Abstract

Four-wave mixing in the form of Bragg scattering (BS) has been predicted to enable quantum noise-less frequency conversion by analytic quantum approaches. Using a semi-classical description of quantum noise that accounts for loss and stimulated and spontaneous Raman scattering, which are not currently described in existing quantum approaches, we quantify the impacts of these effects on the conversion efficiency and on the quantum noise properties of BS in terms of an induced noise figure (NF). We give an approximate closed-form expression for the BS conversion efficiency that includes loss and stimulated Raman scattering, and we derive explicit expressions for the Raman-induced NF from the semi-classical approach used here. We find that Raman scattering induces a NF in the BS process that is comparable to the 3-dB NF associated with linear amplifiers.

© 2017 Optical Society of America

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References

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2016 (1)

2015 (1)

2013 (2)

2012 (5)

2011 (2)

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, 430–436 (2011).
[Crossref]

B. J. Eggleton, B. Luther-Davies, and K. Richardson, “Chalcogenide photonics,” Nat. Photonics 5, 141–148 (2011).

2010 (3)

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[Crossref] [PubMed]

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics‘,” Nat. Photonics 4, 535–544 (2010).
[Crossref]

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, 2884–2893 (2010).
[Crossref] [PubMed]

2008 (3)

2007 (1)

2006 (4)

2005 (3)

2004 (2)

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

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E,  70, 036604 (2004).
[Crossref]

2003 (1)

2002 (1)

1999 (1)

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photonics Technol. Lett. 11, 530–532 (1999).
[Crossref]

1996 (1)

1994 (1)

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

1989 (2)

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
[Crossref]

K. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[Crossref]

1970 (1)

R. H. Stolen, “Raman scattering and infrared absorption from low lying modes in vitreous SiO2, GeO2, and B2O3,” Phys. Chem. Glasses 11, 83–87 (1970).

1963 (1)

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

Agrawal, G. P.

Alibart, O.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
[Crossref] [PubMed]

Andrekson, P. A.

Baldi, P.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
[Crossref] [PubMed]

Begleris, I.

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. Photonics 5, 430–436 (2011).
[Crossref]

Blow, K.

K. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[Crossref]

Bogris, A.

Bromage, J.

Cantrell, C. D.

Centanni, J. C.

Chen, Y.

Choi, D.

Clark, A. S.

Clemmen, S.

A. Farsi, S. Clemmen, S. Ramelow, and A. L. Gaeta, “Low-noise quantum frequency translation of single photons,” in Conference on Laser-electronics and Optics, Vol. 2 of 2015 OSA Technical Digest Series (Optical Society of America, 2015), paper FM3A.4.

Coen, S.

Collins, M. J.

Croussore, K.

K. Croussore and G. Li, “Phase and amplitude regeneration of differential phase-shift keyed signals using phase-sensitive amplification,” IEEE J. Sel. Top. Quantum Electron. 14, 648–658 (2008).
[Crossref]

Demas, J.

Eggleton, B. J

Eggleton, B. J.

Farsi, A.

A. Farsi, S. Clemmen, S. Ramelow, and A. L. Gaeta, “Low-noise quantum frequency translation of single photons,” in Conference on Laser-electronics and Optics, Vol. 2 of 2015 OSA Technical Digest Series (Optical Society of America, 2015), paper FM3A.4.

Freude, W.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics‘,” Nat. Photonics 4, 535–544 (2010).
[Crossref]

Friis, S. M. M.

Gaeta, A. L.

A. Farsi, S. Clemmen, S. Ramelow, and A. L. Gaeta, “Low-noise quantum frequency translation of single photons,” in Conference on Laser-electronics and Optics, Vol. 2 of 2015 OSA Technical Digest Series (Optical Society of America, 2015), paper FM3A.4.

Gerry, C. C.

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge, 2005).

Gisin, N.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
[Crossref] [PubMed]

Gnauck, A. H.

Gordon, J. P.

