Effects of Wavelength Filtering on Pulse Dynamics in a Tunable, Actively Q-Switched Fiber Laser
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1 Effects of Wavelength Filtering on Pulse Dynamics in a Tunable, Actively Q-Switched Fiber Laser Abstract Manas Srivastava 1, Deepa Venkitesh, Balaji Srinivasan We present a numerical simulation, validated by experimental analysis, of the effect of wavelength filtering on pulse dynamics of a wavelength-tunable Erbium-doped fiber Q-switched laser in a ring configuration. Travelling wave time-dependent model is implemented using finite difference time domain (FDTD) method to accurately simulate the population dynamics and the pulse evolution in the ring laser. Such a model is experimentally validated for a wavelength tunable Q-switched fiber laser and stable Q-switched pulses are obtained over a wavelength tuning range of 30 nm in the C-band. 1. Introduction Several of industrial, medical, and military applications require high energy, short duration (nanoseconds) laser pulses at specific wavelengths and repetition rates. Q-switching is a popular approach for obtaining such high energy pulses. Specifically, Q-switched fiber lasers have attracted considerable attention due to the relative ease of thermal management issues and better wall-plug efficiency as compared to their bulk counterparts. Such laser sources operating in the wavelength range of nm have found applications in light detection and ranging (LIDAR) [1, 2] and distributed sensing [3, 4]. Q-switched fiber lasers are increasingly preferred as pump sources for second harmonic generation (SHG) and optical parametric oscillation / amplification (OPO/OPA). In such applications, it is critical to precisely control the lasing wavelength so that the required phase matching is established. However, the spectral output of a Q-switched fiber laser typically consists of 1 ee13d057@ee.iitm.ac.in Preprint submitted to Elsevier December 1, 2017
2 multiple wavelengths [5], and an intra-cavity filter is required to demonstrate spectrally clean pulses [6, 7]. As such, the use of an intra-cavity tunable filter facilitates an appropriate control of the output wavelength required for the above applications [8, 9, 10]. A key issue in a Q-switched fiber laser employing an intra-cavity filter is that its population dynamics are very different from that of a filterless cavity. For example, the excited state population gets depleted only through a narrow range of wavelengths in the filtered cavity thereby affecting the Q- switched pulse characteristics. While there have been several reports on the modeling of a filter-less Q-switched fiber laser [11], there is a clear requirement for a comprehensive model for wavelength-tunable Q-switched fiber lasers to study the time-dependent phenomena in such lasers. In this paper, we report the development of a FDTD-based model to accurately simulate the population dynamics and the resultant pulse evolution in the ring laser cavity. Such simulation results are corroborated by experimental demonstration of a fiber-based Q-switched laser that is tunable over nm. We establish the dependence of output pulse characteristics (peak power, pulse width) on the center wavelength of the tunable band-pass filter, both in simulations and experiments. Section 2 discusses the complete simulation model of Q-switched fiber laser in ring configuration, including the pulse evolution in spatial, temporal and spectral domains. The details of the experiment on Q-switched operation and discussion of the corresponding results are provided in the subsequent sections. 2. Numerical model for Wavelength Tunable Q-Switched Fiber Laser A schematic diagram of the Q-switched Erbium-doped fiber laser in a ring configuration is shown in Fig. 1. The Er-doped gain fiber (OFS HG980, peak absorption 17.5 db/m at 1530 nm) is pumped using 450 mw power at 976 nm through a wavelength division multiplexer (WDM). Unidirectional operation of the fiber laser is enforced in the cavity through an optical isolator. The output radiation is extracted using a 3-dB output coupler with an insertion loss of 3.35 db. An acousto-optic modulator (AOM, Brimrose) is used to modulate the loss at the desired frequency. A tunable optical bandpass filter (TBPF, Santec OTF-30) with an insertion loss of 2 db and a tuning range between 1533 nm to 1560 nm with a 3-dB bandwidth of 1 nm is inserted in the cavity to control the lasing wavelength. The relevant parameters used 2
