Analysis Frequency Modulation Mode-Locked Fiber Laser by Using ABCD Rule

In this paper the study of F requency Modulation (FM) mode-locking fiber laser is presented. The time-domain ABCD l aw was employed to clarify the impacts of optical elements on the mode-lock pulses. ABCD matrices formalism in the time domain ha s been developed under a Gaussian paraxial approximation. Also, these matrices apply to the theory of the actively mode-locked fiber laser. The numerical results are obtained by using MATLAB (cid:210) software. The analysis was produced a shorter than 3.16ps duration pulses in laser mode locked with FM modulator driven a t repetition f requency of 10GHz an d cavity has a nomalous dispersion of - 0.015ps 2 /m an d nonlinearity of 0.02W -1 m -1 . The values of chirp are plotted vers us average power with many values of dispersion. All relationship between t he pulse laser parameters a nd other effective para meters like, second order disp ersion, nonlinearity, average input power, repetition frequency an d optical filter ba ndwidth are plot ted and discussed.


Introduction
Mode locking is a phenomenon in which the cavity resonant frequencies of the laser are forced to have matched phase shifts.This effect causes the waves to constructively interfere to form ultra-short pulses.The advance of ultrafast technique in the last decade has been able to develop lasers with a variety of features such as ultrashort pulse duration, high peak intensity and high repetition rate.Ultrashort laser pulse with GHz repetition rate will be useful for many applications in areas such as optical communications [1].
The low-loss transmission window of optical fibers has a bandwidth comparable to that of a 100-fs pulse, and therefore ultrashort-pulse technology may play an important role in optical communications.Sub-picosecond pulses have already been used for laboratory experiments demonstrating fiber optic transmission of data at Tbit/s (10 12 bit/s) rates.Ultrashort pulse sources with a high repetition rate are key components in optical time division multiplexing (OTDM) system [2].Ultrashort pulses may also prove important in wavelength-division-multiplexing (WDM) systems in which the fiber bandwidth is carved up into different wavelength bands or channels [3].Here ultrafast optics technology is important not only for pulse generation but also for signal processing, for data detection, and for the advanced metrology necessary for characterizing and optimizing ultrashort-pulse transmission [3][4][5].
Active mode-locking mechanisms are the result of a time varying optical or electronic exploitation of a material's response.FM modulation is discussed here as one types of active mode-locking techniques.However, actively modelocked lasers need a radio-frequency (RF) signal generator and a phase/intensity modulator, which results in a complicated structure and higher cost.Passively mode-locked lasers based on Er :Yb : glass as a gain medium have high fundamental repetition rate.However, a high fundamental repetition rate fiber laser, in contrast to a glass-or crystal-based laser, would be advantageous due to its compact size, robustness, flexibility, low cost, and compatibility with other optical fiber systems [6].
To examine a complicated optical system with many components (such as an eye, a telescope), it can be cumbersome to calculate object and image distances for each component.When the paraxial regime and one axis are considered, the straightforward matrix method to simplify calculation is used [7].
The analyzing of a fiber laser mode locked simultaneously by FM modulator at higher harmonics of the cavity repetition rate is discussed.The set of ABCD matrices and q-parameters are programmed and analyzed to get predicted values of pulse width and chirp parameter versus the average optical output power.The performance analysis of FM mode locking fiber laser is done by using MATLAB ® software.
This paper discusses FM mode locked fiber laser which is analyzed by using ABCD rules and q-parameters.Section 2 explains the modeling of mode-locked fiber lasers.Section 3 gives characterization of matrices of optical elements.In section 4, deals with simulation results of FM mode locking fiber laser.The summary of paper and present conclusion are discussed in section 5.

