Abstract

A cascaded parametric amplifier consists of a first parametric amplifier, which amplifies an input signal and generates an idler, which is a copy of the signal, a signal processor, which controls the phases of the signal and idler, and a second parametric amplifier, which combines the signal and idler in a phase-sensitive manner. In this paper, cascaded parametric amplification is modeled and the conditions required to maximize the constructive–destructive extinction ratio are determined. The results show that a cascaded parametric amplifier can be operated as a filter: A desired signal–idler pair is amplified, whereas undesired signal–idler pairs are deamplified. For the desired signal and idler, the noise figures of the filtering process (input signal-to-noise ratio divided by the output ratios) are only slightly higher than those of the copying process: Signal-processing functionality can be achieved with only a minor degradation in signal quality.

© 2016 Optical Society of America

1. Introduction

Radio-frequency (RF) photonics is the transmission and processing of RF signals (1–100 GHz and beyond) encoded on optical carrier waves [1–6]. Compared to electrical systems, optical systems have the advantages of high bandwidth, low dispersion, low RF-independent loss, resistance to electromagnetic interference and tunability. The required signal-processing functions include amplification, channelization, filtering, frequency conversion and switching.

Optical parametric amplification is made possible by four-wave mixing (FWM) in a nonlinear waveguide [7]. A cascaded parametric amplifier (CPA) consists of a first parametric amplifier, in which a strong pump amplifies an input signal and generates an idler, which is a (conjugated) copy of the signal, a signal processor, which controls the phases of the pump, signal and idler, and can attenuate them independently, and a second parametric amplifier, in which the pump, signal and idler are combined in a phase-sensitive manner [8]. In this paper, it is shown that a CPA can be operated as a filter: A desired signal–idler pair is amplified, whereas undesired signal–idler pairs are deamplified.

The paper is organized as follows: In Sec. 2, the properties of parametric amplification are reviewed, and formulas are stated for the magnitudes and phases of the signal and idler transfer coefficients. In Sec. 3, a study is made of an ideal CPA, from which losses are absent. Formulas are derived for the transfer coefficients of the composite device, in terms of the transfer coefficients of its constituent devices. By controlling the phase of the signal or idler, or both phases (relative to the pump phase), one can control whether the waves combine constructively in the second amplifier, for maximal amplification, or destructively, for minimal amplification (maximal deamplification). The effects of loss are studied in Sec. 4. For each wave (signal or idler), the action of the signal processor can be modeled by a single parameter that includes the transmission factor (which is the reciprocal of the loss factor) and the phase shift. Loss does not prevent the second amplifier from operating in a phase-sensitive manner. In Sec. 5, the noise properties of cascaded parametric amplification (also CPA) are studied. Formulas are derived for the signal and idler noise figures of the filtering process (input signal-to-noise ratio divided by the output ratios), which depend on the composite transfer coefficients. When operated sensibly, CPA produces an amplified signal–idler pair whose (common) noise figure is about 4 dB, which is only 1-dB more than the noise figure of the first amplification process (copying). Thus, CPA can amplify a desired signal–idler pair, and simultaneously deamplify undesired pairs, without degrading the desired signal quality. Finally, in Sec. 6, the main results of this paper are summarized.

2. Parametric amplification

In parametric amplification by degenerate FWM, a strong pump wave (p) drives weak signal and idler waves (s and i). Degenerate FWM is governed approximately by the equations

dzApi(kp+γK|Ap|2)Ap,
dzAs=i(ks+2γK|Ap|2)As+iγKAp2Ai*,
dzAi=i(ki+2γK|Ap|2)Ai+iγKAp2As*,
where dz = d/dz is a distance derivative, Aj is a wave amplitude, kj is a wavenumber and γK is the Kerr nonlinearity coefficient [7]. Pump depletion, and signal- and idler-induced phase modulation were neglected. The squared amplitude |Aj|2 has units of action flux (power divided by frequency), which is proportional to the photon flux of the wave.

By making the substitutions Aj(z) = Bj(z) exp[i(kp + γK|Ap|2)z], one finds that the transformed pump amplitude Bp is constant, and the transformed signal and idler (sideband) amplitudes satisfy the coupled-mode equations

dzBs=iδsBs+iγBi*,dzBi=iδiBi+iγBs*,
where δj = kjkp + |γ| is a relative wavenumber and γ=γKBp2 is the coupling coefficient. By combining the first of Eqs. (4) with the conjugate of the second, one finds that the characteristic wavenumber
k=(δsδi)/2±[(δs+δi)2/4|γ|2]1/2.
This result prompts the substitutions Bs = Cs exp[i(δsδi)z/2] and Bi*=Ci*exp[i(δsδi)z/2], where δsδi = kski. By making these substitutions in Eqs. (4), one obtains the symmetrized equations
dzCs=iδCs+iγCi*,dzCi*=iδCi*iγ*Cs,
where δ = (δs + δi)/2 = (ks + ki)/2 − kp + |γ| is the wavenumber mismatch. The solutions of Eqs. (6) can be written in the input–output (IO) forms
Cs(z)=μ(z)Cs(0)+ν(z)Ci*(0),Ci*(z)=μ*(z)Ci*(0)+ν*(z)Cs(0),
where the transfer functions
μ(z)=cos(kz)+iδsin(kz)/k,ν(z)=iγsin(kz)/k
and the wavenumber k = (δ2 − |γ|2)1/2. Notice that the transfer functions satisfy the auxiliary equation |μ|2 − |ν|2 = 1.

The solutions of Eqs. (4) can be written in the related IO forms

Bs(z)=e(z)μ(z)Bs(0)+e(z)ν(z)Bi*(0),Bi(z)=e*(z)μ(z)Bi(0)+e*(z)ν(z)Bs*(0),
where the phase factor e(z) = exp[i(δsδi)z/2]. Notice that e*(z) = exp[i(δiδs)z/2]. By combining Eqs. (9) with their conjugates and the auxiliary equation, one finds that
|Bs(z)|2|Bi(z)|2=|Bs(0)|2|Bi(0)|2.
Equation (10) is called the Manley–Rowe–Weiss (MRW) equation [9, 10]. The MRW variable |Bs|2 − |Bi|2 is the difference between the action fluxes of the signal and idler, and is proportional to the difference between the associated photon fluxes: Signal and idler photons are created (or destroyed) in pairs. The IO relations for the A-amplitudes need not be stated, because the A-amplitudes differ from the B-amplitudes by a common phase factor which does not affect the current FWM process or subsequent processes (or the associated photon fluxes). Henceforth, the term amplitude will be used as an abbreviation for B-amplitude.

In the presence of second-order dispersion (β2), the mismatch can be written asβ2ω2/2+|γ|, where ω is the (moderate) difference between the signal and pump frequencies. Degenerate FWM is unstable (the signal is amplified) when β2 < 0 (anomalous dispersion) and 0 < ω < |4γ/β2|1/2. In Fig. 1(a) the real and imaginary parts of the characteristic wavenumber (normalized to the maximal growth rate |γ|) are plotted as functions of the frequency difference (normalized to the optimal frequency |2γ/β2|1/2). In Fig. 1(b) the phase of μ is plotted as a function of the normalized frequency for maximal gains [cosh(|γ|l)] of 3 and 10 dB. For most nonoptimal frequencies, this phase is nonzero. The phase of ν (which is not shown) depends on the input pump phase, but does not depend on the other physical parameters.

 figure: Fig. 1

Fig. 1 (a) Real part (solid curve) and imaginary part (dashed curve) of the normalized wavenumber plotted as functions of the normalized frequency. (b) Phase ϕμ plotted as a function of the normalized frequency for maximal gains of 3 dB (dashed curve) and 10 dB (solid curve).

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3. Cascaded parametric amplification

Consider the operation of a cascaded parametric amplifier, which is illustrated in Fig. 2. (The pump is not shown.) The first parametric amplifier, which is based on degenerate FWM, amplifies the signal in a phase-independent (PI) manner and generates an idler, the phase shifter controls the phases of the signal and idler, and the second parametric amplifier combines the (pump) signal and idler in a phase-sensitive (PS) manner. In practice, phase shifts are imposed by an optical signal processor, such as a Finisar Waveshaper, which also can attenuate the signal and idler independently.

 figure: Fig. 2

Fig. 2 Architecture of a cascaded parametric amplifier. A two-mode phase-insensitive amplifier (▹) is followed by a phase shifter (ϕ) and a two-mode phase-sensitive amplifier.

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The first stage of such a device is governed by the IO equations (9). It is convenient to rewrite these equations in the matrix form

B=TA,
where A=[As,Ai*]t is the input amplitude vector, B=[Bs,Bi*]t is the output amplitude vector and the transfer matrix
T=[e1μ1e1ν1e1ν1*e1μ1*].
[In the notation of Eqs. (9), Aj = Bj(0) and Bj = Bj(z).] As stated in Sec. 2, parametric amplification preserves the MRW variable. The second stage is governed by the IO equation (11) and the transfer matrix
T=[es00ei*],
where es = exp(s) and ei = exp(i) are phase factors. Phase shifting also preserves the MRW variable. (The transfer matrices for processes with this property are called symplectic and are discussed further in App. A.) The third stage (second amplification stage) is governed by Eqs. (11) and (12), with the subscript 1 replaced by the subscript 2.

One models concatenated processes by multiplying their transfer matrices. Hence, the composite transfer matrix [11, 12]

[μssνsiνis*μii*]=[e2μ2e2ν2e2ν2*e2μ2*][es00ei*][e1μ1e1ν1e1ν1*e1μ1*]=e2e1[μ2esμ1+ν2ei*ν1*μ2esν1+ν2ei*μ1*μ2*ei*ν1*+ν2*esμ1μ2*ei*μ1*+ν2*esν1].
Equations (11) and (14) show that the composite device functions as a two-mode parametric amplifier. It only remains to determine how the composite gain depends on the component gains and the intermediate phase shifts. (Cascaded parametric amplification without phase shifts is discussed further in App. B.) First, the phase factors e1 and e2 do not affect the composite gain. Henceforth, these factors will be omitted. Second, the transfer coefficient μss (which controls the output signal power) is maximal when
ϕs+ϕi=ϕν2ϕν1ϕμ2ϕμ1.
The same condition also maximizes the other three coefficients. This result reflects the fact that if two or more processes conserve the MRW variable, so also must concatenations of these processes. (The products of symplectic matrices are also symplectic.) The coefficient νis characterizes the output idler produced by an input signal (and |μss|2 − |νis|2 = 1), whereas the coefficients νsi and μii characterize the output signal and idler produced by an input idler (and |μii|2 − |νsi|2 = 1): As the physical parameters change, the output signal and idler powers increase (or decrease) in unison. The minimum condition is similar to Eq. (15), with π added to the right side. In the application (filtering), the goal is to maximize the powers of a signal–idler pair, while minimizing the powers of the neighboring pairs. The phase differences required to do this differ slightly from π, because ϕμ1 and ϕμ2 depend on frequency [Fig. 1(a)]. In principle, one can maximize the powers of an arbitrary set of desired pairs, while minimizing the powers of the complementary set of undesired pairs.

When the extremum conditions are satisfied, the composite gain parameters

μ±=μ2μ1±ν2ν1,ν±=μ2ν1+ν2μ1,
where μj and νj are abbreviations for |μj| and |νj|, respectively, and μ±2ν±2=1. The design challenge is the determination of values of ν1 and ν2 that maximize the constructive signal gain μ+, minimize the destructive signal gain μ and maximize the (amplitude) extinction ratio μ+/μ. The parameters μ+ and ν+ are both increasing functions of ν1 and ν2. Hence, it does not matter how one apportions gain between stages 1 and 2: In the context of constructive gain, more constituent gain is better. However, the dependence of μ and ν on ν1 and ν2 is more complicated. The parameter μ attains its minimal value of 1 when ν = 0 (ν1 = ν2), whereas the parameter ν can be positive, zero or negative, depending on whether ν1 > ν2 (too little deamplification), ν1 = ν2 or ν1 < ν2 (too much deamplification), respectively [Fig. 3(a)]. The extinction ratio
E=(1+ν12)1/2(1+ν22)1/2+ν1ν2(1+ν12)1/2(1+ν22)1/2ν1ν2=1+xy1xy,
where x=ν1/(1+ν12)1/2 and y=ν2/(1+v22)1/2. It is easy to show that the extinction ratio is an increasing function of x and y, which are themselves increasing functions of ν1 and ν2 [Fig. 3(b)]: In the contex of extinction ratio, more constituent gain is also better. For the application, it is best to minimize the gains of the neighboring signal–idler pairs (μ = 1), in which case
E=μ2+ν2,
where μ2 = G is the (common) PI power gain of each amplifier and ν2 = G − 1. In the high-gain regime (μν ≫ 1), the extinction ratio is about 2G. The preceding discussion pertains to the signal extinction ratio. The idler extinction ratio ν+/ν can be infinite, because ν can be zero.

 figure: Fig. 3

Fig. 3 (a) Destructive gain parameter ν and (b) amplitude extinction ratio μ+/μ plotted as functions of the constituent gain parameters ν1 and ν2. Darker shading represents lower values, whereas lighter shading represents higher values. The destructive gain [Eq. (16)] varies from −4 to 4, whereas the extinction ratio [Eq. (17)] varies from 1 to 33. When ν1 = ν2, the destructive gain is zero and the extinction ratio increases monotonically as the (common) gain increases.

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The preceding discussion was based on the assumption that the signal is a continuous wave. However, it also applies approximately to pulsed signals if the transfer coefficients depend weakly on the frequencies of the signal components. If the input signal is chirped (by dispersion, for example), its components have frequency-dependent phase factors. In the first amplifier, such a signal generates an idler whose components have the opposite phase factors. When the signal and idler components are combined in the second amplifier, the sum phase ϕs + ϕi is independent of frequency. Thus, chirped pulses can be amplified and attenuated in the same way as continuous waves.

4. Effects of attenuation on cascaded parametric amplification

The results of the previous section showed that cascaded parametric amplication, with frequency-dependent phase shifts, can amplify a signal-idler pair while deamplifying its neighboring pairs. However, these results were based on the idealization that the composite device (amplifiers and phase-shifter) is lossless. In this section, the effects of loss (attenuation) on the device performance are studied.

In the beam-splitter model of loss, a signal wave interacts with and loses power to a scattered wave (loss mode). This process is governed by Eq. (11), in which the input vector A = [As, Ak]t, the output vector B = [Bs, Bk]t and the transfer matrix

Ts=[τsρsρs*τs*],
where the subscript k denotes the loss mode, and the transfer coefficients τs and ρs satisfy the auxiliary equation |τs|2 + |ρs|2 = 1. Notice that the input and output vectors involve only amplitudes, not conjugates. The loss matrix (19) is unitary (TT = I = TT). For the loss process the MRW variable |As|2 + |Ak|2 is conserved: Signal photons are converted into loss-mode photons.

Suppose that loss is applied to the signal after the phase shift [Fig. 4(a)]. Then the composite IO equation is

[bsbk]=[τsρsρs*τs*][es001][asak]=[τsesρsρs*esτs*][asak].
Equation (20) can be rewritten in the alternative form
[bses*bk]=[τsesρsρs*τs*es*][asak].
By measuring the phase of the output loss-mode relative to ϕs, one obtains a shift-and-loss process that is characterized by the single parameter τses. Now suppose that loss is applied before the phase shift [Fig. 4(b)]. Then the composite IO equation is
[bsbk]=[es001][τsρsρs*τs*][asak]=[esτsesρsρs*τs*][asak]
Equation (22) can be rewritten in the alternative form
[bsbk]=[esτsρsρs*es*τs*][bsesak].
By measuring the phase of the input loss-mode relative to −ϕs, one obtains a loss-and-shift process that is characterized by the single parameter esτs (which equals τses). Thus, the order of loss and phase-shifting does not matter, and one can model the composite process as a single loss process with a transmission coefficient that includes the (controlled) phase shift. Similar results apply to the idler, which interacts with loss-mode l.

 figure: Fig. 4

Fig. 4 Architecture of a cascaded parametric amplifier with internal attenuation. A two-mode phase-insensitive amplifier (▹) is followed by a phase shifter (ϕ) and an attenuator (◃), and a two-mode phase-sensitive amplifier. The order of the attenuator and phase shifter does not matter.

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The three concatenated processes (PI amplification, loss and PS amplification) are also governed by Eq. (11), in which the input vector A=[As,Ak,Ai*,Al*]t, the output vector B=[Bs,Bk,Bi*,Bl*]t and the composite transfer matrix

[μssμskνsiνslμksμkkνkiνklνis*νik*μii*μil*νls*νlk*μli*μll*]=[μ20ν200100ν2*0μ2*00001][τsρs00ρs*τs*0000τi*ρi*00ρiτi][μ10ν100100ν1*0μ1*00001]=[(μ2τsμ1+ν2τi*ν1*)μ2ρs(μ2τsν1+ν2τi*μ1*)ν2ρi*ρs*μ1τs*ρs*ν10(μ2*τi*ν1*+ν2*τsμ1)ν2*ρs(μ2*τi*μ1*+ν2*τsν1)μ2*ρi*ρiν1*0ρiμ1*τi].
Because each of the three constituent matrices is symplectic (App. A), so also is the composite matrix: For each row and column of the composite matrix, the difference between the sums of the squares of the μ- and ν-coefficients equals 1. Of particular interest are the coefficients μss and νis, which characterize the output signal and idler produced by an input signal. These coefficients are in the first column of the matrix, so they satisfy the auxiliary equation
|μss|2+|μks|2|νis|2|νls|2=1.
By combining Eqs. (24) and (25), one finds that
|μss|2|νis|2=1+|ρiν1|2|ρsμ1|2=|τsμ1|2|τiν1|2.
Equation (26) guarantees that the signal and idler powers rise and fall in unison, regardless of the amplification and attenuation coefficients, and the external phase shifts. Although the terms on the left side of Eq. (26) depend on all of these parameters in a complicated way, the terms on the right side do not depend on the phase shifts, or the transfer coefficients of the second amplifier.

4.1. Equal losses

First, consider the case in which the signal-processor losses are equal (|τs| = |τi| = τ). Then the transfer coefficients μss and νis, which are specified in Eq. (24), are maximal when

ϕs+ϕi=ϕν2ϕν1ϕμ2ϕμ1ϕτsϕτi.
Although loss-induced phase shifts can change the values of the external phase shifts required for maximal gain, they do not change the facts that maximal gain can be attained, and the total phase shifts associated with maximal and minimal gain differ by π. When these extremum conditions are satisfied, the composite gain parameters
μ±=τ(μ2μ1±ν2ν1),ν±=τ(μ2ν1±ν2μ1),
where μj and νj are abbreviations for |μj| and |νj|, respectively, and μ±2ν±2=τ2. Parameters (28) differ from parameters (16) only by (common) factors of τ. Hence, the analysis of amplification and deamplification is similar to that of Sec. 3. In particular, the minimal value of μ2 is τ2, so the miminal output-signal power equals the input-signal power divided by the loss factor (1/τ2). The amplitude extinction ratios are unaffected by loss: The signal ratio still is specified by Eq. (17) and the idler ratio still can be infinite, because ν can be zero.

The composite transfer coefficients depend on four independent gain and loss parameters, so there are many ways to operate a cascaded parametric amplifier. In our recent experiments [13, 14], the internal (connection and signal-processor) losses are about 6 dB. To limit the possibilities, suppose that the PI gain of the first amplifier (|μ1|2) is set to 6 dB, to compensate for the aforementioned losses. Then the intermediate signal has the same power as the input signal and the intermediate idler has comparable (but slightly lower) power. The net gain of the composite device is provided by the second amplifier. (This mode of operation is reasonable, but is not necessarily optimal.) In Fig. 5, the composite signal and idler gains are plotted as functions of the PI gain of the second amplifier (|μ2|2). In the constructive case, the signal and idler gains both increase as the second gain increases. Conversely, in the destructive case, the signal gain decreases to its miminal value of 6 dB, then increases, whereas the idler gain decreases to its minimal value of 0, then increases. [See the discussion between Eqs. (16) and (17).] A signal-power extinction ratio (|μ+/μ|2) of 20 dB was demonstrated recently [13]. The idler-power extinction ratio (|ν+/ν|2) was even larger, because the constructive idler gain is comparable to the constructive signal gain, whereas the destructive idler gain is lower than the destructive signal gain [14].

 figure: Fig. 5

Fig. 5 Composite gains of the signal (solid curve) and idler (dashed curve) plotted as functions of the phase-insensitive gain of the second amplifier. The other gain and loss parameters are specified in the text. (a) constructive gains |μ+|2 and |ν+|2, and (b) destructive gains |μ|2 and |ν|2.

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4.2. Compensating losses

Second, consider the case in which the losses are chosen to be unequal, to compensate for the fact that the signal and idler produced by the first amplifier have unequal powers (|τsμ1| = |τiν1| = λ). When the extremum conditions are satisfied, the composite gain parameters

μ±=λ(μ2±ν2)=ν±,
where μ2 and ν2 are abbreviations for |μ2| and |ν2|, respectively and μ±2ν±2=0: Not only are the intermediate signal and idler powers equal, so also are the output powers. The (common) value of μ2 and ν2 is τsτiμ1ν1/(μ2 + ν2)2, so the minimal output power can be lower than the input-signal power divided by the geometric mean of the loss factors. It follows from Eq. (29) that the (common) extinction ratio
E=(μ2+ν2)/(μ2ν2)=(μ2+ν2)2.
In the high-gain regime (μ2ν2 ≫1), the extinction ratio is about 4G2.

In Fig. 6, the composite signal and idler gains are plotted as functions of the PI gain of the second amplifier for the case in which τs=ν11/2/μ13/2 and τi = 1/(μ1ν1)1/2. The PI gain of the first amplifier is 6 dB, and the signal and idler losses are 6.6 and 5.4 dB, respectively. In the constructive case, the signal gain increases to 20 dB, as it did for equal losses. However, in the destructive case, it decreases to −20 dB (rather than 0 or −6 dB), so compensating losses enable lower deamplified powers and larger extinction ratios. In both cases, the signal and idler gains are identical, as they were designed to be. Identical signal and idler extinction ratios enabled by compensating losses were demonstrated recently [13].

 figure: Fig. 6

Fig. 6 Composite gains of the signal (solid curve) and idler (dashed curve) plotted as functions of the phase-insensitive gain of the second amplifier. The other gain and loss parameters are specified in the text. (a) constructive gains |μ+|2 and |ν+|2, and (b) destructive gains |μ|2 and |ν|2. The composite signal and idler gains are identical by design, so the solid and dashed curves are indistinguishable.

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4.3. Overcompensating losses

With equal losses, the mimimal output-idler power is zero, but the minimal output-signal power is nonzero. With compensating losses, the output powers of the signal and idler are equal and nonzero. One can also choose the losses in such a way that the minimal signal power is zero, but the minimal idler power is nonzero (|μ2τsμ1| = |ν2τiν1|). When the extremum conditions are satisfied, the composite gain parameters

μ±=μ2τsμ1±ν2τiν1,μ±=μ2τiν1±ν2τsμ1,
where μj, νj, τs and τi are abbreviations for |μj|, |νj|, |τs| and |τi| respectively, and μ±2ν±2=(τsμ1)2/ν22=(τiν1)2/μ22. Notice that the output idler is always more powerful than the output signal. The value of ν2 is τsτiμ1ν1/(μ2ν2), so the minimal output-idler power is of the order of the input-signal power divided by the geometric mean of the loss factors. It follows from the second of Eqs. (31) that the idler extinction ratio
E=μ22+ν22.
In the high-gain regime, this extinction ratio is about 2G2. The signal extinction ratio is infinite, because μ is zero.

In Fig. 7, the composite signal and idler gains are plotted as functions of the PI gain of the second amplifier for the case in which τs=ν1/μ12 and τi = 1/ν1. The PI gain of the first amplifier is 6 dB and the signal and idler losses are 7.3 and 4.7 dB, respectively. (These losses minimize the signal power when the second gain is 6 dB.) In the constructive case, the signal and idler gains both increase as the second gain increases, as they did for equal losses. However, in the destructive case, the signal gain decreases to its minimal value of 0, then increases, whereas the idler gain decreases to its minimal value of −6 dB, then increases. Thus, the case of overcompensating losses is similar to that of equal losses, with the roles of the signal and idler interchanged. In summary, by controlling the filter losses at the signal and idler frequencies, one can optimize the signal and idler extinction ratios for specific applications.

 figure: Fig. 7

Fig. 7 Composite gains of the signal (solid curve) and idler (dashed curve) plotted as functions of the phase-insensitive gain of the second amplifier. The other gain and loss parameters are specified in the text. (a) constructive gains |μ+|2 and |ν+|2, and (b) destructive gains |μ|2 and |ν|2.

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5. Noise figures of cascaded parametric amplification

The results of the preceding section showed that a realistic cascaded parametric amplifier (with internal losses) can accept (amplify) a signal and idler pair of interest and reject (deamplify) neighboring pairs, and that the associated extinction ratios can be controlled by varying the signal and idler losses. It is also important to determine the effects of this device on the noise properties of the (accepted) signal and idler. Noise is a quantum-mechanical phenomenon [15], but one can model it semi-classically by writing each mode amplitude Aj as αj + δαj, where αj is an amplitude mean and δαj is a random variable [16,17]. The random variables have zero means and Gaussian statistics, and are uncorrelated, so the moments 〈δαj〉 = 0, 〈δαjδαk〉 = 0 and δαjδαk*=δjk/2, where 〈 〉 denotes an ensemble average and δjk is the Kronecker delta. It follows from this ansatz that 〈|Aj|2〉 = |αj|2 + 1/2. Consequently, one describes the semi-classical model by saying that 1/2 noise photon is added to each mode [17]. (In the rest of this paper, |A|2 is proportional to the photon flux. However, when one detetects photons, one integrates this flux over a finite time, so the measurement result is proportional to the photon number. In this section, the latter normalization is used.)

In homodyne detection [15], a two-mode beam splitter is used to combine the signal with a local oscillator (LO). The difference between the output photon numbers is proportional to the input signal quadrature Ps(ϕ1)=(As*eiϕl+Aseiϕl)/21/2, where ϕl is the LO phase. For example, ϕl = 0 corresponds to the real signal quadrature, whereas ϕl = π/2 corresponds to the imaginary quadrature. If the LO phase is matched to the signal phase [ϕl = arg(αs)], the quadrature mean 〈Ps〉 = 21/2|αs|. The quadrature variance δPs2=1/2, independent of the LO phase. For the input idler, 〈Pi〉 = 0 and δPi2=1/2. These semi-classical results are identical to the quantum-mechanical results for a coherent state and a vacuum state, respectively [15].

For homodyne detection, the signal-to-noise ratio (SNR) is the square of the quadrature mean divided by the quadrature variance. For a matched LO phase, the SNR is 4|αs|2. The noise figure of a parametric process is defined to be the input signal SNR divided by the output signal or idler SNR: It is a figure of demerit. The calculations of multiple-mode noise figures are lengthy, but straightforward, and were described in detail elsewhere [12]. In this paper, only the main results are stated. It was shown in [18] that the semi-classical predictions for the homodyne noise figures of parametric processes are identical to the quantum-mechanical results.

It follows from Eq. (24) and the preceding definitions that the mean and variance of the output signal quadrature are

Qs=μssPs,δQs2=(μss2+μsk2+νsi2+νsl2)/2,
respectively, where μ and ν are abbreviations for |μ| and |ν|. The output quadrature variance depends only on the moduli of the composite transfer coefficients because the input signal, idler and loss-mode fluctuations are phase-independent and uncorrelated. (The intermediate signal and idler fluctuations are partially correlated.) In contrast, the relation between the input and output quadrature means is based on the assumption that the input and output LO phases are matched to the input and output signal phases, respectively. By combining Eqs. (33) and their counterparts for the idler (μsjνij and νsjμij), one obtains the noise figures
Fs=(μss2+μsk2+νsi2+νsl2)/μss2,Fi=(νis2+νik2+μii2+μil2)/νis2.
An inspection of Eq. (24) reveals that the coefficients μss and νis involve the products τsμ1 and τiν1, about which assumptions were made in Sec. 4, whereas the coefficients νsi and μii do not. However, by using the auxiliary equations (App. A)
μss2+μsk2νsi2νsl2=1,μii2+μil2νis2νik2=1,
one can rewrite Eqs. (34) in the alternative forms
Fs=[2(μss2+μsk2)1]/μss2,Fi=[2(νis2+νik2)+1]/νis2,
which involve only the desired coefficients.

