Abstract

We present a theoretical model that describes the focusing conditions for second-harmonic generation (SHG) of focused femtosecond pulses as a function of group-velocity mismatch (GVM), with direct application to efficient SHG using a “thick” nonlinear crystal. We observe a direct dependence of the optimal focusing ratio, L/b, on the strength of group-velocity mismatch. Our model also describes the temporal duration of the second-harmonic pulses under these conditions as well as the change in optimal phase mismatch. The theoretical results are compared with an experiment for SHG with focused femtosecond pulses in a “thick” crystal of KNbO3.

© 2004 Optical Society of America

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  1. G. D. Boyd and D. A. Kleinman, “Parametric interactions of focused Gaussian light beams,” J. Appl. Phys. 39, 3897–3641 (1968).
    [CrossRef]
  2. B. Agate, A. J. Kemp, C. T. A. Brown, and W. Sibbett, “Efficient, high repetition-rate femtosecond blue source using a compact Cr:LiSAF laser,” Opt. Express 10, 824–831 (2002).
    [CrossRef] [PubMed]
  3. A. M. Weiner, A. M. Kan’an, and D. E. Leaird, “High-efficiency blue generation by frequency doubling of femtosecond pulses in a thick nonlinear crystal,” Opt. Lett. 23, 1441–1443 (1998).
    [CrossRef]
  4. S. Yu and A. M. Weiner, “Phase-matching temperature shifts in blue generation by frequency doubling of femtosecond pulses in KNbO3,” J. Opt. Soc. Am. B 16, 1300–1304 (1999).
    [CrossRef]
  5. D. Guzun, Y. Q. Li, and M. Xiao, “Blue light generation in single-pass frequency doubling of femtosecond pulses in KNbO3,” Opt. Commun. 180, 367–371 (2000).
    [CrossRef]
  6. Y. Q. Li, D. Guzun, G. Salamo, and M. Xiao, “High-efficiency blue-light generation by frequency doubling of picosecond pulses in a thick KNbO3 crystal,” J. Opt. Soc. Am. B 20, 1285–1289 (2003).
    [CrossRef]
  7. H. Wang and A. M. Weiner, “Second harmonic generation efficiency with simultaneous temporal walkoff, spatial walkoff, and depletion,” Conference on Lasers and Electro-Optics, Vol. 89 of Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2003), paper CThU1.
  8. R. W. Boyd, Nonlinear Optics (Academic, New York, 1992) Chap. II, p. 94.
  9. V. Magni, “Optimum beams for efficient frequency mixing in crystals with second order nonlinearity,” Opt. Commun. 184, 245–255 (2000).
    [CrossRef]
  10. G. D. Xu, T. W. Ren, Y. H. Wang, Y. Y. Zhu, S. N. Zhu, and N. B. Ming, “Third-harmonic generation by use of focused Gaussian beams in an optical superlattice,” J. Opt. Soc. Am. B 20, 360–365 (2003).
    [CrossRef]
  11. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin Optics of Femtosecond Laser Pulses (American Institute of Physics, New York, 1992), Chap. 3, p. 141.
  12. E. Sidick, A. Knoesen, and A. Dienes, “Ultrashort pulse second-harmonic generation in quasi-phase-matched dispersive media,” Opt. Lett. 19, 266–268 (1994).
    [CrossRef] [PubMed]
  13. E. Sidick, A. Knoesen, and A. Dienes, “Ultrashort-pulse second-harmonic generation. I. Transform-limited fundamental pulses,” J. Opt. Soc. Am. B 12, 1704–1712 (1995).
    [CrossRef]
  14. J. T. Manassah and O. R. Cockings, “Induced phase modulation of a generated second-harmonic signal,” Opt. Lett. 12, 1005–1007 (1987).
    [CrossRef] [PubMed]
  15. B. Agate, B. Stormont, A. J. Kemp, C. T. A. Brown, U. Keller, and W. Sibbett, “Simplified cavity designs for efficient and compact femtosecond Cr:LiSAF lasers,” Opt. Commun. 205, 207–213 (2002).
    [CrossRef]
  16. G. G. Gurzadian, V. G. Dmitriev, and D. N. Nikogosian, Handbook of Nonlinear Optical Crystals, 3rd ed., Vol. 64 of Springer Series in Optical Sciences (Springer-Verlag, New York, 1999).
  17. I. Biaggio, P. Kerkoc, L.-S. Wu, P. Günter, and B. Zysset, “Refractive indices of orthorhombic KNbO3. II. Phase matching configurations for nonlinear-optical interactions,” J. Opt. Soc. Am. B 9, 507–517 (1992).
    [CrossRef]
  18. I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Ab-solute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B 14, 2268–2294 (1997).
    [CrossRef]
  19. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
    [CrossRef]
  20. S. M. Saltiel, K. Koynov, B. Agate, and W. Sibbett, “Second harmonic generation with focused beams under conditions of large group velocity mismatch,” presented at the Eleventh Conference on Laser Optics (LO’2003), St. Petersburg, Russia, June 30–July 04, 2003, Poster WeR1–39p.

