## Schmidt decompositions of parametric processes II: Vector four-wave mixing |

Optics Express, Vol. 21, Issue 9, pp. 11009-11020 (2013)

http://dx.doi.org/10.1364/OE.21.011009

Acrobat PDF (837 KB)

### Abstract

In vector four-wave mixing, one or two strong pump waves drive two weak signal and idler waves, each of which has two polarization components. In this paper, vector four-wave mixing processes in a randomly-birefringent fiber (modulation interaction, phase conjugation and Bragg scattering) are studied in detail. For each process, the Schmidt decompositions of the coupling matrices facilitate the solution of the signal–idler equations and the Schmidt decomposition of the associated transfer matrix. The results of this paper are valid for arbitrary pump polarizations.

© 2013 OSA

## 1. Introduction

1. 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] .

2. C. J. McKinstrie, S. Radic, and A. H. Gnauck, “All-optical signal processing by fiber-based parametric devices,” Opt. Photon. News **18**(3), 34–40 (2007) [CrossRef] .

3. H. Cruz-Ramirez, R. Ramirez-Alarcon, M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Spontaneous parametric processes in quantum optics,” Opt. Photon. News **22**(11), 37–41 (2011) [CrossRef] .

4. M. G. Raymer and K. Srinivasan, “Manipulating the color and shape of single photons,” Phys. Today **65**(11), 32–37 (2012) [CrossRef] .

*d*=

_{z}*d/dz*is a space derivative,

*X*

_{1}= [

*x*

_{1j}] and

*X*

_{2}= [

*x*

_{2j}] are

*m*× 1 mode-amplitude vectors,

*J*

_{1},

*J*

_{2}and

*K*are

*m*×

*m*coefficient matrices, and the superscripts * and

*t*denote complex conjugate and transpose, respectively. The self-action (-coupling) matrices

*J*

_{1}and

*J*

_{2}are Hermitian, whereas the cross-coupling matrix

*K*is arbitrary. Equations (1) can be rewritten in the compact form where the 2

*m*× 1 mode vector and 2

*m*× 2

*m*coefficient matrix are respectively. Because Eq. (2) is linear in the mode vector, its solution can be written in the input–output (IO) form where the transfer (Green) matrix satisfies Eq. (2) and the input condition

*T*(0) =

*I*. The mathematical properties of this evolution equation and its solution were studied in detail in [5

5. C. J. McKinstrie and M. Karlsson, “Schmidt decompositions of parametric processes I: Basic theory and simple examples,” Opt. Express **21**, 1374–1394 (2013) and references therein [CrossRef] [PubMed] .

*U*

_{1},

*U*

_{2},

*V*

_{1}and

*V*

_{2}are unitary matrices,

*D*= diag(

_{μ}*μ*) is a positive diagonal matrix,

_{j}*D*diag(

_{ν}*ν*) is a non-negative diagonal matrix and

_{j}*j*= 1,...,

*m*. The columns of

*U*are input Schmidt mode-vectors, the columns of

_{j}*V*are output Schmidt mode-vectors, and the entries of

_{j}*D*and

_{μ}*D*are Schmidt coefficients that satisfy the auxiliary equations

_{ν}*U*

_{1}and

*X*

_{1}(0) and

*V*

_{1}and

*X*

_{1}(

*z*) and

*x̄*

_{1j}and

*X*

_{1}and

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

7. C. M. Caves, “Quantum limits on noise in linear ampifiers,” Phys. Rev. D **26**, 1817–1839 (1982) [CrossRef] .

5. C. J. McKinstrie and M. Karlsson, “Schmidt decompositions of parametric processes I: Basic theory and simple examples,” Opt. Express **21**, 1374–1394 (2013) and references therein [CrossRef] [PubMed] .

8. K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron. **28**, 883–894 (1992) [CrossRef] .

10. H. Kogelnik and C. J. McKinstrie, “Dynamic eigenstates of parametric interactions in randomly birefringent fibers,” IEEE Photon. Technol. Lett. **21**, 1036–1038 (2009) [CrossRef] .

