Open Access
Add to BibSonomy
Abstract
A doughnut mode microchip laser was demonstrated by introducing a monolithic ceramic Nd:YAG/Cr4+:YAG chip in an unstable resonator to deliver laser pulses with an energy of 13.2 mJ and a pulse width of 476 ps, corresponding to a record peak power of 27.7 MW. The laser beam quality was characterized by M2∼6 at 10 Hz repetition rate. No significant degradation or change of beam pattern, pulse width, and M2 was confirmed during energy scaling in the case of the unstable cavity, promising for further brightness improving. In comparison with a flat-flat cavity, pulse broadening and M2 increase was observed up to ∼1.2 ns and ∼10, respectively, during energy scaling up to 18 mJ due to the beam pattern degradation. The doughnut beam was observed to have an Airy disk at the focal point, which was suitable for laser induced breakdown in air. The measured breakdown threshold of doughnut beam was comparable to a near-Gaussian beam (M2=1.3).
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Microchip lasers (MCL) with giant pulse emission have the attractive characteristics of a compactness, a sub-nanosecond (ns) pulse width, and operation at a single axial frequency [1–3]. Additional advantages such as simplicity, stability, and low-cost can be achieved by introducing a monolithic cavity-coated structure combining the gain medium and saturable absorber based on ceramic due to the availability for high quality mass production [4–6]. The combination of the compactness and a high peak power of over MW due to the short sub-ns pulse can open new research and application fields [5–8]. Basically, MCLs tailed with an optical fiber and optics to end-pump using diode lasers have the almost same structural advantage of flexibility like fiber-delivered lasers, which allows various applications such as laser manufacturing for complex geometry processing [9], laser medicine [10], laser peening [11], remote laser induced breakdown spectroscopy (LIBS) [12], and so on. However, conventional fiber-delivered laser has a trade-off between the peak power P0 and the beam quality M2. A single mode fiber allows a good output beam quality but limits a launchable peak power due to the damage threshold of fiber. On the other hand, a large core multi-mode fiber degrades the output beam quality due to the modal dispersion in the fiber. Therefore, there is a challenge of developing high peak-power resistant optical fibers such as hollow core and photonic crystal fibers. On the other hand, the laser pulse of MCL doesn’t need to be fiber-delivered because a compact MCL itself can be positioned or controlled in the front line for LIB as keeping both P0 and M2 [6].
For laser ignition, a high brightness (=P0/(λM2)2) of laser is required for a focal intensity to reach the breakdown threshold of gas (Ith, air ${\sim 1}$ TW/cm2 [8]) and the minimum ignition energy (MIE) [13]. MCL having a comparable size to conventional electric spark plugs has realized laser ignition for internal combustion engine vehicles [6,14]. It is notable that sub-ns pulse width can be an optimally efficient time-scale to create the plasma breakdown of gas [8] due to the dominant avalanche (impact) ionization.
However, there is a challenge of scaling the brightness in passively Q-switched giant-pulse MCLs. For a higher power, pulse width can be reduced by shortening cavity length and reducing initial transmittance of saturable absorber in principle, but the methods have a limit of effectiveness for pulse width reduction. Most common effective method is to enlarge the mode size by pumping in a larger size for energy scaling, however, the large pump can allow additional oscillations of higher modes, which results in the degradation of beam pattern leading pulse broadening and poor beam quality M2 [15]. This limit of beam degradation is inevitable even using cryogenic cooling [15]. However, an unstable cavity can be an alternative for brightness improving because it permits a wider and controllable mode size as keeping the uniform beam pattern of doughnut shape [16,17]. Therefore, we combined MCL with unstable cavity for both compactness and brightness.
