FM Mode-Locking Fiber Laser

: In this paper the study of Frequency Modulation Harmonic Mode-locking for Ytterbium Doped Fiber Laser is presented. The model studied, uses ytterbium-doped, single mode fiber pumped by 976 nm laser source is used with 150 mW pumping power to produce 1055 nm output laser and Frequency Modulation Harmonically Mode-Locked by MZI optical modulator. The effect of both normal and anomalous dispersion regimes on output pulses is investigated. Also, modulation frequency effect on pulse parameters is investigated by driving the modulator into different frequencies values. This study shows the stability of working in anomalous dispersion regime and the pulse compression effect is better than counterpart normal regime, due to the combination effect of both negative(Group velocity dispersion), GVD and nonlinearity. Also it shows the great effect of modulation frequency on pulse parameters and stability of the system. Model-locking fiber laser master equation is introduced, and using the assumed pulse shapes for both dispersion regimes after modifying (Ginzburg-Landau equation), GLE and by applying the moment method, a set of five ordinary differential equations are introduced describing pulse parameters evolution during each roundtrip.To solve these equations numerically using fourth-fifth order, Runge-Kutta method is performed through MatLab 7.0 program.


‫ﺍﻟﻤﻘﻔﻠﺔ‬ ‫ﺫﻭﺍﻷﻨﻤﺎﻁ‬ ‫ﺍﻟﺒﺼﺭﻴﺔ‬ ‫ﺍﻷﻟﻴﺎﻑ‬ ‫ﻟﻠﻴﺯﺭ‬ ‫ﺍﻟﺘﺭﺩﺩﻱ‬ ‫ﺍﻟﺘﻀﻤﻴﻥ‬
Mode-locked lasers are routinely used for a wide variety of applications since they can provide optical pulses ranging in widths from a few femtoseconds to hundreds of picoseconds.As early as 1970, an analytic theory was developed for determining pulse parameters and shape in actively mode-locked solid-state lasers by considering the effects of the mode-locker and gain filtering and then imposing a selfconsistency criterion in the time domain.In many cases, it is possible to include the effect of chromatic dispersion on the pulse shape as well; however, once the nonlinear effects within the cavity become important, analytic investigations begin to falter.The term A (t, z), represents the slowly varying envelope of the electric field and the pulse slot is calculated by following equation: Where r F is the frequency at which the laser is mode-locked, which is often denoted m F as modulation frequency.N is an integer ) 1 N ( ≥ representing the harmonic at which the laser will mode locked.R T , is the roundtrip.The gain medium's finite bandwidth is assumed to have a parabolic filtering effect with a spectral full width at half-maximum (FWHM) which is given by following relation: [5]

T g i
, results from the gain.T he physical origin of this contribution is related to the finite gain band width of the doping fiber and is referred to as gain dispersion ) T g ( 2 2 since it originates from the frequency dependence of the gain.[5] The term (A) is the slowly varying envelope of the electric field in term of T, the propagation time, which is given by the following relation: In the master equation, Eq. (1), there are two time scales which represent: 1.The time (t), measured in the frame of the moving pulse.closed form solution, passive mode-locking mechanisms, such as nonlinear polarization rotation, nonlinear fiber-loop mirror, and saturable absorption, produce "soliton-like" pulses.
The purpose is to obtain the basic equation that satisfies propagation of optical pulses in single-mode fibers.Then the equations that concern the evolution of pulse parameters during each roundtrip will be introduced.These equations will be solved numerically using fourth-fifth order Runge-Kutta.

Mode-Locking Fiber Laser Master Equation:
A general "master" equation used to model mode-locking fiber laser system is introduced.This equation, is in fact a Generalized Non-Linear Schrödinger Equation (GNLSE) or (Ginzburg-Landau equation) [5,6] which, generally describes all types of mode-locking fiber lasers by just changing the term M (A, t) that represents the mode-locker technique.The mode-Lock master equation is: [1,7,8] ( ) Where, the saturation gain g , could be approximated as in following relation: [7,6,8] ) Where, sat P represents the saturation power of the gain medium, o g the average small- signal gain, and, ave P the average power over one pulse slot of duration m T , which be calculated as in the following equation: [9,10]

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To solve it numerically by one of numerical solution method such as split or wavelet method [3], all parameters need to be considered including TOD and modelocker effect.The last two parameters have great effect on pulse shape and stability.
As a result, to solve such third order partial deferential equation, initial conditions and boundary conditions need to be known.Moreover, longtime required for computer program to implement such numerical solution.
It is useful to convert this third order partial differential equation to a set of ordinary differential equations which describe the evolution of pulse parameters during each roundtrip.[7] These pulse parameters evolution equations are obtained using so-called moment method.[10,13] Using moment method pulse parameters equations, it is possible to study the pulse evolution process under the effect of Eq. ( 1), with no need to full numerical simulation.

Pulse Parameters Evolution Equation:
Depending on master equation Eq. (1) and using the assumed pulse shapes for both dispersion regimes after modifying Ginzburg-Landau Equation, GLE, by adding TOD and mode-locker effects, the extended solution will be as in the following relations: [7,9,10,13].a.Normal regime:

The propagation time (T).
Since an average over a single roundtrip is considered, (T) is measured in terms of the roundtrip time: Where: : L The cavity length n : Refractive Index The pulse time scale is assumed to be sufficiently smaller than R T and hence, the two times are essentially decoupled.[11] This treatment is valid for most mode-locked lasers for which R T exceeds 1 ns and pulse widths are typically less than ps.
The effect of FM mode-locker on the field is sinusoidal and as in the following expression: [ represent the second-order dispersion, third order dispersion (TOD), loss, and nonlinearity, respectively, are averaged over the cavity length.

