Abstract

Following a fiber optical parametric amplifier, used as a wavelength converter or in the phase-sensitive mode, by a phase-insensitive amplifier (PIA) can significantly reduce four-wave mixing between signals in broadband systems. We derive the quantum mechanical noise figures (NF) for these two hybrid configurations, and show that adding the PIA only leads to a moderate increase in NF.

© 2012 OSA

1. Introduction

Broadband optical communication systems with multi-terabits per second capacity are currently being deployed [1], and over 100 Tb/s transmission through a singlemode fiber has been demonstrated in the laboratory [2]. Hence devices being developed for insertion into future broadband systems should also be able to handle such high capacities.

Fiber optical parametric amplifiers (OPAs) have been under development for the past two decades, as they can exhibit a number of interesting properties [3]. In particular, when used as phase-sensitive amplifiers (PSAs), they can exhibit noise figures (NFs) as low as 0 dB [4]; this aspect has been the object of intense research in recent years [57]. When used as phase-insensitive amplifiers (PIAs), they can be used for spectral inversion/phase conjugation which can potentially compensate fiber dispersion and some nonlinear effects. These two aspects could be useful if they could be implemented in broadband communication systems.

However, the performance of fiber OPAs in WDM systems is limited by four-wave mixing (FWM) crosstalk [814]. To date, the highest transmission rate demonstrated for a fiber OPA (used as PIA) is about 1 Tb/s. The performance was not quite as good as for an EDFA, being primarily limited by FWM crosstalk [15].

In order to address this crosstalk problem, we recently proposed using a hybrid amplifier formed by a fiber OPA with moderate gain, followed by a (non-fiber-OPA) PIA, as a novel and effective means for reducing the FWM crosstalk [16]. This is based on the fact that FWM between signals occurs primarily where signals are the strongest, i.e. near the amplifier output; hence using an amplifier less prone to FWM as a booster will reduce FWM crosstalk (for the same overall gain). We provided the results of a quantum mechanical NF calculation for the PSA-PIA combination, predicting moderate NF increases compared to the fiber PSA alone [16]

In this Letter we present the detailed calculation of the quantum NF calculation for the PSA-PIA case. We also perform a similar calculation for the case of a parametric wavelength converter (WC) followed by a PIA. We perform the calculations by starting from first principles rather than by using a priori Friis’ formula for cascaded optical amplifiers [17], since the validity of the latter is well established for doped-fiber amplifiers [18], but has not been systematically established for parametric devices [19,20]. For these two combinations, we confirm the benefits to be derived from this hybrid approach.

2. PSA-PIA

Figure 1 shows the PSA-PIA case [16]. It has both signal and idler present at its input, at different wavelengths (i.e. we consider a non-degenerate or two-mode PSA). The two fields are in the same initial quantum state, and therefore if the operator associated with the signal at the PSA input is a^ that associated with the idler is a^. (This would need to be modified if the two fields were not in the same quantum state [21]).

 

Fig. 1 PSA-PIA combination. The dashed line is for the idler.

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We now assume that both input fields are in the same coherent state (CS) with complex amplitudeα, and wavefunction ψ, i.e. such thata^ψ=αψ; the mean photon number for this CS is Na=αα*. Under optimum phase-matching conditions, the PSA output signal is b^=g1a^, where g1 is the PSA signal field gain; the maximum phase-sensitive power gain is G1=|g1|2. Considered by itself, the PSA has the NF F1=1 (0 dB). c^ represents the vacuum field at the upper PIA input, responsible for its noise factor [22]. Letting g2 denote the field gain of the PIA, its NF isF2=21/G2, where G2=|g2|2is the power gain. Then the top PIA output operator is given by d^=Aa^+Cc^ where:A=g1g2; C=h2 is the field conversion gain of the PIA, and |h2|2=G21 is its conversion power gain. Then the photon number operator at the output is

N^d=d^d^=(A*a^+C*c^)(Aa^+Cc^)=|A|2N^a+|C|2(1+N^c)+A*Ca^c^+C*Aa^c^
where N^a=a^a^ and N^c=c^c^ are the photon number operators at the PSA signal input and at the noise input, respectively.

The average number of photons at the PIA output is

Nd=N^d=|A|2Na+|C|2
where Na=N^a is the average number of signal photons at the PSA input. We used the fact that the input noise is in a vacuum state (CS with zero mean photon number).

We now calculate the variance of the output photon number (ΔNd)2=N^d2Nd2. We have

N^d2=(|A|2a^a^+|C|2c^c^+A*Ca^c^+C*Aa^c^)(|A|2a^a^+|C|2c^c^+A*Ca^c^+C*Aa^c^)=|A|4a^a^a^a^+|C|4c^c^c^c^+(A*C)2a^a^c^c^+(AC*)2a^a^c^c^+2|AC|2a^a^c^c^+|A|2A*C(a^a^a^c^+a^a^a^c^)+|A|2C*A(a^a^a^c^+a^a^a^c^)+|C|2A*C(a^c^c^c^+a^c^c^c^)+|C|2AC*(a^c^c^c^+a^c^c^c^)+|AC|2(a^a^c^c^+a^a^c^c^)
and so
(N^d)2=|A|4(Na2+Na)+|C|4+2|AC|2Na+|AC|2(Na+1)=|A|4Na2+(|A|4+3|AC|2)Na+|AC|2+|C|4.
Also
Nd2=(|A|2Na+|C|2)2=|A|4Na2+|C|4+2|AC|2Na.
Therefore
(ΔNd)2=(|A|4+|AC|2)Na+|AC|2=|A|2[(|A|2+|C|2)Na+|C|2].
The NF is then
F=SNRinSNRout=(NaNd)2(ΔNd)2(ΔNa)2=(Na|A|2Na+|C|2i)2|A|2[(|A|2+|C|2)Na+|C|2]Na=Na|A|2(|A|2+|C|2)Na+|C|2(|A|2Na+|C|2)2=NaG1G2(G1G2+G21)Na+G21(G1G2Na+G21)2
where we used the fact that since the input field is in a CS, its variance is (ΔNa)2=Na. We verify that when the PIA has unit gain (G2=1), F = 1 as expected for the PSA alone. (Note that for the calculation of SNRin we considered only the power of the signal, and did not include that of the idler.)