Two limits of Eqs. (36) deserve brief discussions. For the case in which τs = τi = 1 and μ2 = 1 (two-mode amplification), Fs=21/μ12 and Fi=2+1/ν12. As the gain parameter μ12 increases, the signal noise figure varies from 1 (0 dB) to 2 (3 dB), whereas the idler noise figure decreases from ∞ to 2. The initial idler noise figure is infinite because the output idler SNR, which is zero in the absence of gain, is compared to the input signal SNR, which is nonzero. For the case in which μ1 = μ2 = 1 (attenuation), Fs=1/τs2 and Fi = ∞. The signal noise figure equals the signal loss factor. The idler noise figure is infinite for the reason stated above. (Had we compared the output idler SNR to the input idler SNR, the noise figure would equal the loss factor 1/τi2.)

5.1. Equal losses

The noise figures associated with the cases considered in Sec. 4 now can be discussed. First, consider the case in which τs = τi = τ. Then the quadrature mean and variance

Qs=μ±Ps,δQs2=[(μ±2+ν±2)+ρ2(μ22+ν22)]/2,
where μ± and ν± were defined in Eqs. (28). Suppose that μjνj ≫ 1 and τμ1 = 1 (so τ ≪ 1), in which case the first two stages function as a zero-net-gain copier and the third stage functions as a PS amplifier. Then μ+ ≈ 2μ2, ρ ≈ 1 and Fs10μ22/4μ22=2.5(4dB): For the + mode, the composite device is only slightly noisier than a one-stage PI amplifier, which has a noise figure of 3 dB. Now suppose that ν2 = ν1, so that ν = 0 and μ = τ. Then the composite device functions as an attenuator and Fs2μ22/τ2=2μ141: For the − mode, the composite device is very noisy. This noisiness is not a problem, because the desired + mode is retained, whereas the neighboring − modes are discarded. These results and the corresponding idler results [second of Eqs. (36)] are illustrated in Fig. 8, for the same parameters as Fig. 5. Notice that the idler noise figure is infinite in the destructive case when the output idler SNR is zero.

 figure: Fig. 8

Fig. 8 Composite noise figures of the signal (solid curve) and idler (dashed curve) plotted as functions of the phase-insensitive gain of the second amplifier. The other gain and loss parameters are specified in the text. (a) constructive noise figure and (b) destructive noise figure.

Download Full Size | PPT Slide | PDF

5.2. Compensating losses

Second, consider the case in which τsμ1 = τiν1 = λ (balanced copying). Then μss = λ(μ2 ± ν2), but there is no simple expression for νsi. This problem can be avoided by using the first of Eqs. (35). The quadrature mean and variance

Qs=λ(μ2±ν2)Ps,δQs2=[2λ2(μ2±ν2)2+2μ22ρs21]/2.
Suppose that μj, νj ≫ 1 and λ ≈ 1 (so τj ≪ 1), in which case the first two stages function approximately as a zero-net-gain copier. Then ρs ≈ 1 and Fs[2(μ2±ν2)2+2μ221]/(μ2±ν2)2. For the + mode, in the high-gain regime Fs10μ22/4μ22=2.5 and the composite device is only slightly noisier than a PI amplifier, whereas for the − mode, Fs2μ22/(μ2ν2)28μ241 and the composite device is very noisy. These results and the corresponding idler results are illustrated in Fig. 9, for the same parameters as Fig. 6. Notice that the noise figures of the signal and idler are different, because they experience different gains and losses during copying. Notice also that neither noise figure is infinite in the destructive case, because neither output SNR is zero.

 figure: Fig. 9

Fig. 9 Composite noise figures of the signal (solid curve) and idler (dashed curve) plotted as functions of the phase-insensitive gain of the second amplifier. The other gain and loss parameters are specified in the text. (a) constructive noise figure and (b) destructive noise figure. Even though the composite gains of the signal and idler are identical, their noise figures are not.

Download Full Size | PPT Slide | PDF

5.3. Overcompensating losses

Third, consider the case in which τiν1 = λ and τsμ1 = λν1/μ1 (so μ2τsμ1 = ν2τiν1 when μ2 = μ1). Then the quadrature mean and variance

Qi=λ(μ2±ν2ν1/μ1)Ps,δQi2=[2λ2(μ2±ν2ν1/μ1)2+2ν22ρs2+1]/2.
Suppose that μjνj ≫ 1 and λ = 1 (so τj ≪ 1), in which case the first two stages function approximately as a zero-net-gain copier. Then λ(μ2 + ν2ν1/μ1) ≈ 2μ2, ρs ≈ 1 and Fi10μ22/4μ22=2.5(4dB): For the + mode, the composite device is only slightly noisier than a one-stage PI amplifier. Now suppose that ν2 = ν1. Then λ(μ2ν2ν1/μ1) = 1/μ1 and Fi2μ22μ12=2μ141: For the − mode, the composite device is very noisy. These results and the corresponding signal results [first of Eqs. (36)] are illustrated in Fig. 10, for the same parameters as Fig. 7. Notice that the signal noise figure is infinite in the destructive case, whereas the idler noise figure is not.

 figure: Fig. 10

Fig. 10 Composite noise figures of the signal (solid curve) and idler (dashed curve) plotted as functions of the phase-insensitive gain of the second amplifier. The other gain and loss parameters are specified in the text. (a) constructive noise figure and (b) destructive noise figure.

Download Full Size | PPT Slide | PDF

Overall, the noise properties of the composite device are similar in the three cases considered. In particular, varying the signal and idler losses to increase the signal or idler extinction ratios does not increase the noise figures of the accepted modes, which remain below 4 dB.

6. Summary

In this paper, the properties of cascaded parametric amplification (CPA) were studied in detail. The properties of parametric amplification were reviewed in Sec. 2, and formulas were stated for the signal and idler transfer coefficients. For the optimal signal frequency, at which the signal gain is maximal, the pump, signal and idler wavenumbers are matched, and the signal transfer coefficient is real. For any other signal frequency, the wavenumbers are not matched and transfer coefficient is complex, with a phase that depends on the frequency. The idler transfer coefficients are imaginary, for all idler frequencies. In Sec. 3, a study was made of an ideal cascaded parametric amplifier (also CPA), from which losses are absent. Formulas were derived for the transfer coefficients of the composite device, in terms of the transfer coefficients of its constituent devices (amplifiers and phase shifter). By choosing judiciously the phase shift imposed upon the signal or idler, or both phase shifts, one can control whether the waves combine constructively in the second amplifier, for maximal amplification, or destructively, for minimal amplification (maximal deamplification). The power extinction ratio is the maximal power gain divided by the minimal power gain. It was shown that high extinction ratios can be obtained if the amplifier gains are equal. In this case, the extinction ratio is is about 4G2, where G is the (common) phase-insensitive gain. The effects of loss were studied in Sec. 4. For each wave (signal or idler), the action of the signal processor (attenuator and phase shifter) can be modeled by a single parameter that includes the transmission factor (which is the reciprocal of the loss factor) and the phase shift. If the signal and idler losses are equal, the output signal and idler powers are reduced relative to their ideal values, but the extinction ratio remains equal to 4G2. One can also vary the loss factors independently. If one equalizes the signal and idler powers that are input to the second amplifier, the extinction ratio is 16G22. In Sec. 5, formulas were derived for the signal and idler noise figures (input signal-to-noise ratio divided by the output ratios) associated with amplification and deamplifcation. These formulas depend in a complicated way on the gain, loss and phase-shift parameters. The symplectic properties of the transfer matrix (App. A) were used to simplify the formulas. When operated sensibly, CPA produces an amplified signal–idler pair whose (common) noise figure is about 4 dB, which is only 1-dB more than the noise figure of the first amplification process (copying). Lower noise figures are possible. Thus, CPA can amplify a desired signal–idler pair, and simultaneously deamplify undesired pairs, without degrading the desired signal quality significantly. CPA can also increase the sensitivity of phase measurements in quantum metrology [19, 20].

Appendix A: Symplectic transfer matrices

Consider the interaction of n modes (group 1) with n conjugate modes (group 2). Let A1 and A2 be n × 1 mode-amplitude vectors and suppose that their evolution is governed by the quadratic Hamiltonian

H=A1J1A1+A2J2A2+A1KA2*+A2tKA1,
where J1, J2 and K are coefficient matrices. These matrices can be constants, or they can vary with distance. J1 and J2 must be Hermitian, because H is real, but K is arbitrary. By applying the Hamilton equations
dzAj=iH/Aj
the Hamiltonian (40) and using the fact that (A1KA2*)t=A2KtA1*, one obtains the vector coupled-mode equations
dzA1=iJA1+iKA2*,dzA2=iJ2A2+iKtA1*.
Notice that in the first equation the coupling matrix is K, whereas in the second equation it is Kt. For the special case in which n = 1 (two-mode parametric amplification), J1 = δs, J2 = δi and K = γ, as stated in Sec. 2.

