Abstract

A passively Q-switched Nd:YAG/Cr4+:YAG micro-laser with three-beam output was realized. A single active laser source made of a composite, all-ceramics Nd:YAG/Cr4+:YAG monolithic cavity was pumped by three independent lines. At 5 Hz repetition rate, each line delivered laser pulses with ~2.4 mJ energy and 2.8-MW peak power. The M2 factor of a laser beam was 3.7, and stable air breakdowns were realized. The increase of pump repetition rate up to 100 Hz improved the laser pulse energy by 6% and required ~6% increase of the pump pulse energy. Pulse timing of the laser-array beams can by adjusted by less than 5% tuning of an individual line pump energy, and therefore simultaneous multi-point ignition is possible. This kind of laser can be used for multi-point ignition of an automobile engine.

© 2011 OSA

1. Introduction

Extensive research has been performed during last years on laser-induced ignition of air-fuel mixtures in internal combustion engines. In the beginning, experiments were made with sized and robust, commercial available lasers that delivered pulses with energy in the range of tens to a few hundreds of mJ and several ns pulse duration [13]. These investigations revealed that laser-induced ignition offers significant advantages over a conventional spark-ignition system, such as higher probability to ignite leaner mixtures, reduction of erosion effects, increases of engine efficiency, or shorter combustion time.

Subsequent research concluded that suitable laser configuration for engine ignition is a Nd:YAG laser, passively Q-switched by Cr4+:YAG saturable absorber (SA) [4]. Thus, using an end-pumping scheme, Q-switched laser pulses with energy up to 6 mJ and 1.5-ns pulse duration were obtained from a 210-mm long Nd:YAG/Cr4+:YAG laser. Furthermore, side-pumping technique was employed to realize a Nd:YAG laser passively Q-switched by Cr4+:YAG SA with 25 mJ energy per pulse and pulse duration around 3 ns [5]. The laser resonator was also long, around 170 mm. Therefore, with these lasers is difficult to achieve the small device size required by an electrical spark plug used in the automotive industry.

In recent works our group has realized passively Q-switched Nd:YAG/Cr4+:YAG micro-lasers and demonstrated laser ignition of an automobile engine with improved performances in comparison with ignition induced by a conventional spark plug [6,7]. The strategy was to shorten the pulse duration by decreasing the resonator length, and to maximize the laser pulse energy by optimizing the pump conditions, the Nd:YAG doping level and length, as well as Cr4+:YAG initial transmission and the output mirror reflectivity [8]. An Nd:YAG/Cr4+:YAG micro-laser with 2.7-mJ energy per pulse and 600-ps pulse duration was realized [7]. This laser included optics for pumping, the laser resonator of 11-mm length, as well as optics that collimated and focused the beam to a spot that assure fuel ignition, and was assembled in a device that matched the dimensions of an electrical spark plug.

Various papers have also reported that multi-point ignition increases significantly the combustion pressure and shortens the combustion time compared to single-point ignition [2,9,10]. The experiments employed combustion chambers in which two laser beams were inserted through different windows, and thus distance between the ignition points was easily adjusted. However, the use of a single-laser beam that was focused in three points with a diffractive lens failed to show improved combustion [2], in comparison to the two-point ignition experiments. The result was attributed to the short distance between the ignition points. Therefore, study of the influence of multi-point ignition on the performances of a real car engine would require realization of passively Q-switched Nd:YAG/Cr4+:YAG lasers with multiple-beam output and with size close to that of an electrical spark plug.

In this work we report a passively Q-switched Nd:YAG/Cr4+:YAG micro-laser with three-beam output, each beam inducing air-breakdown in points at adjustable distance. Opposite to the previous realized lasers that used discrete Nd:YAG and Cr4+:YAG single-crystals components [48], this laser consist of a composite, all-ceramics Nd:YAG/Cr4+:YAG monolithic medium that were pumped by independent lines.

2. The composite, all-ceramics Nd:YAG/Cr4+:YAG monolithic micro-laser

Figure 1(a) is a sketch of the experimental set-up used for the passively Q-switched Nd:YAG/Cr4+:YAG laser with three-beam output. One could choose to use one pump line, and then to divide the high-energy laser beam into three fascicles, which has to be directed at necessary angle and then focused. This choice increases probability of damaging the laser media (due to the laser beam high intensity, or due to thermal effects), and would complicate the guiding line. Our solution was to employ independent, similar pumping lines, and to use a composite, all-ceramics Nd:YAG/Cr4+:YAG medium that has a diameter imposed by the spark dimension. In this way, the heat deposited in the unit volume decreases, the alignment is made simple, while the laser medium can be realized by ceramics techniques. Furthermore, the Nd:YAG/Cr4+:YAG monolithic structure assures robustness and compactness of the laser, which has to be mounted on the engine, like an electric spark plug.

 

Fig. 1 (a) Schematic of a passively Q-switched, all-ceramics, composite, Nd:YAG/Cr4+:YAG monolithic laser with three-beam output. (b) A photo of a composite medium is shown.

