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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 14 — Jul. 5, 2010
  • pp: 14820–14835
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Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier

Zhi Tong, Adonis Bogris, Carl Lundström, C. J. McKinstrie, Michael Vasilyev, Magnus Karlsson, and Peter A. Andrekson  »View Author Affiliations


Optics Express, Vol. 18, Issue 14, pp. 14820-14835 (2010)
http://dx.doi.org/10.1364/OE.18.014820


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Abstract

Semi-classical noise characteristics are derived for the cascade of a non-degenerate phase-insensitive (PI) and a phase-sensitive (PS) fiber optical parametric amplifier (FOPA). The analysis is proved to be consistent with the quantum theory under the large-photon number assumption. Based on this, we show that the noise figure (NF) of the PS-FOPA at the second stage can be obtained via relative-intensity-noise (RIN) subtraction method after averaging the signal and idler NFs. Negative signal and idler NFs are measured, and <2 dB NF at >16 dB PS gain is estimated when considering the combined signal and idler input, which is believed to be the lowest measured NF of a non-degenerate PS amplifier to this date. The limitation of the RIN subtraction method attributed to pump transferred noise and Raman phonon induced noise is also discussed.

© 2010 OSA

1. Introduction

One unique advantage of parametric amplifiers is that they can operate in the phase-sensitive (PS) mode, which will lead to the highly attractive noiseless amplification [1

1. C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields 26(8), 1817–1839 (1982). [CrossRef]

]. Two types of PS amplifiers (PSAs) have been investigated so far, which are frequency degenerate (signal and idler frequencies are identical, which can be realized in an interferometer-based [2

2. M. E. Marhic, C. H. Hsia, and J. M. Jeong, “Optical amplification in a nonlinear fibre interferometer,” Electron. Lett. 27(3), 210–211 (1991). [CrossRef]

] or a four-wave-mixing based [3

3. K. Croussore and G. Li, “Phase regeneration of NRZ-DPSK based on symmetric-pump phase-sensitive amplification,” IEEE Photon. Technol. Lett. 19(11), 864–866 (2007). [CrossRef]

] scheme) and non-degenerate (frequencies are different) cases. The first nearly noiseless fiber PSA was demonstrated by Levandovsky et. al [4

4. D. Levandovsky, M. Vasilyev, and P. Kumar, “Amplitude squeezing of light by means of a phase-sensitive fiber parametric amplifier,” Opt. Lett. 24(14), 984–986 (1999). [CrossRef]

, 5

5. D. Levandovsky, M. Vasilyev, and P. Kumar, “Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” PRAMANA–, J. Phys. 56, 281–285 (2001). [CrossRef]

]. Imajuku et. al have measured a 1.8 dB NF (however, at 16 GHz electrical frequency to avoid the guided acoustic-wave Brillouin scattering, or GAWBS) at 16 dB gain in a degenerate PSA [6

6. W. Imajuku, A. Takada, and Y. Yamabayashi, “Low-noise amplification under the 3dB noise figure in high-gain phase-sensitive fibre amplifier,” Electron. Lett. 35(22), 1954–1955 (1999). [CrossRef]

], and also demonstrated the error-free amplification [7

7. W. Imajuku and A. Takada, “Error-free operation of in-line phase-sensitive amplifier,” Electron. Lett. 34(17), 1673–1674 (1998). [CrossRef]

]. However, the inherent single-channel amplification provided by a degenerate PSA, limits dramatically its potential applications. In contrast, non-degenerate PSAs can realize exponential gain [8

8. M. Vasilyev, “Phase-sensitive amplification in optical fibers,” in Frontiers in Optics, Technical Digest (Optical Society of America, 2005), paper FThB1.

] and multi-channel amplification [9

9. R. Tang, P. S. Devgan, V. S. Grigoryan, P. Kumar, and M. Vasilyev, “In-line phase-sensitive amplification of multi-channel CW signals based on frequency nondegenerate four-wave-mixing in fiber,” Opt. Express 16(12), 9046–9053 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-12-9046. [CrossRef] [PubMed]

] without suffering from GAWBS, which makes them more promising in WDM systems. Unfortunately, rigorous phase- and frequency-locking among the pump, signal and idler are required for a non-degenerate PSA. To date, two different methods based on electrically modulated side-band generation [10

10. R. Tang, P. Devgan, V. S. Grigoryan, and P. Kumar, “Inline frequency-non-degenerate phase-sensitive fibre parametric amplifier for fibre-optic communication,” Electron. Lett. 41(19), 1072–1074 (2005). [CrossRef]

] and parametric phase-insensitive amplification [11

11. R. Tang, J. Lasri, P. S. Devgan, V. S. Grigoryan, P. Kumar, and M. Vasilyev, “Gain characteristics of a frequency nondegenerate phase-sensitive fiber-optic parametric amplifier with phase self-stabilized input,” Opt. Express 13(26), 10483–10493 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-26-10483. [CrossRef] [PubMed]

], respectively, have been proposed to provide the stably-locked input waves. Particularly the latter one, which uses a phase-insensitive amplifier (PIA) as the first stage and then forms a cascaded PIA + PSA scheme, is practically more attractive since the gain band of the former method is severely limited by the electrical modulation bandwidth, and the latter scheme is also compatible with the existing optical systems which use single-carrier signals (no need to use two separate input wavelengths carrying the same information [10

10. R. Tang, P. Devgan, V. S. Grigoryan, and P. Kumar, “Inline frequency-non-degenerate phase-sensitive fibre parametric amplifier for fibre-optic communication,” Electron. Lett. 41(19), 1072–1074 (2005). [CrossRef]

]).

In this paper, we derive the general output noise formula of a cascaded non-degenerate PS-FOPA in Section 2 based on the semi-classical theory, which is consistent with the exact quantum theory under the large-photon-number assumption. According to this theory, in Section 3 we evaluate the RIN subtraction method which is widely used to measure the PIA NF with excess input noise [21

21. G. Obarski, “Precise calibration for optical amplifier noise figure measurement using the RIN subtraction method,” in Optical Fiber Communications Conference, paper ThZ3 (2003).

], and the results show that, even though RIN subtraction cannot give an accurate NF of either the signal or idler wave alone in a cascaded PS-FOPA, a good estimate can be obtained by taking the average of the signal and idler NFs. Moreover, the limits of the RIN subtraction method are also discussed. In Section 4, negative signal and idler NFs at the PSA stage have been measured after subtracting the PIA excess noise, and a <2 dB ‘real’ NF (>1 dB lower than the PIA quantum limit) is estimated at >16 dB PSA gain, which is the lowest NF ever measured in a non-degenerate PSA.