2. Experimental methods
2.1 Unstable cavity
For the positive branch confocal cavity as shown in Fig. 1(a), the radii of curvature of the mirrors are given by ${R_0} ={-} 2{L_c}/({m - 1} )$ for the output cavity mirror and ${R_b} = 2m{L_c}/({m - 1} )$ for the back cavity mirror, where Lc is the empty cavity length and m is the magnification which is the ratio of the mode size b to output mirror size a ($m = b/a$). The size of output cavity mirror determines the hole size of doughnut mode. The radii of curvatures of the cavity mirrors were designed by choosing ${L_c} = 10\,{mm}$ for a short sub-ns pulse, selecting $m = \sqrt 2 $ for a proper round-trip loss ($= 1 - {m^{ - 2}})$ of 50%, and choosing $a = 2\, \mbox{mm}$ for a realizable mode size ($= a \times m$) with a 600 W level pump diode. One can approximately calculate the performance of unstable cavity such as a pulse energy and required pump energy by assuming it as a stable cavity with a common output cavity mirror with a reflectance of ${m^{ - 2}}$ and with a laser mode size of b using well developed Q-switched laser models [3,18–21].
Fig. 1. (a) Schematic of the positive branch confocal cavity, where Mb (o) and Rb (o) is the back (output) cavity mirror and its radius curvature, respectively, Lc is the cavity length, a is the size of Mo or the hole size of doughnut mode, and b is the size of doughnut mode. (b) Schematic of the experimental cavity, where l is the length of the monolithic ceramic, LMo and L is the substrate lens for Mo and the lens for collimating the divergent beam, respectively.
We first tested a flat back cavity mirror because of the simplicity and considering end-pump induced thermal lens. The flat back cavity mirror also can make the comparison to a flat-flat cavity easier by only exchanging the curved output cavity mirror for a flat one as keeping other conditions. Figure 1(b) shows the schematic diagram of experimental setup. A half inch plano-convex lens (LMo) with a radius of curvature of ∼50 mm was used as a substrate to fabricate the output cavity mirror. The center part of the convex surface was high-reflection (HR) coated in a spot diameter of about 2 mm and the other parts are anti-reflection (AR) coated for laser respectively (Optoquest Co., Ltd., Japan). A smooth variation of reflectance at the boundary was considered in the design for HR coating, which can reduce the diffraction problem at a sharp edge. The HR spot diameter, i.e., the output mirror size a of 2 mm was selected by considering the fabrication availability and a realizable mode size. A monolithic Nd:YAG/Cr4+:YAG ceramic (Konoshima Chemical, Japan) with a dimension of 6×6×7(l) mm3 was used for laser gain medium and passive Q-switching. The Nd3+ doping rate and the initial transmittance of Cr4+:YAG was 1.1% and 30%, respectively. The end surfaces are AR coated for both pump and laser. A fiber (ϕ =0.8 mm) coupled quasi continuous-wave (QCW) diode laser (Dilas diodenlaser GmbH, Germany; λ0 =808 nm, P0∼700 W, τmax =500 µs) was used to end-pump using a telescope composed of two lenses. The divergent output laser beam was collimated using a single convex lens (L) with focal lengths of 100∼150 mm in case of the lens positions.
2.2 Comparison of laser performance with flat-flat cavity
The performances of MCL with unstable resonator using the Nd:YAG/Cr4+:YAG chip were compared with those obtained in a flat-flat cavity, which was built by replacing the curved output cavity mirror of unstable cavity with a flat mirror. All the other conditions (pumping, cooling, and so on) were identical for both resonators. A flat mirror with a reflectance of 50% was used, which is the same round-trip loss for an ideal confocal cavity with m of $\sqrt 2 $. To compare brightness of lasers, we measured pulse energy, pulse width, and beam quality M2. The pulse energy was measured using a pyroelectric energy sensor (Ophir Optronics Solutions Ltd., Israel). The pulse width was measured using a photodetector with a rise time of ∼30 ps and a 13 GHz oscilloscope (Keysight Technologies, USA). A small portion energy using a sampler was focused into the photodetector. A beam quality M2 tool (Cinogy technologies, Germany) and an analysis software (RayCi) according to ISO 11146 were used to determine 2nd moment or 86.5% power-content beam size and M2 value.