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The pump was made at 807 nm (λp) with three fiber-coupled (600-μm diameter and numerical aperture NA = 0.22) diode lasers (JOLD-120-QPXF-2P, Jenoptik, Germany). Pump repetition rate and pump pulse duration were 5 Hz and 250 μs, respectively. Few composite, all-ceramics Nd:YAG/Cr4+:YAG media, each with a 9-mm diameter, were prepared for the experiments by diffusion bonding at elevated temperature (Baikowski Japan Co., Ltd.). While the Cr4+:YAG SA has initial transmission T0 = 0.30 (and a thickness of ~2.5 mm), the influence of Nd-doping level on Q-switch laser characteristics was investigated by using Nd:YAG with 1.1-at.% Nd (~7.5-mm thick), as well as highly-doped 1.5-at.% Nd (thickness of 5 mm) and 2.0-at.% Nd (thickness of 3.5 mm). The Nd:YAG length yields ~95% absorption at λp, thus avoiding bleaching effect of Cr4+:YAG by the pump beam. Surface S1 of Nd:YAG was coated high reflectivity, HR (reflectivity R>0.998) at the 1.06-μm wavelength of lasing (λem) and high transmission, HT (transmission T>0.98) at λp. The out-coupling mirror (OCM) with T = 0.50 at λem was coated on surface S2 of Cr4+:YAG. A photo of the composite 1.1-at.% Nd:YAG/Cr4+:YAG ceramics is shown in Fig. 1(b). The optical pumping was realized through three independent, similar, and compact pumping lines (marked by 1 to 3 in Fig. 1(a)), each line containing a pair of an aspheric collimating lens and an aspheric focusing lens with short focal length and high NA. Furthermore, the influence of pump-beam dimension on laser pulse performances was evaluated by using, for each pumping line, two designs. The pump-beam diameter into Nd:YAG was 0.84 mm for the first design (#A) and 1.0 mm for the second design (#B).

The characteristics of the Q-switched laser pulses yielded by the Nd:YAG/Cr4+:YAG ceramics are given in Table 1 . Using pump line design “#A”, the energy of the laser pulse yielded by the 1.1-at.% Nd:YAG was 1.55 mJ, with a pulse peak power of 1.8 MW. Air breakdown was realized with lenses of focal length up to 6.5 mm. The laser pulse energy increased at 2.37 mJ and pulse peak power improved at 2.8 MW when pump line design “#B” was used. The laser beam M2 factor, which was measured by the knife-edge method, was 3.7. Air breakdown with lenses of focal length up to 14 mm was obtained.

Tables Icon

Table 1. Characteristics of the Q-switched Laser Pulses Obtained with Composite Nd:YAG/Cr4+:YAG Ceramics

In order to explain the influence of pump-beam spot size on the Q-switched laser performances, we used a rate equation model [1113] in which the pump beam was assumed to have a top-hat distribution of radius wp and the laser beam was taken as Gaussian with a spot size of radius wg. Both wp and wg were considered constant along the Nd:YAG/Cr4+:YAG medium. The laser pulse energy is given by general relation:

Ep=hνAg2γgσglnRln(ngfngi)
where is the photon energy at λem, σg represents Nd:YAG stimulated emission, γg is the inversion reduction factor, Ag is the effective area of the laser beam in Nd:YAG, and the OCM reflectivity is R = (1-T). The initial population inversion density, ngi is:
ngi=lnR+LlnT022σgg[1exp(2a2)]
with parameter a = wp/wg. Here L represents the resonator round-trip residual loss and ℓg is the Nd:YAG length. The final population inversion density, ngf and ngi are related by equation:
(1ngfngi)+[1+(1δ)lnT02β]ln(ngfngi)+1α(1δ)lnT02β[1(ngfngi)α]=0
where δ = σESASA, with σSA and σESA the absorption cross section and excited-state absorption cross section of Cr4+:YAG, respectively. Parameter β is β = (-lnR + L - lnT02)/[1-exp(−2a2)] and parameter α is α = (γSAσSA)/(γgσg) × (Ag/ASA); γSA is the inversion reduction factor for Cr4+:YAG and ASA represents the laser-beam effective area in Cr4+:YAG.

Figure 2 presents parameter ngf/ngi versus ratio a = wp/wg. In simulation losses were L = 0.06 (0.01 for Nd:YAG and 0.05 for Cr4+:YAG final transmission), and spectroscopic parameters of Nd:YAG and Cr4+:YAG were σg = 2.63 × 10−19 cm2, σSA = 4.3 × 10−18 cm2, and σESA = 8.2 × 10−19 cm2. If wp<wg the overlap between pump and laser beam is good. However, ngf/ngi increases when wp/wg decreases: Although the initial inversion of population ngi is high, a small fraction of it used for lasing and therefore Q-switched laser pulse energy is low. If wp/wg has a large value, the central part of the inversion of population interacts with laser mode, whereas some outside part could be depleted by spontaneous emission. Increasing wp/wg decreases ngf/ngi: The final inversion of population ngf is low and a pulse laser with high energy is obtained. The expected values of the Q-switched laser pulse at various sizes wg of the laser mode were also shown in Fig. 2.

 

Fig. 2 Ratio ngf/ngi versus wp/wg and Q-switched laser pulse energy for various laser beam radii wg. Symbols are the experimental values measured with pump line “#A” (●) and pump line “#B” (■).

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A plane-plane resonator operates due to thermal effects induced by optical pumping in the active media. According to the model developed in [14], the focal length f of Nd:YAG thermal lens can be evaluated by relation: f = (πKcwg2)/[Ph⋅(dn/dT)]. For Nd:YAG, the thermal conductivity is Kc = 10.1 Wm−1K−1 [15], thermal coefficient of the refraction index is dn/dT = 0.73 × 10−5 K−1, while ~0.24 of the absorbed pump power is transformed into heat (Ph) under efficient laser emission at 1.06 μm. Furthermore, as it will be discussed later, an increase of the pump repetition rate up to 100 Hz changed little the laser pulse energy. Based on these observations, the average thermal lens of the 1.1-at.% Nd:YAG was evaluated as ~3.8 m for the pump with line “#A”, while pump line design “#B” induced in Nd:YAG a thermal lens with ~6.5 m focal length. Corresponding values of wg were determinate by an ABCD description of the resonator as ~340 μm and ~420 μm, respectively. The symbols in Fig. 2 are the experimental values of the laser pulse energies obtained from the 1.1-at.% Nd:YAG. Agreement with theoretical modeling is good, especially if uncertainties in evaluation of thermal focal lens or of other parameters (such as losses L) are considered. Furthermore, the model can be improved by taking into account variation of pump beam radius wp and of laser beam spot size wg along the resonator length.

