## 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

*d*=

_{z}*d*/

*dz*is a distance derivative,

*A*is a wave amplitude,

_{j}*k*is a wavenumber and

_{j}*γ*is the Kerr nonlinearity coefficient [7]. Pump depletion, and signal- and idler-induced phase modulation were neglected. The squared amplitude |

_{K}*A*|

_{j}^{2}has units of action flux (power divided by frequency), which is proportional to the photon flux of the wave.

By making the substitutions *A _{j}*(

*z*) =

*B*(

_{j}*z*) exp[

*i*(

*k*+

_{p}*γ*|

_{K}*A*|

_{p}^{2})

*z*], one finds that the transformed pump amplitude

*B*is constant, and the transformed signal and idler (sideband) amplitudes satisfy the coupled-mode equations

_{p}*δ*=

_{j}*k*−

_{j}*k*+ |

_{p}*γ*| is a relative wavenumber and $\gamma ={\gamma}_{K}{B}_{p}^{2}$ is the coupling coefficient. By combining the first of Eqs. (4) with the conjugate of the second, one finds that the characteristic wavenumber

*B*=

_{s}*C*exp[

_{s}*i*(

*δ*−

_{s}*δ*)

_{i}*z*/2] and ${B}_{i}^{*}={C}_{i}^{*}\text{exp}\left[i\left({\delta}_{s}-{\delta}_{i}\right)z/2\right]$, where

*δ*−

_{s}*δ*=

_{i}*k*−

_{s}*k*. By making these substitutions in Eqs. (4), one obtains the symmetrized equations

_{i}*δ*= (

*δ*+

_{s}*δ*)/2 = (

_{i}*k*+

_{s}*k*)/2 −

_{i}*k*+ |

_{p}*γ*| is the wavenumber mismatch. The solutions of Eqs. (6) can be written in the input–output (IO) forms

*k*= (

*δ*

^{2}− |

*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

*e*(

*z*) = exp[

*i*(

*δ*−

_{s}*δ*)

_{i}*z*/2]. Notice that

*e*

^{*}(

*z*) = exp[

*i*(

*δ*−

_{i}*δ*)

_{s}*z*/2]. By combining Eqs. (9) with their conjugates and the auxiliary equation, one finds that

*B*|

_{s}^{2}− |

*B*|

_{i}^{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 (*β*_{2}), the mismatch can be written as*β*_{2}*ω*^{2}/2+|*γ*|, where *ω* is the (moderate) difference between the signal and pump frequencies. Degenerate FWM is unstable (the signal is amplified) when *β*_{2} < 0 (anomalous dispersion) and 0 < *ω* < |4*γ*/*β*_{2}|^{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 |2*γ*/*β*_{2}|^{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.

## 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.

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

where $A={\left[{A}_{s},\hspace{0.17em}{A}_{i}^{*}\right]}^{t}$ is the input amplitude vector, $B={\left[{B}_{s},\hspace{0.17em}{B}_{i}^{*}\right]}^{t}$ is the output amplitude vector and the transfer matrix*A*=

_{j}*B*(0) and

_{j}*B*=

_{j}*B*(

_{j}*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 where

*e*= exp(

_{s}*iϕ*) and

_{s}*e*= exp(

_{i}*iϕ*) 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.

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

*e*

_{1}and

*e*

_{2}do not affect the composite gain. Henceforth, these factors will be omitted. Second, the transfer coefficient

*μ*(which controls the output signal power) is maximal when

_{ss}*ν*characterizes the output idler produced by an input signal (and |

_{is}*μ*|

_{ss}^{2}− |

*ν*|

_{is}^{2}= 1), whereas the coefficients

*ν*and

_{si}*μ*characterize the output signal and idler produced by an input idler (and |

_{ii}*μ*|

_{ii}^{2}− |

*ν*|

_{si}^{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

*ϕ*

_{μ1}and

*ϕ*

_{μ2}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*and

_{j}*ν*are abbreviations for |

_{j}*μ*| and |

_{j}*ν*|, respectively, and ${\mu}_{\pm}^{2}-{\nu}_{\pm}^{2}=1$. The design challenge is the determination of values of