We tested the capability of doughnut beam practically by comparing the threshold of laser induced breakdown (LIB) in air with a near-Gaussian beam (M2=1.3). Pulse width tunable laser [8] and amplifier [22,23] were used for the near-Gaussian beam. To fairly compare the threshold, we used the same conditions between doughnut and near-Gaussian beam. The pulse width and collimated beam size on focusing lens of near-Gaussian beam was matched to those of doughnut beam by adjusting the cavity length and using a telescope, respectively. The beam size on focusing lens was measured by using a camera positioned at the same distance from laser. We used linear polarization of both lasers and focused them in laboratory air using seven focusing lenses with focal lengths of 8–62 mm. Air-breakdown was confirmed by observing the visible white spark at the right angle using a collecting lens and a Si-photodetector. Low pass filters were used to block the scattered laser light. The breakdown threshold was defined as the minimum energy for 100% breakdown success of the laser pulse train. Detailed experiment on the laser induced air-breakdown was reported elsewhere [8].
3. Results and discussion
Figure 2(a) shows the measured pulse shape and beam pattern of laser. The laser beam has an almost symmetric doughnut pattern with a center Poisson spot due to the diffraction from the boundary of output cavity mirror. The measured pulse width was 476 ps at full width half maximum (FWHM). Figure 2(b) shows a typical stability of the pulse energy in a short term of 5 minutes. The mean energy was 13.2 mJ with a root mean square (RMS) stability as small as 1.0% at a repetition rate of 10 Hz. To the best of our knowledge, the corresponding peak power of 27.7 MW is the highest peak power level using microchip laser. By the way, a stable linear polarization oscillation was observed at the 10 Hz repetition rate. The inset of Fig. 2(b) shows a typical polarization status, where the measured (symbol) and calculated (line) transmittance was plotted as a function of angle of polarizer. The polarization ratio $P = \frac{{{I_{max}} - {I_{min}}}}{{{I_{max}} + {I_{min}}}} = 0.997$, where Imax (min) is the maximum (minimum) transmitted intensity. The polarization of YAG ceramic lasers was turned out to be the same as that of [111]-cut YAG single crystals [24]. A linear polarization could be selected by pump power induced birefringence and depolarization of the isotropic gain materials due to stress and heat.
Fig. 2. (a) Measured pulse shape with a FWHM of 476 ps. Inset: the measured doughnut beam pattern. (b) Measured pulse energy during a short term of 5 minutes, showing a mean energy of 13.2 mJ with a RMS stability of 1%. Inset: the measured and calculated transmittance as a function of angle of polarizer.
To estimate beam quality of doughnut beam, we first estimated second-moment based M2 values [25–27]. Because the second-moment beam width defines the M2 factor mathematically, one can compare the M2 values between any laser beams in principle. However, because the second moment beam width does not provide a constant value for the encircled power between different beam patterns such as Gaussian, flat top beam, and so on, the International Organization for Standardization allows for an alternative based on 86.5% power-content beam width for practical usefulness [28]. Therefore, we also estimated 86.5% power-content based M2 values $M_{pc}^2$ for reference. Figure 3 shows the measured beam patterns and radii around the focal point. It is important to note that the far-field pattern at the focal point is an Airy disk and Airy pattern. The M2 ($M_{pc}^2)$ value of 6.8 (6.5) and 5.3 (5.2) was estimated for the major and minor axis, respectively. The average $M_{ave}^2 = \sqrt {M_{maj}^2M_{min}^2} $ was 6 (5.8). From now on we will use the average value for M2.
Fig. 3. Measured beam radius around the focal point using a lens with a focal length of 300 mm. Inset: typical beam patterns around the focal point.