Modeling of FM Mode-locked Fiber Lasers
Mode locking occurs when phases of various longitudinal modes are synchronized such that the phase difference between any two neighboring modes is locked to a constant value φ such that (φ m -φ m+1 = φ).Such a phase relationship implies that (φ m = mφ + φ 0 ).The mode frequency ω m can be written as (ω m =ω 0 +2mπ∆ν) .For simplicity, it can assume that all modes have the same amplitude E 0 , the sum can be performed analytically and the result is given by [8] In an effort to both understand and predict the behavior of fiber laser, it is felt that the insight obtained by developing a solid model for such laser system is invaluable.Modeling the FM mode locking fiber laser shown in Fig. (1) is suggested to understand mention behavior.For such a model to be physically sound it must include [4]: i-The frequency modulator (FM) has a transfer function is: The optical band pass filter, the transfer function is: Where, Β f is the optical filter bandwidth.ii-effect of dispersion on fiber is represent in frequency domain by: Where, β 2 is a second order dispersion and z is a transmission distance.iii-fiber with nonlinearity which has transfer function as follow:

Characterization of Matrices of Optical Elements
The advantage of this matrix formalism is that any ray can be tracked during its propagation through the optical system by successive matrix multiplications.The transformation of the q-parameter of the Gaussian beam when it passes through the optical system is essential.This qtransformation is already given by applying Fourier optics but can also be derived by simple matrix multiplication [9,10] Compared with other complex numerical models, it is very convenient for easy derivation of analytical solution for understanding pulses characteristics.The pulse shape was assumed to be Gaussian which is defined by [2,11]: Where, P o is the peak power, τ is the 1/e pulse width, C is the linear chirp parameter.Therefore; the complex beam parameter is a complex number that specifies the properties of a Gaussian beam.It is usually denoted by q.
) 7 ( iC 1 q If a Gaussian pulse passes through optical element, the q -parameter at the output (q out ) in term of input q parameter (q in ), will satisfy the following law [11]: Where, ∆ FM is the phase modulation depth.ii-The ABCD matrix for optical band pass filter is [2,11] : iii-The ABCD matrix for dispersion on fiber is: iv-The ABCD matrix for fiber with nonlinearity is: For the Gaussian profiled pulses, the relation between the peak power and the average power is as followed: Where, f is the repetition (modulation) frequency of the pulse train.
Similar to the idea of the split-step algorithm in numeric simulations [11], it can divide the cavity fiber into many segments and consider the dispersion and the nonlinearity separately in each segment.The length of one segment is L/n, for cavity length is L and number of segment is n.
The ABCD matrix represent whole elements of fiber laser are expressed in equation ( 14).It can predict many parameters of FM mode locking fiber laser by compute this ABCD matrix.
From equation (7), we can write an expression for By using equation ( 8), we can express

Results and discussion
By solving the ABCD matrix and qparameter of the FM mode-locking fiber laser in equations (14-16) numerically, the evolution of steady-state pulse parameters was predicted.This numerical analysis is done by using MATLAB ® software.3) demonstrates the strength of ABCD law by studying the effect of both anomalous and normal dispersions on pulse width.Data points marked with symbols were obtained using the qparameter and ABCD rules for four cavity lengths.The pulse width is increasing slightly with increasing both the dispersion and cavity length in case of normal dispersion.While, the pulse width is decreasing both anomalous dispersion and cavity length are increasing.
The shortest pulses are obtained in the case of anomalous dispersion for small values of second order dispersion ( β 2 ).For anomalous dispersion as shown in   4) demonstrates how the second order cavity dispersion ( β 2 ) affect the chirp parameter (C) with various cavity length.The chirp parameter varies with β 2 in the anomalous and normal dispersion regimes.The chirp parameter (C) is positive with negative β 2 and vice versa.In case of large cavity length (L=2000m) and negative β 2, the chirp parameter of the pulse becomes small (0.12).
The effect of nonlinear parameter γ on the pulse width and chirp parameter are shown in Fig (5) and Fig.( 6) respectively.The pulse width ( τ ) is decrease with increase nonlinear parameter ( γ) in both anomalous and normal dispersion regimes.Also the pulse chirp parameter is increase with increase nonlinear parameter ( γ) in both anomalous and normal dispersion regimes The relationships of the pulse width and the pulse chirp versus the average power (P) are shown in Fig. (7) and Fig. (8), respectively.From Fig. (7), It can find that the greater laser power is favorable for obtaining shorter pulses in the anomalous dispersion region.In the normal dispersion region, the pulse width does not always decrease with the increasing of average power, but increases under a small average laser power.The relationship of the laser pulse chirp versus the average power (P) is shown in Fig. (8).It can note that, the pulse chirp parameter is becomes larger for average power more than 15mW in anomalous dispersion region.While, the pulse chirp parameter is always reduced when the average power (P) is increasing.
Fig. (9) shows the change of the pulse width and pulse chirp as a function of change the cavity length at various optical filter band width.It can see that the pulse width is decreasing with increase cavity length at normal dispersion of β 2 =0.015ps 2 /m and nonlinearity γ = 0.02W -1 m -1 .When the cavity length is increase, the chirp parameter began increase from certain value then it reach to a maximum value of 0.5 then return to decrease as cavity length increase.
Fig. (10) shows the effect of repetition (modulation) frequency on the pulse width of the stable output pulses.Larger modulation frequency leads to shorter pulse width.When, repetition frequency varies from 1GHz to 45GHz, the pulse width change from 3.16ps to 2.49ps at modulation depth of ∆ FM = 0.9.The effect of bandwidth of optical filter is illustrated in Fig. (11).It seems that large bandwidth filter is preferable.When larger band width of filter is used, the output pulses become shorter.Therefore, it can be consider that the filter band width as main factor to influence on pulse width.

