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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 9 — May. 6, 2013
  • pp: 11009–11020
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Schmidt decompositions of parametric processes II: Vector four-wave mixing

C. J. McKinstrie, J. R. Ott, and M. Karlsson  »View Author Affiliations


Optics Express, Vol. 21, Issue 9, pp. 11009-11020 (2013)
http://dx.doi.org/10.1364/OE.21.011009


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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

Parametric (wave-mixing) processes provide a variety of signal-processing functions required by classical communication systems [1

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

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] .

] and quantum information experiments [3

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

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

]. Such processes are governed by coupled-mode equations (CMEs) of the forms
dzX1=iJ1X1+iKX2*,dzX2=iJ2X2+iKtX1*,
(1)
where dz = d/dz is a space derivative, X1 = [x1j] and X2 = [x2j] are m × 1 mode-amplitude vectors, J1, J2 and K are m × m coefficient matrices, and the superscripts * and t denote complex conjugate and transpose, respectively. The self-action (-coupling) matrices J1 and J2 are Hermitian, whereas the cross-coupling matrix K is arbitrary. Equations (1) can be rewritten in the compact form
dzX=iLX,
(2)
where the 2m × 1 mode vector and 2m × 2m coefficient matrix are
X=[X1X2*],L=[J1KKJ2*],
(3)
respectively. Because Eq. (2) is linear in the mode vector, its solution can be written in the input–output (IO) form
X(z)=T(z)X(0),
(4)
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] .

] and papers cited therein. It was shown that the transfer matrix has the Schmidt decomposition
T(z)=[V1DμU1V1DνU2tV2*DνU1V2*DμU2t],
(5)
where U1, U2, V1 and V2 are unitary matrices, Dμ = diag(μj) is a positive diagonal matrix, Dν diag(νj) is a non-negative diagonal matrix and j = 1,..., m. The columns of Uj are input Schmidt mode-vectors, the columns of Vj are output Schmidt mode-vectors, and the entries of Dμ and Dν are Schmidt coefficients that satisfy the auxiliary equations μj2νj2=1. By using the columns of U1 and U2* as bases for the input vectors X1(0) and X2*(0), respectively, and the columns of V1 and V2* as bases for the output vectors X1(z) and X2*(z), one obtains the CMEs
x¯1j(0)=μj(z)x¯1j(0)+νj(z)x¯2j*(0),x¯2j*(0)=νj(z)x¯1j(0)+μj(z)x¯2j*(0),
(6)
where the Schmidt mode-amplitudes 1j and x¯2j* are the components of X1 and X2* 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

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

, 7

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

].

2. Modulation interaction

Light-wave propagation in a randomly-birefringent fiber is governed by the vector nonlinear Schrödinger equation (NSE)
zA=iβ(it)A+iγ(AA)A,
(7)
where z and t are space and time derivatives, respectively, A = [x, y]t is the two-component amplitude vector, γ = 8γK/9 is proportional to the Kerr nonlinearity coefficient γK and the superscript † denotes Hermitian conjugate. In the frequency domain, the dispersion function β(ω)=n=1knωn/n!, where the kn 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

11. S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

17

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] .

]. 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

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] .

], which can reduce the FWM efficiency [19

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

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] .

].

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 ansatz
A(z,t)=Ap(z)exp(iωpt)+Ar(z)exp(iωrt)+As(z)exp(iωst)
(8)
in Eq. (7) and collecting terms of like frequency, one obtains the MI equations
dzAp=iβpAp+iγ(ApAp)Ap,
(9)
dzAr=iβrAr+iγ(ApAp+ApAp)Ar+iγ(ApApt)As*,
(10)
dzAs=iβsAs+iγ(ApAp+ApAp)As+iγ(ApApt)Ar*,
(11)
where the wavenumbers βj = β(ωj) and 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

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] .