3 Figure 1: Schematic of wavelength tunable, Q-switched Erbium-doped fiber laser. in the simulation model are listed in Table. 1. The absorption and emission cross-sections of the EDF as provided by the manufacturer (OFS) are shown in Fig. 2. Parameter description Symbol Value Dopant concentration (m 3 ) N Velocity of light in silica glass (m/s) v Pump wavelength (nm) λ p 976 Maximum pump power (W) P p 0.45 Signal wavelength (nm) λ s Wavelength spacing (nm) λ 1 Fluorescence lifetime (s) Excited state lifetime (s) τ [12] Length of EDF (m) L 3.5 Core radius (m) a Numerical aperture N A 0.29 Table 1: Parameter values used in the simulation model Prior to the initiation of the Q-switching process, a steady state condition (continuous wave lasing or amplified spontaneous emission, depending on the cavity configuration) is usually established. Thus, the complete simulation model consists of two parts - a steady state model for evaluating the initial conditions of Q-switching and FDTD model for simulating the transient Q- switching process. The simulations assume a uniform doping profile of Erions in the radial and longitudinal directions. Although Er 3+ doped in silica is a three-level system, it can be approximated as a two-level system owing to the fact that the non-radiative decay rate of the excited state is high 3
4 Figure 2: Absorption and emission cross-sections of the Erbium-doped silica fiber used in simulations. compared to the pumping rate and the spontaneous emission rate of the metastable state [13, 14]. It is found that for the range of pump power used in this implementation, two-level and three-level models yield the same results, both in static and dynamic conditions. This is because the lifetime of the excited state (τ 32 ) is much smaller than the fluorescence lifetime of the metastable state ( ), and the pump power is not sufficient to populate the excited state faster than the non-radiative decay. The differences are found to be significant only for pump power greater than 5 W. Hence, the Erbiumdoped fiber is modeled as a two-level lasing system, and the travelling-wave model is realized using the standard rate equations. The model does not include the excited state (two-photon) absorption, clustering of Er-ions and inhomogeneous broadening in the doped fiber. The steady-state model is ratified by comparing the simulation results with the experimental results of a continuous-wave (CW) laser under different operating conditions, and the comparison helps in estimating the dopant concentration of the EDF, which is found out to be m 3 [15]. The cavity parameters used for the simulations are the same as those measured for the experimental implementation. The loss of AOM is 52.5 db and 9 db in low-q and high-q states respectively and the switching transient duration is 150 ns (verified experimentally) with a sinusoidal variation. The spectral characteristics of the EDF are considered for the range 1501 nm to 1600 nm with a wavelength resolution of 1 nm. The spontaneous emission 4
5 noise in each wavelength band is considered with a noise bandwidth of 1 nm. The high-q duration for all simulations and experiments is 1 µs. The optical filter is modelled to have a Gaussian response with 3-dB transmission bandwidth of 1 nm, insertion loss of 2 db and out-of-band noise suppression of 36 db (as per the specification sheet). Figure 3: Simulation results illustrating the Q-switched laser output for various values of EDF length in a filterless cavity for pump power of 450 mw and repetition rate of 10 khz. (a) Output pulse shapes in time domain, (b) corresponding optical spectra. Optimum length of gain fiber in a filterless configuration is estimated through simulations such that the emission could be tuned through the C- band. The temporal and spectral outputs of the simulated laser for different lengths of EDF are shown in Fig. 3. The output spectrum is broadband with a peak at 1530 nm for EDF length 3 m. For EDF length of 3.5 m, power at 1530 nm is seen to be re-absorbed resulting in emission at 1560 nm predominantly, which indicates the availability of sufficient gain required to facilitate the Q-switched operation over the entire C-band. Hence, the EDF length is chosen to be 3.5 m for further studies. The characteristics of the output pulse (peak power and pulse width) are compared for a filterless cavity and a cavity with filter with its centre wavelength tuned to 1550 nm. The variations of peak power and pulse width with respect to the repetition rate for both the configurations are shown in Fig. 4. Higher peak powers and shorter pulse widths are obtained for low repetition rates in both the cases. This trend is primarily due to the large amount of population inversion accumulated in the long low-q duration for these repetition rates, which is rapidly depleted during the high-q duration. Peak power is higher for a filterless cavity because the feedback to the gain fiber contains broadband ASE spanning over the entire emission bandwidth 5
