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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 16 — Aug. 3, 2009
  • pp: 14362–14366
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High power picosecond vortex laser based on a large-mode-area fiber amplifier

Yuichi Tanaka, Masahito Okida, Katsuhiko Miyamoto, and Takashige Omatsu  »View Author Affiliations


Optics Express, Vol. 17, Issue 16, pp. 14362-14366 (2009)
http://dx.doi.org/10.1364/OE.17.014362


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Abstract

We present the production of picosecond vortex pulses from a stressed large-mode-area fiber amplifier for the first time. 8.5 W picosecond output with a peak power of ~12.5 kW was obtained at a pump power of 29 W. 2009 Optical Society of America

© 2009 OSA

1. Introduction

Optical vortices [1

1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]

,2

2. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40(1), 73–87 (1993). [CrossRef]

] that have doughnut-shaped spatial profiles in the far-field and orbital angular momentum due to a phase singularity have been intensely investigated because they have the potential to be applied in various technologies, including optical tweezers [3

3. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef] [PubMed]

,4

4. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1–6), 169–175 (2002). [CrossRef]

] and super-resolution microscopes that use stimulated emission [5

5. S. Bretschneider, C. Eggeling, and S. W. Hell, “Breaking the diffraction barrier in fluorescence microscopy by optical shelving,” Phys. Rev. Lett. 98(21), 218103 (2007). [CrossRef] [PubMed]

] or up-conversion depletion [6

6. T. Watanabe, Y. Iketaki, T. Omatsu, K. Yamamoto, S. Ishiuchi, M. Sakai, and M. Fujii, “Two-Color Far-Field Super-Resolution Microscope using a Doughnut Beam,” Chem. Phys. Lett. 371(5–6), 634–639 (2003). [CrossRef]

]. In particular, high intense optical vortex pulses can open up various fields including high quality material processing [7

7. J. Hamazaki, R. Morita, Y. Kobayashi, S. Tanda, and T. Omatsu, “Laser Ablation using a Nanosecond Optical Vortex Pulse,” in Proceedings of IEEE Conference on CLEO/Europe-EQEC (IEEE, 2009), CC1.5 THU.

], controllable specificity of chiral matter [8

8. D. L. Andrews, L. C. Dávila Romero, and M. Babiker, “On optical vortex interactions with chiral matter,” Opt. Commun. 237(1–3), 133–139 (2004). [CrossRef]

], and nonlinear frequency conversion [9

9. K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54(5), R3742–R3745 (1996). [CrossRef] [PubMed]

]. And ultra-fast vortex pulses will also be potentially utilized to investigate the influence of optical orbital angular momentum in intense-field ionization processes [10

10. I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal, “Creation of optical vortices in femtosecond pulses,” Opt. Express 13(19), 7599–7608 (2005). [CrossRef] [PubMed]

].

Several methods for generating vortex beams have been successfully demonstrated including a holographic technique [11

11. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992). [CrossRef] [PubMed]

,12

12. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45(6), 1231–1237 (1998). [CrossRef]

] and a mode-conversion technique based on a pair of cylindrical lenses [13

13. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1–3), 123–132 (1993). [CrossRef]

]. In addition, a vortex beam can be produced by mode conversion in an optical fiber, such as in a stressed optical fiber [14

14. D. McGloin, N. B. Simpson, and M. J. Padgett, “Transfer of orbital angular momentum from a stressed fiberoptic waveguide to a light beam,” Appl. Opt. 37(3), 469–472 (1998). [CrossRef]

] or a hollow fiber [15

15. M.-L. Hu, C.-Y. Wang, Y.-J. Song, Y.-F. Li, L. Chai, E. E. Serebryannikov, and A. M. Zheltikov, “A hollow beam from a holey fiber,” Opt. Express 14(9), 4128–4134 (2006). [CrossRef] [PubMed]

]; this technique is robust and cost efficient since it does not require any additional phase elements. Passive fibers in the continuous-wave regime have mostly been used giving outputs of less than 1 W.

In this present paper, we describe, for the first time, the production of intense picosecond vortex pulses by selectively coupling a picosecond master laser to LP11 modes in a stressed large-mode-area fiber amplifier by off-axis injection into the fiber. The vortex output had a maximum power of 8.5 W and a corresponding peak power of 12.5 kW.