It is instructive to consider the evolution of A1A1 and A2A2, which are the total action fluxes of the modes in groups 1 and 2, respectively. For each mode, the action flux is the power divided by the frequency and is proportional to the photon flux of the mode. By combining Eqs. (42) with their conjugates, one finds that

dz(A1A1)=i(A1KA2*A2tKA1),dz(A2A2)=i(A2KtA1*A1tK*A2).
By combining Eqs. (43) and using the fact that the terms on the right sides are scalars, one finds that
dz(A1A1A2A2)=0.
Equation (44) is called the Manley–Rowe–Weiss (MRW) equation and signifies that photons from groups 1 and 2 are produced in pairs [9, 10]. For special cases in which group 2 is absent (examples of which are two-mode beam splitting and frequency conversion), the total number of photons in group 1 is conserved.

For reasons that will become clear, it is useful to rederive the MRW equation in a more formal way. Equations (42) can be rewritten as the single equation

dzA=iLA,
where the 2n × 1 amplitude vector and 2n × 2n coefficient matrix are
A=[A1A2*],L=[J1KKJ2*],
respectively. The MRW variable can be written in the compact form ASA, where S = diag(I, −I) is a 2n × 2n diagonal matrix. It is easy to verify that S2 = I and SL = LS. By using the latter identity, one finds that
dz(ASA)=i[AS(LA)(AL)SA]=0.
Thus, ASA is conserved, even if the coefficient matrices vary with distance.

Now let B = TA, where A is an input vector, B is the associated output vector and T is the transfer matrix. Because the MRW variable is conserved locally, it must also be conserved globally. Hence, BSB = ATSTA = ASA. Because this statement is true for an arbitrary input vector, it must also be true that

TST=S.
A matrix that satisfies Eq. (48) is called symplectic. Notice that a symplectic matrix multiplied by a phase factor is still symplectic and the product of symplectic matrices is also symplectic. For the special case in which group 2 is absent, the transfer matrix is unitary (TT = I = TT).

It follows from Eq. (48) that STS = T−1. (To be precise, the symplectic condition implies that STS is the left inverse of T, but Hamiltonian evolution is reversible, so the right inverse must also exist and equal the left inverse.) The inverse process must also be symplectic, so S = (T−1)ST−1 = (STS)S(STS) = STSTS. By multiplying both sides of this equation by S on the left and S on the right, one finds that

TST=S.
One can also use Eq. (49) to define a symplectic matrix [21, 22], then deduce Eq. (48), but the two approaches are equivalent.

Equations (48) and (49) impose numerous constraints on the elements of the transfer matrix, all of which are consequences of Hamiltonian evolution. The first equation involves only the columns of this matrix, whereas the second equation involves only the rows. It is useful to write the transfer matrix and its inverse in the block forms

T=[M11N12N21*M22*],T1=[M11N21tN12M22t],
respectively. The first of Eqs. (50) defines the block matrices, whereas the second is a consequence of Eq. (48) or (49). By combining Eqs. (50), one finds that
T1T=[M11M11N21tN21*M11N12N21tM22*M22tN21*N12M11M22tM22*N12N12]=[I00I],
TT1=[M11M11N12N12N12M22tM11N21tN21*M11M22*N12M22*M22tN21*N21t]=[I00I].

For the two-mode processes discussed in Secs. 2 and 3, M11 = , N12 = , N21 = e*ν and M22 = e*μ. Hence,

T1T=[|μ|2|ν|2μ*ννμ*μν*ν*μ|μ|2|ν|2],TT1=[|μ|2|ν|2νμμνν*μ*μ*ν*|μ|2|ν|2].
The off-diagonal terms in Eqs. (53) are identically zero, whereas the diagonal terms lead to the auxiliary equation |μ|2 − |ν|2 = 1.

For the four-mode process discussed in Sec. 4, group 1 consists of the signal and loss-mode k, whereas group 2 consists of the idler and loss-mode l. For this process, the block matrices are

M11=[μssμskμksμkk],N12=[νsiνslνkiνkl],
N21=[νisνikνlsνlk],M22=[μiiμilμliμll].
To illustrate the usefuleness of Eqs. (51) and (52), we consider three examples briefly. The 11-component of Eq. (51) implies that the elements in the first column of the transfer matrix satisfy the auxiliary equation
|μss|2+|μks|2|νis|2|νls|2=1,
which relates the changes in the signal and idler photon-fluxes [Eqs. (25) and (26)]. The 11- and 33-components of Eq. (52) imply that the elements in the first and third rows satisfy the auxiliary equations
|μss|2+|μsk|2|νsi|2=1,|μii|2+|μil|2|νis|2|νik|2=1,
respectively, which allow the signal and idler noise figures to be rewritten in alternative (simpler) forms [Eqs. (34)(36)].

To make the transition from classical to quantal mechanics, one replaces the (scalar) mode amplitudes Aj by the mode operators âj, which satisfy the boson commutation relations [âj, âk] = 0 and [a^j,a^k]=δjk, where [x, y] = xyyx is a commutator and δjk is the Kronecker delta. The MRW variable also becomes an operator. The modes from group 1 contribute terms of the form a^ka^j, which are number operators, whereas the modes from group 2 contribute terms of the form [a^k,a^k]=a^k,a^k+1, which differ from number operators only by constants. For systems with quadratic Hamiltonians, the Heisenberg equations for the mode operators are linear in those operators and have the same forms as the Hamilton equations for the mode amplitudes, so the same transfer matrices describe the classical and quantal processes. Hence, the MRW operator is conserved. The symplectic properties of the transfer matrices were used to derive many other important results about parametric processes in [21].

Appendix B: Cascaded parametric amplification

In parametric amplification, the rate at which the signal and idler (sidebands) grow depends on the relation between the pump, signal and idler phases [23, 24]. If there is no input idler, an idler is created with a phase that maximizes the growth rate. In the absence of (wavenumber) mismatch, the sidebands continue to grow rapidly. However, in the presence of mismatch, the phase relation is changed in such a way that the sideband growth is slowed (unstable regime) or slowed and reversed (stable regime). This process is called dephasing.

Cascaded parametric amplification without intermediate phase shifts is governed by Eq. (11) and the composite transfer coefficients

μss=μ1μ2+ν2ν1*,νsi=μ2ν1+ν2μ1*,
where μj and νj are specified by Eqs. (8). The MRW equation (App. A) ensures that |μss|2 = 1 + |νsi|2, so the moduli of the transfer coefficients are maximal (or minimal) simultaneously. The maximization condition is
ϕμ2+ϕμ1=ϕν2ϕν1.
When this condition is satisfied, |νsi| attains its maximal value |μ2||ν1| + |ν2||μ1|.

For example, consider two identical fibers of lengths l1 and l2 (and the same pump), or one fiber of length l1 + l2 with a virtual cut. Then

μss=(c2+iδs2/k)(c1+iδs1/k)+|γ|2s2s1/k2=c2c1+(|γ|2δ2)s2s1/k2+iδ(c2s1+s2c1)/k=c21+iδs21/k,
νsi=(c2+iδs2/k)iγs1/k+iγs2/k(c1iδs1/k)=iγs21/k,
where c1 = cos(kl1), s1 = sin(kl1), c21 = cos[k(l1 + l2)] and s21 = sin[k(l1 + l2)]. Equations (60) and (61) make sense, because a virtual cut (at an arbitrary point) in a fiber does not change the (composite) gain associated with that fiber. However, the gain is not necessarily maximal: In this example, ϕν2 = ϕν1, but ϕμ2 + ϕμ1 ≠ 0 unless δ = 0, because ϕμ = tan−1[(δ/k) tan(kl)]. For a matched composite amplifier, the outputs from the first amplifier are optimally phased as inputs to the second amplifier, whereas for a mismatched amplifier they are not. One can improve the performance of a mismatched amplifier by rephasing the sidebands (choosing the intermediate phase shifts es and ei judiciously).

References and links

1. R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE Trans. Microw. Theory Tech. 54, 832–846 (2006). [CrossRef]  

2. A. J. Seeds and K. J. Williams, “Microwave photonics,” J. Lightwave Technol. 24, 4628–4641 (2006). [CrossRef]  

3. J. Capmany and D. Nowak, “Microwave photonics combines two worlds,” Nat. Photon. 1, 319–330 (2007). [CrossRef]  

4. J. Yao, “Microwave photonics,” J. Lightwave Technol. 27, 314–335 (2009). [CrossRef]  

5. J. Capmany, J. Mora, I. Gasulla, J. Sancho, J. Lloret, and S. Sales, “Microwave photonic signal processing,” J. Lightwave Technol. 31, 571–586 (2013). [CrossRef]  

6. R. A. Minasian, “Ultra-wideband and adaptive photonic signal processing of microwave signals,” IEEE J. Quantum Electron. 52, 0600813 (2016). [CrossRef]  

7. M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge, 2007). [CrossRef]  

8. Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Putnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photon. 5, 430–436 (2011). [CrossRef]  

9. J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956). [CrossRef]  

10. M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc IRE 45, 1012–1013 (1957).

11. Z. Tong, A. Bogris, C. Lundstrom, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier,” Opt. Express 18, 14820–14835 (2010). [CrossRef]   [PubMed]  

12. C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number correlations produced by parametric processes,” Opt. Express 18, 19792–19823 (2010). [CrossRef]   [PubMed]  

13. J. M. Dailey, A. Agarwal, C. J. McKinstrie, and P. Toliver, “Optical filtering through frequency-selective phase-sensitive amplification and deamplification,” Optical Fiber Communication conference, Anaheim, California, 20–24 March 2016, paper W4D.2.

