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INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF OPTICS B: QUANTUM AND SEMICLASSICAL OPTICS

J. Opt. B: Quantum Semiclass. Opt. 6 (2004) S762–S770

PII: S1464-4266(04)76441-1

Raman-effect induced noise limits on χ(3) parametric amplifiers and wavelength converters Paul L Voss1 and Prem Kumar Center for Photonic Communication and Computing, Department of Electrical and Computer Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3118, USA E-mail: [email protected]

Received 20 February 2004, accepted for publication 23 April 2004 Published 27 July 2004 Online at stacks.iop.org/JOptB/6/S762 doi:10.1088/1464-4266/6/8/021

Abstract The non-zero response time of the Kerr (χ (3) ) nonlinearity determines the quantum-limited noise figure of χ (3) parametric amplifiers and wavelength converters. This non-zero response time of the nonlinearity requires coupling of the parametric amplification process to a molecular-vibration phonon bath, causing the addition of excess noise through Raman gain or loss. The effect of this excess noise on the noise figure of the amplifier can be significant. We derive analytical expressions for this quantum-limited noise figure in the case of non-degenerate phase-insensitive operation of a χ (3) parametric amplifier and show excellent agreement with experiment without any fitting parameter. We also derive analytical expressions for the quantum limited noise figure for χ (3) -based wavelength converters. Keywords: optical amplification, quantum fluctuation, optical parametric amplification, Raman amplification, four-wave mixing

1. Introduction Among the many contributions made by Professor Hermann Haus are those pioneering papers treating quantum fluctuations in optical amplifiers [1] and, following the prediction of soliton squeezing [2], the quantum noise of soliton transmission systems [3–5]. Particularly relevant to this paper is his work with Boivin and K¨artner [6, 7] on the quantum theory of dispersionless self-phase modulation when the response of the χ (3) nonlinearity is not instantaneous. That work provides analytical solutions in integral form for continuoustime (i.e. multi-temporal mode) optical fields. In honor of that important work, we present a quantum theory of non-degenerate phase-insensitive parametric amplification in optical fibres. Here the non-degeneracy is with respect to the signal and idler beams, which have frequencies different from each other and from a single-frequency pump. Our model includes the non-instantaneous response time of the fibre medium, which causes Raman loss and gain. We also include 1 Author to whom any correspondence should be addressed.

1464-4266/04/080762+09$30.00 © 2004 IOP Publishing Ltd

dispersion, which permits phase-matching of the nonlinear process between the pump, signal, and idler. We show that the noise figure of a χ (3) phase-insensitive parametric amplifier exceeds that of an ideal phase-insensitive amplifier (3 dB). Our previous work [8] provides a brief derivation of this quantum theory. In this paper we provide further details and and give noise figure (NF) results for the case of phase-insensitive fibre amplifiers, which are currently of great interest for use in wavelength conversion [9] and efficient broadband amplification [10]. Parametric amplifiers are also candidates for performing alloptical networking functions [11–13]. Recent advances in pumping techniques have permitted improvements of the NF [9, 14] and the manufacture of high-nonlinearity and microstructure fibres has improved the gain slope [15, 16] of fibre parametric amplifiers. To date, the lowest published NF measurements in phase insensitive operation of a χ (3) amplifier have been 3.7 dB [17], 3.8 dB [9] and 4.2 dB [14]. Our theory demonstrates excellent agreement with previously published experimental work [17].

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Raman-effect induced noise limits on χ (3) parametric amplifiers and wavelength converters

2. χ(3) nonlinear response The nonlinear refractive index of the Kerr interaction can be written as 3χ (3) , (1) n2 = 40 n 20 c where n 0 is the linear refractive index of the nonlinear material, 0 is the permittivity of free space and c is the speed of light in free space. For clarity, we state that
2 m (3) . (2) χ (3) ≡ χ1111 V2 The χ (3) (t) nonlinear response is composed of a timedomain-delta-function-like electronic response (1 fs) that is constant in the frequency domain over the bandwidths of interest and a time-delayed Raman response (≈50 fs) that varies over frequencies of interest and is caused by back action of nonlinear nuclear vibrations on electronic vibrations. When χ (3) (t) is real and causal, a Kramers–Kronig relation exists between the real and imaginary parts of the nonlinear response in the frequency domain. This implies that the real part of the nonlinear response in the frequency-domain is symmetric about the pump frequency and the imaginary part in the frequency-domain is anti-symmetric about the pump frequency [18]. Recent experimental results demonstrate that the nonlinear response function χ (3) (t) can be treated as if it were real in the time domain, provided the pump-signal detuning does not exceed a few terahertz [19]. The assumption of a real response function needs to be relaxed above several terahertz of pump-signal frequency detuning, as the asymmetry of the imaginary part of the Fourier transform can be as large as 30% at the Raman gain peak of 13 THz [19]. This asymmetry depends on the type and quantity of dopants introduced into the core of the fibre. In what follows, the analytical treatment of the mean fields allows for the more general case of asymmetry in the Raman gain spectrum. However, other results, including graphs, assume a symmetric Raman spectrum. This approximation is valid for our comparison to experimental work, which occurs at pump-signal detunings less than 1.5 THz. Although a nonlinear response is also present in the polarization orthogonal to the pump, this cross-polarized nonlinear interaction is ignored because we assume that the pump, Stokes and anti-Stokes fields of interest stay copolarized as their polarization state evolves during propagation through the FOPA. Parametric fluorescence and Raman spontaneous emission are present in small amounts in the polarization perpendicular to the pump, but do not affect the NF of the amplifier. For the assumed response function, the refractive index n(t) of the fibre can be written as  ∞ dτ δ(t − τ )| A(τ )|2 n(t) = n 0 + n 2e −∞  t + n 2r dτ f (t − τ )| A(τ )|2 , (3) −∞

