Abstract

Sum-frequency generation of femtosecond pulses by noncollinear phase matching in a micro-structured β-BaB2O4 crystal has been investigated. A 1D relief grating was recorded in the surface of a β-BaB2O4 crystal by direct irradiation with femtosecond pulses. Groove dimensions of 1 µm width and 200 nm depth were obtained for the recording parameters. This micro-structured crystal was used for generating ultraviolet pulses (265 nm) by noncollinear sum-frequency generation of the fundamental and the second harmonic of a Ti:Sapphire femtosecond laser system. Effective group-velocity matching between the incident pulses can be achieved for certain noncollinear processes in a compact way. The application of such device to intensity cross-correlation measurements of the second-harmonic pulse was shown.

© 2008 Optical Society of America

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References

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  1. See, for instance, R. W. Boyd, Nonlinear Optics (Academic Press, San Diego, 2003).
  2. D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, "Optical, mechanical and thermal properties of barium borate," J. Appl. Phys. 62, 1968 (1987).
    [CrossRef]
  3. M. Nisoli, S. De Silvestri, V. Magni, O. Svelto, R. Danielus, A. Piskarkas, G. Valiulis, and A. Varanavicius, "Highly efficient parametric conversion of femtosecond Ti :sapphire laser pulses at 1kHz," Opt. Lett. 19, 1973-1975 (1994).
    [CrossRef] [PubMed]
  4. S. R. Greenfield and M. R. Wasielewski, "Near-transform-limited visible and near-IR femtosecond pulses from optical parametric amplification using Type II β-barium borate," Opt. Lett. 20, 1394-1396 (1995).
    [CrossRef] [PubMed]
  5. P. Banks, M. Feit, and M. Perry, "High-intensity third-harmonic generation in beta barium borate through second-order and third-order susceptibilities," Opt. Lett. 24, 4-6 (1999).
    [CrossRef]
  6. V. Krylov, J. Gallus, U. Wild, A. Kalintsev, and A. Rebane, "Femtosecond noncollinear and collinear parametric generation and amplification in BBO crystal,l" Appl. Phys. B 70, 163-168 (2000).
    [CrossRef]
  7. H. Liu, J. Yao, and A. Puri, "Second and third harmonic generation in BBO by femtosecond Ti:sapphire laser pulses," Opt. Commun. 109, 139-144 (1994).
    [CrossRef]
  8. T. Zhang, H. Choo, and M. Downer, "Phase and group velocity matching for second harmonic generation of femtosecond pulses," Appl. Opt. 29, 3927-3933 (1990).
    [CrossRef] [PubMed]
  9. C. Radzewicz, Y. Band, G. Pearson, and J. Krasinski, "Short pulse nonlinear frequency conversion without group-velocity-mismatch broadening," Opt. Commun. 117, 295-302 (1995).
    [CrossRef]
  10. A. Smith, "Group-velocity-matched three-wave mixing in birefringent crystals," Opt. Lett. 26, 719-721 (2001).
    [CrossRef]
  11. G. Arisholm, J. Biegert, P. Schlup, C. P. Hauri, and U. Keller, "Ultra-broadband chirped-pulse optical parametric amplifier with angularly dispersed beams," Opt. Express 12, 518-530 (2004).
    [CrossRef] [PubMed]
  12. L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, "Writing of permanent birefringent microlayers in bulk fused silica with femtosecond laser pulses" Opt. Commun. 171, 279-284 (1999).
    [CrossRef]
  13. X. Wang, H. Guo, H. Yang, H. Jiang, and Q. Gong, "Fabrication of beam shapers in the bulk of fused silica by femtosecond laser pulses," Appl. Opt. 43, 4571-4574 (2004).
    [CrossRef] [PubMed]
  14. N. Takeshima, Y. Narita, S. Tanaka, Y. Kuroiwa, and K. Hirao, "Fabrication of high-efficiency diffraction gratings in glass," Opt. Lett. 30, 352-354 (2005).
    [CrossRef] [PubMed]
  15. K. Kawamura, T. Ogawa, N. Sarukura, M. Hirano, and H. Hosono, "Fabrication of surface relief gratings on transparent dielectric materials by two-beam holographic method using infrared femtosecond laser pulses," Appl. Phys. B 71, 119-121 (2000).
    [CrossRef]
  16. C. Méndez, J. R. Vázquez de Aldana, G. A. Torchia, and L. Roso, "Integrated-grating-induced control of second-harmonic beams in frequency-doubling crystals," Opt. Lett. 30, 2763-2765 (2005).
    [CrossRef] [PubMed]
  17. G. A. Torchia, C. Méndez, I. Arias, L. Roso, A. Ródenas, and D. Jaque, "Laser gain in femtosecond microstructured Nd:MgO:LiNbO3," Appl. Phys. B 83, 559-563 (2006).
    [CrossRef]
  18. Y. Li, P. Lu, N. Dai, X. Wang, Y. Wang, B. Yu, and H. Long, "Surface relief diffraction gratings written on β-BaB2O4 crystal by femtosecond pulses," Appl. Phys. B 88, 227-230 (2007).
    [CrossRef]
  19. N. Barnes and V. Corcoran, "Parametric generation processes: spectral bandwidth and acceptance angles," Appl. Opt. 15, 696-699 (1976).
    [CrossRef] [PubMed]
  20. O. E. Martínez, "Pulse distortions in tilted pulse schemes for ultrashort pulses," Opt. Commun. 59, 229-232 (1986).
    [CrossRef]
  21. J. Zhang, J. Huang, H. Wang, K. Wong, and K. Wong, "Second-harmonic generation from regeneratively amplified femtosecond laser pulses in BBO and LBO crystals," J. Opt. Soc. Am. B 15, 200-209 (1998).
    [CrossRef]

2007

Y. Li, P. Lu, N. Dai, X. Wang, Y. Wang, B. Yu, and H. Long, "Surface relief diffraction gratings written on β-BaB2O4 crystal by femtosecond pulses," Appl. Phys. B 88, 227-230 (2007).
[CrossRef]

2006

G. A. Torchia, C. Méndez, I. Arias, L. Roso, A. Ródenas, and D. Jaque, "Laser gain in femtosecond microstructured Nd:MgO:LiNbO3," Appl. Phys. B 83, 559-563 (2006).
[CrossRef]

2005

2004

2001

2000

K. Kawamura, T. Ogawa, N. Sarukura, M. Hirano, and H. Hosono, "Fabrication of surface relief gratings on transparent dielectric materials by two-beam holographic method using infrared femtosecond laser pulses," Appl. Phys. B 71, 119-121 (2000).
[CrossRef]

V. Krylov, J. Gallus, U. Wild, A. Kalintsev, and A. Rebane, "Femtosecond noncollinear and collinear parametric generation and amplification in BBO crystal,l" Appl. Phys. B 70, 163-168 (2000).
[CrossRef]

1999

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, "Writing of permanent birefringent microlayers in bulk fused silica with femtosecond laser pulses" Opt. Commun. 171, 279-284 (1999).
[CrossRef]

P. Banks, M. Feit, and M. Perry, "High-intensity third-harmonic generation in beta barium borate through second-order and third-order susceptibilities," Opt. Lett. 24, 4-6 (1999).
[CrossRef]

1998

1995

S. R. Greenfield and M. R. Wasielewski, "Near-transform-limited visible and near-IR femtosecond pulses from optical parametric amplification using Type II β-barium borate," Opt. Lett. 20, 1394-1396 (1995).
[CrossRef] [PubMed]

C. Radzewicz, Y. Band, G. Pearson, and J. Krasinski, "Short pulse nonlinear frequency conversion without group-velocity-mismatch broadening," Opt. Commun. 117, 295-302 (1995).
[CrossRef]

1994

1990

1987

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, "Optical, mechanical and thermal properties of barium borate," J. Appl. Phys. 62, 1968 (1987).
[CrossRef]

1986

O. E. Martínez, "Pulse distortions in tilted pulse schemes for ultrashort pulses," Opt. Commun. 59, 229-232 (1986).
[CrossRef]

1976

Arias, I.

G. A. Torchia, C. Méndez, I. Arias, L. Roso, A. Ródenas, and D. Jaque, "Laser gain in femtosecond microstructured Nd:MgO:LiNbO3," Appl. Phys. B 83, 559-563 (2006).
[CrossRef]

Arisholm, G.

Band, Y.

C. Radzewicz, Y. Band, G. Pearson, and J. Krasinski, "Short pulse nonlinear frequency conversion without group-velocity-mismatch broadening," Opt. Commun. 117, 295-302 (1995).
[CrossRef]

Banks, P.

Barnes, N.

Biegert, J.

Choo, H.

Corcoran, V.

Dai, N.

Y. Li, P. Lu, N. Dai, X. Wang, Y. Wang, B. Yu, and H. Long, "Surface relief diffraction gratings written on β-BaB2O4 crystal by femtosecond pulses," Appl. Phys. B 88, 227-230 (2007).
[CrossRef]

Danielus, R.

Davis, L.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, "Optical, mechanical and thermal properties of barium borate," J. Appl. Phys. 62, 1968 (1987).
[CrossRef]

De Silvestri, S.

Downer, M.

Eimerl, D.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, "Optical, mechanical and thermal properties of barium borate," J. Appl. Phys. 62, 1968 (1987).
[CrossRef]

Feit, M.

Franco, M.

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, "Writing of permanent birefringent microlayers in bulk fused silica with femtosecond laser pulses" Opt. Commun. 171, 279-284 (1999).
[CrossRef]

Gallus, J.

V. Krylov, J. Gallus, U. Wild, A. Kalintsev, and A. Rebane, "Femtosecond noncollinear and collinear parametric generation and amplification in BBO crystal,l" Appl. Phys. B 70, 163-168 (2000).
[CrossRef]

Gong, Q.

Graham, E. K.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, "Optical, mechanical and thermal properties of barium borate," J. Appl. Phys. 62, 1968 (1987).
[CrossRef]

Greenfield, S. R.

Guo, H.

Hauri, C. P.

Hirano, M.

K. Kawamura, T. Ogawa, N. Sarukura, M. Hirano, and H. Hosono, "Fabrication of surface relief gratings on transparent dielectric materials by two-beam holographic method using infrared femtosecond laser pulses," Appl. Phys. B 71, 119-121 (2000).
[CrossRef]

Hirao, K.

Hosono, H.

K. Kawamura, T. Ogawa, N. Sarukura, M. Hirano, and H. Hosono, "Fabrication of surface relief gratings on transparent dielectric materials by two-beam holographic method using infrared femtosecond laser pulses," Appl. Phys. B 71, 119-121 (2000).
[CrossRef]

Huang, J.

Jaque, D.

G. A. Torchia, C. Méndez, I. Arias, L. Roso, A. Ródenas, and D. Jaque, "Laser gain in femtosecond microstructured Nd:MgO:LiNbO3," Appl. Phys. B 83, 559-563 (2006).
[CrossRef]

Jiang, H.

Kalintsev, A.

V. Krylov, J. Gallus, U. Wild, A. Kalintsev, and A. Rebane, "Femtosecond noncollinear and collinear parametric generation and amplification in BBO crystal,l" Appl. Phys. B 70, 163-168 (2000).
[CrossRef]

Kawamura, K.

K. Kawamura, T. Ogawa, N. Sarukura, M. Hirano, and H. Hosono, "Fabrication of surface relief gratings on transparent dielectric materials by two-beam holographic method using infrared femtosecond laser pulses," Appl. Phys. B 71, 119-121 (2000).
[CrossRef]

Keller, U.

Krasinski, J.

C. Radzewicz, Y. Band, G. Pearson, and J. Krasinski, "Short pulse nonlinear frequency conversion without group-velocity-mismatch broadening," Opt. Commun. 117, 295-302 (1995).
[CrossRef]

Krylov, V.

V. Krylov, J. Gallus, U. Wild, A. Kalintsev, and A. Rebane, "Femtosecond noncollinear and collinear parametric generation and amplification in BBO crystal,l" Appl. Phys. B 70, 163-168 (2000).
[CrossRef]

Kuroiwa, Y.

Li, Y.

Y. Li, P. Lu, N. Dai, X. Wang, Y. Wang, B. Yu, and H. Long, "Surface relief diffraction gratings written on β-BaB2O4 crystal by femtosecond pulses," Appl. Phys. B 88, 227-230 (2007).
[CrossRef]

Liu, H.