14. A. Agarwal, J. M. Dailey, C. J. McKinstrie, and P. Toliver, “Optical filter optimization using phase-sensitive amplification with unequalized inputs,” Conference on Lasers and Electro-Optics, San Jose, California, 5–10 June 2016, paper STh3F.5.

15. R. Loudon, The Quantum Theory of Light, 3rd Ed. (Oxford, 2000).

16. W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1654 (1961). [CrossRef]  

17. J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev. 129, 481–485 (1963). [CrossRef]  

18. C. J. McKinstrie, N. Alic, Z. Tong, and M. Karlsson, “Higher-capacity communication links based on two-mode phase-sensitive amplifiers,” Opt. Express 19, 11977–11991 (2011). See the appendix. [CrossRef]   [PubMed]  

19. Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 023815 (2012). [CrossRef]  

20. F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014). [CrossRef]   [PubMed]  

21. H. P. Yuen, “Multimode two-photon coherent states and unitary representation of the symplectic group,” Nucl. Phys. B (Proc. Supp.) 6, 309–313 (1989). [CrossRef]  

22. C. J. McKinstrie and M. Karlsson, “Schmidt decompositions of parametric processes I: Basic theory and simple examples,” Opt. Express 21, 1374–1394 (2013). [CrossRef]   [PubMed]  

23. C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B 10, 1856–1869 (1993). See Eqs. (2.2)–(2.4). [CrossRef]  

24. M. E. Marhic, “Analytical solutions for the phases of waves coupled by degenerate or nondegenerate four-wave mixing,” J. Opt. Soc. Am. B 30, 62–70 (2013). [CrossRef]  

References

  • View by:

  1. R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE Trans. Microw. Theory Tech. 54, 832–846 (2006).
    [Crossref]
  2. A. J. Seeds and K. J. Williams, “Microwave photonics,” J. Lightwave Technol. 24, 4628–4641 (2006).
    [Crossref]
  3. J. Capmany and D. Nowak, “Microwave photonics combines two worlds,” Nat. Photon. 1, 319–330 (2007).
    [Crossref]
  4. J. Yao, “Microwave photonics,” J. Lightwave Technol. 27, 314–335 (2009).
    [Crossref]
  5. J. Capmany, J. Mora, I. Gasulla, J. Sancho, J. Lloret, and S. Sales, “Microwave photonic signal processing,” J. Lightwave Technol. 31, 571–586 (2013).
    [Crossref]
  6. R. A. Minasian, “Ultra-wideband and adaptive photonic signal processing of microwave signals,” IEEE J. Quantum Electron. 52, 0600813 (2016).
    [Crossref]
  7. M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge, 2007).
    [Crossref]
  8. Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Putnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photon. 5, 430–436 (2011).
    [Crossref]
  9. J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
    [Crossref]
  10. M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc IRE 45, 1012–1013 (1957).
  11. Z. Tong, A. Bogris, C. Lundstrom, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier,” Opt. Express 18, 14820–14835 (2010).
    [Crossref] [PubMed]
  12. C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number correlations produced by parametric processes,” Opt. Express 18, 19792–19823 (2010).
    [Crossref] [PubMed]
  13. J. M. Dailey, A. Agarwal, C. J. McKinstrie, and P. Toliver, “Optical filtering through frequency-selective phase-sensitive amplification and deamplification,” Optical Fiber Communication conference, Anaheim, California, 20–24 March 2016, paper W4D.2.
  14. A. Agarwal, J. M. Dailey, C. J. McKinstrie, and P. Toliver, “Optical filter optimization using phase-sensitive amplification with unequalized inputs,” Conference on Lasers and Electro-Optics, San Jose, California, 5–10 June 2016, paper STh3F.5.
  15. R. Loudon, The Quantum Theory of Light, 3rd Ed. (Oxford, 2000).
  16. W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1654 (1961).
    [Crossref]
  17. J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev. 129, 481–485 (1963).
    [Crossref]
  18. C. J. McKinstrie, N. Alic, Z. Tong, and M. Karlsson, “Higher-capacity communication links based on two-mode phase-sensitive amplifiers,” Opt. Express 19, 11977–11991 (2011). See the appendix.
    [Crossref] [PubMed]
  19. Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 023815 (2012).
    [Crossref]
  20. F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
    [Crossref] [PubMed]
  21. H. P. Yuen, “Multimode two-photon coherent states and unitary representation of the symplectic group,” Nucl. Phys. B (Proc. Supp.) 6, 309–313 (1989).
    [Crossref]
  22. C. J. McKinstrie and M. Karlsson, “Schmidt decompositions of parametric processes I: Basic theory and simple examples,” Opt. Express 21, 1374–1394 (2013).
    [Crossref] [PubMed]
  23. C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B 10, 1856–1869 (1993). See Eqs. (2.2)–(2.4).
    [Crossref]
  24. M. E. Marhic, “Analytical solutions for the phases of waves coupled by degenerate or nondegenerate four-wave mixing,” J. Opt. Soc. Am. B 30, 62–70 (2013).
    [Crossref]

2016 (1)

R. A. Minasian, “Ultra-wideband and adaptive photonic signal processing of microwave signals,” IEEE J. Quantum Electron. 52, 0600813 (2016).
[Crossref]

2014 (1)

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

2013 (3)

2012 (1)

Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 023815 (2012).
[Crossref]

2011 (2)

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Putnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photon. 5, 430–436 (2011).
[Crossref]

C. J. McKinstrie, N. Alic, Z. Tong, and M. Karlsson, “Higher-capacity communication links based on two-mode phase-sensitive amplifiers,” Opt. Express 19, 11977–11991 (2011). See the appendix.
[Crossref] [PubMed]

2010 (2)

2009 (1)

2007 (1)

J. Capmany and D. Nowak, “Microwave photonics combines two worlds,” Nat. Photon. 1, 319–330 (2007).
[Crossref]

2006 (2)

R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE Trans. Microw. Theory Tech. 54, 832–846 (2006).
[Crossref]

A. J. Seeds and K. J. Williams, “Microwave photonics,” J. Lightwave Technol. 24, 4628–4641 (2006).
[Crossref]

1993 (1)

1989 (1)

H. P. Yuen, “Multimode two-photon coherent states and unitary representation of the symplectic group,” Nucl. Phys. B (Proc. Supp.) 6, 309–313 (1989).
[Crossref]

1963 (1)

J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev. 129, 481–485 (1963).
[Crossref]

1961 (1)

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1654 (1961).
[Crossref]

1957 (1)

M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc IRE 45, 1012–1013 (1957).

1956 (1)

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[Crossref]

Agarwal, A.

A. Agarwal, J. M. Dailey, C. J. McKinstrie, and P. Toliver, “Optical filter optimization using phase-sensitive amplification with unequalized inputs,” Conference on Lasers and Electro-Optics, San Jose, California, 5–10 June 2016, paper STh3F.5.

J. M. Dailey, A. Agarwal, C. J. McKinstrie, and P. Toliver, “Optical filtering through frequency-selective phase-sensitive amplification and deamplification,” Optical Fiber Communication conference, Anaheim, California, 20–24 March 2016, paper W4D.2.

Alic, N.

Andrekson, P. A.

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Putnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photon. 5, 430–436 (2011).
[Crossref]

Z. Tong, A. Bogris, C. Lundstrom, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier,” Opt. Express 18, 14820–14835 (2010).
[Crossref] [PubMed]

Blessing, D. J.

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Putnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photon. 5, 430–436 (2011).
[Crossref]

Bogris, A.

Cao, X. D.

Capmany, J.

Dailey, J. M.

A. Agarwal, J. M. Dailey, C. J. McKinstrie, and P. Toliver, “Optical filter optimization using phase-sensitive amplification with unequalized inputs,” Conference on Lasers and Electro-Optics, San Jose, California, 5–10 June 2016, paper STh3F.5.

J. M. Dailey, A. Agarwal, C. J. McKinstrie, and P. Toliver, “Optical filtering through frequency-selective phase-sensitive amplification and deamplification,” Optical Fiber Communication conference, Anaheim, California, 20–24 March 2016, paper W4D.2.

Gasulla, I.

Gordon, J. P.

J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev. 129, 481–485 (1963).
[Crossref]

Gruner-Nielsen, L.

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Putnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photon. 5, 430–436 (2011).
[Crossref]

Hudelist, F.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

Jing, J.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

Karlsson, M.

Kong, J.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

Li, J. S.

Liu, C.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

Lloret, J.

Loudon, R.

R. Loudon, The Quantum Theory of Light, 3rd Ed. (Oxford, 2000).

Louisell, W. H.

J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev. 129, 481–485 (1963).
[Crossref]

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1654 (1961).
[Crossref]

Lundstrom, C.

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Putnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photon. 5, 430–436 (2011).
[Crossref]

Z. Tong, A. Bogris, C. Lundstrom, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier,” Opt. Express 18, 14820–14835 (2010).
[Crossref] [PubMed]

Manley, J. M.

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[Crossref]

Marhic, M. E.

McKinstrie, C. J.

