## Quantum noise properties of parametric processes

Optics Express, Vol. 13, Issue 13, pp. 4986-5012 (2005)

http://dx.doi.org/10.1364/OPEX.13.004986

Acrobat PDF (343 KB)

### Abstract

In this paper the quantum noise properties of phase-insensitive and phase-sensitive parametric processes are studied. Formulas for the field-quadrature and photon-number means and variances are derived, for processes that involve arbitrary numbers of modes. These quantities determine the signal-to-noise ratios associated with the direct and homodyne detection of optical signals. The consequences of the aforementioned formulas are described for frequency conversion, amplification, monitoring, and transmission through sequences of attenuators and amplifiers.

© 2005 Optical Society of America

## 1. Introduction

1. K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. **49**, 5098–5106 (1978). [CrossRef]

2. R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. **18**, 1062–1072 (1982). [CrossRef]

3. J. Hansryd and P. A. Andrekson, “Broad-band continuous-wave pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency,” IEEE Photon. Technol. Lett. **13**, 194–191 (2001). [CrossRef]

4. S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. **39**, 838–839 (2003). [CrossRef]

5. K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon. Technol. Lett. **6**, 1451–1453 (1994). [CrossRef]

6. K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. **8**560–568 (2002). [CrossRef]

7. T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Tech. Lett. **16**, 551–553 (2004). [CrossRef]

8. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. **8**, 506–520 (2002). [CrossRef]

9. S. Radic and C. J. McKinstrie, “Two-pump fiber parametric amplifiers,” Opt. Fiber Technol. **9**, 7–23 (2003). [CrossRef]

10. S. Radic and C. J. McKinstrie, “Optical parametric amplification and signal processing in highly-nonlinear fibers,” IEICE Trans. Electron. **E88C**, 859–869 (2005). [CrossRef]

11. C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D **26**, 1817–1839 (1982). [CrossRef]

## 2. Description and detection of light

*α*

_{j}and

*j*denotes a mode and † denotes a hermitian conjugate. These dimensionless mode operators obey the boson commutation relations [

*α*

_{j};

*a*

_{k}]=0 and [

*α*

_{j};

*δ*

_{jk}.

*n*

_{j}=

*α*

_{j}. In DD a photodiode is used to measure the photon-number mean 〈

*n*

_{j},〉 where 〈 〉 denotes an expectation value. The uncertainty in such a measurement is the photon-number variance 〈

*n*

_{j}〉

^{2}. For DD the SNR is 〈

*n*

_{j}〉

^{2}/〈

*q*

_{j}(

*ϕ*)=(

*α*

_{j}

*e*

^{-iϕ}+

*e*

^{iϕ})/2, where

*ϕ*is a phase angle, and the second quadrature operator

*p*

_{j}(

*ϕ*)=

*q*

_{j}(

*ϕ*+

*π*/2). If

*α*

_{j}were a complex number, rather than an operator,

*q*

_{j}and

*p*

_{j}would be the real and imaginary parts of that number, measured in a coordinate system rotated by

*ϕ*radians relative to the reference system. The quadrature operators obey the commutation relation [

*q*

_{j};

*p*

_{j}]=

*i*/2, which shows that they are conjugate operators (apart from factors of 2

^{1/2}). The mean of the first quadrature is denoted by

*q*

_{j} and the variance

*q*

_{j}/

^{2}. Similar definitions apply to the second quadrature. It follows from the Heisenberg uncertainty principle that

*a*

_{1};

*a*

_{2};

*µ̄*and

*ν̄*to satisfy the (same) auxiliary condition |

*µ̄*|

^{2}+|

*ν̄*|

^{2}=1. This condition ensures that the total photon-number is conserved. For reference, the functions |

*ν̄*|

^{2}=

*T*and |

*ν̄*|

^{2}=1-

*T*are called the transmittance and reflectance, respectively. By using DD at the output ports, one can measure the photon numbers

*n*

_{j}and

*n*

_{k}, and the photon-number difference

*d*

_{jk}=

*n*

_{j}-

*n*

_{k}. It follows from Eqs. (1) and (2) that

*j*) is combined with a local-oscillator (LO) mode (

*l*) of the same frequency, before detection. It is customary to assume that the LO is a coherent state, for which

*a*

_{l}|

*α*

_{l}〈=

*α*

_{l}/

*α*

_{l}〈, where the coherent-state parameter

*α*

_{l}is a complex number. Let

*ϕ*=

*ϕ*

_{l}-

*ϕ*

_{µ}

*̄*+

*ϕ*

_{ν}

*̄*. Because the LO phase is included explicitly in Eq. (4), the LO eigenvalue equation is

*a*

_{l}|

*α*

_{l}〈=|

*α*

_{l}‖

*α*

_{l}〈. It follows from Eq. (4) and the boson commutation relations that

*l*in the last term in Eq. (6) indicates that it originates from the commutation relation [

*a*

_{l};

*α*

_{l}|

^{2}≫〈

*n*

_{j}〉), the last term can be neglected. Thus, balanced HD measures the quadratures of mode

*j*, with uncertainties that are characteristics of mode

*j*, rather than the LO. For balanced HD the SNR is 〈

*d*

_{j}〉

^{2}/〈

*q*

_{j}〉

^{2}/〈δ

*q*

_{j}〉

^{2}. Henceforth, for simplicity,

*ϕ*will be called the LO phase. The measurement of optical fields is discussed in detail in [14].

## 3. Examples of parametric processes

*ω*

_{1+}=

*ω*

_{1}+

*ω*, where

*ω*is the modulation frequency, and let

*γ*denote a photon. Then the modulation interaction (MI) in which 2

*γ*

_{1}→

*γ*

_{1}_+

*γ*

_{1+}produces an idler with frequency

*ω*

_{1}=

*ω*

_{1}-

*ω*, the phase-conjugation (PC) process in which

*γ*

_{1}+

*γ*

_{2}!

*γ*

_{1}++

*γ*

_{2}produces an idler with frequency

*ω*

_{2}_=

*ω*

_{2}-

*ω*and the Bragg scattering (BS), or FC, process in which

*γ*

_{1+}+

*γ*

_{2}→

*γ*

_{1}+

*γ*

_{2+}produces an idler with frequency

*ω*

_{2+}=

*ω*

_{2}+

*ω*. It is customary to use a classical model for the strong (constant-amplitude) pumps and a quantal model for the weak (variable-amplitude) products (sidebands). In this approach, each of the preceding FWM processes involves two (interaction-picture) mode operators (one for each sideband). MI is characterized by the input-output relation

*a*

_{1}+;

*µ*and

*ν*to satisfy the auxiliary equation |

*µ*|

^{2}-|

*ν*|

^{2}=1. This condition ensures that transformation (7) is unitary.

*µ*|

^{2}=

*G*and |

*ν*|

^{2}=

*G*-1 are called the signal and idler gain, respectively. In quantum optics, transformation (7) is called a two-mode squeezing transformation [15

15. R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. **34**, 709–759 (1987). [CrossRef]

*µ*and

*ν*also satisfy the auxiliary equation |

*µ*|

^{2}-|

*ν*|

^{2}=1. A similar relation characterizes the generation of the 2- idler. Notice that Eq. (8) has the same form as Eq. (7).

