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
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 formsEquations (1) can be rewritten in the compact form Eq. (2) is linear in the mode vector, its solution can be written in the input–output (IO) form Eq. (2) and the input condition T (0) = I. The mathematical properties of this evolution equation and its solution were studied in detail in  and papers cited therein. It was shown that the transfer matrix has the Schmidt decomposition 6, 7].
In a previous paper , 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 [8–10].
2. Modulation interaction
Light-wave propagation in a randomly-birefringent fiber is governed by the vector nonlinear Schrödinger equation (NSE)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]. 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 , 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 2ωp = ωr + ωs, which is illustrated in Fig. 1(a). By substituting the three-frequency ansatzEq. (7) and collecting terms of like frequency, one obtains the MI equations 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 , which produces self-phase modulation (PM) and cross-PM, and the tensor operator , which produces cross-polarization rotation (PR). Notice that , 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 , as required by Eqs. (1). Because the pump vector Ap depends on z, so also do the coupling matrices.
It is convenient to define the operator Op, which satisfies the evolution equationEq. (12) is constant. It is also Hermitian. Hence, the operator 18] is a rotation about the Stokes vector of the pump by the angle 2γ|Ap|2z[9, 21].
It is also convenient to define the transformed amplitude vectorsEq. (9) and using Eq. (13), one finds that dzBp = 0: The transformed pump vector is constant. By substituting the other definitions in Eqs. (10) and (11), one obtains the transformed MI equations Eqs. (15) and (16).
Every complex matrix 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 is symmetric, it has the simpler Schmidt decomposition K = VDγVt. Let E‖ and E⊥ denote unit vectors that are parallel and perpendicular (orthogonal) to the pump vector Bp. Then, in the context of MI, the columns of V are E‖ and E⊥, and the diagonal entries of Dγ are γ|Bp|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 substituting the decompositionsEqs. (15) and (16), one obtains the scalar equations Equations (18) describe two-mode stretching and squeezing. Their solutions, which are well known, can be written in the IO forms Eqs. (15) and (16) in the vector IO forms Equations (25) and (26) can be rewritten in the compact form Eq. (27) is similar to the matrix in Eq. (5). It is in Schmidt-like form, rather than Schmidt form, because the diagonal matrices eDμ, , eDν and 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 ϕe = arg(e), ϕμ = arg(μ) and ϕν = arg(ν), and define the phase average ϕa = (ϕμ + ϕν)/2 and phase difference ϕd = (ϕν − ϕμ)/2, which depend implicitly on j. Furthermore, define the column vectors Uj = Vj exp(iϕd), Vrj = Vj exp[i(ϕa + ϕe)] and Vsj = Vj exp[i(ϕa − ϕe)]. Then, by using this notation, one can rewrite Eq. (27) in the (canonical) Schmidt form 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 ϕe and −ϕe, respectively, this difference would disappear and decomposition (28) would involve only two unitary matrices (U and V).
3. 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 ωp + ωq = ωr + ωs, which is illustrated in Fig. 2. By substituting the four-frequency ansatzEq. (7) and collecting terms of like frequency, one obtains the PC equations Eqs. (30)–(33) contain the scalar operators and , which produce PM, and the tensor operators and , which produce PR. Notice that in Eq. (30) one can replace by and in Eq. (31) one can replace by . Notice also that in Eqs. (32) and (33) the self-coupling matrices are Hermitian, and the cross-coupling matrices satisfy the equation , as required by Eqs. (1). Because the pump vectors Ap and Aq depend on z, so also do the coupling matrices.
It is convenient to define the operators Op and Oq, which satisfy the evolution equations9, 21].
It is also convenient to define the transformed amplitude vectors(38) into Eqs. (30) and (31), and using Eqs. (36) and (37), one finds that dzBp = 0 and dzBq = 0: The transformed pump vectors are constant. By substituting definitions (38) and (39) in Eqs. (32) and (33), and using the facts that , , , and are scalar operators, one obtains the transformed PC equations
The transformed PC equations are similar to their MI counterparts. The self-coupling matrices are diagonal, with (repeated) entries δr = βr − βp + γ|Bp|2 and δs = βs − βq + γ|Bq|2, and the (common) cross-coupling matrix 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  and , respectively. The latter formulas are more compact. Let 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, where18]. This configuration, for which Eq. (42) is indeterminate, is discussed in . The associated Schmidt coefficients (entries of Dγ) are Fig. 3. Parallel pumps produce strong sideband-polarization-dependent coupling (γ+ = 2|BpBq| and γ− = 0), whereas perpendicular pumps provide moderate polarization-independent coupling (γ+ = γ− = |BpBq|). Notice that γ+ + γ− = 2|BpBq|.Eq. (23). By substituting these definitions in Eqs. (40) and (41), one obtains the alternative (symmetrized) PC equations 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), ωr = ωs and the pumps drive only a single sideband (s), subject to the frequency-matching condition ωp + ωq = 2ωs, which is illustrated in Fig. 1(b). For this degenerate process, the pump equations (30) and (31) are unchanged, and the signal equation isEqs. (32) and (33) have this common limit. It is convenient to define the unitary operator Eqs. (39), one obtains the transformed signal equation 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 e (because δd = 0).
