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Optics Express

  • Editor: J. H. Eberly
  • Vol. 9, Iss. 11 — Nov. 19, 2001
  • pp: 579–585

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Second - harmonic generation of ultra-high intensity femtosecond pulses with a KDP crystal

Makoto Aoyama, Tetsuo Harimoto, J. Ma, Yutaka Akahane, and Koichi Yamakawa  »View Author Affiliations


Optics Express, Vol. 9, Issue 11, pp. 579-585 (2001)
http://dx.doi.org/10.1364/OE.9.000579


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Abstract

We investigated second harmonic generation with ultrahigh intensity femtosecond laser pulses from a terawatt Ti: sapphire laser system. Energy conversion efficiency of about 80 % for a type I potassium dideuterium phosphate crystal was obtained with 130 fs laser pulses at an intensity as high as 192 GW/cm2.

© Optical Society of America

1. Introduction

Since the appearance of chirped pulse amplification (CPA) laser systems, there has been rapid progress in generating high peak power, ultrashort laser pulses. Such lasers can realize pulses with a peak intensity of >1020 W/cm2 [1

K. Yamakawa, M. Aoyama, S. Matsuoka, T. Kase, Y. Akahane, and H. Takuma, “100-TW, sub 20-fs Ti : Sapphire laser system operating at a 10 Hz repetition rate,” Opt. Lett. 23, 1468–1470 (1998). [CrossRef]

] and are therefore useful for a variety of high-field applications such as the generation of ultrafast x-ray radiation [2

T. Guo, Ch. Spielmann, B. C. Walker, and C. P. J. Barty, “Generation of hard x rays by ultrafast terawatt lasers,” Rev. Sci. Instrum. 72, 41–47 (2001) [CrossRef]

] and high harmonic generation [3

P. Gibbon, “High-order harmonic generation in plasmas,” IEEE J. Quantum Electron 33, 1915–1924 (1997). [CrossRef]

] from solid targets. In general the ultrashort laser pulses from a CPA laser system have amplified spontaneous emission (ASE) background associated with the main laser pulse and pre- and post-pulses. In a laser - matter interaction experiment, such pre-pulses and ASE would create a low density plasma in advance of the main laser pulse and thus significantly alter the physics of the laser - matter interaction [4

P. B. Corkum, F. Brunel, N. K. Sherman, and T. Srinivasan-Rao, “Thermal response of metals to ultrashort-pulse laser excitation,” Phys. Rev. Lett. 61, 2886–2889 (1988). [CrossRef] [PubMed]

]. It is significantly important for the study of the high field laser - solid target interaction experiments. Frequency conversion of femtosecond pulses can improve intensity contrast of laser pulses. Simple experimental configuration of second-harmonic generation (SHG) and improvement of intensity contrast are so attractive for some applications of intense femtosecond laser pulses. However it is well known that there are problems [5

I. V. Tomov, R. Fedosejevs, and A. A. Offenberger, “Up-conversion of subpicosecond light pulses,” IEEE J. Quantum Electron. 18, 2048–2056(1982). [CrossRef]

–6

R. C. Eckard and J. Reintjes, “Phase matching limitations of high efficiency second harmonic generation,” IEEE J. Quantum Electron. 20, 1178–1187 (1984). [CrossRef]

] in frequency conversion such as SHG with sub-picosecond and femtosecond laser pulses. The effectiveness of the SHG with ultrashort laser pulses can be greatly reduced by group-velocity mismatch between the fundamental and generated second harmonic (SH) pulses. At high intensities the local refractive index of the crystals changes with the laser intensity, primarily because of the third-order nonlinear susceptibility, resulting in phase modulation and self-focusing effects. These effects will also influence the conversion efficiency and pulse duration of the SH pulse. In order to reduce effects of the dispersion and the higher-order nonlinear processes, a crystal with a thin thickness is generally necessary.