Energies of 2.03 mJ and 1.37 mJ were measured from the 1.5-at.% Nd:YAG and 2.0-at.% Nd:YAG ceramics, respectively. The pump pulse energy was 26.7 mJ for the 1.1-at.% Nd:YAG, 33 mJ for the 1.5-at.% Nd:YAG, and 32 mJ for the 2.0-at.% Nd:YAG. The decrease of the 4F3/2 upper-level lifetime with Nd-doping could be a reason for lower laser performances recorded with the highly-doped Nd:YAG compared with the 1.1-at.% Nd:YAG. The OCM transmission has also to be optimized for the highly-doped Nd:YAG ceramics.

Very important for performances of the monolithic laser is the uniformity of the composite Nd:YAG/Cr4+:YAG ceramics. Table 1 presents also average values of the laser pulse energy determined along Ox and Oz axes (Fig. 1(b)), estimated by scanning each medium at a 0.5-mm step. The laser pulse energies were very close to those measured at the media center. Standard deviation was small, below 3% for 1.1-at.% and 1.5-at.% Nd:YAG, and less than 4% for the 2.0-at.% Nd:YAG. The results indicate a very good homogeneity as well as quality of every composite Nd:YAG/Cr4+:YAG ceramics, in spite of the high, 9-mm diameter.

The three-beam output laser was realized with the composite, all-ceramics 1.1-at.% Nd:YAG/Cr4+:YAG ceramics. As shown in Fig. 1(a), the guiding line consisted of two sections. In the first step, each laser beam was expanded and then collimated (the expander part). Next, each laser beam was bent with a prism (patent pending), and finally it was focused to a dimension that assured air breakdown. Taking into account the size of an electrical spark used in an automobile engine, as well as engine chamber dimension, the guiding line was designed such to assure a distance between a focusing point and the laser axis, φc of 4.5 mm and a depth of the focusing point inside the combustion chamber, bc of 9 mm. These parameters can be easily modified by changing the prism angle. A photo of the all-ceramics, composite Nd:YAG/Cr4+:YAG monolithic laser with three-beam output is presented in Fig. 3 , and air breakdown is illustrated. An automobile electrical spark plug is shown for comparison.

 

Fig. 3 A photo of the composite, all-ceramics passively Q-switched Nd:YAG/Cr4+:YAG monolithic laser with three-beam output is shown.

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Various other characteristics of a Q-switched laser pulse, such us the delay time (i.e. the time between the moment when the pump pulse begins and the moment when the laser pulse start to build up), or the pulse jitter and standard deviation were determined. Figure 4 presents these parameters function of the pump pulse energy. As expected, delay time decreases when the pump pulse energy increases. Therefore, the use of independent pumping lines allows control of the air breakdown timing by changing the pump energy of each line. Furthermore, real simultaneous ignition in all three points can be obtained by a small (less than 5%) tuning of individual line pump energy. Time jitter is low (2.1 μs at the pump pulse energy of 26.7 mJ and 1.0 μs at 32-mJ energy of the pump pulse) and thus it would not have a negative impact on an automobile engine that is ignited by the laser. A second laser pulse was not observed. Nevertheless, increasing the pump pulse duration would enable obtaining of multiple laser pulses [7], which are useful for car ignition especially if lean fuel-air mixtures are used.

 

Fig. 4 Time delay of the Q-switched laser pulse, time jitter and standard deviation function of pump pulse energy (5-Hz pump repetition rate, 250-μs pump pulse duration).

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Previous experiments were performed at 5-Hz pump repetition, whereas temperature of the composite Nd:YAG/Cr4+:YAG ceramics was not controlled. However, repetition rates up to 60 Hz are necessary for operation of a car engine. Variation of laser pulse characteristics was therefore investigated versus the pump repetition rate. Experiments concluded that an increase from 5 to 100 Hz of the pump repetition rate improves the Q-switched pulse energy from 2.37 mJ to 2.51 mJ, i.e. by a 6% fraction. This change required a small (~6%) increase of the minimal pump pulse energy, from 26.7 mJ at 5 Hz to 28.4 mJ at 100-Hz pump repetition rate. The variation of the laser performances with temperature was beyond the scope of this study, but would be considered in future investigations. Nevertheless, in previous experiments [1618] we have measured only a slight increase of the laser pulse energy when temperature of a Nd:YAG laser passively Q-switched by Cr4+:YAG SA (build of discrete, single-crystals components) was increased to 150°C. Future experiments would also consider testing of the laser to shock and vibration conditions that are similar to those experienced in a car engine.

3. Conclusions

A compact, passively Q-switched Nd:YAG/Cr4+:YAG giant-pulse emitting micro-laser with three-beam output has been realized. This prototype laser incorporates a composite, all-ceramics Nd:YAG/Cr4+:YAG monolithic structure that was pumped by independent lines. The laser size is comparable to that of an electrical spark plug, being the first demonstration of this kind of device to the best of our knowledge. Laser pulses with energy of ~2.4 mJ and 2.8-MW peak power at 5-Hz repetition rate were obtained from a 10-mm thick Nd:YAG/Cr4+:YAG ceramics, just such as “giant micro-photonics”. Increasing pump repetition rate up to 100 Hz improved the laser pulse energy by 6% and required an ~6% increase of the pump pulse energy compared with operation at 5 Hz. Pulse timing of the laser-array beams can by controlled by changing the pump energy of each individual line. On the other hand, simultaneous multi-point ignition is possible by less than 5% tuning of the individual pump line energy This kind of laser will enable studies on the performance of internal combustion engines with multi-point ignition.