2. Noise characteristics of a cascaded non-degenerate PS-FOPA

The general diagram of a cascaded PS-FOPA is shown in Fig. 1
Fig. 1 Schematic of a non-degenerate PIA+ATT+PSA cascade, and the principle diagram of the RIN subtraction measurement. Sin and Sout are the noise PSD measured at the PSA input and output.
. The first PIA stage will generate an almost equalized idler which has a self-stabilized phase relationship with signal and pump waves, and the following PSA stage will provide the PS amplification. Since the output noise level of the PIA stage is monitored, an inter-stage attenuation (ATT) is introduced to account for the passive component like a coupler (where T and T a represent the main path loss and the bypass loss of the coupler, respectively, as shown in Fig. 1). Moreover, the inter-stage ATT is important for measuring a low PSA NF, which will be discussed below.

For a conventional PIA+ATT+PIA cascade, the output noise formulas have been discussed in detail in Ref [22

22. E. Desuvire, Erbium-doped fiber amplifiers (John Wiley and Sons, 1994), Chap. 2.

]. However, things become more complicated when a cascaded PS-FOPA with mid-stage ATT is considered, since both correlated (from the parametric amplification) and uncorrelated (from the attenuation) amplitude fluctuations will be produced through the cascade. Quantum theory is usually adopted to obtain the accurate noise characteristics of a PS-FOPA [13

13. C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express 12(21), 5037–5066 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-21-5037. [CrossRef] [PubMed]

,14

14. C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13(13), 4986–5012 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-13-13-4986. [CrossRef] [PubMed]

,16

16. C. J. McKinstrie, M. G. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257(1), 146–163 (2006). [CrossRef]

]. Here we carry out a semi-classical analysis to derive the noise performance of a PIA+ATT+PSA cascade, which is consistent with the quantum theory under the high-photon-number assumption, but easier to understand and apply.

As is well known, the general input-output relation of a non-degenerate (two-mode) parametric amplifier in the undepleted pump approximation is given by [13

13. C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express 12(21), 5037–5066 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-21-5037. [CrossRef] [PubMed]

16

16. C. J. McKinstrie, M. G. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257(1), 146–163 (2006). [CrossRef]

]:
[AsAi*]=[μνν*μ*][As0Ai0*]=G[As0Ai0*],
(1)
where A represents the complex amplitude, subscripts s and i denote signal and idler waves, respectively, As 0 is the input signal amplitude, superscript * represents the conjugation operation, μ and ν are the complex transfer coefficients, which depend on both pump and phase-matching conditions, as defined in Ref [15

15. M. Vasilyev, “Distributed phase-sensitive amplification,” Opt. Express 13(19), 7563–7571 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-13-19-7563. [CrossRef] [PubMed]

], and also satisfy the auxiliary equation |μ|2|ν|2=1. Eq. (1) applies to both PI- and PS-FOPAs with single- or dual-pumping. In this paper we only consider the co-polarized pump-signal-idler case to achieve the maximum performance [16]. By including the vacuum fluctuations at the input, we can re-write the input-output relation for the PIA stage as
[As,PIAAi,PIA*]=[μ1ν1ν1*μ1*][As0+δAsδAi*]=G1[As0+δAsδAi*],
(2)
where δAs, δAi denote the uncorrelated vacuum noise fields at signal and idler frequencies. The statistics of the vacuum noise is assumed to be complex Gaussian, and we have <δAs,i>=0, <δAs,i2>=0 and <|δAs,i|2>=hvs,i/2 [23

23. P. Kylemark, P.-O. Hedekvist, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Noise characteristics of fiber optical parametric amplifiers,” J. Lightwave Technol. 22(2), 409–416 (2004). [CrossRef]

], where <> denotes the expectation operation, h is the Planck’s constant and v is the optical frequency. The noise field at vs,ican be classically expressed as |δAs,i|exp[j(2πvs,it+θ)] (within 1Hz bandwidth), where θ is the random phase. Subsequently, the attenuation process can be semi-classically modeled as [24

24. A. Mecozzi and P. Tombesi, “Parametric amplification and signal-to-noise ratio in optical transmission lines,” Opt. Commun. 75(3-4), 256–262 (1990). [CrossRef]

]
[As,PIA+ATTAi,PIA+ATT*]=[Ts00Ti][As,PIAAi,PIA*]+[1TsδAs'1TiδAi']=T^[As,PIAAi,PIA*],
(3)
where T^ represents the attenuation operator, which behaves like a 4-port beam-splitter coupling the vacuum fields to the signal and the idler, T is the transmittance, and δAs,i´ represents the newly-introduced uncorrelated vacuum fluctuations through the loss process. For simplicity, we assume that Ts = Ti = T. In addition, relative phase shift between the signal and idler will be induced by the dispersion of the attenuator unless perfect dispersion compensation is conducted, which means that the signal and idler fields should be multiplied by the additional phase shifts ejφ1 and ejφ2, respectively. Similarly from Eq. (1), the input-output relation of the PSA stage is
[As,PIA+ATT+PSAAi,PIA+ATT+PSA*]=[μ2ν2ν2*μ2*][ejφ100ejφ2][As,PIA+ATTAi,PIA+ATT*]=G2D[As,PIA+ATTAi,PIA+ATT*],
(4)
where D represents the additional phase shift induced by the mid-stage dispersion. By combining Eqs. (2)-(4), we have the output fields of the whole cascade as
[As,PIA+ATT+PSAAi,PIA+ATT+PSA*]=G2DT^G1[As0+δAsδAi*]=T[μs12νs12νi12*μi12*][As0+δAsδAi*]+1T[μ2ejφ1ν2ejφ2ν2*ejφ1μ2*ejφ2][δAs'δAi*'],=T[μs12νs12νi12*μi12*][As0+δAsδAi*]+1T[μ2ejφ1ν2ejφ2ν2*ejφ1μ2*ejφ2][δAs'δAi*'],
(5)
where
μs12=μ1μ2ejφ1+ν1*ν2ejφ2,νs12=ν1μ2ejφ1+μ1*ν2ejφ2,μi12=μ1μ2ejφ2+ν1*ν2ejφ1,νi12=ν1μ2ejφ2+μ1*ν2ejφ1.
(6)
The first term at the right side of Eq. (5) denotes the output signal/idler plus the PSA amplified correlated noise components generated from the PIA stage, while the second term represents the PSA amplified uncorrelated noise induced by the mid-stage attenuation. It can be deduced from Eq. (6) that
G12(ϕ)=|μs12|2=|μi12|2=|μ1μ2|2+|ν1ν2|2+2|μ1μ2ν1ν2|cos(ϕ)=G1G2+(G11)(G21)+2G1G2(G11)(G21)cos(ϕ),
(7)
G12(ϕ)1=|νs12|2=|νi12|2=|ν1μ2|2+|μ1ν2|2+2|μ1μ2ν1ν2|cos(ϕ)=(G11)G2+G1(G21)+2G1G2(G11)(G21)cos(ϕ),
(8)
where subscripts 1 and 2 denote the PIA and PSA, respectively, G12 represents the total signal gain of the PIA + PSA cascade (without ATT), G1=|μ1|2 is the signal parametric gain of the PIA stage and G11=|ν1|2 is the idler conversion efficiency, while G2=|μ2|2 denotes the PI gain of the PSA stage (when no idler is launched), and similarly we have G21=|ν2|2. Thus the PSA gain of the second stage can be expressed as
Gs,PSA(ϕ)=G12(ϕ)/G1,Gi,PSA(ϕ)=[G12(ϕ)1]/(G11).
(9)
From Eq. (9) we note that the PS gain depends on the relative phase ϕ, which satisfies the relation ϕ=ϕμ1+ϕμ2+ϕν1ϕν2+φ1+φ2 (ϕμ,ν represent the phase angles of μ and ν, respectively). By optimizing the relative phase among the pump, signal and idler waves (e.g. by choosing the proper inter-stage phase shifts [11

