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

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 Optical Society of America OCIS codes: (060.2320) Fiber optics, amplifiers and oscillators; (190.2620) Frequency conversion; (190.4380) Nonlinear optics, four-wave mixing; (270.6570) Squeezed states. 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. 8, 506–520 (2002). 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). 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). 4. M. G. Raymer and K. Srinivasan, “Manipulating the color and shape of single photons,” Phys. Today 65 (11), 32–37 (2012). 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. 6. H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976). 7. C. M. Caves, “Quantum limits on noise in linear ampifiers,” Phys. Rev. D 26, 1817–1839 (1982). 8. K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron. 28, 883–894 (1992). 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). 10. H. Kogelnik and C. J. McKinstrie, “Dynamic eigenstates of parametric interactions in randomly birefringent fibers,” IEEE Photon. Technol. Lett. 21, 1036–1038 (2009). 11. S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974). 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. 16, 1735–1737 (1991). 13. S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992). 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. 14, 148–157 (1996). #182805 $15.00 USD Received 3 Jan 2013; revised 18 Apr 2013; accepted 21 Apr 2013; published 26 Apr 2013 (C) 2013 OSA 6 May 2013 | Vol. 21, No. 9 | DOI:10.1364/OE.21.011009 | OPTICS EXPRESS 11009 15. T. I. Lakoba, “Concerning the equations governing nonlinear pulse propagation in randomly birefringent fibers,” J. Opt. Soc. Am. B 13, 2006–2011 (1996). 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. 15, 1735–1746 (1997). 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). 18. J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci. 97, 4541–4550 (2000). 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). 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). 21. M. Karlsson and H. Sunnerud, “Effects of nonlinearities on PMD-induced system impairments,” J. Lightwave Technol. 24, 4127–4137 (2006). 22. C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express 12, 4973–4979 (2004). 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). 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). 25. S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam splitter,” Opt. Commun. 62, 139–145 (1987). 26. H. Fearn and R. Loudon, “Quantum theory of the lossless beam splitter,” Opt. Commun. 64, 485–490 (1987).


Introduction
Parametric (wave-mixing) processes provide a variety of signal-processing functions required by classical communication systems [1,2] and quantum information experiments [3,4].Such processes are governed by coupled-mode equations (CMEs) of the forms where d z = d/dz is a space derivative, X 1 = [x 1 j ] and X 2 = [x 2 j ] 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 2m × 1 mode vector and 2m × 2m 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] and papers cited therein.It was shown that the transfer matrix has the Schmidt decomposition Schmidt mode-vectors, the columns of V j are output Schmidt mode-vectors, and the entries of D μ and D ν are Schmidt coefficients that satisfy the auxiliary equations μ 2 j − ν 2 j = 1.By using the columns of U 1 and U * 2 as bases for the input vectors X 1 (0) and X * 2 (0), respectively, and the columns of V 1 and V * 2 as bases for the output vectors X 1 (z) and X * 2 (z), one obtains the CMEs where the Schmidt mode-amplitudes x1 j and x * 2 j are the components of X 1 and X * 2 relative to the aforementioned bases.The physical significance of this result is that every parametric processes (no matter how complicated), can be decomposed into a collection of independent two-mode (stretching and squeezing) processes, about which much is known [6,7].
In a previous paper [5], two specific examples were discussed: Scalar (inverse) modulation interaction (MI) and phase conjugation (PC).Although these examples were sufficient to illustrate the general results, they involved only one or two complex modes: For such processes, the Schmidt decomposition is an elegant, but unnecessary, tool.This paper is the first in a sequence of papers on four-mode parametric processes.Such processes are more complicated than their one-and two-mode counterparts, and their analyses showcase the benefits of Schmidt decompositions.In this paper, vector four-wave mixing (FWM) in a randomly-birefringent fiber is considered [8][9][10].