2. Experiments

2.1 Experimental setup

Figures 1(a) and 1(b) show a schematic diagram of the experimental setup and the basic concept behind generating a vortex mode. A home-made continuous-wave mode-locked Nd:Gd0.6Y0.4VO4 laser [16

16. J. Liu, Z. Wang, X. Meng, Z. Shao, B. Ozygus, A. Ding, and H. Weber, “Improvement of passive Q-switching performance reached with a new Nd-doped mixed vanadate crystal Nd:Gd0.64Y0.36VO4.,” Opt. Lett. 28(23), 2330–2332 (2003). [CrossRef] [PubMed]

,17

17. Y. F. Chen, M. L. Ku, L. Y. Tsai, and Y. C. Chen, “Diode-pumped passively Q-switched picosecond Nd:GDxY1-xVO4 self-stimulated raman laser,” Opt. Lett. 29(19), 2279–2281 (2004). [CrossRef] [PubMed]

] that has a semiconductor saturable absorber mirror [18

18. J.-L. He, Y.-X. Fan, J. Du, Y. G. Wang, S. Liu, H. T. Wang, L. H. Zhang, and Y. Hang, “4-ps passively mode-locked Nd:Gd0.5Y0.5VO4 laser with a semiconductor saturable-absorber mirror,” Opt. Lett. 29(23), 2803–2805 (2004). [CrossRef] [PubMed]

,19

19. Y. G. Wang, X. Y. Ma, Y. X. Fan, and H. T. Wang, “Passively mode-locking Nd:Gd0.5Y0.5VO4 laser with an In0.25Ga0.75As absorber grown at low temperature,” Appl. Opt. 44(20), 4384–4387 (2005). [CrossRef] [PubMed]

] with a modulation depth of ∆R = 2% was used as the master laser. The output was linearly polarized and it had a lasing frequency of 1064.4 nm, a pulse width of 4.5 ps, a pulse repetition frequency (PRF) of 150 MHz, and an output power of 300 mW. The collimated master laser output was delivered by relay optics to a polarization-maintained large-mode-area Yb3+-doped double-clad fiber (core diameter: 30 μm; core NA: 0.06; cladding diameter: 400 μm, cladding NA: 0.46) amplifier. It was off-axially injected into the fiber amplifier using an objective lens ( × 10, NA: 0.25) resulting in highly efficient in-phase coupling with the two orthogonal LP11 modes. The optical coupling efficiency of the master laser to the fiber amplifier was measured to be ~30%. The cutoff value of the fiber amplifier was estimated to be 5.3, thus preventing higher-order LPlm modes with indices of l>1 and m>1 from lasing [20

20. A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University Press, 1997), Chap. 3.

]. The output from the fiber amplifier had a TEM01-like profile, as shown in Fig. 2(a).

Stress was appropriately applied to the fiber amplifier at 45° to the vertical (TEM01 mode profile) so that the two orthogonal LP11 modes were 90° or -90° out of phase with each other at the end of the fiber. The output from the fiber amplifier was then converted to a circular annular beam, as shown in Fig. 2(b). The fiber amplifier was also pumped by a 400-μm-diameter fiber coupled to a 975 nm laser diode with an output power of 30 W. To prevent self-lasing, a 8° wedge was formed on both facets of the fiber amplifier.

Fig. 1. (a) Experimental setup and (b) the scheme for mode conversion in the fiber amplifier.
Fig. 2. Spatial forms of the output from the fiber amplifier.

2.2 Experimental results

Figure 3 shows the vortex output power as a function of the pump diode power. The output power reached 8.5 W at the maximum pump power of 29 W, corresponding to an optical efficiency from the diode to the output of 29%.

Fig. 3. TEM00 and vortex output powers as a function of pump power.

The pulse width (4.5 ps) of the amplified vortex pulses was almost identical to that of the master laser, while the lasing spectrum of the amplified vortex output was slightly shifted to lower wavelengths relative to that of the master laser (Figs. 4(a) and 4(b)). The maximum peak power of the vortex output was estimated to be 12.5 kW.

Fig. 4. (a) Autocorrelation traces of the seed laser (Seed) and the amplified vortex output (Amp) and (b) lasing spectra of the seed laser and the amplified vortex output.

