Signal replication by multiple sum- or difference-frequency generation

C. J. McKinstrie; A. Agarwal; T. C. Banwell; J. M. Dailey

doi:10.1364/OE.23.024675

1. Introduction

In four-wave mixing (FWM), which occurs in a third-order nonlinear medium (such as a fiber), one or two strong pump waves drive weak signal and idler waves (sidebands). Parametric devices based on FWM can amplify, frequency-convert and phase-conjugate optical signals in communication systems [1, 2]. They also enable high-performance sampling, in both the time and frequency domains [3]. Suppose that a train of similar signal pulses is illuminated by a train of short pump pulses. Then each pump pulse generates a short idler pulse (by degenerate FWM), whose peak power is proportional to the signal power at the instant it was sampled (and whose energy can be measured by a moderate-speed detector). If the separation of the pump pulses differs slightly from that of the signals, each pump samples a different time-slice of its signal, so the average shape (waveform) of the signal pulses can be inferred [4].

Now suppose that multiple copies of the signal are made, each with a different carrier frequency. Then, by passing the copies (idlers) through a dispersive medium, one can delay them relative to one another, so that a short pump pulse illuminates different time-slices of the idlers. In this way, one can measure the shape of an individual signal pulse [5]. Alternatively, one can send the idlers through a periodic frequency filter. If the spacing between the passband frequencies differs slightly from that of the idler frequencies, each passband transmits (samples) a different part of its idler spectrum. In this way, one can measure the complete spectrum of a broad-bandwidth signal or separate the spectra of synchronous narrow-bandwidth signals [6]. Thus, one can use optical preprocessing to replace high-speed electrical sampling by multiple instances of moderate-speed sampling and broad-bandwidth electrical sampling by multiple instances of moderate-bandwidth sampling. Applications of these sampling schemes include the analog-to-digital conversion [7] and channelization [8] of radio-frequency signals.

What enables the aforementioned schemes is the generation of faithful copies of the signal: The idlers should have the same shapes (spectra) as the signal, and should not be polluted with excess noise, which makes them harder to read (measure) than the signal. In the standard copying (replication) scheme [9, 10], two strong pumps and a weak signal are launched into a fiber. Multiple FWM processes produce new pumps and idlers (which are copies of the signal). Although this scheme works well (and produces outputs that are amplified versions of the input signal), it also has drawbacks. First, the number of FWM processes increases faster than the square of the number of pumps or sidebands, so it is difficult to model replication analytically. Strategies for optimizing the operation of a FWM-based copier (for example, equalizing the output idler powers) must be determined by doing numerical simulations based on the nonlinear Schrodinger equation and using numerical search algorithms. Second, in multiple-sideband mixing the noise figure of each idler-generation process usually scales linearly with the number of sidebands, so making more copies usually results in lower-quality idlers [11, 12]. Fortunately, numerical simulations revealed specific dispersion conditions under which the idler noise figures can be limited to about 6 dB [12]. This limit is 3-dB higher than the noise figure of two-sideband amplification and 6-dB higher than the noise figure of two-sideband frequency conversion [13–16]. Because of the aforementioned issues, it is useful to consider other replication schemes.

In three-wave mixing (TWM), which occurs in a second-order nonlinear medium, a strong pump wave drives weak signal and idler waves. In difference-frequency generation (DFG), the signal is amplified and an idler is generated (π_p → π_s + π_i, where π_j represents a photon with frequency ω_j), whereas in sum-frequency generation (SFG) the signal power is transferred to the idler (π_p + π_s → π_i). If the single pump is replaced by multiple pumps (from a frequency comb, perhaps), multiple TWM processes occur. These processes are illustrated in Fig. 1, which is based on the assumption that the signal frequency is less than one half of the lowest pump frequency. (A similar figure can be drawn based on the opposite assumption.) TWM driven by multiple pumps generates multiple copies of the signal. Each DFG idler is a conjugated copy of the signal, whereas each SFG idler is a direct (unconjugated) copy.

Fig. 1 Frequency diagrams for DFG and SFG driven by one pump (top) and three pumps (bottom). The black arrows denote the pumps and the red arrows denote the signal. The green arrows denote the DFG idlers, whereas the blue arrows denote the SFG idlers. The tick marks denote the zero frequency.

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This paper is organized as follows: In Sec. 2 the properties (quadrature means and variances) of the input signal and idler modes are described briefly. The input–output (IO) equations for SFG and DFG are stated, and used to determine the noise properties (quadrature variances and correlation) of the output signal and idler modes [13–16]. In Sec. 3, the coupled-mode equations (CMEs) for SFG driven by multiple pumps are stated and solved analytically. The solutions (transfer coefficients) are used to determine the covariance matrix (quadrature variances and correlations) of the output signal and idlers, and the noise figures of the idler-generation processes. In Sec. 4, the CMEs for DFG driven by multiple pumps are also stated and solved. The transfer coefficients are used to determine the covariance matrix of the output signal and idlers, and the noise figures of the idler-generation processes. For both multiple-mode processes, which are complicated, the mathematical results are explained physically in terms of superposition modes and simpler processes. In Sec. 5, the physical limitations of the aforementioned CMEs are discussed briefly. Finally, in Sec. 6 the main results of this paper are summarized.