*µ̄*and

*ν̄*satisfy the auxiliary condition |

*µ̄*|

^{2}+|

*ν̄*|

^{2}=1, which ensures that transformation (9) is unitary. A similar transformation characterizes the generation of the 2+ idler. For reference, the BS relations have the same form as the beam-splitter relations (1) and (2). Formulas for the MI, PC and BS transfer functions are stated in [16

16. C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express **12**, 5037–5066 (2004). [CrossRef] [PubMed]

*µ*

_{21}|

^{2}+|

*µ*

_{22}|

^{2}-|

*ν*

_{23}|

^{2}+|

*µ*

_{24}|

^{2}=1. Similar relations characterize the idler-generation processes. Formulas for the four-mode transfer functions are stated in [16

16. C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express **12**, 5037–5066 (2004). [CrossRef] [PubMed]

*p*pumps couple the evolution of (at least) 2

*p*sidebands. Although one can increase the number of coupled modes by increasing the number of pumps, the extent to which one does this is limited in practice (by fiber dispersion). One can couple many more modes by concatenation [17]. Consider a communication link that consists of

*s*stages, in each of which fiber attenuation (loss) is followed by PA (gain). The architecture of a typical stage (

*r*) is illustrated in Fig. 2.

*r*is characterized by the input-output relation

*r*+1 is the scattered mode associated with the loss mechanism,

*r*-1 (beginning of stage

*r*) and

*r*. The attenuator transfer functions

*µ̄*and

*ν̄*satisfy the auxiliary equation |

*µ̄*|

^{2}+|

*ν̄*|

^{2}=1. A similar relation characterizes the generation of the scattered mode, but is of lesser interest. For reference, the input-output relations for attenuators are identical to the relations for beam-splitters and frequency converters.

*r*is characterized by the input-output relation

*r*is the idler,

*r*and |

*µ*|

^{2}-|

*ν*|

^{2}=1. A similar relation characterizes the idler-generation process, but is of lesser interest. Equation (12) has the same form as Eqs. (7) and (8).

*µ*

_{11}=

*µµ̄*,

*ν*

_{12r}=

*ν*and

*µ*

_{12r+1}=

*µν̄*. By using the auxiliary equations for the loss and gain processes, one can show that |

*µ*

_{11}|

^{2}-|

*ν*

_{12r}|

^{2}+|

*µ*

_{12r+1}|

^{2}=1: The product of unitary transformations is also unitary. By iterating Eq. (13), one finds that signal transmission through the entire link is characterized by the many-mode input-output relation

*z*

_{s}and the input points of the idlers and scattered modes were denoted by 0. (One can formalize this relabeling of the input points by using step functions to extend the domains of the transfer functions.)

*G*

_{2}=1).

2. R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. **18**, 1062–1072 (1982). [CrossRef]

18. C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express **12**, 4973–4979 (2004). [CrossRef] [PubMed]

18. C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express **12**, 4973–4979 (2004). [CrossRef] [PubMed]

*γ*

_{1}+

*γ*

_{2}→

*γ*

_{1}+

*γ*

_{2-}involves waves whose frequencies satisfy the matching condition

*ω*

_{1}+

*ω*

_{2}=

*ω*

_{1}+

*ω*

_{2}. Equation (8) shows that the signal operator

*a*

_{1+}is coupled to the idler operator

*ω*

_{1}+

*ω*

_{2}=2

*ω*

_{1+}, which is illustrated in Fig. 3, the idler coincides with the signal and

*a*

_{1+}is coupled to

18. C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express **12**, 4973–4979 (2004). [CrossRef] [PubMed]

*µ*and

*ν*satisfy the auxiliary equation |

*µ*|

^{2}-|

*ν*|

^{2}=1. Formulas for these transfer functions are stated in [18

**12**, 4973–4979 (2004). [CrossRef] [PubMed]

15. R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. **34**, 709–759 (1987). [CrossRef]

*a*

_{1+}to

*γ*

_{1}+

*γ*

_{1+}→

*γ*

_{1}+

*γ*

_{2}also couples

*a*

_{1+}to

*a*

_{1-}. Likewise, the MI of pump 2, in which 2

*γ*

_{2}→

*γ*

_{1+}+

*γ*

_{2+}, couples

*a*

_{1+}to

*γ*

_{1+}+

*γ*

_{2}→

*γ*

_{1}+

*γ*

_{2+}couples

*a*

_{1+}to

*a*

_{2+}. If one relabels the 1-, 1+/2- and 2+ modes as 1, 2 and 3, respectively, one can write the three-mode input-output relation in the form

*ν*

_{21}|

^{2}+|

*µ*

_{21}|

^{2}-|

*ν*

^{22}|

^{2}+|

*µ*

_{22}|

^{2}-|

*ν*

_{23}|

^{2}+|

*µ*

_{23}|

^{2}=1. The relations that characterize the idler-generation processes are similar, but not identical (

*a*

_{1}is coupled to

*a*

_{3}, and

*a*

_{3}is coupled to

*a*

_{1}).

*s*stages, in each of which fiber loss is compensated by PS gain (produced by degenerate PC, for example). The architecture of a typical stage (

*r*) is illustrated in Fig. 4. (The branching and rejoining of the signal line in the amplifier symbolizes the interaction of the signal with itself, rather than an idler.) The loss process at the beginning of stage

*r*is characterized by the input-output relation

*r*is characterized by the input-output relation

*µ*

_{11}=

*µµ̄*,

*ν*

_{11}=

*νµ̄**,

*µ*

_{1r+1}=

*µν̄*and

*ν*

_{1r+1}=

*νν̄**. By using the auxiliary equations for the loss and gain processes, one can show that |

*µ*

_{11}|

^{2}-|

*ν*

_{11}|

^{2}+ |

*µ*

_{1r+1}|

^{2}-|

*ν*

_{1r+1}|

^{2}=1: Once again, the composite transformation is unitary. By iterating Eq. (19), one finds that signal transmission through the entire link is characterized by the many-mode input-output relation

*z*

_{s}and the input points of the scattered modes were denoted by 0.

## 4. Properties of parametric processes

11. C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D **26**, 1817–1839 (1982). [CrossRef]

*µ*

_{jk}and

*ν*

_{jk}couple the output annihilation operator of mode

*j*to the input annihilation and creation operators of mode

*k*, respectively. The boson commutation relations, which are valid for all distances, imply that

*α*

_{i}(0). Then the input state |

*α*

_{i}〈=

*D*(

*α*

_{i})|0〈, where

*D*(

*α*

_{i}) is the displacement operator [13] and |0‹ is the vacuum state. It is customary to use the vacuum state as a virtual input state and incorporate the displacement required to produce the actual input state in the input-output relations associated with the process. By using the identity

*D*

^{†}(

*α*

_{i})

*α*

_{j}

*D*(

*α*

_{i})=

*α*

_{j}+

*α*

_{i}

*δ*

_{ij}, one can rewrite the input-output relation (21) in the form

*j*and the operator

*j*. Because the

*v*

_{j}operators only differ from the

*α*

_{j}operators by complex numbers, the

*v*

_{j}operators also must satisfy the boson commutation relations. The quadrature and number means and variances of mode

*j*depend on the first-, second- and fourth-order moments of

*α*

_{j}. The first-order moment is Eq. (25), from which it follows that

*α*

_{j}, one must first calculate the moments of

*v*

_{j}.

