Factors Effect on the Saturation Parameter S and there Influences on the Gain Behavior of Ytterbium Doped Fiber Amplifier

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1 Australian Journal of Basic and Alied Sciences, 5(12): , 2011 ISSN Factors Effect on the Saturation Parameter S and there Influences on the Gain Behavior of Ytterbium Doed Fiber Amlifier O. Mahran, M. Salem, M.S. Helmi and G.D. Roston Faculty of Science and Physics Deartment, Alexandria University, Egyt. Abstract: A theoretical aroach for modeling the saturated single ass gain in a three level Ytterbium doed fiber amlifier is resented, relevant to the behavior of rare earth doed silica fibers. A basic aroximation considering the stimulated emission rate as a dimensionless arameter S, indeendent of the satial and frequency variables, allows obtaining analytic exressions for inut and outut um, amlified sontaneous emission and signal owers inside the fiber core. Many factors effect on the saturation arameter allowing us after comutation of a few analytic equations to redict the behavior of the gain Ytterbium doed fiber amlifier. Comaring the results of this model with two other models with the assumtion of an averaged inversion oulation, we obtain a good descrition of the gain behavior for Ytterbium doed fiber amlifier. Key words: Saturation arameter-ytterbium doed fiber amlifier-otical fiber amlifier-three level laser systems. INTRODUCTION Lasers with high outut owers are required for a number of alications, e.g. for material rocessing, large scale laser dislays, remote sensing, medical alications, military alications and laser induced nuclear fusion. In fiber otics to deliver the outut ower for long distance such in free sace communications, a laser system consisting of a seed laser and a laser amlifier for boosting the outut ower. Master oscillator ower amlifier (MOPA) is a device which directly roduces the required outut ower (Dr. Rüdiger Paschotta, 2008). High ower fiber lasers and fiber amlifiers are nearly always realized with rare earth doed double clad fibers, which are umed with fiber couled high ower diode bars or other kinds of laser diodes. The um light is launched into an inner cladding rather than into the fiber core. MOPA is usually uses a cladding umed high ower amlifier, often based on an ytterbium doed fiber. The descrition of high ower amlifiers such as side umed fibered amlifiers consists of both um and signal saturation. Several analytical or semi-analytical have been develoed, and the commonly used model considers an averaged inversion oulation of the metastable level. H. Coïc and et.al (2002), (2003) had develoed a new analytical model deended on the saturated single ass gain in a three level fiber amlifier, relevant to the behavior of rare earth doed silica fibers. A basic aroximation considering an averaged value of the deleted um over the length of the fiber allows us to obtain analytic exressions for inut and outut um ower, amlified sontaneous emission (ASE) and signal ower in the fiber. We show that, these exressions can be described with the hel of a unique arameter S shown to be fully reresentative of the saturation in the medium. This model is very useful for many alications because it gives a good value for the effective gain under saturation. In this aer, a theoretical aroach for modeling the saturated single ass gain in a three level Ytterbium doed fiber amlifier is resented, relevant to the behavior of rare earth doed silica fibers. A basic aroximation considering the stimulated emission rate as a dimensionless arameter S, indeendent of the satial and frequency variables, allows obtaining analytic exressions for inut and outut um, amlified sontaneous emission and signal owers inside the fiber core.we show the S arameter is determined by solving the hoton balance equation. The model takes a simle analytical form, which can be searated into two arts, below and above the saturated absortion length, which is a function of S. Comare to other models with the assumtion of an averaged inversion oulation, we obtain a good descrition of the gain behavior of Ytterbium doed fiber amlifier. 2. Model: a- (Basic Equations): The behavior of rare earth doed fiber devices can be described in terms of rate equations for the oulation inversion densityn 2 (z), the um field P(z), the signal field P s (z) and the ASE P f (z), where z is the location along the fiber. In the set of differential equations detailed below, we make aroximations in the modal overlaing between um and signal with the fiber core consisting of bringing overla factors which are indeendent of ower and geometrical arameters. If we assume that the relaxation time τ f is fast comared with the metastable level and that the doant concentration and field distribution are uniform across the fiber core, Corresonding Author: O. Mahran, Faculty of Science and Physics Deartment, Alexandria University, Egyt. o_mahran2003@yahoo.com 2010

