Analytical model for optimizing the parameters of an external passive Q-switch in a fiber laser
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(2) approximation of the uniform pumping of the gain medium, the intracavity optical intensity as axially uniform, and the complete recovery of the saturable absorber [17–19]. Including the focusing effect, the coupled equations for a four-level saturable absorber are given by [16,20] dϕ ϕ ¼ 2σnl − 2σ gs ngs ls − 2σ es nes ls dt tr 1 þL ; ð1Þ − ln R dn ¼ −γcσϕn; dt. ð2Þ. dngs A ¼ − cσ gs ϕngs ; dt As. ð3Þ. ngs þ nes ¼ nso ;. ð4Þ. where ϕ is the intracavity photon density with respect to the cross-sectional area of the laser beam in the gain medium, n is the population density of the gain medium, ls is the length of the saturable absorber, A=As is the ratio of the average area in the gain medium and in the saturable absorber, nso is the total population density of the saturable absorber, ngs and nes are the instantaneous population densities in the ground and excited states of the saturable absorber, respectively, σ gs and σ es are the ground-state absorption and excited-state absorption cross sections of the saturable absorber, respectively, R is the reflectivity of the output mirror, L is the round-trip dissipative optical losses, γ is the inversion reduction factor (γ ¼ 1 and γ ¼ 2 correspond to, respectively, four-level and three-level systems; see Ref. [17]), and tr ¼ 2l0 =c is the round-trip transit time of light in the cavity optical length l0 , where c is the speed of light. Note that although the approximation of uniform population inversion and photon density is not rigorously accurate, it has been confirmed to provide an excellent design guideline for solidstate lasers [16–18]. The experimental results shown in Section 5 validate the uniform approximation to be also applicable in fiber lasers. Dividing Eq. (3) by Eq. (2) yields a separable firstorder ordinary differential equation: dngs A 1 σ gs ngs ¼ : dn As γ σ n. ð5Þ. Equation (5) can be solved to obtain ngs ¼ nso. α n ; ni. ð6Þ. where ni is the initial threshold value of the population inversion density in the gain medium at the start of Q-switching and the quantity α is given by 2298. APPLIED OPTICS / Vol. 47, No. 13 / 1 May 2008. α¼. 1 σ gs A : γ σ As. ð7Þ. We denote the quantity α to be the bleaching rate parameter (BRP) because its value determines the bleaching rate after the population inversion density reaches the initial threshold value. With the coupled rate of Eqs. (1)–(4), the expression for the output pulse energy of the passively Q-switched laser has been derived to be given by [16,19] hvA 1 E¼ ln x; 2σγ R. ð8Þ. where hv is the laser photon energy, R is the reflectivity of the output mirror, and the parameter x represents the extraction efficiency of the energy stored in the gain medium through the lasing process. In terms of the BRP, the equation for the parameter x is given by 1 − σ es =σ gs ln 1=T 2o 1 − e−x − x þ ln 1=T 2o þ ln 1=R þ L 1 − e−αx ¼ 0: × x− α. ð9Þ. Note that in the calculation of the pulse energy with Eq. (8) the effect of cavity losses is included in the process of solving Eq. (9) with the parameter L. Furthermore, the nonlinear effects are not included in the present model because of the property of large mode areas of the fiber lasers studied here. Equation (6) manifests that the larger the BRP, the faster the saturable absorber is bleached. As a consequence, the larger the BRP, the higher is the output efficiency of a passively Q-switching laser [16–19]. As indicated in Eq. (7), the BRP is proportional to the effective ratio A=As for a given gain medium and a given saturable absorber. Therefore, the optimization of the effective ratio A=As is essentially critical for developing an efficient passively Qswitched laser. For a given fiber laser, the maximization of the effective ratio A=As is directly related to the minimum As , as illustrated in Fig. 1. Since the minimum As is governed by the reimaging magnification M a of the extended cavity, it is of practical usefulness to derive the optimum magnification in terms of the physical parameters of the laser fiber and the saturable absorber. Itis worthwhile to mention that the use of an external passive Q-switch and a high reflector is usually sufficient to completely prevent the CW lasing between the fiber end facets because the threshold is considerably reduced by the external feedback. The detailed discussion will be given in Section 5..
