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

C. J. McKinstrie; J. R. Ott; M. Karlsson

doi:10.1364/OE.21.011009

1. 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

d_{z} X_{1} = i J_{1} X_{1} + i K X_{2}^{*}, d_{z} X_{2} = i J_{2} X_{2} + i K^{t} X_{1}^{*},

where d_z = d/dz is a space derivative, X₁ = [x_1j] and X₂ = [x_2j] 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

d_{z} X = i L X,

where the 2m × 1 mode vector and 2m × 2m coefficient matrix are

X = [\begin{matrix} X_{1} \\ X_{2}^{*} \end{matrix}], L = [\begin{matrix} J_{1} & K \\ - K^{†} & - J_{2}^{*} \end{matrix}],

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),

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

T (z) = [\begin{matrix} V_{1} D_{μ} U_{1}^{†} & V_{1} D_{ν} U_{2}^{t} \\ V_{2}^{*} D_{ν} U_{1}^{†} & V_{2}^{*} D_{μ} U_{2}^{t} \end{matrix}],

where U₁, U₂, V₁ and V₂ are unitary matrices, D_μ = diag(μ_j) is a positive diagonal matrix, D_ν diag(ν_j) is a non-negative diagonal matrix and j = 1,..., m. The columns of U_j are input 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_{j}^{2} - ν_{j}^{2} = 1

. By using the columns of U₁ and

U_{2}^{*}

as bases for the input vectors X₁(0) and

X_{2}^{*} (0)

, respectively, and the columns of V₁ and

V_{2}^{*}

as bases for the output vectors X₁(z) and

X_{2}^{*} (z)

, one obtains the CMEs

{\bar{x}}_{1 j} (0) = μ_{j} (z) {\bar{x}}_{1 j} (0) + ν_{j} (z) {\bar{x}}_{2 j}^{*} (0), {\bar{x}}_{2 j}^{*} (0) = ν_{j} (z) {\bar{x}}_{1 j} (0) + μ_{j} (z) {\bar{x}}_{2 j}^{*} (0),

where the Schmidt mode-amplitudes x̄_1j and

{\bar{x}}_{2 j}^{*}

are the components of X₁ 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 [8–10].

2. Modulation interaction

Light-wave propagation in a randomly-birefringent fiber is governed by the vector nonlinear Schrödinger equation (NSE)

\partial_{z} A = i β (i \partial_{t}) A + i γ (A^{†} A) A,

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

β (ω) = \sum_{n = 1}^{\infty} k_{n} ω^{n} / n!

, 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 [11–17]. 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 (- i ω_{p} t) + A_{r} (z) exp (- i ω_{r} t) + A_{s} (z) exp (- i ω_{s} t)

in Eq. (7) and collecting terms of like frequency, one obtains the MI equations

d_{z} A_{p} = i β_{p} A_{p} + i γ (A_{p}^{†} A_{p}) A_{p},

d_{z} A_{r} = i β_{r} A_{r} + i γ (A_{p}^{†} A_{p} + A_{p} A_{p}^{†}) A_{r} + i γ (A_{p} A_{p}^{t}) A_{s}^{*},

d_{z} A_{s} = i β_{s} A_{s} + i γ (A_{p}^{†} A_{p} + A_{p} A_{p}^{†}) A_{s} + i γ (A_{p} A_{p}^{t}) A_{r}^{*},

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 self-phase modulation (PM) and cross-PM, and the tensor operator

A_{p} A_{p}^{†}

, which produces cross-polarization 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_{p}^{t} = {(A_{p} A_{p}^{t})}^{t}

, as required by Eqs. (1). Because the pump vector A_p 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.

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It is convenient to define the operator O_p, which satisfies the evolution equation

d_{z} O_{p} = i (β_{p} + γ A_{p} A_{p}^{†}) O_{p}

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

O_{p} (z) = exp [i (β_{p} + γ A_{p} A_{p}^{†}) z],

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|²z[9, 21].

It is also convenient to define the transformed amplitude vectors

A_{j} (z) = O_{p} (z) B_{j} (z) .

By substituting the first of these definitions in Eq. (9) and using Eq. (13), one finds that d_zB_p = 0: The transformed pump vector is constant. By substituting the other definitions in Eqs. (10) and (11), one obtains the transformed MI equations

d_{z} B_{r} = i (β_{r} - β_{p} + γ {| B_{p} |}^{2}) B_{r} + i γ (B_{p} B_{p}^{t}) B_{s}^{*},

d_{z} B_{s} = i (β_{s} - β_{p} + γ {| B_{p} |}^{2}) B_{s} + i γ (B_{p} B_{p}^{t}) B_{r}^{*} .