To confirm that the annular output had a phase singularity, we analyzed interferometric fringes formed by the interference between the laser output beam and a spherical reference wavefront [21

21. V. Yu, Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992). [CrossRef]

,22

22. G. B. Jung, K. Kanaya, and T. Omatsu, “Highly efficient phase-conjugation of a 1 μm pico-second Laguerre-Gaussian beam,” Opt. Express 14(6), 2250–2255 (2006). [CrossRef] [PubMed]

]. The output had a single spiral indicating that the vortex output has a phase singularity. Selective control of the rotational direction of the spiral could be achieved by varying the stress applied to the fiber amplifier, as shown in Figs. 5(a) and 5(b).

Fig. 5. Interferometric fringes formed by the amplified vortex and a spherical reference wavefront. The vortex outputs (a) and (b) exhibit clockwise and counterclockwise single arms, respectively.

When the master laser was axially injected into the fiber amplifier, the coupling efficiency between the master laser and the fiber amplifier increased to 70%. The fiber amplifier was wound into a circle with a diameter of ~200 mm to suppress higher-order modes, such as the LP11 mode. The output from the fiber amplifier had a near TEM00 profile (Fig. 2(c) and it had a power of 11 W at the maximum pump power, corresponding to a peak power of ~16 kW. An energy extraction efficiency was measured to be 38%. The peak intensity of the output near the exit facet of the fiber amplifier was 2.3 GW/cm2. To avoid damage to the surfaces of the facets of the fiber amplifier, the facets were not anti-reflection coated. The frequency dispersion of the fiber as well as nonlinear effects such as self-phase modulation frequently induce pulse broadening of the amplified output, which will also limit power scaling of the system. The product of the frequency dispersion of the silica fiber (-30 ps/nm/km) [23

23. G. Ghosh, M. Endo, and T. Iwasaki, “Temperature-dependent Sellmeier coefficient and chromatic dispersions for some optical fiber glasses,” J. Lightwave Technol. 12(8), 1338–1342 (1994). [CrossRef]

], the frequency bandwidth of the pulse (0.8 nm), and the fiber length (3 m) is estimated to be 0.07 ps. The B integral [24

24. A. E. Siegman, Lasers (University Science Books, Mill Valley, 1986) pp. 386.

] (defined as the product of the pulse intensity, the nonlinear refractive index of the fiber, and the fiber length) is estimated to be 1.02 × 10-3. Thus, the effects of pulse broadening in this system are expected to be negligible

3. Conclusions

We have generated intense picosecond vortex pulses by selectively coupling a picosecond master laser to the LP11 mode in a stressed large-mode-area fiber amplifier by using off-axis fiber coupling. The maximum power of the vortex output was 8.5 W with a corresponding peak power of 12.5 kW at a pump power of 29 W. The optical-optical efficiency from the pump diode was 29%. The sign of the vortex can also be adjusted by varying the strength and the direction of the applied stress.

This method can directly produce intense picosecond vortex pulses from a simple fiber amplifier without using any phase elements. Further power scaling of the system is possible by improving the coupling efficiency of the mater laser to the amplifier and by using a more powerful pump diode. Frequency extension of this system is also possible by using second-order nonlinear materials. High intensity picosecond optical vortex pulses based on a simple fiber amplifier have the potential to be used in new applications including laser ablation.

Acknowledgements

The authors acknowledge support from a Scientific Research Grant-in-Aid (16032202, 18360031) and a support program for improving graduate school education from the Ministry of Education, Science and Culture of Japan and from the Japanese Society for the Promotion of Science. The authors would also like to thank Dr. Hidetsugu Yoshida in Institute of Laser Engineering, Osaka University, for his constructive suggestion.

References and links

1.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]

2.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40(1), 73–87 (1993). [CrossRef]

3.

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef] [PubMed]

4.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1–6), 169–175 (2002). [CrossRef]

5.

S. Bretschneider, C. Eggeling, and S. W. Hell, “Breaking the diffraction barrier in fluorescence microscopy by optical shelving,” Phys. Rev. Lett. 98(21), 218103 (2007). [CrossRef] [PubMed]

6.

