# EDFA SIMULINK MODEL FOR ANALYZING GAIN SPECTRUM AND ASE. Stephen Z. Pinter

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2 Fig. 1. Simulink EDFA Module for EDFA dynamics shown below. t N 2 = P S (0, t[1 e B SN 2 C S ] + P P (0, t[1 e B P N 2 C P ] N 2 τ Equation (1 is the key equation for studying gain dynamics in an EDFA and is implemented in Simulink as shown in Fig. 1. The co-directional input pump power is P P (0, t and the input signal power is P S (0, t. These input powers are in photons/second and are related to the power in Watts by P P,S = P P,S (hν, where ν is the frequency in Hertz and h is Plank s constant in units of J/Hz [5]. For our purposes where we must consider the wavelength, the equation is rewritten as P hc P,S = P P,S λ, where c is the speed of light in m/s and λ is the wavelength in m (note: in [4], this relationship was incorrectly implemented and the results of the gain simulations were inaccurate. The output pump and signal powers are P P (L, t = P P (0, te B P N 2 C P (1 P S (L, t = P S (0, te B SN 2 C S (2 In equation (2, quantities B and C characterize the physical EDFA and are given by [B P, B S ] = α + β ρA [C P, C S ] = αl The scale factor 1/ converts decibels to log base e. This is important because the gain produced by the EDFA model is in base e and must be multiplied by to convert it to decibels. Multiple channels are shown in Fig. 1 by the numbers beside the block connecting lines. In this case there are 26 signal wavelengths and 1 pump wavelength. The EDFA module in Fig. 1 is called from the main Simulink model shown in Fig. 2. In this model, the input signal power is -30 dbm and the input pump power is 17.8 dbm Optimum Length Before performing simulations, the optimum length (L opt of the EDFA model was determined. The parameter [C P, C S ] is proportional to the length of the fiber and had to be recalculated for each fiber length. Fig. 3 shows the gain as the length of the amplifier varies from 0 m to 50 m. It is clear that the gain varies significantly from wavelength to wavelength, and that there are two distinct peaks, one (3

3 Fig. 2. Simulink Model Gain (db nm λ s 1570nm length (m Fig. 3. Dependance of gain on amplifier length for the range of wavelength s indicated Fig. 4. Emission and Absorption Coefficients [7] 3. HANDLING MULTIPLE WAVELENGTHS around 12 m and the other around 30 m. The wavelengths that reach maximum at around 12 m on Fig. 3 correspond to the wavelengths on the emission and absorption spectra (Fig. 4 where the peak occurs. The effect of the emission/absorption peak is of key interest so the optimum length is chosen as 12 m. This length is optimal only in the sense that the signal gain of the amplifier is maximized. When ASE is included, the optimum length is determined by maximizing the signal gain in the presence of ASE, which is also a function of the length [6]. The ASE dependance on length will be discussed in section 5. Multiple signal wavelengths are handled by allowing B S and C S in equation (3, as well as the input signal P S (0, t, to be multidimensional [3]. The input signal is wavelength dependent as shown by the formula P P,S the input signal power to the EDFA module had to calculated for each wavelength. Furthermore, the parameters that determine the wavelength dependency of B S and C S are α and β, the emission and absorption cross-section coefficients, respectively. For this project, the emission and absorption coefficients used were those from Fig. 4. It is important to note that simulating one wavelength at a time will give different results than simulating all the wavelengths simultaneously. EDFAs are designed to work in the nonlinear regime, so properties like linear superpo- = P P,S hc λ. So,

4 Gain (db Pump gain (db wavelength (nm wavelength (nm Fig. 5. Gain versus wavelength for 12 m length, λ P at 980 nm, pump power of 18 dbm and signal power of -30 dbm Fig. 6. Pump gain required for a flat gain of 30 db 4. GAIN FLATTENING sition don t hold. This is because when there are several channels in an EDFA there is an effect called gain stealing [8]. How much of the energy each of the channels takes from the pump depends on the details of the emission and absorption spectra [8]. The Simulink model in [3] uses only one channel along with the pump. For this project the signal was expanded to 26 wavelength channels. This provided an adequate representation of gain versus wavelength, however, more channels can be used to get a more accurate representation. So, an important relationship for EDFA s is found, i.e. gain versus wavelength. Using a length of 12 m, the small signal gain over 1520 nm λ S 1570 nm is plotted as shown in Fig. 5. An input signal power of -30 dbm is used because a large signal would drive the EDFA into saturation causing the difference in the gains at different wavelengths to be small. The ratio between the absorption and emission cross-sections at a particular wavelength is critical in determining the gain for the amplifier[6]. The shape of the gain in Fig. 5 is highly dependant on the parameters α and β, particularly the peak value. So, different variations in gain can be produced when using different cross-sectional data Simulation Time In regards to simulations, a certain amount of time is needed for the system to reach steady state ( 1 ms. This is important because when simulating over multiple wavelengths, a 1 ms time frame is allowed before the gain of the EDFA is recorded. To be cautious, the gain is recorded after 3 ms of run time. It is important to compensate for the non-uniform gain spectrum in WDM applications so that each wavelength experiences approximately the same gain. A different approach to gain flattening is considered in this paper. Usually the gain is flattened using a notch filter or a fiber Bragg grating, however in this paper I considered how gain flattening can be done using the pump signal only. If the gain can be flattened by varying the pump signal (according to a certain relationship, then there is no need for external filters. A relationship between the pump gain and the signal gain can be derived using equations (2 and (3 as follows. ln ( PP (L, t P P (0, t N 2 = C ( P PP (L, t + ln, B P P P (0, t (4 ( PS (L, t ln = B S N 2 C S P S (0, t (5 substitute equation 4 into 5 ( = ln(g S + C S B SC P B P ( BP B S Therefore, the final equation relating the pump gain to the signal gain can be represented as follows. ln(g P = ( ln(g S + C S B ( SC P BP B P B S In equation (6, the signal gain (G S is chosen to be 30 db, B P & C P are constant, and B S & C S vary with wavelength. The equation is plotted versus wavelength and is shown in Fig. 6. In Fig. 6 the pump gain is negative because the pumps energy gets transferred to the signal resulting in the amplification. This figure shows how the pump gain should vary over wavelength in order to achieve a flat signal gain. (6

5 ASE power (mw ASE Power (dbm length (m wavelength (nm Fig. 7. ASE power versus amplifier length for λ S of 1530 nm and λ P of 980 nm Comparing Fig. 5 and 6 it is clear that the location of the large peak in Fig. 5 (around 1530 nm is where the pump gain should be slightly larger than for the rest of the wavelengths. In a practical sense it might be difficult to obtain a different pump power for many different wavelengths so this approach to gain flattening is something to be further researched. 5. AMPLIFIED SPONTANEOUS EMISSION The forward ASE is now considered for the EDFA model. The forward ASE power is given in [6] by P ASE = 2n sp hν ν(g 1 (7 1 where, n sp = 1 β P α S α P β S (8 The ASE power is in Watts, G is the gain, ν and ν refer to the wavelength deviation of the ASE power around λ, h is Plank s constant, and n sp is the population-inversion factor which is dimensionless. In an EDFA, complete inversion can only be obtained when being pumped at 980 nm, at 980 nm β P = 0 and therefore n sp = 1. The ASE power to length relationship is shown in Fig. 7. In this figure it is clear that the ASE power builds up as the length of the fiber increases. This is an expected result because as spontaneously emitted photons travel down the fiber they get amplified and they also stimulate the emission of more photons [6]. From Fig. 7 it is observed that the ASE power is relatively small at lengths around 4 m. In this case the optimum length is chosen to be 4 m compared to the 12 m obtained in section 2.1. However, at 4 m the EDFA Fig. 8. ASE power versus wavelength for λ S of 1530 nm and λ P of 980 nm for a 4 m amplifier length gain is reduced as shown in Fig. 3. Essentially, this is one of the compromises that has to be considered when making a selection of amplifier length. The output spectrum of the ASE is shown in Fig. 8 for a signal wavelength of 1530 nm. The ASE spectrum is very similar to that of the gain spectrum shown above in Fig. 5. This is expected because of the relationship in equation (7. Also, the ASE is present over the whole operational range of the EDFA, thereby reducing the overall gain of the system. The obvious difference between the ASE spectrum and the gain spectrum is the output power. In the EDFA usable range of 1530 nm λ S 1560 nm, the ASE spectrum varies from dbm to dbm. It is clear that ASE is a dominant noise generated in the amplifier. 6. SUMMARY This paper presented a method for simulating the gain spectrum and forward ASE of an EDFA using the Simulink model implemented by Novak and Gieske [3]. The optimum simulation length was determined and the model in [3] was expanded to include multiple wavelengths. Forward ASE was also added to the model and the simulation results verified with [6]. The resulting EDFA model accurately represented EDFA gain dynamics and ASE. The advantages of this model are that an accurate EDFA model for Simulink is now available and that numerous EDFA parameters can be changed and the effect on the output easily observed. An interesting approach to gain flattening was also presented in this paper. Using the equations presented in [3], an equation relating the pump gain to the signal gain was derived. The signal gain was held constant (flat gain and the distribution of the pump power over wavelength was ob-

6 tained. Further investigation into this approach can be done using this model. 7. APPENDIX Symbol Description Value Dimension N 2 total erbium ions in excited state variable τ spontaneous emission 10.5 ms L length of erbium-doped fiber variable m (L opt = 12 r radius of fiber core m A area of fiber core πr 2 m 2 ρ density of erbium ions m 3 α P pump absorption coefficient 3.31 db/m β P pump emission coefficient 0 db/m α S signal absorption coefficient varies with λ db/m β S signal emission coefficient varies with λ db/m λ P pump wavelength 980 nm λ S signal wavelength nm Table 1. EDFA Parameters 8. REFERENCES [1] G. Keiser, Optical Fiber Communications, [Third Edition]. New York: McGraw-Hill, 2000, pp [2] X. Fernando, Ryerson University, EE8114 Fall 2003: Suggested Project Titles, September 2003, Projects/project outline.htm [3] S. Novak and R. Gieske, Simulink Model for EDFA Dynamics Applied to Gain Modulation, Journal of Lightwave Technology, vol. 20, No. 6, pp , June [4] J. Jiang, Simulink Model for EDFA Dynamics in a WDM System, EE8114 Course Project, Department of Electrical and Computer Engineering, Ryerson University, [5] S. Novak and A. Moesle, Analytic Model for Gain Modulation in EDFAs, Journal of Lightwave Technology, vol. 20, No. 6, pp , June [6] P.C. Becker, N.A. Olsson, and J.R. Simpson, Erbium- Doped Fiber Amplifiers Fundamentals and Technology. San Diego: Academic Press, [7] Ö. Mermer, Ege University, EDFA Gain Flattening By Using Optical Fiber Grating Techniques, October 2001, wwweee/ docs/seminer.pdf [8] S. Novak ( communication, 2003.

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