Abstract

Traditionally, by means of full quantum theory, we present the intensity noise transfer function of an Er-doped fiber laser, on the basis of which we analyze the spectrum of the intensity noise. Our theoretical results are in agreement with the existing experiment results. This model explains not only how the noise is produced, but also how the spontaneous emission and dipole fluctuations have an effect on the output noise, which cannot be explained via rate equation theory. We analyze the physical sources of various contributions to the noise spectrum as well. The simulation results show that the noise of the Er-doped fiber laser mainly consists of the vacuum noise resulting from the output coupling, dipole fluctuation noise, the pump source intensity noise, and the spontaneous emission from the upper level to the ground level, which provides the theoretical basis for noise suppression. Compared to the solid laser, the Er-doped fiber laser shows lower resonant relaxation oscillation frequency.

© 2013 Optical Society of America

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References

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  1. G. A. Cranch, M. A. Englund, and C. K. Kirkendall, “Intensity noise characteristics of erbium-doped distributed-feedback fiber lasers,” IEEE J. Quantum Electron. 39, 1579–1586 (2003).
    [CrossRef]
  2. T. C. Ralph, C. C. Harb, and H. Bachor, “Intensity noise of injection locked laser: quantum theory using a linearized input–output method,” Phys. Rev. A 54, 4359–4369 (1996).
    [CrossRef]
  3. C. C. Harb, T. C. Ralph, and E. H. Huntingon, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
    [CrossRef]
  4. C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
    [CrossRef]
  5. Z. Meng, G. Stewart, and G. Whitenett, “Stable single-mode operation of a narrow-linewidth, linearly, polarized, erbium-fiber ring laser using a saturable absorber,” J. Lightwave Technol. 24, 2179–2183 (2006).
    [CrossRef]
  6. S. Sanders, J. W. Dawson, N. Park, and K. J. Vahala, “Measurement of intensity noise of a broadly tunable, erbium-doped fiber ring laser, relative to standard quantum limit,” Appl. Phys. Lett. 60, 2583–2585 (1992).
    [CrossRef]
  7. W. K. Marshall, B. Crosignani, and A. Yariv, “Laser phase noise to intensity noise conversion by lowest-order group-velocity dispersion in optical fiber: exact theory,” Opt. Lett. 25, 165–167 (2000).
    [CrossRef]
  8. H. L. An, E. Y. B. Pun, X. Z. Lin, and H. D. Liu, “Effects of ion-clusters on the intensity noise of heavily erbium-doped fiber lasers,” IEEE Photon. Technol. Lett. 11, 803–805(1999).
    [CrossRef]
  9. K. Croussore, D. C. Kilper, and M. Y. A. Raja, “Polarization-resolved intensity noise in erbium-ytterbium codoped fiber lasers,” J. Opt. Soc. Am. B 21, 865–870 (2004).
    [CrossRef]
  10. L. Ma, Y. Hu, S. Xiong, Z. Meng, and Z. Hu, “Intensity noise and relaxation oscillation of a fiber-laser sensor array integrated in a single fiber,” Opt. Lett. 35, 1795–1797 (2010).
    [CrossRef]
  11. S. Sanders, N. Park, J. W. Dawson, and K. J. Vahala, “Reduction of the intensity noise from an erbium-doped fiber laser to the standard quantum limit by intracavity spectral filtering,” Appl. Phys. Lett. 61, 1889–1891 (1992).
    [CrossRef]
  12. C. Spiegelberg, J. Geng, Y. Hu, Y. Kaneda, S. Jiang, and N. Peyghambarian, “Low-noise narrow-linewidth fiber laser at 1550 nm,” J. Lightwave Technol. 22, 57–62 (2004).
    [CrossRef]
  13. S. Pradhan, G. E. Town, D. Wilson, and K. J. Grant, “Intensity noise reduction in a multiwavelength distributed Bragg reflector fiber laser,” Opt. Lett. 31, 2963–2965 (2006).
    [CrossRef]
  14. M. B. Gray, J. H. Chow, K. McKenzie, and D. E. McClelland, “Using a passive fiber ring cavity to generate shot-noise-limited laser light for low-power quantum optics applications,” IEEE Photon. Technol. Lett. 19, 1063–1065 (2007).
    [CrossRef]

2010 (1)

2007 (1)

M. B. Gray, J. H. Chow, K. McKenzie, and D. E. McClelland, “Using a passive fiber ring cavity to generate shot-noise-limited laser light for low-power quantum optics applications,” IEEE Photon. Technol. Lett. 19, 1063–1065 (2007).
[CrossRef]

2006 (2)

2004 (2)

2003 (1)