If the input CS has a large amplitude, we can assume that Na is much larger thanG1, G2 and 1. Under these circumstances,

FF'=(G1G2+G21)(G1G2)=1+1G11G1G2.
We can compare this result to that obtained by Friis’ formula. The latter was originally developed for electronic amplifiers [17], but it has also been found to apply for optical amplifiers such as EDFAs in some circumstances [18]. Friis’ formula yields the NF value F''F1+(F21)/G1, where Fk is the NF of the kth stage, k = 1,2.. Since F1=1, and F2=21/G2 (because the second amplifier is an ideal PIA), we find that F''=F'. Hence the exact approach presented here yields the same result as Friis’ formula, under the reasonable assumptions that we have made.

Let us consider an example to see the extent of the NF degradation due to the PIA stage. We compare two ways for obtaining the same overall 20 dB gain: (a) only with a PSA; (b) with the gain evenly divided between a PSA front end and a PIA booster. For (a) the overall NF is Fa=1 (0 dB), while for (b) it is Fb=1.09 (0.37 dB). Hence we see that this combination contributes only a modest 0.37 dB increase in NF, which may well be worthwhile for reducing nonlinear crosstalk.

3. Wavelength converter-PIA

We now consider the case of a parametric WC followed by a PIA, Fig. 2 . a^is the input signal, v^1is the vacuum field at the idler wavelength, b^is the converted output, v^2is the input vacuum field of the PIA, and c^ is the output field. giis the WC field conversion gain, gsis its signal (field) gain, g2is the PIA field gain, and h2 its conversion field gain. The corresponding power gains are Gk=|gk|2, k = s,i,2, and H2=G21. We have b^=gia^+gsv^1, c^=g2(gia^+gsv^1)+h2v^2=U^+V^+W^,where U^=g2gia^ V^=g2gsv^1 W^=h2v^2. Hence

N^c=c^c^=(U^+V^+W^)(U^+V^+W^)=U^U^+U^V^+U^W^+V^U^+V^V^+V^W^+W^U^+W^V^+W^W^..
To calculate its variance we need N^c2 which contains 81 terms. However, most of them make no contribution to the expectation value, which becomes
N^c2=U^U^+U^V^+U^W^+V^U^+V^V^+V^W^+W^U^+W^V^+W^W^=U^U^U^U^+U^U^W^W^+U^V^V^U^+W^U^U^W^+W^V^V^W^+W^W^U^U^+W^W^W^W^=Aa^a^a^a^+2(B+C)a^a^+Da^a^+E+H
where A=(GiG2)2, B=D=GiG2(G21), C=GsGiG2(G21), E=GsG2(G21), H=(G21)2. Evaluating the expectation values we find that
N^c2=ANa2+Na(3A+2B+C+D)+A+2B+C+E+H.
We also have
N^c=U^U^+W^W^=GiG2(Na+1)+G21,
N^c2=ANa2+2(A+B)Na+A+2B+H.
Then the variance of N^c is
(ΔNc)2=(A+C+D)Na+C+E
and the NF is
F=(A+C+D)Na2+(C+E)NaANa2+2(A+B)Na+A+2B+H.
For a large-amplitude CS at the input, this is well approximated by
F=1+C+DA=2+2Gi1GiG2
which is again in agreement with Friis’ formula, since the NF of the WC is Fi=2+1/Gi.

 

Fig. 2 WC-PIA combination.

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Let us consider an example to see how the use of a PIA booster affects the NF of the WC. We want to obtain an overall 20 dB conversion gain, and we compare three ways of obtaining it: (a) only a WC, with 20 dB conversion gain; (b) a WC with 10 dB conversion gain, followed by a PIA with 10 dB gain; (c) a WC with unit conversion gain (0 dB), followed by a PIA with 20 dB gain. We find that:Fa=2.01 (3.03 dB); Fb=2.19 (3.4 dB).; Fc=3.99 (6 dB). Of course (a) would be best, however making a fiber WC with such a high conversion gain is difficult, because of nonlinear crosstalk, as well as of the need for avoiding pump phase modulation (PM) to reduce stimulated Brillouin scattering (SBS). (b) alleviates these problems, as 10 dB gain can be achieved without pump PM, and crosstalk is greatly reduced; the modest 0.37 NF penalty may well be worthwhile in practice. (c) would be best from the point of view of SBS suppression, and excellent suppression of FWM crosstalk, but the 3 dB penalty incurred may be too large.

4. Discussion

Several types of amplifiers can potentially be considered for the booster stage. Two PIAs are shown in Fig. 1, one for the signal and one for the idler. This is one possibility, which could for example be implemented by means of a C-band EDFA and an L-band one. On the other hand, the two PIAs could also be replaced by a single PIA, which should have a large enough bandwidth to simultaneously amplify signals and idlers. The PIAs could in principle also be implemented by means of SOAs, discrete Raman amplifiers, etc. These amplifiers have their own advantage and disadvantages, which will have to be considered. At this time it appears that using EDFAs for feasibility demonstrations is a logical choice. Of course there are some tradeoffs involved with the use of this architecture. In particular, the large bandwidth and flexible spectral region aspects of fiber OPAs are now limited by the characteristics of the PIA.