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
[Crossref]

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

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. Photonics 5, 430–436 (2011).
[Crossref]

Halder, M.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
[Crossref] [PubMed]

Harvey, J. D.

Haus, H. A.

He, J.

Headley, C.

Hollenbeck, D.

Horak, P.

Hsieh, A. S. Y.

Inoue, K.

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

Jopson, R. M.

Judge, A. C.

Jung, Y.

Karlsson, M.

Kidorf, H.

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photonics Technol. Lett. 11, 530–532 (1999).
[Crossref]

Knight, P. L.

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge, 2005).

Kolesik, M.

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E,  70, 036604 (2004).
[Crossref]

Koos, C.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics‘,” Nat. Photonics 4, 535–544 (2010).
[Crossref]

Köprülü, K. G.

Kumar, P.

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, 598–609 (2006).
[Crossref]

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

Leng, L.

Leonhardt, R.

Leuthold, J.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics‘,” Nat. Photonics 4, 535–544 (2010).
[Crossref]

Li, G.

K. Croussore and G. Li, “Phase and amplitude regeneration of differential phase-shift keyed signals using phase-sensitive amplification,” IEEE J. Sel. Top. Quantum Electron. 14, 648–658 (2008).
[Crossref]

Lin, Q.

Lines, M. E.

Louisell, W. H.

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

Lundström, C.

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, 430–436 (2011).
[Crossref]

Luther-Davies, B.

Ma, M.

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photonics Technol. Lett. 11, 530–532 (1999).
[Crossref]

Madden, S. J.

Mägi, E. C.

McGuinness, H. J.

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[Crossref] [PubMed]

McKinstrie, C. J.

S. M. M. Friis, K. Rottwitt, and C. J. McKinstrie, “Raman and loss induced quantum noise in depleted fiber optical parametric amplifiers,” Opt. Express 21, 29320–29331 (2013).
[Crossref]

L. Mejling, C. J. McKinstrie, M. G. Raymer, and K. Rottwitt, “Quantum frequency translation by four-wave mixing in a fiber: low-conversion regime,” Opt. Express 20, 8367–8396 (2012).
[Crossref] [PubMed]

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A 85, 053829 (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. Photonics 5, 430–436 (2011).
[Crossref]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[Crossref] [PubMed]

A. H. Gnauck, R. M. Jopson, C. J. McKinstrie, J. C. Centanni, and S. Radic, “Demonstration of low-noise frequency conversion by Bragg scattering in a fiber,” Opt. Express 14, 8989–8994 (2006)
[Crossref] [PubMed]

D. Méchin, R. Provo, J. D. Harvey, and C. J. McKinstrie, “180-nm wavelength conversion based on Bragg scattering in an optical fiber,” Opt. Express 14, 8995–8999 (2006).
[Crossref] [PubMed]

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

C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005).
[Crossref] [PubMed]

Méchin, D.

Mejling, L.

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A 85, 053829 (2012).
[Crossref]

L. Mejling, C. J. McKinstrie, M. G. Raymer, and K. Rottwitt, “Quantum frequency translation by four-wave mixing in a fiber: low-conversion regime,” Opt. Express 20, 8367–8396 (2012).
[Crossref] [PubMed]

Moloney, J. V.

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E,  70, 036604 (2004).
[Crossref]

Murdoch, S. G.

Nissov, M.

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photonics Technol. Lett. 11, 530–532 (1999).
[Crossref]

Oda, S.

Parmigiani, F.

Petropoulos, P.

Poletti, F.

Provo, R.

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. Photonics 5, 430–436 (2011).
[Crossref]

Rabarijaona, E.

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photonics Technol. Lett. 11, 530–532 (1999).
[Crossref]

Radic, S.

Ramachandran, S.

Ramelow, S.

A. Farsi, S. Clemmen, S. Ramelow, and A. L. Gaeta, “Low-noise quantum frequency translation of single photons,” in Conference on Laser-electronics and Optics, Vol. 2 of 2015 OSA Technical Digest Series (Optical Society of America, 2015), paper FM3A.4.