A number of groups have investigated SHG with intense femtosecond laser pulses with intensities exceeding ~100 GW/cm2. Chien et al. [7

C. Y. Chien, G. Korn, J. S. Coe, J. Squier, G. Mourou, and R.S. Craxton, “Highly efficient second-harmonic generation of ultraintense Nd : glass laser pulses,” Opt. Lett. 20, 353–355(1995). [CrossRef] [PubMed]

] reported an energy conversion efficiency of 80 % for 4 mm thick type I potassium dideuterium phosphate (KDP) crystals using 500 fs, 1053 nm laser pulses at an intensity as high as 400 GW/cm2. Krylov et al. [8

V. Krylov, A. Rebane, A. G. Kalintsev, H. Schwoerer, and U. P. Wild, “Second-harmonic generation of amplified femtosecond Ti : sapphire laser pulses,” Opt. Lett. 20, 198–200 (1995). [CrossRef] [PubMed]

] investigated SHG of type I KDP crystals with crystal lengths of 3, 5, 10, and 40 mm, respectively, using 150 fs, 780 nm laser pulses at an intensity as high as 150 GW/cm2. They concluded that energy conversion efficiency could not exceed 50 % due to the modulation of the phase of the fundamental pulses. Tamaki et al. [9

Y. Tamaki, M. Obara, and K. Midorikawa, “Second harmonic generation from intense, 100-fs Ti Sapphire laser pulses in Potassium dihydrogen phosphate, Cesium lithium borate and β-barium metaborate,” Jpn. J. Appl. Phys. 37, 4801–4805 (1998). [CrossRef]

] described SHG of intense femtosecond pulses using three types of crystals. They obtained an energy conversion efficiency of approximately 20 % for a 1 mm thick type I KDP crystal and for a 1 mm thick type I cesium lithium borate (CLBO) crystal using 100 fs, 800 nm laser pulses at intensities as high as 200 GW/cm2 and 30 GW/cm2, respectively. Neely et al. [10

D. Neely, C. N. Danson, R. Allott, F. Amiranoff, J. L. Collier, A. E. Dangor, C. B. Edwards, P. Flintoff, P. Hatton, M. Harman, M. H. R. Hutchinson, Z. Najmudin, D. A. Pepler, I. N. Ross, M. Salvati, and T. Winstone, “Frequency doubling of multi-terawatt picosecond pulses,” Laser and Particle Beams. 17, 281–286 (1999). [CrossRef]

] investigated frequency conversion of intense picosecond laser pulses with 4 mm and 2 mm long KDP crystals. They observed the break-up of the far field of the SH pulse due to the thickness of the crystal and the high intensity. More recently, Queneuille et al. [11

J. Queneuille, F. Druon, A. Maksimchuk, G. Cheriaux, G. Mourou, and K. Nemoto, “Second-harmonic generation and wave-front correction of a terawatt laser system,” Opt. Lett. 25, 508–510 (2000). [CrossRef]

] discussed SHG and wavefront correction for high intensity laser pulses with a deformable mirror. They obtained an energy conversion efficiency of approximately 55 % for a 4 mm thick type I KDP crystal and using 400 fs, 1053 nm laser pulses at an intensity as high as 200 GW/cm2. They estimated an obtained intensity contrast of ~109 with SHG.

As we mentioned above, frequency conversion with picosecond or sub-picosecond laser pulses accomplished more than 50% efficiencies. However conversion efficiencies with ~100 fs laser pulses are lower than those with sub-picosecond laser pulses. As Krylov et al. [8

V. Krylov, A. Rebane, A. G. Kalintsev, H. Schwoerer, and U. P. Wild, “Second-harmonic generation of amplified femtosecond Ti : sapphire laser pulses,” Opt. Lett. 20, 198–200 (1995). [CrossRef] [PubMed]

] pointed out, energy conversion efficiency should be lower than 50 % using thick crystals with 150 fs laser pulses. They showed that phase modulation of the fundamental pulses decreases the frequency conversion efficiency for intensities less than 150 GW/cm2. Ditmire et al. [12

T. Ditmire, A. M. Rubenchik, D. Eimerl, and M. D. Perry, “Effects of cubic nonlinearity on frequency doubling of high-power laser pulses,” J. Opt. Soc. Am. B 13, 649–655 (1996). [CrossRef]

] numerically discussed the precompensation for a phase shift due to the third-order susceptibility χ(3) by affording an initial phase mismatch upon the doubling crystal on SHG with ultrahigh intensity laser pulses. Their calculated results on some experimental conditions showed the possibility for obtaining an energy conversion efficiency of about 70–80 % with ultrahigh intensity laser pulses of ~100 fs and incorporating this precompensation into their calculations.