2 and moreover if we have monochromatic uming, the comlete rate equations for the um field, the signal field and ASE can be written as (Jarabo, S. and M.A. Rebolledo, 1995). dp ( z ) absp ( z) N tot (1 ) N 2( z) esap( z) N 2( z) (1) dz dps ( z, ) s e( ) Ps ( z, ) (1 ( )) N 2( z) ( ) N tot (2) dz dps ( z, ) s e( ) Ps ( z, ) (1 ( )) N 2( z) ( ) N tot (3) dz dpf ( z, ) s e( ) Pf ( z, ) (1 ( )) N 2( z) ( ) N tothn 2( z) dz dpf ( z, ) s e( ) Pf ( z, ) (1 ( )) N 2( z) ( ) N tothn 2( z) dz (4) (5) Here, the overlaing factors η and η s are resectively the roortion of the um and signal roagated within the fiber core, P(z)is the ower of the um, P s + (z,υ), P s - (z,υ), P f + (z, υ) and P f (z,υ)are resectively the ower sectral densities of the injected co-roagating and counter-roagating signals and ASE. N tot is the doant concentration, N 2 (z) is the metastable level density oulation. σ abs, σ esa,σ e (υ) are the um absortion, um ESA and stimulated emission cross section. δ=σ e (υ )/σ abs is the ratio between the stimulated emission and absortion cross section at the um wavelength, and α=σ a (υ)/σ e (υ) is the ratio between the cross section of the signal absortion and stimulated emission. α (υ)=1for an ideal three level system, and 0 for a ure four level scheme. In the ASE ower exression, the factor hυδυ is the noise ower corresonding to one hoton er mode in bandwidth δυ; if the two olarizations can be roagated in the fiber, the noise ower becomes 2hυδυ.The um and stimulated emission rates are given by (Coïc, H., 2002). abs W ( z) P( z) (6) h a and se( ) WS( z, ) Ps ( z, ) Ps ( z, ) Pf ( z, ) Pf ( z, ) (7) ha where a is the core area of the fiber. The last arameter used to describe the medium is the density of the metastable level N 2 (z), which is given, for equilibrium, by (Coïc, H., 2002). N ( z) N 2 tot W ( z) ( ) Ws( z, ) d W ( z)[1 ] (1 ( )) W ( z, ) d1 s f (8) We introduce a new variable S with no dimension, and no deendence on z or υ. we can relace in the exression of N 2 (z) the quantities Ws(,) z fd and () W (,) s z fd by S and α m S resectively, where α m is the mean value of α(υ)over the signal emission, and S is the new arameter describing the saturation of the medium. We can introduce some normalized arameters in order to solve the differential equations. We now define the following arameters in order to give simlified forms of the exressions (Coïc, H., 2002): ( ) ( ) 0 se N, tot absn, (1 S) ms tot 1, 1 (1 m ) S 1, N esa 2, S m With the hel of these arameters, we can write a reduced form of the initial set of equations (1)-(5) governing the evolution of um, signal and ASE. To achieve the normalization of the set of equations, we shall use the exression of the um P (z) defined as 2011