2. Sum- and difference-frequency generation

In this section, the noise properties of SFG and DFG are reviewed briefly. In both processes a strong pump wave, which does not evolve, drives weak signal and idler waves (modes), which do evolve [13–16]. Let a_s and a_i be the signal- and idler-mode amplitude operators, respectively. These operators satisfy the boson commutation relations [a_j,a_k] = 0 and $[a_{j}, a_{k}^{†}] = δ_{j k}$ , where [x,y] = xy − yx is a commutator, † is a Hermitian conjugate and δ_jk is the Kronecker delta [17]. The real and imaginary parts of the mode amplitudes (quadratures) can be measured by homodyne detection, which involves a local oscillator (LO). Each quadrature operator $p_{j} (ϕ_{j}) = (a_{j}^{†} e^{i ϕ_{j}} + a_{j} e^{- i ϕ_{j}}) / 2^{1 / 2}$ , where φ_j is a LO phase. By combining this definition with the amplitude commutation relations, one obtains the quadrature commutation relations [p_j(ϕ_j), p_k(ϕ_k + π/2)] = iδ_jk. The input signal and idler are characterized by their quadrature means ⟨p_j(ϕ_j)⟩, where ⟨⟩ is an expectation value, and the quadrature covariance matrix C_p(ϕ_j, ϕ_k) = ⟨δp_j(ϕ_j) δp_k (ϕ_k)⟩, where δp_j(ϕ_j) = p_j(ϕ_j) − ⟨p_j(ϕ_j)⟩ is a quadrature-fluctuation operator.

For a coherent-state (CS) signal and a vacuum-state (VS) idler, the quadrature means

〈 p_{s} (ϕ_{s}) 〉 = 2^{1 / 2} | α | \cos (ϕ_{s} - ϕ_{α}), 〈 p_{i} (ϕ_{i}) 〉 = 0,

where

α = | α | e^{i ϕ α}

is the CS amplitude. The measured signal quadrature is maximal when the LO phase equals the signal phase (ϕ_s = ϕ_α). The covariance matrix

C_{p} (ϕ_{s}, ϕ_{i}) = σ [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}],

where σ = 1/2. Equation (2) shows that the input signal and idler have uncorrelated vacuum-level fluctuations, which do not depend on the LO phases. For reference, the signal-to-noise ratio (SNR) of the signal is defined to be the square of the quadrature mean divided by the quadrature variance. The SNR attains its maximal value 4|α|² when the LO phase equals the signal phase.

The CMEs for SFG [13,18] are linear in the signal and idler operators. Hence, their solutions can be written in the IO forms

b_{s} = τ a_{s} + ρ a_{i}, b_{i} = - ρ^{*} a_{s} + τ^{*} a_{i},

where a_j is an input operator and b_j is an output operator. The transfer coefficients τ and ρ depend on the pump power, wavenumber mismatch and medium length, and satisfy the auxiliary equation |τ|²+|ρ|² = 1, which ensures that the total number of signal and idler photons is conserved. By choosing the phase references of the input and output modes judiciously, one can replace Eqs. (3) by similar equations, in which the transfer coefficients are real. By combining Eqs. (3) with their conjugates, one obtains the quadrature IO equations

q_{s} (ϕ_{s}) = τ p_{s} (ϕ_{s}) + ρ p_{i} (ϕ_{s}), q_{i} (ϕ_{i}) = - ρ p_{s} (ϕ_{i}) + τ p_{i} (ϕ_{i}),

where the output quadratures are related to the output amplitudes in the same ways as the input quantities. It follows from Eqs. (4) that the output means

〈 q_{s} (ϕ_{s}) 〉 = τ 〈 p_{s} (ϕ_{s}) 〉, 〈 q_{i} (ϕ_{i}) 〉 = - ρ 〈 p_{s} (ϕ_{i}) 〉 .

Each output quadrature is proportional to the input signal quadrature. It also follows that the output covariance matrix

C_{q} (ϕ_{s}, ϕ_{i}) = σ [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] .

Thus, SFG produces outputs that are diminished copies of the input signal, and whose fluctuations are uncorrelated and phase-independent. The output SNRs are defined in the same way as the input SNR, and the SFG noise figures are defined to be the input signal SNR divided by the output signal and idler SNRs. By combining the preceding definitions and results (and assuming that the LO phases are optimal), one obtains the noise figures

F_{s} = 1 / τ^{2}, F_{i} = 1 / ρ^{2} .

In the high-conversion regime (ρ ≈ 1), the idler noise figure is approximately 1 (0 dB), which means that the output idler is a perfect copy of the input signal: Frequency conversion by SFG does not add noise to the output idler.