11. R. Tang, J. Lasri, P. S. Devgan, V. S. Grigoryan, P. Kumar, and M. Vasilyev, “Gain characteristics of a frequency nondegenerate phase-sensitive fiber-optic parametric amplifier with phase self-stabilized input,” Opt. Express 13(26), 10483–10493 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-26-10483. [CrossRef] [PubMed]

], i.e. φ 1 and φ 2, to ensure ϕ=2mπ,m=0,1,2,...), one can obtain the maximum PSA gains for signal and idler as
Gs,PSA(2mπ)=(|μ1μ2|+|ν1ν2|)2/|μ1|2=[G1G2+(G11)(G21)]2/G1,
(10)
Gi,PSA(2mπ)=(|ν1μ2|+|μ1ν2|)2/|ν1|2=[(G11)G2+G1(G21)]2/(G11).
(11)
One can easily find that when G1 and G2 are much larger than 1,
Gs,PSA(2mπ)Gi,PSA(2mπ)4G2,
(12)
which is consistent with the reported 6 dB gain advantage of a non-degenerate PS-FOPA [11

11. R. Tang, J. Lasri, P. S. Devgan, V. S. Grigoryan, P. Kumar, and M. Vasilyev, “Gain characteristics of a frequency nondegenerate phase-sensitive fiber-optic parametric amplifier with phase self-stabilized input,” Opt. Express 13(26), 10483–10493 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-26-10483. [CrossRef] [PubMed]

,17

17. C. Lundström, J. Kakande, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Experimental comparison of gain and saturation characteristics of a parametric amplifier in phase-sensitive and phase-insensitive mode,” in European Conference on Optical Communications, paper Mo. 1.1.1 (2009).

,18

18. J. Kakande, C. Lundström, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Detailed characterization of afiber-optic parametric amplifier in phase-sensitive and phase-insensitive operation,” Opt. Express 18(5), 4130–4137 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-4130. [CrossRef] [PubMed]

]. In fact, the idler has a slightly larger PS gain than the signal. On the other hand, when ϕ=(2m+1)π the maximum PS attenuation can also be derived as
Gs,PSA[(2m+1)π])(G1+G2)2/4G12G2,Gi,PSA[(2m+1)π](G1G2)2/4G12G2,
(13)
which means that the signal and idler have different maximum attenuations and the gain-attenuation product GPSA(2mπ)GPSA[(2m+1)π]1. This result which deviates from the conclusions in Refs [9

9. R. Tang, P. S. Devgan, V. S. Grigoryan, P. Kumar, and M. Vasilyev, “In-line phase-sensitive amplification of multi-channel CW signals based on frequency nondegenerate four-wave-mixing in fiber,” Opt. Express 16(12), 9046–9053 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-12-9046. [CrossRef] [PubMed]

, 14

14. C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13(13), 4986–5012 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-13-13-4986. [CrossRef] [PubMed]

, 15

15. M. Vasilyev, “Distributed phase-sensitive amplification,” Opt. Express 13(19), 7563–7571 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-13-19-7563. [CrossRef] [PubMed]

]. is due to the un-equalized signal and idler powers input to the PSA. Actually, the maximum PSA attenuation is sensitive to the input power difference.

To achieve the output intensity noise after the square-law detection, we need to obtain both the expectation and variance of the photocurrent Is,i, which satisfies Is,i=R|As,i|2Bo, where R = q / hv is the responsivity of an ideal detector (q is the electron charge, and hv is the photon energy) and B o is the optical bandwidth. According to the NF definition, an ideal photodetector should be assumed, and the imperfect responsivity can be compensated through a calibration process. Based on the semi-classical method used in Ref [25

25. N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7(7), 1071–1082 (1989). [CrossRef]

], the single-sided power spectral densities (PSD) of the output noise at the signal and idler frequencies are
Ss,out=4R2Ps0TG12(ϕ)hvs2{[2G12(ϕ)1]T+(2G21)(1T)},
(14)
Si,out=4R2Ps0T[G12(ϕ)1]hvs2{[2G12(ϕ)1]T+(2G21)(1T)},
(15)
where P s0 = | A s0 |2 B o is the input signal power at the PIA stage. In Eqs. (14)-(15) only the signal-noise beat term is considered, which is reasonable since the noise-noise beat term will become negligible provided that the signal (idler) power is much larger than the noise power along the FOPA (which is the case for most practical communication systems, as discussed in Ref [26

26. E. Desuvire, “Comments on ‘The noise figure of optical amplifiers’,” IEEE Photon. Technol. Lett. 11(5), 620–621 (1999). [CrossRef]

].). However, the above results are not valid if the PS gain is much lower than 1 (PS attenuation), where the noise-noise beat term will dominate. The first term in the parentheses of Eq. (14) [or Eq. (15)] originates from the correlated noise induced by the first PIA stage, while the second term comes from the uncorrelated noise introduced by the inter-stage attenuator. The physical meaning of Eqs. (14) and (15) is that the correlated noise and the uncorrelated noise will experience different gain [i.e. (2G121)/(2G11) for correlated noise vs. 2G21 for uncorrelated] in the following PSA. Equations (14)-(15) have been verified by using a full quantum theory in the Appendix, under the large-photon-number assumption. In this paper we will mainly focus on the NF of the PSA stage, while a detailed noise analysis of the whole cascade will be discussed in another paper [27

27. Z. Tong, C. J. McKinstrie, C. Lundström, M. Karlsson, and P. A. Andrekson, “Noise performance of optical fiber transmission links that use non-degenerate cascaded phase-sensitive amplifiers,” Opt. Express (submitted to). [PubMed]

].