_{j}*ν*

_{1}and

*ν*

_{2}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

*ν*

_{1}and

*ν*

_{2}. 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

*ν*

_{1}and

*ν*

_{2}is more complicated. The parameter

*μ*

_{−}attains its minimal value of 1 when

*ν*

_{−}= 0 (

*ν*

_{1}=

*ν*

_{2}), whereas the parameter

*ν*

_{−}can be positive, zero or negative, depending on whether

*ν*

_{1}>

*ν*

_{2}(too little deamplification),

*ν*

_{1}=

*ν*

_{2}or

*ν*

_{1}<

*ν*

_{2}(too much deamplification), respectively [Fig. 3(

*a*)]. The extinction ratio

*x*and

*y*, which are themselves increasing functions of

*ν*

_{1}and

*ν*

_{2}[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 where

*μ*

^{2}=

*G*is the (common) PI power gain of each amplifier and

*ν*

^{2}=

*G*− 1. In the high-gain regime (

*μ*≈

*ν*≫ 1), the extinction ratio is about 2

*G*. The preceding discussion pertains to the signal extinction ratio. The idler extinction ratio

*ν*

_{+}/

*ν*

_{−}can be infinite, because

*ν*

_{−}can be zero.

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 *ϕ _{s}* +

*ϕ*is independent of frequency. Thus, chirped pulses can be amplified and attenuated in the same way as continuous waves.

_{i}## 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* = [*A _{s}*,

*A*]

_{k}*, the output vector*

^{t}*B*= [

*B*,

_{s}*B*]

_{k}*and the transfer matrix*

^{t}*k*denotes the loss mode, and the transfer coefficients

*τ*and

_{s}*ρ*satisfy the auxiliary equation |

_{s}*τ*|

_{s}^{2}+ |

*ρ*|

_{s}^{2}= 1. Notice that the input and output vectors involve only amplitudes, not conjugates. The loss matrix (19) is unitary (

*T*

^{†}

*T*=

*I*=

*TT*

^{†}). For the loss process the MRW variable |

*A*|

_{s}^{2}+ |

*A*|

_{k}^{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

*ϕ*, one obtains a shift-and-loss process that is characterized by the single parameter

_{s}*τ*. Now suppose that loss is applied before the phase shift [Fig. 4(

_{s}e_{s}*b*)]. Then the composite IO equation is

*ϕ*, one obtains a loss-and-shift process that is characterized by the single parameter

_{s}*e*(which equals

_{s}τ_{s}*τ*). 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

_{s}e_{s}*l*.

The three concatenated processes (PI amplification, loss and PS amplification) are also governed by Eq. (11), in which the input vector $A={\left[{A}_{s},\hspace{0.17em}{A}_{k},{A}_{i}^{*},{A}_{l}^{*}\right]}^{t}$, the output vector $B={\left[{B}_{s},\hspace{0.17em}{B}_{k},{B}_{i}^{*},{B}_{l}^{*}\right]}^{t}$ and the composite transfer matrix

*μ*- and

*ν*-coefficients equals 1. Of particular interest are the coefficients

*μ*and

_{ss}*ν*, 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

_{is}#### 4.1. Equal losses

First, consider the case in which the signal-processor losses are equal (|*τ _{s}*| = |

*τ*| =

_{i}*τ*). Then the transfer coefficients

*μ*and

_{ss}*ν*, which are specified in Eq. (24), are maximal when

_{is}*π*. When these extremum conditions are satisfied, the composite gain parameters

*μ*and

_{j}*ν*are abbreviations for |

_{j}*μ*| and |

_{j}*ν*|, respectively, and ${\mu}_{\pm}^{2}-{\nu}_{\pm}^{2}={\tau}^{2}$. Parameters (28) differ from parameters (16) only by (common) factors of

_{j}*τ*. Hence, the analysis of amplification and deamplification is similar to that of Sec. 3. In particular, the minimal value of ${\mu}_{-}^{2}$ is

*τ*

^{2}, so the miminal output-signal power equals the input-signal power divided by the loss factor (1/

*τ*

^{2}). 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 (|*μ*_{1}|^{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 (|*μ*_{2}|^{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].

#### 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 (|*τ _{s}μ*

_{1}| = |

*τ*

_{i}ν_{1}| =

*λ*). When the extremum conditions are satisfied, the composite gain parameters

*μ*

_{2}and

*ν*

_{2}are abbreviations for |

*μ*

_{2}| and |

*ν*

_{2}|, respectively and ${\mu}_{\pm}^{2}-{\nu}_{\pm}^{2}=0$: Not only are the intermediate signal and idler powers equal, so also are the output powers. The (common) value of ${\mu}_{-}^{2}$ and ${\nu}_{-}^{2}$ is

*τ*

_{s}τ_{i}μ_{1}

*ν*

_{1}/(

*μ*

_{2}+

*ν*

_{2})

^{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 In the high-gain regime (

*μ*

_{2}≈

*ν*

_{2}≫1), the extinction ratio is about 4

*G*

_{2}.