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 = 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 $K = γ B_{p} B_{p}^{t}$ is symmetric, it has the simpler Schmidt decomposition K = VD_γV^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 substituting the decompositions

B_{r} = \sum_{j} b_{r j} V_{j} B_{s} = \sum_{j} b_{s j} V_{j},

which are based on the same Schmidt vectors, in Eqs. (15) and (16), one obtains the scalar equations

d_{z} b_{r j} = i δ_{r} b_{r j} + i γ_{j} b_{s j}^{*}, d_{z} b_{s j} = i δ_{s} b_{s j} + i γ_{j} b_{r j}^{*},

where 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

b_{r j} (z) = e (z) μ_{j} (z) b_{r j} (0) + e (z) ν_{j} (z) b_{s j}^{*} (0),

b_{s j} (z) = e^{*} (z) μ_{j} (z) b_{s j} (0) + e^{*} (z) ν_{j} (z) b_{r j}^{*} (0),

where the transfer functions and phase factor are

μ_{j} (z) = cos (k_{j} z) + i δ_{a} sin (k_{j} z) / k_{j},

ν_{j} (z) = i γ_{j} sin (k_{j} z) / k_{j},

e (z) = exp (i δ_{d} z),

respectively. In these formulas, the mismatch average δ_a = (δ_r + δ_s)/2, the mismatch difference δ_d = (δ_r − δ_s)/2 and the MI wavenumbers

k_{j} = {(δ_{a}^{2} - γ_{j}^{2})}^{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_⊥ = δ_a, ν_⊥(z) = 0, e(z)μ_⊥(z) = exp(iδ_rz) and e^*(z)μ_⊥(z) = exp(iδ_sz).

By combining Eqs. (19) and (20) with Eqs. (17) and their inverses

b_{r j} = V_{j}^{†} B_{r}, b_{s j} = V_{j}^{†} B_{s},

one can write the solutions of Eqs. (15) and (16) in the vector IO forms

B_{r} (z) = \sum_{j} V_{j} e (z) μ_{j} (z) V_{j}^{†} B_{r} (0) + \sum_{j} V_{j} e (z) ν_{j} (z) V_{j}^{t} B_{s}^{*} (0),

B_{s}^{*} (z) = \sum_{j} V_{j}^{*} e (z) ν_{j}^{*} (z) V_{j}^{†} B_{r} (0) + \sum_{j} V_{j}^{*} e (z) μ_{j}^{*} (z) V_{j}^{t} B_{s}^{*} (0) .

Equations (25) and (26) can be rewritten in the compact form

[\begin{matrix} B_{r} (z) \\ B_{s}^{*} (z) \end{matrix}] = [\begin{matrix} V e D_{μ} V^{†} & V e D_{ν} V^{t} \\ V^{*} e D_{ν}^{*} V^{†} & V^{*} e D_{μ}^{*} V^{t} \end{matrix}] [\begin{matrix} B_{r} (0) \\ B_{s}^{*} (0) \end{matrix}] .

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,

e D_{μ}^{*}

, eD_ν and

e D_{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_rj = V_j exp[i(ϕ_a + ϕ_e)] and V_sj = V_j exp[i(ϕ_a − ϕ_e)]. Then, by using this notation, one can rewrite Eq. (27) in the (canonical) Schmidt form

[\begin{matrix} B_{r} (z) \\ B_{s}^{*} (z) \end{matrix}] = [\begin{matrix} V_{r} | D_{μ} | U^{†} & V_{r} | D_{ν} | U^{t} \\ V_{s}^{*} | D_{ν} | U^{†} & V_{s}^{*} | D_{μ} | U^{t} \end{matrix}] [\begin{matrix} B_{r} (0) \\ B_{s}^{*} (0) \end{matrix}],

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).

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 ω_p + ω_q = ω_r + ω_s, which is illustrated in Fig. 2. By substituting the four-frequency ansatz

\begin{array}{l} A (z, t) = A_{p} (z) exp (- i ω_{p} t) + A_{q} (z) exp (- i ω_{q} t) \\ + A_{r} (z) exp (- i ω_{r} t) + A_{s} (z) exp (- i ω_{s} t) \end{array}

in Eq. (7) and collecting terms of like frequency, one obtains the PC equations

d_{z} A_{p} = i β_{p} A_{p} + i γ (A_{p}^{†} A_{p} + A_{q}^{†} A_{q} + A_{q} A_{q}^{†}) A_{p},