3. Evaluation of the RIN subtraction method

Contrary to the conventional PIAs and degenerate PSAs, non-degenerate PS-FOPAs are two-mode devices, the ‘real’ NF of which can only be derived by measuring both the signal and idler fields simultaneously (e.g. through nonlinear mixing) according to Ref [15

15. M. Vasilyev, “Distributed phase-sensitive amplification,” Opt. Express 13(19), 7563–7571 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-13-19-7563. [CrossRef] [PubMed]

]. However, this type of measurement is difficult and impractical. The most straightforward way is to measure the signal and idler NFs separately. When assuming the two input waves are identical and shot-noise limited (with uncorrelated vacuum noise), one has [14

14. C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13(13), 4986–5012 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-13-13-4986. [CrossRef] [PubMed]

,15

15. M. Vasilyev, “Distributed phase-sensitive amplification,” Opt. Express 13(19), 7563–7571 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-13-19-7563. [CrossRef] [PubMed]

]
NFs,i=(2G1)/GPSA(ϕ),
(16)
where G is the phase-insensitive gain. The maximum PS gain is GPSA(2mπ)=(G+G1)2 under the in-phase condition, where input amplitudes add coherently. Thus, a −3 dB minimum NF can be obtained for either signal or idler wave at high gain regime, which corresponds to a 0 dB ‘real’ NF of a two-mode-squeezed PSA taking both signal and idler into account. However, if the input signal and idler have different signal-to-noise ratios (SNR), the NF of the better one will be degraded while that of the poorer one will be improved. Thus it is necessary to measure both signal and idler NFs to get the full picture.

To measure the PSA NF in a PIA + PSA cascade, one needs to subtract the excess noise generated from the PIA to meet the shot-noise-limited input criterion [22

22. E. Desuvire, Erbium-doped fiber amplifiers (John Wiley and Sons, 1994), Chap. 2.

]. The most widely used technique to measure the NF of a PIA with excess input noise is the RIN subtraction method [21

21. G. Obarski, “Precise calibration for optical amplifier noise figure measurement using the RIN subtraction method,” in Optical Fiber Communications Conference, paper ThZ3 (2003).

,28

28. M. Movassaghi, M. K. Jackson, V. M. Smith, and W. J. Hallam, “Noise figure of erbium-doped fiber amplifiers in saturated operation,” J. Lightwave Technol. 16(5), 1461–1465 (1998). [CrossRef]

], which assumes that the excess noise generated from the first stage will experience the same gain as signal in the following stage. However, it is still not clear whether this method applies to a non-degenerate cascaded PSA. In this section the validity of the RIN subtraction method will be evaluated based on the aforementioned semi-classical theory.

The principle diagram of the RIN subtraction method is shown in Fig. 1, where an inter-stage attenuator is inserted to both account for the coupler and to reduce the excess noise. The signal (or idler) NF of the PSA stage will be [28

28. M. Movassaghi, M. K. Jackson, V. M. Smith, and W. J. Hallam, “Noise figure of erbium-doped fiber amplifiers in saturated operation,” J. Lightwave Technol. 16(5), 1461–1465 (1998). [CrossRef]

]
NFs=1/Gs,PSA+Ps0TG1(Ss,outSs,in)/(2hvsIs,out2),NFi=1/Gi,PSA+Ps0T(G11)(Si,outSi,in)/(2hviIi,out2),
(17)
where subscript s, i denote the signal and the idler waves, respectively, P 0 is the input power to the PIA, while Ps 0 TG 1 and Ps 0 T(G 1 - 1) denote the input signal and idler powers to the PSA, respectively. Iout is the output DC current, Sout and Sin are the output and input noise PSD (noted in Fig. 1) of the PSA as defined in Eqs. (14)-(15), measured at the same detected power level (where ideal photodetectors are assumed). This implies that
Ta=Gk,PSATTb,
(18)
where subscript k = s, i denotes either the signal or the idler wave, Ta and Tb are the variable attenuators before the input and the output noise measurement to adjust the signal/idler power, as marked in Fig. 1. One may also show that

Is,out=RPs0G12TTb,Ii,out=RPs0(G121)TTb.
(19)

First let us consider a hypothetical ideal case in a cascaded PS-FOPA, i.e. G1=|μ1|2=|ν1|2 in Eq. (2), which means that the signal and idler will have the same gain and noise performance after the PIA stage, thus a perfectly equalized signal and idler pair can be prepared before the PSA, and that the same PSA gain can be obtained for both signal and idler at the PSA stage. In theory this condition might be reached at a specific wavelength with the assistance of Raman effect [29

29. P. L. Voss and P. Kumar, “Raman-effect induced noise limits on χ(3) parametric amplifiers and wavelength converters,” J. Opt. B Quantum Semiclassical Opt. 6(8), 762–770 (2004) (and references therein). [CrossRef]

]. Under this assumption, the input and output single-sided noise PSDs can be expressed based on Eqs. (2)-(3) and (14)-(15)
Sk,in=4R2G1TaPs0hvk2[2G1Ta+(1Ta)],
(20)
Sk,out=4R2Ps0TTbG12(ϕ)hvk2{[2G12(ϕ)T+(2G21)(1T)]Tb+(1Tb)},
(21)
where k = s, i denotes either the signal or idler wave. In this ideal condition, both signal and idler have the same output noise expression. By combining Eqs. (17)-(21) and doing some algebra, the PSA NF formula can be expressed as
NFideal(ϕ)=[GPSA(φ)T+(2G21)(1T)]/GPSA(ϕ),
(22)
where GPSA=Gs,PSA=Gi,PSA. From Eq. (22) we can find that the excess noise term generated from the PIA stage, i.e. the first term in the parentheses of Eq. (21), has been totally canceled through the RIN subtraction. At the maximum PS gain [GPSA(2mπ)4G2], Eq. (22) will be reduced to NFideal(2mπ)(T+1)/2 by assuming G2>>1, which means that the minimum PSA NF measured by RIN subtraction depends on the mid-stage loss. The physics behind this effect is that the input noise of the PSA is a mixture of well correlated (result of the first-stage PIA) and completely uncorrelated (result of mid-stage loss) signal-idler noises, as shown in Eqs. (2)-(3). This makes the results measured via RIN subtraction method to deviate from the pure shot-noise-limited-input condition. When T = 1 (no loss), the NF becomes 1 (0 dB) since the signal and idler noise fields are totally correlated, which will lead to an identical PS amplification for both the mean signal (idler) and the noise power at the maximum gain. Whereas when T = 0 (infinite loss), the PSA NF of either signal or idler will be 1/2 (−3 dB), which is exactly the same result from Eq. (16) since a noisy signal light will eventually converge to the coherent state after enough attenuation. Therefore the inter-stage loss has two different functions, which are 1) reducing the excess noise produced by the PIA stage, and 2) introducing uncorrelated vacuum noise resembling to the single-stage PSA case.