_{k}〈 represents a state with 1 photon in mode

*k*and no photons in the other modes. Because different number states are orthogonal, Eqs. (32) and (33) imply that the first-order moments 〈

*v*

_{j}〉 and 〈

_{k}〉 represents a state with 2 photons in mode

*k*and no photons in the other modes, and |1

_{k}1

_{l}〉 represents a state with 1 photon in mode

*k*, one photon in mode

*l*and no photons in the other modes. The second-order moments are all nonzero. Notice that 〈

*v*

_{j}

*v*

_{j}〉=1, as it must. Because each state in Eqs. (34)–(37) differs from the vacuum state by zero or two raising operations, each state in the third-order moment equations must differ from the vacuum state by one or three raising operations. Consequently, the third-order moments must all be zero. Because the operator

*v*

_{j}is hermitian, one can deduce the fourth-order moment 〈(

*v*

_{j})

^{2}〉 from Eq. (36) and the identity ∑

_{k}∑

_{l≠k}

*µ**

_{jl}

*ν*

_{jk}|1

_{k}1

_{l}〉=∑

_{k}∑

_{l>k}(

*µ**

_{jk}

*ν*

_{jl}+

*µ**

_{jl}

*ν*

_{jk})|1

_{k}1

_{l}〉.

*λ*

_{jk}=

*µ*

_{jk}

*e*

^{-iϕ}+

*ν**

_{jk}

*e*

^{iϕ}. Then Eq. (39) can be rewritten in the compact form

_{k}|

*ν*

_{jk}|

^{2}=∑

_{k}|

*µ*

_{jk}|

^{2}-1 depends only on the magnitudes of the transfer coefficients. In contrast, the number variance depends on the phases of the output amplitude and the transfer coefficients. Let

*λ*

_{jk}.) Then Eq. (42) can be rewritten in the compact form In the context of our model, which was described at the beginning of Section 3, Eqs. (38)–(43) are exact.

*α*

_{i}}〉=∏

_{i}

*D*(

*α*

_{i})|0〉. The output amplitude is defined by the equation

## 5. Selected applications

*q*

_{j}〉

^{2}/〈

*n*

_{j}〉

^{2}/〈

*q*

_{i}〉=|

*α*

_{i}|cos(

*ϕ*

_{i}-

*ϕ*), where

*ϕ*

_{i}and

*ϕ*are the input-signal and LO phases, respectively. The quadrature mean attains its maximal value |

*a*

_{i}| when

*ϕ*=

*ϕ*

_{i}, whereas the quadrature variance 〈

*ϕ*. The photon-number mean 〈

*n*

_{i}〉=|

*α*

_{i}|

^{2}and variance 〈

*α*

_{i}|

^{2}. Hence, for HD the maximal input-signal SNR

*j*and

*λ*

_{jk}was defined before Eq. (40). As the following sections demonstrate, the SNR is often maximal when

*ϕ*=

*ϕ*

_{j}. For these cases

*α*

_{i}|

^{2}≫1), the stimulated terms in Eq. (48), which depend on the output strength |

*α*

_{j}|

^{2}, are usually much larger that the spontaneous terms, which do not. For the usual cases

## 5.1 Frequency conversion

16. C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express **12**, 5037–5066 (2004). [CrossRef] [PubMed]

_{1}(0). Then it follows from the results of Section 4 that the output quantities

*α*

_{j}(

*z*)=

*µ*

_{j}

_{1}

*α*

_{1}(0), 〈

*q*

_{j}(

*z*)〉=|

*α*

_{j}(

*z*)|, 〈

*z*)〉=1/4, 〈

*n*

_{j}(

*z*)〉=|

*α*

_{j}(

*z*)|

^{2}and 〈

*z*)〉=|

*α*

_{j}(

*z*)|

^{2}, where

*µ*

_{11}=

*µ̄*,

*µ*

_{21}=-

*ν̄** and the LO phase

*ϕ*=arg[

*α*

_{j}(

*z*)] is optimal. This standard FC process is PI, because the output photon-numbers (powers) of the signal and idler do not depend on the phase of the input signal.

*α*

_{i}replaced by

*α*

_{j}(

*z*): HD is still more sensitive than DD by a factor of 4. The SNRs of the transmitted signal are reduced by the common factor 1=|

*µ̄*|

^{2}, whereas the SNRs of the generated idler are lower than those of the input signal by the common factor 1=|

*ν̄*|

^{2}. Because the SNRs associated with HD and DD are reduced by the same factors, the NFs associated with HD and DD are equal. The common NFs

*T*=|

*µ̄*|

^{2}is a periodic function of distance [16

**12**, 5037–5066 (2004). [CrossRef] [PubMed]

*T*=1,

*F*

_{1}=1 (0 dB): The signal is transmitted perfectly and no idler is generated. Conversely, if

*T*=0,

*F*

_{2}=1 (0 dB): No signal is transmitted and the generated idler is a perfect copy of the input signal.

*α*

_{j}(

*z*)=

*µ*

_{j1}

*α*

_{1}(0)+

*µ*

_{j2}

*α*

_{2}(0), where

*µ*

_{12}=

*ν̄*and

*µ*

_{22}=

*µ̄**. Let

*ξ*=

*ϕ*

_{ν}

*̄*-

*ϕ*

_{µ}

*̄*+

*ϕ*

_{2}(0)-

*ϕ*

_{1}(0). On the right sides of Eqs. (53) and (54),

*α*

_{j}is an abbreviation for

*α*

_{j}(0). This alternative FC process is PS, because the output numbers of both modes depend on the phases of the input modes and transfer functions. However, Eqs. (45) and (46) still apply to HD and DD, respectively, with

*α*

_{i}replaced by the PS

*α*

_{j}(

*z*), and the NFs associated with HD and DD are still equal. In Fig. 6 the signal and idler NFs are plotted as functions of the relative phase for the case in which |

*α*

_{1}|=|

*α*

_{2}| and

*T*=0:5. Because of constructive interference, which allows one of the output amplitudes to be larger that the corresponding input amplitude, the NFs of the FC process can be less than 1 (0 dB). One could also define NFs based on the total input number, which would be greater than (or equal to) 1.

## 5.2 Phase-insensitive parametric amplification

*α*

_{1}(0). Then it follows from the results of Section 4 that the output quantities

*α*

_{1}(

*z*)=

*µ*

_{11}

*α*

_{1}(0),

*µ*

_{2}(

*z*)=

*ν*

_{21}

*a*

_{1}* (0), 〈

*q*

_{j}(

*z*)〉=|

*α*

_{j}(

*z*)|, 〈

*z*)〉=(|

*µ*

_{jj}|

^{2}+|

*ν*

_{jk}|

^{2})/4,〈

*n*

_{j}(

*z*)〉=|

*α*

_{j}(

*z*)|

^{2}+|

*ν*

_{jk}|

^{2}and 〈

*z*)=|

*α*

_{j}(

*z*)|

^{2}(|

*µ*

_{jj}|

^{2}+|

*ν*

_{jk}|

^{2})+|

*µ*

_{jj}

*ν*

_{jk}|

^{2}, where

*µ*

_{11}=

*µ*=

*µ*

_{22},

*ν*

_{12}=

*ν*=

*ν*

_{21},

*k*≠

*j*and the LO phase is optimal. This standard PA process is PI, because the output photon-numbers of the signal and idler do not depend on the phase of the input signal.