]. 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 ApAp=|Ap|2I, which produces self-phase modulation (PM) and cross-PM, and the tensor operator ApAp, which produces cross-polarization rotation (PR). Notice that (ApAp)Ap=(ApAp)Ap, 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 ApApt=(ApApt)t, as required by Eqs. (1). Because the pump vector Ap depends on z, so also do the coupling matrices.

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.

It is convenient to define the operator Op, which satisfies the evolution equation
dzOp=i(βp+γApAp)Op
(12)
and the input condition Op(0) = I. Because the pump equation conserves the products |Ap|2 and ApAp, the operator on the right side of Eq. (12) is constant. It is also Hermitian. Hence, the operator
Op(z)=exp[i(βp+γApAp)z],
(13)
which is unitary. Op describes linear PM and nonlinear PR, which 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] .

] is a rotation about the Stokes vector of the pump by the angle 2γ|Ap|2z[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

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

].

It is also convenient to define the transformed amplitude vectors
Aj(z)=Op(z)Bj(z).
(14)
By substituting the first of these definitions in Eq. (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
dzBr=i(βrβp+γ|Bp|2)Br+iγ(BpBpt)Bs*,
(15)
dzBs=i(βsβp+γ|Bp|2)Bs+iγ(BpBpt)Br*.
(16)
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 Bp, Br and Bs relative to a common reference phase (which could be the input phase of one of the components of Bp), one can remove common phase factors from 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 MM, 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 MM and MM. Because the cross-coupling matrix K=γBpBpt 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 decompositions
Br=jbrjVjBs=jbsjVj,
(17)
which are based on the same Schmidt vectors, in Eqs. (15) and (16), one obtains the scalar equations
dzbrj=iδrbrj+iγjbsj*,dzbsj=iδsbsj+iγjbrj*,
(18)
where j = ‖ or ⊥. The wavenumber mismatches δr = βrβp + γ|Bp|2 and δs = βsβp +γ|Bp|2, and the coupling coefficients γ = γ|Bp|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
brj(z)=e(z)μj(z)brj(0)+e(z)νj(z)bsj*(0),
(19)
bsj(z)=e*(z)μj(z)bsj(0)+e*(z)νj(z)brj*(0),
(20)
where the transfer functions and phase factor are
μj(z)=cos(kjz)+iδasin(kjz)/kj,
(21)
νj(z)=iγjsin(kjz)/kj,
(22)
e(z)=exp(iδdz),
(23)
respectively. In these formulas, the mismatch average δa = (δr + δs)/2, the mismatch difference δd = (δrδs)/2 and the MI wavenumbers kj=(δa2γj2)1/2. Notice that k can be imaginary, so the parallel process is conditionally unstable (as required for amplification). For the perpendicular process γ = 0, so k = δa, ν(z) = 0, e(z)μ(z) = exp(rz) and e*(z)μ(z) = exp(sz).