6 Figure 4: Comparison of peak power (top) and pulse width (bottom) as a function of pulse repetition rate for a filterless cavity and a cavity with filter. of the EDF, whereas in case of a cavity with filter, the feedback is limited to the pass-band of the filter. Since the power of the feedback is smaller in the latter case, the rate of build-up of the pulse is smaller, leading to slower rise time and hence a larger pulse width compared to the filterless case. Figure 5: Simulation results for various values of centre wavelength of filter for pump power of 200 mw and repetition rate of 2 khz (a) pulse shapes along with their energy at different filter wavelength (inset), and (b) the corresponding spectrum. Simulations are also carried out for a cavity with filter and the temporal and spectral features of the pulse are compared as the center wavelength of the filter is tuned across the C-band. Figure 5(a) shows that the peak power of the laser output pulses varies with respect to wavelength according to the shape of the EDF gain spectrum. This variation is dependent on the gain spectrum built up at the end of the low-q duration and the initial ASE feed- 6
7 back power at the wavelength of operation, which are in turn dependent on the pump power and the repetition rate. Moreover, the larger gain coefficient at shorter wavelengths causes the signal power to increase rapidly compared to the same at longer wavelengths, as indicated by the slope corresponding to the rising edge of the pulse. However, the total pulse energy is smaller for shorter wavelengths, because the rate of re-absorption in the chosen length of EDF is larger in case of shorter wavelengths, as explained above. Figure 5(b) shows that the output spectrum is clearly defined by the pass-band of tunable filter. Another interesting aspect of the simulation results shown in Fig. 5(a) is the ripples observed on the pulse envelope. These are found to occur with 75 ns period corresponding to the round-trip duration of our 15 m long cavity [12]. The above results are consistent across the entire spectrum shown in Fig. 5(b), which confirms that the Q-switching operation can be carried out over the entire tuning range of the TBPF. 3. Experimental implementation of Wavelength Tunable Q-Switched Fiber Laser As mentioned earlier, the above simulation results are validated experimentally. The ring configuration described in Sec. 2 is used for the experimental demonstration of the Er-doped Q-switched fiber laser [16]. The acousto-optic modulator is driven by a signal generator, whose output is modulated at the desired repetition rate for Q-switching. We use a fiber coupled thin-film filter (Santec OTF-30) as the TBPF in the cavity. The optical pulses from the Q-switched laser are suitably attenuated before they are detected using a PIN-based optical receiver of bandwidth 50 MHz and a 200 MHz oscilloscope. An optical spectrum analyzer (Agilent 86141B) with a resolution of 0.07 nm is used to measure the output spectrum Effect of filter In order to study the effect of wavelength filtering on the pulse dynamics experimentally, the Q-switched operation of the cavity is performed with and without the filter. In the former configuration, the TBPF is tuned to 1533 nm, which is very close to the peak of the EDF gain spectrum. The AOM is modulated at various repetition rates ranging from 100 Hz to 10 khz while maintaining a high-q duration of 1 µs. 7
8 Figure 6: (a) Comparison of experimental output spectrum and pulse shapes (inset) at the output of the Q-switched laser with and without filter for repetition rate of 100 Hz (b) Peak power (top) and pulse width (bottom) for different repetition rates of the Q-switched fiber laser with and without filter Figure 6(a) shows the output pulses obtained from the Q-switched laser in both the cases with and without filter for repetition rate of 100 Hz. As expected, the spectrum for the Q-switched laser with filter is much cleaner and narrower compared to the same without filter. However, as predicted by the simulations, the peak output power is found to be smaller for the cavity with filter. This has been investigated thoroughly at different repetition rates and the results are presented in Fig. 6(b). The results indicate that pulses with higher peak power (30 W) and shorter pulse duration (120 ns) are obtained for the filter-less cavity compared to the cavity with filter. This may be attributed to the relatively small round-trip loss for the cavity without filter. Moreover, for such a cavity the emission is stimulated by the ASE photons over a broad wavelength range corresponding to the emission spectrum of Erbium. As discussed earlier, such high level of stimulation results not only