For any pump alignment, there are two signal polarizations for which the signal experiences (one-mode) phase-sensitive amplification. The most useful configuration involves perpendicular pumps, for which the amplification strength is signal-polarization independent. If the pump vectors are used as basis vectors, the signal-polarization vectors are [1, eiϕ]t/21/2 and [1, − eiϕ]t/21/2, 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, 23].
4. Bragg scattering
In the nondegenerate FWM process called Bragg scattering (BS), two strong pumps (p and q) drive weak sidebands (r and s), subject to the frequency-matching condition ωp + ωs = ωq + ωr, 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 equationsEquations (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 . Notice that Ar is coupled to As, rather than . This type of coupling differentiates BS from MI and PC.
The sideband equations can be written in the compact formEquation (54) is both a special case of Eq. (2), in which J1 = H and the other block matrices are absent, and an equation worthy of study in its own right.24]. The physical significance of this result is that every BS process, no matter how complicated, can be decomposed into a collection of independent beam-splitter-like processes, about which much is known [25, 26].
Because the pump equations for BS are identical to those for PC, the pump evolution (linear and nonlinear PM, and nonlinear PR) is described by Eqs. (34)–(38). By substituting definitions (38) and (39) in Eqs. (52) and (53), and using the facts that and are scalar operators, one obtains the transformed BS equations9] and , respectively. The Stokes representation of the idler and signal Schmidt vectors are ±r⃗ and ±s⃗, respectively, where 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− = |BpBq|.
By substituting the decompositionsEqs. (57) and (58), one obtains the scalar equations Equations (62) describe two-mode beam splitting (frequency conversion). Their solutions, which are well known, can be written in the IO forms Eqs. (57) and (58) in the vector IO forms Eq. (71) is in Schmidt-like form, because the diagonal matrices and are complex. Nonetheless, Eq. (71) shows that the polarization properties of BS are determined by only two unitary matrices (U and V), rather than the four matrices allowed by Eq. (56). Let ϕτ = arg(τ) and ϕρ = arg(ρ), and define the phase average ϕa = (ϕτ + ϕρ)/2 and phase difference ϕd = (ϕρ − ϕτ)/2, which depend implicitly on j. Furthermore, define the column vectors Urj = Uj exp(iϕd), Vrj = Uj exp[i(ϕe + ϕa)], Usj = Vj exp(−iϕd) and Vsj = Vj exp[i(ϕe − ϕa)]. Then, by using this notation, one can rewrite Eq. (71) in the Schmidt form
In this paper, vector four-wave mixing in a randomly-birefringent fiber was studied for arbitrary pump polarizations. The coupled-mode equations for (inverse) modulation interaction, phase conjugation and Bragg scattering were derived from the Manakov equation (7) and solved analytically. For each process, one can reduce a complicated system of four coupled equations to two simple systems of two coupled equations by using the Schmidt vectors of the cross-coupling matrix as basis vectors. Not only do these Schmidt vectors facilitate the solution of the coupled-mode equations and the Schmidt decomposition of the associated transfer matrix, they also determine completely the polarization properties of each process. This simplification is not required by the Schmidt decomposition theorem. It is a consequence of the facts that the dispersion term in the Manakov equation does not depend on the wave polarizations and the nonlinearity term depends on the polarizations in a relatively simple way.
JRO was supported by the Danish Council for Independent Research.
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) [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] .
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] .
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] .
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] .
11. S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).
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) [CrossRef] .
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) [CrossRef] .
15. T. I. Lakoba, “Concerning the equations governing nonlinear pulse propagation in randomly birefringent fibers,” J. Opt. Soc. Am. B 13, 2006–2011 (1996) [CrossRef] .
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) [CrossRef] .
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] .
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] .
21. M. Karlsson and H. Sunnerud, “Effects of nonlinearities on PMD-induced system impairments,” J. Lightwave Technol. 24, 4127–4137 (2006) [CrossRef] .
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] .
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] .