## 2. Modulation interaction

*∂*and

_{z}*∂*are space and time derivatives, respectively,

_{t}*A*= [

*x*,

*y*]

*is the two-component amplitude vector,*

^{t}*γ*= 8

*γ*/9 is proportional to the Kerr nonlinearity coefficient

_{K}*γ*and the superscript † denotes Hermitian conjugate. In the frequency domain, the dispersion function

_{K}*k*are dispersion coefficients evaluated at some reference (carrier) frequency, and

_{n}*ω*is the difference between the actual frequency and this carrier frequency. One converts from the frequency domain to the time domain by replacing

*ω*with

*i∂*. Equation (7) is the simplest equation that models the effects of convection, dispersion, nonlinearity and polarization, and is sometimes called the Manakov equation [11–17

_{t}17. C. J. McKinstrie, H. Kogelnik, G. G. Luther, and L. Schenato, “Stokes-space derivations of generalized Schrödinger equations for wave propagation in various fibers,” Opt. Express **15**, 10964–10983 (2007) [CrossRef] [PubMed] .

18. J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci. **97**, 4541–4550 (2000) [CrossRef] [PubMed] .

19. P. O. Hedekvist, M. Karlsson, and P. A. Andrekson, “Polarization dependence and efficiency in a fiber four-wave mixing phase conjugator with orthogonal pump waves,” IEEE Photon. Technol. Lett. **8**, 776–778 (1996) [CrossRef] .

20. F. Yaman, Q. Lin, and G. P. Agrawal, “Effects of polarization-mode dispersion in dual-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. **16**, 431–433 (2004) [CrossRef] .

*p*) drives weak signal (

*s*) and idler (

*r*) waves (sidebands), subject to the frequency-matching condition 2

*ω*=

_{p}*ω*+

_{r}*ω*, which is illustrated in Fig. 1(a). By substituting the three-frequency ansatz in Eq. (7) and collecting terms of like frequency, one obtains the MI equations where the wavenumbers

_{s}*β*=

_{j}*β*(

*ω*) and

_{j}*j*=

*p*,

*r*or

*s*. For reference, this procedure is described in [9

9. C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express **12**, 2033–2055 (2004) [CrossRef] [PubMed] .

10. H. Kogelnik and C. J. McKinstrie, “Dynamic eigenstates of parametric interactions in randomly birefringent fibers,” IEEE Photon. Technol. Lett. **21**, 1036–1038 (2009) [CrossRef] .

*A*depends on

_{p}*z*, so also do the coupling matrices.

*O*, which satisfies the evolution equation and the input condition

_{p}*O*(0) =

_{p}*I*. Because the pump equation conserves the products |

*A*|

_{p}^{2}and

*O*describes linear PM and nonlinear PR, which in Stokes space [18

_{p}18. J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci. **97**, 4541–4550 (2000) [CrossRef] [PubMed] .

*γ*|

*A*|

_{p}^{2}

*z*[9

9. C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express **12**, 2033–2055 (2004) [CrossRef] [PubMed] .

21. M. Karlsson and H. Sunnerud, “Effects of nonlinearities on PMD-induced system impairments,” J. Lightwave Technol. **24**, 4127–4137 (2006) [CrossRef] .

*d*= 0: The transformed pump vector is constant. By substituting the other definitions in Eqs. (10) and (11), one obtains the transformed MI equations Notice that the self-coupling matrices are still Hermitian and the (common) cross-coupling matrix is still symmetric, but all three matrices are now constant. By measuring the phases of

_{z}B_{p}*B*,

_{p}*B*and

_{r}*B*relative to a common reference phase (which could be the input phase of one of the components of

_{s}*B*), one can remove common phase factors from Eqs. (15) and (16).

