Abstract

The quantum-limited noise figure (NF) for a general combination of linear optical elements is obtained in a compact form. The entries in this expression are obtained from the classical input–output transmission coefficient of the signal, and of the various transmission coefficients from the noise inputs to the output. The same result is also obtained by means of a semiclassical derivation. The linear elements can be: population-inversion amplifiers, parametric amplifiers and/or wavelength converters, beam splitters, lossy fibers, etc. They form a network with a simple directed graph (no loops). We identify a class of networks for which Friis’ formula for concatenated elements holds. We verify the general formula for a variety of well-known cases. We also verify that the NF for an optical communication link periodically amplified by nondegenerate phase-sensitive parametric amplifiers (PSAs) is only about half of that for a similar system using degenerate PSAs.

© 2013 Optical Society of America

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References

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  1. H. Haus, “The noise figure of optical amplifiers,” IEEE Photon. Technol. Lett. 10, 1602–1604 (1998).
    [CrossRef]
  2. E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (Wiley, 1994).
  3. Y. Yamamoto and K. Inoue, “Noise in amplifiers,” J. Lightwave Technol. 21, 2895–2915 (2003).
    [CrossRef]
  4. R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
    [CrossRef]
  5. M. Vasilyev, “Distributed phase-sensitive amplification,” Opt. Express 13, 7563–7571 (2005).
    [CrossRef]
  6. R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” IEEE J. Quantum Electron. 21, 766–773 (1985).
    [CrossRef]
  7. C. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005).
    [CrossRef]
  8. M. E. Marhic, “Noise figure of hybrid optical parametric amplifiers,” Opt. Express 20, 28752–28757 (2012).
    [CrossRef]
  9. W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes. I,” Phys. Rev. 124, 1646–1654 (1961).
    [CrossRef]
  10. 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]
  11. H. Friis, “Noise figures of radio receivers,” Proc. IRE 32, 419–422 (1944).
    [CrossRef]
  12. M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge University, 2007).
  13. U. Leonhardt, “Quantum physics of simple optical instruments,” Rep. Prog. Phys. 66, 1207–1249 (2003).
    [CrossRef]
  14. C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number correlations produced by parametric processes,” Opt. Express 18, 19792–19823 (2010).
    [CrossRef]
  15. Z. Tong, A. Bogris, C. Lundström, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier,” Opt. Express 18, 14820–14835 (2010).
    [CrossRef]
  16. Z. Tong, A. O. J. Wiberg, E. Myslivets, B. P. P. Kuo, N. Alic, and S. Radic, “Broadband parametric multicasting via four-mode phase-sensitive interaction,” Opt. Express 20, 19363–19373 (2012).
    [CrossRef]

2012 (2)

2010 (2)

2005 (3)

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

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
[CrossRef]

M. Vasilyev, “Distributed phase-sensitive amplification,” Opt. Express 13, 7563–7571 (2005).
[CrossRef]

2003 (2)

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

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

1998 (1)

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

1985 (1)

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

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]

1961 (1)

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes. I,” Phys. Rev. 124, 1646–1654 (1961).
[CrossRef]

1944 (1)

H. Friis, “Noise figures of radio receivers,” Proc. IRE 32, 419–422 (1944).
[CrossRef]

Alic, N.

Andrekson, P. A.

Bogris, A.

Desurvire, E.

E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (Wiley, 1994).

Devgan, P.

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
[CrossRef]

Friis, H.

H. Friis, “Noise figures of radio receivers,” Proc. IRE 32, 419–422 (1944).
[CrossRef]

Gordon, J. P.

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]

Grigoryan, V. S.

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
[CrossRef]

Haus, H.

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

Inoue, K.

Karlsson, M.

Kumar, P.

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
[CrossRef]

Kuo, B. P. P.

Leonhardt, U.

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

Loudon, R.

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

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]

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes. I,” Phys. Rev. 124, 1646–1654 (1961).
[CrossRef]

Lundström, C.

Marhic, M. E.

M. E. Marhic, “Noise figure of hybrid optical parametric amplifiers,” Opt. Express 20, 28752–28757 (2012).
[CrossRef]

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

McKinstrie, C.

McKinstrie, C. J.

Myslivets, E.

Radic, S.

Raymer, M. G.

Siegman, A. E.

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes. I,” Phys. Rev. 124, 1646–1654 (1961).
[CrossRef]

Tang, R.