The performance between the microchip lasers with unstable and flat-flat cavity was compared at different laser energy levels. Three different pump sizes were tuned by adjusting the distance from the telescope, composed of two lenses for pumping, to the gain medium in several mm range. A short distance of ∼5 mm between the mount of telescope and the gain medium, and a high energy of pump made a direct pump beam imaging difficult. Therefore, we used a 0.1-mm-thick Nd:YAG ceramics to be excited by the pump at the same distance, and then a fluorescence from the ceramics was imaged to estimate the pump beam size and distribution [29]. The three pump beam diameters are roughly estimated to be ∼2.6 mm in 10% difference.
The only output cavity mirror was exchanged between the curved and flat one for every pump condition. Figure 4(a) shows the measured beam patterns for both cavities. The pulse energies of 8.8, 11.5, and 13.2 mJ for unstable cavity are paired with 10.2, 14.3, and 18 mJ for flat-flat cavity by every pump condition, respectively. The energy difference is attributed to the divergent smaller mode of unstable cavity due to the flat back cavity mirror. The beam patterns from the both cavities showed different aspects as expected. The almost same doughnut patterns were kept at the different energies for unstable cavity but random poor beam patterns were observed for flat-flat cavity due to higher mode oscillations. Moreover, the oscillation of the higher modes in the flat-flat cavity resulted in the pulse broadening [Fig. 4(b)]. The pulses of flat-flat cavity had about 2.5 times larger pulse widths (FWHM) of ∼1.25 ns than those of ∼0.5 ns of unstable cavity, as shown in Fig. 4(c). The pulse widths of ∼0.5 ns of unstable cavity can be comparable to those of flat-flat cavity at low pulse energies. Pulse widths of 0.4–0.53 ns were reported in MCLs with the pulse energies of 0.7–3.2 mJ using Cr:YAG saturable absorbers with the initial transmittances of 25–50% and cavity lengths of 6.3–8 mm [3,8,15]. Guo et. al. also reported the pulse broadening with pulse energy increasing for flat-flat cavity in Ref. 15. Consequently, the uniform doughnut beam pattern permits a high peak power of MCL due to a short pulse width. The maximum peak power of 27.7 MW is the highest peak power level using MCL because of the pulse width broadening for stable cavities [15]. In addition, the beam quality M2 was also affected by the random poor beam pattern. Figure 4(e) shows the measured M2 ($\mbox{and}\; M_{pc}^2)$ value as a function of pulse energy for both cavities. For unstable cavity, the M2 ($M_{pc}^2)$ values were moved between 5.7 (5.2) to 6.7 (6.1) but didn’t increase with the energy increase. However, the M2 values for flat-flat cavity were higher than those for unstable cavity at the pulse energies over 10 mJ and increased with the energy increase. The uniform doughnut beam pattern allowed to keep a stable beam quality M2 for energy scaling. Consequently, the uniform beam pattern permits a high brightness $[{B = {P_0}/{{({\lambda {M^2}} )}^2}} ]$ scaling. Figure 4(f) shows the measured brightness of laser for both cavities. We believe that about 1 order higher brightness of unstable cavity can be achieved at the energy around 20 mJ because of the increasing both pulse width and M2 of flat-flat cavity.
Fig. 4. Measured beam patterns (a) and pulse shapes (b) of unstable and flat-flat cavity at different pulse energies, where the energies of 8.8, 11.5, and 13.2 mJ for unstable cavity are paired with 10.2, 14.3, and 18 mJ for flat-flat cavity by every pump condition, respectively. Measured pulse width (FWHM) (c), peak power (d), M2 (e), and brightness (f) of unstable and flat-flat cavity as a function of pulse energy.