However, in practice |μ|2 and |ν|2 are not equal in a parametric amplifier, which makes the input signal and idler SNRs to the PSA not perfectly equalized. By using the equation |μ|2|ν|2=1, the input and output noise PSD at the signal frequency can be obtained as
Ss,in=4R2G1TaPs0hvs2[(2G11)Ta+(1Ta)],
(23)
Ss,out=4R2Ps0TTbG12(ϕ)hvs2{[(2G12(ϕ)T+2G2(1T)1]Tb+(1Tb)}.
(24)
Substituting Eq. (23)-(24) into (17) and combining (17)-(19), we have the signal PSA NF as

NFs(ϕ)=[2G22G2T+2Gs,PSA(ϕ)T1]/Gs,PSA(ϕ).
(25)

At the maximum PS gain Eq. (25) is reduced to NFs(2mπ)(3T+1)/2, when assuming G2>>1. In the same way, the idler PSA NF can also be deduced as
NFi(ϕ)=[2G22G2T+2T1]/Gi,PSA(ϕ),
(26)
where Gi,PSA(ϕ) has been defined in Eq. (9). At the maximum PS gain, the minimum idler NF can be expressed as NFi(2mπ)(1T)/2. Apparently in practical conditions, the NFs measured by using RIN subtraction at both signal and idler waves will diverge from the ideal case. The reason for this deviation is that in a real cascaded parametric amplifier, the mean signal (idler) and the excess noise from the first stage will experience different gain at the PSA due to the unbalanced PIA amplification. In fact, the signal will experience a lower gain (Gs,PSA) than the excess noise [(2G121)/(2G11), according to Eq. (14)], whereas the idler will have a larger gain [(G121)/(G11), according to Eq. (9)] than the excess noise, which means that the RIN subtraction tends to overestimate the signal NF (by subtracting less noise power) but underestimate the idler NF (by subtracting more noise power), according to the basic formula Eq. (17). However, by using Gs,PSAGi,PSA (when both G1 and the PS gain are much larger than 1), and combining Eq. (25)-(26), we have the equivalent NF of the PSA after taking average of both the signal and idler NFs as
NFeq=(NFs+NFi)/2NFideal,
(27)
which represents the actual PSA NF with an equalized signal and idler input. Equation (27) implies that the NF measurement error induced by RIN subtraction can be compensated by averaging the separately-measured signal and idler NFs. In Fig. 2
Fig. 2 Comparison of the minimum ideal and equivalent NF vs. PSA gain, according to Eq. (21) and (26), respectively. The PIA gain is fixed at 10 dB.
, we compare the minimum ideal NF and the equivalent NF measured by RIN subtraction at different T and PSA gain. According to our calculations, the relative error between the NFideal and NFeq will be negligible when G1 > 7 dB. This conclusion also applies to an un-optimized PSA (where cos(ϕ)1) provided that the signal/idler power is high enough to meet the large-photon-number requirement. As a result, though the RIN subtraction method does not apply to non-degenerate cascaded PSAs when considering only signal or idler channel, accurate NF estimation for the equalized-input case can still be obtained by taking the average of signal and idler NFs. The large mid-stage attenuation de-correlates the signal and idler waves and provides a decent approximation of the case where the two inputs are two independent shot-noise limited waves. Thus, a large mid-stage attenuator is required to measure a low PSA NF.

Until now we only considered the amplified quantum noise (AQN) [30

30. P. Kylemark, P. –O. Hedekvist, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Noise characteristics of fiber optical parametric amplifiers,” J. Lightwave Technol. 22, 409–416 (2004) and 23, 2192 (2005). [CrossRef]

] in the NF analysis, however, other noise contributions [31

31. Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18(3), 2884–2893 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-3-2884. [CrossRef] [PubMed]

] such as pump transferred noise (PTN) [32

32. P. Kylemark, M. Karlsson, and P. A. Andrekson, “Gain and wavelength dependence of the noise-figure in fiber optical parametric amplifiers,” IEEE Photon. Technol. Lett. 18(11), 1255–1257 (2006). [CrossRef]

] and Raman induced excess noise [12

12. P. L. Voss, K. G. Köprülü, and P. Kumar, “Raman-noise-induced quantum limits for χ(3) nondegenerate phase-sensitive amplification and quadrature squeezing,” J. Opt. Soc. Am. B 23(4), 598–610 (2006). [CrossRef]

, 29

29. P. L. Voss and P. Kumar, “Raman-effect induced noise limits on χ(3) parametric amplifiers and wavelength converters,” J. Opt. B Quantum Semiclassical Opt. 6(8), 762–770 (2004) (and references therein). [CrossRef]

] will degrade the PIA and PSA noise performance as well. First, we consider the PTN which originates from the ultrafast response of the parametric process. For simplicity we only consider the ideal case where the signal and the idler have identical gains at the PIA. According to Ref [30

30. P. Kylemark, P. –O. Hedekvist, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Noise characteristics of fiber optical parametric amplifiers,” J. Lightwave Technol. 22, 409–416 (2004) and 23, 2192 (2005). [CrossRef]

]. and Eqs. (20)-(21), the excess noise PSDs induced by the PTN at the input and output of the PSA stage are
ΔSk,inPTN=4[RPs0(G1PP)TaPP]2/(2OSNRΔv),
(28)
ΔSk,outPTN=4[RPs0(G12(ϕ)PP)TTbPP]2/(2OSNRΔv),
(29)
where PP is the launched pump power into the PIA, OSNR is the pump optical signal-noise-ratio measured in Δv bandwidth, and factor 2 implies the single-polarization nature of the beat noise, whereas the OSNR measurement collects noise from both polarizations. By substituting Eqs. (28)-(29) into (17) and combining (18)-(19), we have
ΔNFkPTN=Ps0PP2T[(GPSAPP)2G1+2(G1PP)(GPSAPP)GPSA]OSNRGPSA2hvkΔv,
(30)
where ΔNFkPTN is the PTN induced additional NF of the PSA stage after RIN subtraction. In Eq. (30), the first term in the numerator describes the exact PTN-induced additional NF of a single-stage PSA (without PIA stage), while the second term is the error term induced by the RIN subtraction, which is due to the nonlinear transfer function of the PTN. As a result, RIN subtraction tends to overestimate the PTN impact on the PSA NF, especially when G1/PP and GPSA/PP become large (e.g. close to the gain edges).