3 Pz ( ) PP ( z) with hpa Psat P sat abs P f P (z) is a dimensionless arameter and we can exress the set of differential equations as functions of P (z), β, η, ε and γ 0 (υ). Because the arameter S does not deend on z, the set of differential equations (1)-(5) describing the amlification can be integrated analytically without any restrictions on the values of um, signal and ASE owers; we only suose that there is no excited absortion ( b = 0). b. Analytical Solutions to the Rate Equations: b.1. Analytic Exressions for Signal and ASE Power: The integration of the differential equations (2)-(5) for the signal and ASE has been already erformed by Jarabo and Rebolledo (1995). Using the reduced arameters, the signal and ASE owers can be exressed as a function of the gain G (z,υ). P ( z, ) P (0, ) G( z, ) (9) s s G( L, ) Ps ( z, ) Ps (0, ) (10) G( z, ) 1 uz G( z, ) Pf ( z, ) P0 G( z, ) 1 0( ) ( ) Pf (0, ) G( z, ) 1 ( ) u 0 Gu (, ) (11) 1 G( L, ) ul G( u, ) G( L, ) Pf ( z, ) P0 1 0( ) ( ) Pf ( L, ) 1 ( ) G( z, ) u zg( z, ) G( z, ) (12) with 0 ( )(1 ( )) (1 ) 1 ( ) P ( ) z 0 z 1 P (0) G( z, ) ex ( ) (13) These exressions are functions of the two unevaluated comonents S and P (z). b.2. Analytic Exression for Pum Power: The equation governing the um absortion with ESA (1) can be exressed with the normalized arameters by (Coïc, H., 2002) (14) The integration of this equation is straightforward and rovides a transcendental equation which allows us to calculate the um distribution along the fiber, but this exression is not easy to maniulate. The results of these calculations give us the analytical form of um ower equation P ( z) = hc( z). (15) with The ower series exansion show that the um distribution can be searated into two arts, below and above the saturated absortion length L0 defined by h + P (0) L0 = (16) b 2012

4 c. Simlified Procedure of Resolution the Gain: A faster method to find the arameter S is also ossible by using a model where the inversion of oulation N 2 (z) is considered uniform along the fiber (Georges, T. and E. Delevaque, 1992; Nielsen, A.O., 1991). In this case, the um ower P (z) from (1) becomes urely exonential. Introducing the value of P (z) in exression (13), we find (17) Here 1 z= L N 2( z) = N 2( z) dz. L òz = 0 (18) The exressions (9)-(12) can be easily described with exonential functions, but, as we have mentioned before, the co-roagating and counter-roagating ASE ower exressions are identical in this model. In any case, the hoton balance equation to solve takes a simle form and gives a numerical value of N 2 (z) which is in fairly good agreement with the numerical calculus. To obtain the arameter S which allows us to make discrimination between co- and counter-roagating ASE, we just have to solve the equation (19) The calculation of the averaged value of the oulation inversion <N 2 (z)> becomes necessary in the case where the averaged three level coefficient α m is not known a riori. RESULTS AND DISCUSSION Introducing the analytic exressions of the um, signal, ASE as described in mathematical model, the comutation of the hoton balance equation gives the value of S. Relacing the value of S inside the exressions of um, signal, ASE co-roagating, ASE counter-roagating and gain, we obtain the sectral distribution of the gain. The numerical solution of the couled differential equations is achieved using the Runge-Kutta method, this art is erformed using Matlab (version7) rogram for gain calculations. A faster method to calculate the sectral distribution of the gain, by determining the average value of oulation inversion along the fiber (19), which can be described as semi-analytical solution model or <N 2 > model. Finally, we make a comarison between three different models, which are numerical model, S model and <N 2 > model using the arameters as listed in table 1. Table 1: Yb 3+ : SiO 2 arameters used to our simulation (Mahran, O., 2010; Coïc, H., 2002). Physical quantity Symbol Value Pum wavelength λ 974 nm Signal wavelength λ s 1053 nm Length of fiber L 20 m Overla factor of the um η 0.65 Overla factor of the signal η s 0.65 Fluorescence time τ f 0.77 ms Ratio between σ a/σ e α Ratio between σ e/σ abs δ 1 Noise ower P µw Saturation ower P sat 1.5 mw By using the hoton balance equation the values of S, and the otimal length are resectively S=1.183 and L 0 =11.9 m. This is done taking into consideration the comarison between S, numerical and <N 2 > models. Fig. 5.5 describes the evolution of the gain in db as a function of the saturation arameter S. We observe that for a low um ower, the gain decreases exonentially with S. this is due to the saturation of the stored energy. For a high um ower, the behavior of the gain is not exonential because, for a low value of S, the medium is totally inverted and the gain is equal to η s σ e (υ f ) N tot. In reality, for a high uming ower, S is always greater than zero. This is due to the existence of ASE Signal Inut Power Effects: For each inut signal ower, we calculating the hoton balance equation as described in the mathematical model of the value of S, which is imortant for analytical solution. The value of S inserted into the exressions of signal outut ower, ASE+ ower, ASE- ower and signal gain to evaluate their values for each inut signal 2013