G. A. Cranch, M. A. Englund, and C. K. Kirkendall, “Intensity noise characteristics of erbium-doped distributed-feedback fiber lasers,” IEEE J. Quantum Electron. 39, 1579–1586 (2003).
[CrossRef]

2000 (1)

1999 (1)

H. L. An, E. Y. B. Pun, X. Z. Lin, and H. D. Liu, “Effects of ion-clusters on the intensity noise of heavily erbium-doped fiber lasers,” IEEE Photon. Technol. Lett. 11, 803–805(1999).
[CrossRef]

1996 (2)

T. C. Ralph, C. C. Harb, and H. Bachor, “Intensity noise of injection locked laser: quantum theory using a linearized input–output method,” Phys. Rev. A 54, 4359–4369 (1996).
[CrossRef]

C. C. Harb, T. C. Ralph, and E. H. Huntingon, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef]

1992 (2)

S. Sanders, J. W. Dawson, N. Park, and K. J. Vahala, “Measurement of intensity noise of a broadly tunable, erbium-doped fiber ring laser, relative to standard quantum limit,” Appl. Phys. Lett. 60, 2583–2585 (1992).
[CrossRef]

S. Sanders, N. Park, J. W. Dawson, and K. J. Vahala, “Reduction of the intensity noise from an erbium-doped fiber laser to the standard quantum limit by intracavity spectral filtering,” Appl. Phys. Lett. 61, 1889–1891 (1992).
[CrossRef]

1985 (1)

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[CrossRef]

An, H. L.

H. L. An, E. Y. B. Pun, X. Z. Lin, and H. D. Liu, “Effects of ion-clusters on the intensity noise of heavily erbium-doped fiber lasers,” IEEE Photon. Technol. Lett. 11, 803–805(1999).
[CrossRef]

Bachor, H.

T. C. Ralph, C. C. Harb, and H. Bachor, “Intensity noise of injection locked laser: quantum theory using a linearized input–output method,” Phys. Rev. A 54, 4359–4369 (1996).
[CrossRef]

Chow, J. H.

M. B. Gray, J. H. Chow, K. McKenzie, and D. E. McClelland, “Using a passive fiber ring cavity to generate shot-noise-limited laser light for low-power quantum optics applications,” IEEE Photon. Technol. Lett. 19, 1063–1065 (2007).
[CrossRef]

Collett, M. J.

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[CrossRef]

Cranch, G. A.

G. A. Cranch, M. A. Englund, and C. K. Kirkendall, “Intensity noise characteristics of erbium-doped distributed-feedback fiber lasers,” IEEE J. Quantum Electron. 39, 1579–1586 (2003).
[CrossRef]

Crosignani, B.

Croussore, K.

Dawson, J. W.

S. Sanders, J. W. Dawson, N. Park, and K. J. Vahala, “Measurement of intensity noise of a broadly tunable, erbium-doped fiber ring laser, relative to standard quantum limit,” Appl. Phys. Lett. 60, 2583–2585 (1992).
[CrossRef]

S. Sanders, N. Park, J. W. Dawson, and K. J. Vahala, “Reduction of the intensity noise from an erbium-doped fiber laser to the standard quantum limit by intracavity spectral filtering,” Appl. Phys. Lett. 61, 1889–1891 (1992).
[CrossRef]

Englund, M. A.

G. A. Cranch, M. A. Englund, and C. K. Kirkendall, “Intensity noise characteristics of erbium-doped distributed-feedback fiber lasers,” IEEE J. Quantum Electron. 39, 1579–1586 (2003).
[CrossRef]

Gardiner, C. W.

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[CrossRef]

Geng, J.

Grant, K. J.

Gray, M. B.

M. B. Gray, J. H. Chow, K. McKenzie, and D. E. McClelland, “Using a passive fiber ring cavity to generate shot-noise-limited laser light for low-power quantum optics applications,” IEEE Photon. Technol. Lett. 19, 1063–1065 (2007).
[CrossRef]

Harb, C. C.

C. C. Harb, T. C. Ralph, and E. H. Huntingon, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef]

T. C. Ralph, C. C. Harb, and H. Bachor, “Intensity noise of injection locked laser: quantum theory using a linearized input–output method,” Phys. Rev. A 54, 4359–4369 (1996).
[CrossRef]

Hu, Y.

Hu, Z.

Huntingon, E. H.

C. C. Harb, T. C. Ralph, and E. H. Huntingon, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef]

Jiang, S.

Kaneda, Y.

Kilper, D. C.

Kirkendall, C. K.