_{p}*M*has the Schmidt decomposition

*M*=

*VDU*

^{†}, where

*U*and

*V*are unitary matrices and

*D*is a non-negative diagonal matrix. The columns of

*U*(input Schmidt vectors) are the eigenvectors of

*M*

^{†}

*M*, the columns of

*V*(output Schmidt vectors) are the eigenvectors of

*MM*

^{†}, and the entries of

*D*(Schmidt coefficients) are the square roots of the (common) eigenvalues of

*M*

^{†}

*M*and

*MM*

^{†}. Because the cross-coupling matrix

*K*=

*VD*. Let

_{γ}V^{t}*E*

_{‖}and

*E*

_{⊥}denote unit vectors that are parallel and perpendicular (orthogonal) to the pump vector

*B*. Then, in the context of MI, the columns of

_{p}*V*are

*E*

_{‖}and

*E*

_{⊥}, and the diagonal entries of

*D*are

_{γ}*γ*|

*B*|

_{p}^{2}and 0 (parallel sidebands couple to the pump, whereas perpendicular sidebands do not couple). The self-coupling matrices are proportional to the identity matrix, which has the unitary decomposition

*I*=

*VV*

^{†}. Notice that the polarization properties of MI are determined completely by the Schmidt vectors of the cross-coupling matrix.

*j*= ‖ or ⊥. The wavenumber mismatches

*δ*=

_{r}*β*−

_{r}*β*+

_{p}*γ*|

*B*|

_{p}^{2}and

*δ*=

_{s}*β*−

_{s}*β*+

_{p}*γ*|

*B*|

_{p}^{2}, and the coupling coefficients

*γ*

_{‖}=

*γ*|

*B*|

_{p}^{2}and

*γ*

_{⊥}= 0. Equations (18) describe two-mode stretching and squeezing. Their solutions, which are well known, can be written in the IO forms where the transfer functions and phase factor are respectively. In these formulas, the mismatch average

*δ*= (

_{a}*δ*+

_{r}*δ*)/2, the mismatch difference

_{s}*δ*= (

_{d}*δ*−

_{r}*δ*)/2 and the MI wavenumbers

_{s}*k*

_{‖}can be imaginary, so the parallel process is conditionally unstable (as required for amplification). For the perpendicular process

*γ*

_{⊥}= 0, so

*k*

_{⊥}=

*δ*,

_{a}*ν*

_{⊥}(

*z*) = 0,

*e*(

*z*)

*μ*

_{⊥}(

*z*) = exp(

*iδ*) and

_{r}z*e*

^{*}(

*z*)

*μ*

_{⊥}(

*z*) = exp(

*iδ*).

_{s}z*eD*,

_{μ}*eD*and

_{ν}*V*, rather than the four matrices allowed by the general theory of parametric processes. Let

*ϕ*= arg(

_{e}*e*),

*ϕ*= arg(

_{μ}*μ*) and

*ϕ*= arg(

_{ν}*ν*), and define the phase average

*ϕ*= (

_{a}*ϕ*+

_{μ}*ϕ*)/2 and phase difference

_{ν}*ϕ*= (

_{d}*ϕ*−

_{ν}*ϕ*)/2, which depend implicitly on

_{μ}*j*. Furthermore, define the column vectors

*U*=

_{j}*V*exp(

_{j}*iϕ*),

_{d}*V*=

_{rj}*V*exp[

_{j}*i*(

*ϕ*+

_{a}*ϕ*)] and

_{e}*V*=

_{sj}*V*exp[

_{j}*i*(

*ϕ*−

_{a}*ϕ*)]. Then, by using this notation, one can rewrite Eq. (27) in the (canonical) Schmidt form in which the diagonal matrices |

_{e}*D*| and |

_{μ}*D*| are non-negative. Notice that in Eq. (28) the output Schmidt vectors of the signal and idler are different. However, if one were to measure the output signal and idler phases relative to

_{ν}*ϕ*and −

_{e}*ϕ*, respectively, this difference would disappear and decomposition (28) would involve only two unitary matrices (

_{e}*U*and

*V*).

## 3. Phase conjugation

*p*and

*q*) drive weak sidebands (

*r*and

*s*), subject to the frequency-matching condition

*ω*+

_{p}*ω*=

_{q}*ω*+

_{r}*ω*, which is illustrated in Fig. 2. By substituting the four-frequency ansatz in Eq. (7) and collecting terms of like frequency, one obtains the PC equations The right sides of Eqs. (30)–(33) contain the scalar operators

_{s}*A*and

_{p}*A*depend on

_{q}*z*, so also do the coupling matrices.

