Abstract

We theoretically studied temporal properties of the second-harmonic generation of a chirped short pulse. Transient coupling-wave equations are used in describing the time-dependent nonlinear process. The pulse distortion of a fundamental wave occurs at the leading or trailing edge, which is determined by the difference of the group velocity between fundamental and harmonic waves. Pulse chirp and phase mismatch disturb the pulse shape and cause phase distortion. Even in exact phase matching, phase distortion is expected because of pulse chirp. Both distortion of pulse shape and phase shift are dependent on the sign and the quantity of the chirp.

© 1998 Optical Society of America

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References

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  1. J. J. Bakker, P. C. Planken, L. Kuipers, and A. Lagendijk, “Phase modulation in second-order nonlinear-optical processes,” Phys. Rev. A 42, 4085–4101 (1990).
    [CrossRef] [PubMed]
  2. P. C. M. Planken, H. J. Bakker, L. Kuipers, and A. Lagendijk, “Frequency chirp in optical parametric amplification with large phase mismatch in noncentrosymmetric crystals,” J. Opt. Soc. Am. B 7, 2150–2154 (1990).
    [CrossRef]
  3. R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992).
    [CrossRef] [PubMed]
  4. A. L. Belostotsky, A. S. Leonov, and A. V. Meleshko, “Nonlinear phase change in type II second-harmonic generation under exact phase-matched conditions,” Opt. Lett. 19, 856–858 (1994).
    [CrossRef] [PubMed]
  5. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  6. Tso Yee Fan, C. E. Huang, B. Q. Hu, R. C. Eckardt, Y. X. Fan, R. L. Byer, and R. S. Feigelson, “Second harmonic generation and accurate index of refraction measurements in flux-grown KTiOPO4,” Appl. Opt. 26, 2390–2394 (1987).
    [CrossRef] [PubMed]
  7. A. Yariv and P. Yeh, Optical Waves in Crystal: Propagation and Control of Laser Radiation (Wiley, Toronto), p. 526.
  8. P. D. Lax, “Weak solutions of nonlinear hyperbolic equations and their numerical computation,” Commun. Pure Appl. Math. 7, 159–193 (1954).
    [CrossRef]
  9. W. H. Knox, R. L. Fork, M. C. Downer, R. H. Stolen, C. V. Shank, and J. A. Valdmanis, “Optical pulse compression to 8 fs at a 5 kHz repetition rate,” Appl. Phys. Lett. 46, 1120–1121 (1985).
    [CrossRef]
  10. R. L. Fork, C. H. Brito Cruz, and C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compensation,” Opt. Lett. 12, 483–485 (1987).
    [CrossRef] [PubMed]

1994 (1)

1992 (1)

1990 (2)

1987 (2)

1985 (1)

W. H. Knox, R. L. Fork, M. C. Downer, R. H. Stolen, C. V. Shank, and J. A. Valdmanis, “Optical pulse compression to 8 fs at a 5 kHz repetition rate,” Appl. Phys. Lett. 46, 1120–1121 (1985).
[CrossRef]

1954 (1)

P. D. Lax, “Weak solutions of nonlinear hyperbolic equations and their numerical computation,” Commun. Pure Appl. Math. 7, 159–193 (1954).
[CrossRef]

Bakker, H. J.

Bakker, J. J.

J. J. Bakker, P. C. Planken, L. Kuipers, and A. Lagendijk, “Phase modulation in second-order nonlinear-optical processes,” Phys. Rev. A 42, 4085–4101 (1990).
[CrossRef] [PubMed]

Belostotsky, A. L.

Brito Cruz, C. H.

Byer, R. L.

DeSalvo, R.

Downer, M. C.

W. H. Knox, R. L. Fork, M. C. Downer, R. H. Stolen, C. V. Shank, and J. A. Valdmanis, “Optical pulse compression to 8 fs at a 5 kHz repetition rate,” Appl. Phys. Lett. 46, 1120–1121 (1985).
[CrossRef]

Eckardt, R. C.

Fan, Tso Yee

Fan, Y. X.

Feigelson, R. S.

Fork, R. L.

R. L. Fork, C. H. Brito Cruz, and C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compensation,” Opt. Lett. 12, 483–485 (1987).
[CrossRef] [PubMed]

W. H. Knox, R. L. Fork, M. C. Downer, R. H. Stolen, C. V. Shank, and J. A. Valdmanis, “Optical pulse compression to 8 fs at a 5 kHz repetition rate,” Appl. Phys. Lett. 46, 1120–1121 (1985).
[CrossRef]

Hagan, D. J.

Hu, B. Q.

Huang, C. E.

Knox, W. H.

W. H. Knox, R. L. Fork, M. C. Downer, R. H. Stolen, C. V. Shank, and J. A. Valdmanis, “Optical pulse compression to 8 fs at a 5 kHz repetition rate,” Appl. Phys. Lett. 46, 1120–1121 (1985).
[CrossRef]

Kuipers, L.

Lagendijk, A.

Lax, P. D.

P. D. Lax, “Weak solutions of nonlinear hyperbolic equations and their numerical computation,” Commun. Pure Appl. Math. 7, 159–193 (1954).
[CrossRef]

Leonov, A. S.

Meleshko, A. V.