G. A. Cranch, M. A. Englund, and C. K. Kirkendall, “Intensity noise characteristics of erbium-doped distributed-feedback fiber lasers,” IEEE J. Quantum Electron. 39, 1579–1586 (2003).
[CrossRef]

Lin, X. Z.

H. L. An, E. Y. B. Pun, X. Z. Lin, and H. D. Liu, “Effects of ion-clusters on the intensity noise of heavily erbium-doped fiber lasers,” IEEE Photon. Technol. Lett. 11, 803–805(1999).
[CrossRef]

Liu, H. D.

H. L. An, E. Y. B. Pun, X. Z. Lin, and H. D. Liu, “Effects of ion-clusters on the intensity noise of heavily erbium-doped fiber lasers,” IEEE Photon. Technol. Lett. 11, 803–805(1999).
[CrossRef]

Ma, L.

Marshall, W. K.

McClelland, D. E.

M. B. Gray, J. H. Chow, K. McKenzie, and D. E. McClelland, “Using a passive fiber ring cavity to generate shot-noise-limited laser light for low-power quantum optics applications,” IEEE Photon. Technol. Lett. 19, 1063–1065 (2007).
[CrossRef]

McKenzie, K.

M. B. Gray, J. H. Chow, K. McKenzie, and D. E. McClelland, “Using a passive fiber ring cavity to generate shot-noise-limited laser light for low-power quantum optics applications,” IEEE Photon. Technol. Lett. 19, 1063–1065 (2007).
[CrossRef]

Meng, Z.

Park, N.

S. Sanders, J. W. Dawson, N. Park, and K. J. Vahala, “Measurement of intensity noise of a broadly tunable, erbium-doped fiber ring laser, relative to standard quantum limit,” Appl. Phys. Lett. 60, 2583–2585 (1992).
[CrossRef]

S. Sanders, N. Park, J. W. Dawson, and K. J. Vahala, “Reduction of the intensity noise from an erbium-doped fiber laser to the standard quantum limit by intracavity spectral filtering,” Appl. Phys. Lett. 61, 1889–1891 (1992).
[CrossRef]

Peyghambarian, N.

Pradhan, S.

Pun, E. Y. B.

H. L. An, E. Y. B. Pun, X. Z. Lin, and H. D. Liu, “Effects of ion-clusters on the intensity noise of heavily erbium-doped fiber lasers,” IEEE Photon. Technol. Lett. 11, 803–805(1999).
[CrossRef]

Raja, M. Y. A.

Ralph, T. C.

T. C. Ralph, C. C. Harb, and H. Bachor, “Intensity noise of injection locked laser: quantum theory using a linearized input–output method,” Phys. Rev. A 54, 4359–4369 (1996).
[CrossRef]

C. C. Harb, T. C. Ralph, and E. H. Huntingon, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef]

Sanders, S.

S. Sanders, J. W. Dawson, N. Park, and K. J. Vahala, “Measurement of intensity noise of a broadly tunable, erbium-doped fiber ring laser, relative to standard quantum limit,” Appl. Phys. Lett. 60, 2583–2585 (1992).
[CrossRef]

S. Sanders, N. Park, J. W. Dawson, and K. J. Vahala, “Reduction of the intensity noise from an erbium-doped fiber laser to the standard quantum limit by intracavity spectral filtering,” Appl. Phys. Lett. 61, 1889–1891 (1992).
[CrossRef]

Spiegelberg, C.

Stewart, G.

Town, G. E.

Vahala, K. J.

S. Sanders, N. Park, J. W. Dawson, and K. J. Vahala, “Reduction of the intensity noise from an erbium-doped fiber laser to the standard quantum limit by intracavity spectral filtering,” Appl. Phys. Lett. 61, 1889–1891 (1992).
[CrossRef]

S. Sanders, J. W. Dawson, N. Park, and K. J. Vahala, “Measurement of intensity noise of a broadly tunable, erbium-doped fiber ring laser, relative to standard quantum limit,” Appl. Phys. Lett. 60, 2583–2585 (1992).
[CrossRef]

Whitenett, G.

Wilson, D.

Xiong, S.

Yariv, A.