where f (t) is a real-valued function normalized so that its integral is equal to 1, n 2e is the electronic portion of the Kerrnonlinearity refractive index, n 2r is the nuclear-vibrational

portion and A(t) is the field amplitude such that | A(t)|2 has units of Watts. The electronic nonlinear response is constant in the frequency domain over the bandwidths of interest and to good approximation can be represented by a delta function. It is only approximately a delta function in the time domain because a true delta function response leads to a singular quantum theory [21, 22]; such a nonlinear response would couple an infinite number of frequencies to each other. In order to obtain the nonlinear coefficient N2 () in the frequency domain, we write equation (3) in the Fourier domain as N () = n 0 δ() + n 2e I () + n 2r F()I (), (4) where I () is the Fourier transform of the intensity I (t) ≡ | A(t)|2 , and take the I ()-proportional terms to represent N2 (): (5) N2 () = n 2e + n 2r F(). Published measurements of the real part of the Kerr nonlinearity in common optical fibres, though widely varying, yield N2 (0) when nonlinear interaction times in the measurements are of much longer duration than the Raman response time, but are of much shorter duration than the electrostriction response time (typically in the nanosecond range). Along with measurement of the Raman gain profile, one may, by means of the Kramers–Kronig transformation, obtain N2 () at the frequencies of interest [18]. We next explain the relation between the published spectra of the Raman-gain coefficient and the nonlinear coefficients used in this paper. Typical measurements of the counterpropagating pump-and-signal Raman-gain spectrum yield the polarization averaged power-gain coefficient gr (−) = [g (−) + g⊥ (−)]/2. At the Raman-gain peak, g⊥  0, as can be seen from figure 1 of Dougherty et al [20]. We define a nonlinearity coefficient H () =

2π N2 () , λAeff

(6)

where λ is the pump wavelength and Aeff is the fibre effective area. Thus our H (0) is equivalent to the nonlinear coupling coefficient γ used by Agrawal [23]. For co-propagating, copolarized, optical waves Im{H (−)} = g (−)/2. We estimate the spectrum of g , normalized to its maximum value, from [20] for both dispersion-shifted fibre (DSF) and standard single-mode fibre (SMF). From this normalized Raman gain spectrum we obtain via the Kramers–Kronig transform the normalized real part of the spectrum of the Raman response, Re{Hr ()}. In figure 1 the real (dots) and imaginary parts (line) of this spectrum are shown. In figure 2, we show in arbitrary units the response function obtained from the sine transform of the parallel Raman gain spectrum. We take the magnitude of the Raman gain spectrum from [24] for both DSF and SMF. For N2 (0), we use measurements from Boskovic et al [25]. The nonlinear coefficients are then calculated as follows: Im{H ()} =

sgn()gnormalized ()g peak 2

(7)

Re{Hr (0)}g peak Re{Hr ()}g peak + , 2 2 (8) where Re{Hr ()} is the Kramers–Kronig transform of g normalized (). Re{H ()} = H (0)−

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Figure 1. Raman contribution Im{HRaman ()} (solid curve) and Re{HRaman ()} (dotted curve) normalized to the peak of Im{HRaman }. Calculated from [20].

Figure 2. Response function in arbitrary units calculated from the sine transform of the parallel Raman gain spectrum Im{HRaman ()}. Calculated from [20].

3. Input–output quantum mode transformations

equation (9), m(z, ˆ t) is a Hermitian phase-noise operator √  ∞ W () ˆ† d m(z, ˆ t) = {id (z)eit − idˆ (z)e−it }, (10) 2π 0

Having briefly discussed the nonlinear response of a χ (3) medium, we turn now to a discussion of the quantum mechanics of the χ (3) parametric amplifier. The goal of this section is to derive input–output mode transformations suitable for predicting the noise figure of continuous-wave χ (3) parametric amplifiers and wavelength converters. Previously, a self-consistent quantum theory of light propagation in a dispersionless non-zero χ (3) -response-time medium has been developed [6], and the associated Ramannoise limit on the generation of squeezing in such a medium via fully frequency-degenerate four-wave mixing has been found [26]. This theory is consistent with the classical mean-field solutions and preserves the continuous-time field commutators. While the theory in Boivin et al [6] provides integral-form expressions for propagation of a multimode total field, dispersion was not explicitly included. Further work from the same group [7] introduced dispersion, but did not perform a detailed analysis of the effect of dispersion on the quantum noise of the system. That paper also explicitly treated the coupling of the phonon modes to a heat bath, eliminated the degrees of freedom and arrived at Boivin’s expressions. In what follows, gain and quantum fluctuations occur at a point z in the fibre and it is necessary to solve for the total field and fluctuations at the output point L. We first briefly review Boivin’s paper [6] before discussing parametric amplification. The fields propagate in a dispersionless, lossless, polarization-preserving, singletransverse-mode fibre under the slowly-varying-envelope approximation. Here the frequency deviation from the pump frequency is  = ωa − ωp = ωp − ωs . The quantum equation of motion for the total field can be written as [6, 7]: 
 ˆ ∂ A(t) ˆ ) A(t) ˆ + im(z, ˆ =i dτ h(t − τ ) Aˆ † (τ ) A(τ ˆ t) A(t), ∂z (9) wherein h(t) is the causal response function of the nonlinearity; i.e. the inverse Fourier transform of H () in equation (6). In S764

