. Karlsson  »View Author Affiliations

Optics Express, Vol. 21, Issue 2, pp. 1374-1394 (2013)

View Full Text Article

Enhanced HTML    Acrobat PDF (940 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



Parametric devices based on four-wave mixing in fibers perform many signal-processing functions required by optical communication systems. In these devices, strong pumps drive weak signal and idler sidebands, which can have one or two polarization components, and one or many frequency components. The evolution of these components (modes) is governed by a system of coupled-mode equations. Schmidt decompositions of the associated transfer matrices determine the natural input and output mode vectors of such systems, and facilitate the optimization of device performance. In this paper, the basic properties of Schmidt decompositions are derived from first principles and are illustrated by two simple examples (one- and two-mode parametric amplification). In a forthcoming paper, several nontrivial examples relevant to current research (including four-mode parametric amplification) will be discussed.

© 2013 OSA

OCIS Codes
(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(270.6570) Quantum optics : Squeezed states

ToC Category:
Nonlinear Optics

Original Manuscript: September 11, 2012
Revised Manuscript: December 5, 2012
Manuscript Accepted: December 30, 2012
Published: January 14, 2013

C. J. McKinstrie and M. Karlsson, "Schmidt decompositions of parametric processes I: Basic theory and simple examples," Opt. Express 21, 1374-1394 (2013)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge, 2007). [CrossRef]
  2. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002). [CrossRef]
  3. J. H. Lee, “All-optical signal processing devices based on holey fiber,” IEICE Trans. Electron.E88-C, 327–334 (2005). [CrossRef]
  4. S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly nonlinear optical fiber,” IEICE Trans. Electron.E88-C, 859–869 (2005). [CrossRef]
  5. P. A. Andrekson and M. Westlund, “Nonlinear optical fiber based high resolution all-optical waveform sampling,” Laser Photon. Rev.1, 231–248 (2007). [CrossRef]
  6. S. Radic, “Parametric signal processing,” IEEE J. Sel. Top. Quantum Electron.18, 670–680 (2012). [CrossRef]
  7. R. Loudon, The Quantum Theory of Light, 3rd Ed. (Oxford, 2000).
  8. M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett.14, 983–985 (2002). [CrossRef]
  9. M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express17, 4670–4676 (2009). [CrossRef] [PubMed]
  10. O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett.102, 123603 (2009). [CrossRef] [PubMed]
  11. K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron.28, 883–894 (1992). [CrossRef]
  12. C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express12, 2033–2055 (2004). [CrossRef] [PubMed]
  13. M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optic parametric amplifiers with lineary or circularly polarized waves,” J. Opt. Soc. Am. B20, 2425–2433 (2003). [CrossRef]
  14. C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express14, 8516–8534 (2006). [CrossRef] [PubMed]
  15. G. W. Stewart, “On the early history of the singular value decomposition,” SIAM Rev.35, 551–566 (1993). [CrossRef]
  16. G. J. Gbur, Mathematical Methods for Optical Physics and Engineering (Cambridge, 2011), Sec. 5.4.
  17. A. K. Ekert and P. L. Knight, “Relationship between semiclassical and quantum-mechanical input-output theories of optical response,” Phys. Rev. A43, 3934–3938 (1991). [CrossRef] [PubMed]
  18. S. L. Braunstein, “Squeezing as an irreducible resource,” Phys. Rev. A71, 055801 (2005). [CrossRef]
  19. H. P. Yuen, “Two-photon states of the radiation field,” Phys. Rev. A13, 2226–2243 (1976). [CrossRef]
  20. C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D26, 1817–1839 (1982). [CrossRef]
  21. C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: Effective finite Hilbert space and entropy control,” Phys. Rev. Lett.84, 5304–5307 (2000). [CrossRef] [PubMed]
  22. W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A64, 063815 (2001). [CrossRef]
  23. M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun.283, 747–752 (2010). [CrossRef]
  24. C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical pulse reshaping and multiplexing by four-wave mixing in fibers,” Phys. Rev. A85, 053829 (2012). [CrossRef]
  25. C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express13, 4986–5012 (2005). [CrossRef] [PubMed]
  26. Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.18, 1016–1032 (2012). [CrossRef]
  27. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Euro. Trans. Telecom.10, 585–595 (1999). [CrossRef]
  28. C. J. McKinstrie and N. Alic, “Information efficiencies of parametric devices,” J. Sel. Top. Quantum Electron.18, 794–811 (2012). [CrossRef]
  29. C. J. McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun.282, 583–593 (2009). [CrossRef]
  30. D. A. Edwards, J. D. Fehribach, R. O. Moore, and C. J. McKinstrie, “An application of matrix theory to the evolution of coupled modes,” to appear in SIAM Rev.
  31. C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express12, 4973–4979 (2004). [CrossRef] [PubMed]
  32. K. Croussore and G. Li, “Phase and amplitude regeneration of differential phase-shift keyed signals using phase-sensitive amplification,” IEEE J. Sel. Top. Quantum Electron.14, 648–658 (2008). [CrossRef]
  33. J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE44, 904–913 (1956). [CrossRef]
  34. M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE45, 1012–1013 (1957).
  35. A. I. Lvovsky, W. Wasilewski, and K. Banaszek, “Decomposing a pulsed optical parametric amplifier into independent squeezers,” J. Mod. Opt.54, 721–733 (2007). [CrossRef]
  36. C. J. McKinstrie, M. G. Raymer, and H. J. McGuinness, “Spatial-temporal evolution of asymmetrically-pumped phase conjugation I: General formalism,” Alcatel-Lucent ITD-09-48636Q, available upon request.
  37. H. Goldstein, Classical Mechanics, 2nd Ed. (Addison-Wesley, 1980).
  38. V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd Ed. (Springer, 2000).
  39. D. H. Sattinger and O. L. Weaver, Lie groups and Algebras with Applications to Physics, Geometry and Mechanics (Springer, 1986). [CrossRef]
  40. M. Hamermesh, Group Theory and its Application to Physical Problems (Dover, 1989).
  41. H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nou. Rev. Opt.4, 37–41 (1973). [CrossRef]
  42. Y. S. Kim and M. E. Noz, “Illustrative examples of the symplectic group,” Am. J. Phys.51, 368–375 (1983). [CrossRef]
  43. A. Mufti, H. A. Schmitt, and M. Sargent, “Finite-dimensional matrix representations as calculational tools in quantum optics,” Am. J. Phys.61, 729–733 (1993). [CrossRef]
  44. C. C. Gerry, “Remarks on the use of group theory in quantum optics,” Opt. Express8, 76–85 (2001). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


Fig. 1

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited