Abstract

We present a full quantum mechanical model for a composite-ring erbium-doped fiber laser with a feedback loop, with a view to predict the achievable intensity noise reduction. By adding an electronic feedback term to the noise transfer function of the free running composite-ring fiber laser, an analytical expression is developed for the intensity noise spectrum of the current system. For ring fiber lasers, the model shows that the subcavity can suppress the noise at the high frequency and reduce the cutoff frequency to the quantum noise limit, and the feedback loop has a significant impact on suppressing the intensity noise near the relaxation oscillation.

© 2013 Optical Society of America

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References

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  1. Z. Meng, G. Stevant, 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]
  2. T. C. Ralph, C. C. Harb, and H. Bachor, “Intensity noise of injection locked laser: quantum theory using a linearied input-output method,” Phys. Rev. A 54, 4359–4369 (1996).
    [CrossRef]
  3. B. C. Buchler, E. H. Huntington, C. C. Harb, and T. C. Ralph, “Feedback control of laser intensity noise,” Phys. Rev. A 57, 1286–1294 (1998).
    [CrossRef]
  4. J. Zhang, C. Xie, and K. Peng, “Electronic feedback control of the intensity noise of a single-frequency intra-cavity doubled laser,” J. Opt. Soc. Am. B 19, 1910–1916 (2002).
    [CrossRef]
  5. W. J. Yue, Y. X. Wang, C.-D. Xiong, Z.-Y. Wang, and Q. Qiu, “Intensity noise of erbium doped fiber laser based on full quantum theory,” J. Opt. Soc. Am. B 30, 275–281 (2013).
    [CrossRef]
  6. K. Lee, S. D. Lim, C. H. Kim, and J. H. Lee, “Noise reduction in multiwave-length SOA-based ring laser by coupled dual cavities for WDM applications,” J. Lightwave Technol. 5, 739–745 (2010).
    [CrossRef]

2013 (1)

2010 (1)

K. Lee, S. D. Lim, C. H. Kim, and J. H. Lee, “Noise reduction in multiwave-length SOA-based ring laser by coupled dual cavities for WDM applications,” J. Lightwave Technol. 5, 739–745 (2010).
[CrossRef]

2006 (1)

2002 (1)

1998 (1)

B. C. Buchler, E. H. Huntington, C. C. Harb, and T. C. Ralph, “Feedback control of laser intensity noise,” Phys. Rev. A 57, 1286–1294 (1998).
[CrossRef]

1996 (1)

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

Bachor, H.

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

Buchler, B. C.

B. C. Buchler, E. H. Huntington, C. C. Harb, and T. C. Ralph, “Feedback control of laser intensity noise,” Phys. Rev. A 57, 1286–1294 (1998).
[CrossRef]

Harb, C. C.

B. C. Buchler, E. H. Huntington, C. C. Harb, and T. C. Ralph, “Feedback control of laser intensity noise,” Phys. Rev. A 57, 1286–1294 (1998).
[CrossRef]

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

Huntington, E. H.

B. C. Buchler, E. H. Huntington, C. C. Harb, and T. C. Ralph, “Feedback control of laser intensity noise,” Phys. Rev. A 57, 1286–1294 (1998).
[CrossRef]

Kim, C. H.

K. Lee, S. D. Lim, C. H. Kim, and J. H. Lee, “Noise reduction in multiwave-length SOA-based ring laser by coupled dual cavities for WDM applications,” J. Lightwave Technol. 5, 739–745 (2010).
[CrossRef]

Lee, J. H.

K. Lee, S. D. Lim, C. H. Kim, and J. H. Lee, “Noise reduction in multiwave-length SOA-based ring laser by coupled dual cavities for WDM applications,” J. Lightwave Technol. 5, 739–745 (2010).
[CrossRef]

Lee, K.

K. Lee, S. D. Lim, C. H. Kim, and J. H. Lee, “Noise reduction in multiwave-length SOA-based ring laser by coupled dual cavities for WDM applications,” J. Lightwave Technol. 5, 739–745 (2010).
[CrossRef]

Lim, S. D.

K. Lee, S. D. Lim, C. H. Kim, and J. H. Lee, “Noise reduction in multiwave-length SOA-based ring laser by coupled dual cavities for WDM applications,” J. Lightwave Technol. 5, 739–745 (2010).
[CrossRef]

Meng, Z.

Peng, K.

Qiu, Q.

Ralph, T. C.

B. C. Buchler, E. H. Huntington, C. C. Harb, and T. C. Ralph, “Feedback control of laser intensity noise,” Phys. Rev. A 57, 1286–1294 (1998).
[CrossRef]

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

Stevant, G.

Wang, Y. X.

Wang, Z.-Y.

Whitenett, G.

Xie, C.

Xiong, C.-D.

Yue, W. J.

Zhang, J.

J. Lightwave Technol. (2)

Z. Meng, G. Stevant, 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]

K. Lee, S. D. Lim, C. H. Kim, and J. H. Lee, “Noise reduction in multiwave-length SOA-based ring laser by coupled dual cavities for WDM applications,” J. Lightwave Technol. 5, 739–745 (2010).
[CrossRef]

J. Opt. Soc. Am. B (2)

Phys. Rev. A (2)

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

B. C. Buchler, E. H. Huntington, C. C. Harb, and T. C. Ralph, “Feedback control of laser intensity noise,” Phys. Rev. A 57, 1286–1294 (1998).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Configuration of the composite-ring Er-doped fiber laser with a feedback loop. LD, laser diode; PD, photodiode.

Fig. 2.
Fig. 2.

Diagram of the features included in the linearized quantum model of the laser.