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
[CrossRef]

Tong, Z.

Vasilyev, M.

Voss, P. L.

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
[CrossRef]

Walker, L. R.

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]

Wiberg, A. O. J.

Yamamoto, Y.

Yariv, A.

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes. I,” Phys. Rev. 124, 1646–1654 (1961).
[CrossRef]

Yu, M.

IEEE J. Quantum Electron. (1)

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

IEEE Photon. Technol. Lett. (2)

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

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
[CrossRef]

J. Lightwave Technol. (1)

Opt. Express (6)

Phys. Rev. (2)

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes. I,” Phys. Rev. 124, 1646–1654 (1961).
[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]

Proc. IRE (1)

H. Friis, “Noise figures of radio receivers,” Proc. IRE 32, 419–422 (1944).
[CrossRef]

Rep. Prog. Phys. (1)

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

Other (2)

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

E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (Wiley, 1994).

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Figures (11)

Fig. 1.
Fig. 1.

Schematic diagram of part of linear network whose NF can be calculated by Friis’ formula.

Fig. 2.
Fig. 2.

Schematic diagram of a PIA with a signal entering one input but only vacuum fluctuations present at the other input.

Fig. 3.
Fig. 3.

Schematic diagram of a degenerate (one-mode) PSA.

Fig. 4.
Fig. 4.

Nondegenerate PSA, with the same signal at both inputs. The expressions for the output fields are shown in the figure.

Fig. 5.
Fig. 5.

Nondegenerate PSA, with conjugate signals at the two inputs. The expressions for the output fields are shown in the figure.

Fig. 6.
Fig. 6.

Combination of a PIA followed by a PSA. The PIA serves as a copier, generating from the input signal an output signal and an idler with nearly the same amplitude. Some noise is added in the process, due to the open idler input of the PIA.

Fig. 7.
Fig. 7.

Schematic diagram of an amplifier based on population inversion. Both input fields are at the same frequency.

Fig. 8.
Fig. 8.

Schematic diagram of a free-space beamsplitter, represented by the diagonal line. Coupling between waves occurs only at the beamsplitter, not at the other intersections.

Fig. 9.
Fig. 9.

Schematic of an asymmetric frequency converter. The device is again represented by a diagonal line, by analogy with the preceding beamsplitter. However, this device couples frequencies, which is not the case with the beamsplitter.

Fig. 10.
Fig. 10.

Fiber transmission system periodically amplified by degenerate PSAs. T is the transmittance of each fiber segment, and G is the power gain of each PSA.

Fig. 11.
Fig. 11.

Fiber transmission system periodically amplified by nondegenerate PSAs. T is the transmittance of each fiber segment, and G is the power gain of each PSA.

Equations (68)