T. Watanabe, Y. Iketaki, T. Omatsu, K. Yamamoto, S. Ishiuchi, M. Sakai, and M. Fujii, “Two-Color Far-Field Super-Resolution Microscope using a Doughnut Beam,” Chem. Phys. Lett. 371(5–6), 634–639 (2003). [CrossRef]

7.

J. Hamazaki, R. Morita, Y. Kobayashi, S. Tanda, and T. Omatsu, “Laser Ablation using a Nanosecond Optical Vortex Pulse,” in Proceedings of IEEE Conference on CLEO/Europe-EQEC (IEEE, 2009), CC1.5 THU.

8.

D. L. Andrews, L. C. Dávila Romero, and M. Babiker, “On optical vortex interactions with chiral matter,” Opt. Commun. 237(1–3), 133–139 (2004). [CrossRef]

9.

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54(5), R3742–R3745 (1996). [CrossRef] [PubMed]

10.

I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal, “Creation of optical vortices in femtosecond pulses,” Opt. Express 13(19), 7599–7608 (2005). [CrossRef] [PubMed]

11.

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992). [CrossRef] [PubMed]

12.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45(6), 1231–1237 (1998). [CrossRef]

13.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1–3), 123–132 (1993). [CrossRef]

14.

D. McGloin, N. B. Simpson, and M. J. Padgett, “Transfer of orbital angular momentum from a stressed fiberoptic waveguide to a light beam,” Appl. Opt. 37(3), 469–472 (1998). [CrossRef]

15.

M.-L. Hu, C.-Y. Wang, Y.-J. Song, Y.-F. Li, L. Chai, E. E. Serebryannikov, and A. M. Zheltikov, “A hollow beam from a holey fiber,” Opt. Express 14(9), 4128–4134 (2006). [CrossRef] [PubMed]

16.

J. Liu, Z. Wang, X. Meng, Z. Shao, B. Ozygus, A. Ding, and H. Weber, “Improvement of passive Q-switching performance reached with a new Nd-doped mixed vanadate crystal Nd:Gd0.64Y0.36VO4.,” Opt. Lett. 28(23), 2330–2332 (2003). [CrossRef] [PubMed]

17.

Y. F. Chen, M. L. Ku, L. Y. Tsai, and Y. C. Chen, “Diode-pumped passively Q-switched picosecond Nd:GDxY1-xVO4 self-stimulated raman laser,” Opt. Lett. 29(19), 2279–2281 (2004). [CrossRef] [PubMed]

18.

J.-L. He, Y.-X. Fan, J. Du, Y. G. Wang, S. Liu, H. T. Wang, L. H. Zhang, and Y. Hang, “4-ps passively mode-locked Nd:Gd0.5Y0.5VO4 laser with a semiconductor saturable-absorber mirror,” Opt. Lett. 29(23), 2803–2805 (2004). [CrossRef] [PubMed]

19.

Y. G. Wang, X. Y. Ma, Y. X. Fan, and H. T. Wang, “Passively mode-locking Nd:Gd0.5Y0.5VO4 laser with an In0.25Ga0.75As absorber grown at low temperature,” Appl. Opt. 44(20), 4384–4387 (2005). [CrossRef] [PubMed]

20.

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University Press, 1997), Chap. 3.

21.

V. Yu, Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992). [CrossRef]

22.

G. B. Jung, K. Kanaya, and T. Omatsu, “Highly efficient phase-conjugation of a 1 μm pico-second Laguerre-Gaussian beam,” Opt. Express 14(6), 2250–2255 (2006). [CrossRef] [PubMed]

23.

G. Ghosh, M. Endo, and T. Iwasaki, “Temperature-dependent Sellmeier coefficient and chromatic dispersions for some optical fiber glasses,” J. Lightwave Technol. 12(8), 1338–1342 (1994). [CrossRef]

24.

