## 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

Active harmonic mode-locking is a powerful technique useful for producing ultra-short and high repetition rate pulse trains. The theory of active harmonic mode-locking could be traced back to that of active fundamental mode-locking established by Haus [1] and Kuizenga [2]. From standard laser textbooks, it is well-known that for practical lasers with certain amount of cavity dispersions, the cavity longitudinal mode spacing is dependent on the lasing wavelength. By requiring that the active modulation frequency equals the harmonics of the cavity longitudinal mode spacing measured at the gain peak wavelength, exactly-tuned condition for active harmonic mode-locking can be defined. When the mode-locking is exactly-tuned, every time the circulating pulse passes through the cavity loss modulator, the modulator is at its transmittance peak as can be conjectured easily from the symmetry considerations.

When the modulation frequency deviates from the exact harmonics of the cavity mode spacing measured at the gain peak wavelength, detuned active harmonic mode-locking happens. Depending on the degree of the cavity dispersion, the problem of detuned active harmonic mode-locking can be sorted into three different cases: the zero dispersion case, the strong dispersion case and the medium dispersion case. The differences among these three cases can be briefly discussed as follows:

(a) The zero dispersion case: The zero dispersion case has been previously studied by Haus [1] and Li [3]. Since when no dispersion is present, the cavity mode spacing or the group velocity of the circulating pulse is independent on the carrier wavelength, therefore, in their treatment, they have assumed that the carrier wavelength is fixed at the gain peak position. When detuning happens, they concluded that there will appear certain amount of phase lag between the pulse arrival time (with respect to the modulator) and the modulation transmittance peak. This pulse-modulator phase lag will then provide either “pull” or “push” functions to compensate the round trip delay of the circulating pulse in the time domain [3,4]. Besides these, they also concluded that this phase lag is linearly proportional to the modulation frequency detuning. However, experimentally, it has been observed that the linear relationship is valid only for the small signal detuning regime. When the applied detuning exceeds a certain level, the measured phase lag will tend to saturate. In order to theoretically show how the phase lag saturation happens, extra cares must be taken during the analysis. In this paper, we will develop an improved version of the time domain theory which can predict the phase lag saturation phenomenon.

(b) The strong dispersion case: In contrary to the case of zero dispersion, if strong dispersion exists, there is no phase lag between the pulse arrival time and the modulator transmittance peak when the modulation frequency is detuned [5]. In other words, the pulse will always be synchronized to the modulator transmittance peak. Instead, the optical carrier wavelength will shift as a function of the detuning [6]. The above situation can be easily understood from the point of view that the laser system will always prefer to operate in a state where cavity loss is the minimum. The minimum cavity loss requirement can be satisfied if the pulse chooses to pass through the modulator at its transmittance peak. With strong cavity dispersion being present, this synchronization is done by dynamically changing the carrier wavelength so that at the new wavelength the pulse can adjust its group velocity to compensate the round trip delay and catch up the detuned modulator transmittance peak.

(c) The medium dispersion case: For the case where only medium cavity dispersion is present, according to the reasons outlined above, we then expect to see both the pulse-modulator phase lag and the carrier wavelength shift phenomenon to occur at the same time. In the following part of this paper, we will analyze how the cavity dispersion and the modulation frequency detuning play together to form the active harmonic mode-locking and what are their effects on the properties of the generated pulse train. Note that from a practical point of view, since the modulation frequency detuning can always be compensated using the feedback control technique, the pulse train properties studied in this paper will therefore be confined to the pulse-modulator phase lag [7] and carrier wavelength shift [8] as a function of detuning, upon which the current cavity length thermal stabilization techniques are based.

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

In this section, we will first develop a master equation for the detuned active harmonic mode-locking and then solve the master equation self-consistently. The model considered here is a ring type erbium doped fiber laser (EDFL) actively harmonic mode-locked to an external RF source. To simplify the problem, the ring cavity of the EDFL is assumed to be composed of a dispersive erbium doped fiber amplifier (EDFA) followed by an optical band-pass filter and an intensity modulator (please refer to Fig. 1(a)). We assume that the gain spectrum of the EDFA is quite flat, therefore, the actual gain bandwidth of the cavity is determined by the intracavity optical band-pass filter, of which the transmission peak optical frequency and the filter bandwidth (FWHM) are denoted as *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.

Before we start the circulating pulse analysis, let us first recapitulate how are the effects of dispersive EDFA, band-pass gain filter, and intensity modulator mathematically treated. Considering a pulse propagating through a dispersive EDFA [9]

where *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}=${\beta}_{1}^{*}$+${\beta}_{2}^{*}$
*ω*_{d}
, and *β*
_{2}=${\beta}_{2}^{*}$, where a star denotes the value measured at *ω*_{f}
(the optical filter center frequency).

