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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 8 — Apr. 16, 2007
  • pp: 5172–5178
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Power scaling of semiconductor laser pumped Praseodymium-lasers

A. Richter, E. Heumann, G. Huber, V. Ostroumov, and W. Seelert  »View Author Affiliations


Optics Express, Vol. 15, Issue 8, pp. 5172-5178 (2007)
http://dx.doi.org/10.1364/OE.15.005172


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Abstract

We report on efficient lasing of Pr-doped fluoride materials with cw output powers up to 600 mW in the visible spectral range. Praseodymium doped LiYF4 and LiLuF4 crystals were pumped either by an intracavity frequency doubled optically pumped semiconductor laser with output powers up to 1.6 W and nearly diffraction limited beam quality or by a multimode GaN-laser diode with an output power of about 370 mW. Furthermore, intracavity frequency doubling of the red Pr-laser radiation to 320 nm reaching output powers of more than 360 mW with a conversion efficiency of 61% and an optical-to-optical efficiency of 22% are presented.

© 2007 Optical Society of America

1. Introduction

The trivalent Praseodymium ion (Pr3+) offers different fluorescence transitions in the visible spectral range covering the blue (around 485 nm), green (around 525 nm), orange (around 605 nm), red (around 640 nm), and deep red (695 nm, 720 nm) region (Fig. 1).

Fig. 1. Energy level scheme of Pr with transitions in the visible spectral region

Combined with high cross-sections (in the order of 10-19 cm2), Pr3+ is a favourable candidate for realizing lasers emitting visible radiation. Except the transition in the blue spectral region, the mentioned transitions are 4-level-laser transitions. While the first Pr3+-lasers realized were pumped by pulsed dye laser systems [1

1. L. Esterowitz, R. Allen, M. Kruer, F. Bartoli, L S. Goldberg, H. P. Jenssen, A. Linz, and V. O. Nicolai, “Blue light emission by a Pr:LiYF4-laser operated at room temperature,” J. Appl. Phys. 48, 650–652, (1977). [CrossRef]

] or flashlamps [2

2. A. A. Kaminskii, A. I. Lyashenko, N. P. Isaev, V. N. Karlov, V. L. Pavlovich, S. N. Bagaev, A. V. Butashin, and L. E. Li, “Quasi-cw Pr3+:LiYF4 laser with λ = 0.395 μm and an average output power of 2.3 W,” Quantum Electron. 28, 187–188 (1998). [CrossRef]

], different approaches for direct blue laser pumping of Pr3+-ions were performed for example by Ar-ion lasers [3

3. T. Sandrock, T. Danger, E. Heumann, G. Huber, and B. T. H. Chai, “Efficient continuous-wave laser emission of Pr3+-doped fluorides at room temperature,” Appl. Phys. B 58, 149–151, (1994). [CrossRef]

] or frequency doubled Nd:YAG ground state lasers [4

4. E. Heumann, C. Czeranowski, T. Kellner, and G. Huber, “An efficient all-solid-state Pr3+:LiYF4 laser in the visible spectral range” in Technical Digest CLEO ’99, paper CTuG1, p86, (Baltimore, 1999).

] in the past years. However, only a minor part of the pump radiation was absorbed in the Pr3+-doped material due to the lack of overlap between the pump wavelengths and the absorption peaks of Pr3+. During the last three years significant progress has been made in development of high power optically pumped semiconductor lasers (OPS) [5

5. A. C. Tropper and S. Hoogland, “Extended cavity surface emitting semiconductor lasers,” Prog. Quantum Electron. 30, 1–43 (2006). [CrossRef]

]. OPS-chips can be designed in principle for any fundamental wavelength in the region between 900 nm and 1000 nm. In 2004 a frequency doubled OPS of suitable wavelength (~480 nm) was available in our laboratory to demonstrate a compact and efficient cw Pr3+:LiYF4 (Pr:YLF)-laser for the first time [6

6. E. Osiac, E. Heumann, A. Richter, G. Huber, A. Diening, and W. Seelert, “Red Pr3+:YLiF4 laser excited by 480 nm optically pumped semiconductor laser” in Technical Digest CLEO ’04, paper CFE2, (San Francisco, 2004).