(3) where la is the thickness of the saturable absorber and the weighting function e−nso σgs z comes from the absorption effect. For the single-pass approximation, the effective beam area factor can be expressed as hω2s i. Fig. 1. Schematic illustration of the external cavity in a passively Q-switched fiber laser. ωo is the beam waist of the laser mode; zo is the position of the beam waist.. 3. Optimization of Reimaging Magnification. The laser mode size in the saturable absorber is generally given by [21] ωs ðzÞ ¼ ωo. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ½Cðz − zo Þ 2 ;. ð10Þ. with C¼. M2 λ ; πnr ωo2. ð11Þ. where ωo is the beam waist of the laser mode, zo is the position of the beam waist, λ is the laser wavelength, M 2 is the beam-quality factor, nr is the refractive index of the saturable absorber, and the point z ¼ 0 is set at the incident surface of the saturable absorber. For a given core radius rc, the beam waist in the saturable absorber is directly related to the reimaging magnification M a by ωo ¼ M a rc :. ð12Þ. With the brightness theorem, the beam-quality factor M 2 can be given by M 2 ¼ ðNA · rc Þðπ=λÞ;. ð13Þ. ¼. R la 0. ω2s ðzÞe−nso σgs z dz : R la −n σ z so gs dz 0 e. ð17Þ. Figure 2 shows a comparison for the calculated results hω2s i obtained with Eqs. (16) and (17) for a typical case with the parameters of NA ¼ 0:04, rc ¼ 12:5 μm, nr ¼ 1:82, T o ¼ 0:4, and la ¼ 2 mm. It can be seen that the optimum focal position is shifted by approximately 0:2 mm when the round-trip effect is taken into account. Therefore, the influence of the round-trip effect on the effective beam area can be clearly found to be insignificant. Since Eq. (17) for the single-pass approximation leads the derivation to be more concise, it is used for optimizing the effective beam area. On the other hand, the standing-wave effect is omitted because the multilongitudinal mode operation of long fibers reduces this effect. Substituting (10) into (17), the integration can be exactly carried out and the average mode area is expressed as As ¼. πω2o C2 . . z2o. 1 To − − 2zo la lnð1=T o Þ 1 − T o. . 1 2la2 þ 2 ðlnð1=T o ÞÞ2 C To 2 2 1þ ; − la 1 − To lnð1=T o Þ. þ. ð18Þ. where T o ¼ e−nso σ gs la represents the initial transmission of the saturable absorber. Note that the expression of Eq. (18) is in term of the initial transmission T o instead of nso σ gs because T o is a macroscopic property of the saturable absorber and can be definitely measured.. where NA is the numerical aperture of the laser fiber. Substituting Eqs. (12) and (13) into Eq. (11), the factor C can be expressed as C¼. NA : nr M a 2 rc. ð14Þ. To take the round-trip effect into account, the average mode area in the saturable absorber, As , can be properly in terms of the mean square of the average mode size as As ¼ πhω2s i. ð15Þ. with hω2s i. ¼. R la 0. ω2s ðzÞ½e−nso σgs z þ e−nso σ gs ð2la −zÞ dz ; R la −n σ z so gs þ e−nso σ gs ð2la −zÞ dz 0 ½e. ð16Þ. Fig. 2. A comparison for the calculated results hωs2 i obtained with Eqs. (16) and (17) for a typical case with the parameters of NA ¼ 0:04, rc ¼ 12:5 μm, nr ¼ 1:82, T o ¼ 0:4, and la ¼ 2 mm. 1 May 2008 / Vol. 47, No. 13 / APPLIED OPTICS. 2299.