2003 (2)

2002 (2)

B. Agate, A. J. Kemp, C. T. A. Brown, and W. Sibbett, “Efficient, high repetition-rate femtosecond blue source using a compact Cr:LiSAF laser,” Opt. Express 10, 824–831 (2002).
[CrossRef] [PubMed]

B. Agate, B. Stormont, A. J. Kemp, C. T. A. Brown, U. Keller, and W. Sibbett, “Simplified cavity designs for efficient and compact femtosecond Cr:LiSAF lasers,” Opt. Commun. 205, 207–213 (2002).
[CrossRef]

2000 (2)

D. Guzun, Y. Q. Li, and M. Xiao, “Blue light generation in single-pass frequency doubling of femtosecond pulses in KNbO3,” Opt. Commun. 180, 367–371 (2000).
[CrossRef]

V. Magni, “Optimum beams for efficient frequency mixing in crystals with second order nonlinearity,” Opt. Commun. 184, 245–255 (2000).
[CrossRef]

1999 (1)

1998 (1)

1997 (1)

1995 (1)

1994 (1)

1992 (2)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

I. Biaggio, P. Kerkoc, L.-S. Wu, P. Günter, and B. Zysset, “Refractive indices of orthorhombic KNbO3. II. Phase matching configurations for nonlinear-optical interactions,” J. Opt. Soc. Am. B 9, 507–517 (1992).
[CrossRef]

1987 (1)

1968 (1)

G. D. Boyd and D. A. Kleinman, “Parametric interactions of focused Gaussian light beams,” J. Appl. Phys. 39, 3897–3641 (1968).
[CrossRef]

Agate, B.

B. Agate, B. Stormont, A. J. Kemp, C. T. A. Brown, U. Keller, and W. Sibbett, “Simplified cavity designs for efficient and compact femtosecond Cr:LiSAF lasers,” Opt. Commun. 205, 207–213 (2002).
[CrossRef]

B. Agate, A. J. Kemp, C. T. A. Brown, and W. Sibbett, “Efficient, high repetition-rate femtosecond blue source using a compact Cr:LiSAF laser,” Opt. Express 10, 824–831 (2002).
[CrossRef] [PubMed]

Biaggio, I.

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, “Parametric interactions of focused Gaussian light beams,” J. Appl. Phys. 39, 3897–3641 (1968).
[CrossRef]

Brown, C. T. A.

B. Agate, B. Stormont, A. J. Kemp, C. T. A. Brown, U. Keller, and W. Sibbett, “Simplified cavity designs for efficient and compact femtosecond Cr:LiSAF lasers,” Opt. Commun. 205, 207–213 (2002).
[CrossRef]

B. Agate, A. J. Kemp, C. T. A. Brown, and W. Sibbett, “Efficient, high repetition-rate femtosecond blue source using a compact Cr:LiSAF laser,” Opt. Express 10, 824–831 (2002).
[CrossRef] [PubMed]

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Cockings, O. R.