d_{z} A_{q} = i β_{q} A_{q} + i γ (A_{q}^{†} A_{q} + A_{p}^{†} A_{p} + A_{p} A_{p}^{†}) A_{q},

\begin{array}{l} d_{z} A_{r} = i β_{r} A_{r} + i γ (A_{p}^{†} A_{p} + A_{p} A_{p}^{†} + A_{q}^{†} A_{q} + A_{q} A_{q}^{†}) A_{r} \\ + i γ (A_{p} A_{q}^{t} + A_{q} A_{p}^{t}) A_{s}^{*}, \end{array}

\begin{array}{l} d_{z} A_{s} = i β_{s} A_{s} + i γ (A_{p}^{†} A_{p} + A_{p} A_{p}^{†} + A_{q}^{†} A_{q} + A_{q} A_{q}^{†}) A_{s} \\ + i γ (A_{p} A_{q}^{t} + A_{q} A_{p}^{t}) A_{r}^{*} . \end{array}

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 cross-coupling matrices satisfy the equation

A_{p} A_{q}^{t} + A_{q} A_{p}^{t} = {(A_{p} A_{q}^{t} + A_{q} A_{p}^{t})}^{t}

, as required by Eqs. (1). Because the pump vectors A_p and A_q 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.

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It is convenient to define the operators O_p and O_q, which satisfy the evolution equations

d_{z} O_{p} = i [β_{p} + γ {| A_{q} |}^{2} + γ (A_{p} A_{p}^{†} + A_{q} A_{q}^{†})] O_{p},

d_{z} O_{q} = i [β_{q} + γ {| A_{p} |}^{2} + γ (A_{p} A_{p}^{†} + A_{q} A_{q}^{†})] O_{q},

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

O_{p} (z) = exp {i [β_{p} + γ {| A_{q} |}^{2} + γ (A_{p} A_{p}^{†} + A_{q} A_{q}^{†})] z},

O_{q} (z) = exp {i [β_{q} + γ {| A_{p} |}^{2} + γ (A_{p} A_{p}^{†} + A_{q} A_{q}^{†})] z} .

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

A_{p} (z) = O_{p} (z) B_{p} (z), A_{q} (z) = O_{q} (z) B_{q} (z),

A_{r} (z) = O_{p} (z) B_{r} (z), A_{s} (z) = O_{q} (z) B_{s} (z) .

By substituting definitions (38) into Eqs. (30) and (31), and using Eqs. (36) and (37), one finds that d_zB_p = 0 and d_zB_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_{p}^{t} O_{q}^{*}

,

O_{q}^{†} O_{p}

, and

O_{q}^{t} O_{p}^{*}

are scalar operators, one obtains the transformed PC equations

d_{z} B_{r} = i (β_{r} - β_{p} + γ {| B_{p} |}^{2}) B_{r} + i γ (B_{p} B_{q}^{t} + B_{q} B_{p}^{t}) B_{s}^{*},

d_{z} B_{s} = i (β_{s} - β_{q} + γ {| B_{q} |}^{2}) B_{s} + i γ (B_{p} B_{q}^{t} + B_{q} B_{p}^{t}) B_{r}^{*} .

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 $γ (B_{p} B_{q}^{t} + B_{q} B_{p}^{t})$ 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 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

\vec{r} = \vec{s} = (\vec{p} + \vec{q}) / {(2 + 2 \vec{p} \cdot \vec{q})}^{1 / 2} .

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]. This configuration, for which Eq. (42) is indeterminate, is discussed in [10]. The associated Schmidt coefficients (entries of D_γ) are

γ_{\pm}^{2} = [3 + \vec{p} \cdot \vec{q} \pm 2 {(2 + 2 \vec{p} \cdot \vec{q})}^{1 / 2}] {| B_{p} B_{q} |}^{2} / 2,

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 (p⃗ ·q⃗) are illustrated in Fig. 3. Parallel pumps produce strong sideband-polarization-dependent coupling (γ₊ = 2|B_pB_q| and γ₋ = 0), whereas perpendicular pumps provide moderate polarization-independent coupling (γ₊ = γ₋ = |B_pB_q|). Notice that γ₊ + γ₋ = 2|B_pB_q|.