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
${\tau}_{s}={\nu}_{1}^{1/2}/{\mu}_{1}^{3/2}$ and *τ _{i}* = 1/(

*μ*

_{1}

*ν*

_{1})

^{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].

#### 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 (|*μ*_{2}*τ _{s}μ*

_{1}| = |

*ν*

_{2}

*τ*

_{i}ν_{1}|). When the extremum conditions are satisfied, the composite gain parameters

*μ*,

_{j}*ν*,

_{j}*τ*and

_{s}*τ*are abbreviations for |

_{i}*μ*|, |

_{j}*ν*|, |

_{j}*τ*| and |

_{s}*τ*| respectively, and ${\mu}_{\pm}^{2}-{\nu}_{\pm}^{2}=-{\left({\tau}_{s}{\mu}_{1}\right)}^{2}/{\nu}_{2}^{2}=-{\left({\tau}_{i}{\nu}_{1}\right)}^{2}/{\mu}_{2}^{2}$. Notice that the output idler is always more powerful than the output signal. The value of ${\nu}_{-}^{2}$ is

_{i}*τ*

_{s}τ_{i}μ_{1}

*ν*

_{1}/(

*μ*

_{2}

*ν*

_{2}), 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 In the high-gain regime, this extinction ratio is about 2

*G*

_{2}. 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
${\tau}_{s}={\nu}_{1}/{\mu}_{1}^{2}$ and *τ _{i}* = 1/

*ν*

_{1}. 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.

## 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 *A _{j}* as

*α*+

_{j}*δα*, where

_{j}*α*is an amplitude mean and

_{j}*δα*is a random variable [16,17]. The random variables have zero means and Gaussian statistics, and are uncorrelated, so the moments 〈

_{j}*δα*〉 = 0, 〈

_{j}*δα*〉 = 0 and $\u3008\delta {\alpha}_{j}\delta {\alpha}_{k}^{*}\u3009={\delta}_{jk}/2$, where 〈 〉 denotes an ensemble average and

_{j}δα_{k}*δ*is the Kronecker delta. It follows from this ansatz that 〈|

_{jk}*A*|

_{j}^{2}〉 = |

*α*|

_{j}^{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
${P}_{s}\left({\varphi}_{1}\right)=\left({A}_{s}^{*}{e}^{i{\varphi}_{l}}+{A}_{s}{e}^{-i{\varphi}_{l}}\right)/{2}^{1/2}$, where *ϕ _{l}* is the LO phase. For example,

*ϕ*= 0 corresponds to the real signal quadrature, whereas

_{l}*ϕ*=

_{l}*π*/2 corresponds to the imaginary quadrature. If the LO phase is matched to the signal phase [

*ϕ*= arg(

_{l}*α*)], the quadrature mean 〈

_{s}*P*〉 = 2

_{s}^{1/2}|

*α*|. The quadrature variance $\u3008\delta {P}_{s}^{2}\u3009=1/2$, independent of the LO phase. For the input idler, 〈

_{s}*P*〉 = 0 and $\u3008\delta {P}_{i}^{2}\u3009=1/2$. These semi-classical results are identical to the quantum-mechanical results for a coherent state and a vacuum state, respectively [15].

_{i}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|*α _{s}*|

^{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

*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 (

*μ*→

_{sj}*ν*and

_{ij}*ν*→

_{sj}*μ*), one obtains the noise figures

_{ij}*μ*and

_{ss}*ν*involve the products

_{is}*τ*

_{s}μ_{1}and

*τ*

_{i}ν_{1}, about which assumptions were made in Sec. 4, whereas the coefficients

*ν*and

_{si}*μ*do not. However, by using the auxiliary equations (App. A)

_{ii}Two limits of Eqs. (36) deserve brief discussions. For the case in which *τ _{s}* =

*τ*= 1 and

_{i}*μ*

_{2}= 1 (two-mode amplification), ${F}_{s}=2-1/{\mu}_{1}^{2}$ and ${F}_{i}=2+1/{\nu}_{1}^{2}$. As the gain parameter ${\mu}_{1}^{2}$ 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

*μ*

_{1}=

*μ*

_{2}= 1 (attenuation), ${F}_{s}=1/{\tau}_{s}^{2}$ and

*F*= ∞. 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 $1/{\tau}_{i}^{2}$.)