in higher power, but also shorter pulses. On the other hand, for the cavity configuration including the filter the stimulation is limited to the pass band of the filter and it also undergoes loss due to the filter. Due to these factors, the rate of pulse build-up is slow in this case resulting in lower peak power (15 W) and longer pulses (160 ns) for the cavity with filter. From Fig. 6(b), it is also clear that high peak powers and smaller pulse widths are obtained at lower repetition rates in both configurations. This is attributed to the larger population inversion built up in the EDF during the long OFF period (low Q state) at low repetition rates. As repetition rate increases, the low-q duration and hence the population inversion decreases (high-q duration being constant). This leads to a decrease in peak power 8
9 and increase in pulse width. It should also be noted that filter-less laser yields larger peak power and smaller pulse widths for all the repetition rates, as discussed earlier. It is to be noted that all these trends are consistent with the simulations, thereby validating our numerical model Wavelength tunability Figure 7: Q-switched laser output features obtained experimentally for various settings of TBPF with pump power of 450 mw at repetition rate of 10 khz (a) pulse shapes (Inset) variation of pulse energy with filter wavelength and (b) corresponding spectrum We also studied the tuning characteristics of the laser incorporating the tunable bandpass filter as its center wavelength is tuned across the C-band [15]. The output characteristics obtained for various settings of the filter are observed in spectral and temporal domains, as shown in Fig. 7. It is observed that Q-switching operation is sustained over the entire C-band, and is limited only by the filter characteristics. We observe that the spectral width of the Q-switched laser is 1 nm (corresponding to our tunable filter bandwidth) and that the peak power and total energy of the pulse is smaller for shorter wavelength. As explained previously, the lower power at shorter wavelength may be attributed to the higher re-absorption at those wavelengths. Pulse width is smaller for shorter wavelengths because of high gain available at these wavelengths. Moreover, as seen in Fig. 7(a), ripples are observed in the pulse output, corresponding to the cavity length with the inclusion of TBPF. It is important to note that these trends observed in our experiments are similar to those predicted by the simulation results shown in Fig. 5. The discrepancy in the power levels between those observed in simulations and experiments is most likely due to the fact that our model ignores the nonuniform doping profile, inhomogeneous broadening and other possible effects of clustering. 9
10 Figure 8: Variation of experimental pulse characteristics with repetition rate for different settings of filter wavelength (a) peak power (b) pulse width Figure 8(a) describes the dependence of output peak power for various setting of filter wavelength and different repetition rates. It is seen that for any fixed filter wavelength, the peak power is constant over a range of repetition rate ( khz). For further higher repetition rates, the peak power reduces due to a smaller amount of population inversion accumulated in the EDF. Peak power is found to be higher for longer wavelengths as predicted in simulations and explained in Section 2. The dependence of pulse width on repetition rate and filter wavelength is shown in Fig. 8(b). For a fixed filter wavelength, uniform pulse widths are obtained upto a repetition rate of 12.5 khz. Beyond this value, the pulse width is found to increase due to smaller amount of gain in the EDF. Pulses with shorter duration are obtained for shorter wavelengths, as explained in Section Maximum achievable power The variation of the signal peak power with pump power for various repetition rates but operated with the same high-q duration (1 µs ) is shown in Fig. 9 for a filter wavelength of 1550 nm. For a given repetition rate, the signal peak power increases with pump power because the amount of population inversion accumulated during the low-q duration is high. As the repetition rate is increased, the low-q duration decreases because the high-q duration is maintained constant. So, the inversion built up during the low-q duration is less, resulting in lower peak power. For lower repetition rates the peak power value saturates beyond certain pump power because the low-q duration is long enough to achieve the maximum possible inversion in the EDF. 10