In order to obtain higher conversion efficiency, we investigated SHG with fundamental laser pulses of ~100 fs in more than the hundreds of GW/cm2 intensity regime with a technique of the precompensation described by Ditmire et al.. Although frequency conversion of sub-picosecond laser pulses with thick crystals enables high efficiency conversion, the thick crystals degrade the beam quality of SH pulses. In our experiment a thin crystal was used for SHG in order to reduce effects of dispersion, beam-breakup and third-order nonlinear processes. This method with intense femtosecond pulses also makes the experimental arrangement of frequency conversion simple. Therefore, it is easy to apply to any kind of CPA laser system.

In this paper, we describe frequency conversion with a thin crystal and ~100 fs high intensity laser pulses from a terawatt Ti: sapphire laser system. We have achieved an energy conversion efficiency of about 80 % for a type I KDP crystal with 130 fs laser pulses for intensities as high as 192 GW/cm2 and have found no optical fatal damage of the nonlinear crystal.

2. Experimental setup

We used a part of our Ti: sapphire CPA laser system [1

K. Yamakawa, M. Aoyama, S. Matsuoka, T. Kase, Y. Akahane, and H. Takuma, “100-TW, sub 20-fs Ti : Sapphire laser system operating at a 10 Hz repetition rate,” Opt. Lett. 23, 1468–1470 (1998). [CrossRef]

] in this study. The system is seeded with pulses from an all-solid-state mirror-dispersion-controlled Ti: sapphire oscillator capable of producing 10-fs pulses. These pulses are temporally stretched by a factor of 100,000 in an all-reflective, cylindrical mirror-based pulse expander. Then, stretched pulses are amplified with two amplifier stages. The Ti: sapphire amplifier chain consists of a regenerative amplifier, and a four-pass amplifier. These amplifiers are pumped by a fraction of the 532 nm radiation at 10 Hz from Q-switched Nd: YAG lasers. We also controlled the central wavelength and spectral bandwidth of amplified pulses using angle-tuned thin etalons in the regenerative amplifier. Then, the Ti: sapphire amplifier chain laser pulses are compressed with a parallel grating pair which uses 1200 lines/mm. The intensity of the fundamental pulses can be adjusted by using the half-wave plates and thin film polarizers before the compressor. The compressed pulses have a spectral width of 14.7 nm at full-width at half maximum (FWHM) and a center wavelength of 800 nm. The output pulse duration used in this experiment was about 130 fs.

Fig.1: Experimental setup for SHG with femtosecond laser pulses. SM: silver mirror, DM: dichroic mirror, PM1: power meter for SH laser pulses, L1: lens for SH laser pulses, C1: CCD camera for SH laser pulses, PM2: power meter for fundamental laser pulses, L2: lens for fundamental laser pulses, C2: CCD camera for fundamental laser pulses.

The optical layout of SHG is depicted in Fig. 1. The fundamental pulses were focused with a 1000 mm focal-length concave mirror on a SHG crystal. SHG was achieved in the Type I 1-mm-thick KDP crystal which has a sol-gel antireflective coating and a transmission spatial quality of λ/7.5 @ 800 nm. The crystal was positioned 56 cm away from the focusing mirror. The laser pulses focused on the KDP crystal had an elliptical shape with a major diameter of 5.76 mm and a minor diameter of 2.73 mm. The generated SH and fundamental pulses were separated with a dichroic mirror. Spectral properties of the dichroic mirror were calibrated with a spectrophotometer (U-400, Hitachi Ltd.). Images of the SH and fundamental pulses were taken with charge-coupled device (CCD) cameras that were placed after the nonlinear crystal. We observed the images of a point on the nonlinear crystal.