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b^=αa^+βa^+k=1N(λkv^k+μkv^k)=A^+V^,
N^b=b^b^=(A^+V^)(A^+V^)=A^A^+A^V^+V^A^+V^V^.
Nb=N^b=A^A^+V^V^=αα*Na+αβ*Na+α*βNa+ββ*(Na+1)+k=1Nμkμk*=|α+β|2Na+|β|2+D,
Nb2=|α+β|4Na2+2|α+β|2(|β|2+D)Na+(|β|2+D)2.
N^b2=(A^A^+A^V^+V^A^+V^V^)(A^A^+A^V^+V^A^+V^V^),
N^b2=A^A^A^A^+2A^A^V^V^+A^V^A^V^+A^V^V^A^+V^A^A^V^+V^A^V^A^+V^V^V^V^=A^A^A^A^+2A^A^V^V^+A^A^V^V^+A^A^V^V^+A^A^V^V^+A^A^V^V^+V^V^V^V^.
var(N^b)=N^b2N^b2=var(A^A^)+var(V^V^)+A^A^V^V^+A^A^V^V^+A^A^V^V^+A^A^V^V^.
A^A^A^A^=|α+β|4Na2+2|αβ|2+|β|4+(α2α*2+2αβα*2+2α*β*α2+8αα*ββ*+4α*β*β2+4αββ*2+3β2β*2)Na
A^A^2=|α+β|4Na2+2(αα*ββ*+α*β*β2+αββ*2+β2β*2)Na+|β|4.
var(A^A^)=[|α+β|4(αβ*α*β)2]Na+2|αβ|2.
A^A^=|α+β|2Na+|α|2,
A^A^=|α+β|2Na+|β|2,
A^A^=(α+β)2Na+αβ,
A^A^=A^A^*=(α*+β*)2Na+α*β*.
V^V^=k=1Nλkμk=B,
V^V^=k=1Nλk*μk*=B*,
V^V^=k=1N|λk|2=C,
V^V^=k=1N|μk|2=D,
V^V^V^V^=k=1N(|μk|4+2|λkμk|2)=E.
var(N^b)=[|α+β|4(αβ*α*β)2]Na+2|αβ|2+ED2+B((α*+β*)2Na+α*β*)+B*((α+β)2Na+αβ)+C(|α+β|2Na+|β|2)+D(|α+β|2Na+|α|2)=[|α+β|4(αβ*α*β)2+B(α*+β*)2+B*(α+β)2+(C+D)|α+β|2]Na+G,
F=SNRaSNRb=[|α+β|4(αβ*α*β)2+B(α*+β*)2+B*(α+β)2+(C+D)|α+β|2]Na2+GNa|α+β|4Na2+2|α+β|2(|β|2+D)Na+(|β|2+D)2.
F=1(αβ*α*β)2|α+β|4+B(α+β)2+B*(α*+β*)2+C+D|α+β|2
F=1(αβ*α*β)2|α+β|4+k=1N|λkα+β+μk*α*+β*|2,
Nb2=AA*2+2AA*VV*+VV*2.
Nb2=(AA*+AV*+A*V+V*V)2=(AA*)2+(AV*)2+(A*V)2+(V*V)2+2[AA*AV*+AA*A*V+AA*V*V+AV*A*V+AV*V*V+A*VV*V]
Nb2=(AA*)2+A2V*2+A*2V2+(V*V)2+4AA*VV*.
(ΔNb)2=(AA*)2+A2V*2+A*2V2+(V*V)2+4AA*VV*(AA*2+2AA*VV*+VV*2)=(ΔAA*)2+A2V*2+A*2V2+2AA*VV*+(V*V)2VV*2,
V2=k=1N(λkvk+μkvk*)l=1N(λlvl+μlvl*)=k=1Nλkμk,
VV*=k=1N(λkvk+μkvk*)l=1N(λl*vl*+μl*vl)=12k=1N(λkλk*+μkμk*).
(ΔNb)2(ΔAA*)2+k=1N[A2λk*μk*+A*2λkμk+AA*(λkλk*+μkμk*)]=(ΔAA*)2+k=1N|Aλk*+A*μk|2.
(AA*)2=|A0|4+A02α*β*+(A0*)2αβ+2|A0|2(αα*+ββ*)
AA*2=|A0|4+|A0|2(αα*+ββ*).
(ΔAA*)2=A02α*β*+(A0*)2αβ+|A0|2(αα*+ββ*)2.
(ΔAA*)2a02[(α+β)2α*β*+(α*+β*)2αβ+(α+β)(α*+β*)(αα*+ββ*)]=a02[|α+β|4(αβ*α*β)2].
|Aλk*+A*μk|2=a02|(α+β)λk*+(α+β)*μk|2.
(ΔNb)2a02[|α+β|4(αβ*α*β)2]+a02k=1N[|(α+β)λk*+(α+β)*μk|2].
Nb2=AA*2+2AA*VV*+VV*2a04[(α+β)(α+β)*]2=a04|α+β|4.