In addition, we compared the practical capability of doughnut beam to near-Gaussian beam for LIB by measuring its threshold in air. Because a collimated beam size on focusing lens, wavelength, M2, and pulse width determine a focal fluence and intensity, we set the same beam size on focusing lens, wavelength, and pulse width, then compared the capability of LIB between the doughnut (M2=6) and near-Gaussian beam (M2=1.3). If the other conditions except M2 values are same, the ratio between focal beam waist of doughnut (wd) and near-Gaussian (wnG) beam $\frac{{{w_d}}}{{{w_{nG}}}} = \frac{{M_d^2}}{{M_{nG}^2}} = \frac{6}{{1.3}} \approx 4.6$.
Figure 5(a) shows the measured beam and pulse shape having a diameter of 7.5 mm and a width of 570 ps, respectively. Figure 5(b) shows the measured breakdown threshold energy of doughnut (open circles) and near-Gaussian beam (closed circles) and the ratio of the threshold energy of doughnut beam Eth, d to that of near-Gaussian beam Eth, nG (blue squares) as a function of focal length. The ratio $\frac{{{E_{th,\; d}}}}{{{E_{th, nG}}}}$ had a mean value of 1.08 with a standard deviation of 0.22 for the focal length range. The comparable capability for air-breakdown between the doughnut and the near-Gaussian beam could be strange because of the ∼4.6 times different focal beam sizes, i.e., ∼21 times different fluences or intensities. This phenomenon can be explained by the Airy disk of doughnut beam at the focal point. Figure 6(a) shows a typical beam pattern at the focal point which was measured for M2 estimation. The Y-axis cross sectional intensity distribution at the center was plotted in the blue symbols of Fig. 6(b). The 2nd moment beam diameter 2wy of the Airy disk and Airy pattern was 0.29 mm. The black line indicates a Gaussian distribution with the same beam diameter of 0.29 mm. On the other hand, the size of only Airy disk was estimated to be ∼0.2 times the whole size of 0.29 mm. The effective beam waist of doughnut beam or the radius of Airy disk ${w_{Airy}} \approx 0.2{w_{d({{M^2} = 6} )}} \approx 0.92{w_{nG({{M^2} = 1.3} )}} \approx 1.2{w_{G({{M^2} = 1} )}}$. The effective ∼0.2 times smaller beam size makes the doughnut beam (M2=6) to be comparable to the near-Gaussian beam (M2=1.3) for air-breakdown. The far-field pattern of doughnut beam can be approximately calculated by assuming the plane wave focused by an annular aperture lens [30]. The intensity distribution at the focal plane is given by
Fig. 5. (a) Compared pulse shapes between doughnut (red line) and near-Gaussian beam (black line) with a width (FWHM) of 570 ps, respectively. Inset: measured beam patterns on focusing lens for doughnut (up) and near-Gaussian (down) with a diameter of 7.5 mm, respectively. (b) Measured breakdown threshold energy of doughnut (open circles) and near-Gaussian beam (closed circles) and the ratio of the threshold energy of doughnut beam Eth, d to that of near-Gaussian beam Eth, nG (blue squares) as a function of focal length.
Fig. 6. (a) Measured beam pattern of doughnut beam at a focal point showing an Airy disk and Airy pattern. (b) Cross sectional intensity distribution of the Airy disk and Airy pattern (blue symbol) compared with a Gaussian distribution with the same beam size (black line) and 0.2 times smaller beam size (red line). The Airy disk and Airy pattern were fitted by Eq. (1) (blue solid line).
4. Conclusion
In conclusion, we demonstrated 27.7 MW peak power Nd:YAG/Cr4+:YAG microchip laser with unstable cavity. The uniform doughnut beam permitted no significant degradation of pulse width and beam quality M2 for energy scaling, which could be attractive more for further brightness improving. We also confirmed that the doughnut beam can be comparable to a near-Gaussian beam (M2=1.3) for laser induced breakdown in gas (air) because of the Airy disk at focal point. For further power and high-repetition scaling, a distributed face cooled (DFC) tiny integrated laser (TILA) [33] can be introduced with unstable cavity using a larger pump size and a higher magnification m.