Raymer, M. G.

L. Mejling, C. J. McKinstrie, M. G. Raymer, and K. Rottwitt, “Quantum frequency translation by four-wave mixing in a fiber: low-conversion regime,” Opt. Express 20, 8367–8396 (2012).
[Crossref] [PubMed]

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A 85, 053829 (2012).
[Crossref]

M. G. Raymer and K. Srinivasan, “Manipulating the color and shape of single photons,” Phys. Today 65, 32–37 (2012).
[Crossref]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[Crossref] [PubMed]

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

C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005).
[Crossref] [PubMed]

Richardson, D. J.

Richardson, K.

B. J. Eggleton, B. Luther-Davies, and K. Richardson, “Chalcogenide photonics,” Nat. Photonics 5, 141–148 (2011).

Rishøj, L.

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Figures (6)

Fig. 1
Fig. 1 Sketches of selected frequency configurations performing down-conversion from signal (s) to idler (i); the two pumps, p and q, need not have equal magnitudes. δ is the frequency separation between the two side-bands in either of (a)–(d), and Δ is the frequency separation between the zero-dispersion frequency and the closest wave component on either side. By changing ωsωi and ωpωq, up-conversion is achieved
Fig. 2
Fig. 2 Rate of SpRS versus real frequency shift νij = Ωnm/2π from a pump p or q for temperatures T = 300 K (solid blue curve) and T = 77 K (dashed red curve). The inset shows the Raman gain coefficient g R ( n m ) for silica core fibers (solid black curve) and the phonon equilibrium numbers n T ( n m ) for the same temperatures as in the main plot (with same line styles).
Fig. 3
Fig. 3 (i) A coherent state ensemble visualized in a phase-space diagram, (ii) how the ensemble is affected by loss in the classical equations, and (iii) the effect of adding loss fluctuations.
Fig. 4
Fig. 4 (i) CE versus fiber length with optimal phase matching and without Raman scattering. The red lines visually illustrate the effects of attenuation on the CE. (ii) The same for the conversion NF. In the simulation, α = 1 dB/km, Pp = Pq = 0.2 W, γs = 9.89 (W km)−1, β4 = 0 ps4/km, δ/2π = Δ/2π = 1 THz, and Δz = 20 m.
Fig. 5
Fig. 5 (i) CE versus fiber length for Case (a) (top) and (b) (bottom); both simulation (dotted blue) and analytic result of Eq. (15) are shown; the legend applies to both plots, and the thick red line is the Raman amplification term. (ii) The same for the NF; the analytic Raman NF (solid black) is Eq. (27) and Eq. (28) for the top and bottom plots, respectively; the dashed green line is the Raman NF at 0 K. The parameters are the same as in Fig. 4, but α = 0 dB/km and T = 300 K.
Fig. 6
Fig. 6 (i) and (ii) CE of Eq. (15) of Cases (a) and (b) of Fig. 1, respectively; (iii) and (iv) NFs of Eqs. (27) and (28), respectively, color scales are in dB.

Equations (36)