Acknowledgments

This work was financed by Japan Science and Technical Agency (JST), and partially supported by DENSO Company, Japan. The authors thank Mr. Mizutani of the IMS Equipment Development Division for the help with the laser module design.

References and links

1. J. X. Ma, D. R. Alexander, and D. E. Poulain, “Laser spark ignition and combustion characteristics of methane-air mixtures,” Combust. Flame 112(4), 492–506 (1998). [CrossRef]  

2. M. Weinrotter, H. Kopecek, M. Tesch, E. Wintner, M. Lackner, and F. Winter, “Laser ignition of ultra-lean methane/hydrogen/air mixtures at high temperature and pressure,” Exp. Therm. Fluid Sci. 29(5), 569–577 (2005). [CrossRef]  

3. M. Weinrotter, H. Kopecek, and E. Wintner, “Laser ignition of engines,” Laser Phys. 15(7), 947–953 (2005).

4. H. Kofler, J. Tauer, G. Tartar, K. Iskra, J. Klausner, G. Herdin, and E. Wintner, “An innovative solid-state laser for engine ignition,” Laser Phys. Lett. 4(4), 322–327 (2007). [CrossRef]  

5. G. Kroupa, G. Franz, and E. Winkelhofer, “Novel miniaturized high-energy Nd:YAG laser for spark ignition in internal combustion engines,” Opt. Eng. 48(1), 014202 (2009). [CrossRef]  

6. M. Tsunekane, T. Inohara, A. Ando, K. Kanehara, and T. Taira, “High peak power, passively Q-switched Cr:YAG/Nd:YAG micro-laser for ignition of engines,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2008), paper MB4.

7. M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010). [CrossRef]  

8. H. Sakai, H. Kan, and T. Taira, “>1 MW peak power single-mode high-brightness passively Q-switched Nd 3+:YAG microchip laser,” Opt. Express 16(24), 19891–19899 (2008). [CrossRef]   [PubMed]  

9. T. X. Phuoc, “Single-point versus multi-point laser ignition: experimental measurements of combustion times and pressures,” Combust. Flame 122(4), 508–510 (2000). [CrossRef]  

10. M. H. Morsy, Y. S. Ko, S. H. Chung, and P. Cho, “Laser-induced two-point ignition of premixture with a single-shot laser,” Combust. Flame 124(4), 724–727 (2001). [CrossRef]  

11. J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. 31(11), 1890–1901 (1995). [CrossRef]  

12. N. Pavel, J. Saikawa, S. Kurimura, and T. Taira, “High average power diode end-pumped composite Nd:YAG laser passively Q-switched by Cr4+:YAG saturable absorber,” Jpn. J. Appl. Phys. 40(Part 1, No. 3A), 1253–1259 (2001). [CrossRef]  

13. S. T. Li, X. Y. Zhang, Q. P. Wang, P. Li, J. Chang, X. L. Zhang, and Z. H. Cong, “Modeling of Q-switched lasers with top-hat pump beam distribution,” Appl. Phys. B 88(2), 221–226 (2007). [CrossRef]  

14. M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990). [CrossRef]  

15. Y. Sato and T. Taira, “The studies of thermal conductivity in GdVO4, YVO4, and Y3Al5O12 measured by quasi-one-dimensional flash method,” Opt. Express 14(22), 10528–10536 (2006). [CrossRef]   [PubMed]  

16. M. Tsunekane and T. Taira, “Temperature and polarization dependences of Cr:YAG transmission for passive Q-switching,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuD8.

17. T. Dascalu and N. Pavel, “High-temperature operation of a diode-pumped passively Q-switched Nd:YAG/Cr4+:YAG laser,” Laser Phys. 19(11), 2090–2095 (2009). [CrossRef]  

18. N. Pavel, M. Tsunekane, and T. Taira, “Enhancing performances of a passively Q-switched Nd:YAG/Cr4+:YAG microlaser with a volume Bragg grating output coupler,” Opt. Lett. 35(10), 1617–1619 (2010). [CrossRef]   [PubMed]  