Modulation interaction
Light-wave propagation in a randomly-birefringent fiber is governed by the vector nonlinear Schrödinger equation (NSE) where ∂ z and ∂ t are space and time derivatives, respectively, A = [x, y] t is the two-component amplitude vector, γ = 8γ K /9 is proportional to the Kerr nonlinearity coefficient γ K and the superscript † denotes Hermitian conjugate.In the frequency domain, the dispersion function where the k n are dispersion coefficients evaluated at some reference (carrier) frequency, and 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∂ t .Equation (7) is the simplest equation that models the effects of convection, dispersion, nonlinearity and polarization, and is sometimes called the Manakov equation [11][12][13][14][15][16][17].It is written in a frame that rotates with the birefringence axes of the fiber, and is based on the assumption that the FWM length is much longer than the length over which the birefringence strength and axes change due to random fiber nonuniformities (1-100 m).Although this condition is barely satisfied for fibers shorter than 1 Km, the predictions of the Manakov equation agree with the results of many recent FWM experiments.The Manakov equation does not account for polarization-mode dispersion [18], which can reduce the FWM efficiency [19,20].
In the degenerate FWM process called modulation interaction (MI), one strong pump wave (p) drives weak signal (s) and idler (r) waves (sidebands), subject to the frequency-matching condition 2ω p = ω r + ω s , which is illustrated in Fig. 1(a).By substituting the three-frequency ansatz A(z,t) = A p (z) exp(−iω p t) + A r (z) exp(−iω r t) + A s (z) exp(−iω s t) (8) in Eq. ( 7) and collecting terms of like frequency, one obtains the MI equations where the wavenumbers β j = β (ω j ) and j = p, r or s.For reference, this procedure is described in [9,10].Notice that the weak sidebands do not affect the strong pump, which is undepleted.The right sides of Eqs. ( 9)-( 11) contain the scalar operator A † p A p = |A p | 2 I, which produces selfphase modulation (PM) and cross-PM, and the tensor operator A p A † p , which produces crosspolarization rotation (PR).Notice that (A † p A p )A p = (A p A † p )A p , so one can write the operator in Eq. ( 9) as a PM or a PR operator, whichever is more convenient.Notice also that in Eqs.(10) and ( 11) the self-coupling operators (matrices) are Hermitian, and the cross-coupling operators (matrices) satisfy the equation A p A t p = (A p A t p ) t , as required by Eqs.(1).Because the pump vector A p depends on z, so also do the coupling matrices.It is convenient to define the operator O p , which satisfies the evolution equation and the input condition O p (0) = I.Because the pump equation conserves the products |A p | 2 and A p A † p , the operator on the right side of Eq. ( 12) is constant.It is also Hermitian.Hence, the operator which is unitary.O p describes linear PM and nonlinear PR, which in Stokes space [18] is a rotation about the Stokes vector of the pump by the angle 2γ|A p | 2 z [9,21].
It is also convenient to define the transformed amplitude vectors By substituting the first of these definitions in Eq. ( 9) and using Eq. ( 13), one finds that d z B p = 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 B p , B r and B s relative to a common reference phase (which could be the input phase of one of the components of B p ), one can remove common phase factors from Eqs. ( 15) and ( 16).
Every complex matrix M has the Schmidt decomposition M = V DU † , 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 = γB p B t p is symmetric, it has the simpler Schmidt decomposition K = V D γ V t .Let E and E ⊥ denote unit vectors that are parallel and perpendicular (orthogonal) to the pump vector B p .Then, in the context of MI, the columns of V are E and E ⊥ , and the diagonal entries of D γ are 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.
By combining Eqs. ( 19) and (20) with Eqs.(17) and their inverses one can write the solutions of Eqs. ( 15) and ( 16) in the vector IO forms Equations ( 25) and ( 26) can be rewritten in the compact form The transfer matrix in Eq. ( 27) is similar to the matrix in Eq. ( 5).It is in Schmidt-like form, rather than Schmidt form, because the diagonal matrices eD μ , eD * μ , eD ν and eD * ν are complex, rather than non-negative.Nonetheless, Eq. ( 27) is useful: It shows that the polarization properties of MI are determined by the single unitary matrix V , rather than the four matrices allowed by the general theory of parametric processes.Let φ e = arg(e), φ μ = arg(μ) and φ ν = arg(ν), and define the phase average φ a = (φ μ + φ ν )/2 and phase difference φ d = (φ ν − φ μ )/2, which depend implicitly on j.Furthermore, define the column vectors U j = V j exp(iφ d ), V r j = V j exp[i(φ a + φ e )] and V s j = V j exp[i(φ a − φ e )].Then, by using this notation, one can rewrite Eq. ( 27) in the (canonical) Schmidt form in which the diagonal matrices |D μ | and |D ν | 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 φ e and −φ e , respectively, this difference would disappear and decomposition (28) would involve only two unitary matrices (U and V ).