3. Experimental and numerical results

Experimental and numerical energy conversion efficiencies of the fundamental laser intensities from 0 to 500 GW/cm2 are shown in Fig. 2 and Fig. 4. The energy of the fundamental and generated SH pulses was measured with a power meter (PE - 10 and PE-25, Ophir Optronics Inc.). All experimental points were obtained by averaging over 300 laser pulses. The conversion efficiency was corrected for loss from the mirrors and the KDP crystal. The experimental results and the calculated results are shown in Fig. 2. This calculation model is based on the set of coupled equations describing the propagation of the fundamental and SH pulses, which includes a frequency chirp, group velocity mismatch, group velocity dispersion of the fundamental and the SH pulses. These equations also take other physical effects into account : a phase shift due to the third-order susceptibility χ (3), second-order nonlinear susceptibility [13

T. Harimoto, M. Aoyama, K. Yamakawa, and M. Yonemura, “Suppression of cubic nonlinearity in second-harmonic generation of ultrahigh intensity laser pulses by initial frequency chirp,” Jpn. J. Appl. Phys. 41(2002). [CrossRef]

]. This numerical result in Fig.2 was obtained for fundamental pulses which had no frequency chirp. The experimental energy conversion efficiency was about 80 % at the intensity of 192 GW/cm2. The calculated and experimental results of the energy conversion efficiency behave similarly. Although pulse duration of the generated SH pulses were not measured, the calculated result shows the SH pulse duration was shorter than the pulse duration of the fundamental pulses at 192 GW/cm2. At this intensity the broadening of the SH spectrum was not observed with self-phase modulation as shown in Fig.3. Therefore, we conjecture that such phase modulation doesn’t almost affect the SH pulse duration at the optimum intensity of the energy conversion efficiency. We had however observed spectral broadening of SH pulses due to self-phase modulation in the more than ~230 GW/cm2.

Fig. 2. Experimental and numerical energy conversion efficiencies of SHG as a function of fundamental intensity. Circles indicate experimental results. The solid curve shows the numerical result where the calculation includes a phase shift due to the third-order susceptibility χ (3). The dot curve shows the numerical result of the SH pulse duration.
Fig.3: Measured spectra of second harmonic pulses. The solid curve and the dot curve show the SH spectrum at an intensities of the fundamental pulses as high as 219 GW/cm2 and 455 GW/cm2, respectively.

On the other hand, a calculated energy conversion efficiency with fundamental pulses which have an appropriate frequency chirp is shown in Fig.4. When the fundamental pulses have an appropriate frequency chirp, this method can also be used to shift the laser intensity for optimum conversion efficiency towards the more than ~500 GW/cm2 intensity regime and at the same time suppress the effect of the third-order susceptibility. However undesirable frequency chirp of the fundamental pulses reduces the energy conversion efficiency. We found this case in Fig.4. The experimental fundamental pulses in Fig.4 had a positive group delay dispersion of 6.06×102 fs2/nm. The experimental conversion efficiency decreases to 60 %. We also found that the laser intensity for optimum conversion efficiency depends on the initial frequency chirp of the fundamental laser pulse. Beyond the optimum laser intensity the conversion efficiency decreases. The phase shift due to χ (3) causes this phenomenon. This means that the optimum laser intensity can be controlled with the initial phase shift such as by the frequency chirp of the fundamental laser pulse or the initial phase mismatch[13

T. Harimoto, M. Aoyama, K. Yamakawa, and M. Yonemura, “Suppression of cubic nonlinearity in second-harmonic generation of ultrahigh intensity laser pulses by initial frequency chirp,” Jpn. J. Appl. Phys. 41(2002). [CrossRef]

].