(4) Figure 3 shows a calculated example with Eq. (18) and the parameters of NA ¼ 0:04, rc ¼ 12:5 μm, nr ¼ 1:82, and la ¼ 2 mm to demonstrate the dependence of the average mode area on the focal position for several T o values. It can be seen that there is an optimum focal position for minimizing the average mode area. The optimum focal position zopt for the minimum mode area can be analytically determined by partially differentiating Eq. (18) with respect to zo and setting the resulting equation equal to zero:. ∂As. ¼ 2πω2o C2 ∂zo zo ¼zopt × zopt − la. 1 To − lnð1=T o Þ 1 − T o. ¼ 0:. ð19Þ. Equation (19) leads to the zopt to be given by . zopt. 1 To − ¼ la : lnð1=T o Þ 1 − T o. ð20Þ. Equation (20) indicates that the optimum focal position depends only on la and T o , i.e., the properties of the saturable absorber. Substituting Eqs. (12), (14), and (20) into Eq. (18), the average mode area at the optimum focal position is then given by. ∂As. 2ðNAÞ2 l2 ¼ π 2M opt r2c − 2 3 a. ∂M a M a ¼Mopt nr M opt 1 To ¼ 0: × − ðlnð1=T o ÞÞ2 ð1 − T o Þ2. ð22Þ. Consequently, the M opt is given by M opt. 1=4 ðNAÞ2 l2a 1 To ¼ − : ð23Þ n2r r2c ðlnð1=T o ÞÞ2 ð1 − T o Þ2. Equation (23) indicates that the optimum reimaging magnification M opt can be straightforwardly determined with the numerical aperture and core radius of the laser fiber as well as the thickness and initial transmission of the saturable absorber. Figure 4 depicts a calculated example with Eq. (22) and the parameters of NA ¼ 0:04, rc ¼ 12:5 μm, nr ¼ 1:82, and T o ¼ 0:5 to reveal the dependence of the average mode area on the magnification M a for several la values. The dashed line in Fig. 4 shows the minimum average mode areas corresponding to the optimum magnification M opt . 4.. Experimental Results. The optimum magnification, M opt , for minimizing the mode area can be analytically determined by partially differentiating Eq. (21) with respect to M a and setting the resulting equation equal to zero:. To illustrate the utility of the present model, a Ybdoped fiber laser with a Cr4þ :YAG crystal as a saturable absorber is considered. Figure 5 shows the plot of the experimental setup that consists of a 1:5 m long fiber with a core diameter of 25 μm and a numerical aperture of 0.04. The fiber end facets were cut to be normal incident for the free-running operation. The pump source was a 13 W 976 nm fiber-coupled laser diode with a core diameter of 400 μm and a numerical aperture of 0.22. A focusing lens with 25 mm focal length and 92% coupling efficiency was used to reimage the pump beam into the fiber through a dichroic mirror with high transmission (>90%) at 976 nm and high reflectivity (>99:8%) at 1075 nm. The pump spot radius was approximately 200 μm.. Fig. 3. Dependence of the average mode area on the focal position for several T o values; the results are calculated with Eq. (18) and the parameters of NA ¼ 0:04, rc ¼ 12:5 μm, nr ¼ 1:82, and la ¼ 2 mm.. Fig. 4. Dependence of the average mode area on the magnification for several la values; the results are calculated with Eq. (22) and the parameters of NA ¼ 0:04, rc ¼ 12:5 μm, nr ¼ 1:82, and T o ¼ 0:5.. ðNAÞ2 l2 1 To As ¼ π M a 2 r2c þ 2 2a − : nr M a ðlnð1=T o ÞÞ2 ð1 − T o Þ2 ð21Þ. 2300. APPLIED OPTICS / Vol. 47, No. 13 / 1 May 2008.