Dienes, A.

Fejer, M. M.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Günter, P.

Guzun, D.

Y. Q. Li, D. Guzun, G. Salamo, and M. Xiao, “High-efficiency blue-light generation by frequency doubling of picosecond pulses in a thick KNbO3 crystal,” J. Opt. Soc. Am. B 20, 1285–1289 (2003).
[CrossRef]

D. Guzun, Y. Q. Li, and M. Xiao, “Blue light generation in single-pass frequency doubling of femtosecond pulses in KNbO3,” Opt. Commun. 180, 367–371 (2000).
[CrossRef]

Ito, R.

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Kan’an, A. M.

Keller, U.

B. Agate, B. Stormont, A. J. Kemp, C. T. A. Brown, U. Keller, and W. Sibbett, “Simplified cavity designs for efficient and compact femtosecond Cr:LiSAF lasers,” Opt. Commun. 205, 207–213 (2002).
[CrossRef]

Kemp, A. J.

B. Agate, B. Stormont, A. J. Kemp, C. T. A. Brown, U. Keller, and W. Sibbett, “Simplified cavity designs for efficient and compact femtosecond Cr:LiSAF lasers,” Opt. Commun. 205, 207–213 (2002).
[CrossRef]

B. Agate, A. J. Kemp, C. T. A. Brown, and W. Sibbett, “Efficient, high repetition-rate femtosecond blue source using a compact Cr:LiSAF laser,” Opt. Express 10, 824–831 (2002).
[CrossRef] [PubMed]

Kerkoc, P.

Kitamoto, A.

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, “Parametric interactions of focused Gaussian light beams,” J. Appl. Phys. 39, 3897–3641 (1968).
[CrossRef]

Knoesen, A.

Kondo, T.

Leaird, D. E.

Li, Y. Q.

Y. Q. Li, D. Guzun, G. Salamo, and M. Xiao, “High-efficiency blue-light generation by frequency doubling of picosecond pulses in a thick KNbO3 crystal,” J. Opt. Soc. Am. B 20, 1285–1289 (2003).
[CrossRef]

D. Guzun, Y. Q. Li, and M. Xiao, “Blue light generation in single-pass frequency doubling of femtosecond pulses in KNbO3,” Opt. Commun. 180, 367–371 (2000).
[CrossRef]

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Magni, V.

V. Magni, “Optimum beams for efficient frequency mixing in crystals with second order nonlinearity,” Opt. Commun. 184, 245–255 (2000).
[CrossRef]

Manassah, J. T.

Ming, N. B.

Ren, T. W.

Salamo, G.

Shirane, M.

Shoji, I.

Sibbett, W.

B. Agate, A. J. Kemp, C. T. A. Brown, and W. Sibbett, “Efficient, high repetition-rate femtosecond blue source using a compact Cr:LiSAF laser,” Opt. Express 10, 824–831 (2002).
[CrossRef] [PubMed]

B. Agate, B. Stormont, A. J. Kemp, C. T. A. Brown, U. Keller, and W. Sibbett, “Simplified cavity designs for efficient and compact femtosecond Cr:LiSAF lasers,” Opt. Commun. 205, 207–213 (2002).
[CrossRef]

Sidick, E.

Stormont, B.

B. Agate, B. Stormont, A. J. Kemp, C. T. A. Brown, U. Keller, and W. Sibbett, “Simplified cavity designs for efficient and compact femtosecond Cr:LiSAF lasers,” Opt. Commun. 205, 207–213 (2002).
[CrossRef]

Wang, Y. H.

Weiner, A. M.

Wu, L.-S.

Xiao, M.