DFG is governed by the IO equations [13,18]

b_{s} = μ a_{s} + v a_{i}^{†}, b_{i} = v a_{s}^{†} + μ a_{i},

which involve the signal and idler operators, and their conjugates. The transfer coefficients µ and ν satisfy the auxiliary equation |µ|²−|v|² = 1, which ensures that signal and idler photons are produced in pairs. By choosing the phase references of the input and output modes judiciously, one can replace Eqs. (8) by similar equations, in which µ and ν are real. By combining Eqs. (8) with their conjugates, one obtains the quadrature IO equations

q_{s} (ϕ_{s}) = μ p_{s} (ϕ_{s}) + v p_{i} (- ϕ_{s}), q_{i} (ϕ_{i}) = v p_{s} (- ϕ_{i}) + μ p_{i} (ϕ_{i}) .

It follows from Eqs. (9) that the output means

〈 q_{s} (ϕ_{s}) 〉 = μ 〈 p_{s} (ϕ_{s}) 〉, 〈 q_{i} (ϕ_{i}) 〉 = v 〈 p_{s} (- ϕ_{i}) 〉 .

The output signal quadrature is proportional to the input signal quadrature, measured with the same LO phase, whereas the output idler quadrature is proportional to the input signal quadrature, measured with the opposite LO phase. This opposite-phase relation is a consequence of the conjugates in Eqs. (8) and signifies that the output idler is a conjugated copy of the input signal. (If one were to derive IO equations for the real and imaginary quadratures separately, one would find that the real equations are identical to the preceding ones, whereas the imaginary equations differ only in the sign of ν.) It also follows that the output covariance matrix

C_{q} (ϕ_{s}, ϕ_{i}) = σ [\begin{matrix} (μ^{2} + v^{2}) & 2 μ v \cos (ϕ_{s} + ϕ_{i}) \\ 2 μ v \cos (ϕ_{i} + ϕ_{s}) & (μ^{2} + v^{2}) \end{matrix}] .

Thus, DFG produces outputs that are amplified copies of the signal, and whose fluctuations are correlated and phase-dependent. The correlations are maximimal when φ_i = −φ_s, in which case each LO phase is aligned with the approriate output phase. (The real fluctuations are correlated, whereas the imaginary ones are anti-correlated.) In the high-gain regime (µ ≈ ν ≫ 1), the output fluctuations are correlated completely: The output idler is a perfect (conjugated) copy of the output signal. By combining the preceding results, one obtains the noise figures

F_{s} = 2 - 1 / μ^{2}, F_{i} = 2 + 1 / v^{2} .

In the high-gain regime, the signal and idler noise figures are approximately 2 (3 dB): Parametric amplification by DFG adds excess noise to the output signal and idler (so neither output is a perfect copy of the input signal).

3. Multiple sum-frequency generation

SFG driven by a comb of pumps (multiple SFG) is governed by the CMEs

d_{z} a_{s} = i \sum_{i} γ_{i}^{*} a_{i}, d_{z} a_{i} = i γ_{i} a_{s},

where d_z = d/dz is a distance derivative, a_s is the signal-mode operator, a_i is an idler-mode operator, and γ_i is the product of the second-order nonlinearity coefficient and a pump amplitude. (The effects of secondary TWM on SFG are discussed briefly in Sec. 5.) By writing γ_i = |γ|exp(iϕ_i) and

a_{i} = i a_{i}^{'} exp (i ϕ_{i})

, one obtains the transformed CMEs

d_{z} a_{s} = - | γ | \sum_{i} a_{i}^{'}, d_{z} a_{i}^{'} = | γ | a_{s},

where |γ| is the (common) coupling coefficient. (The assumption of equal pump powers is convenient, but unnecessary.) The transformed operators

a_{i}^{'}

also satisfy the commutation relations. Henceforth, the modulus signs and primes will be omitted. By combining Eqs. (14) with their (Hermitian) conjugates, one finds that

d_{z} a_{s}^{†} a_{s} = - γ \sum_{i} (a_{s}^{†} a_{i} + a_{i}^{†} a_{s}), d_{z} a_{i}^{†} a_{i} = γ (a_{i}^{†} a_{s} + a_{s}^{†} a_{i}),

and by combining Eqs. (15) with each other, one obtains the Manley–Rowe–Weiss (MRW) equation [19,20]

d_{z} (a_{s}^{†} a_{s} + \sum_{i} a_{i}^{†} a_{i}) = 0.

Equation (16) reflects the fact that the total number of signal and idler photons is conserved.

For the symmetric case in which a_s(0) ≠ 0 and a_i(0) = 0, the solutions of Eqs. (14) are

a_{s} (z) = a_{s} (0) \cos (γ n^{1 / 2} z), a_{i} (z) = a_{s} (0) \sin (γ n^{1 / 2} z) / n^{1 / 2},

where n is the number of idlers. The idlers start equal and stay equal, so the signal power (photon flux) is split evenly among the idlers. For the asymmetric case in which a_s(0) = 0, a_i(0) ≠ 0 and a_j(0) = 0 for j ≠ i, the solutions of Eqs. (14) are

a_{s} (z) = - a_{i} (0) \sin (γ_{n}^{1 / 2} z) / n^{1 / 2}, a_{j} (z) = a_{i} (0) {δ_{i j} + [\cos (γ n^{1 / 2} z) - 1] / n} .