(5) Fig. 5. Schematic setup of a diode-pumped passively Q-switched Yb-doped double-clad fiber laser. HR: high reflection; HT: high transmission.. The Cr4þ :YAG crystal has a thickness of 1:57 mm with 40% initial transmission at 1075 nm. Both sides of the Cr4þ :YAG crystal were coated for antireflection at 1075 nm (R < 0:2%). The saturable absorber was wrapped with indium foil and mounted in a copper block without active cooling. Substitution of the experimental parameters of NA ¼ 0:04, rc ¼ 12:5 μm, nr ¼ 1:82, T o ¼ 0:4, and la ¼ 1:57 mm into Eq. (22) yields M opt ¼ 0:88. With the available optics, we setup an extended cavity to obtain a reimaging magnification of M a ¼ 0:9 that nearly achieves the optimum value of M opt ¼ 0:88. A translation stage was used to adjust the longitudinal position of the Cr4þ :YAG crystal for investigating the influence of the focal position on the average mode area as well as the output performance. Figure 6 shows the experimental results for the dependence of the output pulse energy on the focal position at an incident pump power of 10 W. The theoretical calculations based on Eqs. (8) and (9) and the parameters of σ ¼ 2:4 × 10−21 cm2 [22], σ gs ¼ 8:7 × 10−19 cm2 [23], σ es ¼ 2:2 × 10−19 cm2 [23], R ¼ 0:04, and L ¼ 0:04 are also shown in Fig. 6 for comparison. It can be seen that the output energy is significantly influenced by the focal position and the optimum focal position agrees very well with the theoretical analysis of zopt ¼ 0:7–0:9 mm.. Fig. 6. Experimental and theoretical results for the output pulse energy as a function of the focal position. Symbols are experimental data. Solid line is calculated results using Eqs. (8) and (9).. Fig. 7. AverageoutputpowersversustheincidentpumppowerforCW lasingbetweenfacets,CWlasingwiththeexternalhigh-reflectioncavity, andpassiveQ-switchingoperationwiththeexternalhigh-reflectioncavity and the saturable absorber at the optimum focal position.. Although most of the fiber lasers can get CW lasing between facets without the external high-reflection cavity, the threshold of the fiber laser with an extended highreflection cavity is considerably lower than that of the fiber laser without external feedback. Experimental results reveal that the threshold of the passive Q-switched fiber laser with an extended cavity is also usually lower than that of CW free-running operation between facets, as shown in Fig. 7. Since the extended cavity dominates the lasing, the couple-cavity effect arising from facets is insignificant in the performance of the passive Qswitching operation. Figure 8 shows the pulse repetition rate and the pulse energy versus the incident pump powerattheoptimumfocalposition.Thepulserepetition rate initially increases with pump power, and is approximately up to 22 kHz at an incident pump power of 10 W. Like typically passively Q-switched lasers, the pulse energies weakly depend on the pump power and their values are found to be approximately 210 μJ. The pulse width is found to be in the range of 60–70 ns, as shown in the inset of Fig. 8.. Fig. 8. Pulse repetition rate and pulse energy versus the incident pump power at the optimum focal position. 1 May 2008 / Vol. 47, No. 13 / APPLIED OPTICS. 2301.