which describes coupling of the field to a collection of localized, independent, medium oscillators (optical phonon modes) that are required to preserve the continuous-time commutators ˆ [ A(t), Aˆ † (t )] = δ(t − t ),

(11)

ˆ ˆ )] = 0. [ A(t), A(t

(12)

Note that the time t is in a reference frame travelling at group velocity vg , i.e. t = tstationary frame − vzg . The weighting function W () must be positive for  > 0 so that the molecular vibration oscillators absorb energy from the mean fields rather than providing energy to the mean fields. The operators dˆ (z) and dˆ† (z) obey the commutation relation [dˆ (z), dˆ† (z )] = δ( −  )δ(z − z )

(13)

and each phonon mode is in thermal equilibrium:

dˆ† (z)dˆ (z ) = δ( −  )δ(z − z )n th

(14)

with a mean phonon number n th = [exp(¯h /kT )−1]−1 . Here h¯ is Planck’s constant over 2π, k is Boltzmann’s constant, and T is the temperature. Looking ahead to a Fourier domain treatment of the parametric amplifier, note that the creation operator at each , dˆ† (z), oscillates as eit (a Stokes detuning frequency) and that the annihilation operator at each , dˆ (z), oscillates as e−it (an anti-Stokes detuning frequency). Because the nonlinear index operator
  † ˆ ˆ dτ h(t − τ ) A (τ ) A(τ ) + m(z, ˆ t) at each location commutes at equal t, it is possible to integrate equation (9) as if it were a complex number equation. The

Raman-effect induced noise limits on χ (3) parametric amplifiers and wavelength converters

solution of equation (9) is then    L ˆ dz m(z, ˆ t) A(L , t) = exp i    ∞0 ˆ ) A(0, ˆ t). dτ h(t − τ ) Aˆ † (τ ) A(τ × exp iL

[dˆ (z), dˆ (z )] = 0,

(15)

0

Defining φˆ NL (t) = L



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ˆ ) dτ h(t − τ ) Aˆ † (τ ) A(τ

(16)

0

as the quantum self-phase modulation operator and φˆ Noise (t) =



L

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(17)

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as the Raman phase-noise operator, it can be shown after some algebra that the output field commutators are both preserved if † (L , t )] = iL{h(t − t) − h(t − t )}. (18) [φˆ Noise (L , t), φˆNoise

It can also be shown that W () = 4π Im{H ()}. With the accumulated quantum noise added by the Raman effect accounted for in a way that preserves the commutation relations for the multimode field exiting the fibre, it is a matter of calculation to find the expectation values of interest. The reader may consult Boivin [6] for further details. We now discuss parametric amplification. The fields propagate in a dispersive, lossless, polarizationpreserving, single-transverse-mode fibre under the slowlyvarying-envelope approximation. Consider the field operator ˆ = Aˆ p + Aˆ s exp(it) + Aˆ a exp(−it) A(t)

ˆ ∂ A(t) =i ∂z




 ˆ ) A(t) ˆ + im(z, ˆ dτ h(t − τ ) Aˆ † (τ ) A(τ ˆ t) A(t),

(20) where the operator m(z, ˆ t) has the same form as equation (10). Because the involved waves (Stokes, anti-Stokes, and pump) are CW, the usual group-velocity dispersion term does not explicitly appear in equation (20). However, dispersion is included; its effect is simply to modify the wavevector of each CW component. The pump, Stokes, and anti-Stokes fields are treated as separate frequency modes, implying that the required commutators are [ Aˆ j (z), Aˆ †k (z )] = δ j k δ(z − z ),

(21)

[ Aˆ j (z), Aˆ k (z )] = 0,

(22)

for j, k ∈ { p, a, s}. The commutator relations for the Fourier components of the noise that couple with the pump, Stokes and anti-Stokes fields, dˆ (z), become [dˆ (z), dˆ† (z )] = δ(z − z )δ( −  ),

so that the noise oscillators are modelled as localized and independent, a suitable assumption for molecular-vibration oscillators. Taking the Fourier transform of equation (20) and separating into frequency-shifted components that are capable of phase-matching, we obtain the following differential equations for the mean fields (denoted by overbars) [27]: d A¯ p = iH (0)| A¯ p |2 A¯ p , dz d A¯ a = i [H (0) + H ()] | A¯ p |2 A¯ a dz + iH () A¯ 2p A¯ ∗s exp(−i kz), d A¯ s = i [H (0) + H (−)] | A¯ p |2 A¯ s dz + iH (−) A¯ 2p A¯ ∗a exp(−i kz).