*G*=|

*µ*|

^{2}is a monotonically-increasing function of distance [16

**12**, 5037–5066 (2004). [CrossRef] [PubMed]

*n*

_{1}〉=|

*α*

_{1}(0)|

^{2}. By combining Eqs. (45), (55) and (56), one finds that the NFs

*n*

_{1}〉≫1. In this limit, the SNR formulas simplify and the NFs

*G*≫1) the signal and idler NFs are both about 2 (3 dB). The degradation in signal quality is caused by the coupling of the signal to the (amplified) vacuum fluctuations associated with the idler.

*a*

_{i}(0). Then it follows from the results of Section 4 that the output quantities

*α*

_{j}(

*z*)=

*k*

_{ji}[(1-σ

_{ji})

*α*

_{i}(0)+

*σ*

_{ji}

*α*

_{i}* (0)], 〈

*q*

_{j}(

*z*)〉=|

*α*

_{j}(

*z*)|, 〈

*z*)i=∑

_{k}|

*k*

_{jk}|

^{2}/4, 〈

*n*

_{j}(

*z*)〉=|

*α*

_{j}(

*z*)|

^{2}+∑

_{k}|

*k*

_{jk}|

^{2}σ

_{jk}and 〈

*z*)〉=|

*α*

_{j}(

*z*)|

^{2}∑

_{k}|

*k*

_{jk}|

^{2}+∑

_{k}∑

_{l>k}|

*k*

_{jk}

*k*

_{jl}|

^{2}σ

_{kl}, where

*k*

_{jk}=

*µ*

_{jk}if

*j*and

*k*are both odd, or even,

*k*

_{jk}=

*ν*

_{jk}if one of

*j*and

*k*is odd and the other is even,

*σ*

_{jk}=0 if

*j*and

*k*are both odd, or even,

*σ*

_{jk}=1 if one of

*j*and

*k*is odd and the other is even, and the LO phase is optimal. The formulas for the photon-number mean and variance are consistent with Eqs. (122) and (123) of [16

**12**, 5037–5066 (2004). [CrossRef] [PubMed]

*n*

_{1}〉=|

*α*

_{i}(0)|

^{2}. For DD the SNRs

*n*

_{i}〉≫1), the SNR formulas simplify and the NFs

**12**, 5037–5066 (2004). [CrossRef] [PubMed]

*k*

_{jk}|

^{2}) depend on the physical parameters associated with the pumps and signal, and the fiber. In some applications, such as optical switching [10

10. S. Radic and C. J. McKinstrie, “Optical parametric amplification and signal processing in highly-nonlinear fibers,” IEICE Trans. Electron. **E88C**, 859–869 (2005). [CrossRef]

*k*

_{jk}|

^{2}are also comparable , and (in fibers with random birefringence) the signal and idler NFs are closer to 6 dB than 0 dB (two-mode FC) or 3 dB (two-mode PA): Extra frequency diversity comes at the price of extra noise. However, if the pump frequencies are tuned in ways such that the PA or FC bandwidths are maximized, the signal is coupled strongly to the primary idler (which is generated by PC or BS, respectively), and is coupled weakly to the (other) secondary idlers. The signal and primary-idler NFs are only slightly higher than the NFs associated with the limiting two-mode processes: PA with signal and primary-idler NFs of about 3 dB, and FC with a primary-idler NF of about 0 dB, are possible [16

**12**, 5037–5066 (2004). [CrossRef] [PubMed]

## 5.3 Transmission through a phase-insensitive link

*α*

_{1}(

*z*)=

*µ*

_{11}

*α*

_{1}(0), 〈

*q*

_{1}(

*z*)〉=|

*α*

_{1}(

*z*)|, 〈

*z*)〉=(|

*µ*

_{11}|

^{2}+|

*ν*

_{12}|

^{2}+|

*µ*

_{13}|

^{2})/4, 〈

*n*

_{1}(

*z*)〉=|

*α*

_{1}(

*z*)|

^{2}+|

*ν*

_{12}|

^{2}and 〈

*z*)〉=|

*α*

_{1}(

*z*)|

^{2}(|

*µ*

_{11}|

^{2}+|

*ν*

_{12}|

^{2}+|

*µ*

_{13}|

^{2}) +|

*µ*

_{11}

*ν*

_{12}|

^{2}+|

*ν*

_{12}

*µ*

_{13}|

^{2}, where

*µ*

_{11}=

*µ*

*µ̄*,

*ν*

_{12}=

*ν*,

*µ*

_{13}=

*µν̄*and the LO phase is optimal.

*G*=|

*µ*|

^{2}, the attenuation (transmittance)

*T*=|

*µ̄*|

^{2}and the input photon-number 〈

*n*

_{1}〉=|

*a*

_{1}(0)|

^{2}. Formula (67) has a simple physical interpretation: Attenuation transforms a coherent state with number 〈

*n*

_{1}〉 into a coherent state with number

*T*〈

*n*

_{1}〉, which is amplified subsequently. Hence, Eq. (67) is like Eq. (55), with a modified input number. For DD the SNR

*n*

_{1}〉 replaced by Th

*n*

_{1}i. For many-photon signals (

*T*〈

*n*

_{1}〉≫1), the NF

*GT*=1),

*F*

_{1}=2

*G*-1. In the high-gain limit (

*G*≫1) the NF is about 2

*G*.

*α*

_{1}(0). Then it follows from the results of Section 4 that the output quantities

*α*

_{1}(

*z*)=

*k*

_{1}

*α*

_{1}(0), 〈

*q*

_{1}(

*z*)〉=|

*α*

_{1}(z)|, 〈

*z*)〉=∑

_{k}|

*k*

_{k}|

^{2}/4, 〈

*n*

_{1}(

*z*)〉=|

*α*

_{1}(

*z*)|

^{2}+∑

_{k}|

*k*

_{k}|

^{2}σ

_{1k}and 〈

*z*)〉=|

*α*

_{1}(

*z*)|

^{2}+∑

_{k}|

*k*

_{k}|

^{2}+∑

_{k}∑

_{l>k}|

*k*

_{k}

*k*

_{l}|

^{2}σ

_{kl}, where

*k*

_{k}=

*µ*

_{1k}if

*k*is odd,

*k*

_{k}=

*ν*

_{1k}if

*k*is even, σ

_{jk}=0 if

*j*and

*k*are both odd, or even, σ

_{jk}=1 if one of

*j*and

*k*is odd and the other is even, and the LO phase is optimal. The formulas for the photon-number mean and variance are consistent with Eqs. (122) and (123) of [16

**12**, 5037–5066 (2004). [CrossRef] [PubMed]

*µ*

_{11}=(

*µµ̄*)

^{s},

*ν*

_{12}

_{r}=(

*µµ̄*)

^{s-r}

*ν*and

*µ*

_{12r+1}=(

*µµ̄*)

^{s-r}

*µν̄*(as shown in Appendix B). For a balanced link |

*µ*

_{11}|

^{2}=1, |

*ν*

_{12r}|

^{2}=

*G*-1 and |

*µ*

_{12r+1}|

^{2}=

*G*-1.