*O*and

_{p}*O*, which satisfy the evolution equations together with the input conditions

_{q}*O*(0) =

_{p}*I*and

*O*(0) =

_{q}*I*. Because the pump equations conserve the products |

*A*|

_{p}^{2}, |

*A*|

_{q}^{2}and

9. C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express **12**, 2033–2055 (2004) [CrossRef] [PubMed] .

21. M. Karlsson and H. Sunnerud, “Effects of nonlinearities on PMD-induced system impairments,” J. Lightwave Technol. **24**, 4127–4137 (2006) [CrossRef] .

*d*= 0 and

_{z}B_{p}*d*= 0: The transformed pump vectors are constant. By substituting definitions (38) and (39) in Eqs. (32) and (33), and using the facts that

_{z}B_{q}*δ*=

_{r}*β*−

_{r}*β*+

_{p}*γ*|

*B*|

_{p}^{2}and

*δ*=

_{s}*β*−

_{s}*β*+

_{q}*γ*|

*B*|

_{q}^{2}, and the (common) cross-coupling matrix

**12**, 2033–2055 (2004) [CrossRef] [PubMed] .

10. H. Kogelnik and C. J. McKinstrie, “Dynamic eigenstates of parametric interactions in randomly birefringent fibers,” IEEE Photon. Technol. Lett. **21**, 1036–1038 (2009) [CrossRef] .

*p⃗*and

*q⃗*denote the (unit) Stokes vectors of pumps

*p*and

*q*, respectively. Then the Stokes representations of the idler and signal (unit) Schmidt vectors are ±

*r⃗*and ±

*s⃗*, respectively, where For reference, if a Jones vector has the Stokes representation (

*v*

_{1},

*v*

_{2},

*v*

_{3}), the conjugate vector has the representation (

*v*

_{1}, −

*v*

_{2}, −

*v*

_{3}). Pump vectors that are perpendicular in Jones space are anti-parallel in Stokes space [18

18. J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci. **97**, 4541–4550 (2000) [CrossRef] [PubMed] .

**21**, 1036–1038 (2009) [CrossRef] .

*D*) are where

_{γ}*p⃗*·

*q⃗*) are illustrated in Fig. 3. Parallel pumps produce strong sideband-polarization-dependent coupling (

*γ*

_{+}= 2|

*B*| and

_{p}B_{q}*γ*

_{−}= 0), whereas perpendicular pumps provide moderate polarization-independent coupling (

*γ*

_{+}=

*γ*

_{−}= |

*B*|). Notice that

_{p}B_{q}*γ*

_{+}+

*γ*

_{−}= 2|

*B*|.

_{p}B_{q}*δ*,

_{r}*δ*and

_{s}*V*), so no further analysis is required. Nonetheless, it is instructive to define the alternative amplitudes where

*δ*was defined after Eq. (23). By substituting these definitions in Eqs. (40) and (41), one obtains the alternative (symmetrized) PC equations where

_{d}*δ*also was defined after Eq. (23). In Eqs. (45) and (46) the mismatches are equal, so the phase factor

_{a}*e*(

*z*) does not appear in the associated Schmidt-like decomposition (27) and only two unitary matrices (

*U*and

*V*) appear in the associated Schmidt decomposition (28), as stated previously.

*ω*=

_{r}*ω*and the pumps drive only a single sideband (

_{s}*s*), subject to the frequency-matching condition

*ω*+

_{p}*ω*= 2

_{q}*ω*, which is illustrated in Fig. 1(b). For this degenerate process, the pump equations (30) and (31) are unchanged, and the signal equation is It is only because the cross-coupling matrix is symmetric that Eqs. (32) and (33) have this common limit. It is convenient to define the unitary operator which is a symmetric combination of the operators

_{s}*O*and

_{p}*O*. By using

_{q}*O*in the second of Eqs. (39), one obtains the transformed signal equation where the mismatch