By combining Eqs. (19) and (20) with Eqs. (17) and their inverses
brj=VjBr,bsj=VjBs,
(24)
one can write the solutions of Eqs. (15) and (16) in the vector IO forms
Br(z)=jVje(z)μj(z)VjBr(0)+jVje(z)νj(z)VjtBs*(0),
(25)
Bs*(z)=jVj*e(z)νj*(z)VjBr(0)+jVj*e(z)μj*(z)VjtBs*(0).
(26)
Equations (25) and (26) can be rewritten in the compact form
[Br(z)Bs*(z)]=[VeDμVVeDνVtV*eDν*VV*eDμ*Vt][Br(0)Bs*(0)].
(27)
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 eDμ, eDμ*, eDν and eDν* 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 = (ϕνϕμ)/2, which depend implicitly on j. Furthermore, define the column vectors Uj = Vj exp(d), Vrj = Vj exp[i(ϕa + ϕe)] and Vsj = Vj exp[i(ϕaϕe)]. Then, by using this notation, one can rewrite Eq. (27) in the (canonical) Schmidt form
[Br(z)Bs*(z)]=[Vr|Dμ|UVr|Dν|UtVs*|Dν|UVs*|Dμ|Ut][Br(0)Bs*(0)],
(28)
in which the diagonal matrices |Dμ| and |Dν| 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 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 ansatz
A(z,t)=Ap(z)exp(iωpt)+Aq(z)exp(iωqt)+Ar(z)exp(iωrt)+As(z)exp(iωst)
(29)
in Eq. (7) and collecting terms of like frequency, one obtains the PC equations
dzAp=iβpAp+iγ(ApAp+AqAq+AqAq)Ap,
(30)
dzAq=iβqAq+iγ(AqAq+ApAp+ApAp)Aq,
(31)
dzAr=iβrAr+iγ(ApAp+ApAp+AqAq+AqAq)Ar+iγ(ApAqt+AqApt)As*,
(32)
dzAs=iβsAs+iγ(ApAp+ApAp+AqAq+AqAq)As+iγ(ApAqt+AqApt)Ar*.
(33)
The right sides of Eqs. (30)(33) contain the scalar operators ApAp and AqAq, which produce PM, and the tensor operators ApAp and AqAq, which produce PR. Notice that in Eq. (30) one can replace (ApAp)Ap by (ApAp)Ap and in Eq. (31) one can replace (AqAq)Aq by (AqAq)Aq. Notice also that in Eqs. (32) and (33) the self-coupling matrices are Hermitian, and the cross-coupling matrices satisfy the equation ApAqt+AqApt=(ApAqt+AqApt)t, as required by Eqs. (1). Because the pump vectors Ap and Aq depend on z, so also do the coupling matrices.

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.

It is convenient to define the operators Op and Oq, which satisfy the evolution equations
dzOp=i[βp+γ|Aq|2+γ(ApAp+AqAq)]Op,
(34)
dzOq=i[βq+γ|Ap|2+γ(ApAp+AqAq)]Oq,
(35)
together with the input conditions Op(0) = I and Oq(0) = I. Because the pump equations conserve the products |Ap|2, |Aq|2 and ApAp+AqAq, the operators
Op(z)=exp{i[βp+γ|Aq|2+γ(ApAp+AqAq)]z},
(36)
Oq(z)=exp{i[βq+γ|Ap|2+γ(ApAp+AqAq)]z}.
(37)
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

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

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

].

It is also convenient to define the transformed amplitude vectors
Ap(z)=Op(z)Bp(z),Aq(z)=Oq(z)Bq(z),
(38)
Ar(z)=Op(z)Br(z),As(z)=Oq(z)Bs(z).
(39)
By substituting definitions (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 OpOq, OptOq*, OqOp, and OqtOp* are scalar operators, one obtains the transformed PC equations
dzBr=i(βrβp+γ|Bp|2)Br+iγ(BpBqt+BqBpt)Bs*,
(40)
dzBs=i(βsβq+γ|Bq|2)Bs+iγ(BpBqt+BqBpt)Br*.
(41)
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 δr = βrβp + γ|Bp|2 and δs = βsβq + γ|Bq|2, and the (common) cross-coupling matrix γ(BpBqt+BqBpt) 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

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] .

] and [10

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] .

], 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, where
r=s=(p+q)/(2+2pq)1/2.
(42)
For reference, if a Jones vector has the Stokes representation (v1, v2, v3), the conjugate vector has the representation (v1, − v2, − v3). 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] .

]. This configuration, for which Eq. (42) is indeterminate, is discussed in [10

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] .

]. The associated Schmidt coefficients (entries of Dγ) are
γ±2=[3+pq±2(2+2pq)1/2]|BpBq|2/2,
(43)
where |BpBq|=(BpBpBqBq)1/2. The dependences of these coefficients (coupling strengths) on the polarization alignment of the pumps (p⃗ ·q⃗) are illustrated in 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|.

