## Dispersion effects on the detuning properties of actively harmonic mode-locked fiber lasers

Optics Express, Vol. 13, Issue 7, pp. 2688-2698 (2005)

http://dx.doi.org/10.1364/OPEX.13.002688

Acrobat PDF (183 KB)

### 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

## 1. Introduction

1. H.A. Haus, “A theory of forced mode-locking,” IEEE J. Quantum Electron. **QE-11**, 323–330 (1975). [CrossRef]

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

1. H.A. Haus, “A theory of forced mode-locking,” IEEE J. Quantum Electron. **QE-11**, 323–330 (1975). [CrossRef]

3. Y. Li, C. Luo, and Y. Gao, “Detuning characteristics of the AM mode-locked fiber laser,” Opt. Quantum Electron. **33**, 589–597, (2001). [CrossRef]

3. Y. Li, C. Luo, and Y. Gao, “Detuning characteristics of the AM mode-locked fiber laser,” Opt. Quantum Electron. **33**, 589–597, (2001). [CrossRef]

4. J.S. Wey, J. Goldhar, and G.L. Burdge, “Active harmonic mode-locking of an erbium fiber laser with intra-cavity FP filters,” J. Lightwave Technol. **15**, 1171–1180 (1997). [CrossRef]

5. K. Tamura and M. Nakazawa, “Dispersion tuned harmonically mode-locked fiber ring laser for self-synchronization to an external clock,” Opt. Lett. **21**, 1984–1986 (1996). [CrossRef] [PubMed]

6. R.H. Stolen, C. Lin, and R.K. Jain, “A time dispersion tuned fiber Raman oscillator,” Appl. Phys. Lett. **30**, 340–342 (1977). [CrossRef]

7. X. Shan, D. Cleland, and A. Ellis, “Stabilizing erbium fiber soliton laser with pulse phase locking,” Electron. Lett. , **28**, 182–184, (1992). [CrossRef]

8. 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). [CrossRef]

## 2. Theoretical analysis for detuned active harmonic mode-locking

*ω*

_{f}and Ω, respectively. Since we are studying the general case of detuned active harmonic mode-locking with the presence of cavity dispersion, from the discussion of the introduction section, we understand that the carrier frequency will in general be different from the cavity filter peak frequency. Denoting the carrier frequency as

*ω*

_{0}, the frequency shift

*ω*

_{d}from the carrier frequency to the cavity filter center frequency can then be written as

*ω*

_{d}=

*ω*

_{0}-

*ω*

_{f}(please refer to Fig. 1(b)). Note that

*ω*

_{d}is a unknown function of the modulation frequency detuning and must be determined self-consistently.

9. G. P. Agrawal, “Optical pulse propagation in doped fiber amplifiers,” Phys. Rev. A **44**, 7493–7501 (1991). [CrossRef] [PubMed]

*β*

_{1}and

*β*

_{2}are the propagation constant and dispersion constant,

*g*is the EDFA gain per unit length,

*A*(

*z*,

*t*) is defined from

*E*(

*z*,

*t*)=exp(

*iβ*

_{o}

*z*-

*iω*

_{0}

*t*)·

*A*(

*z, t*) as the pulse envelope. It should be noted that

*β*

_{1}and

*β*

_{2}used in (1) are the values measured at

*ω*

_{0}(the pulse carrier frequency) instead of

*ω*

_{f}(the optical filter center frequency). Using the technique of Taylor expansion and only keeping second order dispersion,

*β*

_{1}and

*β*

_{2}can then be written as

*β*

_{1}=

*ω*

_{d}, and

*β*

_{2}=

*ω*

_{f}(the optical filter center frequency).

*A*(

*z,t*) from

*E*(

*z, t*)=exp(

*iβ*

_{o}

*z*-

*iω*

_{0}

*t*)·

*A*(

*z, t*) as the envelop, we have implicitly assumed that

*A*(

*z,t*) does not contain any linear oscillating part like exp(-

*iδω*·

*t*), since term like this belongs to the carrier part and by the defination of envelop has already been absorbed. This assumption will be used later as the

**first criteria**for self-consistent calculations.

*τ*=

*t*-

*β*

_{1}

*z*, in the moving reference frame, we have

*L*, from (2), for a given input pulse, the envelope of the pulse at the EDFA output port (

*z*=

*L*) can be derived as

*f*(

*t*)=

*A*(0,

*t*) is the envelop of the initial pulse at the input port of the EDFA.

*F*(

*ω*)=exp[-(

*ω*-

*ω*

_{f})

^{2}/Ω

^{2}] as

*ω*-

*ω*

_{0}(in the frequency domain) as

*i∂*/

*∂t*(in the time domain), the reshaped envelope becomes

*M*is the modulation depth,

*ω*

_{m}is the modulation frequency, and

*α*=

*ω*

_{m}

*t*

_{0}represents the relative phase lag between the pulse arrival time and the modulator transmittance peak (please refer to Fig. 1(c)). Note that the phase lag

*α*is an unknown function of the modulation frequency detuning and must be determined self-consistently.