_{s}*δ*=

_{s}*β*− (

_{s}*β*+

_{p}*β*)/2 +

_{q}*γ*(|

*B*|

_{p}^{2}+ |

*B*|

_{q}^{2})/2 depends symmetrically on the pump wavenumbers and powers. Thus, the cross-coupling matrix for inverse MI is the same as that for PC, so it remains true that

*K*=

*VD*, where the Schmidt vectors (columns of

_{γ}V^{t}*V*) and coefficients (entries of

*D*) were defined by Eqs. (42) and (43), respectively. The equations for the signal vector and its conjugate are similar to Eqs. (45) and (46), so the IO relations for these quantities can be written in the form of Eq. (27), but without the phase factor

_{γ}*e*(because

*δ*= 0).

_{d}*e*]

^{iϕ}*/2*

^{t}^{1/2}and [1, −

*e*]

^{iϕ}*/2*

^{t}^{1/2}, where

*ϕ*is an arbitrary phase. For example, if the pumps are polarized linearly along reference axes,

*ϕ*= 0 corresponds to signals polarized linearly at ±45° to these axes, whereas

*ϕ*=

*π*/2 corresponds to left- and right-circularly-polarized signals. If the pumps are circularly polarized,

*ϕ*= 0 corresponds to signals polarized linearly along the axes, whereas

*ϕ*=

*π*/2 corresponds to signals polarized linearly at ±45° to the axes. The preceding results generalize those of [22

22. C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express **12**, 4973–4979 (2004) [CrossRef] [PubMed] .

23. C. J. McKinstrie, M. G. Raymer, S. Radic, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. **257**, 146–163 (2006) [CrossRef] .

## 4. Bragg scattering

*p*and

*q*) drive weak sidebands (

*r*and

*s*), subject to the frequency-matching condition

*ω*+

_{p}*ω*=

_{s}*ω*+

_{q}*ω*, which is illustrated in Fig. 4. By substituting the four-frequency ansatz (29) in Eq. (7) and collecting terms of like frequency, one obtains the BS equations Equations (50) and (51) are identical to Eqs. (30) and (31), respectively. In Eqs. (52) and (53), the self-coupling matrices are Hermitian, and the coupling matrices satisfy the equation

_{r}*A*is coupled to

_{r}*A*, rather than

_{s}*J*and

_{r}*J*are the aforementioned self-coupling matrices and

_{s}*K*is the (common) coupling matrix. Notice that

*H*is Hermitian. Equation (54) is both a special case of Eq. (2), in which

*J*

_{1}=

*H*and the other block matrices are absent, and an equation worthy of study in its own right.

*U*and

_{j}*V*are unitary matrices,

_{j}*D*= diag(

_{τ}*τ*) and

_{j}*D*= diag(

_{ρ}*ρ*) are non-negative diagonal matrices whose entries satisfy the auxiliary equations

_{j}*j*= 1 or 2 [24

24. 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] .

25. S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam splitter,” Opt. Commun. **62**, 139–145 (1987) [CrossRef] .

26. H. Fearn and R. Loudon, “Quantum theory of the lossless beam splitter,” Opt. Commun. **64**, 485–490 (1987) [CrossRef] .

*K*=

*UD*

_{γ}V^{†}, whereas the self-coupling matrices are proportional to the identity matrix, which has the unitary decompositions

*I*=

*UU*

^{†}=

*VV*

^{†}. Hence, the polarization properties of BS are determined completely by the Schmidt vectors of the coupling matrix. Specific formulas for these vectors are stated in terms of the pump components and Stokes vectors in [9

**12**, 2033–2055 (2004) [CrossRef] [PubMed] .

**21**, 1036–1038 (2009) [CrossRef] .

*r⃗*and ±

*s⃗*, respectively, where and the associated Schmidt coefficients are The dependences of these coefficients on the pump-polarization alignment is illustrated in Fig. 5. For any pump alignment, there are strongly- and weakly-coupled sideband polarizations: The coupling is always sideband-polarization dependent. Notice that

*γ*

_{+}−

*γ*

_{−}= |

*B*|.