H. Liu, J. Yao, and A. Puri, "Second and third harmonic generation in BBO by femtosecond Ti:sapphire laser pulses," Opt. Commun. 109, 139-144 (1994).
[CrossRef]

Long, H.

Y. Li, P. Lu, N. Dai, X. Wang, Y. Wang, B. Yu, and H. Long, "Surface relief diffraction gratings written on β-BaB2O4 crystal by femtosecond pulses," Appl. Phys. B 88, 227-230 (2007).
[CrossRef]

Lu, P.

Y. Li, P. Lu, N. Dai, X. Wang, Y. Wang, B. Yu, and H. Long, "Surface relief diffraction gratings written on β-BaB2O4 crystal by femtosecond pulses," Appl. Phys. B 88, 227-230 (2007).
[CrossRef]

Magni, V.

Martínez, O. E.

O. E. Martínez, "Pulse distortions in tilted pulse schemes for ultrashort pulses," Opt. Commun. 59, 229-232 (1986).
[CrossRef]

Méndez, C.

G. A. Torchia, C. Méndez, I. Arias, L. Roso, A. Ródenas, and D. Jaque, "Laser gain in femtosecond microstructured Nd:MgO:LiNbO3," Appl. Phys. B 83, 559-563 (2006).
[CrossRef]

C. Méndez, J. R. Vázquez de Aldana, G. A. Torchia, and L. Roso, "Integrated-grating-induced control of second-harmonic beams in frequency-doubling crystals," Opt. Lett. 30, 2763-2765 (2005).
[CrossRef] [PubMed]

Mysyrowicz, A.

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, "Writing of permanent birefringent microlayers in bulk fused silica with femtosecond laser pulses" Opt. Commun. 171, 279-284 (1999).
[CrossRef]

Narita, Y.

Nisoli, M.

Ogawa, T.

K. Kawamura, T. Ogawa, N. Sarukura, M. Hirano, and H. Hosono, "Fabrication of surface relief gratings on transparent dielectric materials by two-beam holographic method using infrared femtosecond laser pulses," Appl. Phys. B 71, 119-121 (2000).
[CrossRef]

Pearson, G.

C. Radzewicz, Y. Band, G. Pearson, and J. Krasinski, "Short pulse nonlinear frequency conversion without group-velocity-mismatch broadening," Opt. Commun. 117, 295-302 (1995).
[CrossRef]

Perry, M.

Piskarkas, A.

Prade, B.

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, "Writing of permanent birefringent microlayers in bulk fused silica with femtosecond laser pulses" Opt. Commun. 171, 279-284 (1999).
[CrossRef]

Puri, A.

H. Liu, J. Yao, and A. Puri, "Second and third harmonic generation in BBO by femtosecond Ti:sapphire laser pulses," Opt. Commun. 109, 139-144 (1994).
[CrossRef]

Radzewicz, C.

C. Radzewicz, Y. Band, G. Pearson, and J. Krasinski, "Short pulse nonlinear frequency conversion without group-velocity-mismatch broadening," Opt. Commun. 117, 295-302 (1995).
[CrossRef]

Rebane, A.

V. Krylov, J. Gallus, U. Wild, A. Kalintsev, and A. Rebane, "Femtosecond noncollinear and collinear parametric generation and amplification in BBO crystal,l" Appl. Phys. B 70, 163-168 (2000).
[CrossRef]

Ródenas, A.

G. A. Torchia, C. Méndez, I. Arias, L. Roso, A. Ródenas, and D. Jaque, "Laser gain in femtosecond microstructured Nd:MgO:LiNbO3," Appl. Phys. B 83, 559-563 (2006).
[CrossRef]

Roso, L.

G. A. Torchia, C. Méndez, I. Arias, L. Roso, A. Ródenas, and D. Jaque, "Laser gain in femtosecond microstructured Nd:MgO:LiNbO3," Appl. Phys. B 83, 559-563 (2006).
[CrossRef]

C. Méndez, J. R. Vázquez de Aldana, G. A. Torchia, and L. Roso, "Integrated-grating-induced control of second-harmonic beams in frequency-doubling crystals," Opt. Lett. 30, 2763-2765 (2005).
[CrossRef] [PubMed]

Sarukura, N.