*t*=0 is the moment at which the pulse arrives the modulator. If the pulse arrives the modulator at

*t*≠0, we can always change the definition of

*t*

_{0}so that with the new definition of

*t*

_{0}, the pulse will arrive the modulator at

*t*=0. This implicit assumption will be used later as the second criteria for self-consistent calculations.

*f*

_{n}(

*t*) as the envelop of the pulse observed at the output port of the modulator after

*n*

^{th}round trips. According to Fig. 1(a), before the pulse represented by

*f*

_{n}(

*t*) can move back to the modulator input port, it must pass through the dispersive EDFA and the optical filter respectively. From the discussion of (1)~(5), the envelop of the pulse observed at the input port of the modulator after the dispersive EDFA and the optical filter can be written as

*f′*

_{n}(

*t*)=

*e*

^{gL}(1+

*R̂*)

*f*

_{n}(

*t*-

*β*

_{1}

*L*). When the circulating pulse represented by

*f′*

_{n}(

*t*) leaves the modulator, with the help of (7), the envelop

*f*

_{n}

_{+1}(

*t*) for the

*n*+1

^{th}round becomes

*f*

_{n}(

*t*) defined above, since in the steady state, we have

*f*

_{n}

_{+1}(

*t*)=

*f*

_{n}(

*t*-

*NT*

_{m}), i.e., two nearby pulses (generated by the same circulating seed for the case of harmonic mode-locking) are displaced by the time of the modulation period

*T*

_{m}multiplied by

*N*(the order of harmonic mode-locking). This conflict can be easily overcomed by defining a new envelop

*P*

_{n}(

*t*) as

*P*

_{n}(

*t*)=

*f*

_{n}(

*t*+

*n*·

*NT*

_{m}), for which in the steady state

*P*

_{n}

_{+1}(

*t*)=

*P*

_{n}(

*t*). According to these considerations outlined above, in the following part, we will then work with

*P*

_{n}(

*t*) instead.

*β*

_{1}is the value measured at the carrier frequency

*ω*

_{0}(unknown) and

*β*

_{1}=

*ω*

_{d}(see equation (1)). Write the modulation period as

*T*

_{m}=

*T*

_{m}

_{0}+Δ

*T*

_{m}, we get

*T*

_{m}

_{0}has been defined from

*NT*

_{m}

_{0}=

*L*as the

**for active harmonic mode-locking. Further write**

*exactly tuned modulation period**D*=

*L*Ω

^{2}/2,

*x*=

*ω*

_{d}/Ω,

*δ*=Ω·

*N*·Δ

*T*

_{m}to represent the normalized cavity dispersion, carrier frequency shift, and modulation frequency detuning respectively, Eq. (11) then becomes

*R̂*is an operator defined in Eq. (6).

*M*cos(

*ω*

_{m}

*t*+

*α*) is expanded to second order using the Taylor series. The result is

*P*

_{n}(

*t*) per round trip is small, therefore, the iteration equation (17) reduces to differential equation [1

1. H.A. Haus, “A theory of forced mode-locking,” IEEE J. Quantum Electron. **QE-11**, 323–330 (1975). [CrossRef]

*τ*]=0. Since if Im[Δ

*τ*]≠0, then the pulse envelop will have oscillating term proportional to exp[2

*i*·Im[Δ

*τ*]·Re[1/

*τ*

^{2}]·

*t*], by the assumption of the envelop function, this term must be absorbed into the carrier part. Second, we require that Re[Δ

*τ*]=0. Since if Re[Δ

*τ*]≠0, then the pulse will reach the modulator at

*t*=-Re[Δ

*τ*]. However, this temporal displacement of the pulse arrival time can be and has already been absorbed into the phase lag parameter

*α*defined in Eq. (7).

*δ*, the normalized carrier frequency shift

*x*can be first solved from Eq. (23), and then the pulse-modulator phase lag

*α*can be solved from Eq. (22) after plugging both

*δ*and

*x*. Due to the existence of the square root in Eqs. (22) and (23), an explicit solution can not be obtained for the general case. However, simple formulas can still be obtained under two extreme conditions: the strong dispersion case where

*D*≫1 and the weak dispersion case where

*D*≪1. In the following part, we will discuss them separately.

*D*≫1, we then expect the carrier wavelength shift effect to be large enough to compensate the round trip delay, indicating that (

*δ*-2

*D*·

*x*)≪1 is usually satisfied. Using this condition in Eq. (23), we obtain

*ω*

_{d}=

*N*·Δ

*T*

_{m}/

*L*. Identifying

*N*·Δ

*T*

_{m}as the mismatch of timing for the circulating pulse per round trip, and

*L*·

*ω*

_{d}as the adjustment of round trip time through the change of carrier frequency, we therefore see that physically Eq. (24-1) means the mismatch of timing caused by the detuning is completely compensated owning to the change of the group velocity of the circulating pulse.