_{i}#### 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 *τ _{s}* =

*τ*=

_{i}*τ*. Then the quadrature mean and variance

*μ*

_{±}and

*ν*

_{±}were defined in Eqs. (28). Suppose that

*μ*≈

_{j}*ν*≫ 1 and

_{j}*τμ*

_{1}= 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*

_{+}≈ 2

*μ*

_{2},

*ρ*≈ 1 and ${F}_{s}\approx 10{\mu}_{2}^{2}/4{\mu}_{2}^{2}=2.5\left(4\hspace{0.17em}\text{dB}\right)$: 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

*ν*

_{2}=

*ν*

_{1}, so that

*ν*

_{−}= 0 and

*μ*

_{−}=

*τ*. Then the composite device functions as an attenuator and ${F}_{s}\approx 2{\mu}_{2}^{2}/{\tau}^{2}=2{\mu}_{1}^{4}\gg 1$: 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.

#### 5.2. Compensating losses

Second, consider the case in which *τ _{s}μ*

_{1}=

*τ*

_{i}ν_{1}=

*λ*(balanced copying). Then

*μ*=

_{ss}*λ*(

*μ*

_{2}±

*ν*

_{2}), but there is no simple expression for

*ν*. This problem can be avoided by using the first of Eqs. (35). The quadrature mean and variance

_{si}*μ*,

_{j}*ν*≫ 1 and

_{j}*λ*≈ 1 (so

*τ*≪ 1), in which case the first two stages function approximately as a zero-net-gain copier. Then

_{j}*ρ*≈ 1 and ${F}_{s}\approx \left[2{\left({\mu}_{2}\pm {\nu}_{2}\right)}^{2}+2{\mu}_{2}^{2}-1\right]/{\left({\mu}_{2}\pm {\nu}_{2}\right)}^{2}$. For the + mode, in the high-gain regime ${F}_{s}\approx 10{\mu}_{2}^{2}/4{\mu}_{2}^{2}=2.5$ and the composite device is only slightly noisier than a PI amplifier, whereas for the − mode, ${F}_{s}\approx 2{\mu}_{2}^{2}/{\left({\mu}_{2}-{\nu}_{2}\right)}^{2}\approx 8{\mu}_{2}^{4}\gg 1$ 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.

_{s}#### 5.3. Overcompensating losses

Third, consider the case in which *τ _{i}ν*

_{1}=

*λ*and

*τ*

_{s}μ_{1}=

*λν*

_{1}/

*μ*

_{1}(so

*μ*

_{2}

*τ*

_{s}μ_{1}=

*ν*

_{2}

*τ*

_{i}ν_{1}when

*μ*

_{2}=

*μ*

_{1}). Then the quadrature mean and variance

*μ*≈

_{j}*ν*≫ 1 and

_{j}*λ*= 1 (so

*τ*≪ 1), in which case the first two stages function approximately as a zero-net-gain copier. Then

_{j}*λ*(

*μ*

_{2}+

*ν*

_{2}

*ν*

_{1}/

*μ*

_{1}) ≈ 2

*μ*

_{2},

*ρ*≈ 1 and ${F}_{i}\approx 10{\mu}_{2}^{2}/4{\mu}_{2}^{2}=2.5\hspace{0.17em}(4\hspace{0.17em}\text{dB})$: For the + mode, the composite device is only slightly noisier than a one-stage PI amplifier. Now suppose that

_{s}*ν*

_{2}=

*ν*

_{1}. Then

*λ*(

*μ*

_{2}−

*ν*

_{2}

*ν*

_{1}/

*μ*

_{1}) = 1/

*μ*

_{1}and ${F}_{i}\approx 2{\mu}_{2}^{2}{\mu}_{1}^{2}=2{\mu}_{1}^{4}\gg 1$: 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.

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 4*G*^{2}, 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 4*G*^{2}. 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
$16{G}_{2}^{2}$. 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 *A*_{1} and *A*_{2} be *n* × 1 mode-amplitude vectors and suppose that their evolution is governed by the quadratic Hamiltonian

*J*

_{1},

*J*

_{2}and

*K*are coefficient matrices. These matrices can be constants, or they can vary with distance.