Conclusion
This paper discussed the predicted results of FM mode locking fiber laser by using ABCD matrix and q parameter.The shortest pulses are obtained in the PDF created with pdfFactory Pro trial version www.pdffactory.comcase of anomalous dispersion rather than for normal dispersion.Also, the pulse chirp always decreases while the average dispersion varies from the normal region to the anomalous region.In both dispersion regimes, the increase average power lead to output pulse with shorter width.It is obvious that the repetition frequency affect strongly on pulse width where the pulse width is decrease with increase repetition frequency.PDF created with pdfFactory Pro trial version www.pdffactory.com Fig.(2) illustrated the flow chart of proposed method to calculate pulse parameter.The true strength of ABCD rule lies in predicting the steady-state values for mode-locked pulses and chirp parameter.The results are presented in both anomalous (D<0) and normal (D>0) dispersion regimes.Fig.( pdfFactory Pro trial version www.pdffactory.com

Fig.( 3
Fig.(3-a), the pulse width becomes equal 1.76 ps for β 2 of -0.025 ps 2 /m and cavity length of 2000m.But it can note the pulse width has larger value for smaller cavity length.Fig.(3-b) shows the pulse width is exceeds 3.16ps in the normal dispersion case.Also the pulse width becomes very high with larger cavity length and dispersion.It can note the pulse width approximately equal to 12ps at dispersion of β 2 = 0.025ps 2 /m and cavity length of 2000m.It can remember that the dispersion (D) is calculated by [12]: ) 17 ( c 2 D 2 2 β λ π − = In This work, the effect of cavity length, second order dispersion, repetition frequency, input power and modulation depth on the output laser pulse are studied.Fig.(4) demonstrates how the second order cavity dispersion ( β 2 ) affect the chirp parameter (C) with various cavity length.The chirp parameter varies with β 2 in the anomalous and normal dispersion regimes.The chirp parameter (C) is positive with negative β 2 and vice versa.In case of large cavity length (L=2000m) and negative β 2, the chirp parameter of the pulse becomes small (0.12).The effect of nonlinear parameter γ on the pulse width and chirp parameter are shown in Fig(5) and Fig.(6) respectively.The pulse width ( τ ) is decrease with increase nonlinear parameter ( γ) in both anomalous and normal dispersion regimes.Also the pulse chirp parameter is increase with increase nonlinear parameter ( γ) in both anomalous and normal dispersion regimesThe relationships of the pulse width and the pulse chirp versus the average power (P) are shown in Fig.(7) and Fig.(8), respectively.From Fig.(7), It can find that the greater laser power is

Figure ( 3 )Figure ( 4 )
Figure (3): Influence of second order dispersion on pulse width with various cavity length.a) Anomalous dispersion b) Normal dispersion

Figure ( 5 )
Figure (5): Affect of nonlinear parameter on the pulse width atanomalous and normal dispersion regimes.

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): Flow chart of proposed program to plot the pulse width and chirp versus average power PDF
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