Notice that idler i shares only the fraction 1/n of its power with the signal.

Define the amplitude vector X = [a_s,a₁,…,a_n]^t. Then the general solutions of Eqs. (14) can be written in the matrix input-output (IO) form

X (z) = T (z) X (0),

where the transfer (Green) matrix

T (z) = [\begin{array}{l} c & - s^{'} & \dots & - s^{'} \\ s^{'} & 1 + c^{'} & \dots & c^{'} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ s^{'} & c^{'} & \dots & 1 + c^{'} \end{array}],

and the entries c = cos(γn¹^/²z), c′ = (c − 1)/n and s′ = sin(γn¹^/²z)/n¹^/². In the lower-right block of this matrix, only the diagonal elements include 1s. It is easy to verify that the square of the elements in each row and column add to 1, and that T is orthogonal for arbitrary z (T^tT = I).

By combining the amplitude IO equations (19) with their conjugates, one obtains the quadrature IO equations

q_{i} (ϕ_{i}) = \sum_{k} τ_{i k} p_{k} (ϕ_{i}),

where τ_ik is a transfer coefficient (which depends implicitly on z) and the subscripts i and k represent any (signal or idler) mode. Thus, the quadratures transform in the same ways as the amplitudes. The quadrature correlation

〈 δ q_{i} (ϕ_{i}) δ q_{j} (ϕ_{j}) 〉 = \sum_{k} \sum_{l} τ_{i k} τ_{j l} 〈 δ p_{k} (ϕ_{i}) δ p_{l} (ϕ_{j}) 〉 = σ \sum_{k} τ_{i k} τ_{j k},

because the input fluctuations have vacuum levels and are uncorrelated [Eq. (2)]. Hence, the output covariance matrix

C_{q} = σ T T^{t} = σ I .

Equations (21) and (23) show that multiple SFG acts in the same way as multiple-mode beam splitting: The signal is divided (or idlers are combined) in a power-preserving manner [Eq. (16) or (20)]. Furthermore, if the input fluctuations are quantum-limited, so also are the output fluctuations. An alternative derivation of Eq. (23) is given in App. A.

In the preceding paragraphs, the components of the amplitude vector X were defined relative to the physical modes (basis vectors). It is sometimes useful to define them relative to other basis vectors (superposition modes). Let U be an arbitrary unitary matrix (U^†U = I). Then the column vectors U_j form an orthonormal set $(U_{j}^{†} U_{k} = δ_{j k})$ . Now let X′ = U^†X (or X = UX′). Then the entries of X′ are the components of X defined relative to the basis vectors U_j. The alternative amplitude vector satisfies the IO equation X′(z) = T′(z)X′ (0), where the alternative transfer matrix T′ = U^†TU. It is sometimes possible to choose U in such a way that T′ is simpler than T. A real unitary matrix is orthogonal (O^t O = I).

For example, let O_s be the signal vector [1,0,…,0]^t, O₁ be the idler sum-vector [0,1,…,1]^t/n¹^/² and O₂, …, O_n be any other collection of idler vectors that are orthogonal to O_s, O₁ and each other. Then the second condition (O^t₁O_j = 0) implies that the sum of the entries of O_j equals zero. By using this fact, one can show that the alternative transfer matrix

T^{'} (z) = [\begin{array}{l} c & - s & 0 & \dots & 0 \\ s & c & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 1 \end{array}],

where s = sin(γn¹^/²z). In the lower-right block of this matrix, only the diagonal entries are 1. The alternative transfer matrix shows clearly that n − 1 idler-superposition modes are inert, and one idler-superposition mode (the sum mode) undergoes a beam-splitter-like transformation with the signal, as does the (single) idler in SFG. It also shows why, in solution (18), only a fraction of the input idler power is shared with the signal: The stated input condition corresponds to an idler-sum mode with 1/n of the input power, which is shared completely with the signal. The remaining idler power stays with the other idler-superposition modes, which are inert. It is easy to verify that

C_{q}^{'} = σ T^{'} {(T^{'})}^{t} = σ I

. Thus, the output fluctuations are also quantum-limited in the alternative basis.

Recall that the SNR of the (ideal) input signal is ⟨p_s⟩²/σ. The preceding results show that the output signal has the SNR (c⟨p_s⟩)²/σ and each output idler has the SNR (s′⟨p_s⟩)²/σ. By combining these results, one obtains the noise figures

F_{s} = 1 / c^{2}, F_{i} = 1 / {(s^{'})}^{2} .

If the signal is depleted completely, c² = 0 and (s′)² = 1/n. Hence, the (common) idler-generation noise figure is n. One can interpret this result as idler sum-mode generation, with a noise figure of 1 [Eqs. (7)], followed by multiple-idler-mode beam splitting, with a (common) noise figure of n. If the input signal is very noisy $(〈 δ p_{s}^{2} 〉 ≫ σ)$ , one can ignore the vacuum fluctuations associated with the input idlers. In this case, which is analyzed in App. B, one finds that the signal is divided into n equal parts, each of which has the same SNR as the input. However, this result is a Pyrrhic victory, because the output idlers are also very noisy.