20. R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” IEEE J. Quantum Electron. **21**, 766–773 (1985). [CrossRef]

*s*(

*G*-1) comes from the |

*ν*

_{12r}|

^{2}terms, whereas the other comes from the |

*α*

_{12r+1}|

^{2}terms. Because the latter terms are only nonzero if

*ν̄*is nonzero (|

*µ̄*|<1), one can conclude that attenuation and amplification both degrade the signal quality. For DD the SNR

20. R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” IEEE J. Quantum Electron. **21**, 766–773 (1985). [CrossRef]

*n*

_{1}〉≫

*s*(

*G*-1)], the NF

*sG*.

## 5.4 Phase-insensitive parametric monitoring.

*α*

_{1}(

*z*)=

*k*

_{1}

*α*

_{1}(0), 〈

*q*

_{1}(

*z*)〉=|

*α*

_{1}(

*z*)|, 〈

*z*)〉=(|

*k*

_{1}|

^{2}+|

*k*

_{2}|

^{2}+|

*k*

_{3}|

^{2}+|

*k*

_{5}|

^{2})=4, 〈

*n*

_{1}(

*z*)〉=|

*α*

_{1}(

*z*)|

^{2}+|

*k*

_{2}|

^{2}and 〈

*z*)〉=|

*α*

_{1}(

*z*)|

^{2}(|

*k*

_{1}|

^{2}+|

*k*

_{2}|

^{2}+|

*k*

_{3}|

^{2}+|

*k*

_{5}|

^{2})+|

*k*

_{1}

*k*

_{2}|

^{2}+|

*k*

_{2}

*k*

_{3}|

^{2}+|

*k*

_{2}

*k*

_{5}|

^{2}, where

*k*

_{1}=

*µ̄*

_{f}

*µµ̄*〉

_{i},

*k*

_{2}=

*µ̄*

_{f}

*ν*,

*k*

_{3}=

*µ̄*

_{f}

*µ*

*ν̄*

_{f}〉,

*k*

_{5}=

*ν̄*

_{f}and the LO phase is optimal. The subscripts

*i*and

*f*denote the initial loss (in stage 1) and the final loss (in stage 2), respectively.

*G*=|

*µ*|

^{2}, the transmittances

*T*

_{i}=|

*µ̄*

_{i}|

^{2}and

*T*

_{f}=|

*µ̄*

_{f}|

^{2}, and the input photon-number 〈

*n*

_{1}〉=|

*α*

_{1}(0)|

^{2}. For DD the SNR

*T*

_{i}〈

*n*

_{1}〉≫1), the SNR formula simplifies and the NF

*T*

_{f}

*GT*

_{i}=1),

*F*

_{1}=1+2(1=

*T*

_{i}-

*T*

_{f})>1. In contrast, coupling losses have little effect on the performance of distant monitors, because the SNR degradations associated with them are insignificant compared to the SNR degradations associated with the fiber losses in typical links [17].

## 5.5 Phase-sensitive parametric amplification

15. R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. **34**, 709–759 (1987). [CrossRef]

**12**, 4973–4979 (2004). [CrossRef] [PubMed]

*α*

_{1}(

*z*)=

*µα*

_{1}(0)+

*να**

_{1}(0), 〉

*q*

_{1}(

*z*)〉=|

*α*

_{1}(

*z*)|cos[

*ϕ*

_{1}(

*z*)-

*ϕ*], 〈

*λ*|

^{2}/4, 〈

*ν*

_{1}(

*z*)〉=|

*α*

_{1}(

*z*)|

^{2}+|

*ν*|

^{2}and 〈δ

*z*)〉=|

*α*

_{1}(

*z*)|

^{2}|

*λ*

^{′}|

^{2}+2|

*µν*|

^{2}, where

*ϕ*

_{1}(

*z*)=arg[

*α*

_{1}(

*z*)] is the output phase,

*ϕ*is the LO phase,

*λ*=

*µe*

^{-iϕ}+

*ν**

*e*

^{iϕ}and

*G*=|

*µ*|

^{2}is a monotonically-increasing function of distance [18

**12**, 4973–4979 (2004). [CrossRef] [PubMed]

*ϕ*

_{ν}-

*ϕ*

_{µ}-2

*ϕ*

_{1}(0). It is convenient to define the PS gain-function

*H*(

*ξ*)=2

*G*-1+2[

*G*(

*G*-1)]

^{1/2}cosξ. The output photon-number

*n*

_{1}〈=|

*α*

_{1}(0)|

^{2}, and the output quadrature- and number-variances

*ϕ*

_{v}+

*ϕ*

_{µ}-2

*ϕ*and ζ=

*ϕ*

_{v}+

*ϕ*

_{µ}-2

*ϕ*

_{1}(

*z*). Notice that ξ-

*η*=2[

*ϕ*-

*ϕ*

_{1}(

*z*)]. It follows from Eq. (80) and the definition of ζ that

*ϕ*

_{ν}-

*ϕ*

_{µ}-2

*ϕ*

_{1}(0), which depends on the coupling phases

*ϕ*

_{µ}and

*ν*

_{v}, and the input phase

*ϕ*

_{1}(0).

*ϕ*

_{ν}+

*ϕ*

_{µ}-2

*ϕ*, which depends on the coupling phases and the LO phase

*ϕ*, but not the input phase. These phase dependences were illustrated in [19].

*n*

_{1}≫1), the SNR formula simplifies and the NF

*η*by replacing the LO phase with the output phase, in which case formula (86) reduces to formula (88): One can consider DD as self-homodyning detection.

*η*, for the case in which

*G*=10. The region of Fig. 8

*a*in which the contours merge (

*η*≈

*π*) is magnified in Fig. 8

*b*.

*π*). For the same reason, the in-phase axis of the probability cloud that characterizes the (initially-coherent) signal fluctuations is stretched, whereas the out-of-phase axis is squeezed. As one varies the LO phase (quadrature-measurement axis), the measured signal-strength decreases at almost the same rate as the (squared) width of the probability cloud. Hence, the NF is almost independent of

*η*. For most values of

*η*, the NF is maximal when ξ≈

*π*, because the signal strength is minimal. In Fig. 9 the NF associated with HD is plotted as a function ξ, and as a function of

*η*, for cases in which

*G*=10.

*G*=10 and 〈

*n*

_{1}〉=100. The approximate formula (88) predicts the NF accurately, provided that the signal is amplified significantly (ξ≠

*π*).

*α*

_{j}(

*z*)=

*µ*

_{jj}

*α*

_{j}(0)+

*ν*

_{jk}

*α**

_{k}(0), where

*µ*

_{jj}=

*µ*,

*ν*

_{jk}=

*ν*and

*k*≠

*j*. It follows from the results of Section 4 that 〈

*q*

_{j}(

*z*)〈=|

*α*

_{j}(

*z*)|cos[

*ϕ*

_{j}(

*z*)-

*ϕ*], 〈

*z*)〉=(2

*G*-1)/4, 〈

*n*

_{j}(

*z*)〉=|

*α*

_{j}(

*z*)|

^{2}+

*G*-1 and 〈

*z*)〉=|

*α*

_{j}(

*z*)

^{2}(2

*G*-1)+

*G*(

*G*-1), where the PI gain

*G*=|

*µ*|

^{2}. This alternative PA process is PS [2

2. R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. **18**, 1062–1072 (1982). [CrossRef]

*α*

_{j}(

*z*)| when

*ϕ*=

*ϕ*

_{j}(

*z*), whereas the quadrature variance does not depend on

*ϕ*. The output strengths

*ϕ*

_{ν}-

*ϕ*

_{µ}-

*ϕ*

_{2}(0)-

*ϕ*

_{1}(0). On the right sides of Eqs. (89) and (90),

*α*

_{j}is an abbreviation for

*α*

_{j}(0).