*J*

_{1}and

*J*

_{2}must be Hermitian, because

*H*is real, but

*K*is arbitrary. By applying the Hamilton equations the Hamiltonian (40) and using the fact that ${\left({A}_{1}^{\u2020}K{A}_{2}^{*}\right)}^{t}={A}_{2}^{\u2020}{K}^{t}{A}_{1}^{*}$, one obtains the vector coupled-mode equations

*K*, whereas in the second equation it is

*K*. For the special case in which

^{t}*n*= 1 (two-mode parametric amplification),

*J*

_{1}=

*δ*,

_{s}*J*

_{2}=

*δ*and

_{i}*K*=

*γ*, as stated in Sec. 2.

It is instructive to consider the evolution of ${A}_{1}^{\u2020}{A}_{1}$ and ${A}_{2}^{\u2020}{A}_{2}$, 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

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

where the 2*n*× 1 amplitude vector and 2

*n*× 2

*n*coefficient matrix are

*A*

^{†}

*SA*, where

*S*= diag(

*I*, −

*I*) is a 2

*n*× 2

*n*diagonal matrix. It is easy to verify that

*S*

^{2}=

*I*and

*SL*=

*L*

^{†}

*S*. By using the latter identity, one finds that

*A*

^{†}

*SA*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, *B*^{†}*SB* = *A*^{†}*T*^{†}*STA* = *A*^{†}*SA*. Because this statement is true for an arbitrary input vector, it must also be true that

*T*

^{†}

*T*=

*I*=

*TT*

^{†}).

It follows from Eq. (48) that *ST*^{†}*S* = *T*^{−1}. (To be precise, the symplectic condition implies that *ST*^{†}*S* 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} = (*ST*^{†}*S*)^{†}*S*(*ST*^{†}*S*) = *STST*^{†}*S*. By multiplying both sides of this equation by *S* on the left and *S* on the right, one finds that

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

For the two-mode processes discussed in Secs. 2 and 3, *M*_{11} = *eμ*, *N*_{12} = *eν*, *N*_{21} = *e ^{*}ν* and

*M*

_{22}=

*e*. Hence,

^{*}μ*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

To make the transition from classical to quantal mechanics, one replaces the (scalar) mode amplitudes *A _{j}* by the mode operators

*â*, which satisfy the boson commutation relations [

_{j}*â*,

_{j}*â*] = 0 and $\left[{\widehat{a}}_{j},{\widehat{a}}_{k}^{\u2020}\right]={\delta}_{jk}$, where [

_{k}*x*,

*y*] =

*xy*−

*yx*is a commutator and

*δ*is the Kronecker delta. The MRW variable also becomes an operator. The modes from group 1 contribute terms of the form ${\widehat{a}}_{k}^{\u2020}{\widehat{a}}_{j}$, which are number operators, whereas the modes from group 2 contribute terms of the form $\left[{\widehat{a}}_{k},{\widehat{a}}_{k}^{\u2020}\right]={\widehat{a}}_{k}^{\u2020},{\widehat{a}}_{k}+1$, 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].

_{jk}## 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

*μ*and

_{j}*ν*are specified by Eqs. (8). The MRW equation (App. A) ensures that |

_{j}*μ*|

_{ss}^{2}= 1 + |

*ν*|

_{si}^{2}, so the moduli of the transfer coefficients are maximal (or minimal) simultaneously. The maximization condition is When this condition is satisfied, |

*ν*| attains its maximal value |

_{si}*μ*

_{2}||

*ν*

_{1}| + |

*ν*

_{2}||

*μ*

_{1}|.

For example, consider two identical fibers of lengths *l*_{1} and *l*_{2} (and the same pump), or one fiber of length *l*_{1} + *l*_{2} with a virtual cut. Then

*c*

_{1}= cos(

*kl*

_{1}),

*s*

_{1}= sin(

*kl*

_{1}),

*c*

_{21}= cos[

*k*(

*l*

_{1}+

*l*

_{2})] and

*s*

_{21}= sin[

*k*(

*l*

_{1}+

*l*

_{2})]. 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,

*ϕ*

_{ν2}=

*ϕ*

_{ν1}, but

*ϕ*

_{μ2}+

*ϕ*

_{μ1}≠ 0 unless

*δ*= 0, because

*ϕ*= tan

_{μ}^{−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

*e*and

_{s}*e*judiciously).

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