Funding
Japan Science and Technology Agency (JPMJMI17A1); JST-Mirai ProgramCouncil for Science, Technology and Innovation; Cabinet Office, Government of Japan(ImPACT Program of Council for Science, Technology and Innovation); New Energy and Industrial Technology Development Organization.
Disclosures
The authors declare no conflict of interest.
References
1. T. Taira, A. Mukai, Y. Nozawa, and T. Kobayashi, “Single-mode oscillation of laser-diode-pumped Nd:YVO4 microchip lasers,” Opt. Lett. 16(24), 1955–1957 (1991). [CrossRef]
2. J. J. Zayhowski and C. Dill, “Diode-pumped passively Q-switched picosecond microchip lasers,” Opt. Lett. 19(18), 1427–1429 (1994). [CrossRef]
3. H. Sakai, H. Kan, and T. Taira, “>1 MW peak power single-mode high-brightness passively Q-switched Nd3+:YAG microchip laser,” Opt. Express 16(24), 19891–19899 (2008). [CrossRef]
4. T. Taira, “RE3+-ion-doped YAG ceramic lasers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 798–809 (2007). [CrossRef]
5. N. Pavel, M. Tsunekane, and T. Taira, “Composite, all-ceramics, high-peak power Nd:YAG/Cr4+:YAG monolithic micro-laser with multiple-beam output for engine ignition,” Opt. Express 19(10), 9378–9384 (2011). [CrossRef]
6. T. Taira, S. Morishima, K. Kanehara, N. Taguchi, A. Sugiura, and M. Tsunekane, “World first laser ignited gasoline engine vehicle,” The 1st Laser Ignition Conference (LIC'13), Yokohama, Japan, Apr. 23–25, 2013.
7. S. Hayashi, K. Nawata, T. Taira, J. Shikata, K. Kawase, and H. Minamide, “Ultrabright continuously tunable terahertz-wave generation at room temperature,” Sci. Rep. 4(1), 5045 (2015). [CrossRef]
8. H. H. Lim and T. Taira, “Sub-nanosecond laser induced air-breakdown with giant-pulse duration tuned Nd: YAG ceramic micro-laser by cavity-length control,” Opt. Express 25(6), 6320–6334 (2017). [CrossRef]
9. K. A. Ghany, H. A. Rafea, and M. Newishy, “Using a Nd-YAG laser and six axes robot to cut zinc-coated steel,” Int. J. Adv. Manuf. Technol. 28(11-12), 1111–1117 (2006). [CrossRef]
10. Z. Wang and N. Chocat, “Fiber-optic technologies in laser-based therapeutics: Threads for a cure,” Curr. Pharm. Biotechnol. 11(4), 384–397 (2010). [CrossRef]
11. N. Mukai, Y. Sano, M. Yoda, I. Chida, T. Uehara, and T. Yamamoto, “Preventive maintenance against stress corrosion cracking in nuclear power reactors by laser peening. Remote delivery of high-power pulsed laser,” Reza Kenkyu 33(7), 444–451 (2005). [CrossRef]
12. V. N. Lednev, A. E. Dormidonov, P. A. Sdvizhenskii, M. Y. Grishin, A. N. Fedorov, A. D. Savvin, E. S. Safronova, and S. M. Pershin, “Compact diode-pumped Nd:YAG laser for remote analysis of low-alloy steels by laser-induced breakdown spectroscopy,” J. Anal. At. Spectrom. 33(2), 294–303 (2018). [CrossRef]
13. J. L. Beduneau, B. Kim, L. Zimmer, and Y. Ikeda, “Measurements of minimum ignition energy in premixed laminar methane/air flow by using laser induced spark,” Combust. Flame 132(4), 653–665 (2003). [CrossRef]
14. N. Pavel, T. Dascalu, G. Salamu, M. Dinca, N. Boicea, and A. Birtas, “Ignition of an automobile engine by high-peak power Nd:YAG/Cr4+:YAG laser-spark devices,” Opt. Express 23(26), 33028–33037 (2015). [CrossRef]