*α*

_{j}(

*z*)|

^{2}≫

*G*-1), the SNR formulas simplify and the NFs

*G*=10 and |

*α*

_{1}(0)|

^{2}=100=|

*α*

_{2}(0)|

^{2}(and the signal and idler NFs are equal). Because of constructive interference, the common NF of this alternative PA process can be less than 1 (0 dB). The approximate formula (94) predicts the NF accurately, provided that the input modes are amplified significantly (ξ≠

*π*). The dashed curve in Fig. 11 also represents the common NF associated with HD. The noise properties of four-mode PA with two nonzero input amplitudes are similar.

## 5.6 Transmission through a phase-sensitive link

*α*

_{1}(0). Then it follows from the results of Section 4 that the output quantities

*α*

_{1}(

*z*)=

*µ*

_{11}

*α*

_{1}(0) +

*ν*

_{11}

*α**

_{1}(0), 〈

*q*

_{1}(

*z*)〉=|

*α*

_{1}(

*z*)|cos[

*ϕ*

_{1}(

*z*)-

*ϕ*], 〈

*z*)〉=(|

*λ*

_{11}|

^{2}+|

*λ*

_{12}|

^{2})/4, 〈

*ν*

_{1}(

*z*)〉=|

*α*

_{1}(

*z*)|

^{2}+|

*ν*

_{11}|

^{2}+|

*ν*

_{12}|

^{2}, 〈

*z*)〉=|

*α*

_{1}(

*z*)|

^{2}(|

^{2}+|

^{2})+

^{2}(|

*µ*

_{11}

*ν*

_{11}|

^{2}+|

*µ*

_{12}

*ν*

_{12}|

^{2})+|

*µ**

_{11}

*ν*

_{12}+

*µ**

_{12}

*ν*

_{11}|

^{2}, where

*µ*

_{11}=

*µµ̄*,

*ν*

_{11}=

*νµ̄**,

*µ*

_{12}=

*µν̄*and

*ν*

_{12}=

*νν̄**,

*ϕ*

_{1}(

*z*)=arg[

*α*

_{1}(

*z*)] is the output phase,

*ϕ*is the LO phase,

*λ*

_{1j}=

*µ*

_{1}

*j*

^{e}

^{-iϕ}+

*ν**

_{1j}

*e*

^{iϕ}and

*G*=|

*µ*|

^{2}, the attenuation (transmittance)

*T*=|

*µ̄*|

^{2}and the relative phase ξ=

*ϕ*

_{ν}-

*ϕ*

_{µ}-2[

*ϕ*

_{1}(0)+

*ϕ*

_{µ}

*̄*]. It is convenient to define the PS gain-function

*H*(

*ξ*)=2

*G*-1+2[

*G*(

*G*-1)]

^{1/2}cosξ. The output photon-number

*n*

_{1}〉=|

*α*

_{1}(0)|

^{2}, and the output quadrature- and number-variances

*n*

_{1}≫1), the SNR formula simplifies and the NF

*α*

_{1}(0). Then the output quantities are given by Eqs. (26), (38), (40), (41) and (43). Because these formulas are complicated, we illustrate their consequences for a simple case. The preceding analysis of a one-stage link showed that the quadrature and number variances do not depend on the phase shift

*ϕ*

_{µ}

*̄*imposed on the signal by the fiber, or the phase

*ϕ*

_{v}

*̄*. Furthermore, the optimal value of the input phase ξ=0, in which case the amplitude contributions

*µα*

_{1}(0) and

*νa**

_{1}(0) add constructively, the output phase ζ=0, and the optimal value of the LO phase

*η*=0. Consequently, in our discussion of a many-stage link, we assume, without significant loss of generality, that

*µ̄*,

*ν̄*,

*µ*and

*ν*are all real, and that ξ=0 and

*η*=0. Under these conditions, and the assumption that every stage in the link is identical,

*µ*

_{11}=

*µ̄*

^{s}

*ps*,

*ν*

_{11}=

*µ̄*

^{s}

*qs*,

*µ*

_{1r+1}=

*µ̄*

^{s-r}

*ν̄*

*p*

_{s+1-r}and

*ν*

_{1r+1}=

*µ̄*

^{s-r}

*ν̄*

*q*

_{s+1-r}, where

*p*

_{s}and

*q*

_{s}are polynomial functions of

*µ*and

*ν*with the property

*p*

_{s}+

*q*

_{s}=(

*µ*+

*ν*)

^{s}(as shown in Appendix C). For a balanced link [

*µ̄*(

*µ*+

*ν*)=1], |

*µ*

_{11}+

*ν*

_{11}|

^{2}=1 and |

*µ*

_{1r+1}+

*ν*

_{1r+1}|

^{2}=

*L*-1.

*µ*

_{1k}+

*ν*

_{1k}|

^{2}, one finds that for HD the SNR and NF

20. R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” IEEE J. Quantum Electron. **21**, 766–773 (1985). [CrossRef]

*µ*

_{1k}and

*ν*

_{1k}(as detailed in Appendix C), one finds that for DD the SNR

*sL*and (

*sL*)

^{2}, respectively. Equation (106) shows that, in a PS link, the spontaneous contributions grow in the same ways. However, in a PS link the coefficients are smaller by factors of 4 and 8, respectively. For many-photon signals [|

*α*

_{1}(0)|

^{2}≫

*s*(

*L*-1)], the SNR formula simplifies and the NF

*s*) because loss adds PI uncertainty at each stage.

## 6. Summary

**12**, 5037–5066 (2004). [CrossRef] [PubMed]

*L*. For a balanced PI link (Section 5.3), the signal NF is approximately equal to 2

*sL*. One factor of

*sL*comes from the amplifiers, whereas the other comes from the attenuators: Both types of component degrade the signal quality.

*L*

_{i}-1/

*L*

_{f})>1 (0 dB), where

*L*

_{i}and

*L*

_{f}are the initial and final losses, respectively. Coupling losses have little effect on the performance of distant monitors, because the SNR degradations associated with them are insignificant compared to those associated with the losses in typical links [17].

*sL*. Although the PS amplifiers do not degrade the signal quality, the PI attenuators do, so the NF of a PS link is lower than that of PI link by a factor of only 2 (3 dB).