K. Kawamura, T. Ogawa, N. Sarukura, M. Hirano, and H. Hosono, "Fabrication of surface relief gratings on transparent dielectric materials by two-beam holographic method using infrared femtosecond laser pulses," Appl. Phys. B 71, 119-121 (2000).
[CrossRef]

Schlup, P.

Smith, A.

Sudrie, L.

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, "Writing of permanent birefringent microlayers in bulk fused silica with femtosecond laser pulses" Opt. Commun. 171, 279-284 (1999).
[CrossRef]

Svelto, O.

Takeshima, N.

Tanaka, S.

Torchia, G. A.

G. A. Torchia, C. Méndez, I. Arias, L. Roso, A. Ródenas, and D. Jaque, "Laser gain in femtosecond microstructured Nd:MgO:LiNbO3," Appl. Phys. B 83, 559-563 (2006).
[CrossRef]

C. Méndez, J. R. Vázquez de Aldana, G. A. Torchia, and L. Roso, "Integrated-grating-induced control of second-harmonic beams in frequency-doubling crystals," Opt. Lett. 30, 2763-2765 (2005).
[CrossRef] [PubMed]

Valiulis, G.

Varanavicius, A.

Vázquez de Aldana, J. R.

Velsko, S.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, "Optical, mechanical and thermal properties of barium borate," J. Appl. Phys. 62, 1968 (1987).
[CrossRef]

Wang, H.

Wang, X.

Y. Li, P. Lu, N. Dai, X. Wang, Y. Wang, B. Yu, and H. Long, "Surface relief diffraction gratings written on β-BaB2O4 crystal by femtosecond pulses," Appl. Phys. B 88, 227-230 (2007).
[CrossRef]

X. Wang, H. Guo, H. Yang, H. Jiang, and Q. Gong, "Fabrication of beam shapers in the bulk of fused silica by femtosecond laser pulses," Appl. Opt. 43, 4571-4574 (2004).
[CrossRef] [PubMed]

Wang, Y.

Y. Li, P. Lu, N. Dai, X. Wang, Y. Wang, B. Yu, and H. Long, "Surface relief diffraction gratings written on β-BaB2O4 crystal by femtosecond pulses," Appl. Phys. B 88, 227-230 (2007).
[CrossRef]

Wasielewski, M. R.

Wild, U.

V. Krylov, J. Gallus, U. Wild, A. Kalintsev, and A. Rebane, "Femtosecond noncollinear and collinear parametric generation and amplification in BBO crystal,l" Appl. Phys. B 70, 163-168 (2000).
[CrossRef]

Wong, K.

Yang, H.

Yao, J.

H. Liu, J. Yao, and A. Puri, "Second and third harmonic generation in BBO by femtosecond Ti:sapphire laser pulses," Opt. Commun. 109, 139-144 (1994).
[CrossRef]

Yu, B.

Y. Li, P. Lu, N. Dai, X. Wang, Y. Wang, B. Yu, and H. Long, "Surface relief diffraction gratings written on β-BaB2O4 crystal by femtosecond pulses," Appl. Phys. B 88, 227-230 (2007).
[CrossRef]

Zalkin, A.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, "Optical, mechanical and thermal properties of barium borate," J. Appl. Phys. 62, 1968 (1987).
[CrossRef]

Zhang, J.

Zhang, T.

Appl. Opt.

Appl. Phys. B

V. Krylov, J. Gallus, U. Wild, A. Kalintsev, and A. Rebane, "Femtosecond noncollinear and collinear parametric generation and amplification in BBO crystal,l" Appl. Phys. B 70, 163-168 (2000).
[CrossRef]

K. Kawamura, T. Ogawa, N. Sarukura, M. Hirano, and H. Hosono, "Fabrication of surface relief gratings on transparent dielectric materials by two-beam holographic method using infrared femtosecond laser pulses," Appl. Phys. B 71, 119-121 (2000).
[CrossRef]

G. A. Torchia, C. Méndez, I. Arias, L. Roso, A. Ródenas, and D. Jaque, "Laser gain in femtosecond microstructured Nd:MgO:LiNbO3," Appl. Phys. B 83, 559-563 (2006).
[CrossRef]