It must be pointed out that by defining *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.

Starting from (1), with *τ*=*t*-*β*
_{1}
*z*, in the moving reference frame, we have

Assuming that the EDFA length is *L*, from (2), for a given input pulse, the envelope of the pulse at the EDFA output port (*z*=*L*) can be derived as

where *f*(*t*)=*A*(0,*t*) is the envelop of the initial pulse at the input port of the EDFA.

After the pulse exits the EDFA, it is spectrally reshaped by the optical filter. Writing the normalized filter transmission *F*(*ω*)=exp[-(*ω*-*ω*_{f}
)^{2}/Ω^{2}] as

and identifying *ω*-*ω*
_{0}(in the frequency domain) as *i∂*/*∂t* (in the time domain), the reshaped envelope becomes

with

Once the pulse leaves the gain filter, it will enter the intensity modulator, of which the transfer function for the field can be effectively modeled as

where Γ represents the combined loss including contributions from both the modulator and the output coupler, *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.

It should be pointed out that in writing the modulator transmittance as in (7), we have actually assumed that *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.

We now move to the discussion of the circulating pulse analysis. Let us first denote *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

Equation (8) is the basic equation for the iteration analysis. Note that the idea of iteration requires $\underset{n->\infty}{\mathit{Lim}}=\left[{f}_{n+1}\left(t\right)-{f}_{n}\left(t\right)\right]=0$. However, this requirement cannot be satisfied for *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.

From the definition of *P*_{n}
(*t*) and using Eq. (8), we obtain

Note that

$$={P}_{n}\left(t\right)+\left(N{T}_{m}-{\beta}_{1}L\right)\frac{d}{\mathit{dt}}{P}_{n}\left(t\right)+\frac{1}{2}{\left(N{T}_{m}-{\beta}_{1}L\right)}^{2}\frac{{d}^{2}}{\mathit{dt}}{P}_{n}\left(t\right).$$

To explicitly introduce the condition of exactly tuned harmonic mode-locking, recall that *β*
_{1} is the value measured at the carrier frequency *ω*
_{0}(unknown) and *β*
_{1}=${\beta}_{1}^{*}$+${\beta}_{2}^{*}$
*ω*_{d}
(see equation (1)). Write the modulation period as *T*_{m}
=*T*_{m}
_{0}+Δ*T*_{m}
, we get

where *T*_{m}
_{0} has been defined from *NT*_{m}
_{0}=${\beta}_{1}^{*}$
*L* as the ** exactly tuned modulation period** for active harmonic mode-locking. Further write

*D*=${\beta}_{2}^{*}$

*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

Substituting Eq. (12) into Eq. (10), we obtain

Combining Eqs. (9) and (13), we are then led to

$$=\Gamma {e}^{\mathit{gL}}\left\{1+M\phantom{\rule{.2em}{0ex}}\mathrm{cos}\left({\omega}_{m}t+\alpha \right)\right\}\left\{1+\hat{R}\right\}\left\{1+\frac{\left(\delta -2D\xb7x\right)}{\Omega}\frac{d}{\mathit{dt}}+\frac{1}{2}\frac{{\left(\delta -2D\xb7x\right)}^{2}}{{\Omega}^{2}}\frac{{d}^{2}}{d{t}^{2}}\right\}{P}_{n}\left(t\right),$$

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

To further simplify Eq. (14), the modulator transfer function 1+*M*cos(*ω*_{m}*t*+*α*) is expanded to second order using the Taylor series. The result is

$$=\left(1+M\phantom{\rule{.2em}{0ex}}\mathrm{cos}\alpha \right)\left\{1-\frac{M\phantom{\rule{.2em}{0ex}}\mathrm{sin}\alpha}{1+M\phantom{\rule{.2em}{0ex}}\mathrm{cos}\alpha}{\omega}_{m}t-\frac{M\phantom{\rule{.2em}{0ex}}\mathrm{cos}\alpha}{2\left(1+M\phantom{\rule{.2em}{0ex}}\mathrm{cos}\alpha \right)}{\omega}_{m}^{2}{t}^{2}\right\}.$$

From Eqs. (14) and (15), the iteration equation for *P*_{n}
(*t*) is finally derived as

with

$$+\frac{1+\frac{1}{2}{\left(\delta -2D\xb7x\right)}^{2}-iD}{{\Omega}^{2}}\frac{{d}^{2}}{d{t}^{2}}+\frac{\left(\delta -2D\xb7x-2i\xb7x\right)}{\Omega}\frac{d}{\mathit{dt}}.$$

When the system is close to equilibrium, the change of *P*_{n}
(*t*) per round trip is small, therefore, the iteration equation (17) reduces to differential equation [1], which reads

In the steady state, we have *∂P*_{n}
(*t*)/*∂n*=0, Eq. (18) then becomes

with

Note that the master equation (19) has the solution

where $\Delta \tau =\mathrm{sin}\alpha \u2044{\omega}_{m}\mathrm{cos}\alpha +d\u2044\sqrt{4\mathit{bc}}$, and ${\tau}^{2}=\sqrt{4c\u2044b}$.