## Appendix A: Arbitrary input states

*v*

_{j}were evaluated in Section 4. Because the (expectation values of the) first-and third-order moments of

*v*

_{j}are zero, the first-order moment of

*α*

_{j}is determined solely by

*u*

_{j}, the contributions of

*u*

_{j}and

*v*

_{j}to the second-order moments add independently, and the fourth, fifth, seventh and eighth terms on the right side of Eq. (114) are zero. It follows from Eq. (108), Eqs. (111)–(113) and these observations, that the quadrature mean and variance

*δq*

^{2}(

*u*

_{j})〉=〈

*q*

^{2}(

*u*

_{j})〉-〈

*q*(

*u*

_{j})〉

^{2}is the variance contribution from

*u*

_{j}and

*λ*

_{jk}=

*µ*

_{jk}

*e*

^{-iϕ}+

*ν**

_{jk}

*e*

^{iϕ}, is the other variance contribution. It follows from Eqs. (113) and (114), and the preceding observations, that the number mean and variance

*δn*

^{2}(

*u*

_{j})〉=〈

*n*

^{2}(

*u*

_{j})〉-〈

*n*(

*u*

_{j})〉

^{2}is the variance contribution from

*u*

_{j}, the second-order moments

*u*

_{j}, with

*k*=1. It follows from this observation that

*α*

_{j}=

*µ*

_{j}

_{1}

*α*

_{1}+

*ν*

_{j1}

*α**

_{1}and

*λ*

_{j1}=

*µ*

_{j1}

*e*

^{-iϕ}+

*ν**

_{j1}

*e*

^{iϕ}. Equation (125) is identical to Eq. (38). By combining Eqs. (116), (117) and (126), one obtains Eq. (40). It also follows from the preceding observation that

*ϕ*

_{j}=arg(

*α*

_{j}). By combining Eqs. (118), (122) and (127), one obtains Eq. (41). It follows from Eqs. (119)–(124) and Eq. (128) that

*i*. Then one can rewrite the input-output relation in the form of Eq. (108), in which the output operators

*v*

_{j}are given by truncated versions of Eqs. (120)–(124). Once again, further progress requires the specification of the input states.

## Appendix B: Phase-insensitive link

*s*times, one finds that

*µ*

_{11}=(

*µµ̄*)

^{s},

*ν*

_{12r}=(

*µµ̄*)

^{s-r}

*ν*and

*µ*

_{12r+1}=(

*µµ̄*)

^{s-r}

*µν̄*, as stated in Section 5.3.

_{k}|

*k*

_{k}|

^{2}and ∑

_{k}∑

_{l>k}|

*k*

_{k}

*k*

_{l}|

^{2}σ

_{kl}, where σ

_{kl}=0 if

*k*and

*l*are both odd, or even, and σ

_{kl}=1 if one of

*k*and

*l*is odd and the other is even. For a balanced link, |

*k*

_{1}|

^{2}=1 and |

*k*

_{k}|

^{2}=

*G*-1 if

*k*>1. It follows from these facts that the first sum ∑

_{k}|

*k*

_{k}|

^{2}=1+2

*s*(

*G*-1), as stated in Eqs. (71) and (73). The second sum ∑

_{k}∑

_{l>k}|

*k*

_{k}

*k*

_{l}|

^{2}σ

_{kl}is evaluated as follows: For stage 0 (

*k*=1) there are

*s*contributions of the form |

*k*

_{1}

*k*

_{l}|

^{2}=1(

*G*-1), which sum to

*s*(

*G*-1). For stage 1 (

*k*=2 and 3) there are

*s*contributions of the form |

*k*

_{2}

*k*

_{l}|

^{2}=(

*G*-1)

^{2}and

*s*-1 contributions of the form |

*k*

_{3}

*k*

_{l}|

^{2}=(

*G*-1)

^{2}, which sum to (2

*s*-1)(

*G*-1)

^{2}. For stage 2 (

*k*=4 and 5) there are

*s*-1 contributions of the form |

*k*

_{4}

*k*

_{l}|

^{2}=(

*G*-1)

^{2}and

*s*-2 contributions of the form |

*k*

_{5}

*k*

_{l}|

^{2}=(

*G*-1)

^{2}, which sum to (2

*s*-3)(

*G*-1)

^{2}. By continuing this counting process, one finds that the second sum is

*s*(

*G*-1)+

*s*-

*r*)+1](

*G*-1)

^{2}=

*s*(

*G*-1)[1+

*s*(

*G*-1)], as stated in Eq. (73).

## Appendix C: Phase-sensitive link

*µ*,

*ν*,

*µ̄*and

*ν̄*are real. It follows from Eqs. (19) and (137) that, for a two-stage link,

*s*times, one finds that

*p*

_{s}and

*q*

_{s}are defined recursively:

*p*

_{1}=

*µ*,

*q*

_{1}=

*ν*,

*p*

_{s+1}=

*µ p*

_{s}+

*νq*

_{s}and

*q*

_{s+1}=

*µq*

_{s}+

*νp*

_{s}. Thus,

*µ*

_{11}=

*µ̄*

^{s}

*p*

_{s},

*ν*

_{11}=

*µ̄*

^{s}

*q*

_{s},

*µ*

_{1r+1}=

*µ̄*

^{s-r}

*ν̄*

*p*

_{s+1-r}and

*ν*

_{1r+1}=

*µ̄*

^{s-r}

*ν̄*

*q*

_{s+1-r}, as stated in Section 5.6. It follows from the preceding definitions that

*p*

_{s}±

*q*

_{s}=(

*µ*±

*ν*)

^{s},

*µ*

_{11}±

*ν*

_{11}=[

*µ̄*(

*µ*±

*ν*)]

^{s}and

*µ*

_{1r+1}±

*ν*

_{1r+1}=[

*µ̄*(

*µ*±

*ν*)]

^{s-r}

*ν̄*(

*µ*±

*ν*), where 1·

*r*·

*s*. For a balanced link [

*µ̄*(

*µ*+

*ν*)=1],

*µ*=

*L*

^{1/2}(1+

*T*)/2 and

*ν*=

*L*

^{1/2}(1-

*T*)/2, where

*T*=|

*µ̄*|

^{2}and

*L*=1/

*T*. It follows from the preceding results that

*µ*

_{11}=(1+

*T*

^{s})/2,

*ν*

_{11}=(1-

*T*

^{s})/2,

*µ*

_{1r+1}=(

*L*-1)

^{1/2}(1+

*T*

^{s+1-r})/2 and

*ν*

_{1r+1}=(

*L*-1)

^{1/2}(1-

*T*

^{s+1-r})/2.

_{k}|

*λ*

_{1k}|

^{2}and ∑

_{k}|

^{2}, where

*λ*

_{1k}=

*µ*

_{1k}

*e*

^{-iϕ}+

*ν**

_{1k}

*e*

^{iϕ}and

*ϕ*=0. It follows from the results of the preceding paragraph that |

*λ*

_{11}|

^{2}=1 and |

*λ*

_{1r+1}|

^{2}=

*L*-1. Thus,

*λ*

_{1r+1}(0)|

^{2}=1+

*s*(

*L*-1), as stated in Eq. (104): The in-phase sum (quadrature variance) increases monotonically as the number of stages increases. For the case considered, in which

*ϕ*

_{1}(

*z*)=0,

*λ*

_{1r+1}, so

^{2}=1+

*s*(

*L*-1), as stated in Eq. (106). For the out-of-phase quadrature

*ϕ*=

*π*/2. It follows from the results of the preceding paragraph that |

*λ*

_{11}|

^{2}=

*T*

^{2s}and |

*λ*

_{1r+1}|

^{2}=

*T*(1-

*T*)

*T*

^{2s-2r}. Thus,

*λ*

_{1r+1}(

*π*/2)|

^{2}=(

*T*+

*T*

^{2s})/(1+

*T*): As the number of stages increases, the out-of-phase sum (quadrature variance) tends quickly to its limit

*T*/(1+

*T*). It follows from Eq. (40), the definition of

*λ*

_{1k}, and the assumption that

*µ*

_{1k}and

*ν*

_{1k}are real, that 〈

*ϕ*)〉=〈

^{2}

*ϕ*+〈

*π*/2)sin

^{2}

*ϕ*. Because the in-phase variance is much larger then the out-of-phase variance, 〈

*ϕ*)〉≈〈

^{2}

*ϕ*. Recall that 〈

*q*

_{1}(

*ϕ*)〉

^{2}=|

*α*

_{1}(0)|

^{2}cos

^{2}

*ϕ*. By combining these results, one finds that the SNR associated with HD depends weakly on

*ϕ*, and in-phase measurement (

*ϕ*=0) is optimal.