15. X. Guo, S. Tokita, and J. Kawanaka, “High beam quality and high peak power Yb:YAG/Cr:YAG microchip laser,” Opt. Express 27(1), 45–54 (2019). [CrossRef]
16. A. E. Siegman, “Unstable optical resonators,” Appl. Opt. 13(2), 353–367 (1974). [CrossRef]
17. R. L. Herbst, H. Komine, and R. L. Byer, “A 200 mJ unstable resonator Nd:YAG oscillator,” Opt. Commun. 21(1), 5–7 (1977). [CrossRef]
18. J. Dong, “Numerical modeling of CW-pumped repetitively passively Q-switched Yb:YAG lasers with Cr:YAG as saturable absorber,” Opt. Commun. 226(1-6), 337–344 (2003). [CrossRef]
19. 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]
20. W. Yu-Ye, X. De-Gang, X. Jing-Ping, W. Zhuo, W. Peng, and Y. Jian-Quan, “Numerical modelling of QCW-pumped passively Q-switched Nd:YAG lasers with Cr4+:YAG as saturable absorber,” Chin. Phys. Lett. 25(8), 2880–2883 (2008). [CrossRef]
21. A. Kausas and T. Taira, “Giant-pulse Nd:YVO4 microchip laser with MW-level peak power by emission cross-sectional control,” Opt. Express 24(4), 3137–3149 (2016). [CrossRef]
22. V. Yahia and T. Taira, “High brightness energetic pulses delivered by compact microchip-MOPA system,” Opt. Express 26(7), 8609–8618 (2018). [CrossRef]
23. T. Kawasaki, V. Yahia, and T. Taira, “100 Hz operation in 10 PW/sr·cm2 class Nd:YAG micro-MOPA,” Opt. Express 27(14), 19555–19561 (2019). [CrossRef]
24. I. Shoji, Y. Sato, S. Kurimura, V. Lupei, T. Taira, A. Ikesue, and K. Yoshida, “Thermal-birefringence-induced depolarization in Nd:YAG ceramics,” Opt. Lett. 27(4), 234–236 (2002). [CrossRef]
25. G. E. Slobodzian, “Beam profiling: second-moment method characterizes higher-order beam modes,” Laser Focus World 51(7), 35–38 (2015).
26. ISO 11146-1:2005 “Test methods for laser beam widths, divergence angles and beam propagation ratios Part 1: Stigmatic and simple astigmatic beams,”
27. ISO 11146–2:2005, “Lasers and laser-related equipment - Test methods for laser beam widths, divergence angles and beam propagation ratios - Part 2: General astigmatic beams,”
28. ISO 11146:1999, “Lasers and laser-related equipment - Test methods for laser beam parameters – Beam widths, divergence angle and beam propagation factor,”
29. M. Tsunekane and T. Taira, “New method of in-situ measurement of 2D laser beam profile,” in 35th Annual Meeting of the Laser Society of Japan, Tokyo, Japan, 11-12 Jan. 2015, paper 12pIX03 (in Japanese).
30. B. Lü, X. Ji, and S. Luo, “The beam quality of annular lasers and related problems,” J. Mod. Opt. 48(7), 1171–1178 (2001). [CrossRef]
31. A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS Lasers: Applications and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (OSA, 1998), pp. 184–199.
32. A. Garay, “Continuous wave deuterium fluoride laser beam diagnostic system,” Proc. SPIE 0888, 17–22 (1988). [CrossRef]
33. L. Zheng, A. Kausas, and T. Taira, “Drastic thermal effects reduction through distributed face cooling in a high power giant-pulse tiny laser,” Opt. Mater. Express 7(9), 3214–3221 (2017). [CrossRef]