SNRb=(ΔNb)2Nb2a02|α+β|4|α+β|4(αβ*α*β)2+k=1N|(α+β)λk*+(α+β)*μk|2.
F=SNRaSNRb|α+β|4(αβ*α*β)2+k=1N|(α+β)λk*+(α+β)*μk|2|α+β|4=1(αβ*α*β)2|α+β|4+k=1N|λkα+β+μk*α*+β*|2.
F1=1(α+β)2k=1N|λk+μk*|2.
F1=1k=1NGkk=1N(Hkl=kNGl)=k=1N(Hkl=1k1Gl)=H1+H2G1+H3G1G2++HNG1G2GN1.
F1=k=1NFk1l=1k1Gl=F11+F21G1+F31G1G2++FN1G1G2GN1.
Fs=1+gi2gs2=1+gs21gs2=21gs2=21Gs,
Fi=1+gs2gi2=1+gi2+1gi2=2+1gi2=2+1Gi,
F=1|αβ|2(e2iφe2iφ)2||α|eiφ+|β|eiφ|4.
F=1+[sin(2φ)coth(2Φ)+cos(2φ)]2.
F=1|αβ|2(e2iφe2iφ)2||α|eiφ+|β|eiφ|4+|μ1|2||α|eiφ+|β|eiφ|2,
F=1+s2(c+s)2=1+[sinh(Φ)]2exp(2Φ)=1+14(1e2Φ)2.
F=1+s2(c+s)2=1+14(1e2Φ).
Fk1,l=SNR(ak1)SNR(al).
Fk1,l=SNR(ak1)SNR(al)=SNR(ak1)SNR(a0)SNR(a0)SNR(al)=F0,lF0,k1.
F1,2=F0,2F0,1=1+[tanh(Φ1+Φ2)]21+[tanh(Φ1)]2.
Fk=SNRakSNRbk.
Fk=1+λ2α2=1+1TT=1T,
bk=cos(Φ)ak+sin(Φ)a3k,
a1=T[cosh(Φ)a0+sinh(Φ)a0*],
a2=T[cosh(Φ)a1+sinh(Φ)a1*]=T{cosh(Φ)[cosh(Φ)a0+sinh(Φ)a0*]+sinh(Φ)[cosh(Φ)a0*+sinh(Φ)a0]}=T{[cosh2(Φ)+sinh2(Φ)]a0+2sinh(Φ)cosh(Φ)a0*}=T[cosh(2Φ)a1+sinh(2Φ)a1*],
aN=TN/2[cosh(NΦ)a0+sinh(NΦ)a0*].
λk=1TTT(Nk)/2cosh[(Nk)Φ]μk=1TTT(Nk)/2sinh[(Nk)Φ]
λk+μk=1TTT(Nk)/2{cosh[(Nk)Φ]+sinh[(Nk)Φ]}=1TTT(Nk)/2e(Nk)Φ=1TT.
F=1+k=1N[(λk+μk)2]=1+N(1T)T=1+N(G1),
M(Φ)=[cosh(Φ)sinh(Φ)sinh(Φ)cosh(Φ)].
[vk,Ns,svk,Ns,i]=1TT(Nk)/2MNk+1[vks0]=1TT(Nk)/2[cosh[(Nk+1)Φ]sinh[(Nk+1)Φ]sinh[(Nk+1)Φ]cosh[(Nk+1)Φ]][vks0].=1TT(Nk)/2[cosh[(Nk+1)Φ]sinh[(Nk+1)Φ]]vks.
(λk,s)2+(μk,i)2=(1T)TNk{cosh2[(Nk+1)Φ]+sinh2[(Nk+1)Φ]}=(1T)TNkcosh[2(Nk+1)Φ]=(1T)TNk(e2(Nk+1)Φ+e2(Nk+1)Φ2)=(1T)2[G+T2(Nk)+1].
F=1+14+k=1N[(λk,s)2+(μk,i)2]=54+(1T)2k=1N[G+T2(Nk)+1]=54+N2(G1)+(1T)2k=1NT2(Nk)+1=54+N2(G1)+1G2N2(G+1).
F54+N2(G1).
a^=a,a^=a*,a^a^=aa*,a^a^=aa*+1,a^a^=a2,a^a^=(a*)2a^a^a^=a3,a^a^a^=a2a*+2a,a^a^a^=a2a*+a,a^a^a^=a(a*)2+2a*,a^a^a^=a2a*,a^a^a^=a(a*)2+a*,a^a^a^=a(a*)2,a^a^a^=(a*)3
a^a^a^a^=a4,a^a^a^a^=a3a*+3a2,a^a^a^a^=a3a*+2a2,a^a^a^a^=(aa*)2+4aa*+2a^a^a^a^=a3a*+a2,a^a^a^a^=(aa*)2+3aa*+1,a^a^a^a^=(aa*)2+2aa*,a^a^a^a^=a(a*)3+3(a*)2,a^aaa=a3a*,a^a^a^a^=(aa*)2+2aa*,a^a^a^a^=(aa*)2+aa*,a^a^a^a^=a(a*)3+2(a*)2,a^a^a^a^=(aa*)2a^a^a^a^=a(a*)3+(a*)2,a^a^a^a^=a(a*)3,a^a^a^a^=(a*)4

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