11 Figure 9: Variation of peak power vs pump power with repetition rates when the laser is tuned to 1550 nm wavelength 4. Conclusion In this paper, we have investigated the effect of wavelength filtering on pulse dynamics in a wavelength tunable Q-switched fiber laser using a comprehensive time-dependent model. Simulations have been carried out to study the wavelength-dependent pulse dynamics through a finite difference time domain (FDTD) technique. Specifically, we have investigated the population dynamics involved in the Q-switching process with and without an intra-cavity filter for different pump powers and pulse repetition rates. We observe that the broadband ASE generated in the gain medium in the absence of the intra-cavity filter results in a rapid saturation of the gain medium, thereby providing shorter pulses with higher peak powers. In the presence of the intra-cavity filter, the ASE spectrum supported in the cavity is limited to the pass-band of the filter resulting in slightly longer pulses with smaller peak power. However, the output spectrum is fairly well-defined and tunable. The above simulation results have been validated experimentally using an Er-doped fiber laser in the ring configuration, employing an acousto-optic modulator for Q-switching. We find good agreement between the experimental and simulation results. Even though we use a filter of bandwidth 1 nm in these experiments, similar desired bandwidth at the output can be achieved with the use of appropriate filters. We have demonstrated a Q-switched fiber laser with pulse repetition rate up to 12.5 khz, whose center wavelength is tunable between 1533 nm to 1560 nm. 11
12 Acknowledgements We acknowledge the financial support from Defence Research and Development Organisation, Government of India. We also acknowledge the constructive comments from the reviewers and the editor, which have improved the quality of our work. Appendix A. Simulation model Erbium doped fiber is a 3-level system, with the energy levels and the absorption and emission bands as shown in Fig. A.10 [12]. R 13 denotes the absorption rate at pump wavelength, and R 12, R 21 denote the absorption and emission rates respectively in the signal band. The traveling wave model is Figure A.10: Three level energy band diagram of Erbium doped fiber used for simulating the fiber laser system. The model is described through population rate equations (Eqns. A.1-A.3) and power propagation equations including the ASE (Eqns. A.4 and A.5), as shown below. These differential equations are coupled through the gain coefficients b s and b p. The various parameters involved in the equations and their values used for simulations are described in Table A.2. dn 1 (z) dt = R 13 (z)n 1 (z) R 12 (z)n 1 (z) + R 21 (z)n 2 (z) + N 2(z) (A.1) 12
13 dn 2 (z) dt = R 12 (z)n 1 (z) R 21 (z)n 2 (z) N 2(z) dn 3 (z) = R 13 (z)n 1 (z) N 3(z) dt + N 3(z) τ 32 (A.2) τ 32 (A.3) P p z + 1 P p = b p P p (A.4) v t P s z + 1 P s v t = b sp s + σ es N 2 PASE 0 P 0 ASE = 2hν ν (A.5) (A.6) The boundary conditions for the population inversion, signal power and pump power in the steady state when the cavity is open, are necessary to initiate the dynamic simulations corresponding to the Q-switched laser. To obtain this, the time derivatives in the rate equations are set to zero resulting in Eqns. A.7-A.11. These coupled equations are now solved numerically; at z = 0 the pump power is initialized to the value used in the experiment and the signal power is initialized to zero since the cavity is open. The ASE power generated at the output is now fed back to the input of the fiber after accounting for the loss in the intra-cavity elements. The signal and pump powers are further evaluated in multiple iterations, until these values at the output of the fiber converges, corresponding to a steady state condition. N 1 = R R R 12 + R 13 + τ 32 R 13 ( R )N 0 (A.7) N 2 = N 3 = R 12 + R 13 ( )N R R 12 + R 13 + τ 32 R 13 R ( ) τ 32 R 13 R ( )N R R 12 + R 13 + τ 32 R 13 R (A.8) (A.9)
14 Symbol Description N 1 (z) Ground state population density (m 3 ) N 2 (z) Metastable state population density (m 3 ) N 3 (z) Excited state population density (m 3 ) N 0 = N 1 + N 2 + N 3 Dopant concentration (m 3 ) P p (z) Pump power (W) λ s Signal wavelength (nm) P s (λ, z) Signal power at wavelength λ(w) σ ap, σ as Absorption cross section at pump and signal wavelengths (m 2 ) σ ep, σ es Emission cross section at pump and signal wavelengths (m 2 ) Γ R 13 (z) pσ ap hν pa p(z) (s 1 ) Γ R 12 (z) sσ as(λ) hν sa s(λ, z) (s 1 ) λ Γ R 21 (z) sσ es hν sa s(λ, z) (s 1 ) λ Fluorescence lifetime (s) τ 32 Excited state lifetime (s) Γ p, Γ s Overlap integrals for pump and signal wavelengths b p (z) Γ p [σ ep N 2 σ ap N 1 ](m 1 ) b s (z) Γ s [σ es N 2 σ as N 1 ](m 1 ) λ Wavelength spacing (nm) ν Frequency spacing = c λ (Hz) λ 2 Table A.2: Parameters used in the simulation model dp p dz = b p(z)p p (z) (A.10) dp s dz = b s(z)p s (z) + σ es N 2 (z)p 0 (A.11) The FDTD model of the Erbium doped fiber ring laser is implemented to observe and analyze the effects of spatial distribution of powers and population densities, and also the propagation delays involved in the Q-switching dynamics. The pump power and population inversion evolve only over the gain fiber length, and hence, Eqns. A.1 - A.4 are solved only in the gain fiber region. On the other hand, the signal power evolves through the entire cavity and hence eqn. A.5 is solved for the entire length of the fiber. The space 14