_{k}|

*ν*

_{1k}|

^{2}, ∑

_{k}|

*µ*

_{1k}

*ν*

_{1k}|

^{2}and ∑

_{k}∑

_{l>k}|

*µ*

_{1k}

*ν*

_{1l}+

*µ*

_{1l}

*ν*

_{1k}|

^{2}. The first sum

*L*≫1 and

*s*≫1), ∑

_{k}|

*ν*

_{1k}|

^{2}≈[

*s*(

*L*-1)-1]=4. The second sum

_{k}|

*µ*

_{1k}

*ν*

_{1k}|

^{2}≈[

*s*(

*L*-1)2-1]=16. It is convenient to split the third sum into two parts. The first part

_{l>1}|

*µ*

_{11}

*ν*

_{1l}+

*µ*

_{1l}

*ν*

_{11}|

^{2}≈

*s*(

*L*-1)/4. The second part involves contributions of the form

*q*≤

*s*-1. It is not difficult to sum these contributions. For a long link, ∑

_{k}>1∑

_{l>k}|

*µ*

_{1k}

*ν*

_{1l}+

*µ*

_{1l}

*ν*

_{1k}|

^{2}≈

*s*(

*s*-1)(

*L*-1)

^{2}=8. By combining the preceding results, one finds that

## Acknowledgment

## References and links

1. | K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. |

2. | R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. |

3. | J. Hansryd and P. A. Andrekson, “Broad-band continuous-wave pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency,” IEEE Photon. Technol. Lett. |

4. | S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. |

5. | K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon. Technol. Lett. |

6. | K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. |

7. | T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Tech. Lett. |

8. | J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. |

9. | S. Radic and C. J. McKinstrie, “Two-pump fiber parametric amplifiers,” Opt. Fiber Technol. |

10. | S. Radic and C. J. McKinstrie, “Optical parametric amplification and signal processing in highly-nonlinear fibers,” IEICE Trans. Electron. |

11. | C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D |

12. | W. H. Louisell, |

13. | R. Loudon, |

14. | M. G. Raymer and M. Beck, “Experimental quantum state tomography of optical fields and ultrafast statistical sampling,” in |

15. | R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. |

16. | C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express |

17. | C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” submitted to Opt. Commun. |

18. | C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express |

19. | C. J. McKinstrie, M. G. Raymer, S. Radic, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” submitted to Opt. Commun. |

20. | R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” IEEE J. Quantum Electron. |

**OCIS Codes**

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(270.2500) Quantum optics : Fluctuations, relaxations, and noise

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 16, 2005

Revised Manuscript: June 14, 2005

Published: June 27, 2005

**Citation**

C. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, "Quantum noise properties of parametric processes," Opt. Express **13**, 4986-5012 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-13-4986

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### References

- K. O. Hill, D. C. Johnson, B. S. Kawasaki and R. I. MacDonald, �??CW three-wave mixing in single-mode optical fibers,�?? J. Appl. Phys. 49, 5098�??5106 (1978). [CrossRef]
- R. H. Stolen and J. E. Bjorkholm, �??Parametric amplification and frequency conversion in optical fibers,�?? IEEE J. Quantum Electron. 18, 1062�??1072 (1982). [CrossRef]
- J. Hansryd and P. A. Andrekson, �??Broad-band continuous-wave pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency,�?? IEEE Photon. Technol. Lett. 13, 194�??191 (2001). [CrossRef]
- S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin and G. P. Agrawal, �??Record performance of a parametric amplifier constructed with highly-nonlinear fiber,�?? Electron. Lett. 39, 838�??839 (2003). [CrossRef]
- K. Inoue, �??Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,�?? IEEE Photon. Technol. Lett. 6, 1451�??1453 (1994). [CrossRef]
- K. Uesaka, K. K. Y. Wong, M. E. Marhic and L. G. Kazovsky, �??Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,�?? IEEE J. Sel. Top. Quantum Electron. 8 560�??568 (2002). [CrossRef]
- T. Tanemura, C. S. Goh, K. Kikuchi and S. Y. Set, �??Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,�?? IEEE Photon. Tech. Lett. 16, 551�??553 (2004). [CrossRef]
- J. Hansryd, P. A. Andrekson, M. Westlund, J. Li and P. O. Hedekvist, �??Fiber-based optical parametric amplifiers and their applications,�?? IEEE J. Sel. Top. Quantum Electron. 8, 506�??520 (2002). [CrossRef]
- S. Radic and C. J. McKinstrie, �??Two-pump fiber parametric amplifiers,�?? Opt. Fiber Technol. 9, 7�??23 (2003). [CrossRef]
- S. Radic and C. J. McKinstrie, �??Optical parametric amplification and signal processing in highly-nonlinear fibers,�?? IEICE Trans. Electron. E88C, 859�??869 (2005). [CrossRef]
- C. M. Caves, �??Quantum limits on noise in linear amplifiers,�?? Phys. Rev. D 26, 1817�??1839 (1982). [CrossRef]
- W. H. Louisell, Radiation and Noise in Quantum Electronics (McGraw-Hill, 1964).
- R. Loudon, The Quantum Theory of Light, 3rd Ed. (Oxford University Press, 2000).
- M. G. Raymer and M. Beck, �??Experimental quantum state tomography of optical fields and ultrafast statistical sampling,�?? in Lecture Notes in Physics, Vol. 649, edited by M. Paris and J. Rehacek (Springer-Verlag, 2004), pp. 235�??295.
- R. Loudon and P. L. Knight, �??Squeezed light,�?? J. Mod. Opt. 34, 709�??759 (1987). [CrossRef]
- C. J. McKinstrie, S. Radic and M. G. Raymer, �??Quantum noise properties of parametric amplifiers driven by two pump waves,�?? Opt. Express 12, 5037�??5066 (2004). [CrossRef] [PubMed]
- C. J. McKinstrie, S. Radic, R. M. Jopson and A. R. Chraplyvy, �??Quantum noise limits on optical monitoring with parametric devices,�?? submitted to Opt. Commun.
- C. J. McKinstrie and S. Radic, �??Phase-sensitive amplification in a fiber,�?? Opt. Express 12, 4973�??4979 (2004). [CrossRef] [PubMed]
- C. J. McKinstrie, M. G. Raymer, S. Radic and M. V. Vasilyev, �??Quantum mechanics of phase-sensitive amplification in a fiber,�?? submitted to Opt. Commun.
- R. Loudon, �??Theory of noise accumulation in linear optical-amplifier chains,�?? IEEE J. Quantum Electron. 21, 766�??773 (1985). [CrossRef]

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