A. E. Siegman, Lasers (University Science Books, Mill Valley, 1986) pp. 386.

OCIS Codes
(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators
(080.4865) Geometric optics : Optical vortices

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: May 28, 2009
Revised Manuscript: July 24, 2009
Manuscript Accepted: July 24, 2009
Published: July 31, 2009

Citation
Yuichi Tanaka, Masahito Okida, Katsuhiko Miyamoto, and Takashige Omatsu, "High power picosecond vortex laser based on a large-mode-area fiber amplifier," Opt. Express 17, 14362-14366 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14362


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References

  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]
  2. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40(1), 73–87 (1993). [CrossRef]
  3. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef] [PubMed]
  4. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002). [CrossRef]
  5. S. Bretschneider, C. Eggeling, and S. W. Hell, “Breaking the diffraction barrier in fluorescence microscopy by optical shelving,” Phys. Rev. Lett. 98(21), 218103 (2007). [CrossRef] [PubMed]
  6. T. Watanabe, Y. Iketaki, T. Omatsu, K. Yamamoto, S. Ishiuchi, M. Sakai, and M. Fujii, “Two-Color Far-Field Super-Resolution Microscope using a Doughnut Beam,” Chem. Phys. Lett. 371(5-6), 634–639 (2003). [CrossRef]
  7. J. Hamazaki, R. Morita, Y. Kobayashi, S. Tanda, and T. Omatsu, “Laser Ablation using a Nanosecond Optical Vortex Pulse,” in Proceedings of IEEE Conference on CLEO/Europe-EQEC (IEEE, 2009), CC1.5 THU.
  8. D. L. Andrews, L. C. Dávila Romero, and M. Babiker, “On optical vortex interactions with chiral matter,” Opt. Commun. 237(1-3), 133–139 (2004). [CrossRef]
  9. K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54(5), R3742–R3745 (1996). [CrossRef] [PubMed]
  10. I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal, “Creation of optical vortices in femtosecond pulses,” Opt. Express 13(19), 7599–7608 (2005). [CrossRef] [PubMed]
  11. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992). [CrossRef] [PubMed]
  12. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45(6), 1231–1237 (1998). [CrossRef]
  13. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993). [CrossRef]
  14. D. McGloin, N. B. Simpson, and M. J. Padgett, “Transfer of orbital angular momentum from a stressed fiber-optic waveguide to a light beam,” Appl. Opt. 37(3), 469–472 (1998). [CrossRef]
  15. M.-L. Hu, C.-Y. Wang, Y.-J. Song, Y.-F. Li, L. Chai, E. E. Serebryannikov, and A. M. Zheltikov, “A hollow beam from a holey fiber,” Opt. Express 14(9), 4128–4134 (2006). [CrossRef] [PubMed]
  16. J. Liu, Z. Wang, X. Meng, Z. Shao, B. Ozygus, A. Ding, and H. Weber, “Improvement of passive Q-switching performance reached with a new Nd-doped mixed vanadate crystal Nd:Gd0.64Y0.36VO4.,” Opt. Lett. 28(23), 2330–2332 (2003). [CrossRef] [PubMed]
  17. Y. F. Chen, M. L. Ku, L. Y. Tsai, and Y. C. Chen, “Diode-pumped passively Q-switched picosecond Nd:GDxY1-xVO4 self-stimulated raman laser,” Opt. Lett. 29(19), 2279–2281 (2004). [CrossRef] [PubMed]
  18. J.-L. He, Y.-X. Fan, J. Du, Y. G. Wang, S. Liu, H. T. Wang, L. H. Zhang, and Y. Hang, “4-ps passively mode-locked Nd:Gd0.5Y0.5VO4 laser with a semiconductor saturable-absorber mirror,” Opt. Lett. 29(23), 2803–2805 (2004). [CrossRef] [PubMed]
  19. Y. G. Wang, X. Y. Ma, Y. X. Fan, and H. T. Wang, “Passively mode-locking Nd:Gd0.5Y0.5VO4 laser with an In0.25Ga0.75As absorber grown at low temperature,” Appl. Opt. 44(20), 4384–4387 (2005). [CrossRef] [PubMed]
  20. A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University Press,1997), Chap. 3.
  21. V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992). [CrossRef]
  22. G. B. Jung, K. Kanaya, and T. Omatsu, “Highly efficient phase-conjugation of a 1 μm pico-second Laguerre-Gaussian beam,” Opt. Express 14(6), 2250–2255 (2006). [CrossRef] [PubMed]
  23. G. Ghosh, M. Endo, and T. Iwasaki, “Temperature-dependent Sellmeier coefficient and chromatic dispersions for some optical fiber glasses,” J. Lightwave Technol. 12(8), 1338–1342 (1994). [CrossRef]
  24. A. E. Siegman, Lasers (University Science Books, Mill Valley, 1986) pp. 386.

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