*D*≪1, we then expect the carrier frequency shift effect to be small, therefore, we have

*D*·

*x*≪

*δ*. With the help of this, from equation (23), we obtain

*δ*is small, both the carrier frequency shift and the phase lag are linearly proportional to the detuning. This conclusion agrees with the results obtained by Haus [1

**QE-11**, 323–330 (1975). [CrossRef]

3. Y. Li, C. Luo, and Y. Gao, “Detuning characteristics of the AM mode-locked fiber laser,” Opt. Quantum Electron. **33**, 589–597, (2001). [CrossRef]

*δ*for different cavity dispersion parameter

*D*. In our calculation, the modulation depth

*M*is assumed to be unity. From these figures, it can be seen that the value of the normalized cavity dispersion strongly affects the properties of the detuned active harmonic mode-locking. Note that in plotting Fig. 2 and Fig. 3, all the curves are generated by numerically solving the coupled Eqs. (22) and (23) as a general case.

## 3. Numerical and experimental study of detuned active harmonic mode-locking

*M*used in the simulation is chosen to be one.

*D*=5. The nearly perfect agreement for

*D*=5 is due to the reason that when strong cavity dispersion exists, there is almost no phase lag between the pulse arrival time and modulator transmittance peak even for large detuning values. Therefore, the expansion of the modulator transmission function up to the second order in Eq. (15) remains as a very good approximation even for large detuning values. However, for the case of small cavity dispersion such as

*D*=0.1, when the modulation frequency detuning is large, the resulted large pulse-modulator phase lag will destroy the validity of the second order Taylor expansion for the modulator transmittance function in Eq. (15). As a result, in order to be accurate, one needs to keep more terms when doing the Taylor expansion.

*M*=1 are valid. In our measurement, we have chosen the modulator bias to be at 0.7

*V*

_{π}, and the RF modulation signal amplitude to be at 0.3

*V*

_{π}, where

*V*

_{π}represents the extinction voltage of the intensity modulator.

*δ*~5 in the analytic and numerical plots. The normalized cavity dispersion for the constructed fiber laser is also estimated to be

*D*~1. When comparing our experimental results (Fig. 6,7) with the

*D*~1 case of the analytic (Fig. 2,3) and numerical (Fig. 4,5) plots, reasonable agreements can be found.

## 4. Conclusion

## References and links

1. | H.A. Haus, “A theory of forced mode-locking,” IEEE J. Quantum Electron. |

2. | D.I. Kuizenga and A.E. Siegman, “FM and AM mode-locking in homogeneous laser. Part I: theory,” IEEE J. Quantum Electron. |

3. | Y. Li, C. Luo, and Y. Gao, “Detuning characteristics of the AM mode-locked fiber laser,” Opt. Quantum Electron. |

4. | J.S. Wey, J. Goldhar, and G.L. Burdge, “Active harmonic mode-locking of an erbium fiber laser with intra-cavity FP filters,” J. Lightwave Technol. |

5. | K. Tamura and M. Nakazawa, “Dispersion tuned harmonically mode-locked fiber ring laser for self-synchronization to an external clock,” Opt. Lett. |

6. | R.H. Stolen, C. Lin, and R.K. Jain, “A time dispersion tuned fiber Raman oscillator,” Appl. Phys. Lett. |

7. | X. Shan, D. Cleland, and A. Ellis, “Stabilizing erbium fiber soliton laser with pulse phase locking,” Electron. Lett. , |

8. | 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. |

9. | G. P. Agrawal, “Optical pulse propagation in doped fiber amplifiers,” Phys. Rev. A |

**OCIS Codes**

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(140.4050) Lasers and laser optics : Mode-locked lasers

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 1, 2004

Revised Manuscript: March 8, 2005

Published: April 4, 2005

**Citation**

G. Zhu and N. K. Dutta, "Dispersion effects on the detuning properties of actively harmonic mode-locked fiber lasers," Opt. Express **13**, 2688-2698 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-7-2688

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### References

- 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]
- Y. Li, C. Luo, Y. Gao, �??Detuning characteristics of the AM mode-locked fiber laser,�?? Opt. Quantum Electron. 33, 589-597, (2001). [CrossRef]
- 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]
- K. Tamura, M. Nakazawa, �??Dispersion tuned harmonically mode-locked fiber ring laser for self-synchronization to an external clock,�?? Opt. Lett. 21, 1984-1986 (1996). [CrossRef] [PubMed]
- R.H. Stolen, C. Lin, and R.K. Jain, �??A time dispersion tuned fiber Raman oscillator,�?? Appl. Phys. Lett. 30, 340-342 (1977). [CrossRef]
- X. Shan, D. Cleland, and A. Ellis, �??Stabilizing erbium fiber soliton laser with pulse phase locking,�?? Electron. Lett., 28, 182-184, (1992). [CrossRef]
- 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) [CrossRef]
- G. P. Agrawal, �??Optical pulse propagation in doped fiber amplifiers,�?? Phys. Rev. A 44, 7493-7501 (1991). [CrossRef] [PubMed]

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