According to Eq. (30) and (33), the accuracy of the RIN subtraction is significantly limited by the PTN and Raman noise generated from the PIA stage. To reduce the measurement inaccuracy, both PTN and Raman noise from the first stage should be kept at a very low level. To achieve low PTN, a very high pump OSNR, a low input signal and a small signal-pump wavelength separation (within ±10 nm) are required [30

30. P. Kylemark, P. –O. Hedekvist, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Noise characteristics of fiber optical parametric amplifiers,” J. Lightwave Technol. 22, 409–416 (2004) and 23, 2192 (2005). [CrossRef]

]. To reduce the Raman induced excess noise, the best way is to move the signal close to the pump wavelength [12

12. P. L. Voss, K. G. Köprülü, and P. Kumar, “Raman-noise-induced quantum limits for χ(3) nondegenerate phase-sensitive amplification and quadrature squeezing,” J. Opt. Soc. Am. B 23(4), 598–610 (2006). [CrossRef]

, 29

29. P. L. Voss and P. Kumar, “Raman-effect induced noise limits on χ(3) parametric amplifiers and wavelength converters,” J. Opt. B Quantum Semiclassical Opt. 6(8), 762–770 (2004) (and references therein). [CrossRef]

]. By carefully designing the cascaded PS-FOPA, the measurement error of the RIN subtraction induced by the PTN and Raman noise will become less important, since the PTN and Raman induced impacts will be partly canceled by each other around the maximum gain. However, the measurement error will drastically increase with the PTN and Raman noise in the PIA. To accurately characterize the noise performance of a non-degenerate PS-FOPA, the best way is to directly launch the shot-noise-limited signal and idler into a phase-locked single-stage PSA, though up to this date it is still challenging to do.

4. Experimental results and discussions

Figure 3
Fig. 3 Measurement setup. NFA: Noise figure analyzer; ESA: Electrical spectrum analyzer; OSA: Optical spectrum analyzer; PC: Polarization controller; ATT: Variable attenuator. Inset (a) shows the theoretical and measured shot noise level after calibration vs. the detected photocurrent, and Inset (b) shows the electrical noise spectrum measured by the ESA, where the spurious tones are due to the pump phase- to intensity-modulation transfer.
shows the experimental setup. A 60 mW low-noise DFB laser (1554.4 nm) was used as the pump laser, which was phase-modulated by four tones to suppress the stimulated Brillouin scattering (SBS). After an 8.5 W EDFA booster followed by two cascaded 2 nm filters, the amplified pump was combined with the signal by a 10 dB coupler. We use 50 m and 500 m highly nonlinear fibres (HNLF, parameters are λ0=1552 nm, γ=11.8 W−1km−1 and S0=0.02 ps/nm2 ּkm) as the PIA and PSA, respectively. The launched pump power into the PIA and the PSA stages are about 5W and 0.4W, respectively. In between the two stages, another SMF-based 10 dB coupler (SMF length = 7 m) was inserted as the mid-stage attenuator as well as the signal/idler/noise monitor. A polarization controller was also connected with the coupler to align the PIA output waves with the principle axes of the PSA HNLF. We connected the 10% port to the PSA input. A 20 dB coupler was spliced after the PSA to monitor the output spectrum, and two cascaded filters were used to effectively filter out the residual pump and the amplified noise. Finally the amplified signal and idler were detected by the NF analyzer, as shown in Fig. 3.

In Figs. 6
Fig. 6 Measured PSA, PIA gain and PSA NF spectra of the signal wave at a) the anti-Stokes and b) Stokes bands, and the PSA gain and NF spectra of the idler wave at c) the anti-Stokes and d) Stokes bands.
, the measured PSA gain and NF spectra are shown for the signal and idler waves, respectively, with less than ±0.55 dB measurement error, and the PIA gain spectrum is also shown. The NF measurement error is determined through the differential form of the NF equation [(Eq. (17)]. We can find that the uncertainty of the NF will increase at low output power. In our setup, 10 dB average PIA gain was achieved, which leads to almost balanced (less than 0.5 dB power difference) signal and idler powers input to the PSA, and larger than 16 dB PSA gain can be obtained. To keep both the Raman noise and PTN at low levels, we only measured the NF spectra around the PS gain peaks within ±10 nm pump separation. The input signal and idler powers of the PSA are −19.9 dBm and −20.2 dBm, respectively. According to simulations based on the above parameters, the PTN and Raman induced additional noise increase from the first stage are less than 0.2 dB and 0.6 dB, respectively. The former value will cause a less than 0.5 dB PSA NF overestimation, while the latter one will approximately induce a 0.2-0.4 dB NF underestimation (by assuming that the uncorrelated-to-total Raman noise ratio ranges from 50% to 100%, as a safe estimation), thus the NF measurement error induced by RIN subtraction will be less than 0.3 dB (i.e. NF is slightly overestimated). It should also be mentioned that our measurements provide good estimate of the NF when PS gain is close to its maximum value according to Eq. (27).

From Figs. 6, it is easy to observe an almost symmetrical PSA NF spectrum for both signal and idler at the Stokes and anti-Stokes bands. The slight NF asymmetry observed in Ref [20

20. Z. Tong, C. Lundström, A. Bogris, M. Karlsson, P. Andrekson, and D. Syvridis, “Measurement of sub-1 dB noise figure in a non-degenerate cascaded phase-sensitive fiber parametric amplifier,” in European Conference on Optical Communications, paper Mo. 1.1.2 (2009).

]. is believed to be due to the measurement uncertainty. Both the signal and idler NFs will increase as the PS gains deviate from the maximum, which agrees well with the theory in [14

14. C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13(13), 4986–5012 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-13-13-4986. [CrossRef] [PubMed]

16

16. C. J. McKinstrie, M. G. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257(1), 146–163 (2006). [CrossRef]

]. Moreover the idler NF is more than 1dB lower than the signal, which is consistent with the results that the RIN subtraction tends to underestimate the idler NF but overestimate the signal NF. We also calculated the averaged NF in Fig. 7
Fig. 7 Averaged PSA gain and equivalent NF spectra by taking average of the signal and idler gains and NFs [Eq. (27)] at a) the anti-Stokes and b) the Stokes bands.
, which shows approximately a −1 dB equivalent NF can be estimated at 1547.2 nm and 1561.7 nm. The relatively higher NFs measured at the first two gain lobes are due to the large pump residual noise from the booster. To further look for the optimal PSA NF, in Fig. 8
Fig. 8 Measured signal and idler NF as a function of input power, and the estimated equivalent NF at averaged input power. Minimum −1dB equivalent NF can be observed.
, the signal, idler and equivalent NFs as functions of the input power are shown at 1561.7 nm signal wavelength. The lowest equivalent NF (signal NF at 1561.7 nm: −0.4 dB, idler NF at 1547.2 nm: −1.7 dB) is measured to be −1 dB in Fig. 8 at about −22 dBm average input. According to Eq. (22), the minimum ideal NF which can be measured by using RIN subtraction is −2.6 dB for T = 0.1 for both signal and idler, which means that the measured minimum equivalent NF is 1.6 dB larger than the ideal value. As a result, a 1.6 dB ‘real’ PSA NF can be deduced by considering both signal and idler inputs, which is approximately 1.4 dB below the PIA quantum limit. The 1.6 dB NF increase is mainly due to the PTN and Raman induced excess noise, and the former can be clearly seen from the NF increase with the input signal/idler power, as shown in Fig. 8. The impact of polarization mis-alignment on the PSA noise performance is negligible in our setup.