Moment Method:
As shown in previous section, mode-locked fiber lasers are governed by nonlinear partial differential equation Eq. ( 1), which generally, does not posses analytic solution [10] and, hence to model the pulse that produces.Another drawback, it gives the final pulse shape, it does not explain how does the pulse evolutes during each roundtrip.

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Since the accuracy of this approach depends upon the knowledge of exact pulse shape, it needs for master equation Eq. ( 1) to be solved numerically for more accuracy.A numerical solution for these five equations, using the fourth order Runge-Kutta method demonstrates that steady-state values obtained are deviate from the values obtained by direct solution of Eq. (1) by: (less than 3%) in the anomalous dispersion regime and (less than 12%) in the normal dispersion regime.[7,10] Hence, these results justify the use of the moment method and the pulse shapes assumed.
Where pulse parameters for both profiles are: (a) : pulse amplitude, (ξ) : Temporal shift, (τ) : pulse width, (q) : chirp, (Ω) : frequency shift, 0 i ikT ϕ + : represents the phase and rarely is of physical interest in lasers producing picoseconds pulses, which will be ignored.[14] A set of five equations are introduced (Appendix A) describing pulse parameters evolution during each roundtrip: [7,10,14] A numerical solution will be done for both normal and anomalous dispersion regime.In normal dispersion regime, GVD has positive values i.e. 2 β > 0 and solution will be done for both cases of TOD:

M2 External Mirror
Polarizing Beam Splitter Since the work is below zero dispersion wavelength (1.22 µm), (the used laser wavelength is 1.055 µm) where normal dispersion GVD must compensated, a grating pair usually used to derive the system into anomalous regime.[15,16] This will make the balancing between negative GVD and nonlinearity leading to pulse compression [17].See Fig. (1) describing the simulation.Where: WDM: Wavelength Division Multiplexing PBS: Polarizing Beam Splitter.

Results And Discussions:
Five equations are introduced using moment method which describes pulse parameters evolution at each round trip.To solve these equations numerically, a Mat-lab program has been written using fourth-fifth order Runge-Kutta method which uses the function ODE-45.This method is used to solve ordinary differential equations numerically.Ytterbium doped fiber pumped by 976 nm laser source is used with 150mW pumping power to produce 1055 nm output laser.[9] Mat-Lab 7.0 program uses the constants in Table (1), with the following initial values for pulse parameters:

Effect Of Changing Modulation Frequency Of Pulse Parameters:
Since our model considers FM-Mode-locking fiber laser type, we will study the effect of changing modulation frequency on pulse parameters evolution using the same privies (table 1) but for different values of Fr (Fr=2.5-30GHz).In both dispersion regimes the effect of changing modulation frequency, on pulse parameters is as shown in

Conclusion And Future Work:
In the present study and its numerical results could be concluded regarding comparison between both dispersion regimes, and modulation frequency effect as follow: • For variable frequency modulation, it is obvious that it affected on all pulse parameters in addition to the system stability without exception.

:
The delay between the center of the modulation cycle and the temporal window in which the pulses are viewed, and m ω : Modulation frequency (assumed to be identical to that of the mode-locked pulse train in this work), i.e.The over bar in Eq. (1) refer to the averaged value of the corresponding parameter.For example, ...( .......... .......... .........In the case of an auto solution, 1.

Fig ( 1 )
Fig (1) Block diagram model of roundtrips is needed 4000 RT ss >> ..From the plot of pulse chirp versus pulse width as in Fig. (3), it is clear that state is far to achieve unless large number of roundtrips has to introduced.In Fig. (4b), pulse chirp and width evolution are shown during first roundtrips.

3 β.F
p s), so it is necessary to include the 3 β parameter, since it distorts the pulse by broadening it asymmetrically, thus producing a temporal and frequency shift.As shown in Fig.(2) (a and b), in the absence of TOD, no temporal shift (ξ = 0), while in the presence of TOD, at temporal shift is introduced with positive and negative oscillation , not symmetrical around zero axis, 90 ≈ ξ ∆ fs (variation between maximum and minimum) finally it converges to zero.The same effect for TOD on pulse frequency shifthas a value, a negative frequency shift is introduced with negative oscillation, then decreasing with increasing roundtrips, converges to zero steady state.As shown in Fig. (2) for frequency shift plot, most frequency shift is negative however, GHz at RT = 4000.While to achieve zero frequency shift, 4000 RT ss >> .In Fig. (4a), pulse frequency shift evolution during first roundtrips is shown.As shown in Fig. (3) (a) and (b), no effect for TOD on pulse chirp.In fact normal dispersion produces positive pulse chirp, oscillating around zero chirp axis until reaches zero, its steady-state value.Oscillation is symmetrical around zero chirp axes From plots of pulse width evolution as in Fig. (3) (a) and (b), for pulse width plots, a broadening in pulse width is introduced with maximum width changes much more than in anomalous regime, and unaffected for high r F values, i.e. 5 F r > GHz.