5 ower value. After calculating S value, we take into account the effect of signal inut ower on the saturation arameter S as shown in Fig. 2. Fig. 1: Gain in db versus saturation arameter S for different values of dimensionless inut um ower arameter P (0). Fig. 2: Effect of signal inut ower on saturation arameter S. From this figure we can see as the inut signal ower increases for the same inut um ower, the values of saturation arameter S is increasing. This increasing in S arameter values is due to the deendence of S arameter on the rate of uming and the rate of stimulated emission. So, as inut signal ower increases the rate of stimulated emission increases and also the value of S arameter increases. Table 2: S values over signal inut ower. Signal inut ower S value 1 μw μw μw μw μw μw μw μw μw μw

6 Fig. 3: Effect of signal inut ower on otimal length. Fig. 3 reresents the relation between the otimal length and signal inut ower. As the inut signal ower increases the otimal length decreases. The decrease of the otimal length is due to the deletion of the um when the saturation arameter increases. So, we take the same inut um ower for all signal inut ower. The increase of the inut signal ower causes an increase in saturation arameter and this reduces the otimal length to comensate the deletion of the um for increasing the S arameter values. It is obvious that as the inut signal ower increases the outut signal ower also increases. This doesn t mean that the gain (db) must be increase. So, the next figure is devoted the behavior of gain in db with changing the inut signal ower. Fig. 4: Effect of signal inut ower on signal gain for numerical, analytical and average models. For the gain behavior versus inut signal ower, the three models show good agreement for the calculated values of the signal gain. Table 3 reresents some imortant values of the gain for different models. Table 3: Comarison between the gain of different models over signal inut ower. Signal inut ower Numerical model S model <N 2> model 1 μw db db db 5 μw db db db 10 μw db db db 3.2. Inut Pum Power Effects: After comutation of hoton balance equation to calculate S arameter for different inut um owers from 50 mw to 1000 mw and also for different inut signal ower, we obtain Fig

7 Fig. 5: Effect of um ower inut ower on saturation arameter S. S arameter versus inut um ower shows an increase in the arameter S value with increasing inut ower. It is seen also, that the increase of signal ower leads to the increase of S values. So, the most imortant arameter which affects the saturation arameter S is the values of owers for both um and signal. Table 4 gives some imortant values of S arameter. Table 4: Some S values for different signal ower at different inut um ower. Inut um ower S value (signal 1μW) S value (signal 5μW) S value (signal 10μW) 200 mw mw mw mw mw Fig. 6 shows the effect of uming ower on the otimal length. For each inut signal ower, the inut um owers increase and also the otimal length. When inut um ower exceeds a certain value, in our case 350 mw, the change in otimal lengths are very close. This is because, at this inut um ower, the gain reaches the saturation. For other signal inut ower (i.e, 5 μw and 10 μw), we notice that otimal length values are decreasing. This behavior is due to the increase of the S arameter values and to comensate deletion of the um. Fig. 6: Effect of um ower inut ower on otimal length for different signal inut ower. To comlete resolution on the effect of changing inut um ower, we must take into the account the effect of uming ower on signal gain. In Fig. 7, the grah reresents the relations between inuts um ower and signal gain. The figure shows as the inut um ower increases the gain increasing over a range of inut owers. Then the gain reaches the saturation at inut um ower aroximately equal to 350 mw. At this inut ower gain is 31.8 db. Maximum gain with our simulation at 1000 mw is about db. 2016

8 Fig. 7: Effect of inut um ower on signal gain for numerical, analytical and average models. The S model shows identical gain behavior with the numerical one. Also, <N 2 > model has a good agreement with other models. The table 5 reresents some results of the gain comaring with different models. Table 5: Comarison between the gain of different models over inut um ower. Inut um ower Numerical model S model <N 2> model 50 mw db db db 200 mw db db db 500 mw db db db 1000 mw db db db This behavior of the gain over inut um ower of Yb 3+, likes the behavior of the gain of Er 3+. A similarity of gain behavior of Yb 3+ and Er 3+ is due to that they have aroximately the same system energy levels (three level system) Core Radius Effects: After comutation the hoton balance equation and taking into account the effect of changing core radius, we obtained the following values in the table 6. Table 6: Some S arameter for different core radius. Core radius S value 1 μm μm μm μm μm Fig. 8: Effect of core radius on saturation arameter S. 2017