5. Conclusions

Appendix: Quantum mechanical derivation of the noise of a PIA+ATT+PSA cascade

Two-mode parametric amplification is governed by the input-output (IO) relations [14

14. C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13(13), 4986–5012 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-13-13-4986. [CrossRef] [PubMed]

]

bs=μas+νai,
(35)
bi=μai+νas,
(36)

where a and b are the operators at input and output, s and i represent the signal- and idler-mode, respectively, and † denotes a Hermitian conjugate. The transfer functions μ and ν satisfy the auxiliary equation |μ|2|ν|2=1, which ensures that the transformation is canonical. Attenuation is modeled as two-mode beam-splitting, which is governed by

bs=τas+ρal,
(37)
bl=ρ*as+τ*al,
(38)

where s and l represent the signal- and loss-mode, respectively. The transfer functions τ and ρ satisfy the auxiliary equation |τ|2+|ρ|2=1. For direct detection, the relevant quantities are the means (photocurrent) and variances (noise power) of the photon numbers. By defining the number and variance operators nj=ajaj and δnj2=nj2nj2, respectively, the noise characteristics can be determined through the IO relations.

First we consider a PIA followed by parallel ATTs (for both signal and idler modes). The IO relations can be obtained according to Eqs. (35)-(38) as

cs=(τsμ1)as+(τsν1)ai+ρsal,
(39)
ci=(τiμ1)ai+(τiν1)as+ρiam,
(40)

where c denotes the output mode, μ1 and ν1 are the transfer coefficients for the PIA stage, τ and ρ are the transfer coefficients for the following ATTs, and the subscripts s, i, l, m denote signal, idler and two loss modes, respectively. Here we assume that the attenuations for the signal and idler modes are identical, which means that τs=τi=τ and ρs=ρi=ρ. For PI operation, the input signal mode can be assumed to be a coherent state, for which δns2=ns, while the idler input is a vacuum state. Equations (39) and (40) have the same forms as the IO relations considered in [13

13. C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express 12(21), 5037–5066 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-21-5037. [CrossRef] [PubMed]

, 14

14. C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13(13), 4986–5012 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-13-13-4986. [CrossRef] [PubMed]

], so the general formulas derived therein apply to PIA followed by an ATT. The output variances of the signal and idler modes are

δns_PIA+ATT2=|τμ1|4ns+|τμ1|2(|τν1|2+|ρ|2)ns+|τν1|2(|τμ1|2+|ρ|2),
(41)
δni_PIA+ATT2=|τν1|4ns+|τν1|2(|τμ1|2+|ρ|2)ns+|τν1|2(|τμ1|2+|ρ|2),
(42)

where the first two terms on the right sides of Eq. (41) and (42) represent the signal-noise beat terms, whereas the third terms are the noise-noise beat terms. In practice, the noise-noise terms can be neglected when ns>>1 (large-photon-number assumption). Therefore by defining |τ|2=T, |ρ|2=1T, |μ1|2=G1, |ν1|2=G11, and Ps0=nshvBo, we find that Eq. (41) can be expressed in exactly the same form as Eq. (23) except for the factor 2R2(hv)2 B o. This factor comes from the photon detection and the vacuum fluctuations. Considering the well-known semi-classical shot-noise expression ΔIshot2=2qIsBe=4R2ns(hv)2BoBe/2 (single-sided PSD), where q is the electron charge and Is=RnshvBo is the detected photocurrent, we can see this factor is necessary to relate the semi-classical to the quantum results (δns2=ns). Thus Eq. (23) can be verified by Eq. (41) in a quantum way under the large-photon-number condition. Similarly, the output idler noise formula of the PIA + ATT cascade can also be confirmed by Eq. (42).

ds=μ11as+ν12ai+μ13al+ν14am,
(43)
di=ν21as+μ22ai+ν23al+μ24am.
(44)

The composite transfer coefficients are

μ11=μ2μ1τejφ1+ν2ν1*τ*ejφ2,μ22=μ2μ1τejφ2+ν2ν1*τ*ejφ1,ν12=μ2ν1τejφ1+ν2μ1*τ*ejφ2,ν21=μ2ν1τejφ2+ν2μ1*τ*ejφ1,μ13=μ24=μ2ρ,ν14=ν23=ν2ρ*,
(45)

where μ2 and ν2 are the transfer coefficients for the PSA stage. It is easy to verify that |μ11|2|ν12|2+|μ13|2|ν14|2=1, which ensures that the composite transformation is canonical. Finally by using the aforementioned definitions and the results of [13

13. C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express 12(21), 5037–5066 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-21-5037. [CrossRef] [PubMed]

, 14

14. C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13(13), 4986–5012 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-13-13-4986. [CrossRef] [PubMed]

], one can write the output variances of the signal and idler modes as

δns_PIA+ATT+PSA2=|μ11|4ns+|μ11|2(|ν12|2+|μ13|2+|ν14|2)ns+(|μ11|2+|μ13|2)(|ν12|2+|ν14|2),
(46)
δni_PIA+ATT+PSA2=|ν21|4ns+|ν21|2(|μ22|2+|ν23|2+|μ24|2)ns+(|μ22|2+|μ24|2)(|ν21|2+|ν23|2),
(47)

where the first two terms on the right sides of Eq. (46) and (47) represent the signal-noise beat terms, while the third terms are the noise-noise beat terms. After neglecting the noise-noise terms under the large-photon-number assumption, and by defining |μ11|2=G12T, |ν12|2=(G121)T, |μ13|2=G2(1T) and |ν14|2=(G21)(1T), we can find that Eq. (46) and (47) are in exactly the same form as Eq. (14) and (15) after multiplying the 2R2 (hv)2 B o factor, which clearly proves the validity of the semi-classical method.

Acknowledgement

This research leading to these results has received funding from the European Communities 7th Framework Programme FP/2007-2013 under grant agreement 224547(STREP PHASORS), and also from the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant number FA8655-09-1-3076. Z. Tong would like to thank Per-Olof Hedekvist for help with error analysis.

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A. Mecozzi and P. Tombesi, “Parametric amplification and signal-to-noise ratio in optical transmission lines,” Opt. Commun. 75(3-4), 256–262 (1990). [CrossRef]

25.