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

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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

B_{r} (z) = C_{r} (z) exp (i δ_{d} z), B_{s} (z) = C_{s} (z) exp (- i δ_{d} z),

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

d_{z} C_{r} = i δ_{a} C_{r} + i γ (B_{p} B_{q}^{t} + B_{q} B_{p}^{t}) C_{s}^{*},

d_{z} C_{s} = i δ_{a} C_{s} + i γ (B_{p} B_{q}^{t} + B_{q} B_{p}^{t}) C_{r}^{*},

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

\begin{array}{l} d_{z} A_{s} = i β_{s} A_{s} + i γ (A_{p}^{†} A_{p} + A_{p} A_{p}^{†} + A_{q}^{†} A_{q} + A_{q} A_{q}^{†}) A_{s} \\ + i γ (A_{p} A_{q}^{t} + A_{q} A_{p}^{t}) A_{s}^{*} . \end{array}

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

\begin{array}{l} O_{s} (z) = exp {i [(β_{p} + β_{q}) / 2 + γ ({| A_{p} |}^{2} + {| A_{q} |}^{2}) / 2 \\ + γ (A_{p} A_{p}^{†} + A_{q} A_{q}^{†})] z}, \end{array}

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

d_{z} B_{s} = i δ_{s} B_{s} + i γ (B_{p} B_{q}^{t} + B_{q} B_{p}^{t}) B_{s}^{*},

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 as that for PC, so it remains true that K = VD_γV^t, where the Schmidt vectors (columns of V) and coefficients (entries of D_γ) 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 δ_d = 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^iϕ]^t/2^1/2 and [1, − e^iϕ]^t/2^1/2, 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, 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 ω_p + ω_s = ω_q + ω_r, 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

d_{z} A_{p} = i β_{p} A_{p} + i γ (A_{p}^{†} A_{p} + A_{q}^{†} A_{q} + A_{q} A_{q}^{†}) A_{p},

d_{z} A_{q} = i β_{q} A_{q} + i γ (A_{q}^{†} A_{q} + A_{p}^{†} A_{p} + A_{p} A_{p}^{†}) A_{q},

\begin{array}{l} d_{z} A_{r} = i β_{r} A_{r} + i γ (A_{p}^{†} A_{p} + A_{p} A_{p}^{†} + A_{q}^{†} A_{q} + A_{q} A_{q}^{†}) A_{r} \\ + i γ (A_{p} A_{q}^{†} + A_{q}^{†} A_{p}) A_{s}, \end{array}

\begin{array}{l} d_{z} A_{s} = i β_{s} A_{s} + i γ (A_{p}^{†} A_{p} + A_{p} A_{p}^{†} + A_{q}^{†} A_{q} + A_{q} A_{q}^{†}) A_{s} \\ + i γ (A_{p}^{†} A_{q} + A_{q} A_{p}^{†}) A_{r} . \end{array}

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

A_{p}^{†} A_{q} + A_{q} A_{p}^{†} = {(A_{p} A_{q}^{†} + A_{q}^{†} A_{p})}^{†}

. Notice that A_r is coupled to A_s, rather than

A_{s}^{*}

. 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.

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The sideband equations can be written in the compact form

d_{z} X = i H X,

where the mode vector and coefficient matrix are

X = [\begin{matrix} A_{r} \\ A_{s} \end{matrix}], H = [\begin{matrix} J_{r} & K \\ K^{†} & J_{s} \end{matrix}],

respectively. J_r and J_s 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) = [\begin{matrix} V_{1} D_{τ} U_{1}^{†} & V_{1} D_{ρ} U_{2}^{†} \\ - V_{2} D_{ρ} U_{1}^{†} & V_{2} D_{τ} U_{2}^{†} \end{matrix}],

where U_j and V_j are unitary matrices, D_τ = diag(τ_j) and D_ρ = diag(ρ_j) are non-negative diagonal matrices whose entries satisfy the auxiliary equations

τ_{j}^{2} + ρ_{j}^{2} = 1

, and j = 1 or 2 [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 $O_{p}^{†} O_{q}$ and $O_{q}^{†} O_{p}$ are scalar operators, one obtains the transformed BS equations

d_{z} B_{r} = i (β_{r} - β_{p} + γ {| B_{p} |}^{2}) B_{r} + i γ (B_{p} B_{q}^{†} + B_{q}^{†} B_{p}) B_{s},

d_{z} B_{s} = i (β_{s} - β_{q} + γ {| B_{q} |}^{2}) B_{s} + i γ (B_{p}^{†} B_{q} + B_{q} B_{p}^{†}) B_{r} .

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 = UD_γV^†, 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] and [10], respectively. The Stokes representation of the idler and signal Schmidt vectors are ±r⃗ and ±s⃗, respectively, where

\vec{r} = (2 \vec{p} + \vec{q}) / {(5 + 4 \vec{p} \cdot \vec{q})}^{1 / 2}, \vec{s} = (\vec{p} + 2 \vec{q}) / {(5 + 4 \vec{p} \cdot \vec{q})}^{1 / 2},

and the associated Schmidt coefficients are

γ_{\pm}^{2} = [3 + 2 \vec{p} \cdot \vec{q} \pm {(5 + 4 \vec{p} \cdot \vec{q})}^{1 / 2}] {| B_{p} B_{q} |}^{2} / 2.