(23)

(25)

(26)

(27)

Here k = ka + ks − 2kp is the phase mismatch caused by dispersion. Expanding the wavevectors in a Taylor series around the pump frequency, one obtains k = β2 2 to second order, where β2 is the group-velocity dispersion coefficient. Equations (25)–(27) are valid when the pump remains essentially undepleted by the Stokes and anti-Stokes waves and is much stronger than the Stokes and anti-Stokes waves. The solution to equations (25) and (27) can be expressed as [27]

(19)

for the total field propagating through a FOPA having a frequency and polarization degenerate pump. No matter which field is considered to be the input signal field, the lower frequency field is called the Stokes field, Aˆ s ; the higher frequency field is called the anti-Stokes field, Aˆ a . The quantum equation of motion for the total field is the same as equation (9):

(24)

A¯ a (z) = µa (z) A¯ a (0) + νa (z) A¯ ∗s (0),

(28)

A¯ s (z) = µs (z) A¯ s (0) + νs (z) A¯ ∗a (0),

(29)

where

    i k − [2H (0) + H () − H ∗ (−)]Ip z µa (z) = exp − 2   iκ × sinh(gz) + cosh(gz) , (30) 2g     i k − [2H (0) + H (−) − H ∗ ()]Ip z µs (z) = exp − 2  ∗  iκ × sinh(g ∗ z) + cosh(g ∗ z) , (31) 2g ∗     i k − [2H (0) + H () − H ∗ (−)]Ip z νa (z) = exp − 2 iH ()Ip × sinh(gz), (32) g     i k − [2H (0) + H (−) − H ∗ ()]Ip z νs (z) = exp − 2 iH (−)Ip × sinh(g ∗ z). (33) g∗ Here Ip = | A¯ p |2 is the pumppower in Watts, κ = k + [H () + H ∗ (−)]Ip and g = −(κ/2)2 + H ()H ∗ (−)Ip2 is the complex gain coefficient. The mean optical power gain of the parametric amplifier is G = |µ j |2 , where j ∈ a, s. This mean gain is shown in figure 3 for a typical 1 km long DSFbased fibre parametric amplifier with and without the Raman effect. The fibre coefficients are as explained in the caption. S765

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Figure 3. Gain and NF spectra for 1 km-long FOPA made with DSF and pumped at 1537.6 nm with 1.5 W power. The fibre’s dispersion zero is at 1537 nm, the dispersion slope is 0.064 ps nm−2 km−1 and the nonlinear coefficient H (0) = 1.8 W−1 km−1 . For our four-wave-mixing plus Raman theory (dotted curves), Im{H ()} is calculated from the measurements in [20, 24, 25]. For the four-wave-mixing theory excluding the Raman effect (solid curves), Im{H ()} = 0. Temperature is 300 K.

Linearization of the Fourier transformation of equation (20) shows that the quantum fluctuations of the pump contribute negligibly small amounts of fluctuations to the Stokes and anti-Stokes waves when the pump is strong. These fluctuations are neglected by replacing the pump field operators with their mean fields. Under the strong pump approximation it is also acceptable to neglect the fluctuation operators at all frequencies except the Stokes and anti-Stokes frequencies because the pump mean-field will interact only with these modes of interest to a non-negligible degree, as is also shown by linearization of the quantum fluctuations around the mean fields. As a result, we obtain d A¯ p (34) = iH (0)| A¯ p |2 A¯ p , dz d Aˆ a = i [H (0) + H ()] | A¯ p |2 Aˆ a + iH () A¯ 2p Aˆ †s exp(−i kz) dz (35) + C()dˆ (z) A¯ p exp[i(kp − ka )z], d Aˆ s = i [H (0) + H (−)] | A¯ p |2 Aˆ s dz + iH (−) A¯ 2p Aˆ †a exp(−i kz) C()dˆ† (z) A¯ p

exp[i(kp − ks )z], (36) − where C() remains an unspecified constant that is necessary to preserve the commutation relations. Its value will be derived and its relationship to W () elucidated. In equations (35) and (36), respectively, we identify i[H (0) + Re{H (±)}]| A¯ p |2 Aˆ a(s) as pump cross-phase modulation terms, −Im{H (±)}| A¯ p |2 Aˆ a(s) as Raman loss (gain) terms, i Re{H (±)} A¯ 2p Aˆ †a(s) exp(−i kz) as electronic and in-phase Raman-mediated four-wave-mixing terms, and S766

as quadrature Raman-mediated four-wave-mixing terms. Under the undepleted pump approximation, wherein the pump is treated essentially classically, it is also permissible to relax the commutation relations so that only the commutators at Stokes and anti-Stokes frequencies are required to be preserved. Even when the pump is treated quantum mechanically, changes to the commutators of the pump field are of second order in this linearized first-order theory. Therefore the commutators that are required to be obeyed are: [ Aˆ j (z), Aˆ †k (z )] = δ j k δ(z − z ),

(37)