N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7(7), 1071–1082 (1989). [CrossRef]

26.

E. Desuvire, “Comments on ‘The noise figure of optical amplifiers’,” IEEE Photon. Technol. Lett. 11(5), 620–621 (1999). [CrossRef]

27.

Z. Tong, C. J. McKinstrie, C. Lundström, M. Karlsson, and P. A. Andrekson, “Noise performance of optical fiber transmission links that use non-degenerate cascaded phase-sensitive amplifiers,” Opt. Express (submitted to). [PubMed]

28.

M. Movassaghi, M. K. Jackson, V. M. Smith, and W. J. Hallam, “Noise figure of erbium-doped fiber amplifiers in saturated operation,” J. Lightwave Technol. 16(5), 1461–1465 (1998). [CrossRef]

29.

P. L. Voss and P. Kumar, “Raman-effect induced noise limits on χ(3) parametric amplifiers and wavelength converters,” J. Opt. B Quantum Semiclassical Opt. 6(8), 762–770 (2004) (and references therein). [CrossRef]

30.

P. Kylemark, P. –O. Hedekvist, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Noise characteristics of fiber optical parametric amplifiers,” J. Lightwave Technol. 22, 409–416 (2004) and 23, 2192 (2005). [CrossRef]

31.

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18(3), 2884–2893 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-3-2884. [CrossRef] [PubMed]

32.

P. Kylemark, M. Karlsson, and P. A. Andrekson, “Gain and wavelength dependence of the noise-figure in fiber optical parametric amplifiers,” IEEE Photon. Technol. Lett. 18(11), 1255–1257 (2006). [CrossRef]

33.

X. Li, J. Chen, P. Voss, J. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications: Improved generation of correlated photons,” Opt. Express 12(16), 3737–3744 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-16-3737. [CrossRef] [PubMed]

OCIS Codes
(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: April 12, 2010
Revised Manuscript: June 9, 2010
Manuscript Accepted: June 16, 2010
Published: June 28, 2010

Citation
Zhi Tong, Adonis Bogris, Carl Lundström, C. J. McKinstrie, Michael Vasilyev, Magnus Karlsson, and Peter A. Andrekson, "Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier," Opt. Express 18, 14820-14835 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-14820


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References

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  11. R. Tang, J. Lasri, P. S. Devgan, V. S. Grigoryan, P. Kumar, and M. Vasilyev, “Gain characteristics of a frequency nondegenerate phase-sensitive fiber-optic parametric amplifier with phase self-stabilized input,” Opt. Express 13(26), 10483–10493 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-26-10483 . [CrossRef] [PubMed]
  12. P. L. Voss, K. G. Köprülü, and P. Kumar, “Raman-noise-induced quantum limits for χ(3) nondegenerate phase-sensitive amplification and quadrature squeezing,” J. Opt. Soc. Am. B 23(4), 598–610 (2006). [CrossRef]
  13. C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express 12(21), 5037–5066 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-21-5037 . [CrossRef] [PubMed]
  14. C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13(13), 4986–5012 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-13-13-4986 . [CrossRef] [PubMed]
  15. M. Vasilyev, “Distributed phase-sensitive amplification,” Opt. Express 13(19), 7563–7571 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-13-19-7563 . [CrossRef] [PubMed]
  16. C. J. McKinstrie, M. G. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257(1), 146–163 (2006). [CrossRef]
  17. C. Lundström, J. Kakande, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Experimental comparison of gain and saturation characteristics of a parametric amplifier in phase-sensitive and phase-insensitive mode,” in European Conference on Optical Communications, paper Mo. 1.1.1 (2009).
  18. J. Kakande, C. Lundström, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Detailed characterization of afiber-optic parametric amplifier in phase-sensitive and phase-insensitive operation,” Opt. Express 18(5), 4130–4137 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-4130 . [CrossRef] [PubMed]
  19. O. K. Lim, V. S. Grigoryan, M. Shin, and P. Kumar, “Ultra-low-noise inline fiber-optic phase-sensitive amplifier for analog optical signals,” in Optical Fiber Communications Conference, paper OML3 (2008).
  20. Z. Tong, C. Lundström, A. Bogris, M. Karlsson, P. Andrekson, and D. Syvridis, “Measurement of sub-1 dB noise figure in a non-degenerate cascaded phase-sensitive fiber parametric amplifier,” in European Conference on Optical Communications, paper Mo. 1.1.2 (2009).
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  22. E. Desuvire, Erbium-doped fiber amplifiers (John Wiley and Sons, 1994), Chap. 2.
  23. P. Kylemark, P.-O. Hedekvist, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Noise characteristics of fiber optical parametric amplifiers,” J. Lightwave Technol. 22(2), 409–416 (2004). [CrossRef]
  24. A. Mecozzi and P. Tombesi, “Parametric amplification and signal-to-noise ratio in optical transmission lines,” Opt. Commun. 75(3-4), 256–262 (1990). [CrossRef]
  25. N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7(7), 1071–1082 (1989). [CrossRef]
  26. E. Desuvire, “Comments on ‘The noise figure of optical amplifiers’,” IEEE Photon. Technol. Lett. 11(5), 620–621 (1999). [CrossRef]
  27. Z. Tong, C. J. McKinstrie, C. Lundström, M. Karlsson, and P. A. Andrekson, “Noise performance of optical fiber transmission links that use non-degenerate cascaded phase-sensitive amplifiers,” Opt. Express (submitted to). [PubMed]
  28. M. Movassaghi, M. K. Jackson, V. M. Smith, and W. J. Hallam, “Noise figure of erbium-doped fiber amplifiers in saturated operation,” J. Lightwave Technol. 16(5), 1461–1465 (1998). [CrossRef]
  29. P. L. Voss and P. Kumar, “Raman-effect induced noise limits on χ(3) parametric amplifiers and wavelength converters,” J. Opt. B Quantum Semiclassical Opt. 6(8), 762–770 (2004) (and references therein). [CrossRef]
  30. P. Kylemark, P. –O. Hedekvist, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Noise characteristics of fiber optical parametric amplifiers,” J. Lightwave Technol. 22, 409–416 (2004) and 23, 2192 (2005). [CrossRef]
  31. Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18(3), 2884–2893 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-3-2884 . [CrossRef] [PubMed]
  32. P. Kylemark, M. Karlsson, and P. A. Andrekson, “Gain and wavelength dependence of the noise-figure in fiber optical parametric amplifiers,” IEEE Photon. Technol. Lett. 18(11), 1255–1257 (2006). [CrossRef]
  33. X. Li, J. Chen, P. Voss, J. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications: Improved generation of correlated photons,” Opt. Express 12(16), 3737–3744 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-16-3737 . [CrossRef] [PubMed]

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