C. J. McKinstrie and M. Karlsson, “Schmidt decompositions of parametric processes I: Basic theory and simple examples,” Opt. Express 21, 1374–1394 (2013).
[Crossref] [PubMed]

C. J. McKinstrie, N. Alic, Z. Tong, and M. Karlsson, “Higher-capacity communication links based on two-mode phase-sensitive amplifiers,” Opt. Express 19, 11977–11991 (2011). See the appendix.
[Crossref] [PubMed]

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Putnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photon. 5, 430–436 (2011).
[Crossref]

Z. Tong, A. Bogris, C. Lundstrom, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier,” Opt. Express 18, 14820–14835 (2010).
[Crossref] [PubMed]

C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number correlations produced by parametric processes,” Opt. Express 18, 19792–19823 (2010).
[Crossref] [PubMed]

C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B 10, 1856–1869 (1993). See Eqs. (2.2)–(2.4).
[Crossref]

J. M. Dailey, A. Agarwal, C. J. McKinstrie, and P. Toliver, “Optical filtering through frequency-selective phase-sensitive amplification and deamplification,” Optical Fiber Communication conference, Anaheim, California, 20–24 March 2016, paper W4D.2.

A. Agarwal, J. M. Dailey, C. J. McKinstrie, and P. Toliver, “Optical filter optimization using phase-sensitive amplification with unequalized inputs,” Conference on Lasers and Electro-Optics, San Jose, California, 5–10 June 2016, paper STh3F.5.

Minasian, R. A.

R. A. Minasian, “Ultra-wideband and adaptive photonic signal processing of microwave signals,” IEEE J. Quantum Electron. 52, 0600813 (2016).
[Crossref]

R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE Trans. Microw. Theory Tech. 54, 832–846 (2006).
[Crossref]

Mora, J.

Nowak, D.

J. Capmany and D. Nowak, “Microwave photonics combines two worlds,” Nat. Photon. 1, 319–330 (2007).
[Crossref]

Ou, Z. Y.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 023815 (2012).
[Crossref]

Putnam, B. J.

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Putnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photon. 5, 430–436 (2011).
[Crossref]

Rowe, H. E.

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[Crossref]

Sales, S.

Sancho, J.

Seeds, A. J.

Siegman, A. E.

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1654 (1961).
[Crossref]

Tipsuwannakul, E.

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Putnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photon. 5, 430–436 (2011).
[Crossref]

Toda, H.

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Putnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photon. 5, 430–436 (2011).
[Crossref]

Toliver, P.

J. M. Dailey, A. Agarwal, C. J. McKinstrie, and P. Toliver, “Optical filtering through frequency-selective phase-sensitive amplification and deamplification,” Optical Fiber Communication conference, Anaheim, California, 20–24 March 2016, paper W4D.2.

A. Agarwal, J. M. Dailey, C. J. McKinstrie, and P. Toliver, “Optical filter optimization using phase-sensitive amplification with unequalized inputs,” Conference on Lasers and Electro-Optics, San Jose, California, 5–10 June 2016, paper STh3F.5.

Tong, Z.

Vasilyev, M.

Walker, L. R.

J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev. 129, 481–485 (1963).
[Crossref]

Weiss, M. T.

M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc IRE 45, 1012–1013 (1957).

Williams, K. J.

Yao, J.

Yariv, A.

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1654 (1961).
[Crossref]

Yuen, H. P.

H. P. Yuen, “Multimode two-photon coherent states and unitary representation of the symplectic group,” Nucl. Phys. B (Proc. Supp.) 6, 309–313 (1989).
[Crossref]

Zhang, W.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

IEEE J. Quantum Electron. (1)

R. A. Minasian, “Ultra-wideband and adaptive photonic signal processing of microwave signals,” IEEE J. Quantum Electron. 52, 0600813 (2016).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE Trans. Microw. Theory Tech. 54, 832–846 (2006).
[Crossref]

J. Lightwave Technol. (3)

J. Opt. Soc. Am. B (2)

Nat. Commun. (1)

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

Nat. Photon. (2)

J. Capmany and D. Nowak, “Microwave photonics combines two worlds,” Nat. Photon. 1, 319–330 (2007).
[Crossref]

Z. Tong, C. Lundstrom, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Putnam, H. Toda, and L. Gruner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photon. 5, 430–436 (2011).
[Crossref]

Nucl. Phys. B (Proc. Supp.) (1)

H. P. Yuen, “Multimode two-photon coherent states and unitary representation of the symplectic group,” Nucl. Phys. B (Proc. Supp.) 6, 309–313 (1989).
[Crossref]

Opt. Express (4)

Phys. Rev. (2)

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1654 (1961).
[Crossref]

J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev. 129, 481–485 (1963).
[Crossref]

Phys. Rev. A (1)

Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 023815 (2012).
[Crossref]

Proc IRE (1)

M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc IRE 45, 1012–1013 (1957).

Proc. IRE (1)

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[Crossref]

Other (4)

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge, 2007).
[Crossref]

J. M. Dailey, A. Agarwal, C. J. McKinstrie, and P. Toliver, “Optical filtering through frequency-selective phase-sensitive amplification and deamplification,” Optical Fiber Communication conference, Anaheim, California, 20–24 March 2016, paper W4D.2.

A. Agarwal, J. M. Dailey, C. J. McKinstrie, and P. Toliver, “Optical filter optimization using phase-sensitive amplification with unequalized inputs,” Conference on Lasers and Electro-Optics, San Jose, California, 5–10 June 2016, paper STh3F.5.

R. Loudon, The Quantum Theory of Light, 3rd Ed. (Oxford, 2000).

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Figures (10)

Fig. 1
Fig. 1 (a) Real part (solid curve) and imaginary part (dashed curve) of the normalized wavenumber plotted as functions of the normalized frequency. (b) Phase ϕμ plotted as a function of the normalized frequency for maximal gains of 3 dB (dashed curve) and 10 dB (solid curve).
Fig. 2
Fig. 2 Architecture of a cascaded parametric amplifier. A two-mode phase-insensitive amplifier (▹) is followed by a phase shifter (ϕ) and a two-mode phase-sensitive amplifier.
Fig. 3
Fig. 3 (a) Destructive gain parameter ν and (b) amplitude extinction ratio μ+/μ plotted as functions of the constituent gain parameters ν1 and ν2. Darker shading represents lower values, whereas lighter shading represents higher values. The destructive gain [Eq. (16)] varies from −4 to 4, whereas the extinction ratio [Eq. (17)] varies from 1 to 33. When ν1 = ν2, the destructive gain is zero and the extinction ratio increases monotonically as the (common) gain increases.
Fig. 4
Fig. 4 Architecture of a cascaded parametric amplifier with internal attenuation. A two-mode phase-insensitive amplifier (▹) is followed by a phase shifter (ϕ) and an attenuator (◃), and a two-mode phase-sensitive amplifier. The order of the attenuator and phase shifter does not matter.
Fig. 5
Fig. 5 Composite gains of the signal (solid curve) and idler (dashed curve) plotted as functions of the phase-insensitive gain of the second amplifier. The other gain and loss parameters are specified in the text. (a) constructive gains |μ+|2 and |ν+|2, and (b) destructive gains |μ|2 and |ν|2.
Fig. 6
Fig. 6 Composite gains of the signal (solid curve) and idler (dashed curve) plotted as functions of the phase-insensitive gain of the second amplifier. The other gain and loss parameters are specified in the text. (a) constructive gains |μ+|2 and |ν+|2, and (b) destructive gains |μ|2 and |ν|2. The composite signal and idler gains are identical by design, so the solid and dashed curves are indistinguishable.
Fig. 7
Fig. 7 Composite gains of the signal (solid curve) and idler (dashed curve) plotted as functions of the phase-insensitive gain of the second amplifier. The other gain and loss parameters are specified in the text. (a) constructive gains |μ+|2 and |ν+|2, and (b) destructive gains |μ|2 and |ν|2.
Fig. 8
Fig. 8 Composite noise figures of the signal (solid curve) and idler (dashed curve) plotted as functions of the phase-insensitive gain of the second amplifier. The other gain and loss parameters are specified in the text. (a) constructive noise figure and (b) destructive noise figure.
Fig. 9
Fig. 9 Composite noise figures of the signal (solid curve) and idler (dashed curve) plotted as functions of the phase-insensitive gain of the second amplifier. The other gain and loss parameters are specified in the text. (a) constructive noise figure and (b) destructive noise figure. Even though the composite gains of the signal and idler are identical, their noise figures are not.
Fig. 10
Fig. 10 Composite noise figures of the signal (solid curve) and idler (dashed curve) plotted as functions of the phase-insensitive gain of the second amplifier. The other gain and loss parameters are specified in the text. (a) constructive noise figure and (b) destructive noise figure.