_{p}B_{q}*δ*=

_{r}*β*−

_{r}*β*+

_{p}*γ*|

*B*|

_{p}^{2}and

*δ*=

_{s}*β*−

_{s}*β*+

_{q}*γ*|

*B*|

_{q}^{2}, and

*j*= + or −. Equations (62) describe two-mode beam splitting (frequency conversion). Their solutions, which are well known, can be written in the IO forms where the transfer functions and phase factor are respectively. In these formulas, the mismatches

*δ*= (

_{a}*δ*+

_{r}*δ*)/2 and

_{s}*δ*= (

_{d}*δ*−

_{r}*δ*)/2, and the BS wavenumbers

_{s}*τ*and

_{j}*ρ*depend on

_{j}*δ*, rather than

_{d}*δ*, and the

_{a}*k*are real, so BS is always stable.

_{j}*U*and

*V*), rather than the four matrices allowed by Eq. (56). Let

*ϕ*= arg(

_{τ}*τ*) and

*ϕ*= arg(

_{ρ}*ρ*), and define the phase average

*ϕ*= (

_{a}*ϕ*+

_{τ}*ϕ*)/2 and phase difference

_{ρ}*ϕ*= (

_{d}*ϕ*−

_{ρ}*ϕ*)/2, which depend implicitly on

_{τ}*j*. Furthermore, define the column vectors

*U*=

_{rj}*U*exp(

_{j}*iϕ*),

_{d}*V*=

_{rj}*U*exp[

_{j}*i*(

*ϕ*+

_{e}*ϕ*)],

_{a}*U*=

_{sj}*V*exp(−

_{j}*iϕ*) and

_{d}*V*=

_{sj}*V*exp[

_{j}*i*(

*ϕ*−

_{e}*ϕ*)]. Then, by using this notation, one can rewrite Eq. (71) in the Schmidt form where the diagonal matrices |

_{a}*D*| and |

_{τ}*D*| are non-negative.

_{ρ}## 5. Summary

## Acknowledgment

## References and links

1. | 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. |

2. | C. J. McKinstrie, S. Radic, and A. H. Gnauck, “All-optical signal processing by fiber-based parametric devices,” Opt. Photon. News |

3. | H. Cruz-Ramirez, R. Ramirez-Alarcon, M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Spontaneous parametric processes in quantum optics,” Opt. Photon. News |

4. | M. G. Raymer and K. Srinivasan, “Manipulating the color and shape of single photons,” Phys. Today |

5. | C. J. McKinstrie and M. Karlsson, “Schmidt decompositions of parametric processes I: Basic theory and simple examples,” Opt. Express |

6. | H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A |

7. | C. M. Caves, “Quantum limits on noise in linear ampifiers,” Phys. Rev. D |

8. | K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron. |

9. | C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express |

10. | H. Kogelnik and C. J. McKinstrie, “Dynamic eigenstates of parametric interactions in randomly birefringent fibers,” IEEE Photon. Technol. Lett. |

11. | S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP |

12. | P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Effects of randomly varying birefringence on soliton interactions in optical fibers,” Opt. Lett. |

13. | S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. |

14. | P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. |

15. | T. I. Lakoba, “Concerning the equations governing nonlinear pulse propagation in randomly birefringent fibers,” J. Opt. Soc. Am. B |

16. | D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. |

17. | C. J. McKinstrie, H. Kogelnik, G. G. Luther, and L. Schenato, “Stokes-space derivations of generalized Schrödinger equations for wave propagation in various fibers,” Opt. Express |

18. | J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci. |

19. | P. O. Hedekvist, M. Karlsson, and P. A. Andrekson, “Polarization dependence and efficiency in a fiber four-wave mixing phase conjugator with orthogonal pump waves,” IEEE Photon. Technol. Lett. |

20. | F. Yaman, Q. Lin, and G. P. Agrawal, “Effects of polarization-mode dispersion in dual-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. |

21. | M. Karlsson and H. Sunnerud, “Effects of nonlinearities on PMD-induced system impairments,” J. Lightwave Technol. |

22. | C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express |

23. | C. J. McKinstrie, M. G. Raymer, S. Radic, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. |