In this paper we have investigated the quantum mechanical NF of two amplifying structures, assuming that all the elements used had ideal characteristics. However, in practice all these elements will in fact be non-ideal, and exhibit a variety of features which will limit their performance. This in turn will lead to practically achievable NFs which will be larger than those predicted here. Among the issues that will need to be examined for practical systems are: the spontaneous emission factor nsp of EDFAs; the losses introduced by the pump regeneration requirements of PSAs; phase stability and cost when using two serarate EDFAs,

We note that PSA-EDFA combinations have been used in experiments [6,23]. The motivation for using the EDFAs was not for reducing nonlinear effects, but for boosting the PSA gain which was kept low (to avoid pump PM). However, the fact that good overall performance was obtained by these combinations is consistent with their ability to retain a low NF, theoretically investigated here.

5. Conclusion

In summary, we have shown that the combination of a fiber PSA or WC front-end followed by a PIA booster can in principle exhibit a quantum-limited noise figure which is almost as low as that of the front end with the same overall gain. In both cases it is found that the NF expressions obtained conform to Friis’ formula for cascaded amplifiers. The PIA can be designed to exhibit reduced levels of detrimental FWM which are otherwise generated near the output end of a fiber OPA. This new architecture can provide tens of decibels of reduction of these FWM effects, which should considerably improve the performance of such amplifiers in broadband optical communication systems, including WDM systems.

The overall conclusion is that hybrid fiber OPA-nonfiber OPA combinations should be considered whenever it is desired to exploit the key features of fiber OPAs (low NF amplification, or wavelength conversion) in broadband communication systems. Such hybrid combinations could be used at the transmitter and/or inline in such systems. The gain of the fiber OPA front end should be sufficient to provide good overall noise performance, but not so large as to generate strong nonlinear crosstalk. These two constraints will guide the design of such combinations.

Acknowledgments

This work was supported in part by Huawei Technologies, Shenzhen, China, and by grant EP/J009709/1 from the UK’s EPSRC. Thanks are due to S. Radic for reading and commenting on the manuscript.

References and links

1. A. E. Willner, Z. Pan, and M. I. Hayee, “Major accomplishments in 2010 on optical fiber communications,” IEEE Photon. J. 3(2), 320–324 (2011). [CrossRef]  

2. D. Qian, M.-F. Huang, E. Ip, Y.-K. Huang, Y. Shao, J. Hu, and T. Wang, “101.7-Tb/s (370×294-Gb/s) PDM-128QAM-OFDM transmission over 3x55-km SSMF using pilot-based phase noise mitigation,” in Proceedings of OFC/NFOEC 2011, Anaheim, CA, March 6–10, 2011, paper PDPB5.

3. M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge University Press, 2007).

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

5. P. A. Andrekson, C. Lundstrom, and Z. Tong, “Phase-sensitive fiber-optic parametric amplifiers and their applications,” in Proceedings of ECOC2010, 19–23 September 2010, Turin, Italy, We.6.E.1.

6. Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011). [CrossRef]  

7. Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron. 18(2), 1016–1032 (2012). [CrossRef]  

8. J. L. Blows, “Design strategy for controlling four-wave mixing-induced crosstalk between channels in a fibre optical parametric amplifier,” Opt. Commun. 236(1-3), 115–122 (2004). [CrossRef]  

9. J. L. Blows and P.- Hu, “Cross-talk-induced limitations of two-pump optical fiber parametric amplifiers,” J. Opt. Soc. Am. B 21(5), 989–995 (2004). [CrossRef]  

10. P.- Hu and J. L. Blows, “Four-wave mixing crosstalk in optical fibre parametric amplifiers with orthogonal pumps,” Opt. Commun. 250(4-6), 421–427 (2005). [CrossRef]  

11. J. M. C. Boggio, J. D. Marconi, and H. L. Fragnito, “Crosstalk in double-pumped fiber optic parametric amplifiers for wavelength division multiplexing systems,” Opt. Commun. 259(1), 94–103 (2006). [CrossRef]  

12. J. M. C. Boggio, E. A. M. Fagotto, M. E. Marhic, F. A. Callegari, and H. L. Fragnito “Amplification of 12 × 10 Gb/s WDM signals with negligible FWM crosstalk in a double-pumped fiber optical parametric amplifier,” Opt. Commun. 280(2), 468–471 (2007). [CrossRef]  

13. B. P.-P. Kuo, P. C. Chui, and K. K.-Y. Wong, “A comprehensive study on crosstalk suppression techniques in fiber optical parametric amplifier by modulation format,” IEEE J. Sel. Top. Quantum Electron. 14(3), 659–665 (2008). [CrossRef]  

14. M. Jamshidifar, A. Vedadi, and M. E. Marhic, “Reduction of four-wave mixing crosstalk in a short fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 21(17), 1244–1246 (2009). [CrossRef]  

15. N. E. Dahdah, D. S. Govan, M. Jamshidifar, N. J. Doran, and M. E. Marhic, “Fiber optical parametric amplifier performance in a 1-Tb/s DWDM communication system,” IEEE J. Sel. Top. Quantum Electron. 18(2), 950–957 (2012). [CrossRef]  

16. M. E. Marhic, “Hybrid phase-sensitive and phase-insensitive amplifiers for optical communication,” in Proceedings of 17th OECC, Busan, Korea, July 2–6 2012.