[ Aˆ j (z), Aˆ k (z )] = 0,

(38)

for j, k = a, s only. The reason that there is only one term proportional to C() in equations (35) and (36) is as follows: the Fourier transformation of ˆ C()[dˆ† (z)eit + dˆ (z)e−it ] A(t) for the chosen A(t) in equation (19) contains terms at five frequencies. However, under the strong pump approximation, a linearization around the mean field of the pump shows that only the terms proportional to A¯ p contribute meaningfully to the nonlinear interaction. Of these two terms, the one proportional to exp(−it) contributes to the anti-Stokesfield, equation (35), and the other proportional to exp(it) to equation (36). Also, a linear phase shift of the Stokes and anti-Stokes waves relative to the pump wave occurs because the phase velocity is different at the pump, Stokes and anti-Stokes frequencies. Thus, for example, at the Stokes frequency the result is C()dˆ† (z) A¯ p exp[i(kp − ks )z] as in equation (36). The solution of equations (35) and (36) is Aˆ a (L) = µa (L) Aˆ a (0) + νa (L) Aˆ †s (0)  L + C() dz A¯ p (z) exp[i(kp − ka )z] 0

× [µa (L − z) − νa (L − z)] dˆ (z), ˆ As (L) = µs (L) Aˆ s (0) + νs (L) Aˆ †a (0)  L dz A¯ p (z) exp[i(kp − ks )z] + C()

(39)

0

× [−µs (L − z) + νs (L − z)] dˆ† (z).

(40)

We next find the value of C(). One can show that C() must be real in order for the total nonlinear phase operator in equation (20) to be Hermitian. In addition, the previously mentioned positivity requirement on W () in equation (10) implies that C() must also be positive in order for the Raman oscillators to absorb net energy from the field. We then find the value of the unknown constant C() by requiring that the output commutator [ Aˆ a (L), Aˆ †a (L)] = 1

(41)

be satisfied. This gives  L C()2 | A¯ p |2 dz [µa (L − z) − νa (L − z)] 0

× [µ∗a (L − z) − νa∗ (L − z)] = −|µa (L)|2 + |νa (L)|2 + 1.

Raman-effect induced noise limits on χ (3) parametric amplifiers and wavelength converters 2.5

(42)

One may verify similarly that

and

[ Aˆ s (L), Aˆ †s (L)] = 1

(43)

[ Aˆ a (L), Aˆ s (L)] = 0

(44)

are satisfied by the above value of C() for any positive fibre √ length√L. Then C() is related to Boivin’s W () by W () = 2π C(). In this section, we have presented a thorough derivation of the input–output mode transformations that govern a lossless χ (3) parametric amplifier. In the following two sections, we use these input–output mode transformations to obtain the noise figure of χ (3) parametric amplifiers and wavelength converters.

4. Noise figure of χ(3) phase-insensitive parametric amplification In this section, the noise figure of a fibre parametric amplifier in the phase-insensitive configuration is analysed. In such a configuration a coherent-state input signal is injected at the Stokes (anti-Stokes) frequency while the input at the antiStokes (Stokes) frequency remains in the vacuum state. The NF is then defined as NFPIA =

SNRin, j , SNRout, j

(45)

where SNRin, j =

n¯ j (0)2 ,

nˆ j (0)2 

SNRout, j =

n¯ j (L)2 (46)

nˆ j (L)2 

with nˆ j (ξ ) = Aˆ †j (ξ ) Aˆ j (ξ ) for ξ = 0 or L, n¯ j = nˆ j , and nˆ j = nˆ j − n¯ j for j ∈ {a, s}. The gain of the amplifier is, therefore, G = n¯ j (L)/n¯ j (0) = |µ j |2 . For a coherentstate anti-Stokes input, the mean number of photons exiting the amplifier at the anti-Stokes frequency is n¯ a (L) = n¯ a (0)|µa (L)|2 + |νa (L)|2 + n th (−|µa (L)|2 + |νa (L)|2 + 1).

(47)

Similarly, for a coherent-state Stokes input, the mean number of photons exiting the amplifier at the Stokes frequency is n¯ s (L) = n¯ s (0)|µs (L)|2 + |νs (L)|2 + (n th + 1)(|µs (L)|2 − |νs (L)|2 − 1).

(48)

When the input-signal photon number is much greater than the amplifier gain, n¯ j (0)  G, equations (39) and (40) lead to the following expression for the NF: NF j, PIA = 1 +

|ν j |2 + (1 + 2n th )| − 1 + |µ j |2 − |ν j |2 | , (49) |µ j |2

where j = s(a) if the signal frequency is on the Stokes (antiStokes) side. It should be noted that in the limit where Im{H ()} → 0, the above NF expression is that of the ideal PIA. Also,

(c) Noise Figure (Linear Scale)

Solving for C(), we see after some calculation that C() = 2 Im{H ()}.

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Figure 4. NF versus gain obtained from equation (49) for a phase-matched amplifier (Re{κ} = 0). (a) Dotted curve, an ideal PIA; (b) dashed curve, a PIA made with SMF-28 fibre; (c) solid curve, a PIA made with DSF. All curves are for 1.38 THz pump-signal detuning and 300 K. Experimental data points for the DSF case are from Voss et al [17]. No fitting parameters have been used and H () is calculated from measured nonlinear coefficients [20, 24, 25].