Equations (61)

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d z A p i ( k p + γ K | A p | 2 ) A p ,
d z A s = i ( k s + 2 γ K | A p | 2 ) A s + i γ K A p 2 A i * ,
d z A i = i ( k i + 2 γ K | A p | 2 ) A i + i γ K A p 2 A s * ,
d z B s = i δ s B s + i γ B i * , d z B i = i δ i B i + i γ B s * ,
k = ( δ s δ i ) / 2 ± [ ( δ s + δ i ) 2 / 4 | γ | 2 ] 1 / 2 .
d z C s = i δ C s + i γ C i * , d z C i * = i δ C i * i γ * C s ,
C s ( z ) = μ ( z ) C s ( 0 ) + ν ( z ) C i * ( 0 ) , C i * ( z ) = μ * ( z ) C i * ( 0 ) + ν * ( z ) C s ( 0 ) ,
μ ( z ) = cos ( k z ) + i δ sin ( k z ) / k , ν ( z ) = i γ sin ( k z ) / k
B s ( z ) = e ( z ) μ ( z ) B s ( 0 ) + e ( z ) ν ( z ) B i * ( 0 ) , B i ( z ) = e * ( z ) μ ( z ) B i ( 0 ) + e * ( z ) ν ( z ) B s * ( 0 ) ,
| B s ( z ) | 2 | B i ( z ) | 2 = | B s ( 0 ) | 2 | B i ( 0 ) | 2 .
B = TA ,
T = [ e 1 μ 1 e 1 ν 1 e 1 ν 1 * e 1 μ 1 * ] .
T = [ e s 0 0 e i * ] ,
[ μ s s ν s i ν i s * μ i i * ] = [ e 2 μ 2 e 2 ν 2 e 2 ν 2 * e 2 μ 2 * ] [ e s 0 0 e i * ] [ e 1 μ 1 e 1 ν 1 e 1 ν 1 * e 1 μ 1 * ] = e 2 e 1 [ μ 2 e s μ 1 + ν 2 e i * ν 1 * μ 2 e s ν 1 + ν 2 e i * μ 1 * μ 2 * e i * ν 1 * + ν 2 * e s μ 1 μ 2 * e i * μ 1 * + ν 2 * e s ν 1 ] .
ϕ s + ϕ i = ϕ ν 2 ϕ ν 1 ϕ μ 2 ϕ μ 1 .
μ ± = μ 2 μ 1 ± ν 2 ν 1 , ν ± = μ 2 ν 1 + ν 2 μ 1 ,
E = ( 1 + ν 1 2 ) 1 / 2 ( 1 + ν 2 2 ) 1 / 2 + ν 1 ν 2 ( 1 + ν 1 2 ) 1 / 2 ( 1 + ν 2 2 ) 1 / 2 ν 1 ν 2 = 1 + x y 1 x y ,
E = μ 2 + ν 2 ,
T s = [ τ s ρ s ρ s * τ s * ] ,
[ b s b k ] = [ τ s ρ s ρ s * τ s * ] [ e s 0 0 1 ] [ a s a k ] = [ τ s e s ρ s ρ s * e s τ s * ] [ a s a k ] .
[ b s e s * b k ] = [ τ s e s ρ s ρ s * τ s * e s * ] [ a s a k ] .
[ b s b k ] = [ e s 0 0 1 ] [ τ s ρ s ρ s * τ s * ] [ a s a k ] = [ e s τ s e s ρ s ρ s * τ s * ] [ a s a k ]
[ b s b k ] = [ e s τ s ρ s ρ s * e s * τ s * ] [ b s e s a k ] .
[ μ s s μ s k ν s i ν s l μ k s μ k k ν k i ν k l ν i s * ν i k * μ i i * μ i l * ν l s * ν l k * μ l i * μ l l * ] = [ μ 2 0 ν 2 0 0 1 0 0 ν 2 * 0 μ 2 * 0 0 0 0 1 ] [ τ s ρ s 0 0 ρ s * τ s * 0 0 0 0 τ i * ρ i * 0 0 ρ i τ i ] [ μ 1 0 ν 1 0 0 1 0 0 ν 1 * 0 μ 1 * 0 0 0 0 1 ] = [ ( μ 2 τ s μ 1 + ν 2 τ i * ν 1 * ) μ 2 ρ s ( μ 2 τ s ν 1 + ν 2 τ i * μ 1 * ) ν 2 ρ i * ρ s * μ 1 τ s * ρ s * ν 1 0 ( μ 2 * τ i * ν 1 * + ν 2 * τ s μ 1 ) ν 2 * ρ s ( μ 2 * τ i * μ 1 * + ν 2 * τ s ν 1 ) μ 2 * ρ i * ρ i ν 1 * 0 ρ i μ 1 * τ i ] .
| μ s s | 2 + | μ k s | 2 | ν i s | 2 | ν l s | 2 = 1 .
| μ s s | 2 | ν i s | 2 = 1 + | ρ i ν 1 | 2 | ρ s μ 1 | 2 = | τ s μ 1 | 2 | τ i ν 1 | 2 .
ϕ s + ϕ i = ϕ ν 2 ϕ ν 1 ϕ μ 2 ϕ μ 1 ϕ τ s ϕ τ i .
μ ± = τ ( μ 2 μ 1 ± ν 2 ν 1 ) , ν ± = τ ( μ 2 ν 1 ± ν 2 μ 1 ) ,
μ ± = λ ( μ 2 ± ν 2 ) = ν ± ,
E = ( μ 2 + ν 2 ) / ( μ 2 ν 2 ) = ( μ 2 + ν 2 ) 2 .
μ ± = μ 2 τ s μ 1 ± ν 2 τ i ν 1 , μ ± = μ 2 τ i ν 1 ± ν 2 τ s μ 1 ,
E = μ 2 2 + ν 2 2 .
Q s = μ s s P s , δ Q s 2 = ( μ s s 2 + μ s k 2 + ν s i 2 + ν s l 2 ) / 2 ,
F s = ( μ s s 2 + μ s k 2 + ν s i 2 + ν s l 2 ) / μ s s 2 , F i = ( ν i s 2 + ν i k 2 + μ i i 2 + μ i l 2 ) / ν i s 2 .
μ s s 2 + μ s k 2 ν s i 2 ν s l 2 = 1 , μ i i 2 + μ i l 2 ν i s 2 ν i k 2 = 1 ,
F s = [ 2 ( μ s s 2 + μ s k 2 ) 1 ] / μ s s 2 , F i = [ 2 ( ν i s 2 + ν i k 2 ) + 1 ] / ν i s 2 ,
Q s = μ ± P s , δ Q s 2 = [ ( μ ± 2 + ν ± 2 ) + ρ 2 ( μ 2 2 + ν 2 2 ) ] / 2 ,
Q s = λ ( μ 2 ± ν 2 ) P s , δ Q s 2 = [ 2 λ 2 ( μ 2 ± ν 2 ) 2 + 2 μ 2 2 ρ s 2 1 ] / 2 .
Q i = λ ( μ 2 ± ν 2 ν 1 / μ 1 ) P s , δ Q i 2 = [ 2 λ 2 ( μ 2 ± ν 2 ν 1 / μ 1 ) 2 + 2 ν 2 2 ρ s 2 + 1 ] / 2 .
H = A 1 J 1 A 1 + A 2 J 2 A 2 + A 1 K A 2 * + A 2 t K A 1 ,
d z A j = i H / A j
d z A 1 = i J A 1 + i K A 2 * , d z A 2 = i J 2 A 2 + i K t A 1 * .
d z ( A 1 A 1 ) = i ( A 1 K A 2 * A 2 t K A 1 ) , d z ( A 2 A 2 ) = i ( A 2 K t A 1 * A 1 t K * A 2 ) .
d z ( A 1 A 1 A 2 A 2 ) = 0 .
d z A = i L A ,
A = [ A 1 A 2 * ] , L = [ J 1 K K J 2 * ] ,
d z ( A S A ) = i [ A S ( L A ) ( A L ) S A ] = 0 .
T ST = S .
TST = S .
T = [ M 11 N 12 N 21 * M 22 * ] , T 1 = [ M 11 N 21 t N 12 M 22 t ] ,
T 1 T = [ M 11 M 11 N 21 t N 21 * M 11 N 12 N 21 t M 22 * M 22 t N 21 * N 12 M 11 M 22 t M 22 * N 12 N 12 ] = [ I 0 0 I ] ,
T T 1 = [ M 11 M 11 N 12 N 12 N 12 M 22 t M 11 N 21 t N 21 * M 11 M 22 * N 12 M 22 * M 22 t N 21 * N 21 t ] = [ I 0 0 I ] .
T 1 T = [ | μ | 2 | ν | 2 μ * ν ν μ * μ ν * ν * μ | μ | 2 | ν | 2 ] , T T 1 = [ | μ | 2 | ν | 2 ν μ μ ν ν * μ * μ * ν * | μ | 2 | ν | 2 ] .
M 11 = [ μ s s μ s k μ k s μ k k ] , N 12 = [ ν s i ν s l ν k i ν k l ] ,
N 21 = [ ν i s ν i k ν l s ν l k ] , M 22 = [ μ i i μ i l μ l i μ l l ] .
| μ s s | 2 + | μ k s | 2 | ν i s | 2 | ν l s | 2 = 1 ,
| μ s s | 2 + | μ s k | 2 | ν s i | 2 = 1 , | μ i i | 2 + | μ i l | 2 | ν i s | 2 | ν i k | 2 = 1 ,
μ s s = μ 1 μ 2 + ν 2 ν 1 * , ν s i = μ 2 ν 1 + ν 2 μ 1 * ,
ϕ μ 2 + ϕ μ 1 = ϕ ν 2 ϕ ν 1 .
μ s s = ( c 2 + i δ s 2 / k ) ( c 1 + i δ s 1 / k ) + | γ | 2 s 2 s 1 / k 2 = c 2 c 1 + ( | γ | 2 δ 2 ) s 2 s 1 / k 2 + i δ ( c 2 s 1 + s 2 c 1 ) / k = c 21 + i δ s 21 / k ,
ν s i = ( c 2 + i δ s 2 / k ) i γ s 1 / k + i γ s 2 / k ( c 1 i δ s 1 / k ) = i γ s 21 / k ,

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