4. Multiple difference-frequency generation

DFG driven by a comb of pumps (multiple DFG) is governed by the CMEs

d_{z} a_{s} = i \sum_{i} γ_{i} a_{i}^{†}, d_{z} a_{i} = i γ_{i} a_{s}^{†} .

(The effects of secondary TWM on DFG are discussed briefly in Sec. 5.) By proceeding as described in Sec. 3, one obtains the transformed CMEs

d_{z} a_{s} = γ \sum_{i} a_{i}^{†}, d_{z} a_{i} = γ a_{s}^{†},

where γ is real. In contrast to Eqs. (14)a_s is coupled to

a_{i}^{†}

and vice versa, and there is no − sign in the first equation. By combining Eqs. (27) with their conjugates and using the commutation relations, one finds that

d_{z} a_{s}^{†} a_{s} = γ \sum_{i} (a_{s}^{†} a_{i}^{†} + a_{s} a_{i}), d_{z} a_{i}^{†} a_{i} = γ (a_{s}^{†} a_{i}^{†} + a_{s} a_{i}),

and by combining Eqs. (28) with each other, one obtains the MRW equation

d_{z} (a_{s}^{†} a_{s} - \sum_{i} a_{i}^{†} a_{i}) = 0.

Equation (29) reflects the fact that signal and idler photons are produced in pairs (π_s and π₁, or π_s and π₂, …).

Define the pseudo-amplitude vector $X = {[a_{s}^{†}, a_{1}, \dots, a_{n}]}^{t}$ . Then the general solutions of Eqs. (27) can be written in the matrix IO form

X (z) = T (z) X (0),

where the transfer matrix

T (z) = [\begin{array}{l} c & s^{'} & \dots & s^{'} \\ s^{'} & 1 + c^{'} & \dots & c^{'} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ s^{'} & c^{'} & \dots & 1 + c^{'} \end{array}],

and c = cosh(γn¹^/²z), c′ = (c − 1)/n and s′ = sinh(γn¹^/²z)/n¹^/². In the lower-right block of this matrix, only the diagonal elements include 1s. For the symmetric case in which a_s(0) ≠ 0 and a_i(0) = 0, the output signal power is larger than the input signal power by the factor c², whereas each output idler power is larger than the input signal power by the (common) factor (s′)² = s²/n, where s = sinh(γn¹^/²z). For the asymmetric case in which a_s(0) = 0, a_i(0) ≠ 0 and a_j(0) = 0 for j ≠ i, the output signal power is larger than the input idler power by the factor s²/n. In the high-gain regime, each output idler power is larger than the input idler power by a factor of about c²/n².

By combining the amplitude IO equations (30) with their conjugates, one obtains the quadrature IO equations

q_{s} (ϕ_{s}) = τ_{s s} p_{s} (ϕ_{s}) + \sum_{j} τ_{s}_{j} p_{j} (- ϕ_{s}), q_{i} (ϕ_{i}) = τ_{i s} p_{s} (- ϕ_{i}) + \sum_{j} τ_{i j} p_{j} (ϕ_{i}),

where τ_ss, τ_sj, τ_is and τ_ij are transfer coefficients (which depend implicitly on z). These equations are complicated by the fact that signal quadratures with phase ϕ are coupled to idler quadratures with phase −ϕ and vice versa, so it is convenient to consider the real (ϕ = 0) and imaginary (ϕ = π/2) quadratures separately. The real quadratures satisfy the IO equations

q_{s} = τ_{s s} p_{s} + \sum_{j} τ_{s j} p_{j}, q_{i} = τ_{i s} p_{s} + \sum_{j} τ_{i j} p_{j},

which involve the same transfer coefficients as the amplitude equations. The imaginary quadratures satisfy similar equations, in which τ_sj→−τ_sj and τ_is→−τ_is. Hence, one can derive results for the real quadratures, then deduce the imaginary results from the real ones by changing the sign of s′ [Eq. (31)]. It remains true that C_q = TC_pT^t, but it is no longer true that TT^t = 1. By doing the matrix multiplication, one obtains the output covariance matrix

C_{q} = σ [\begin{array}{l} 1 + 2 s^{2} & 2 c s^{'} & \dots & 2 c s^{'} \\ 2 c s^{'} & 1 + 2 {(s^{'})}^{2} & \dots & 2 {(s^{'})}^{2} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 2 c s^{'} & 2 {(s^{'})}^{2} & \dots & 1 + 2 {(s^{'})}^{2} \end{array}] .

In the lower-right block of this matrix, only the diagonal elements include 1s. Notice that the output fluctuations are not quantum-limited. For long distances, the output idler quadratures are correlated completely. This behavior is an indication that the idlers evolve in concert, as a sum mode. An alternative derivation of Eq. (34) is given in App. A.

One can explain the preceding results by rewriting Eq. (30) in terms of the basis vectors defined before Eq. (24). By doing so, one obtains the alternative transfer matrix

T^{'} (z) = [\begin{array}{l} c & s & 0 & \dots & 0 \\ s & c & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 1 \end{array}] .