The measured far-field profile of the SH pulse at a fundamental intensity of about 190 GW/cm2 indicates that the spatial quality of the SH laser pulse and the fundamental pulses are 1.3 and 1.1 times diffraction limited, respectively. The results show that the thin crystal suppresses degradation of the beam quality at even high fundamental intensity. It should be noted that we have found no optical fatal damage of the KDP crystal from CCD images of the laser pulses.

Fig.4: Experimental and numerical energy conversion efficiencies of SHG as a function of the fundamental intensity. Circles indicate experimental results. The solid and dot curves show the numerical results where the calculation includes a phase shift due to the third-order susceptibility χ (3) and a frequency chirp of 6.18×104 fs2/nm and 3.24×104 fs2/nm, respectively.

This SHG does not require a complicated experimental setup, so it leads to higher efficiency. This method is suitable for SHG of a large CPA system, which generates >100 fs and > TW laser pulses. Under this condition, SHG can also be achieved without temporal broadening of the SH pulses. As mentioned in Section 1, Queneuille et al. [11

J. Queneuille, F. Druon, A. Maksimchuk, G. Cheriaux, G. Mourou, and K. Nemoto, “Second-harmonic generation and wave-front correction of a terawatt laser system,” Opt. Lett. 25, 508–510 (2000). [CrossRef]

] discussed SHG and wavefront correction. They obtained an energy conversion efficiency of approximately 55 % for a 4 mm thick type I KDP crystal using 400 fs, 1053 nm laser pulses at an intensity as high as 200 GW/cm2. They obtained pulses which had focused intensity of 4×1019 W/cm2 with estimated intensity contrast of ~109 with SHG and a deformable mirror. The thickness of the KDP crystal in our experiment is thinner than that on above experiment. sThe thin crystal can suppress degradation of the beam quality compared with thick crystals. Therefore, frequency conversion of intense femtosecond pulses can improve the intensity contrast of laser pulses to some degree even without a deformable mirror. It is important for the study of high intensity laser - matter interaction. The simple experimental configuration of this SHG and improvement of intensity contrast are very attractive for some applications of intense femtosecond laser pulses. Furthermore, as mentioned in Ref. 13, an initial frequency chirp of the fundamental pulse in SHG with ultrahigh intensity laser pulses can be used to suppress third-order nonlinearity in the temporal and spatial profile. This method can also be used to shift the laser intensity for optimum conversion efficiency towards the more than ~500 GW/cm2 intensity regime and at the same time suppress the degradation of the beam quality. At the expense of pulse broadening we obtain higher beam quality of the generated SH pulses and accomplish SHG with a smaller aperture of the laser pulse during the frequency conversion stage by the frequency chirp of the fundamental pulses. In a preliminary calculation, we found that a frequency chirp of the fundamental pulses can suppress the spectral broadening of SH pulses when SHG is accomplished in the more than ~500 GW/cm2 intensity regime. These ultrahigh intensity laser pulses enable a smaller aperture of the laser pulse during the frequency conversion stage. So this SHG method can also be accomplished in a smaller aperture nonlinear crystal. This leads to a higher transmission spatial quality of the crystal, compared with a large aperture crystal. The crystal used in this experiment has a clear aperture of about 40 mm in diameter. We can expect that the fundamental pulses of 5.8 TWat 470 GW/cm2 can be converted into SH pulses of 4.4 TW with this crystal.

4. Conclusion

We have investigated second harmonic generation with ultrahigh intensity femtosecond laser pulses from a terawatt Ti: sapphire laser system. The maximum energy conversion efficiency was about 80 % at a laser intensity of 192 GW/cm2 without optical damage of the type-I KDP crystal. SHG with intense femtosecond pulses makes experimental arrangements of frequency conversion simple. This method can also be used to shift the laser intensity for optimum conversion efficiency towards the more than ~500 GW/cm2 intensity regime and at the same time suppress the degradation of the beam quality. At the expense of pulse broadening we obtain higher beam quality and accomplish SHG with a smaller aperture of the laser pulse during the frequency conversion stage. Furthermore, this can be accomplished with a smaller aperture nonlinear crystal. Therefore, it is feasible to apply this method to any kind of CPA laser system.