Phase conjugation
In the nondegenerate FWM process called phase conjugation (PC), two strong pumps (p and q) drive weak sidebands (r and s), subject to the frequency-matching condition ω p +ω q = ω r +ω s , 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 A † p A p and A † q A q , which produce PM, and the tensor operators A p A † p and A q A † q , which produce PR.Notice that in Eq. ( 30) one can replace (A † p A p )A p by (A p A † p )A p and in Eq. (31) one can replace (A † q A q )A q by (A q A † q )A q .Notice also that in Eqs.(32) and (33) the self-coupling matrices are Hermitian, and the crosscoupling matrices satisfy the equation A p A t q + A q A t p = (A p A t q + A q A t p ) t , as required by Eqs.(1).Because the pump vectors A p and A q depend on z, so also do the coupling matrices.It is convenient to define the operators O p and O q , which satisfy the evolution equations together with the input conditions O p (0) = I and O q (0) = I.Because the pump equations conserve the products |A p | 2 , |A q | 2 and A p A † p + A q A † q , the operators These unitary operators describe linear and nonlinear PM, and nonlinear PR, which in Stokes space is a rotation about the total Stokes vector of the pumps [9,21].
It is also convenient to define the transformed amplitude vectors By substituting definitions (38) into Eqs.( 30) and (31), and using Eqs.( 36) and (37), one finds that d z B p = 0 and d z B q = 0: The transformed pump vectors are constant.By substituting definitions (38) and (39) in Eqs. ( 32) and (33), and using the facts that O † p O q , O t p O * q , O † q O p , and O t q O * p are scalar operators, one obtains the transformed PC 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.The transformed PC equations are similar to their MI counterparts.The self-coupling matrices are diagonal, with (repeated) entries 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 γ(B p B t q + B q B t p ) is symmetric.Hence, the polarization properties of PC are determined completely by the Schmidt vectors of the cross-coupling matrix.Specific formulas for these vectors are stated in terms of the pump components and Stokes vectors in [9] and [10], respectively.The latter formulas are more compact.Let p and q denote the (unit) Stokes vectors of pumps p and q, respectively.Then the Stokes representations of the idler and signal (unit) Schmidt vectors are ± r and ± s, 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].This configuration, for which Eq. ( 42) is indeterminate, is discussed in [10].The associated Schmidt coefficients (entries of D γ ) are where |B p B q | = (B † p B p B † q B q ) 1/2 .The dependences of these coefficients (coupling strengths) on the polarization alignment of the pumps ( p • q) are illustrated in Fig. 3. Parallel pumps produce strong sideband-polarization-dependent coupling (γ + = 2|B p B q | and γ − = 0), whereas perpendicular pumps provide moderate polarization-independent coupling (γ Equations ( 17)-(28) also apply to PC (with the appropriate definitions of δ r , δ s and V ), so no further analysis is required.Nonetheless, it is instructive to define the alternative amplitudes where δ d was defined after Eq. ( 23).By substituting these definitions in Eqs. ( 40) and (41), one obtains the alternative (symmetrized) PC equations where δ a also was defined after Eq. ( 23).In Eqs. ( 45) and ( 46) the mismatches are equal, so the phase factor 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.
In degenerate PC (inverse MI), ω r = ω s and the pumps drive only a single sideband (s), subject to the frequency-matching condition ω p + ω q = 2ω s , which is illustrated in Fig. 1(b).For this degenerate process, the pump equations (30) and (31) are unchanged, and the signal equation is d z A s = iβ s A s + iγ(A † p A p + A p A † p + A † q A q + A q A † q )A s + iγ(A p A t q + A q A t p )A * s . (47) 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 O p and O q .By using O s in the second of Eqs.(39), one obtains the transformed signal equation where the mismatch δ s = β s − (β p + β q )/2 + 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

Fig. 1 .
Fig.1.Frequency diagrams for (a) modulation interaction and (b) inverse modulation interaction.Long arrows denote pumps (p and q), whereas short arrows denote sidebands (r and s).Downward arrows denote modes that lose photons, whereas upward arrows denote modes that gain photons.

Fig. 2 .
Fig.2.Frequency diagrams for (a) outer-band and (b) inner-band phase conjugation.Long arrows denote pumps (p and q), whereas short arrows denote sidebands (r and s).Downward arrows denote modes that lose photons, whereas upward arrows denote modes that gain photons.