17. H. Friis, “Noise figures of radio receivers,” Proc. I.R.E. 32, 419–422 (1944).

18. H. Haus, “The noise figure of optical amplifiers,” IEEE Photon. Technol. Lett. 10(11), 1602–1604 (1998). [CrossRef]  

19. Y. Yamamoto and K. Inoue, “Noise in amplifiers,” J. Lightwave Technol. 21(11), 2895–2915 (2003). [CrossRef]  

20. C. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13(13), 4986–5012 (2005). [CrossRef]   [PubMed]  

21. C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number correlations produced by parametric processes,” Opt. Express 18(19), 19792–19823 (2010). [CrossRef]   [PubMed]  

22. U. Leonhardt, “Quantum physics of simple optical instruments,” Rep. Prog. Phys. 66(7), 1207–1249 (2003). [CrossRef]  

23. B. Corcoran, S. L. I. Olsson, C. Lundström, M. Karlsson, and P. Andrekson, “Phase-sensitive optical pre-amplifier implemented in an 80 km DQPSK link,” in Proceedings of OFC/NFOEC 2012, Los Angeles, CA, March 6–8, 2012, paper PDP5A.4.

References

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  1. A. E. Willner, Z. Pan, and M. I. Hayee, “Major accomplishments in 2010 on optical fiber communications,” IEEE Photon. J.3(2), 320–324 (2011).
    [CrossRef]
  2. D. Qian, M.-F. Huang, E. Ip, Y.-K. Huang, Y. Shao, J. Hu, and T. Wang, “101.7-Tb/s (370×294-Gb/s) PDM-128QAM-OFDM transmission over 3x55-km SSMF using pilot-based phase noise mitigation,” in Proceedings of OFC/NFOEC 2011, Anaheim, CA, March 6–10, 2011, paper PDPB5.
  3. M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge University Press, 2007).
  4. C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields26(8), 1817–1839 (1982).
    [CrossRef]
  5. P. A. Andrekson, C. Lundstrom, and Z. Tong, “Phase-sensitive fiber-optic parametric amplifiers and their applications,” in Proceedings of ECOC2010, 19–23 September 2010, Turin, Italy, We.6.E.1.
  6. Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5(7), 430–436 (2011).
    [CrossRef]
  7. Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron.18(2), 1016–1032 (2012).
    [CrossRef]
  8. J. L. Blows, “Design strategy for controlling four-wave mixing-induced crosstalk between channels in a fibre optical parametric amplifier,” Opt. Commun.236(1-3), 115–122 (2004).
    [CrossRef]
  9. J. L. Blows and P.- Hu, “Cross-talk-induced limitations of two-pump optical fiber parametric amplifiers,” J. Opt. Soc. Am. B21(5), 989–995 (2004).
    [CrossRef]
  10. P.- Hu and J. L. Blows, “Four-wave mixing crosstalk in optical fibre parametric amplifiers with orthogonal pumps,” Opt. Commun.250(4-6), 421–427 (2005).
    [CrossRef]
  11. J. M. C. Boggio, J. D. Marconi, and H. L. Fragnito, “Crosstalk in double-pumped fiber optic parametric amplifiers for wavelength division multiplexing systems,” Opt. Commun.259(1), 94–103 (2006).
    [CrossRef]
  12. J. M. C. Boggio, E. A. M. Fagotto, M. E. Marhic, F. A. Callegari, and H. L. Fragnito “Amplification of 12 × 10 Gb/s WDM signals with negligible FWM crosstalk in a double-pumped fiber optical parametric amplifier,” Opt. Commun.280(2), 468–471 (2007).
    [CrossRef]
  13. B. P.-P. Kuo, P. C. Chui, and K. K.-Y. Wong, “A comprehensive study on crosstalk suppression techniques in fiber optical parametric amplifier by modulation format,” IEEE J. Sel. Top. Quantum Electron.14(3), 659–665 (2008).
    [CrossRef]
  14. M. Jamshidifar, A. Vedadi, and M. E. Marhic, “Reduction of four-wave mixing crosstalk in a short fiber optical parametric amplifier,” IEEE Photon. Technol. Lett.21(17), 1244–1246 (2009).
    [CrossRef]
  15. N. E. Dahdah, D. S. Govan, M. Jamshidifar, N. J. Doran, and M. E. Marhic, “Fiber optical parametric amplifier performance in a 1-Tb/s DWDM communication system,” IEEE J. Sel. Top. Quantum Electron.18(2), 950–957 (2012).
    [CrossRef]
  16. M. E. Marhic, “Hybrid phase-sensitive and phase-insensitive amplifiers for optical communication,” in Proceedings of 17th OECC, Busan, Korea, July 2–6 2012.
  17. H. Friis, “Noise figures of radio receivers,” Proc. I.R.E. 32, 419–422 (1944).
  18. H. Haus, “The noise figure of optical amplifiers,” IEEE Photon. Technol. Lett.10(11), 1602–1604 (1998).
    [CrossRef]
  19. Y. Yamamoto and K. Inoue, “Noise in amplifiers,” J. Lightwave Technol.21(11), 2895–2915 (2003).
    [CrossRef]
  20. C. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express13(13), 4986–5012 (2005).
    [CrossRef] [PubMed]
  21. C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number correlations produced by parametric processes,” Opt. Express18(19), 19792–19823 (2010).
    [CrossRef] [PubMed]
  22. U. Leonhardt, “Quantum physics of simple optical instruments,” Rep. Prog. Phys.66(7), 1207–1249 (2003).
    [CrossRef]
  23. B. Corcoran, S. L. I. Olsson, C. Lundström, M. Karlsson, and P. Andrekson, “Phase-sensitive optical pre-amplifier implemented in an 80 km DQPSK link,” in Proceedings of OFC/NFOEC 2012, Los Angeles, CA, March 6–8, 2012, paper PDP5A.4.

