Abstract

Detuning properties for actively harmonic mode-locked fiber lasers has been studied both theoretically and experimentally while taking into account of finite cavity dispersions. The theoretical work is based on the self-consistent time domain circulating pulse method. By keeping terms which are usually neglected in previous studies, we have derived an analytic formula which can predict the saturation behavior associated with large modulation frequency detuning. It is found that for the case of medium cavity dispersion, both the pulse-modulator RF phase lag and the optical carrier frequency of the circulating pulse will change significantly as a function of the modulation frequency detuning. The analytical results are supported by both the numerical simulations as well as the experimental measurements. Our theory can potentially serve as a design guidance for cavity length feedback control of harmonic mode-locked fiber lasers.

© 2005 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  8. K.S. Abedin, M. Hyodo, and N. Onodera, K.S. Abedin, M. Hyodo, and N. Onodera, �??Active stabilization of a higher-order mode-locked fiber laser operating at a pulse repetition rate of 154GHz,�?? Opt. Lett. 26, 151-153 (2001)
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  9. G. P. Agrawal, �??Optical pulse propagation in doped fiber amplifiers,�?? Phys. Rev. A 44, 7493-7501 (1991).
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Appl. Phys. Lett.

R.H. Stolen, C. Lin, and R.K. Jain, �??A time dispersion tuned fiber Raman oscillator,�?? Appl. Phys. Lett. 30, 340-342 (1977).
[CrossRef]

Electron. Lett.

X. Shan, D. Cleland, and A. Ellis, �??Stabilizing erbium fiber soliton laser with pulse phase locking,�?? Electron. Lett., 28, 182-184, (1992).
[CrossRef]

IEEE J. Quantum Electron.

H.A. Haus, �??A theory of forced mode-locking,�?? IEEE J. Quantum Electron. QE-11, 323-330 (1975).
[CrossRef]

D.I. Kuizenga, A.E. Siegman, �??FM and AM mode-locking in homogeneous laser. Part I: theory,�?? IEEE J. Quantum Electron. QE-6, 694-708 (1970).
[CrossRef]

J. Lightwave Technol.

J.S. Wey, J. Goldhar, G.L. Burdge, �??Active harmonic mode-locking of an erbium fiber laser with intracavity FP filters,�?? J. Lightwave Technol. 15, 1171-1180 (1997).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

Y. Li, C. Luo, Y. Gao, �??Detuning characteristics of the AM mode-locked fiber laser,�?? Opt. Quantum Electron. 33, 589-597, (2001).
[CrossRef]

Phys. Rev. A

G. P. Agrawal, �??Optical pulse propagation in doped fiber amplifiers,�?? Phys. Rev. A 44, 7493-7501 (1991).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1.
Fig. 1.

(a) Block diagram of the simplified active harmonic mode-locked fiber laser for time domain circulating pulse analysis. (b) Illustration on the pulse spectrum profile with respect to the gain curve under the detuned operations. (c) Illustration on the pulse intensity profile with respect to the modulator transmittance curve under the detuned operations. Note that when only medium cavity dispersion is present, for the detuned operations, both the carrier wavelength and the pulse arrival time (with respect to the modulator) will deviate from the gain peak position and the modulation transmittance peak position respectively.

Fig. 2.
Fig. 2.

Normalized pulse carrier frequency shift x as a function of the normalized modulation frequency detuning parameter δ for different dispersion D. The normalized parameters D, x, δ are defined as D=β2* LΩ2/2, x=ωd /Ω, δ=Ω·N·ΔTm , where β2* L is the cavity dispersion, Ω is the cavity filter bandwidth, ωd is the frequency shift between the pulse carrier frequency and the filter peak frequency. N is the order of harmonic mode-locking. ΔTmfm /fm2 represents the modulation period detuning.

Fig. 3.
Fig. 3.

Pulse-modulator phase lag α as a function of the normalized modulation frequency detuning δ for different dispersion parameter D. For the definition of the normalized parameters D, δ, please refer to Fig 2.

Fig. 4.
Fig. 4.

Simulated normalized pulse carrier frequency x as a function of the normalized modulation frequency detuning δ for different dispersion parameter D. Triangle: D=0.1 ; Square: D=1 ; Cross: D=5 ; For the definition of the normalized parameters D, x, δ, please refer to Fig 2.

Fig. 5.
Fig. 5.

Simulated pulse-modulator phase lag α as a function of the normalized modulation frequency detuning δ for different dispersion parameter D. Triangle: D=0.1 ; Square: D=1 ; Cross: D=5 ; For the definition of the normalized parameters D, x, δ, please refer to Fig 2.

Fig. 6.
Fig. 6.

Measured pulse carrier wavelength as a function of modulation frequency detuning. The detuning range 400 kHz (corresponding to round trip delay 8ps) maps to the normalized detuning range δ~5 as in Fig. 2 and Fig. 4.

Fig. 7.
Fig. 7.