15 dependence of b s is considered while it propagates through the gain fiber, while bulk losses in the intra-cavity components are introduced through a space and time-independent b s. The switching of the AOM is modeled as a time dependent b s. The output signal power from the gain fiber is iteratively fed back to the input of the fiber after accounting for the loss in the intracavity elements and the iterations are repeated to obtain the output of the Q-switched laser when it operates in the steady state. The three-level model can be approximated to a two-level model, owing to the rapid non-radiative decay of the excited state. Assuming τ 32 0, Eqns. A.7-A.9 are reduced to the following, thus simplifying the model considerably. N 1 (z) = R R 13 + R 12+ R N 0, N 2 (z) = N 0 N 1 (z), N 3 0 (A.12) Figure A.11: (Left) Comparison of L-I characteristics of a CW laser in a filterless cavity simulated using two-level and three-level models (Inset) L-I characteristics zoomed to indicate lasing threshold (Right) Q-switched laser pulse in steady state simulated using two-level and three-level models for pump power of 450 mw and repetition rate of 10 khz In order to justify the assumption made above, both the models are used to simulate the CW laser and Q-switched laser pulses and the results are compared, as shown in Fig. A.11. CW laser cavity is simulated in a filterless configuration using both the models with pump power varying from 5 mw to 20 W. A comparison of the corresponding L-I curves reveals that the both models yield the same results in terms of power levels, threshold pump power and slope efficiency for pump power less than 5 W. The lasing wavelength and features of the optical spectrum are also identical in that regime. For further higher pump power, the power obtained though the three level model is lower than that from the two level model. Subsequently, Q-switched laser 15
16 cavity with filter tuned to 1550 nm is simulated using both the models for pump power of 450 mw and repetition rate of 10 khz, and the pulse features obtained in steady state are compared. Fig. A.11(right) shows that the pulses obtained through both the models have almost identical pulse shapes and peak power. The above results validate the assumption about the lifetime of the excited state and indicate that the population of excited state (N 3 ) does not have any significant contribution to the amplification in either steady-state or time-varying conditions. Equation A.9 reveals that the population of excited state is dependent on the pump power through the rate coefficient R 13. Hence, it is possible that for higher pumping rates, the excited state may acquire a significant amount of population density, rendering the assumption N 3 0 invalid. However, for the values of pump power used in this implementation, the assumption seems to be valid, as indicated by the above comparisons. So, a two-level model would be preferable due to the simplicity and ease of implementation. The proposed laser model is flexible and can be easily adapted to simulate other gain systems such as Ytterbium, Thulium etc., and various components and configurations in a ring cavity. Figure A.12: Steady state condition obtained from simulations and compared with the corresponding experiments (a) L-I characteristics for the ring laser with and without intracavity filter (b) Output spectrum of the ring laser without the filter in the cavity. A comparison of the spectra for the cavity with filter is shown in the inset EDF laser operating in CW conditions is experimentally implemented and the steady state model is verified by comparing the L-I curves and spectra obtained from numerical and experimental results. Fig. A.12(a) shows the L-I curves - the results obtained from the experiments are shown as discrete points while that obtained from simulations are shown as continuous lines. For a dopant concentration of m 3, the simulation results are found 16