The alternative transfer matrix shows clearly that n − 1 idler-superposition modes are inert, and one idler-superposition mode (the sum mode) undergoes a stretching (squeezing) transformation with the signal, as does the (single) idler in DFG. (The sum of the signal and idler-sum modes is stretched, whereas the difference between the signal and idler-sum modes is squeezed.) This is a general property of parametric processes with unequal numbers of amplitudes and conjugate amplitudes [21]. It also explains why the signal- and some of the idler-generation coefficients in Eq. (31) are proportional to 1/n¹^/². In the symmetric case (nonzero input signal), the output signal and idler-sum powers are larger than the input signal power by the usual factors c² and s², respectively. The sum mode is a symmetric combination of the physical modes, so the output power in each physical mode is larger than the input signal power by a factor of only s²/n. In the asymmetric case, the input condition (one nonzero input idler) corresponds to an idler-sum mode with the fraction 1/n of the input idler power. The output signal and idler-sum powers are larger than the input idler-sum power by the usual factors s² and c², respectively. Hence, the output signal power is larger than the input idler power by the factor s²/n. In the high-gain regime, the contributions of the inert idler-superposition modes to the output idler powers are negligible. Each output idler power is larger than the input idler-sum power by a factor of about c²/n and the input idler power by a factor of about c²/n². It is easy to verify that

C_{q}^{'} = [\begin{matrix} c^{2} + s^{2} & 2 c s & 0 & \dots & 0 \\ 2 c s & c^{2} + s^{2} & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 1 \end{matrix}] .

Equation (36) is consistent with Eq. (11).

The preceding results show that the output signal has the SNR (c⟨p⟩)²/σ[1 + 2s²] and each output idler has the SNR (s′⟨p_s⟩)²/σ[1 + 2(s′)²]. By combining these results, one obtains the noise figures

F_{s} = 2 - 1 / c^{2}, F_{i} = 2 + 1 / {(s^{'})}^{2} .

In the high-gain regime, c² ≫ 1 and (s′)² = s²/n ≫ 1. Hence, the (common) idler-generation noise figure is 2, independent of n! One can interpret this result as idler sum-mode generation, with a noise figure of 2 [Eqs. (12)], followed by multiple-idler-mode beam splitting. In the present context, beam splitting does not reduce the SNRs of the output modes relative to that of the input sum mode, because the amplified fluctuations associated with the sum mode swamp the vacuum fluctuations associated with the other modes.

5. Discussion

The CMEs for multiple SFG [Eqs. (13)] were simplified by the approximation that the idler-generation processes are wavenumber-matched simultaneously and by the omission of secondary TWM processes. SFG driven by three pumps is illustrated in Fig. 2. In the primary TWM processes, a signal with frequency ω_s interacts with pumps with frequencies ω_p, ω_p +ω and ω_p + 2ω to produce idlers with frequencies ω_i, ω_i + ω and ω_i + 2ω, respectively, where ω_i = ω_p + ω_s. In turn, these idlers seed secondary TWM processes (inverse SFG) that are also (approximately) wavenumber-matched. The interactions of the pumps with idler 1 produce signals with frequencies ω_s, ω_s − ω and ω_s − 2ω, the interactions with idler 2 produce signals with ω_s + ω, ω_s and ω_s − ω, and the interactions with idler 3 produce signals with frequencies ω_s + 2ω, ω_s + ω and ω_s. Thus, secondary TWM modifies the input signal, and generates new signals with frequencies ω_s ± ω and ω_s ± 2ω. Modeling these (and higher-order) processes quantitatively is a challenging task, which is outside the scope of this paper. Nonetheless, it is possible to draw some qualitative conclusions.

Fig. 2 Enhanced frequency diagram for SFG driven by three pumps. The black arrows denote the pumps, the long and short red arrows denote the primary and secondary signals, respectively, and the blue arrows denote the SFG idlers.

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Let A_s and A_i be n×1 vectors whose elements are the input signal- and idler-mode operators, respectively, and let B_s and B_i be the associated output vectors. Then the Schmidt decomposition theorem for SFG [22] states that

[\begin{matrix} B_{s} \\ B_{i} \end{matrix}] = [\begin{matrix} V_{s} D_{τ} U_{s}^{†} & V_{s} D_{ρ} U_{i}^{†} \\ - V_{i} D_{ρ} U_{s}^{†} & V_{i} D_{τ} U_{i}^{†} \end{matrix}] [\begin{matrix} A_{s} \\ A_{i} \end{matrix}],