Measured pulse-modulator phase lag as a function of modulation frequency detuning. The detuning range 400 kHz (corresponding to round trip delay 8ps) maps to the normalized detuning range δ~5 as in Fig. 3 and Fig. 5.

Equations (34)

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A z + [ β 1 t + i 2 β 2 2 t 2 g ] A = 0
A z + [ i 2 β 2 2 τ 2 g ] A = 0 .
A ( L , t ) = e gL [ 1 i β 2 L 2 2 t 2 ] f ( t β 1 L ) ,
F ( ω ) 1 ( ω ω f Ω ) 2 = 1 ( ω ω 0 ) 2 Ω 2 2 ω d ( ω ω 0 ) Ω 2 ω d 2 Ω 2
f ( t ) = F ( i t ) A ( L , t ) = e gL ( 1 + R ̂ ) f ( t β 1 L ) ,
R ̂ = 1 Ω 2 2 t 2 2 i ω d Ω 2 t ω d 2 Ω 2 i β 2 L 2 2 t 2 .
T ( t ) = Γ { 1 + M cos [ ω m ( t + t 0 ) ] } = Γ { 1 + M cos ( ω m t + α ) } ,
f n + 1 ( t ) = T ( t ) · f n ( t ) = Γ { 1 + M cos ( ω m t + α ) } e gL { 1 + R ̂ } f n ( t β 1 L ) .
P n + 1 ( t ) = f n + 1 [ t + ( n + 1 ) N T m ] = Γ e gL { 1 + M cos ( ω m t + α ) } { 1 + R ̂ } f n [ t + ( n + 1 ) N T m β 1 L ] .
f n [ t + ( n + 1 ) N T m β 1 L ] = f n ( t + n N T m + N T m β 1 L )
= P n ( t ) + ( N T m β 1 L ) d dt P n ( t ) + 1 2 ( N T m β 1 L ) 2 d 2 dt P n ( t ) .
N T m β 1 L = N ( T m 0 + Δ T m ) ( β 1 * + β 2 * ω d ) L = N · Δ T m β 2 * ω d · L ,
N T m β 1 L = δ 2 D · x Ω .
f n [ t + ( n + 1 ) N T m β 1 L ] = { 1 + ( δ 2 D · x ) Ω d dt + 1 2 ( δ 2 D · x ) 2 Ω 2 d 2 d t 2 } P n ( t ) .
P n + 1 ( t ) = Γ e gL { 1 + M cos ( ω m t + α ) } { 1 + R ̂ } f n [ t + ( n + 1 ) N T m β 1 L ]
= Γ e gL { 1 + M cos ( ω m t + α ) } { 1 + R ̂ } { 1 + ( δ 2 D · x ) Ω d dt + 1 2 ( δ 2 D · x ) 2 Ω 2 d 2 d t 2 } P n ( t ) ,
1 + M cos ( ω m t + α ) = 1 + M { cos α sin α · ω m t 1 2 cos α · ω m 2 t 2 }
= ( 1 + M cos α ) { 1 M sin α 1 + M cos α ω m t M cos α 2 ( 1 + M cos α ) ω m 2 t 2 } .
P n + 1 ( t ) = Γ e gL ( 1 + M cos α ) { 1 + Q ̂ } · P n ( t ) ,
Q ̂ = x 2 + M sin 2 α ( 2 + 2 M cos α ) cos α M cos α · ω m 2 ( t + sin α ω m cos α ) 2 2 + 2 M cos α
+ 1 + 1 2 ( δ 2 D · x ) 2 i D Ω 2 d 2 d t 2 + ( δ 2 D · x 2 i · x ) Ω d dt .
P n ( t ) n = Γ e gL ( 1 + M cos α ) { 1 e gL Γ · ( 1 + M cos α ) + Q ̂ } · P n ( t ) .
{ a b · ( t + sin α ω m cos α ) 2 + c 2 t 2 + d t } P ( t ) = 0 ,
a = 1 e gL Γ · ( 1 + M cos α ) x 2 + M sin 2 α ( 2 + 2 M cos α ) cos α ,
b = M cos α · ω m 2 2 + 2 M cos α ,
c = 1 Ω 2 { 1 + 1 2 ( δ 2 D · x ) 2 i D } ,
d = δ 2 D · x 2 i · x Ω .
P ( t ) = exp [ ( t + Δ τ ) 2 τ 2 ] ,
sin α ω m cos α + Re { δ 2 D · x 2 i · x 4 · M cos α · ω m 2 2 + 2 M cos α [ 1 + 1 2 ( δ 2 D · x ) 2 i D ] } = 0 ,
Im { δ 2 D · x 2 i · x 4 · M cos α · ω m 2 2 + 2 M cos α [ 1 + 1 2 ( δ 2 D · x ) 2 i D ] } = 0 .
x = δ 2 D + 2 · Sign [ D ] δ 2 D .
α = 2 + M D δ 2 D .
x = D · δ 4 + 2 δ 2 .
sin α cos α + 1 + M cos α 2 M cos α δ 1 + δ 2 2 = 0 .

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