Appl. Phys. Lett. (2)

S. Sanders, J. W. Dawson, N. Park, and K. J. Vahala, “Measurement of intensity noise of a broadly tunable, erbium-doped fiber ring laser, relative to standard quantum limit,” Appl. Phys. Lett. 60, 2583–2585 (1992).
[CrossRef]

S. Sanders, N. Park, J. W. Dawson, and K. J. Vahala, “Reduction of the intensity noise from an erbium-doped fiber laser to the standard quantum limit by intracavity spectral filtering,” Appl. Phys. Lett. 61, 1889–1891 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. A. Cranch, M. A. Englund, and C. K. Kirkendall, “Intensity noise characteristics of erbium-doped distributed-feedback fiber lasers,” IEEE J. Quantum Electron. 39, 1579–1586 (2003).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

H. L. An, E. Y. B. Pun, X. Z. Lin, and H. D. Liu, “Effects of ion-clusters on the intensity noise of heavily erbium-doped fiber lasers,” IEEE Photon. Technol. Lett. 11, 803–805(1999).
[CrossRef]

M. B. Gray, J. H. Chow, K. McKenzie, and D. E. McClelland, “Using a passive fiber ring cavity to generate shot-noise-limited laser light for low-power quantum optics applications,” IEEE Photon. Technol. Lett. 19, 1063–1065 (2007).
[CrossRef]

J. Lightwave Technol. (2)

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

Opt. Lett. (3)

Phys. Rev. A (3)

T. C. Ralph, C. C. Harb, and H. Bachor, “Intensity noise of injection locked laser: quantum theory using a linearized input–output method,” Phys. Rev. A 54, 4359–4369 (1996).
[CrossRef]

C. C. Harb, T. C. Ralph, and E. H. Huntingon, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef]

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Energy level scheme of the active atoms.

Fig. 2.
Fig. 2.

Comparison between the (a) theoretical results and (b) experimental results [5].

Fig. 3.
Fig. 3.

Noise spectrum with all kind of sources.

Fig. 4.
Fig. 4.

Relationship between intensity noise and pump noise.

Fig. 5.
Fig. 5.

Relationship between intensity noise and input power.

Fig. 6.
Fig. 6.

Relationship between intensity noise and output coupling ratio. CR, coupling ratio.

Fig. 7.
Fig. 7.

Relationship between intensity noise and cavity length.

Tables (1)

Tables Icon

Table 1. Parameters of Er-doped Fiber Laser

Equations (54)