References

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  1. J. X. Ma, D. R. Alexander, and D. E. Poulain, “Laser spark ignition and combustion characteristics of methane-air mixtures,” Combust. Flame 112(4), 492–506 (1998).
    [CrossRef]
  2. M. Weinrotter, H. Kopecek, M. Tesch, E. Wintner, M. Lackner, and F. Winter, “Laser ignition of ultra-lean methane/hydrogen/air mixtures at high temperature and pressure,” Exp. Therm. Fluid Sci. 29(5), 569–577 (2005).
    [CrossRef]
  3. M. Weinrotter, H. Kopecek, and E. Wintner, “Laser ignition of engines,” Laser Phys. 15(7), 947–953 (2005).
  4. H. Kofler, J. Tauer, G. Tartar, K. Iskra, J. Klausner, G. Herdin, and E. Wintner, “An innovative solid-state laser for engine ignition,” Laser Phys. Lett. 4(4), 322–327 (2007).
    [CrossRef]
  5. G. Kroupa, G. Franz, and E. Winkelhofer, “Novel miniaturized high-energy Nd:YAG laser for spark ignition in internal combustion engines,” Opt. Eng. 48(1), 014202 (2009).
    [CrossRef]
  6. M. Tsunekane, T. Inohara, A. Ando, K. Kanehara, and T. Taira, “High peak power, passively Q-switched Cr:YAG/Nd:YAG micro-laser for ignition of engines,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2008), paper MB4.
  7. M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
    [CrossRef]
  8. H. Sakai, H. Kan, and T. Taira, “>1 MW peak power single-mode high-brightness passively Q-switched Nd 3+:YAG microchip laser,” Opt. Express 16(24), 19891–19899 (2008).
    [CrossRef] [PubMed]
  9. T. X. Phuoc, “Single-point versus multi-point laser ignition: experimental measurements of combustion times and pressures,” Combust. Flame 122(4), 508–510 (2000).
    [CrossRef]
  10. M. H. Morsy, Y. S. Ko, S. H. Chung, and P. Cho, “Laser-induced two-point ignition of premixture with a single-shot laser,” Combust. Flame 124(4), 724–727 (2001).
    [CrossRef]
  11. J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. 31(11), 1890–1901 (1995).
    [CrossRef]
  12. N. Pavel, J. Saikawa, S. Kurimura, and T. Taira, “High average power diode end-pumped composite Nd:YAG laser passively Q-switched by Cr4+:YAG saturable absorber,” Jpn. J. Appl. Phys. 40(Part 1, No. 3A), 1253–1259 (2001).
    [CrossRef]
  13. S. T. Li, X. Y. Zhang, Q. P. Wang, P. Li, J. Chang, X. L. Zhang, and Z. H. Cong, “Modeling of Q-switched lasers with top-hat pump beam distribution,” Appl. Phys. B 88(2), 221–226 (2007).
    [CrossRef]
  14. M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990).
    [CrossRef]
  15. Y. Sato and T. Taira, “The studies of thermal conductivity in GdVO4, YVO4, and Y3Al5O12 measured by quasi-one-dimensional flash method,” Opt. Express 14(22), 10528–10536 (2006).
    [CrossRef] [PubMed]
  16. M. Tsunekane and T. Taira, “Temperature and polarization dependences of Cr:YAG transmission for passive Q-switching,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuD8.
  17. T. Dascalu and N. Pavel, “High-temperature operation of a diode-pumped passively Q-switched Nd:YAG/Cr4+:YAG laser,” Laser Phys. 19(11), 2090–2095 (2009).
    [CrossRef]
  18. N. Pavel, M. Tsunekane, and T. Taira, “Enhancing performances of a passively Q-switched Nd:YAG/Cr4+:YAG microlaser with a volume Bragg grating output coupler,” Opt. Lett. 35(10), 1617–1619 (2010).
    [CrossRef] [PubMed]

2010 (2)

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[CrossRef]

N. Pavel, M. Tsunekane, and T. Taira, “Enhancing performances of a passively Q-switched Nd:YAG/Cr4+:YAG microlaser with a volume Bragg grating output coupler,” Opt. Lett. 35(10), 1617–1619 (2010).
[CrossRef] [PubMed]

2009 (2)

T. Dascalu and N. Pavel, “High-temperature operation of a diode-pumped passively Q-switched Nd:YAG/Cr4+:YAG laser,” Laser Phys. 19(11), 2090–2095 (2009).
[CrossRef]

G. Kroupa, G. Franz, and E. Winkelhofer, “Novel miniaturized high-energy Nd:YAG laser for spark ignition in internal combustion engines,” Opt. Eng. 48(1), 014202 (2009).
[CrossRef]

2008 (1)

2007 (2)

H. Kofler, J. Tauer, G. Tartar, K. Iskra, J. Klausner, G. Herdin, and E. Wintner, “An innovative solid-state laser for engine ignition,” Laser Phys. Lett. 4(4), 322–327 (2007).
[CrossRef]

S. T. Li, X. Y. Zhang, Q. P. Wang, P. Li, J. Chang, X. L. Zhang, and Z. H. Cong, “Modeling of Q-switched lasers with top-hat pump beam distribution,” Appl. Phys. B 88(2), 221–226 (2007).
[CrossRef]

2006 (1)

2005 (2)

M. Weinrotter, H. Kopecek, M. Tesch, E. Wintner, M. Lackner, and F. Winter, “Laser ignition of ultra-lean methane/hydrogen/air mixtures at high temperature and pressure,” Exp. Therm. Fluid Sci. 29(5), 569–577 (2005).
[CrossRef]

M. Weinrotter, H. Kopecek, and E. Wintner, “Laser ignition of engines,” Laser Phys. 15(7), 947–953 (2005).

2001 (2)

M. H. Morsy, Y. S. Ko, S. H. Chung, and P. Cho, “Laser-induced two-point ignition of premixture with a single-shot laser,” Combust. Flame 124(4), 724–727 (2001).
[CrossRef]

N. Pavel, J. Saikawa, S. Kurimura, and T. Taira, “High average power diode end-pumped composite Nd:YAG laser passively Q-switched by Cr4+:YAG saturable absorber,” Jpn. J. Appl. Phys. 40(Part 1, No. 3A), 1253–1259 (2001).
[CrossRef]

2000 (1)

T. X. Phuoc, “Single-point versus multi-point laser ignition: experimental measurements of combustion times and pressures,” Combust. Flame 122(4), 508–510 (2000).
[CrossRef]

1998 (1)

J. X. Ma, D. R. Alexander, and D. E. Poulain, “Laser spark ignition and combustion characteristics of methane-air mixtures,” Combust. Flame 112(4), 492–506 (1998).
[CrossRef]

1995 (1)

J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. 31(11), 1890–1901 (1995).
[CrossRef]

1990 (1)

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990).
[CrossRef]

Alexander, D. R.

J. X. Ma, D. R. Alexander, and D. E. Poulain, “Laser spark ignition and combustion characteristics of methane-air mixtures,” Combust. Flame 112(4), 492–506 (1998).
[CrossRef]

Ando, A.

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[CrossRef]

Chang, J.