9 We see from Fig. 8 that the S arameter values at small core are small, not more than 1.7, but for large core radius S value becomes high values. So, if S value is high, it will reduce the otimal length and causes a decrease in the signal gain as shown in Fig. 9 and Fig. 10. Fig. 9: Effect of core radius on otimal length. Fig. 10: Effect of core radius on signal gain for numerical, analytical and average models. Table 7: Comarison between the gain of different models over core radius. Core radius Numerical model S model <N 2> model 1 μm db db db 2 μm db db db 3 μm db db db 4 μm db db db 5 μm db db db 3.4. Doant Concentration Effects: Doant concentration does not effect on the S arameter. All comutation results occur for the same value of S. Fig. 11 shows that as the doant concentration increases the otimal length decreases. The figure reresents also that increasing inut ower, otimal length will increase but with increasing doant concentration the otimal length decreases to small values with small differences from one ower to another. This decreases of otimal length to comensate deletion of the um which gone to increases ions in uer level er unit volume. 2018

10 Fig. 11: Effect of doant concentration on otimal length for different inut um ower. In Fig. 12 we reresent the effect of doant concentration on the signal gain. From figure we notice that as the doant concentration increasing the gain (db) is also increasing until a certain concentration almost around 350 m and the gain remain constant over a wide range of concentration. Fig. 12: Effect of doant concentration on signal gain for numerical, analytical and average models. The analytical, numerical and average models show good agreement for the calculating gain. The following table 8 shows some gain values for these models. Table 8: Comarison between the gain of different models over doant concentration. Doant concentration Numerical model S model <N 2> model 100 m db db db 500 m db db db 1000 m db db db 4. Conclusion: In this aer, many factors effect on S arameter discussed for Ytterbium doed fiber amlifier. The S arameter varies from zero to 10. We obtained the best gain with small values of S (0 2), which show that the medium is totally inverted and the saturation occurs at this small values of S arameter. The most imortant factor effect on S arameter is the values of inut ower for both um and signal. For signal ower and core radius effects, the S values increase but that reduces the gain. For um ower effect, the S values increase and that increases the gain. S arameter does not deend on doant concentration factor. The results of the comutations resented in this aer show a good agreement among the three models in the case of redicting the behavior of the gain and its values. The most imortant advantage of S model is the short comutation time. The gain value aroriates for amlification of signals with very low inut signals ower. These signals around 1 μw, which roduced in master oscillator ower amlifier as a seed laser 2019

11 amlified by ytterbium as doed fiber amlifier. In our case of Yb 3+ :SiO 2 doed fiber amlifier, we obtained the best gain around 27 db with inut um ower 200 mw and core radius 2 μm for doant concentration around 500 m. REFERENCES Coïc, H., "Analytic modelling of high-gain ytterbium-doed fibre amlifiers", Journal of Otics A: Pure and Alied Otics, : Coïc, H., F. Leconte, J.F. Gleyze, A. Jolly, "Analytic Modeling of Three level fibered amlifiers," Proceedings of SPIE, Dr. Rüdiger Paschotta, Encycloedia of Laser Physics and Technology, Wiley-VCH Verlag GmbH & Co. Georges, T. and E. Delevaque, "Analytic modeling of high-gain erbium-doed fiber amlifiers," Ot. Lett., 17: Jarabo, S. and M.A. Rebolledo, "Analytic modeling of erbium-doed fiber amlifier on the basis of intensity-deendent overlaing factors," Al. Ot., 34: Mahran, O., "Gain and noise figure of ytterbium doed lead fluoroborate otical fiber Amlifiers," Australian Journal of Basic and Alied Sciences, 4(8): Nielsen, A.O., J.H. Povlsen, A. Bjarklev, O. Lumholt, T.P. Rasmussen and K. Rottwitt, "Fast method for accurate rediction of fibre laser oscillation wavelength," Electron. Lett., 27: