Abstract

We consider second harmonic generation (SHG) of ultrashort pulses in the case of strong phase- and group-velocity mismatch. Spectral fringes appear in the second harmonic related to two delayed replicas of the fundamental pulse in the time domain. The fringe separation can be used to evaluate the group-velocity and refractive index of nonlinear crystals at extreme wavelengths. Experimental results with femtosecond pulses in SrB4O7 (SBO) are used to refine the Sellmeier equation describing the nc refractive index down to 160 nm, essential for the use of this unique nonlinear crystal for random quasi-phase-matching in the VUV.

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References

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  1. V. Petrov, F. Noack, D. Shen, F. Pan, G. Shen, X. Wang, R. Komatsu, and V. Alex, “Application of the nonlinear crystal SrB4O7 for ultrafast diagnostics converting to wavelengths as short as 125 nm,” Opt. Lett. 29(4), 373–375 (2004).
    [Crossref] [PubMed]
  2. Yu. S. Oseledchik, A. L. Prosvirnin, A. I. Pisarevskiy, V. V. Starshenko, V. V. Osadchuk, S. P. Belokrys, N. V. Svitanko, A. S. Korol, S. A. Krikunov, and A. F. Selevich, “New nonlinear optical crystals: strontium and lead tetraborates,” Opt. Mater. 4(6), 669–674 (1995).
    [Crossref]
  3. F. Pan, G. Shen, R. Wang, X. Wang, and D. Shen, “Growth, characterization and nonlinear optical properties of SrB4O7,” J. Cryst. Growth 241(1-2), 108–114 (2002).
    [Crossref]
  4. A. S. Aleksandrovsky, A. M. Vyunishev, A. I. Zaitsev, A. V. Zamkov, and V. G. Arkhipkin, “Detection of randomized nonlinear photonic crystal structure in a non-ferroelectric crystal,” J. Opt. A, Pure Appl. Opt. 9(9), S334–S338 (2007).
    [Crossref]
  5. A. I. Zaitsev, A. S. Aleksandrovsky, A. D. Vasiliev, and A. V. Zamkov, “Domain structure in strontium tetraborate single crystal,” J. Cryst. Growth 310(1), 1–4 (2008).
    [Crossref]
  6. M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432(7015), 374–376 (2004).
    [Crossref] [PubMed]
  7. P. Trabs, F. Noack, A. S. Aleksandrovsky, A. M. Vyunishev, A. I. Zaitsev, N. V. Radionov, and V. Petrov, “Generation of fs-pulses down to 121 nm by frequency doubling using random quasi-phase-matching in strontium tetraborate,” in Conference Proceedings of Ultrafast Optics IX, Davos, Switzerland, March 4–8, Conference Program, paper Fr2.4.
  8. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, Academic press, Elsevier, 2nd edition, 2006.
  9. M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, “Second harmonic generation of femtosecond pulses at the boundary of a nonlinear dielectric,” Phys. Rev. Lett. 83(15), 2934–2937 (1999).
    [Crossref]
  10. J. T. Manassah, “Effects of velocity dispersion on a generated second harmonic signal,” Appl. Opt. 27(21), 4365–4367 (1988).
    [Crossref] [PubMed]
  11. K. Kato, “Second-harmonic generation to 2048 Å in β-BaB2O4,” IEEE J. Quantum Electron. 22(7), 1013–1014 (1986).
    [Crossref]

2008 (1)

A. I. Zaitsev, A. S. Aleksandrovsky, A. D. Vasiliev, and A. V. Zamkov, “Domain structure in strontium tetraborate single crystal,” J. Cryst. Growth 310(1), 1–4 (2008).
[Crossref]

2007 (1)

A. S. Aleksandrovsky, A. M. Vyunishev, A. I. Zaitsev, A. V. Zamkov, and V. G. Arkhipkin, “Detection of randomized nonlinear photonic crystal structure in a non-ferroelectric crystal,” J. Opt. A, Pure Appl. Opt. 9(9), S334–S338 (2007).
[Crossref]

2004 (2)

V. Petrov, F. Noack, D. Shen, F. Pan, G. Shen, X. Wang, R. Komatsu, and V. Alex, “Application of the nonlinear crystal SrB4O7 for ultrafast diagnostics converting to wavelengths as short as 125 nm,” Opt. Lett. 29(4), 373–375 (2004).
[Crossref] [PubMed]

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432(7015), 374–376 (2004).
[Crossref] [PubMed]

2002 (1)

F. Pan, G. Shen, R. Wang, X. Wang, and D. Shen, “Growth, characterization and nonlinear optical properties of SrB4O7,” J. Cryst. Growth 241(1-2), 108–114 (2002).
[Crossref]

1999 (1)

M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, “Second harmonic generation of femtosecond pulses at the boundary of a nonlinear dielectric,” Phys. Rev. Lett. 83(15), 2934–2937 (1999).
[Crossref]

1995 (1)

Yu. S. Oseledchik, A. L. Prosvirnin, A. I. Pisarevskiy, V. V. Starshenko, V. V. Osadchuk, S. P. Belokrys, N. V. Svitanko, A. S. Korol, S. A. Krikunov, and A. F. Selevich, “New nonlinear optical crystals: strontium and lead tetraborates,” Opt. Mater. 4(6), 669–674 (1995).
[Crossref]

1988 (1)

1986 (1)

K. Kato, “Second-harmonic generation to 2048 Å in β-BaB2O4,” IEEE J. Quantum Electron. 22(7), 1013–1014 (1986).
[Crossref]

Aleksandrovsky, A. S.

A. I. Zaitsev, A. S. Aleksandrovsky, A. D. Vasiliev, and A. V. Zamkov, “Domain structure in strontium tetraborate single crystal,” J. Cryst. Growth 310(1), 1–4 (2008).
[Crossref]

A. S. Aleksandrovsky, A. M. Vyunishev, A. I. Zaitsev, A. V. Zamkov, and V. G. Arkhipkin, “Detection of randomized nonlinear photonic crystal structure in a non-ferroelectric crystal,” J. Opt. A, Pure Appl. Opt. 9(9), S334–S338 (2007).
[Crossref]

Alex, V.

Arkhipkin, V. G.

A. S. Aleksandrovsky, A. M. Vyunishev, A. I. Zaitsev, A. V. Zamkov, and V. G. Arkhipkin, “Detection of randomized nonlinear photonic crystal structure in a non-ferroelectric crystal,” J. Opt. A, Pure Appl. Opt. 9(9), S334–S338 (2007).
[Crossref]

Baudrier-Raybaut, M.

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432(7015), 374–376 (2004).
[Crossref] [PubMed]

Belokrys, S. P.

Yu. S. Oseledchik, A. L. Prosvirnin, A. I. Pisarevskiy, V. V. Starshenko, V. V. Osadchuk, S. P. Belokrys, N. V. Svitanko, A. S. Korol, S. A. Krikunov, and A. F. Selevich, “New nonlinear optical crystals: strontium and lead tetraborates,” Opt. Mater. 4(6), 669–674 (1995).
[Crossref]

Bloembergen, N.

M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, “Second harmonic generation of femtosecond pulses at the boundary of a nonlinear dielectric,” Phys. Rev. Lett. 83(15), 2934–2937 (1999).
[Crossref]

Haïdar, R.

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432(7015), 374–376 (2004).
[Crossref] [PubMed]

Kato, K.

K. Kato, “Second-harmonic generation to 2048 Å in β-BaB2O4,” IEEE J. Quantum Electron. 22(7), 1013–1014 (1986).
[Crossref]

Komatsu, R.

Korol, A. S.

Yu. S. Oseledchik, A. L. Prosvirnin, A. I. Pisarevskiy, V. V. Starshenko, V. V. Osadchuk, S. P. Belokrys, N. V. Svitanko, A. S. Korol, S. A. Krikunov, and A. F. Selevich, “New nonlinear optical crystals: strontium and lead tetraborates,” Opt. Mater. 4(6), 669–674 (1995).
[Crossref]

Krikunov, S. A.

Yu. S. Oseledchik, A. L. Prosvirnin, A. I. Pisarevskiy, V. V. Starshenko, V. V. Osadchuk, S. P. Belokrys, N. V. Svitanko, A. S. Korol, S. A. Krikunov, and A. F. Selevich, “New nonlinear optical crystals: strontium and lead tetraborates,” Opt. Mater. 4(6), 669–674 (1995).
[Crossref]

Kupecek, P.

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432(7015), 374–376 (2004).
[Crossref] [PubMed]

Lemasson, P.

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432(7015), 374–376 (2004).
[Crossref] [PubMed]

Manassah, J. T.

Mlejnek, M.

M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, “Second harmonic generation of femtosecond pulses at the boundary of a nonlinear dielectric,” Phys. Rev. Lett. 83(15), 2934–2937 (1999).
[Crossref]

Moloney, J. V.

M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, “Second harmonic generation of femtosecond pulses at the boundary of a nonlinear dielectric,” Phys. Rev. Lett. 83(15), 2934–2937 (1999).
[Crossref]

Noack, F.

Osadchuk, V. V.

Yu. S. Oseledchik, A. L. Prosvirnin, A. I. Pisarevskiy, V. V. Starshenko, V. V. Osadchuk, S. P. Belokrys, N. V. Svitanko, A. S. Korol, S. A. Krikunov, and A. F. Selevich, “New nonlinear optical crystals: strontium and lead tetraborates,” Opt. Mater. 4(6), 669–674 (1995).
[Crossref]

Oseledchik, Yu. S.

Yu. S. Oseledchik, A. L. Prosvirnin, A. I. Pisarevskiy, V. V. Starshenko, V. V. Osadchuk, S. P. Belokrys, N. V. Svitanko, A. S. Korol, S. A. Krikunov, and A. F. Selevich, “New nonlinear optical crystals: strontium and lead tetraborates,” Opt. Mater. 4(6), 669–674 (1995).
[Crossref]

Pan, F.

Petrov, V.

Pisarevskiy, A. I.

Yu. S. Oseledchik, A. L. Prosvirnin, A. I. Pisarevskiy, V. V. Starshenko, V. V. Osadchuk, S. P. Belokrys, N. V. Svitanko, A. S. Korol, S. A. Krikunov, and A. F. Selevich, “New nonlinear optical crystals: strontium and lead tetraborates,” Opt. Mater. 4(6), 669–674 (1995).
[Crossref]

Prosvirnin, A. L.

Yu. S. Oseledchik, A. L. Prosvirnin, A. I. Pisarevskiy, V. V. Starshenko, V. V. Osadchuk, S. P. Belokrys, N. V. Svitanko, A. S. Korol, S. A. Krikunov, and A. F. Selevich, “New nonlinear optical crystals: strontium and lead tetraborates,” Opt. Mater. 4(6), 669–674 (1995).
[Crossref]

Rosencher, E.

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432(7015), 374–376 (2004).
[Crossref] [PubMed]

Selevich, A. F.

Yu. S. Oseledchik, A. L. Prosvirnin, A. I. Pisarevskiy, V. V. Starshenko, V. V. Osadchuk, S. P. Belokrys, N. V. Svitanko, A. S. Korol, S. A. Krikunov, and A. F. Selevich, “New nonlinear optical crystals: strontium and lead tetraborates,” Opt. Mater. 4(6), 669–674 (1995).
[Crossref]

Shen, D.

Shen, G.

Starshenko, V. V.

Yu. S. Oseledchik, A. L. Prosvirnin, A. I. Pisarevskiy, V. V. Starshenko, V. V. Osadchuk, S. P. Belokrys, N. V. Svitanko, A. S. Korol, S. A. Krikunov, and A. F. Selevich, “New nonlinear optical crystals: strontium and lead tetraborates,” Opt. Mater. 4(6), 669–674 (1995).
[Crossref]

Svitanko, N. V.

Yu. S. Oseledchik, A. L. Prosvirnin, A. I. Pisarevskiy, V. V. Starshenko, V. V. Osadchuk, S. P. Belokrys, N. V. Svitanko, A. S. Korol, S. A. Krikunov, and A. F. Selevich, “New nonlinear optical crystals: strontium and lead tetraborates,” Opt. Mater. 4(6), 669–674 (1995).
[Crossref]

Vasiliev, A. D.

A. I. Zaitsev, A. S. Aleksandrovsky, A. D. Vasiliev, and A. V. Zamkov, “Domain structure in strontium tetraborate single crystal,” J. Cryst. Growth 310(1), 1–4 (2008).
[Crossref]

Vyunishev, A. M.

A. S. Aleksandrovsky, A. M. Vyunishev, A. I. Zaitsev, A. V. Zamkov, and V. G. Arkhipkin, “Detection of randomized nonlinear photonic crystal structure in a non-ferroelectric crystal,” J. Opt. A, Pure Appl. Opt. 9(9), S334–S338 (2007).
[Crossref]

Wang, R.

F. Pan, G. Shen, R. Wang, X. Wang, and D. Shen, “Growth, characterization and nonlinear optical properties of SrB4O7,” J. Cryst. Growth 241(1-2), 108–114 (2002).
[Crossref]

Wang, X.

Wright, E. M.

M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, “Second harmonic generation of femtosecond pulses at the boundary of a nonlinear dielectric,” Phys. Rev. Lett. 83(15), 2934–2937 (1999).
[Crossref]

Zaitsev, A. I.

A. I. Zaitsev, A. S. Aleksandrovsky, A. D. Vasiliev, and A. V. Zamkov, “Domain structure in strontium tetraborate single crystal,” J. Cryst. Growth 310(1), 1–4 (2008).
[Crossref]

A. S. Aleksandrovsky, A. M. Vyunishev, A. I. Zaitsev, A. V. Zamkov, and V. G. Arkhipkin, “Detection of randomized nonlinear photonic crystal structure in a non-ferroelectric crystal,” J. Opt. A, Pure Appl. Opt. 9(9), S334–S338 (2007).
[Crossref]

Zamkov, A. V.

A. I. Zaitsev, A. S. Aleksandrovsky, A. D. Vasiliev, and A. V. Zamkov, “Domain structure in strontium tetraborate single crystal,” J. Cryst. Growth 310(1), 1–4 (2008).
[Crossref]

A. S. Aleksandrovsky, A. M. Vyunishev, A. I. Zaitsev, A. V. Zamkov, and V. G. Arkhipkin, “Detection of randomized nonlinear photonic crystal structure in a non-ferroelectric crystal,” J. Opt. A, Pure Appl. Opt. 9(9), S334–S338 (2007).
[Crossref]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

K. Kato, “Second-harmonic generation to 2048 Å in β-BaB2O4,” IEEE J. Quantum Electron. 22(7), 1013–1014 (1986).
[Crossref]

J. Cryst. Growth (2)

A. I. Zaitsev, A. S. Aleksandrovsky, A. D. Vasiliev, and A. V. Zamkov, “Domain structure in strontium tetraborate single crystal,” J. Cryst. Growth 310(1), 1–4 (2008).
[Crossref]

F. Pan, G. Shen, R. Wang, X. Wang, and D. Shen, “Growth, characterization and nonlinear optical properties of SrB4O7,” J. Cryst. Growth 241(1-2), 108–114 (2002).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

A. S. Aleksandrovsky, A. M. Vyunishev, A. I. Zaitsev, A. V. Zamkov, and V. G. Arkhipkin, “Detection of randomized nonlinear photonic crystal structure in a non-ferroelectric crystal,” J. Opt. A, Pure Appl. Opt. 9(9), S334–S338 (2007).
[Crossref]

Nature (1)

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432(7015), 374–376 (2004).
[Crossref] [PubMed]

Opt. Lett. (1)

Opt. Mater. (1)

Yu. S. Oseledchik, A. L. Prosvirnin, A. I. Pisarevskiy, V. V. Starshenko, V. V. Osadchuk, S. P. Belokrys, N. V. Svitanko, A. S. Korol, S. A. Krikunov, and A. F. Selevich, “New nonlinear optical crystals: strontium and lead tetraborates,” Opt. Mater. 4(6), 669–674 (1995).
[Crossref]

Phys. Rev. Lett. (1)

M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, “Second harmonic generation of femtosecond pulses at the boundary of a nonlinear dielectric,” Phys. Rev. Lett. 83(15), 2934–2937 (1999).
[Crossref]

Other (2)

P. Trabs, F. Noack, A. S. Aleksandrovsky, A. M. Vyunishev, A. I. Zaitsev, N. V. Radionov, and V. Petrov, “Generation of fs-pulses down to 121 nm by frequency doubling using random quasi-phase-matching in strontium tetraborate,” in Conference Proceedings of Ultrafast Optics IX, Davos, Switzerland, March 4–8, Conference Program, paper Fr2.4.