17 to match with experimental results within 10%. Fig. A.12(b) shows the corresponding spectrum. The experimental results shows some ripples in the spectrum, indicating that more than one wavelength is satisfying the threshold conditions in lasing. These could be attributed to the inhomogeneous emission characteristics of Erbium doped in silica glass. The power in these peaks are significantly lower than the peak of the spectrum. To simulate the wavelength tunable laser, a wavelength tunable band-pass filter is introduced in the cavity. A comparison between the simulation and experimental results of the LI curve and the spectra are shown in Fig. A.12. Thus, the comparison between the simulation and experimental results in the steady-state helps to ratify the simulation model and to estimate all the laser cavity parameters. Figure A.13: (a) Plot of log(error) with respect to stability factor (R) and log of gain coefficient log(b), (b) Pulse output for various numbers of space steps The error analysis of the scheme was carried out in detail. The tolerance limit of absolute error was chosen as 10 2 (1%). The objective of the analysis is to choose R for a given range of b, such that the error is less than 1%. The absolute error between the numerical and analytical solutions to Eqn. A.4 for a constant gain coefficient (b) was estimated for different combinations of b and stability factor (R). The result obtained is shown in Fig. A.13(a). Error is depicted in colour and is plotted for various values of R and b. It is seen that choosing a stability factor of 0.9 satisfies the condition of tolerance limit of 1%. Even though the error is found to be independent of the stability factor, we choose it to be 0.9 so that the numerical dispersion is minimized. Convergence analysis of the FDTD model in terms of space discretization was also carried out in a manner similar to the steady state model, by varying the size of space step. It is found that on reducing size of space step smaller than m, the error is < 1%, as shown in Fig. A.13(b). Hence, a space 17
18 step size of 15 mm is used to carry out the simulations. References [1] U. Sharma, C.-S. Kim, J. U. Kang, N. M. Fried, Highly stable tunable dual-wavelength Q-switched fiber laser for DIAL applications, in: Laser Applications to Chemical and Environmental Analysis, Optical Society of America, 2004, p. MB3. [2] Y. Kaneda, Y. Hu, C. Spiegelberg, J. Geng, S. Jiang, Single-frequency, all-fiber Q-switched laser at 1550 nm, in: Advanced Solid-State Photonics (TOPS), Optical Society of America, 2004, p [3] G. Lees, A. Hartog, A. Leach, T. Newson, 980 nm diode pumped Erbium3+/Ytterbium3+ doped Q-switched fibre laser, Electron. Lett. 31 (21) (1995) [4] K. De Souza, D. Culverhouse, T. Newson, Dual-operation Q-switched Erbium-doped fibre laser for distributed fibre sensing, Electron. Lett. 33 (24) (1997) [5] A. Piper, A. Malinowski, K. Furusawa, D. Richardson, High-power, high-brightness, mj Q-switched Ytterbium-doped fibre laser, Electron. Lett. 40 (15) (2004) [6] Y. Gan, X. Gu, J. Y. Koo, W. Liang, C.-Q. Xu, Second Harmonic Generation Using an All-Fiber Q-Switched Yb-Doped Fiber Laser and MgO:c-PPLN, Advances in OptoElectronics 2008 (2008) (2008) 6. [7] W. Shin, B.-A. Yu, Y. L. Lee, T. J. Yu, T. J. Eom, Y.-C. Noh, J. Lee, D.-K. Ko, Tunable Q-switched Erbium-doped fiber laser based on digital micro-mirror array, Opt. Express 14 (12) (2006) [8] D. Bouyge, A. Crunteanu, V. Couderc, D. Sabourdy, P. Blondy, Synchronized Tunable Q-Switched Fiber Lasers Using Deformable Achromatic Microelectromechanical Mirror, IEEE Photon. Technol. Lett. 20 (12) (2008)
19 [9] P. E. Britton, D. Taverner, K. Puech, D. J. Richardson, P. G. R. Smith, G. W. Ross, D. C. Hanna, Optical parametric oscillation in periodically poled Lithium Niobate driven by a diode-pumped Q-switched Erbium fiber laser, Opt. Lett. 23 (8) (1998) [10] L. E. Myers, W. R. Bosenberg, Periodically Poled Lithium Niobate and Quasi-Phase-Matched Optical Parametric Oscillators, IEEE J. Quantum Electron. 33 (10) (1997) [11] Y. Wang, C.-Q. Xu, Actively Q-switched fiber lasers: Switching dynamics and nonlinear processes, Prog. Quantum Electron. 31 (3 5) (2007) [12] S. Adachi, Y. Koyamada, Analysis and Design of Q-Switched Erbium- Doped Fiber Lasers and Their Application to OTDR, Journal of Lightwave Technology 20 (8) (2002) [13] E. Desurvire, Erbium doped fiber amplifiers : Principles and applications, John Wiley & Sons, Inc., [14] M. J. F. Digonnet, Rare-Earth-Doped Fiber Lasers and Amplifiers, Marcel Dekker, Inc., [15] M. Srivastava, D. Venkitesh, B. Srinivasan, Design and demonstration of tunable Q-switched fiber laser, 2013, pp Q 86012Q 7. [16] M. Srivastava, D. Venkitesh, B. Srinivasan, Demonstration of a Wavelength Tunable Q-Switched Fiber Laser, in: Asia Communications and Photonics Conference, Optical Society of America, 2012, p. ATh3A.5. 19
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