where U and V are unitary matrices, D_τ = diag(τ_j), D_ρ = diag(ρ_j) and

τ_{j}^{2} + ρ_{j}^{2} = 1

. The colums of U_s are the input-signal Schmidt vectors (modes), the colums of V_i are the output-idler Schmidt vectors (modes), and the Schmidt coefficients τ_j and ρ_j are the square roots of the (modal) transmission and reflection (conversion) coefficients, respectively. These quantities all depend on the pump powers, wavenumber mismatches and medium length. In general, ρ₁ ≥ ρ₂ ≥ … ≥ ρ_n. Decomposition (38) asserts that each input-signal Schmidt mode is coupled to an output-idler Schmidt mode and power is transferred between them in the manner described by Eqs. (3). Suppose that the physical parameters can be chosen in such a way that ρ₁ = 1. Then the power of the input signal mode is transferred completely to the output idler mode with a noise figure of 1. The challenge is to design the system in such a way that the input Schmidt mode resembles the physical signal mode and the output Schmidt mode has n physical idler components of comparable magnitude. Failure to do so increases the idler noise figures. Furthermore, unless the other conversion coefficients equal 1 simultaneously, some of the input signal power remains in the signal, which also increases the idler noise figures. These considerations show that 1/n is a lower estimate of the noise figure of idler generation by SFG. This dependence on n is a consquence of the facts that SFG conserves (signal plus idler) power and produces outputs with uncorrelated vacuum-level fluctuations.

The CMEs for multiple DFG [Eqs. (26)] were also simplified. DFG driven by three pumps is illustrated in Fig. 3. By following the procedure described above, one finds that secondary TWM (inverse DFG) modifies the input signal and generates new signals with frequencies ω_s ± ω and ω_s ± 2ω.

Fig. 3 Enhanced frequency diagram for DFG driven by three pumps. The black arrows denote the pumps, the long and short red arrows denote the primary and secondary signals, respectively, and the green arrows denote the DFG idlers.

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The Schmidt decomposition theorem for DFG [21] states that

[\begin{matrix} B_{s} \\ B_{i}^{† t} \end{matrix}] = [\begin{matrix} V_{s} D_{μ} U_{s}^{†} & V_{s} D_{v} U_{i}^{t} \\ V_{i}^{*} D_{v} U_{s}^{†} & V_{i}^{*} D_{μ} U_{i}^{t} \end{matrix}] [\begin{matrix} A_{s} \\ A_{i}^{† t} \end{matrix}],

where D_µ = diag(µ_j), D_ν = diag(ν_j) and

μ_{j}^{2} - v_{j}^{2} = 1

. The Schmidt coefficients µ_j and ν_j characterize signal amplification and idler generation, respectively, and depend on the aforementioned physical parameters. In general, ν₁ ≥ ν₂ ≥… ≥ ν_n. Because the Schmidt coefficients depend exponentially on the pump powers and medium length, if the pump powers are sufficiently high or the medium is sufficiently long, the output is dominated by the first signal and idler Schmidt modes, and the noise figures of the associated amplification and generation processes are 2. Once again, the challenge is to design the system in such a way that the input Schmidt mode resembles the physical signal mode, and the output Schmidt mode has n physical idler components of comparable magnitude. However, DFG produces strong output modes with fluctuations that are stronger than vacuum fluctuations (or the fluctuations associated with the recessive modes). When these Schmidt modes are split into their physical components, the physical noise figures depend inversely on the overlap between the physical signal and the input Schmidt mode, but do not depend on the overlaps between the output Schmidt mode and the physical idlers (because the coherent and incoherent parts of the output mode are split in the same way). In particular, they do not depend on the number of idlers. Hence, multiple DFG is a promising scheme for signal replication that warrants further study.

6. Summary

In this paper, idealized coupled-mode equations for sum-frequency generation (SFG) and difference-frequency generation (DFG) driven by multiple pumps were solved, and the noise figures of idler generation were estimated. For multiple SFG, the (common) noise figure is approximately n, the number of pumps (and idlers), whereas for multiple DFG, the (common) noise figure is approximately 2, independent of n. Thus, multiple DFG enables the generation of multiple low-noise idlers.

One can think of multiple SFG as the interaction of a signal mode with an idler-sum mode, which transfers power from the signal to the sum mode, followed by multiple-idler-mode beam splitting, which divides the sum mode into n physical modes. The sum mode produced by SFG (which has a noise figure of 0 dB) has vacuum-level fluctuations and such a mode is sensitive to the vacuum fluctuations associated with the unused idler ports of the beam splitter. The physical modes also have vacuum-level fluctuations, which are uncorrelated, but each mode has only a fraction of the coherent sum-mode power. Hence, the signal-to-noise ratio (SNR) of each output idler is lower than that of the input signal by the factor n. One can think of multiple DFG in a similar way. However, in DFG (which has a noise figure of 3 dB) the idler-sum mode has a strong coherent field and strong fluctuations. Such a mode is insensitive to the vacuum fluctuations associated with the beam splitter. The physical modes also have strong coherent fields and strong fluctuations, which are correlated. Beam splitting reduces the coherent and incoherent powers by the same factor of n, so each physical mode has the same SNR as the sum mode.

Appendix A: Output correlations

The IO equations (19) and (30) can be rewritten in the (common) canonical form [15,16,23,24]

A (z) = M (z) A (0) + N (z) A^{† t} (0),

where A = [a_s,a₁,…,a_n]^t is an amplitude vector, and M and N are canonical transfer matrices that satisfy the auxiliary equations

M M^{†} - N N^{†} = 1, M N^{t} - N M^{t} = 0.

Equation (10) of [16] specifies the output covariance matrix of an arbitrary parametric process. By using Eqs. (41), one can rewrite this equation in the alternative form

〈 δ q_{i} (ϕ_{i}) δ q_{j} (ϕ_{j}) 〉 / σ = I + 2 Re {{[M^{*} N^{†}]}_{i j} e^{i (ϕ_{i} + ϕ_{j})} + {[N^{*} N^{t}]}_{i j} e^{i (ϕ_{i} - ϕ_{j})}},

which depends on products of the canonical transfer matrices and the LO phases.

For multiple SFG, M = T [Eq. (20)], which is unitary (orthogonal), and N = 0. In this case Eq. (42) implies that the covariance matrix C_q = σI, in agreement with Eq. (23). For multiple DFG,

M (z) = [\begin{matrix} c & 0 & \dots & 0 \\ 0 & 1 + c^{'} & \dots & c^{'} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & c^{'} & \dots & 1 + c^{'} \end{matrix}], N (z) = [\begin{matrix} 0 & s^{'} & \dots & s^{'} \\ s^{'} & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & 0 \\ s^{'} & 0 & \dots & 0 \end{matrix}],

where c = cosh(γn¹^/²z), c′ = (c−1)/n and s′ = sinh(γn¹^/²z)/n¹^/². The products of these (real) matrices are

M^{*} N^{†} = [\begin{matrix} 0 & c s^{'} & \dots & c s^{'} \\ c s^{'} & {(s^{'})}^{2} & \dots & {(s^{'})}^{2} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ c s^{'} & 0 & \dots & 0 \end{matrix}], N^{*} N^{t} = [\begin{matrix} s^{2} & 0 & \dots & 0 \\ 0 & {(s^{'})}^{2} & \dots & {(s^{'})}^{2} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & {(s^{'})}^{2} & \dots & {(s^{'})}^{2} \end{matrix}],

where s = sinh(γn¹^/²z). By combining Eqs. (42) and (44), one obtains the covariance matrix

C_{q} = σ [\begin{matrix} 1 + 2 s^{2} & 2 c s^{'} \cos (ϕ_{s} + ϕ_{1}) & \dots & 2 c s^{'} \cos (ϕ_{s} + ϕ_{n}) \\ 2 c s^{'} \cos (ϕ_{1} + ϕ_{s}) & 1 + 2 {(s^{'})}^{2} & \dots & 2 {(s^{'})}^{2} \cos (ϕ_{1} - ϕ_{n}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 2 c s^{'} \cos (ϕ_{n} + ϕ_{s}) & 2 {(s^{'})}^{2} \cos (ϕ_{n} + ϕ_{1}) & \dots & 1 + 2 {(s^{'})}^{2} \end{matrix}] .

In the lower-right block of this matrix, only the diagonal elements contain 1s. For real quadratures (ϕ_s = ϕ_i = 0), the signal–idler and idler–idler correlations are positive, whereas for imaginary quadratures (ϕ_s = ϕ_i = π/2), the signal–idler correlations are negative and the idler–idler correlations are positive, as stated in Sec. 4. For the case in which n = 1, Eq. (45) reduces to Eq. (11), as it should do.

Appendix B: Noisy signal

In Sec. 3, the input signal was a CS, which has vacuum-level quadrature fluctuations $(〈 δ p_{s}^{2} (ϕ_{s}) 〉 = σ_{v} = 1 / 2)$ . If the input signal is noisy $(〈 δ p_{s}^{2} (ϕ_{s}) 〉 = σ_{v} + σ_{e}, where σ_{e} > 0)$ , where σ_e > 0), the input quadrature correlation

〈 δ p_{i} (ϕ_{i}) δ p_{j} (ϕ_{j}) 〉 = σ_{v} δ_{i j} + σ_{e} δ_{i s} δ_{j s},

where the subscripts i and j represent any (signal or idler) mode. By combining Eq. (22) with Eq. (46), rather than Eq. (2), one finds that the output quadrature correlation

〈 δ q_{i} (ϕ_{i}) δ q_{j} (ϕ_{j}) 〉 = σ_{v} δ_{i j} + σ_{e} τ_{i s} τ_{j s} .

Notice that the excess-noise contribution to the quadrature correlation is similar to the coherent-signal contribution to the second quadrature moment (which is also proportional to τ_isτ_js). The SNR of the input signal is ⟨p_s⟩²/(σ_v +σ_e). It follows from Eq. (47) that the SNR of output idler i is ${(τ_{i s} 〈 p_{s} 〉)}^{2} / (σ_{v} + σ_{e} τ_{i s}^{2})$ . For a very noisy input signal (σ_e ≫ σ_v), the SNR of each output idler is approximately equal to that of the input signal, as stated in Sec. 3.

Acknowledgments

We thank the reviewers for their constructive comments.

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Signal replication by multiple sum- or difference-frequency generation

Abstract

1. Introduction

2. Sum- and difference-frequency generation

3. Multiple sum-frequency generation

4. Multiple difference-frequency generation

5. Discussion

6. Summary

Appendix A: Output correlations

Appendix B: Noisy signal

Acknowledgments

References and links

Cited By

Figures (3)

Equations (47)

Optics Express