S. T. Li, X. Y. Zhang, Q. P. Wang, P. Li, J. Chang, X. L. Zhang, and Z. H. Cong, “Modeling of Q-switched lasers with top-hat pump beam distribution,” Appl. Phys. B 88(2), 221–226 (2007).
[CrossRef]

Cho, P.

M. H. Morsy, Y. S. Ko, S. H. Chung, and P. Cho, “Laser-induced two-point ignition of premixture with a single-shot laser,” Combust. Flame 124(4), 724–727 (2001).
[CrossRef]

Chung, S. H.

M. H. Morsy, Y. S. Ko, S. H. Chung, and P. Cho, “Laser-induced two-point ignition of premixture with a single-shot laser,” Combust. Flame 124(4), 724–727 (2001).
[CrossRef]

Cong, Z. H.

S. T. Li, X. Y. Zhang, Q. P. Wang, P. Li, J. Chang, X. L. Zhang, and Z. H. Cong, “Modeling of Q-switched lasers with top-hat pump beam distribution,” Appl. Phys. B 88(2), 221–226 (2007).
[CrossRef]

Dascalu, T.

T. Dascalu and N. Pavel, “High-temperature operation of a diode-pumped passively Q-switched Nd:YAG/Cr4+:YAG laser,” Laser Phys. 19(11), 2090–2095 (2009).
[CrossRef]

Degnan, J.

J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. 31(11), 1890–1901 (1995).
[CrossRef]

Fields, R. A.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990).
[CrossRef]

Fincher, C. L.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990).
[CrossRef]

Franz, G.

G. Kroupa, G. Franz, and E. Winkelhofer, “Novel miniaturized high-energy Nd:YAG laser for spark ignition in internal combustion engines,” Opt. Eng. 48(1), 014202 (2009).
[CrossRef]

Herdin, G.

H. Kofler, J. Tauer, G. Tartar, K. Iskra, J. Klausner, G. Herdin, and E. Wintner, “An innovative solid-state laser for engine ignition,” Laser Phys. Lett. 4(4), 322–327 (2007).
[CrossRef]

Innocenzi, M. E.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990).
[CrossRef]

Inohara, T.

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[CrossRef]

Iskra, K.

H. Kofler, J. Tauer, G. Tartar, K. Iskra, J. Klausner, G. Herdin, and E. Wintner, “An innovative solid-state laser for engine ignition,” Laser Phys. Lett. 4(4), 322–327 (2007).
[CrossRef]

Kan, H.

Kanehara, K.

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[CrossRef]

Kido, N.

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[CrossRef]

Klausner, J.

H. Kofler, J. Tauer, G. Tartar, K. Iskra, J. Klausner, G. Herdin, and E. Wintner, “An innovative solid-state laser for engine ignition,” Laser Phys. Lett. 4(4), 322–327 (2007).
[CrossRef]

Ko, Y. S.

M. H. Morsy, Y. S. Ko, S. H. Chung, and P. Cho, “Laser-induced two-point ignition of premixture with a single-shot laser,” Combust. Flame 124(4), 724–727 (2001).
[CrossRef]

Kofler, H.

H. Kofler, J. Tauer, G. Tartar, K. Iskra, J. Klausner, G. Herdin, and E. Wintner, “An innovative solid-state laser for engine ignition,” Laser Phys. Lett. 4(4), 322–327 (2007).
[CrossRef]

Kopecek, H.

M. Weinrotter, H. Kopecek, and E. Wintner, “Laser ignition of engines,” Laser Phys. 15(7), 947–953 (2005).

M. Weinrotter, H. Kopecek, M. Tesch, E. Wintner, M. Lackner, and F. Winter, “Laser ignition of ultra-lean methane/hydrogen/air mixtures at high temperature and pressure,” Exp. Therm. Fluid Sci. 29(5), 569–577 (2005).
[CrossRef]

Kroupa, G.

G. Kroupa, G. Franz, and E. Winkelhofer, “Novel miniaturized high-energy Nd:YAG laser for spark ignition in internal combustion engines,” Opt. Eng. 48(1), 014202 (2009).
[CrossRef]

Kurimura, S.

N. Pavel, J. Saikawa, S. Kurimura, and T. Taira, “High average power diode end-pumped composite Nd:YAG laser passively Q-switched by Cr4+:YAG saturable absorber,” Jpn. J. Appl. Phys. 40(Part 1, No. 3A), 1253–1259 (2001).
[CrossRef]

Lackner, M.

M. Weinrotter, H. Kopecek, M. Tesch, E. Wintner, M. Lackner, and F. Winter, “Laser ignition of ultra-lean methane/hydrogen/air mixtures at high temperature and pressure,” Exp. Therm. Fluid Sci. 29(5), 569–577 (2005).
[CrossRef]

Li, P.

S. T. Li, X. Y. Zhang, Q. P. Wang, P. Li, J. Chang, X. L. Zhang, and Z. H. Cong, “Modeling of Q-switched lasers with top-hat pump beam distribution,” Appl. Phys. B 88(2), 221–226 (2007).
[CrossRef]

Li, S. T.

S. T. Li, X. Y. Zhang, Q. P. Wang, P. Li, J. Chang, X. L. Zhang, and Z. H. Cong, “Modeling of Q-switched lasers with top-hat pump beam distribution,” Appl. Phys. B 88(2), 221–226 (2007).
[CrossRef]

Ma, J. X.

J. X. Ma, D. R. Alexander, and D. E. Poulain, “Laser spark ignition and combustion characteristics of methane-air mixtures,” Combust. Flame 112(4), 492–506 (1998).
[CrossRef]

Morsy, M. H.

M. H. Morsy, Y. S. Ko, S. H. Chung, and P. Cho, “Laser-induced two-point ignition of premixture with a single-shot laser,” Combust. Flame 124(4), 724–727 (2001).
[CrossRef]

Pavel, N.