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_pB_q|.

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

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By substituting the decompositions

B_{r} = \sum_{j} b_{r j} U_{j}, B_{s =} \sum_{j} b_{s j} V_{j},

which are based on different Schmidt vectors, in Eqs. (57) and (58), one obtains the scalar equations

d_{z} b_{r j} = i δ_{r} b_{r j} + i γ_{j} b_{s j}, d_{z} b_{s j} = i δ_{s} b_{s j} + i γ_{j} b_{r j},

where the mismatches 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

b_{r j} (z) = e (z) τ_{j} (z) b_{r j} (0) + e (z) ρ_{j} (z) b_{s j} (0),

b_{s j} (z) = - e (z) ρ_{j}^{*} (z) b_{r j} (0) + e (z) τ_{j}^{*} (z) b_{s j} (0),

where the transfer functions and phase factor are

τ_{j} (z) = cos (k_{j} z) + i δ_{d} sin (k_{j} z) / k_{j},

ρ_{j} (z) = i γ_{j} sin (k_{j} z) / k_{j},

e (z) = exp (i δ_{a} z),

respectively. In these formulas, the mismatches δ_a = (δ_r + δ_s)/2 and δ_d = (δ_r − δ_s)/2, and the BS wavenumbers

k_{j} = {(δ_{d}^{2} + γ_{j}^{2})}^{1 / 2}

. Notice that τ_j and ρ_j depend on δ_d, rather than δ_a, and the k_j are real, so BS is always stable.

By combining Eqs. (63) and (64) with Eqs. (61) and their inverses

b_{r j} = U_{j}^{†} B_{r}, b_{s j} = V_{j}^{†} B_{s},

one can write the solutions of Eqs. (57) and (58) in the vector IO forms

B_{r} (z) = \sum_{j} U_{j} e (z) τ_{j} (z) U_{j}^{†} B_{r} (0) + \sum_{j} U_{j} e (z) ρ_{j} (z) V_{j}^{†} B_{s} (0),

B_{s} (z) = - \sum_{j} V_{j} e (z) ρ_{j}^{*} (z) U_{j}^{†} B_{r} (0) + \sum_{j} V_{j} e (z) τ_{j}^{*} (z) V_{j}^{†} B_{s} (0) .

Equations (69) and (70) can be rewritten in the compact form

[\begin{matrix} B_{r} (z) \\ B_{s} (z) \end{matrix}] = [\begin{matrix} U e D_{τ} U^{†} & U e D_{ρ} V^{†} \\ - V e D_{ρ}^{*} U^{†} & V e D_{τ}^{*} V^{†} \end{matrix}] [\begin{matrix} B_{r} (0) \\ B_{s} (0) \end{matrix}] .

The transfer matrix in Eq. (71) is in Schmidt-like form, because the diagonal matrices

e D_{τ}^{(*)}

and

e D_{ρ}^{(*)}

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 ϕ_τ = arg(τ) and ϕ_ρ = arg(ρ), and define the phase average ϕ_a = (ϕ_τ + ϕ_ρ)/2 and phase difference ϕ_d = (ϕ_ρ − ϕ_τ)/2, which depend implicitly on j. Furthermore, define the column vectors U_rj = U_j exp(iϕ_d), V_rj = U_j exp[i(ϕ_e + ϕ_a)], U_sj = V_j exp(−iϕ_d) and V_sj = V_j exp[i(ϕ_e − ϕ_a)]. Then, by using this notation, one can rewrite Eq. (71) in the Schmidt form

[\begin{matrix} B_{r} (z) \\ B_{s} (z) \end{matrix}] = [\begin{matrix} V_{r} | D_{τ} | U_{r}^{†} & V_{r} | D_{ρ} | U_{s}^{†} \\ - V_{s} | D_{ρ} | U_{r}^{†} & V_{s} | D_{τ} | U_{s}^{†} \end{matrix}] [\begin{matrix} B_{r} (0) \\ B_{s} (0) \end{matrix}],

where the diagonal matrices |D_τ| and |D_ρ| 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.

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

Abstract

1. Introduction

2. Modulation interaction

3. Phase conjugation

4. Bragg scattering

5. Summary

Acknowledgment

References and links

Cited By

Figures (5)

Equations (72)

Optics Express