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H1=ig(a^σ^12a^σ^12),
H2=igp(b^σ^13b^σ^13),
Hirrev=idω{κm/2π[A^m(ω)a^A^m(ω)a^]+κl/2π[A^l(ω)a^A^l(ω)a^]+κb/2π[B^(ω)b^B^(ω)b^]+γ/2π[C^(ω)σ^23C^(ω)σ^23]+γt/2π[C^t(ω)σ^12C^t(ω)σ^12]+γp/4π[C^P(ω)C^P(ω)](σ^2σ^1)+γQ/4π[C^Q(ω)C^Q(ω)](σ^3σ^1)},
δA^m(t)=(1/2π)dωexp[iω(tt0)]A^m0(ω),
δA^l(t)=1/2πdωexp[iω(tt0)]A^l0(ω),
B^(t)=1/2πdωexp[iω(tt0)]B^0(ω),
δC^t(t)=1/2πdωexp[iω(tt0)]C^t0(ω),
δC^(t)=1/2πdωexp[iω(tt0)]C^0(ω),
δC^Q(t)=1/2πdωexp[iω(tt0)]C^Q0(ω),
δC^P(t)=1/2πdωexp[iω(tt0)]C^P0(ω),
x^˙=ii=12[x^,Hi]j=17{[x^,a^j](γja^j2γjA^j)(γja^j2γjA^j)[x^,a^j]},
(a^1,a^2,a^3,a^4,a^5,a^6,a^7)(a^,a^,b^,σ^23,σ^12,σ^2σ^1,σ^3σ^1),
(A^1,A^2,A^3,A^4,A^5,A^6,A^7)(δA^m,δA^l,B^,δC^,δC^t,δC^P,δC^Q),
(γ1,γ2,γ3,γ4,γ5,γ6,γ7)(κm,κl,κb,γ,γt,γP/2,γQ/2).
a^˙=gσ^12κa^/2+κmδA^m+κlδA^l,
b^˙=gpσ^13κbb^/2+κbB^,
σ^˙13=ga^σ^23+gp(σ^3σ^1)b^γσ^13/2+γσ^12δC^γtσ^23δC^tγPσ^13/4+γP/2σ^13(δC^PδC^P)γQσ^13+2γQσ^13(δC^QδC^Q),
σ^˙12=g(σ^2σ^1)a^+gpb^σ^23γσ^13δC^γtσ^12/2γtδC^t(σ^2σ^1)γPσ^12+2γPσ^12(δC^PδC^P)γPσ^12/4+γP/2σ^12(δC^QδC^Q),
σ^˙1=g(a^σ^12+a^σ^12)+gp(b^σ^13+b^σ^13)+γtσ^2γt(δC^tσ^12+δC^tσ^12),
σ^˙2=g(a^σ^12+a^σ^12)+γσ^3γ(δC^σ^23+δC^σ^23)γtσ^2+γt(δC^tσ^12+δC^tσ^12),
σ^˙3=gp(b^σ^13+b^σ13)γσ^3+γ(δC^σ^23+δC^σ^23).
a^˙=G/2(σ^2σ^1)a^κa^/2+κmδA^m+κlδA^l+GδΛ^p,
σ^˙1=G(σ^2σ^1)(a^a^+a^a^)/2+G(δΛ^pa^+δΛ^pa^)+γtσ^2γtδΛ^t+(B1+B2+B3)/(2Gpσ^1+κb/2)2,
B1=(B^B^+B^B^)2Gpκbσ^1,B2=2Gpκb(δΛ^qδΛ^q+δΛ^qδΛ^q),
B3=(B^δΛ^q+B^δΛ^q)4Gpκb(κb/22Gpσ^1),
σ^˙2=G(σ^2σ^1)(a^a^+a^a^)/2G(δΛ^pa^+δΛ^pa^)+γσ^3γδΛ^γtσ^2+γtδΛ^t,
σ^˙3=(B1+B2+B3)/(2Gpσ^1+κb/2)2γσ^3+γδΛ^,
δΛ^p=σ^12(δC^PδC^P),δΛ^q=σ^13(δC^QδC^Q),Gp=gp2/2γQ,δΛ^t=δC^tσ^12+δC^tσ^12,δΛ^=δC^σ^23+δC^σ^23,
α˙=G(J2J1)α/2κα/2,
J˙1=G(J2J1)αα*+γtJ2ΓJ1,
J˙2=G(J2J1)αα*+γJ3γtJ2,
J˙3=ΓJ1γJ3.
Γ=B^+B^4Gpκb/(2GpJ1+κb/2)2=Pinηtηdηq/hνpN,
J2=κ/G+J1,J3=ΓJ1/γ,J2=(Γ+Gαα*)J1/(γt+Gαα*).
a^(t)=[|α0|+δa^(t)]exp[iΦ(t)],σ^m(t)=Jm0+δσ^m(t),B^(t)=[|B0|+δB^(t)]exp[iΦb(t)],
δx^˙a=κmδX^Am+κlδX^Al+GδΛ^P+G(δσ^2δσ^1)α0,
δσ^˙1=G(J20J10)α0δx^a+Gα02(δσ^2δσ^1)+γδtσ^2γtδΛ^t+Gα0δΛ^PΓδσ^1ΓηJ10δX^B+Γ(1η)δΛ^Q,
δσ^˙2=G(J20J10)α0δx^aGα02(δσ^2δσ^1)Gα0δΛ^P+γδσ^3γtδσ^2+γtδΛ^tγδΛ^,
δσ^˙3=Γδσ^1+ΓηJ10δX^B+γδΛ^γδσ^3Γ(1η)δΛ^Q,
δx^a=δa^+δa^,δX^Am=δA^m+δA^m,δX^Al=δA^l+δA^l,δΛ^P=δΛ^p+δΛ^p.
δΛ^Q=δΛ^q+δΛ^q,η=4GpκbJ10/(2GpJ10+κb/2)2,δX^B=δB^+δB^.
A^out=κma^δA^m,
δX^out=κmδx^aδX^Am.
δXout={[κm(iω+F1)]δXAm+κmκlδXAlF2κmGα0δΛP+κmGδΛP+F3κmΓηJ10δXB+F2κmγtδΛtF3κmΓ(1η)δΛQF4κmγδΛ}/(iω+F1),
F0=iω[(iω+γ)(iω+Γ+γt+2Gα02)+Γ(Gα02+γt)],
F1=iω(2iω+Γ+2γ)G2α02(J20J10)/F0,
F2=iω(2iω+Γ+2γ)Gα0/F0,
F3=iω(iω+γt+2γ)Gα0/F0,
F4=iω(iω+Γγt)Gα0/F0.
Vout(ω)=(1/|iω+F1|2){|κm(iω+F1)|2Vvac+κmG(J1+J2)|1α0F2|2Vdipole+|F2|2κmγtJ2Vspont21+|F3|2κmΓηJ10VPump+κmκlVloss+|F3|2κmΓ(1η)(J1+J3)Vpa+|F4|2κmγJ3Vspont32}.
δΛ(ω)δΛ(ω)=σ^3δ(ωω),
δΛt(ω)δΛt(ω)=σ^2δ(ωω),
δΛP(ω)δΛP(ω)=σ^1+σ^2δ(ωω),
δΛQ(ω)δΛQ(ω)=σ^1+σ^3δ(ωω).

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