2012 (2)

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron.18(2), 1016–1032 (2012).
[CrossRef]

N. E. Dahdah, D. S. Govan, M. Jamshidifar, N. J. Doran, and M. E. Marhic, “Fiber optical parametric amplifier performance in a 1-Tb/s DWDM communication system,” IEEE J. Sel. Top. Quantum Electron.18(2), 950–957 (2012).
[CrossRef]

2011 (2)

A. E. Willner, Z. Pan, and M. I. Hayee, “Major accomplishments in 2010 on optical fiber communications,” IEEE Photon. J.3(2), 320–324 (2011).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5(7), 430–436 (2011).
[CrossRef]

2010 (1)

2009 (1)

M. Jamshidifar, A. Vedadi, and M. E. Marhic, “Reduction of four-wave mixing crosstalk in a short fiber optical parametric amplifier,” IEEE Photon. Technol. Lett.21(17), 1244–1246 (2009).
[CrossRef]

2008 (1)

B. P.-P. Kuo, P. C. Chui, and K. K.-Y. Wong, “A comprehensive study on crosstalk suppression techniques in fiber optical parametric amplifier by modulation format,” IEEE J. Sel. Top. Quantum Electron.14(3), 659–665 (2008).
[CrossRef]

2007 (1)

J. M. C. Boggio, E. A. M. Fagotto, M. E. Marhic, F. A. Callegari, and H. L. Fragnito “Amplification of 12 × 10 Gb/s WDM signals with negligible FWM crosstalk in a double-pumped fiber optical parametric amplifier,” Opt. Commun.280(2), 468–471 (2007).
[CrossRef]

2006 (1)

J. M. C. Boggio, J. D. Marconi, and H. L. Fragnito, “Crosstalk in double-pumped fiber optic parametric amplifiers for wavelength division multiplexing systems,” Opt. Commun.259(1), 94–103 (2006).
[CrossRef]

2005 (2)

P.- Hu and J. L. Blows, “Four-wave mixing crosstalk in optical fibre parametric amplifiers with orthogonal pumps,” Opt. Commun.250(4-6), 421–427 (2005).
[CrossRef]

C. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express13(13), 4986–5012 (2005).
[CrossRef] [PubMed]

2004 (2)

J. L. Blows, “Design strategy for controlling four-wave mixing-induced crosstalk between channels in a fibre optical parametric amplifier,” Opt. Commun.236(1-3), 115–122 (2004).
[CrossRef]

J. L. Blows and P.- Hu, “Cross-talk-induced limitations of two-pump optical fiber parametric amplifiers,” J. Opt. Soc. Am. B21(5), 989–995 (2004).
[CrossRef]

2003 (2)

Y. Yamamoto and K. Inoue, “Noise in amplifiers,” J. Lightwave Technol.21(11), 2895–2915 (2003).
[CrossRef]

U. Leonhardt, “Quantum physics of simple optical instruments,” Rep. Prog. Phys.66(7), 1207–1249 (2003).
[CrossRef]

1998 (1)

H. Haus, “The noise figure of optical amplifiers,” IEEE Photon. Technol. Lett.10(11), 1602–1604 (1998).
[CrossRef]

1982 (1)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields26(8), 1817–1839 (1982).
[CrossRef]

Andrekson, P. A.

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron.18(2), 1016–1032 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5(7), 430–436 (2011).
[CrossRef]

Blessing, D. J.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5(7), 430–436 (2011).
[CrossRef]

Blows, J. L.

P.- Hu and J. L. Blows, “Four-wave mixing crosstalk in optical fibre parametric amplifiers with orthogonal pumps,” Opt. Commun.250(4-6), 421–427 (2005).
[CrossRef]

J. L. Blows, “Design strategy for controlling four-wave mixing-induced crosstalk between channels in a fibre optical parametric amplifier,” Opt. Commun.236(1-3), 115–122 (2004).
[CrossRef]

J. L. Blows and P.- Hu, “Cross-talk-induced limitations of two-pump optical fiber parametric amplifiers,” J. Opt. Soc. Am. B21(5), 989–995 (2004).
[CrossRef]

Boggio, J. M. C.

J. M. C. Boggio, E. A. M. Fagotto, M. E. Marhic, F. A. Callegari, and H. L. Fragnito “Amplification of 12 × 10 Gb/s WDM signals with negligible FWM crosstalk in a double-pumped fiber optical parametric amplifier,” Opt. Commun.280(2), 468–471 (2007).
[CrossRef]

J. M. C. Boggio, J. D. Marconi, and H. L. Fragnito, “Crosstalk in double-pumped fiber optic parametric amplifiers for wavelength division multiplexing systems,” Opt. Commun.259(1), 94–103 (2006).
[CrossRef]

Bogris, A.

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron.18(2), 1016–1032 (2012).
[CrossRef]

Callegari, F. A.

J. M. C. Boggio, E. A. M. Fagotto, M. E. Marhic, F. A. Callegari, and H. L. Fragnito “Amplification of 12 × 10 Gb/s WDM signals with negligible FWM crosstalk in a double-pumped fiber optical parametric amplifier,” Opt. Commun.280(2), 468–471 (2007).
[CrossRef]

Caves, C. M.