Fig. 3 Normalized Schmidt coefficients (γ±/|BpBq|) plotted as functions of the pump-polarization alignment (p⃗ ·q⃗). The solid and dashed curves represent γ+ and γ, respectively.

Equations (17)(28) also apply to PC (with the appropriate definitions of δr, δs and V), so no further analysis is required. Nonetheless, it is instructive to define the alternative amplitudes
Br(z)=Cr(z)exp(iδdz),Bs(z)=Cs(z)exp(iδdz),
(44)
where δd was defined after Eq. (23). By substituting these definitions in Eqs. (40) and (41), one obtains the alternative (symmetrized) PC equations
dzCr=iδaCr+iγ(BpBqt+BqBpt)Cs*,
(45)
dzCs=iδaCs+iγ(BpBqt+BqBpt)Cr*,
(46)
where δa 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), 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 is
dzAs=iβsAs+iγ(ApAp+ApAp+AqAq+AqAq)As+iγ(ApAqt+AqApt)As*.
(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
Os(z)=exp{i[(βp+βq)/2+γ(|Ap|2+|Aq|2)/2+γ(ApAp+AqAq)]z},
(48)
which is a symmetric combination of the operators Op and Oq. By using Os in the second of Eqs. (39), one obtains the transformed signal equation
dzBs=iδsBs+iγ(BpBqt+BqBpt)Bs*,
(49)
where the mismatch δs = βs − (βp + βq)/2 +γ(|Bp|2 + |Bq|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 = VDγVt, where the Schmidt vectors (columns of V) and coefficients (entries of Dγ) 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 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, e]t/21/2 and [1, − e]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

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

, 23

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

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 equations
dzAp=iβpAp+iγ(ApAp+AqAq+AqAq)Ap,
(50)
dzAq=iβqAq+iγ(AqAq+ApAp+ApAp)Aq,
(51)
dzAr=iβrAr+iγ(ApAp+ApAp+AqAq+AqAq)Ar+iγ(ApAq+AqAp)As,
(52)
dzAs=iβsAs+iγ(ApAp+ApAp+AqAq+AqAq)As+iγ(ApAq+AqAp)Ar.
(53)
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 ApAq+AqAp=(ApAq+AqAp). Notice that Ar is coupled to As, rather than As*. This type of coupling differentiates BS from MI and PC.

Fig. 4 Frequency diagrams for (a) distant and (b) nearby Bragg scattering. 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. The directions of the arrows are reversible.

The sideband equations can be written in the compact form
dzX=iHX,
(54)
where the mode vector and coefficient matrix are
X=[ArAs],H=[JrKKJs],
(55)
respectively. Jr and Js are the aforementioned self-coupling matrices and K is the (common) coupling matrix. Notice that H is Hermitian. Equation (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.

The solution of Eq. (54) can be written in the form of Eq. (4) and the associated transfer matrix has the Schmidt decomposition
T(z)=[V1DτU1V1DρU2V2DρU1V2DτU2],
(56)
where Uj and Vj are unitary matrices, Dτ = diag(τj) and Dρ = diag(ρj) are non-negative diagonal matrices whose entries satisfy the auxiliary equations τj2+ρj2=1, and 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] .

]. 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

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

, 26

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

].

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 OpOq and OqOp are scalar operators, one obtains the transformed BS equations
dzBr=i(βrβp+γ|Bp|2)Br+iγ(BpBq+BqBp)Bs,
(57)
dzBs=i(βsβq+γ|Bq|2)Bs+iγ(BpBq+BqBp)Br.
(58)
Notice that the self-coupling matrices are still Hermitian and cross-coupling is still described by a single matrix (and its Hermitian conjugate), but all three matrices are now constant. The cross-coupling matrix has the Schmidt decomposition K = UDγV, 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

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] .

] and [10

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] .