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E n ( z ) z = i ω n e i β n z 4 N n F n * ( x , y ) P n ( 3 ) ( r ) d x d y ,
E ( r , t ) = 1 2 m F m ( x , y ) N m E n ( z , t ) e i β m z i ω m t + c . c . ,
1 4 [ F m * × H m + F m × H m * ] z ^ d x d y = N m 2
N m 2 = c 0 n m eff 2 F ( x , y ) 2 d x d y ,
P ( 3 ) ( r , t ) = 1 2 m P m ( 3 ) ( r ) e i ω m t + c . c . .
P ( 3 ) ( r , t ) = 0 R ( 3 ) ( t τ 1 , t τ 2 , t τ 3 ) E ( r , τ 1 ) E ( r , τ 2 ) E ( r , τ 3 ) d τ 1 d τ 2 d τ 3 ,
R ( 3 ) ( t 1 , t 2 , t 3 ) = χ ( 3 ) ( [ 1 f R ] δ ( t 1 ) + 3 2 f R h R ( t 1 ) ) δ ( t 1 t 2 ) δ ( t 3 )
E 1 z = α 2 E 1 + i ω 1 n 2 c A eff [ | E 1 | 2 E 1 + ( 2 f R ) E 1 n = 2 4 | E n | 2 + 2 ( 1 f R ) E 2 E 3 E 4 * e i Δ β z + f R E 1 n = 2 4 h ˜ R ( Ω n 1 ) | E n | 2 ]
h R ( t ) = j = 1 13 b j ω j exp ( η j t ) exp ( Γ j 2 t 2 / 4 ) sin ( ω j t ) Θ ( t ) ,
Δ β β 4 12 δ ( 2 Δ + δ ) ( 2 Δ 2 + 2 Δ δ + δ 2 ) .
E p ( z ) = P p exp ( i γ p [ P p + ( 2 f R ) P q ] z ) ,
E q ( z ) = P q exp ( i γ q [ P q + ( 2 f R ) P p ] z ) ,
CE ( z ) = | E i ( z ) | 2 | E s ( 0 ) | 2 = η i 2 ( κ / 2 ) 2 + η i η s sin 2 ( g z )
κ = Δ β ± 1 ( 1 f R ) ( γ p + γ q ) ( P q P p ) ( 1 f R ) ( γ i P q γ s P p ) ,
CE = | E i ( z ) | 2 | E s ( 0 ) | 2 = η i 2 μ 2 exp [ ( f s + f i ) z eff ] exp ( α z ) sin 2 ( μ z eff )
η i 2 μ 2 exp ( ( f s + f i α ) z ) sin 2 ( μ z ) ,
P ASE , S ( z ) = ω m B 0 ( n T ( n m ) + 1 ) g R ( n m ) P n z ,
P ASE , aS ( z ) = ω m B 0 n T n m | g R ( n m ) | P n z .
SNR = | A ens | 2 2 Var ( | A ens | 2 ) ,
A ens = x 0 + δ x + i ( p 0 + δ p ) ,
δ a loss = 0 ,
δ a loss 2 = ω B 0 [ 1 exp ( α Δ z ) ] / 4 ω B 0 α Δ z / 4 ,
δ a Raman , S = 0 ,
δ a Raman , S 2 [ g R ( n m ) P n Δ z ( n T ( n m ) + 1 ) / 2 g R ( n m ) P n Δ z / 4 ] ω m B 0 ,
δ a Raman , aS = 0 ,
δ a Raman , aS 2 [ | g R ( n m ) | P n Δ z n T ( n m ) / 2 + | g R ( n m ) | P n Δ z / 4 ] ω m B 0 .
NF S = 1 G + 2 [ G 1 ] ( n T ( Ω pi ) + 1 ) G ,
NF aS = 1 D + 2 [ 1 D ] n T ( Ω pi ) D ,
A out = G 1 / 2 [ x 0 + δ x + i ( p 0 + δ p ) ] + δ a 1 + i δ a 2 ,
NF = SNR in SNR out 1 + 4 δ a 2 G ω B 0 ,
| A S , out | 2 = G ( x 0 2 + p 0 2 ) + ω S B 0 G / 2 + 2 δ a S 2
| A aS , out | 2 = D ( x 0 2 + p 0 2 ) + ω aS B 0 D / 2 + 2 δ a S 2 ,
δ a S 2 = ( [ G 1 ] ( n T + 1 ) / 2 [ G 1 ] / 4 ) ω S B 0
δ a aS 2 = ( [ 1 D ] n T / 2 + [ 1 D ] / 4 ) ω aS B 0 ,
NF S = 1 G + 2 [ G 1 ] ( n T + 1 ) G 2 ( n T + 1 )
NF aS = 1 D + 2 [ 1 D ] n T D ( 1 + 2 n T ) / D .

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