Acknowledgment

The authors would like to thank Professor M. Nakatsuka of ILE, Osaka University for providing us with a KDP crystal and Y. Inoue, H. Ueda and A. Sagisaka for their technical support. The authors are also thankful to Dr. James K. Koga for many helpful comments.

References and links

1.

K. Yamakawa, M. Aoyama, S. Matsuoka, T. Kase, Y. Akahane, and H. Takuma, “100-TW, sub 20-fs Ti : Sapphire laser system operating at a 10 Hz repetition rate,” Opt. Lett. 23, 1468–1470 (1998). [CrossRef]

2.

T. Guo, Ch. Spielmann, B. C. Walker, and C. P. J. Barty, “Generation of hard x rays by ultrafast terawatt lasers,” Rev. Sci. Instrum. 72, 41–47 (2001) [CrossRef]

3.

P. Gibbon, “High-order harmonic generation in plasmas,” IEEE J. Quantum Electron 33, 1915–1924 (1997). [CrossRef]

4.

P. B. Corkum, F. Brunel, N. K. Sherman, and T. Srinivasan-Rao, “Thermal response of metals to ultrashort-pulse laser excitation,” Phys. Rev. Lett. 61, 2886–2889 (1988). [CrossRef] [PubMed]

5.

I. V. Tomov, R. Fedosejevs, and A. A. Offenberger, “Up-conversion of subpicosecond light pulses,” IEEE J. Quantum Electron. 18, 2048–2056(1982). [CrossRef]

6.

R. C. Eckard and J. Reintjes, “Phase matching limitations of high efficiency second harmonic generation,” IEEE J. Quantum Electron. 20, 1178–1187 (1984). [CrossRef]

7.

C. Y. Chien, G. Korn, J. S. Coe, J. Squier, G. Mourou, and R.S. Craxton, “Highly efficient second-harmonic generation of ultraintense Nd : glass laser pulses,” Opt. Lett. 20, 353–355(1995). [CrossRef] [PubMed]

8.

V. Krylov, A. Rebane, A. G. Kalintsev, H. Schwoerer, and U. P. Wild, “Second-harmonic generation of amplified femtosecond Ti : sapphire laser pulses,” Opt. Lett. 20, 198–200 (1995). [CrossRef] [PubMed]

9.

Y. Tamaki, M. Obara, and K. Midorikawa, “Second harmonic generation from intense, 100-fs Ti Sapphire laser pulses in Potassium dihydrogen phosphate, Cesium lithium borate and β-barium metaborate,” Jpn. J. Appl. Phys. 37, 4801–4805 (1998). [CrossRef]

10.

D. Neely, C. N. Danson, R. Allott, F. Amiranoff, J. L. Collier, A. E. Dangor, C. B. Edwards, P. Flintoff, P. Hatton, M. Harman, M. H. R. Hutchinson, Z. Najmudin, D. A. Pepler, I. N. Ross, M. Salvati, and T. Winstone, “Frequency doubling of multi-terawatt picosecond pulses,” Laser and Particle Beams. 17, 281–286 (1999). [CrossRef]

11.

J. Queneuille, F. Druon, A. Maksimchuk, G. Cheriaux, G. Mourou, and K. Nemoto, “Second-harmonic generation and wave-front correction of a terawatt laser system,” Opt. Lett. 25, 508–510 (2000). [CrossRef]

12.

T. Ditmire, A. M. Rubenchik, D. Eimerl, and M. D. Perry, “Effects of cubic nonlinearity on frequency doubling of high-power laser pulses,” J. Opt. Soc. Am. B 13, 649–655 (1996). [CrossRef]

13.