], respectively. The Stokes representation of the idler and signal Schmidt vectors are ±r⃗ and ±s⃗, respectively, where
r=(2p+q)/(5+4pq)1/2,s=(p+2q)/(5+4pq)1/2,
(59)
and the associated Schmidt coefficients are
γ±2=[3+2pq±(5+4pq)1/2]|BpBq|2/2.
(60)
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 = |BpBq|.

Fig. 5 Normalized Schmidt coefficients (γ±/|BpBq|) plotted as functions of the pump-polarization alignment (p⃗ ·q⃗). The solid and dashed curves represent γ+ and γ, respectively.

By substituting the decompositions
Br=jbrjUj,Bs=jbsjVj,
(61)
which are based on different Schmidt vectors, in Eqs. (57) and (58), one obtains the scalar equations
dzbrj=iδrbrj+iγjbsj,dzbsj=iδsbsj+iγjbrj,
(62)
where the mismatches δr = βrβp +γ|Bp|2 and δs = βsβq + γ|Bq|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
brj(z)=e(z)τj(z)brj(0)+e(z)ρj(z)bsj(0),
(63)
bsj(z)=e(z)ρj*(z)brj(0)+e(z)τj*(z)bsj(0),
(64)
where the transfer functions and phase factor are
τj(z)=cos(kjz)+iδdsin(kjz)/kj,
(65)
ρj(z)=iγjsin(kjz)/kj,
(66)
e(z)=exp(iδaz),
(67)
respectively. In these formulas, the mismatches δa = (δr + δs)/2 and δd = (δrδs)/2, and the BS wavenumbers kj=(δd2+γj2)1/2. Notice that τj and ρj depend on δd, rather than δa, and the kj are real, so BS is always stable.

By combining Eqs. (63) and (64) with Eqs. (61) and their inverses
brj=UjBr,bsj=VjBs,
(68)
one can write the solutions of Eqs. (57) and (58) in the vector IO forms
Br(z)=jUje(z)τj(z)UjBr(0)+jUje(z)ρj(z)VjBs(0),
(69)
Bs(z)=jVje(z)ρj*(z)UjBr(0)+jVje(z)τj*(z)VjBs(0).
(70)

Equations (69) and (70) can be rewritten in the compact form
[Br(z)Bs(z)]=[UeDτUUeDρVVeDρ*UVeDτ*V][Br(0)Bs(0)].
(71)
The transfer matrix in Eq. (71) is in Schmidt-like form, because the diagonal matrices eDτ(*) and eDρ(*) 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 = (ϕρϕτ)/2, which depend implicitly on j. Furthermore, define the column vectors Urj = Uj exp(d), Vrj = Uj exp[i(ϕe + ϕa)], Usj = Vj exp(−d) and Vsj = Vj exp[i(ϕeϕa)]. Then, by using this notation, one can rewrite Eq. (71) in the Schmidt form
[Br(z)Bs(z)]=[Vr|Dτ|UrVr|Dρ|UsVs|Dρ|UrVs|Dτ|Us][Br(0)Bs(0)],
(72)
where the diagonal matrices |Dτ| and |Dρ| are non-negative.

5. Summary

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.

Acknowledgment

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] .

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] .

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) [CrossRef] [PubMed] .

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] .

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] .

21.

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

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] .

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] .

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


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References

  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. News18(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. News22(11), 37–41 (2011). [CrossRef]
  4. M. G. Raymer and K. Srinivasan, “Manipulating the color and shape of single photons,” Phys. Today65(11), 32–37 (2012). [CrossRef]
  5. 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]
  6. H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A13, 2226–2243 (1976). [CrossRef]
  7. C. M. Caves, “Quantum limits on noise in linear ampifiers,” Phys. Rev. D26, 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]
  9. 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]
  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. JETP38, 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). [CrossRef] [PubMed]
  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. B13, 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. Express15, 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]
  21. M. Karlsson and H. Sunnerud, “Effects of nonlinearities on PMD-induced system impairments,” J. Lightwave Technol.24, 4127–4137 (2006). [CrossRef]
  22. C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express12, 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]
  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]

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