µs = µa = µ, νs = νa = ν, with |µ|2 − |ν|2 = 1 when Im{H ()} = 0. The NF becomes |ν j |2 |µ j |2 1 2(G − 1) = + (50) G G with the gain G = |µ j |2 . In the limit of high gain, NF j, PIA (no Raman) = 2, the well-known 3 dB NF limit for an ideal PIA. In the limit where k is large, the four-wave-mixing process is not phase matched and the parametric gain is not able to accumulate. In such a case ν j = 0, and the NF expression at the Stokes frequency becomes NF j, PIA (no Raman) = 1 +

(1 + 2n th )(|µs |2 − 1) (51) |µs |2 1 (2 + 2n th )(G − 1) = + , (52) G G which is the noise figure for Raman amplification in a lossless fibre. In figure 3, we have also plotted the NF versus signal-pump detuning for the experimental setup described in the caption of that figure. This plot shows that the noise figure of the parametric amplifier increases as the pump-signal detuning increases. It also shows the existence of a noise floor near  = 0. In figure 4, we show NF versus gain for (a) an ideal PIA, (b) a PIA made with SMF-28 fibre, and (c) a PIA made with DSF. The experimental data points fit the theoretical prediction without any fitting parameters. The details of the experiment are explained in [17]. One may obtain the limit approached by the NF in equation (49) as the length of the FOPA, and thus the gain, becomes large. Even though the undepleted pump approximation will eventually break down at extremely high gains, for typical amplifiers the NF has essentially converged to its limit when the FOPA gain exceeds 20 dB, a value that is NFs, PIA (no FWM) = 1 +

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Figure 5. High-gain NFPIA and NFWC versus pump-signal detuning for a FOPA phase-matched (Re{κ} = 0) at each detuning. Solid curve, PIA made with DSF (equations (53) and (54)); dashed curve, PIA made with SMF (equations (53) and (54)); dash–dotted curve, WC made with DSF (equations (59) and (60)); dotted curve, WC made with SMF (equations (59) and (60)). The circle is the experimental data point from Voss et al [17]. H () is calculated from measured nonlinear coefficients [20, 24, 25]. Temperature is 300 K.

easily surpassed in experiments in [9, 15], and [16]. In order to calculate this limit, we choose the linear phase-mismatch so that k = −2 Re{H ()}Ip where the domain of  is (0, 1] and the pump power, Ip , is fixed. Assigning a value to  corresponds to adjusting the pump to a wavelength between that for optimal phase-matching ( = 1) and very close to the zero-dispersion wavelength of the fibre. By choosing , we choose k, and then insert equations (30)–(33) into equation (49). Taking the limit as the length L of the amplifier approaches ∞, we obtain solutions that are independent of the pump power Ip . Defining HRe ≡ Re{H ()} and HIm ≡ | Im{H ()}|, we obtain the limit for a Stokes side PIA to be: lim NFs, PIA = 2 + 2n th − 2n th

L→∞

2







H ()

.  ×



i(1 − )HRe + HIm + [1 − (1 − )2 ]H 2 − i2 HRe HIm

Re

(53)

The corresponding limit for the NF on the anti-Stokes side is found to be lim NFa, PIA = −2n th + (2 + 2n th )

L→∞

2







H ()

.  ×



i(1 − )HRe − HIm + [1 − (1 − )2 ]H 2 + i2 HRe HIm

Re

(54)

In figure 5 we show the NF versus signal-pump detuning (lines labelled ‘PIA’) in the case where the gain in the fibre (DSF and SMF) has been phase-matched (i.e.  = 1 or, equivalently, Re{κ} = 0) at each signal frequency. Results are also shown in figure 5 for wavelength conversion (lines labelled ‘WC’), a topic discussed in the next section. The NF limit at a given detuning  is not that of a real fibre, but shows what the quantum limit would be at a particular pump-signal detuning if phase-matched at that detuning. The difference between the quantum-limited NFs for the DSF and S768

Figure 6. Ratio of the Raman to four-wave-mixing spontaneously emitted mean photon number versus the Stokes gain. The FOPA is phase-matched (Re{κ} = 0) in all cases.  = −13.7 THz, solid curve;  = −1.37 THz, dashed curve;  = −137 GHz, dotted curve. H () is calculated from the measured nonlinear coefficients [20, 24, 25]. Temperature is 300 K.