24. | M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. |

25. | S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam splitter,” Opt. Commun. |

26. | H. Fearn and R. Loudon, “Quantum theory of the lossless beam splitter,” Opt. Commun. |

**OCIS Codes**

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(270.6570) Quantum optics : Squeezed states

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: January 3, 2013

Revised Manuscript: April 18, 2013

Manuscript Accepted: April 21, 2013

Published: April 26, 2013

**Citation**

C. J. McKinstrie, J. R. Ott, and M. Karlsson, "Schmidt decompositions of parametric processes II: Vector four-wave mixing," Opt. Express **21**, 11009-11020 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-9-11009

Sort: Year | Journal | Reset

### References

- 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]
- C. J. McKinstrie, S. Radic, and A. H. Gnauck, “All-optical signal processing by fiber-based parametric devices,” Opt. Photon. News18(3), 34–40 (2007). [CrossRef]
- H. Cruz-Ramirez, R. Ramirez-Alarcon, M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Spontaneous parametric processes in quantum optics,” Opt. Photon. News22(11), 37–41 (2011). [CrossRef]
- M. G. Raymer and K. Srinivasan, “Manipulating the color and shape of single photons,” Phys. Today65(11), 32–37 (2012). [CrossRef]
- C. J. McKinstrie and M. Karlsson, “Schmidt decompositions of parametric processes I: Basic theory and simple examples,” Opt. Express21, 1374–1394 (2013) and references therein. [CrossRef] [PubMed]
- H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A13, 2226–2243 (1976). [CrossRef]
- C. M. Caves, “Quantum limits on noise in linear ampifiers,” Phys. Rev. D26, 1817–1839 (1982). [CrossRef]
- K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron.28, 883–894 (1992). [CrossRef]
- 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]
- H. Kogelnik and C. J. McKinstrie, “Dynamic eigenstates of parametric interactions in randomly birefringent fibers,” IEEE Photon. Technol. Lett.21, 1036–1038 (2009). [CrossRef]
- S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP38, 248–253 (1974).
- P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Effects of randomly varying birefringence on soliton interactions in optical fibers,” Opt. Lett.16, 1735–1737 (1991). [CrossRef] [PubMed]
- S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol.10, 28–35 (1992). [CrossRef]
- P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol.14, 148–157 (1996). [CrossRef]
- T. I. Lakoba, “Concerning the equations governing nonlinear pulse propagation in randomly birefringent fibers,” J. Opt. Soc. Am. B13, 2006–2011 (1996). [CrossRef]
- D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15, 1735–1746 (1997). [CrossRef]
- C. J. McKinstrie, H. Kogelnik, G. G. Luther, and L. Schenato, “Stokes-space derivations of generalized Schrödinger equations for wave propagation in various fibers,” Opt. Express15, 10964–10983 (2007). [CrossRef] [PubMed]
- J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci.97, 4541–4550 (2000). [CrossRef] [PubMed]
- P. O. Hedekvist, M. Karlsson, and P. A. Andrekson, “Polarization dependence and efficiency in a fiber four-wave mixing phase conjugator with orthogonal pump waves,” IEEE Photon. Technol. Lett.8, 776–778 (1996). [CrossRef]
- F. Yaman, Q. Lin, and G. P. Agrawal, “Effects of polarization-mode dispersion in dual-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett.16, 431–433 (2004). [CrossRef]
- M. Karlsson and H. Sunnerud, “Effects of nonlinearities on PMD-induced system impairments,” J. Lightwave Technol.24, 4127–4137 (2006). [CrossRef]
- C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express12, 4973–4979 (2004). [CrossRef] [PubMed]
- C. J. McKinstrie, M. G. Raymer, S. Radic, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun.257, 146–163 (2006). [CrossRef]
- 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]
- S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam splitter,” Opt. Commun.62, 139–145 (1987). [CrossRef]
- H. Fearn and R. Loudon, “Quantum theory of the lossless beam splitter,” Opt. Commun.64, 485–490 (1987). [CrossRef]

## 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.

« Previous Article | Next Article »

OSA is a member of CrossRef.