]. In the same year, direct diode pumping of Pr: YLF by a low power (25 mW) GaN laser diode could be realized, too [7

7. A. Richter, E. Heumann, E. Osiac, G. Huber, W. Seelert, and A. Diening, “Diode pumping of a continuous-wave Pr3+-doped LiYF4 laser,” Opt. Lett. 29, 2638–2670 (2004). [CrossRef] [PubMed]

]. Recently, cw UV radiation has been generated by intracavity frequency doubling of the OPS-pumped Pr:YLF laser [8

8. A. Richter, N. Pavel, E. Heumann, G. Huber, D. Parisi, A. Toncelli, M. Tonelli, A. Diening, and W. Seelert, “Continuous-wave unltraviolet generation at 320 nm by intracavity frequency doubling of red-emitting Praseodymium lasers,” Opt. Express 14, 3282–3287 (2006). [CrossRef] [PubMed]

].In comparison to other solid state lasers, Pr lasers need only a single nonlinear step for generation of several wavelengths in the UV.

Compact and efficient Pr-lasers in the visible or UV range may have great potential for several applications like display technology, fluorescence microscopy, biotechnology, and medicine.

In this paper we report on results of first power scaling experiments of visible and UV Pr:YLF and Pr3+:LiLuF4 (Pr:LLF)-lasers pumped by OPS as well as GaN diode lasers.

2. Experimental: Results and discussion

2.1 OPS-pumped Pr: YLF and Pr:LLF

The spectroscopic properties of Pr:YLF and Pr:LLF regarding absorption and emission can be found in Table 1. The emission cross-sections were calculated using the Füchtbauer-Ladenburg formula. Additionally, the effective lifetimes of the upper laser level (3P0) and the Pr3+-ion concentration in these host materials are given.

The samples used in the laser experiments were a 0.65 at-% doped Pr:YLF crystal (ion density: 9.1∙1019 cm-3) with a length of 3.5 mm and a 0.45 at-% doped Pr:LLF crystal (ion density: 6.48∙1019 cm-3) with a length of 3 mm. The Pr-concentration in the crystals was measured using atomic absorption spectroscopy (AAS). The segregation coefficients of Praseodymium ions are 0.22 and 0.15 for YLF and LLF, respectively. LLF is isostructural to YLF, however with slightly smaller lattice constants of 5.13 Å along a and 10.55 Å along c, while for YLF these parameters are 5.16 Å and 10.74 Å.

Table 1. Cross-sections and effective lifetimes of Pr:YLF and of Pr:LLF.

table-icon
View This Table

Both samples were cut along the a⃗ -axis. The facets have been polished in laser quality and had no AR coatings. The pump radiation was linearly polarized and the crystals were oriented for E⃗∥c⃗ (π-polarization) to make use of the large absorption coefficient and thus a high absorption efficiency in the crystal, as can be seen in Fig. 2 in case of Pr-doped LLF.

Fig. 2. Absorption cross-section of Pr:LLF for both polarizations.

The amount of absorbed power in a single pass was determined to be 91% and 73% for Pr:YLF and Pr:LLF, respectively.

A schematic sketch of the setup for the laser experiments is shown in Fig. 3. The OPS-laser delivered pump powers of up to 1.6 W at 479.5 nm with an excellent beam quality of about M 2 ≈ 1.1. The pump radiation was focused into the laser crystals using f= 100 mm lens resulting in a spot size of 75 μm. The laser crystals were mounted on a copper heat sink in a nearly concentric cavity consisting of two mirrors with a radius of curvature of 50 mm each. The resonator length which delivered the highest output power was about 82 mm in both cases. The resulting beam waist diameter was 62 μm, calculated by the ABCD matrix formalism [9

9. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966). [CrossRef] [PubMed]

].

Fig. 3. Schematic setup for the laser experiments.

The laser performance for several transitions in the visible spectral range with respect to the absorbed pump power for the Pr:YLF crystal is displayed in Fig. 4(a). At an absorbed pump power of around 1.4 W, up to 600 mW of coherent green and red radiation was emitted from the laser set-up with slope efficiencies (η) of up to 46%. To the best of our knowledge, this is the highest slope efficiency of a semiconductor laser pumped Pr: YLF laser so far. The realized orange Pr:YLF-laser at 607 nm enabled an output power of about 350 mW with η = 32%.