the SMF arises from the differing ratio Im{H ()}/ Re{H ()} in the two fibres. The ratio varies depending on the dopants introduced into the core of these fibres. Thus in designing ultra-wideband FOPAs, there is a NF advantage in choosing fibres with dopants that minimize the Raman-gain coefficient for a given magnitude of the nonlinear coefficient H (). Even though the effect of Im{H ()} on the mean field is small, the contribution of this Raman-gain-producing coefficient on the NF is significant. This is due to the large mean number of thermal photons, n th , at low frequencies and owing to the larger relative contribution of the Raman noise in the earliest stages of the amplifier. This is illustrated in figure 6 which shows the Stokes-side gain dependence of the ratio of the mean number of spontaneously emitted Stokes-side Raman noise photons to the mean number of spontaneously emitted four-wave-mixing photons for various pump-signal detunings. The large ratio at lower gains shows that the Raman noise dominates in the early stage of the amplifier.  The Raman gain scales linearly ∝ Im{H ()}| A¯ p |2 L , while the parametric gain scales quadratically [∝ [Re{H ()}| A¯ p |2 L]2 ] in the early parts of the amplifier. This is illustrated in figure 7, wherein we show the distance dependence of the spontaneously emitted mean photon number. We note that the number of spontaneously emitted Raman photons is higher in modes on the Stokes side than on the anti-Stokes side. However, the gain is also higher on the Stokes side and thus the noise figure is better on the Stokes side than on the antiStokes side. Thus, when  is near the Raman-loss peak on the anti-Stokes side, the noise figure is very large, which is explained by the competition between the Raman loss and the parametric gain. There is continual Raman-mediated loss with associated back-action noise that contaminates the ideal four-wave-mixing gain. This situation is analogous to that of an erbium-doped-fibre amplifier operating under conditions of imperfect inversion (due either to saturation or inadequate pumping), where loss and gain occur at the same time, harming the noise figure. To summarize the results of this section, we have derived analytical expressions for the quantum-limited noise figure of

Raman-effect induced noise limits on χ (3) parametric amplifiers and wavelength converters

|µs |2 + (1 + 2n th )(|µs |2 − |νs |2 ) |νs |2 2 |νa | + (1 + 2n th )(|νa |2 − |µa |2 ) ≈1+ |µa |2 = NFa, PIA ,

-3

6

x 10

≈1+

Mean Photon Number

5 4 3 2 1 0 0

5

10 15 Position in Fibre (m)

20

25

Figure 7. Spontaneously emitted Raman (circles) and four-wave-mixing (line) mean photon number versus the fibre length for a phase-matched FOPA (Re{κ} = 0) at  = −137 GHz. Pump power is 1.5 W. Raman mean photon number is given by (1 + n th )(|µs |2 − |νs |2 − 1) and the four-wave-mixing mean photon number |νs |2 . H () is calculated from measured nonlinear coefficients [20, 24, 25]. Temperature is 300 K.

(58)

which is the high-gain approximation for the anti-Stokes-side PIA noise figure. Similar calculation for the anti-Stokes-side NF shows that NFa, WC ≈ NFs, PIA in the limit of high gain. This expected symmetry is demonstrated by the high-gain traces in figure 5. The thin curves show the NF for WC, which are essentially mirror images about the pump frequency of the curves for PIA. One may obtain the limit approached by the NF of a WC in equation (56) in a similar manner to that of the PIA. Again we choose the linear phase-mismatch so that k = −2 Re{H ()}Ip where the domain of  is (0, 1] and the pump power, Ip , is fixed. By choosing , we choose k, then insert equations (30)–(33) into equation (56). Taking the limit as the length L of the amplifier approaches ∞, we again obtain solutions that are independent of the pump power Ip . Recalling that HRe ≡ Re{H ()} and HIm ≡ | Im{H ()}|, we obtain the limit for a Stokes side WC to be: lim NFs, WC = −2n th + (2 + 2n th )

L→∞

(3)

χ parametric amplifiers that take into account the non-zero response time of the Kerr nonlinearity, explaining to a large extent why no group has produced parametric amplifiers with a NF below 3.7 dB. As the emerging microstructure fibres permit newfound flexibility in amplifier design, it is important to properly model the nonlinear interaction in order to predict the gain and noise performance of χ (3) amplifiers.

5. Noise figure of χ

(3)

wavelength conversion



2

i(1 − )H + H + [1 − (1 − )2 ]H 2 − i2 H H

Re Im Re Im

Re

.

×

H ()



The corresponding limit for the NF of WC on the anti-Stokes side is found to be lim NFa, WC = 2 + 2n th

L→∞



2

i(1 − )H − H + [1 − (1 − )2 ]H 2 + i2 H H

Re Im Re Im

Re

. − 2n th



H ()



We define the noise figure of wavelength conversion as NFa(s), WC =

SNRs(a), in , SNRa(s), out

(60) (55)

when the Stokes (anti-Stokes) wave in a coherent state is injected into the amplifier at z = 0 and the SNR of the antiStokes (Stokes) wave is measured at the output. Once again, under the condition that the number of input photons is much greater than the parametric gain (n¯ j (0)  |µ j |2 ), the NF of a wavelength converter is:



|µ j |2 + (1 + 2n th ) −1 + |µ j |2 − |ν j |2

NF j, WC = 1 + (56) |ν j |2 for j ∈ {a, s}. Using the fact that at high gains |νa |2 |µs |2 ≈ , 2 |νs | |µa |2

|µs |2 + (1 + 2n th )(|µs |2 − |νs |2 − 1) |νs |2

6. Summary In conclusion, we have presented a thorough derivation of the input–output mode transformations for a χ (3) parametric amplifier. Resulting expressions for the noise figure of phase-insensitively operated fibre parametric amplifiers show excellent agreement with our previously published experimental work [17] without the need for any fitting parameters. We have also derived the noise figure for wavelength conversion and shown that in the limit of high gain it is essentially the same as that for the corresponding PIA. We have also presented high-gain limits on the NF of PIAs and WCs.