In Eq. (21), the pulse envelope function has been obtained for the steady state. We now self-consistently determine the normalized carrier frequency shift and the pulse-modulator phase lag from the following two arguments: First, we require that Im[Δ*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).

The above two arguments then lead to

Generally speaking, for a given *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.

Regarding the strong dispersion case, since *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

Substituting Eq. (24-1) into Eq. (22), the phase lag can be calculated as

Note that from Eq. (24-1), the change of the carrier frequency of the circulating pulse can be expressed as *ω*_{d}
=*N*·Δ*T*_{m}
/${\beta}_{2}^{*}$
*L*. Identifying *N*·Δ*T*_{m}
as the mismatch of timing for the circulating pulse per round trip, and ${\beta}_{2}^{*}$
*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.

Regarding the weak dispersion case, since *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

Substituting (25-1) into (22), the pulse-modulator phase lag then satisfies the following equation

From Eqs. (25-1) and (25-2), we see that for the weak dispersion case, when the detuning *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] and Li [3].

Using the above analytic theory, in Fig. 2 and Fig. 3, we show the calculated carrier frequency shift, and the pulse-modulator phase lag, as a function of the normalized detuning *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

The object of numerical simulation is an active harmonic mode-locked fiber laser, of which the intracavity band-pass filter has a bandwidth of 1nm (FWHM). The applied RF modulation frequency is set to be around 10GHz. Note that in order to make the numerical results consistent with the analytic results, effects such as fiber nonlinearty as well as higher order dispersion are not considered in the numerical model. The modulation depth *M* used in the simulation is chosen to be one.

The simulation steps are as follows: initially, a CW signal with random complex amplitude is used as the input seed, which is then injected into the system and undergoes many circulations until a converged result is obtained. For each circulation, the signal is first transformed from the time domain into the frequency domain using the technique of fast Fourier transformation so that the effect of optical filtering and cavity dispersion can be included. After this step, the signal is then inversely transformed from the frequency domain back to the time domain with the help of inverse fast Fourier transformation. The obtained time domain signal is then time delayed (as caused by the detuning) and amplitude modulated (as caused by the modulator), which completes the last step of the current round pulse circulation.

Using the numerical model described above, and setting the loop dispersion to different values, we have obtained the simulated carrier frequency shift, and the pulse-modulator phase lag as a function of the normalized detuning. The results are plotted in Fig. 4 and Fig. 5. From these figures, we see that they agree with the previous analytic calculations as being plotted in Fig. 2 and Fig. 3. The agreement is especially good for the strong dispersion case where *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.

To further test our theoretical analysis, experimental measurements are also performed in this work. The details of the constructed and measured active harmonic mode-locked EDFL are as follows. A segment of erbium fiber provided by AT&T is used as the gain medium. A tunable filter from New Port is used to introduce a spectral window with FWHM bandwidth of 1nm. A 10GHz Mach-Zehnder intensity modulator from JDSU is used to modulate the cavity loss. Note that since the erbium fiber used in our experiment has a relatively large normal dispersion (-75 ps/km/nm) and the length of the erbium fiber is relatively long (25 meters), the cavity dispersion is therefore dominated by the contribution from the erbium fiber. The cavity longitudinal mode spacing of the constructed fiber laser is measured to be 5MHz. The intra-cavity average optical power is kept to be less than 1mW in order to avoid effects caused by fiber nonlinearty. The active modulation frequency is adjusted around 10GHz to vary the detuning. It is very important that both the modulation signal RF power and the modulator bias must be carefully controlled in order to operate the modulator in the linear regime so that the approximation of the modulator transfer function in Eq. (7) and the assumption that the modulation depth *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.

In Fig. 6 and Fig. 7, we plot the measured carrier wavelength shift, and the pulse-modulator phase lag as a function of the modulation frequency detuning. The applied modulation frequency detuning was changed between -400kHz to 400kHz, corresponding to round trip delay from -8ps to 8ps. Using the definition of the normalized frequency detuning, this 400kHz detuning maps to *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

In conclusion, we have studied the properties of the detuned operation of actively harmonic mode-locked fiber lasers with finite cavity dispersion being considered. Analytical formulas regarding the carrier frequency shift and the pulse-modulator phase lag as a function of the modulation frequency detuning have been obtained. The results of the theoretical analysis agree well with both the numerical simulations and the experimental measurements.

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