T. Harimoto, M. Aoyama, K. Yamakawa, and M. Yonemura, “Suppression of cubic nonlinearity in second-harmonic generation of ultrahigh intensity laser pulses by initial frequency chirp,” Jpn. J. Appl. Phys. 41(2002). [CrossRef]

OCIS Codes
(140.7090) Lasers and laser optics : Ultrafast lasers
(190.2620) Nonlinear optics : Harmonic generation and mixing

ToC Category:
Research Papers

History
Original Manuscript: October 18, 2001
Published: November 19, 2001

Citation
Makoto Aoyama, Tetsuo Harimoto, J. Ma, Yutaka Akahane, and Koichi Yamakawa, "Second - harmonic generation of ultra-high intensity femtosecond pulses with a KDP crystal," Opt. Express 9, 579-585 (2001)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-11-579


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References

  1. K. Yamakawa, M. Aoyama, S. Matsuoka, T. Kase, Y. Akahane, H.Takuma, "100-TW, sub 20-fs Ti :Sapphire laser system operating at a 10 Hz repetition rate," Opt. Lett. 23, 1468-1470 (1998). [CrossRef]
  2. T. Guo, Ch. Spielmann, B. C. Walker, and C. P. J. Barty, " Generation of hard x rays by ultrafast terawatt lasers," Rev. Sci. Instrum. 72, 41-47 (2001) [CrossRef]
  3. P. Gibbon, "High-order harmonic generation in plasmas," IEEE J. Quantum Electron 33, 1915-1924 (1997). [CrossRef]
  4. P. B. Corkum, F. Brunel, N. K. Sherman, and T. Srinivasan-Rao, "Thermal response of metals to ultrashort-pulse laser excitation," Phys. Rev. Lett. 61, 2886-2889 (1988). [CrossRef] [PubMed]
  5. I. V. Tomov, R. Fedosejevs and A. A. Offenberger, "Up-conversion of subpicosecond light pulses," IEEE J. Quantum Electron. 18, 2048-2056 (1982). [CrossRef]
  6. R. C. Eckard and J. Reintjes, "Phase matching limitations of high efficiency second harmonic generation," IEEE J. Quantum Electron. 20, 1178-1187 (1984). [CrossRef]
  7. C. Y. Chien, G. Korn, J. S. Coe, J. Squier, G. Mourou and R.S. Craxton, "Highly efficient second-harmonic generation of ultraintense Nd : glass laser pulses," Opt. Lett. 20, 353-355(1995). [CrossRef] [PubMed]
  8. V. Krylov, A. Rebane, A. G.Kalintsev, H. Schwoerer and U. P. Wild, "Second-harmonic generation of amplified femtosecond Ti : sapphire laser pulses," Opt. Lett. 20, 198-200 (1995). [CrossRef] [PubMed]
  9. Y. Tamaki, M. Obara and K. Midorikawa, " Second harmonic generation from intense, 100-fs Ti Sapphire laser pulses in Potassium dihydrogen phosphate, Cesium lithium borate and ?-bariummetaborate," Jpn. J. Appl. Phys. 37, 4801-4805 (1998). [CrossRef]
  10. D. Neely , C. N. Danson, R. Allott, F. Amiranoff, J. L. Collier, A. E. Dangor, C. B. Edwards, P. Flintoff, P. Hatton, M. Harman, M. H. R. Hutchinson, Z. Najmudin, D. A. Pepler, I. N. Ross, M. Salvati and T. Winstone, "Frequency doubling of multi-terawatt picosecond pulses," Laser and Particle Beams 17, 281-286 (1999). [CrossRef]
  11. J. Queneuille, F. Druon, A. Maksimchuk, G. Cheriaux, G. Mourou and K. Nemoto, "Second-harmonic generation and wave-front correction of a terawatt laser system," Opt. Lett. 25, 508-510 (2000). [CrossRef]
  12. T. Ditmire, A. M. Rubenchik, D. Eimerl, and M. D. Perry, "Effects of cubic nonlinearity on frequency doubling of high-power laser pulses," J. Opt. Soc. Am. B 13, 649-655 (1996). [CrossRef]
  13. T. Harimoto, M. Aoyama, K. Yamakawa, and M. Yonemura, " Suppression of cubic nonlinearity in second-harmonic generation of ultrahigh intensity laser pulses by initial frequency chirp," Jpn. J. Appl. Phys. 41 (2002). [CrossRef]

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