N. Pavel, M. Tsunekane, and T. Taira, “Enhancing performances of a passively Q-switched Nd:YAG/Cr4+:YAG microlaser with a volume Bragg grating output coupler,” Opt. Lett. 35(10), 1617–1619 (2010).
[CrossRef] [PubMed]

T. Dascalu and N. Pavel, “High-temperature operation of a diode-pumped passively Q-switched Nd:YAG/Cr4+:YAG laser,” Laser Phys. 19(11), 2090–2095 (2009).
[CrossRef]

N. Pavel, J. Saikawa, S. Kurimura, and T. Taira, “High average power diode end-pumped composite Nd:YAG laser passively Q-switched by Cr4+:YAG saturable absorber,” Jpn. J. Appl. Phys. 40(Part 1, No. 3A), 1253–1259 (2001).
[CrossRef]

Phuoc, T. X.

T. X. Phuoc, “Single-point versus multi-point laser ignition: experimental measurements of combustion times and pressures,” Combust. Flame 122(4), 508–510 (2000).
[CrossRef]

Poulain, D. E.

J. X. Ma, D. R. Alexander, and D. E. Poulain, “Laser spark ignition and combustion characteristics of methane-air mixtures,” Combust. Flame 112(4), 492–506 (1998).
[CrossRef]

Saikawa, J.

N. Pavel, J. Saikawa, S. Kurimura, and T. Taira, “High average power diode end-pumped composite Nd:YAG laser passively Q-switched by Cr4+:YAG saturable absorber,” Jpn. J. Appl. Phys. 40(Part 1, No. 3A), 1253–1259 (2001).
[CrossRef]

Sakai, H.

Sato, Y.

Taira, T.

N. Pavel, M. Tsunekane, and T. Taira, “Enhancing performances of a passively Q-switched Nd:YAG/Cr4+:YAG microlaser with a volume Bragg grating output coupler,” Opt. Lett. 35(10), 1617–1619 (2010).
[CrossRef] [PubMed]

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[CrossRef]

H. Sakai, H. Kan, and T. Taira, “>1 MW peak power single-mode high-brightness passively Q-switched Nd 3+:YAG microchip laser,” Opt. Express 16(24), 19891–19899 (2008).
[CrossRef] [PubMed]

Y. Sato and T. Taira, “The studies of thermal conductivity in GdVO4, YVO4, and Y3Al5O12 measured by quasi-one-dimensional flash method,” Opt. Express 14(22), 10528–10536 (2006).
[CrossRef] [PubMed]

N. Pavel, J. Saikawa, S. Kurimura, and T. Taira, “High average power diode end-pumped composite Nd:YAG laser passively Q-switched by Cr4+:YAG saturable absorber,” Jpn. J. Appl. Phys. 40(Part 1, No. 3A), 1253–1259 (2001).
[CrossRef]

Tartar, G.

H. Kofler, J. Tauer, G. Tartar, K. Iskra, J. Klausner, G. Herdin, and E. Wintner, “An innovative solid-state laser for engine ignition,” Laser Phys. Lett. 4(4), 322–327 (2007).
[CrossRef]

Tauer, J.

H. Kofler, J. Tauer, G. Tartar, K. Iskra, J. Klausner, G. Herdin, and E. Wintner, “An innovative solid-state laser for engine ignition,” Laser Phys. Lett. 4(4), 322–327 (2007).
[CrossRef]

Tesch, M.

M. Weinrotter, H. Kopecek, M. Tesch, E. Wintner, M. Lackner, and F. Winter, “Laser ignition of ultra-lean methane/hydrogen/air mixtures at high temperature and pressure,” Exp. Therm. Fluid Sci. 29(5), 569–577 (2005).
[CrossRef]

Tsunekane, M.

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[CrossRef]

N. Pavel, M. Tsunekane, and T. Taira, “Enhancing performances of a passively Q-switched Nd:YAG/Cr4+:YAG microlaser with a volume Bragg grating output coupler,” Opt. Lett. 35(10), 1617–1619 (2010).
[CrossRef] [PubMed]

Wang, Q. P.

S. T. Li, X. Y. Zhang, Q. P. Wang, P. Li, J. Chang, X. L. Zhang, and Z. H. Cong, “Modeling of Q-switched lasers with top-hat pump beam distribution,” Appl. Phys. B 88(2), 221–226 (2007).
[CrossRef]

Weinrotter, M.

M. Weinrotter, H. Kopecek, M. Tesch, E. Wintner, M. Lackner, and F. Winter, “Laser ignition of ultra-lean methane/hydrogen/air mixtures at high temperature and pressure,” Exp. Therm. Fluid Sci. 29(5), 569–577 (2005).
[CrossRef]

M. Weinrotter, H. Kopecek, and E. Wintner, “Laser ignition of engines,” Laser Phys. 15(7), 947–953 (2005).

Winkelhofer, E.

G. Kroupa, G. Franz, and E. Winkelhofer, “Novel miniaturized high-energy Nd:YAG laser for spark ignition in internal combustion engines,” Opt. Eng. 48(1), 014202 (2009).
[CrossRef]

Winter, F.

M. Weinrotter, H. Kopecek, M. Tesch, E. Wintner, M. Lackner, and F. Winter, “Laser ignition of ultra-lean methane/hydrogen/air mixtures at high temperature and pressure,” Exp. Therm. Fluid Sci. 29(5), 569–577 (2005).
[CrossRef]

Wintner, E.