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields26(8), 1817–1839 (1982).
[CrossRef]

Chui, P. C.

B. P.-P. Kuo, P. C. Chui, and K. K.-Y. Wong, “A comprehensive study on crosstalk suppression techniques in fiber optical parametric amplifier by modulation format,” IEEE J. Sel. Top. Quantum Electron.14(3), 659–665 (2008).
[CrossRef]

Dahdah, N. E.

N. E. Dahdah, D. S. Govan, M. Jamshidifar, N. J. Doran, and M. E. Marhic, “Fiber optical parametric amplifier performance in a 1-Tb/s DWDM communication system,” IEEE J. Sel. Top. Quantum Electron.18(2), 950–957 (2012).
[CrossRef]

Doran, N. J.

N. E. Dahdah, D. S. Govan, M. Jamshidifar, N. J. Doran, and M. E. Marhic, “Fiber optical parametric amplifier performance in a 1-Tb/s DWDM communication system,” IEEE J. Sel. Top. Quantum Electron.18(2), 950–957 (2012).
[CrossRef]

Fagotto, E. A. M.

J. M. C. Boggio, E. A. M. Fagotto, M. E. Marhic, F. A. Callegari, and H. L. Fragnito “Amplification of 12 × 10 Gb/s WDM signals with negligible FWM crosstalk in a double-pumped fiber optical parametric amplifier,” Opt. Commun.280(2), 468–471 (2007).
[CrossRef]

Fragnito, H. L.

J. M. C. Boggio, E. A. M. Fagotto, M. E. Marhic, F. A. Callegari, and H. L. Fragnito “Amplification of 12 × 10 Gb/s WDM signals with negligible FWM crosstalk in a double-pumped fiber optical parametric amplifier,” Opt. Commun.280(2), 468–471 (2007).
[CrossRef]

J. M. C. Boggio, J. D. Marconi, and H. L. Fragnito, “Crosstalk in double-pumped fiber optic parametric amplifiers for wavelength division multiplexing systems,” Opt. Commun.259(1), 94–103 (2006).
[CrossRef]

Govan, D. S.

N. E. Dahdah, D. S. Govan, M. Jamshidifar, N. J. Doran, and M. E. Marhic, “Fiber optical parametric amplifier performance in a 1-Tb/s DWDM communication system,” IEEE J. Sel. Top. Quantum Electron.18(2), 950–957 (2012).
[CrossRef]

Grüner-Nielsen, L.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5(7), 430–436 (2011).
[CrossRef]

Haus, H.

H. Haus, “The noise figure of optical amplifiers,” IEEE Photon. Technol. Lett.10(11), 1602–1604 (1998).
[CrossRef]

Hayee, M. I.

A. E. Willner, Z. Pan, and M. I. Hayee, “Major accomplishments in 2010 on optical fiber communications,” IEEE Photon. J.3(2), 320–324 (2011).
[CrossRef]

Hu, P.-

P.- Hu and J. L. Blows, “Four-wave mixing crosstalk in optical fibre parametric amplifiers with orthogonal pumps,” Opt. Commun.250(4-6), 421–427 (2005).
[CrossRef]

J. L. Blows and P.- Hu, “Cross-talk-induced limitations of two-pump optical fiber parametric amplifiers,” J. Opt. Soc. Am. B21(5), 989–995 (2004).
[CrossRef]

Inoue, K.

Jamshidifar, M.

N. E. Dahdah, D. S. Govan, M. Jamshidifar, N. J. Doran, and M. E. Marhic, “Fiber optical parametric amplifier performance in a 1-Tb/s DWDM communication system,” IEEE J. Sel. Top. Quantum Electron.18(2), 950–957 (2012).
[CrossRef]

M. Jamshidifar, A. Vedadi, and M. E. Marhic, “Reduction of four-wave mixing crosstalk in a short fiber optical parametric amplifier,” IEEE Photon. Technol. Lett.21(17), 1244–1246 (2009).
[CrossRef]

Karlsson, M.

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron.18(2), 1016–1032 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5(7), 430–436 (2011).
[CrossRef]

C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number correlations produced by parametric processes,” Opt. Express18(19), 19792–19823 (2010).
[CrossRef] [PubMed]

Kuo, B. P.-P.

B. P.-P. Kuo, P. C. Chui, and K. K.-Y. Wong, “A comprehensive study on crosstalk suppression techniques in fiber optical parametric amplifier by modulation format,” IEEE J. Sel. Top. Quantum Electron.14(3), 659–665 (2008).
[CrossRef]

Leonhardt, U.

U. Leonhardt, “Quantum physics of simple optical instruments,” Rep. Prog. Phys.66(7), 1207–1249 (2003).
[CrossRef]

Lundström, C.

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron.18(2), 1016–1032 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5(7), 430–436 (2011).
[CrossRef]

Marconi, J. D.

J. M. C. Boggio, J. D. Marconi, and H. L. Fragnito, “Crosstalk in double-pumped fiber optic parametric amplifiers for wavelength division multiplexing systems,” Opt. Commun.259(1), 94–103 (2006).
[CrossRef]

Marhic, M. E.