(57)

we show that, for a given input signal, the high-gain WC has the same NF as phase-insensitive parametric amplification. For the Stokes-side WC NFs, WC = 1 +

(59)

Acknowledgments This work was supported by the US Office of Naval Research under Grant No. N00014-03-1-0179 and by the US Army Research Office under a MURI grant DAAD19-00-1-0177. The authors thank Sang-Kyung Choi, Kahraman G K¨opr¨ul¨u and Ranjith Nair for useful discussions. S769

P L Voss and P Kumar

References [1] Haus H A and Mullen J A 1962 Quantum noise in linear amplifiers Phys. Rev. 128 2407–13 [2] Carter S J, Drummond P D, Reid M D and Shelby R M 1987 Squeezing of quantum solitons Phys. Rev. Lett. 58 1841–4 [3] Lai Y and Haus H A 1989 Quantum-theory of solitons in optical fibers: 1. Time-dependent Hartree approximation Phys. Rev. A 40 844–53 [4] Lai Y and Haus H A 1989 Quantum-theory of solitons in optical fibers: 2. Exact solution Phys. Rev. A 40 854–66 [5] Haus H A 2000 Electromagnetic Noise and Quantum Optical Measurements (Berlin: Springer) [6] Boivin L, K¨artner F X and Haus H A 1994 Analytical solution to the quantum field theory of self-phase modulation with a finite response time Phys. Rev. Lett. 73 240–3 [7] K¨artner F X, Dougherty D J, Haus H A and Ippen E P 1994 Raman noise and soliton squeezing J. Opt. Soc. Am. B 11 1267–76 [8] Voss P L and Kumar P 2004 Raman-noise induced noise-figure limit for χ (3) parametric amplifiers Opt. Lett. 29 445–7 [9] Wong K K Y, Shimizu K, Marhic M E, Uesaka K, Kalogerakis G and Kazovsky L G 2003 Continuous-wave fiber optical parametric wavelength converter with 40 dB conversion efficiency and a 3.8 dB noise figure Opt. Lett. 28 692–4 [10] Hansryd J, Andrekson P A, Westlund M, Li J and Hedekvist P 2002 Fiber-based optical parametric amplifiers and their applications IEEE J. Sel. Top. Quantum Electron. 8 506–20 [11] Su Y, Wang L, Agarwal A and Kumar P 2000 Wavelength-tunable all-optical clock recovery using a fiber-optic parametric oscillator Opt. Commun. 184 151–6 [12] Wang L, Su Y, Agarwal A and Kumar P 2001 Synchronously mode-locked fiber laser based on parametric gain modulation and soliton shaping Opt. Commun. 194 313–7 [13] Wang L, Agarwal A, Su Y and Kumar P 2002 All-optical picosecond-pulse packet buffer based on four-wave mixing loading and intracavity soliton control IEEE J. Quantum Electron. 38 614–9 [14] Blows J L and French S E 2002 Low-noise-figure optical parametric amplifier with a continuous-wave frequency-modulated pump Opt. Lett. 27 491–3

S770

[15] Tang R, Lasri J, Devgan P, Sharping J E and Kumar P 2003 Microstructure-fibre-based optical parametric amplifier with gain slope of ∼200 dB W−1 km−1 in the telecom range Electron. Lett. 39 195–6 [16] Radic S, McKinstrie C J, Jopson R M, Centanni J C, Lin Q and Agrawal G P 2003 Record performance of parametric amplifier constructed with highly nonlinear fibre Electron. Lett. 39 838–9 [17] Voss P L, Tang R Y and Kumar P 2003 Measurement of the photon statistics and the noise figure of a fiber-optic parametric amplifier Opt. Lett. 28 549–51 [18] Stolen R H, Gordon J P, Tomlinson W J and Haus H A 1989 Raman response function of silica-core fibers J. Opt. Soc. Am. B 6 1159–65 [19] Newbury N R 2002 Raman gain: pump-wavelength dependence in single-mode fiber Opt. Lett. 27 1232–4 [20] Dougherty D J, K¨artner F X, Haus H A and Ippen E P 1995 Measurement of the Raman gain spectrum of optical fibers Opt. Lett. 20 31–3 [21] Joneckis L G and Shapiro J H 1993 Quantum propagation in a Kerr medium: lossless, dispersionless fiber J. Opt. Soc. Am. B 10 1102–20 [22] Joneckis L G and Shapiro J H 1994 Quantum propagation in a Kerr medium: lossless, dispersionless fiber J. Opt. Soc. Am. B 11 150 (erratum) [23] Agrawal G P 2001 Nonlinear Fiber Optics 3rd edn (San Diego, CA: Academic) [24] Koch F, Lewis S A E, Chernikov S V and Taylor J R 2001 Broadband Raman gain characterisation in various optical fibres Electron. Lett. 37 1437–8 [25] Boskovic A, Chernikov S V, Taylor J R, Gruner-Nielsen L and Levring O A 1996 Direct continuous-wave measurement of n 2 in various types of telecommunication fiber at 1.55 mm Opt. Lett. 21 1966–8 [26] Shapiro J H and Boivin L 1995 Raman-noise limit on squeezing in continuous-wave four-wave mixing Opt. Lett. 20 925–7 [27] Golovchenko E, Mamyshev P V, Pilipetskii A N and Dianov E M 1990 Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers IEEE J. Quantum Electron. 26 1815–20