Fig. 3.
Fig. 3.

Noise spectrum of the composite cavity fiber laser without feedback loop.

Fig. 4.
Fig. 4.

Plot of the gain and phase of the pump noise transfer function.

Fig. 5.
Fig. 5.

Effect of the feedback loop on the intensity noise. Parameters used were ε=0.05, ηD=0.9, s=4.472, g=10.

Tables (1)

Tables Icon

Table 1. Parameters of the Composite-Ring Fiber Laser

Equations (41)

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A^in2=κm1a^δA^m1,
c^˙=κ12c^+κm1A^in2+κm1A^insert_loss1+κl1A^l1κ1=κm1+κm1+κl1,
δX^˙c=κ12δX^c+κm1δX^Ain2+κm1δX^Ainsert_loss1+κl1δX^Al1δX^c=δc^+δc^+,δX^Ain2=δA^in2+δA^in2+,etc.
δXc=κm1δXAin2+κm1δXAinsert_loss1+κl1δXAl1iω+κ1/2,
δXc_out=κm1δXcδXAinsert_loss1=κm1δXAin2+(κm1iωκ12)δXAinsert_loss1+κm1κl1δXAl1iω+κ1/2.
Vc_out(ω)=κm12|δXAin2|2+|κm1iωκ1/2|2|δXAinsert_loss1|2+κm1κl1|δXAl1|2|iω+κ1/2|2=κm12Vout+|κm1iωκ1/2|2|δXAinsert_loss1|2+κm1κl1|δXAl1|2|iω+κ1/2|2,
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}.
A^in3=κm1c^δA^m1.
d^˙=κ22d^+κm1A^in3+κm1A^insert_loss1+κm2A^insert_loss2+κlA^lκ2=κm1+κm1+κl+κm2,
δX^˙d=κ22δX^d+κm1δX^Ain3+κm1δX^Ainsert_loss1+κm2δX^Ainsert_loss2+κlδX^AlδX^d=δd^+δd^+,etc.
δXd=κm1δXAin3+κm1δXAinsert_loss1+κm2δXAinsert_loss2+κlδXliω+κ2/2.
δXd_out=κm2κm1δXAin3+κm2κm1δXAinsert_loss1+(κm2iωκ2/2)δXAinsert_loss2+κm2κlδXAliω+κ2/2.
Vd_out(ω)=κm1κm2Vc_out+|κm2iωκ2/2|2|δXAinsert_loss2|2+κm2κl1|δXAinsert_loss1|2+κm2κl|δXAl|2|iω+κ2/2|2.
δXd_out=W×(M0δXAm+M1δXAl+M2δΛP+M3δΛt+M4δΛP+M5δΛB+M6δΛQ+M7δΛ+κm1κm2[(κm1iωκ1/2)δXAinsert_loss1+κm1κl1δXAl1]/(iω+κ2/2)(iω+κ1/2)+[κm2κm1δXAinsert_loss1+(κm2iωκ2/2)δXAinsert_loss2+κm2κlδXAl]/(iω+κ2/2),
A^pump=B^+A^fd,
A^pump=B¯+δB^+δA^fdδA^pump=δB^+δA^fdδX^A_pump=δX^B+δX^A_fd.
δA^fd=β+r(τ)δi^(tτ)dτ1βδν^β,
δX^A_fd=β+r(τ)δX^i(tτ)dτ1βδX^νβ.
i^={εηdA^out+ηd(1ε)δυ^s+1ηdδυ^d}×{εηdA^out+ηd(1ε)δυ^s+1ηdδυ^d},
A^out=km2d^δA^m2=km2d0+km2δd^δA^m2=km2d0+δA^out.
δi^=κm2α0εηd(εηdδX^out+ηd(1ε)δX^νs+1ηdδX^νd).
δX^i=εηdδX^out+ηd(1ε)δX^νs+1ηdδX^νd.
δX^A_fd=βκm2α0εηd+k(τ)δX^i(tτ)dτ1βδX^υβ.
δXd_out=(W×AB)/[1+WM5H(ω)]+D/(iω+κ2/2)[1+WM5H(ω)]+κm1κm2C/[1+WM5H(ω)](iω+κ2/2)(iω+κ1/2),
δX^fd_out=1εδX^outεδX^νs.
Vfd_out(ω)=(1ε)Vd_out(ω)+(1β)|WM5|2Vυβ+|WM5H(ω)|2(1ηDεηD)VυD|1+WM5H(ω)|2+1ε|ε+WM5H(ω)|2|1+WM5H(ω)|2Vνs.
Fpump(ω)=κm1κm2κm1F3κmΓηJ10(iω+κ2/2)(iω+κ1/2)(iω+F1).
H(ω)=gp+iωs2p+iω,
δXAin2={[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.
W=κm1κm2κm1(iω+κ2/2)(iω+κ1/2),M0=[κm(iω+F1)]iω+F1,
M1=κmκliω+F1,M2=κmGiω+F1,M3=F2κmγtiω+F1,M4=F2κmGα0iω+F1,
M5=F3κmΓηJ10iω+F1,M6=F3κmΓ(1η)iω+F1,M7=F4κmγiω+F1,
A=(M0δXAm+M1δXAl+M2δΛP+M3δΛt+M4δΛP+M6δΛQ+M7δΛ+M5δXB),
B=WM5H(ω)(1ε/εδXνs+1ηd/εηdδXνD)+WM51βδXυβ,
C=(κm1iωκ1/2)δXAinsert_loss1+κm1κl1δXAl1,
D=κm2κm1δXAinsert_loss1+(κm2iωκ2/2)δXAinsert_loss2+κm2κlδXAl.

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