Y. Li, P. Lu, N. Dai, X. Wang, Y. Wang, B. Yu, and H. Long, "Surface relief diffraction gratings written on β-BaB2O4 crystal by femtosecond pulses," Appl. Phys. B 88, 227-230 (2007).
[CrossRef]

J. Appl. Phys.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, "Optical, mechanical and thermal properties of barium borate," J. Appl. Phys. 62, 1968 (1987).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

O. E. Martínez, "Pulse distortions in tilted pulse schemes for ultrashort pulses," Opt. Commun. 59, 229-232 (1986).
[CrossRef]

H. Liu, J. Yao, and A. Puri, "Second and third harmonic generation in BBO by femtosecond Ti:sapphire laser pulses," Opt. Commun. 109, 139-144 (1994).
[CrossRef]

C. Radzewicz, Y. Band, G. Pearson, and J. Krasinski, "Short pulse nonlinear frequency conversion without group-velocity-mismatch broadening," Opt. Commun. 117, 295-302 (1995).
[CrossRef]

L. Sudrie, M. Franco, B. Prade, and A. Mysyrowicz, "Writing of permanent birefringent microlayers in bulk fused silica with femtosecond laser pulses" Opt. Commun. 171, 279-284 (1999).
[CrossRef]

Opt. Express

Opt. Lett.

Other

See, for instance, R. W. Boyd, Nonlinear Optics (Academic Press, San Diego, 2003).

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Figures (6)

Fig. 1.
Fig. 1.

(a) Partial view of the grating taken with an optical microscope in reflection mode. (b) Bar chart showing the normalized energy of the diffraction orders when irradiating the grating with a He-Ne laser.

Fig. 2.
Fig. 2.

Experimental setup for SFG. DM: dichroic mirror, SP: short-pass filter, ND: neutral density filter, L: focusing lens.

Fig. 3.
Fig. 3.

(a) Pictures of the emerging beams projected on a paper screen for different values of the crystal tilt ϕ. Red arrows point the ±1 diffraction orders for the SH and yellow arrows point the TH beams generated by noncollinear phase matching whose intensity is maximized for the given tilt. The bright spot in the middle is the order 0 for both frequencies. The orders ±1 for the fundamental are out of the picture frame. (b) Spectra of the TH beams optimized at the tilt angles indicated in the legend.

Fig. 4.
Fig. 4.

(a) Definitions of the axes and angles inside the crystal. Only one half of the diffraction plane (i.e. positive diffraction orders) is shown for clarity. (b) Diagrams showing the diffraction plane for the two noncollinear SFG processes that involve orders 0 and 1. They are phase matched at different crystal tilts (ϕ 1 and ϕ 3); thus the inclination (β(φ)) of the depicted planes is different in the two cases.

Fig. 5.
Fig. 5.

(a) Noncollinear phase matching curves. See text for explanation. (b) Bandwidth of the TH pulses as a function of the angle between the fundamental and the SH beam.

Fig. 6.
Fig. 6.

(a) Diagram of the noncollinear SFG process induced by the grating showing pulse tilting and effective group velocity matching between the incident pulses. (b) Cross-correlation traces obtained with the micro-structured BBO crystal for collinear (red) and noncollinear (blue) SFG processes. In solid line we show a theoretical cross-correlation trace computed by assuming GVM=0 for comparison.

Tables (2)

Tables Icon

Table 1. Measured and calculated angles (external) for the first and second diffraction orders

Tables Icon

Table 2. Calculated (th.) and measured (exp.) crystal tilts (ϕ i , with i=1,2,3,4 as defined in Sect. 4) and emergence angles of the TH pulses χ̃( m , n )).

Equations (3)

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

sin α m ω i = m 2 π c ω i n o ω i d
{ ω 1 + ω 2 = ω 3 k m ω 1 + k n ω 2 = k ( m , n ) ω 3
{ 3 n e ω 3 ( θ ( ϕ , χ ) ) = n o ω 1 cos ( α m ω 1 χ ) + 2 n o ω 2 cos ( χ α n ω 2 ) n o ω 1 sin ( α m ω 1 χ ) + 2 n o ω 2 sin ( α n ω 2 χ ) = 0

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