H. Kofler, J. Tauer, G. Tartar, K. Iskra, J. Klausner, G. Herdin, and E. Wintner, “An innovative solid-state laser for engine ignition,” Laser Phys. Lett. 4(4), 322–327 (2007).
[CrossRef]

M. Weinrotter, H. Kopecek, and E. Wintner, “Laser ignition of engines,” Laser Phys. 15(7), 947–953 (2005).

M. Weinrotter, H. Kopecek, M. Tesch, E. Wintner, M. Lackner, and F. Winter, “Laser ignition of ultra-lean methane/hydrogen/air mixtures at high temperature and pressure,” Exp. Therm. Fluid Sci. 29(5), 569–577 (2005).
[CrossRef]

Yura, H. T.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990).
[CrossRef]

Zhang, X. L.

S. T. Li, X. Y. Zhang, Q. P. Wang, P. Li, J. Chang, X. L. Zhang, and Z. H. Cong, “Modeling of Q-switched lasers with top-hat pump beam distribution,” Appl. Phys. B 88(2), 221–226 (2007).
[CrossRef]

Zhang, X. Y.

S. T. Li, X. Y. Zhang, Q. P. Wang, P. Li, J. Chang, X. L. Zhang, and Z. H. Cong, “Modeling of Q-switched lasers with top-hat pump beam distribution,” Appl. Phys. B 88(2), 221–226 (2007).
[CrossRef]

Appl. Phys. B (1)

S. T. Li, X. Y. Zhang, Q. P. Wang, P. Li, J. Chang, X. L. Zhang, and Z. H. Cong, “Modeling of Q-switched lasers with top-hat pump beam distribution,” Appl. Phys. B 88(2), 221–226 (2007).
[CrossRef]

Appl. Phys. Lett. (1)

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56(19), 1831–1833 (1990).
[CrossRef]

Combust. Flame (3)

J. X. Ma, D. R. Alexander, and D. E. Poulain, “Laser spark ignition and combustion characteristics of methane-air mixtures,” Combust. Flame 112(4), 492–506 (1998).
[CrossRef]

T. X. Phuoc, “Single-point versus multi-point laser ignition: experimental measurements of combustion times and pressures,” Combust. Flame 122(4), 508–510 (2000).
[CrossRef]

M. H. Morsy, Y. S. Ko, S. H. Chung, and P. Cho, “Laser-induced two-point ignition of premixture with a single-shot laser,” Combust. Flame 124(4), 724–727 (2001).
[CrossRef]

Exp. Therm. Fluid Sci. (1)

M. Weinrotter, H. Kopecek, M. Tesch, E. Wintner, M. Lackner, and F. Winter, “Laser ignition of ultra-lean methane/hydrogen/air mixtures at high temperature and pressure,” Exp. Therm. Fluid Sci. 29(5), 569–577 (2005).
[CrossRef]

IEEE J. Quantum Electron. (2)

J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. 31(11), 1890–1901 (1995).
[CrossRef]

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[CrossRef]

Jpn. J. Appl. Phys. (1)

N. Pavel, J. Saikawa, S. Kurimura, and T. Taira, “High average power diode end-pumped composite Nd:YAG laser passively Q-switched by Cr4+:YAG saturable absorber,” Jpn. J. Appl. Phys. 40(Part 1, No. 3A), 1253–1259 (2001).
[CrossRef]

Laser Phys. (2)

M. Weinrotter, H. Kopecek, and E. Wintner, “Laser ignition of engines,” Laser Phys. 15(7), 947–953 (2005).

T. Dascalu and N. Pavel, “High-temperature operation of a diode-pumped passively Q-switched Nd:YAG/Cr4+:YAG laser,” Laser Phys. 19(11), 2090–2095 (2009).
[CrossRef]

Laser Phys. Lett. (1)

H. Kofler, J. Tauer, G. Tartar, K. Iskra, J. Klausner, G. Herdin, and E. Wintner, “An innovative solid-state laser for engine ignition,” Laser Phys. Lett. 4(4), 322–327 (2007).
[CrossRef]

Opt. Eng. (1)

G. Kroupa, G. Franz, and E. Winkelhofer, “Novel miniaturized high-energy Nd:YAG laser for spark ignition in internal combustion engines,” Opt. Eng. 48(1), 014202 (2009).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Other (2)

M. Tsunekane and T. Taira, “Temperature and polarization dependences of Cr:YAG transmission for passive Q-switching,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuD8.

M. Tsunekane, T. Inohara, A. Ando, K. Kanehara, and T. Taira, “High peak power, passively Q-switched Cr:YAG/Nd:YAG micro-laser for ignition of engines,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2008), paper MB4.

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic of a passively Q-switched, all-ceramics, composite, Nd:YAG/Cr4+:YAG monolithic laser with three-beam output. (b) A photo of a composite medium is shown.

Fig. 2
Fig. 2

Ratio ngf/ngi versus wp/wg and Q-switched laser pulse energy for various laser beam radii wg . Symbols are the experimental values measured with pump line “#A” (●) and pump line “#B” (■).

Fig. 3
Fig. 3

A photo of the composite, all-ceramics passively Q-switched Nd:YAG/Cr4+:YAG monolithic laser with three-beam output is shown.

Fig. 4
Fig. 4

Time delay of the Q-switched laser pulse, time jitter and standard deviation function of pump pulse energy (5-Hz pump repetition rate, 250-μs pump pulse duration).

Tables (1)

Tables Icon

Table 1 Characteristics of the Q-switched Laser Pulses Obtained with Composite Nd:YAG/Cr4+:YAG Ceramics

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

E p = h ν A g 2 γ g σ g ln R ln ( n g f n g i )
n g i = ln R + L ln T 0 2 2 σ g g [ 1 exp ( 2 a 2 ) ]
( 1 n g f n g i ) + [ 1 + ( 1 δ ) ln T 0 2 β ] ln ( n g f n g i ) + 1 α ( 1 δ ) ln T 0 2 β [ 1 ( n g f n g i ) α ] = 0

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