(6) Fig. 9. Experimental and theoretical results for the dependence of the output pulse energy on the focal position at an incident pump power of 10 W. Symbols are experimental data. Solid line is calculated results using Eqs. (8) and (9).. To validate the developed models, two more experiments were performed with other saturable absorbers (T o ¼ 0:3, la ¼ 3:0 mm and T o ¼ 0:6, la ¼ 2:6 mm) and a reimaging magnification of M a ¼ 0:7. Figure 9 shows the experimental results for the dependence of the output pulse energy on the focal position at an incident pump power of 10 W. The good agreement between experimental results and theoretical predictions confirms the validity of our physical analysis. 5. Conclusion. We have developed an analytical model for the optimization of the extended cavity with a saturable absorber in a passively Q-switched fiber laser. From the criterion of the minimum average mode area inside the saturable absorber, the optimum focal position was derived to be an analytical function of the thickness and initial transmission of the saturable absorber. With the expression of the optimum focal position, the optimum magnification of the reimaging optics was exactly derived to be a compact close form in terms of the physical properties of the laser fiber as well as the saturable absorber. The present model provides a straightforward procedure to determine the key parameters for optimizing passively Q-switched fiber lasers. Finally, an experiment on the subject of the passively Q-switched fiber laser has been performed to validate the present model and to manifest the utilization. References 1. L. Zenteno, “High-power double-clad fiber lasers,” J. Lightwave Technol. 11, 1435–1446 (1993). 2. M. J. F. Digonnet, Rare-Earth-Doped Fiber Lasers and Amplifiers, 2nd ed. (Marcel Dekker, 2001). 3. Z. J. Chen, A. B. Grudinin, J. 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Fotiadi, A. Kurkov, and I. Razdobreev, “All-fiber passively Q-switched ytterbium laser,” in CLEO/Europe-EQEC 2005, Technical Digest, CJ 2–3 (IEEE, 2005). 9. P. Adel, M. Auerbach, C. Fallnich, S. Unger, H.-R. Müller, and J. Kirchhof, “Passive Q-switching by Tm3þ co-doping of a Yb3þ -fiber laser,” Opt. Express 11, 2730–2735 (2003). 10. M. Laroche, A. M. Chardon, J. Nilsson, D. P. Shepherd, W. A. Clarkson, S. Girard, and R. Moncorgé, “Compact diode-pumped passively Q-switched tunable Er-Yb doubleclad fiber laser,” Opt. Lett. 27, 1980–1982 (2002). 11. V. N. Philippov, A. V. Kiryanov, and S. Unger, “Advanced configuration of erbium fiber passively Q-switched laser with Co2þ :ZnSe crystal as saturable absorber,” IEEE Photonics Technol. Lett. 16, 57–59 (2004). 12. F. Z. Qamar and T. A. King, “Passive Q-switching of the Tmsilica fiber laser near 2 μm by Cr2þ :ZnSe saturable absorber crystal,” Opt. Commun. 248, 501–505 (2005). 13. M. Laroche, H. Gilles, S. Girard, N. Passilly, and K. Aït-Ameur, “Nanosecond pulse generation in a passively Q-switched Ybdoped fiber laser by Cr4þ :YAG saturable absorber,” IEEE Photon. Technol. Lett. 18, 764–766 (2006). 14. R. Paschotta, R. Häring, E. Gini, H. Melchior, U. Keller, H. L. Offerhaus, and D. J. Richardson, “Passively Q-switched 0:1 mJ fiber laser system at 1:53 μm,” Opt. Lett. 24, 388–390 (1999). 15. Y. Wang and C. Q. Xu, “Modeling and optimization of Qswitched double-clad fiber lasers,” Appl. Opt. 45, 2058–2071 (2006). 16. Y. F. Chen, Y. P. Lan, and H. L. Chang, “Analytical model for design criteria of passively Q-switched lasers,” IEEE J. Quantum Electron. 37, 462–468 (2001). 17. J. J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. 31, 1890–1901 (1995). 18. G. Xiao and M. Bass, “A generalized model Q-switched lasers including excited state absorption in the saturable absorber,” IEEE J. Quantum Electron. 33, 41–44 (1997). 19. X. Zhang, S. Zhao, Q. Wang, Q. Zhang, L. Sun, and S. Zhang, “Optimization of Cr4þ -doped saturable-absorber Q-switched lasers,” IEEE J. Quantum Electron. 33, 2286–2294 (1997). 20. M. Hercher, “An analysis of saturable absorbers,” Appl. Opt. 6, 947–954 (1967). 21. A. E. Siegman, Lasers (University Science Books, 1986). 22. K. Lu and N. K. Dutta, “Spectroscopic properties of Yb-doped silica glass,” J. Apl. Phys. 91, 576–581 (2002). 23. Y. Shimony, Z, Burshtein, and Y. Kalisky, “Cr4þ :YAG as passive Q-switch and Brewster plate in a pulsed Nd:YAG laser,” IEEE J. Quantum Electron. 31, 1738–1741 (1995)..
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