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, Academic press, Elsevier, 2nd edition, 2006.

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Figures (4)

Fig. 1
Fig. 1 Non-phase-matched SHG in BBO. Left: Intensity of the fundamental wave and the SH. Right: SH spectrum with fringes, and fundamental and SH pulses at the exit of the crystal.
Fig. 2
Fig. 2 Same as Fig. 1 but for the initial 25 µm propagation in the BBO crystal.
Fig. 3
Fig. 3 Measured spectral fringes for non-phase-matched SHG in BBO and SBO.
Fig. 4
Fig. 4 Experimental data and calculated c/ v z and nz curves with the refined Sellmeier equation for n c n z of SBO.

Equations (12)

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

  E ˜ ( Ω )=  E ˜ ( t ) e iΩt dt 
( z + 1 v F t ) E ˜ F = i χ ( 2 ) ω F 2 2 c 2 k F E ˜ F * E ˜ SH e iΔkz
( z + 1 v SH t ) E ˜ SH = i χ (2) ω SH 2 4 c 2 k SH E ˜ F 2 e iΔkz
E ˜ SH ( Ω,d )=i  χ (2) ω SH 2 d 4 c 2 k SH sinc[ ( MΩΔk ) d 2 ]( E ˜ F * E ˜ F )(Ω),
E ˜ SH ( Ω,d )=i  χ (2) ω SH 2 2 c 2 k SH Δk sin[ ( MΩΔk ) d 2 ]( E ˜ F * E ˜ F )(Ω)
S SH ( Ω,d )= ( χ (2) ω SH 2 2 c 2 k SH Δk ) 2 sin 2 [ ( MΩΔk ) d 2 ] | ( E ˜ F * E ˜ F )( Ω ) | 2
E ˜ SH ( t,d )= 1 2π E ˜ SH ( Ω,d ) e iΩt dΩ=i χ ( 2 ) ω SH 2 4π c 2 k SH Δk sin[ ( MΩΔk ) d 2 ][ ( E ˜ F * E ˜ F )( Ω ) ] e iΩt dΩ =i χ (2) ω SH 2 4π c 2 k SH Δk sin[ ( MΩΔk ) d 2 ] e iΩt dΩ* [ ( E ˜ F * E ˜ F )( Ω ) ] e iΩt dΩ =i χ ( 2 ) ω SH 2 2 c 2 k SH Δk sin[ ( MΩΔk ) d 2 ] e iΩt dΩ* E ˜ F 2 ( t ) = π χ ( 2 ) ω SH 2 2 c 2 k SH Δk [ e iΔkd 2 δ( t+ Md 2 ) e iΔkd 2 δ( t Md 2 ) ]* E ˜ F 2 ( t ) = π χ ( 2 ) ω SH 2 2 c 2 k SH Δk [ e iΔkd 2 E ˜ F 2 ( t+ Md 2 ) e iΔkd 2 E ˜ F 2 ( t Md 2 ) ]
I SH ( t,d ) I F 2 ( t+ Md 2 )+ I F 2 ( t Md 2 )[ e iΔkd E ˜ F 2 ( t Md 2 ) E ˜ F *2 ( t+ Md 2 )+c.c. ]
I SH ( η,d ) I F 2 ( η )+ I F 2 ( ηMd )[ e iΔkd E ˜ F 2 ( ηMd ) E ˜ F *2 ( η )+c.c. ]
I SH ( η,d ) I F 2 ( η )+ I F 2 ( ηMd )2cos( Δkd ) I F ( ηMd ) I F ( η )
I SH ( η,d ) I F 2 ( η ) sin 2 ( πd 2 L c )
v SH =(dΔΩ v F )/(2π v F +dΔΩ)

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