N. E. Dahdah, D. S. Govan, M. Jamshidifar, N. J. Doran, and M. E. Marhic, “Fiber optical parametric amplifier performance in a 1-Tb/s DWDM communication system,” IEEE J. Sel. Top. Quantum Electron.18(2), 950–957 (2012).
[CrossRef]

M. Jamshidifar, A. Vedadi, and M. E. Marhic, “Reduction of four-wave mixing crosstalk in a short fiber optical parametric amplifier,” IEEE Photon. Technol. Lett.21(17), 1244–1246 (2009).
[CrossRef]

J. M. C. Boggio, E. A. M. Fagotto, M. E. Marhic, F. A. Callegari, and H. L. Fragnito “Amplification of 12 × 10 Gb/s WDM signals with negligible FWM crosstalk in a double-pumped fiber optical parametric amplifier,” Opt. Commun.280(2), 468–471 (2007).
[CrossRef]

McKinstrie, C.

McKinstrie, C. J.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5(7), 430–436 (2011).
[CrossRef]

C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number correlations produced by parametric processes,” Opt. Express18(19), 19792–19823 (2010).
[CrossRef] [PubMed]

Pan, Z.

A. E. Willner, Z. Pan, and M. I. Hayee, “Major accomplishments in 2010 on optical fiber communications,” IEEE Photon. J.3(2), 320–324 (2011).
[CrossRef]

Puttnam, B. J.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5(7), 430–436 (2011).
[CrossRef]

Radic, S.

Raymer, M. G.

Tipsuwannakul, E.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5(7), 430–436 (2011).
[CrossRef]

Toda, H.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5(7), 430–436 (2011).
[CrossRef]

Tong, Z.

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron.18(2), 1016–1032 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5(7), 430–436 (2011).
[CrossRef]

C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number correlations produced by parametric processes,” Opt. Express18(19), 19792–19823 (2010).
[CrossRef] [PubMed]

Vedadi, A.

M. Jamshidifar, A. Vedadi, and M. E. Marhic, “Reduction of four-wave mixing crosstalk in a short fiber optical parametric amplifier,” IEEE Photon. Technol. Lett.21(17), 1244–1246 (2009).
[CrossRef]

Willner, A. E.

A. E. Willner, Z. Pan, and M. I. Hayee, “Major accomplishments in 2010 on optical fiber communications,” IEEE Photon. J.3(2), 320–324 (2011).
[CrossRef]

Wong, K. K.-Y.

B. P.-P. Kuo, P. C. Chui, and K. K.-Y. Wong, “A comprehensive study on crosstalk suppression techniques in fiber optical parametric amplifier by modulation format,” IEEE J. Sel. Top. Quantum Electron.14(3), 659–665 (2008).
[CrossRef]

Yamamoto, Y.

Yu, M.

IEEE J. Sel. Top. Quantum Electron. (3)

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron.18(2), 1016–1032 (2012).
[CrossRef]

B. P.-P. Kuo, P. C. Chui, and K. K.-Y. Wong, “A comprehensive study on crosstalk suppression techniques in fiber optical parametric amplifier by modulation format,” IEEE J. Sel. Top. Quantum Electron.14(3), 659–665 (2008).
[CrossRef]

N. E. Dahdah, D. S. Govan, M. Jamshidifar, N. J. Doran, and M. E. Marhic, “Fiber optical parametric amplifier performance in a 1-Tb/s DWDM communication system,” IEEE J. Sel. Top. Quantum Electron.18(2), 950–957 (2012).
[CrossRef]

IEEE Photon. J. (1)

A. E. Willner, Z. Pan, and M. I. Hayee, “Major accomplishments in 2010 on optical fiber communications,” IEEE Photon. J.3(2), 320–324 (2011).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

M. Jamshidifar, A. Vedadi, and M. E. Marhic, “Reduction of four-wave mixing crosstalk in a short fiber optical parametric amplifier,” IEEE Photon. Technol. Lett.21(17), 1244–1246 (2009).
[CrossRef]

H. Haus, “The noise figure of optical amplifiers,” IEEE Photon. Technol. Lett.10(11), 1602–1604 (1998).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (1)

Nat. Photonics (1)

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5(7), 430–436 (2011).
[CrossRef]

Opt. Commun. (4)

P.- Hu and J. L. Blows, “Four-wave mixing crosstalk in optical fibre parametric amplifiers with orthogonal pumps,” Opt. Commun.250(4-6), 421–427 (2005).
[CrossRef]

J. M. C. Boggio, J. D. Marconi, and H. L. Fragnito, “Crosstalk in double-pumped fiber optic parametric amplifiers for wavelength division multiplexing systems,” Opt. Commun.259(1), 94–103 (2006).
[CrossRef]

J. M. C. Boggio, E. A. M. Fagotto, M. E. Marhic, F. A. Callegari, and H. L. Fragnito “Amplification of 12 × 10 Gb/s WDM signals with negligible FWM crosstalk in a double-pumped fiber optical parametric amplifier,” Opt. Commun.280(2), 468–471 (2007).
[CrossRef]

J. L. Blows, “Design strategy for controlling four-wave mixing-induced crosstalk between channels in a fibre optical parametric amplifier,” Opt. Commun.236(1-3), 115–122 (2004).
[CrossRef]

Opt. Express (2)

Phys. Rev. D Part. Fields (1)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields26(8), 1817–1839 (1982).
[CrossRef]

Rep. Prog. Phys. (1)

U. Leonhardt, “Quantum physics of simple optical instruments,” Rep. Prog. Phys.66(7), 1207–1249 (2003).
[CrossRef]

Other (6)

B. Corcoran, S. L. I. Olsson, C. Lundström, M. Karlsson, and P. Andrekson, “Phase-sensitive optical pre-amplifier implemented in an 80 km DQPSK link,” in Proceedings of OFC/NFOEC 2012, Los Angeles, CA, March 6–8, 2012, paper PDP5A.4.