Fig. 4. Laser characteristics of the OPS-pumped (a) Pr:YLF- and (b) Pr:LLF-lasers operating at different transitions in the visible spectral region.

Figure 4(b) shows the laser performance of the OPS-pumped Pr:LLF for the same transitions in the visible spectral range. Using the LLF crystal, more than 550 mW of laser radiation was emitted with a slope efficiency of 52% at the transition 3P03F2 at 640 nm, while the laser power on the transitions 3P13H5 at 522 nm and 3P03F4 at 720 nm were about 500 mW with slope efficiencies of 53% and 56%, respectively. The orange laser transition (3P03H6) of Pr:LLF at 607 nm shows a lower output of about 370 mW with a slope efficiency of 35% with respect to the absorbed pump power, as already observed in YLF. In contrast to former laser results of Pr:YLF lasers under argon ion laser pumping [3

3. T. Sandrock, T. Danger, E. Heumann, G. Huber, and B. T. H. Chai, “Efficient continuous-wave laser emission of Pr3+-doped fluorides at room temperature,” Appl. Phys. B 58, 149–151, (1994). [CrossRef]

] no drop of the output is observed during our laser experiments.

The optimal transmission of the output couplers available of the Pr:YLF (Pr:LLF) laser operating at 523 nm, 607 nm, 640 nm and 720 nm were 3.6% (2.1%), 7.4% (3.7%), 2.0% (3.8%) and 2.7% (5.4%), respectively.

Internal losses of the lasers were estimated by a Findlay-Clay analysis [10

10. D. Findlay and R. A. Clay, “The measurement of internal losses in 4-level lasers,” Phys. Lett. 20, 277–278 (1966). [CrossRef]

] and by the method of Caird et al. [11

11. J. A. Caird, S. A. Payne, P. R. Staber, A. J. Ramponi, L. L Chase, and W. F. Krupke, “Quantum electronics properties of the Na3Ga2Li3F12:Cr3+ laser,” IEEE J. Quantum Electron. QE-24, 1077–1099 (1988). [CrossRef]

]. The internal losses per roundtrip were estimated to be about 0.6% for both Pr:YLF and Pr:LLF. However, the loss estimation for the orange laser transition resulted in values of more than 10% with both methods. We suppose an efficient reabsorption process originating from the Stark level at 213 cm-1 above the lowest state of the 3H4 level and terminating in one of the Stark levels of the 1D2 level. Such absorption could limit the efficiency of this laser transition. Further investigations regarding this process are in progress.

2.2 Intracavity second harmonic generation of the fundamental wavelength of λω = 640 nm

The OPS pumped Pr:YLF laser has been used to generate coherent cw UV radiation by intracavity second harmonic generation (SHG). The set-up used for intracavity SHG was a singly folded cavity consisting of an input coupler (M1), a folding mirror (M2), and an end mirror (M3) with 50 mm, 100 mm and 50 mm radii of curvature, respectively (Fig. 5). The M1-M2 arm’s length was 135 mm, while the length M2-M3 was about 133 mm, depending on the length of the nonlinear crystals (NLCs) resulting in a total resonator length of about 268 mm. The laser crystals were placed in the beam waist of the M1-M2 arm, while the LBO NLCs were placed in the beam waist of the M2-M3 arm of the cavity. The beam diameter inside the laser crystals and the LBO crystals was estimated to be between 66 μm and 69 μm and 74 μm and 69 μm, respectively. The UV radiation generated was coupled out using both mirror M2 and M3 to prevent UV-exposition of the Pr-doped laser crystals.

Fig. 5. Schematic setup for the intracavity frequency doubling experiments

Figures 6(a) and 6(b) display the results of intracavity SHG with LBO crystals of 3 mm, 5 mm, and 8 mm length for Pr:YLF and Pr:LLF, respectively.

Fig. 6. Power characteristics of intracavity SHG of the (a) Pr:YLF and (b) Pr:LLF laser operated at 640 nm. Continuous lines represent simulations (see text)

Using an 8 mm long LBO crystal, up to 364 mW and 261 mW of cw UV radiation could be extracted from the Pr:YLF and Pr:LLF laser resonator, respectively. Regarding the maximum extracted power at the fundamental wave at 640 nm, a conversion efficiency of 61% (Pr:YLF) and 54% (Pr:LLF) was obtained. The overall optical-to-optical efficiency for the UV output was 22% and 16% for Pr:YLF and Pr:LLF, respectively. Neglecting thermal effects in the laser crystal under pumping conditions, the SHG output power was simulated using the model developed by Agnesi et al. [12

12. A. Agnesi, A. Guandalini, and G Reali, “Efficient 671-nm pump source by intracavity doubling of a diode-pumped Nd:YVO4 laser,” J. Opt. Soc. Am. B 19, 1078–1082 (2002) [CrossRef]

]. In Figs. 6(a) and 6(b), Li denotes the total internal round-trip losses of the laser cavity. Further information regarding the simulation process is given in reference [8

8. A. Richter, N. Pavel, E. Heumann, G. Huber, D. Parisi, A. Toncelli, M. Tonelli, A. Diening, and W. Seelert, “Continuous-wave unltraviolet generation at 320 nm by intracavity frequency doubling of red-emitting Praseodymium lasers,” Opt. Express 14, 3282–3287 (2006). [CrossRef] [PubMed]

].

Amplitude fluctuations of the UV-output are in the order of 40% in the range of the measured output powers (like in the so-called “green problem” [13

13. T. Baer, “Large-amplitude fluctuations due to longitudinal mode coupling in diode-pumped intracavity-doubled Nd:YAG lasers,” J. Opt. Soc. Am. B 3, 1175–1180 (1986). [CrossRef]

]). A detailed analysis of the noise of the UV-output will be published elsewhere.

2.4 GaN-diode laser pumping of Pr: YLF and Pr:LLF operated at λω = 640 nm

As mentioned in the introduction, GaN-diode lasers at 444 nm can also be used for pumping Pr in YLF and LLF on the transition 3H43P2 (see Fig. 1). Due to the lower absorption cross-sections compared to the transition 3H43P0, a 5 mm long Pr:LLF single crystal with the same dopant concentration was placed in the setup displayed in Fig. 3. In case of Pr:YLF a 5.66 mm long sample with an ion concentration of 0.98×1020 cm-3 was used in the experiments although the fluorescent lifetime of the upper laser level (3P0) shows strong quenching at this Pr-concentration level. The crystal was anti-reflection (AR) coated for the pump light at 444 nm (Transmission T > 98%) and highly reflecting (HR) at the laser wavelength λω (T < 0.1%). As an output coupler, a 50 mm curved mirror with T = 2.0% was placed in a distance of about 47 mm from the laser crystal, forming a hemispherical cavity setup. The pump laser diode from NICHIA corp. delivered up to 370 mW. The beam quality was determined to be M 2 x = 1 and M2 y = 3. As expected for diode lasers in general, the wavelength of these devices increases when using higher currents through the active layer due to thermal effects. This results in different amounts of absorbed pump power depending on the overlap of the pump light wavelength and the absorption peak of Pr:YLF and Pr:LLF at 444 nm. The amount of absorbed power was determined by monitoring the transmitted pump power behind the laser crystal and varied from 74% to 87% in case of Pr:YLF or 46% to 72% in case of Pr:LLF. Figures 7(a) and 7(b) show the power characteristics of the GaN-diode laser pumped Pr:YLF and Pr:LLF laser with respect to the absorbed pump power. At 293 mW (263 mW) of absorbed pump power, 61 mW (76 mW) of cw laser radiation was measured at a slope efficiency of 28% (36%) originating from the YLF (LLF) laser crystal. The performance of the diode-pumped Pr-lasers was limited because of the reduced beam quality of this pump source in comparison to the OPS laser.

Fig. 7. Power characteristics of the GaN-pumped cw (a) Pr:YLF laser and (b) Pr:LLF laser operating at 640 nm.

3. Summary

In conclusion, laser output powers of up to 600 mW in the green (523 nm) and red (640 nm, 720 nm) as well as more than 300 mW in the orange (607 nm) spectral range were demonstrated under efficient OPS-pumping of Pr:YLF and Pr:LLF single crystals. Slope efficiencies of up to 46% and 56% with respect to the absorbed pump power could be achieved for YLF and LLF, respectively. Intracavity frequency doubling of the laser radiation at 640 nm resulted in cw UV output powers up to 364 mW (Pr:YLF) and 261 mW (Pr:LLF) at 320 nm using an LBO nonlinear crystal with a length of 8 mm. The conversion efficiency was estimated to be 61% (54%) with respect to the fundamental output power and the optical-to-optical efficiency was 22% (16%) for Pr:YLF (Pr:LLF). Additionally, first experiments using a GaN diode laser with more than 370 mW of pump power resulted in output powers of more than 75 mW at a slope efficiency of 36% with respect to the absorbed pump power. Experiments regarding UV generation of 320 nm radiation of Pr-doped materials under GaN diode laser pumping are under way. The results obtained are promising for several applications using visible and UV cw laser sources such as display technology, medicine, and fluorescence microscopy.

References

1.

L. Esterowitz, R. Allen, M. Kruer, F. Bartoli, L S. Goldberg, H. P. Jenssen, A. Linz, and V. O. Nicolai, “Blue light emission by a Pr:LiYF4-laser operated at room temperature,” J. Appl. Phys. 48, 650–652, (1977). [CrossRef]

2.

A. A. Kaminskii, A. I. Lyashenko, N. P. Isaev, V. N. Karlov, V. L. Pavlovich, S. N. Bagaev, A. V. Butashin, and L. E. Li, “Quasi-cw Pr3+:LiYF4 laser with λ = 0.395 μm and an average output power of 2.3 W,” Quantum Electron. 28, 187–188 (1998). [CrossRef]

3.

T. Sandrock, T. Danger, E. Heumann, G. Huber, and B. T. H. Chai, “Efficient continuous-wave laser emission of Pr3+-doped fluorides at room temperature,” Appl. Phys. B 58, 149–151, (1994). [CrossRef]

4.

E. Heumann, C. Czeranowski, T. Kellner, and G. Huber, “An efficient all-solid-state Pr3+:LiYF4 laser in the visible spectral range” in Technical Digest CLEO ’99, paper CTuG1, p86, (Baltimore, 1999).

5.

A. C. Tropper and S. Hoogland, “Extended cavity surface emitting semiconductor lasers,” Prog. Quantum Electron. 30, 1–43 (2006). [CrossRef]

6.

E. Osiac, E. Heumann, A. Richter, G. Huber, A. Diening, and W. Seelert, “Red Pr3+:YLiF4 laser excited by 480 nm optically pumped semiconductor laser” in Technical Digest CLEO ’04, paper CFE2, (San Francisco, 2004).

7.

A. Richter, E. Heumann, E. Osiac, G. Huber, W. Seelert, and A. Diening, “Diode pumping of a continuous-wave Pr3+-doped LiYF4 laser,” Opt. Lett. 29, 2638–2670 (2004). [CrossRef] [PubMed]

8.

A. Richter, N. Pavel, E. Heumann, G. Huber, D. Parisi, A. Toncelli, M. Tonelli, A. Diening, and W. Seelert, “Continuous-wave unltraviolet generation at 320 nm by intracavity frequency doubling of red-emitting Praseodymium lasers,” Opt. Express 14, 3282–3287 (2006). [CrossRef] [PubMed]

9.

H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966). [CrossRef] [PubMed]

10.

D. Findlay and R. A. Clay, “The measurement of internal losses in 4-level lasers,” Phys. Lett. 20, 277–278 (1966). [CrossRef]

11.

J. A. Caird, S. A. Payne, P. R. Staber, A. J. Ramponi, L. L Chase, and W. F. Krupke, “Quantum electronics properties of the Na3Ga2Li3F12:Cr3+ laser,” IEEE J. Quantum Electron. QE-24, 1077–1099 (1988). [CrossRef]

12.

A. Agnesi, A. Guandalini, and G Reali, “Efficient 671-nm pump source by intracavity doubling of a diode-pumped Nd:YVO4 laser,” J. Opt. Soc. Am. B 19, 1078–1082 (2002) [CrossRef]

13.

T. Baer, “Large-amplitude fluctuations due to longitudinal mode coupling in diode-pumped intracavity-doubled Nd:YAG lasers,” J. Opt. Soc. Am. B 3, 1175–1180 (1986). [CrossRef]

OCIS Codes
(140.3480) Lasers and laser optics : Lasers, diode-pumped
(140.3580) Lasers and laser optics : Lasers, solid-state
(140.3610) Lasers and laser optics : Lasers, ultraviolet
(140.7240) Lasers and laser optics : UV, EUV, and X-ray lasers

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: November 27, 2006
Revised Manuscript: January 11, 2007
Manuscript Accepted: January 12, 2007
Published: April 13, 2007

Citation
A. Richter, E. Heumann, G. Huber, V. Ostroumov, and W. Seelert, "Power scaling of semiconductor laser pumped Praseodymium-lasers," Opt. Express 15, 5172-5178 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-5172


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References

  1. L. Esterowitz, R. Allen, M. Kruer, F. Bartoli, L. S. Goldberg, H. P. Jenssen, A. Linz, and V. O. Nicolai, "Blue light emission by a Pr:LiYF4-laser operated at room temperature," J. Appl. Phys. 48, 650-652 (1977). [CrossRef]
  2. A. A. Kaminskii, A. I. Lyashenko, N. P. Isaev, V. N. Karlov, V. L. Pavlovich, S. N. Bagaev, A. V. Butashin, and L. E. Li, "Quasi-cw Pr3+:LiYF4 laser with λ = 0.395 µm and an average output power of 2.3 W," Quantum Electron. 28, 187-188 (1998). [CrossRef]
  3. T. Sandrock, T. Danger, E. Heumann, G. Huber, and B. T. H. Chai, "Efficient continuous-wave laser emission of Pr3+-doped fluorides at room temperature," Appl. Phys. B 58, 149-151 (1994). [CrossRef]
  4. E. Heumann, C. Czeranowski, T. Kellner, and G. Huber, "An efficient all-solid-state Pr3+:LiYF4 laser in the visible spectral range" in Technical Digest CLEO '99, paper CTuG1, p. 86, (Baltimore, 1999).
  5. A. C. Tropper, and S. Hoogland, "Extended cavity surface emitting semiconductor lasers," Prog. Quantum Electron. 30, 1 - 43 (2006). [CrossRef]
  6. E. Osiac, E. Heumann, A. Richter, G. Huber, A. Diening, and W. Seelert, "Red Pr3+:YLiF4 laser excited by 480 nm optically pumped semiconductor laser" in Technical Digest CLEO’04, paper CFE2, (San Francisco, 2004).
  7. A. Richter, E. Heumann, E. Osiac, G. Huber, W. Seelert, and A. Diening, "Diode pumping of a continuous-wave Pr3+-doped LiYF4 laser," Opt. Lett. 29, 2638 - 2670 (2004). [CrossRef] [PubMed]
  8. A. Richter, N. Pavel, E. Heumann, G. Huber, D. Parisi, A. Toncelli, M. Tonelli, A. Diening, and W. Seelert, "Continuous-wave unltraviolet generation at 320 nm by intracavity frequency doubling of red-emitting Praseodymium lasers," Opt. Express 14, 3282-3287 (2006). [CrossRef] [PubMed]
  9. H. Kogelnik and T. Li, "Laser beams and resonators," Appl. Opt. 5, 1550-1567 (1966). [CrossRef] [PubMed]
  10. D. Findlay and R. A. Clay, "The measurement of internal losses in 4-level lasers," Phys. Lett. 20, 277 - 278 (1966). [CrossRef]
  11. J. A. Caird, S. A. Payne, P. R. Staber, A. J. Ramponi, L. L. Chase, and W. F. Krupke, "Quantum electronics properties of the Na3Ga2Li3F12:Cr3+ laser," IEEE J. Quantum Electron. QE-24, 1077-1099 (1988). [CrossRef]
  12. A. Agnesi, A. Guandalini, and G. Reali, "Efficient 671-nm pump source by intracavity doubling of a diode-pumped Nd:YVO4 laser," J. Opt. Soc. Am. B 19, 1078-1082 (2002) [CrossRef]
  13. T. Baer, "Large-amplitude fluctuations due to longitudinal mode coupling in diode-pumped intracavity-doubled Nd:YAG lasers," J. Opt. Soc. Am. B 3, 1175-1180 (1986). [CrossRef]

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