Y. Q. Li, D. Guzun, G. Salamo, and M. Xiao, “High-efficiency blue-light generation by frequency doubling of picosecond pulses in a thick KNbO3 crystal,” J. Opt. Soc. Am. B 20, 1285–1289 (2003).
[CrossRef]

D. Guzun, Y. Q. Li, and M. Xiao, “Blue light generation in single-pass frequency doubling of femtosecond pulses in KNbO3,” Opt. Commun. 180, 367–371 (2000).
[CrossRef]

Xu, G. D.

Yu, S.

Zhu, S. N.

Zhu, Y. Y.

Zysset, B.

IEEE J. Quantum Electron. (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

J. Appl. Phys. (1)

G. D. Boyd and D. A. Kleinman, “Parametric interactions of focused Gaussian light beams,” J. Appl. Phys. 39, 3897–3641 (1968).
[CrossRef]

J. Opt. Soc. Am. B (6)

Opt. Commun. (3)

V. Magni, “Optimum beams for efficient frequency mixing in crystals with second order nonlinearity,” Opt. Commun. 184, 245–255 (2000).
[CrossRef]

D. Guzun, Y. Q. Li, and M. Xiao, “Blue light generation in single-pass frequency doubling of femtosecond pulses in KNbO3,” Opt. Commun. 180, 367–371 (2000).
[CrossRef]

B. Agate, B. Stormont, A. J. Kemp, C. T. A. Brown, U. Keller, and W. Sibbett, “Simplified cavity designs for efficient and compact femtosecond Cr:LiSAF lasers,” Opt. Commun. 205, 207–213 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Other (5)

G. G. Gurzadian, V. G. Dmitriev, and D. N. Nikogosian, Handbook of Nonlinear Optical Crystals, 3rd ed., Vol. 64 of Springer Series in Optical Sciences (Springer-Verlag, New York, 1999).

S. M. Saltiel, K. Koynov, B. Agate, and W. Sibbett, “Second harmonic generation with focused beams under conditions of large group velocity mismatch,” presented at the Eleventh Conference on Laser Optics (LO’2003), St. Petersburg, Russia, June 30–July 04, 2003, Poster WeR1–39p.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin Optics of Femtosecond Laser Pulses (American Institute of Physics, New York, 1992), Chap. 3, p. 141.

H. Wang and A. M. Weiner, “Second harmonic generation efficiency with simultaneous temporal walkoff, spatial walkoff, and depletion,” Conference on Lasers and Electro-Optics, Vol. 89 of Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2003), paper CThU1.

R. W. Boyd, Nonlinear Optics (Academic, New York, 1992) Chap. II, p. 94.

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Figures (7)

Fig. 1
Fig. 1

(a) Value of transient focusing factor, htr, as a function of normalized phase mismatch, ΔkL, for different focusing strengths of m=L/b. The temporal walk-off parameter L/Lnst=30. (b) The optimal values of normalized phase mismatch, ΔkLopt, as a function of m for different values of L/Lnst. For both (a) and (b), the fundamental focal spot is in the center of the nonlinear media μ=0.

Fig. 2
Fig. 2

Dependence of the transient focusing factor, htr, and optimal focusing, mopt=(L/b)opt, on the position, μ, of the fundamental focal spot inside the nonlinear media. Data are calculated for ΔkL=ΔkLopt and L/Lnst=30.

Fig. 3
Fig. 3

(a) Dependence of the transient focusing factor, htr, on focusing strength, m, for different values of the temporal walk-off parameter, L/Lnst. Each curve is normalized to its own maximum; the absolute values of each maxima are shown in brackets. (b) Dependence of optimal focusing strength, mopt=(L/b)opt, on L/Lnst. Focal position lies in the center of the nonlinear medium (μ=0) and ΔkL=(ΔkL)opt.

Fig. 4
Fig. 4

Transient focusing factor, htr, as a function of the temporal walk-off parameter, L/Lnst, for different focusing strengths, m=L/b. Focal position μ=0 and ΔkL=ΔkLopt.