P. A. Andrekson, C. Lundstrom, and Z. Tong, “Phase-sensitive fiber-optic parametric amplifiers and their applications,” in Proceedings of ECOC2010, 19–23 September 2010, Turin, Italy, We.6.E.1.

D. Qian, M.-F. Huang, E. Ip, Y.-K. Huang, Y. Shao, J. Hu, and T. Wang, “101.7-Tb/s (370×294-Gb/s) PDM-128QAM-OFDM transmission over 3x55-km SSMF using pilot-based phase noise mitigation,” in Proceedings of OFC/NFOEC 2011, Anaheim, CA, March 6–10, 2011, paper PDPB5.

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge University Press, 2007).

M. E. Marhic, “Hybrid phase-sensitive and phase-insensitive amplifiers for optical communication,” in Proceedings of 17th OECC, Busan, Korea, July 2–6 2012.

H. Friis, “Noise figures of radio receivers,” Proc. I.R.E. 32, 419–422 (1944).

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Figures (2)

Fig. 1
Fig. 1

PSA-PIA combination. The dashed line is for the idler.

Fig. 2
Fig. 2

WC-PIA combination.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

N ^ d = d ^ d ^ =( A * a ^ + C * c ^ )( A a ^ +C c ^ )= | A | 2 N ^ a + | C | 2 ( 1+ N ^ c )+ A * C a ^ c ^ + C * A a ^ c ^
N d = N ^ d = | A | 2 N a + | C | 2
N ^ d 2 =( | A | 2 a ^ a ^ + | C | 2 c ^ c ^ + A * C a ^ c ^ + C * A a ^ c ^ )( | A | 2 a ^ a ^ + | C | 2 c ^ c ^ + A * C a ^ c ^ + C * A a ^ c ^ ) = | A | 4 a ^ a ^ a ^ a ^ + | C | 4 c ^ c ^ c ^ c ^ + ( A * C ) 2 a ^ a ^ c ^ c ^ + ( A C * ) 2 a ^ a ^ c ^ c ^ +2 | AC | 2 a ^ a ^ c ^ c ^ + | A | 2 A * C( a ^ a ^ a ^ c ^ + a ^ a ^ a ^ c ^ )+ | A | 2 C * A( a ^ a ^ a ^ c ^ + a ^ a ^ a ^ c ^ ) + | C | 2 A * C( a ^ c ^ c ^ c ^ + a ^ c ^ c ^ c ^ )+ | C | 2 A C * ( a ^ c ^ c ^ c ^ + a ^ c ^ c ^ c ^ ) + | AC | 2 ( a ^ a ^ c ^ c ^ + a ^ a ^ c ^ c ^ )
( N ^ d ) 2 = | A | 4 ( N a 2 + N a )+ | C | 4 +2 | AC | 2 N a + | AC | 2 ( N a +1 ) = | A | 4 N a 2 +( | A | 4 +3 | AC | 2 ) N a + | AC | 2 + | C | 4 .
N d 2 = ( | A | 2 N a + | C | 2 ) 2 = | A | 4 N a 2 + | C | 4 +2 | AC | 2 N a .
( Δ N d ) 2 =( | A | 4 + | AC | 2 ) N a + | AC | 2 = | A | 2 [ ( | A | 2 + | C | 2 ) N a + | C | 2 ].
F= SN R in SN R out = ( N a N d ) 2 ( Δ N d ) 2 ( Δ N a ) 2 = ( N a | A | 2 N a + | C | 2 i ) 2 | A | 2 [ ( | A | 2 + | C | 2 ) N a + | C | 2 ] N a = N a | A | 2 ( | A | 2 + | C | 2 ) N a + | C | 2 ( | A | 2 N a + | C | 2 ) 2 = N a G 1 G 2 ( G 1 G 2 + G 2 1 ) N a + G 2 1 ( G 1 G 2 N a + G 2 1 ) 2
FF'= ( G 1 G 2 + G 2 1 ) ( G 1 G 2 ) =1+ 1 G 1 1 G 1 G 2 .
N ^ c = c ^ c ^ =( U ^ + V ^ + W ^ )( U ^ + V ^ + W ^ ) = U ^ U ^ + U ^ V ^ + U ^ W ^ + V ^ U ^ + V ^ V ^ + V ^ W ^ + W ^ U ^ + W ^ V ^ + W ^ W ^ ..
N ^ c 2 = U ^ U ^ + U ^ V ^ + U ^ W ^ + V ^ U ^ + V ^ V ^ + V ^ W ^ + W ^ U ^ + W ^ V ^ + W ^ W ^ = U ^ U ^ U ^ U ^ + U ^ U ^ W ^ W ^ + U ^ V ^ V ^ U ^ + W ^ U ^ U ^ W ^ + W ^ V ^ V ^ W ^ + W ^ W ^ U ^ U ^ + W ^ W ^ W ^ W ^ =A a ^ a ^ a ^ a ^ +2(B+C) a ^ a ^ +D a ^ a ^ +E+H
N ^ c 2 =A N a 2 + N a (3A+2B+C+D)+A+2B+C+E+H.
N ^ c = U ^ U ^ + W ^ W ^ = G i G 2 ( N a +1)+ G 2 1,
N ^ c 2 =A N a 2 +2(A+B) N a +A+2B+H.
( Δ N c ) 2 =( A+C+D ) N a +C+E
F= ( A+C+D ) N a 2 +( C+E ) N a A N a 2 +2(A+B) N a +A+2B+H .
F=1+ C+D A =2+ 2 G i 1 G i G 2

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