Fig. 5
Fig. 5

(a) Ratio of the output SH pulse duration to the fundamental pulse duration as a function of the focusing strength, m=L/b, and different values of L/Lnst. (b) Temporal profiles and relative intensities of the generated SH pulses calculated for several values of m at L/Lnst=30. The fundamental pulse, T(t/τ), is shown for comparison. The SH pulses correspond to (75/4m)|Htr(t/τ)|2. Focal position μ=0 and ΔkL=(ΔkL)opt.

Fig. 6
Fig. 6

(a) Advantage of using a “thick” crystal with respect to a “thin” crystal. Left scale: Relative increase in SHG efficiency, ηSH/ηSH,L=Lnst, as a function of increasing crystal thickness, L/Lnst. A typical “thin” crystal is defined as L=Lnst. Right scale: Relative increase in SH pulse duration, τSH/τSH,L=Lnst, as a function of increasing L/Lnst. Data are calculated at ΔkL=(ΔkL)opt and L/b=(L/b)opt. The values of (L/b)opt are shown in Fig. 3(b). (b) Dependence of the quantity htr/τSH on focusing strength, L/b, for different values of L/Lnst. (See the text for more details). Each curve is normalized to its own maximum; the absolute values of the maxima are shown in brackets. Focal position μ=0 and ΔkL=(ΔkL)opt.

Fig. 7
Fig. 7

(a) Experimental values of SHG efficiency (data points), and the predictions of our model (gray curve) and another model3,4 (black curve) for L/Lnst=30, as a function of the focusing strength m=L/b. The two theoretical curves are normalized to their values for L/b=10, which was the observed optimal focusing strength in the experiment.2 The maximum experimental point corresponds to a SHG efficiency of 30%. (b) Experimentally measured evolution of the SH pulse spectral width relative to the fundamental spectral width (data points), and the theoretical prediction of our model calculated for L/Lnst=30, as a function of the ratio b/L. The curves that represent our model are for focal position μ=0 and ΔkL=(ΔkL)opt.

Tables (1)

Tables Icon

Table 1 Comparison of the Predictions of Our Model with the Experiment in KNbO3, 2 and the Predictions of Other Models3,4 a

Equations (23)

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z+i2k1Δ+1v1 tA1=0,
z+i2k2Δ+1v2 tA2=-iσ2A12 exp(iΔkz),
2x2+2y2.
A1(x, y, z, t)=A(z, t)g1(x, y, u1),
g1(x, y, u1)=11-iu1 exp-x2+y2w012(1-iu1),
u1=2z/b1.
A2(x, y z, t)=S(z, t)g2(x, y, u2),
g2(x, y, u2)=11-iu2 exp-x2+y2w022(1-iu2),
u2=2z/b2.
u2=u1=u=2z/b.
gjz+i2k1Δgj0(j=1, 2),
A(z, t)z+1v1 A(z, t)tA=0,
S(z, t)z+1v2 S(z, t)t=-iσ2A(z, t)2 (g1)2g2 exp(iΔkz).
S(z, q)z+α S(z, q)q=-iσ21-iuA(q)2 exp(iνu),
ν=Δkb2=ΔkL2m,m=Lb.
S(L, p)=-iσ2A02bHtr(m, μ,ν, γ, p),
Htr(m, μ, ν, γ, p)=12 -m(1+μ)m(1-μ) du1-iuT×Tpτ+γu2 exp(iνu),
γ=αb2τ=12m LLnst,
S(L, t)=-iσ2A02L-1/21/2Tt-z/v2+αLxτ2×exp(iΔkLx)dx,
Wsh(L)=18πw012(c0n2)σ22b2|A0|4τ×-+|Htr(m, μ,ν, γ,p)|2dp.
WshWfund=Khtr(m, μ, ν, γ),
K=4σ22n2Lλ1c0n1 Wfund3τ=16π2deff2WfundL3λ13c0n2n1αLnst,
htr(m, μ, ν, γ)=34m -+|Htr(m, μ, ν, γ, p)|2dp.

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