Planken, P. C.

J. J. Bakker, P. C. Planken, L. Kuipers, and A. Lagendijk, “Phase modulation in second-order nonlinear-optical processes,” Phys. Rev. A 42, 4085–4101 (1990).
[CrossRef] [PubMed]

Planken, P. C. M.

Shank, C. V.

R. L. Fork, C. H. Brito Cruz, and C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compensation,” Opt. Lett. 12, 483–485 (1987).
[CrossRef] [PubMed]

W. H. Knox, R. L. Fork, M. C. Downer, R. H. Stolen, C. V. Shank, and J. A. Valdmanis, “Optical pulse compression to 8 fs at a 5 kHz repetition rate,” Appl. Phys. Lett. 46, 1120–1121 (1985).
[CrossRef]

Sheik-Bahae, M.

Stegeman, G.

Stolen, R. H.

W. H. Knox, R. L. Fork, M. C. Downer, R. H. Stolen, C. V. Shank, and J. A. Valdmanis, “Optical pulse compression to 8 fs at a 5 kHz repetition rate,” Appl. Phys. Lett. 46, 1120–1121 (1985).
[CrossRef]

Valdmanis, J. A.

W. H. Knox, R. L. Fork, M. C. Downer, R. H. Stolen, C. V. Shank, and J. A. Valdmanis, “Optical pulse compression to 8 fs at a 5 kHz repetition rate,” Appl. Phys. Lett. 46, 1120–1121 (1985).
[CrossRef]

Van Stryland, E. W.

Vanherzeele, H.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

W. H. Knox, R. L. Fork, M. C. Downer, R. H. Stolen, C. V. Shank, and J. A. Valdmanis, “Optical pulse compression to 8 fs at a 5 kHz repetition rate,” Appl. Phys. Lett. 46, 1120–1121 (1985).
[CrossRef]

Commun. Pure Appl. Math. (1)

P. D. Lax, “Weak solutions of nonlinear hyperbolic equations and their numerical computation,” Commun. Pure Appl. Math. 7, 159–193 (1954).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Lett. (3)

Phys. Rev. A (1)

J. J. Bakker, P. C. Planken, L. Kuipers, and A. Lagendijk, “Phase modulation in second-order nonlinear-optical processes,” Phys. Rev. A 42, 4085–4101 (1990).
[CrossRef] [PubMed]

Other (2)

A. Yariv and P. Yeh, Optical Waves in Crystal: Propagation and Control of Laser Radiation (Wiley, Toronto), p. 526.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

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Figures (6)

Fig. 1
Fig. 1

Temporal variation of normalized amplitudes at different interaction lengths in KTP. The subsidiary peaks are generated on the leading edge of A1 and the trailing edge of A2. The pulse of the harmonic wave broadens monotonically along the interaction length. I10=I20=1.0 GW/cm2, Δt=1.0 ps, Δk=0.0 cm-1, and b1=b2=0. (a) |A1(z, t)|, (b) |A2(z, t)|, and (c) |A3(z, t)|.

Fig. 2
Fig. 2

Normalized amplitude of A1 at different interaction lengths. (a) b1=b2=1; the subsidiary peak disappears and leaves a prolonged leading edge. (b) b1=1 and b2=-1; the subsidiary peak appears when the incident fundamental pulses are inversely chirped. All the parameters are the same as in Fig. 1 except for the chirps.

Fig. 3
Fig. 3

Normalized amplitude of A1 at different interaction length. (a) Δk=1 cm-1 and b1=b2=1; a plateau is on the leading edge. (b) Δk=1 cm-1 and b1=b2=3; the pulse extends to a regular form. (c) Δk=5 cm-1 and b1=b2=3; the weak subsidiary peak appears. I10=I20=1.0 GW/cm2 and Δt =1.0 ps.

Fig. 4
Fig. 4

Calculated phase change φ1 of A1 at a different interaction length. (a) Δk=0 and b1=b2=1. (b) Δk=0, b1=1 and b2=-1. The phase shift is distorted even in the condition of phase matching. I10=I20=1.0 GW/cm2 and Δt=1.0 ps. The dotted lines indicate the location of the pulse peak at the time axis.

Fig. 5
Fig. 5

Calculated phase change φ3 of A3. (a) Δk=0 and b1=b2=1. (b) Δk=0, b1=1, and b2=-1. The phase shift of the harmonic wave keeps a constant value when the incident fundamental pulses are inversely chirped. The dotted lines indicate the location of the pulse peak at time axis.

Fig. 6
Fig. 6

Calculated phase shift φ1 of A1. (a) Δk=1 cm-1 and b1=b2=1; the phase shift decays to an approximately constant value at z=2 mm. (b) Δk=1 cm-1 and b1=b2=3; the phase shift extends to a step form at z=2 mm. (c) Δk =5 cm-1 and b1=b2=3; the distortion of the phase change is preserved. The dotted lines indicate the location of the pulse peak at time axis.

Tables (2)

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Table 1 Parameters for KTP of Type II Phase Matching

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Table 2 Sellmeier Equation Coefficients for KTP

Equations (18)

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Ei(z, t)=Ei(z, t)exp[j(kiz-ωit)](i=1, 2, 3),
z+1v1 tE1=jωdeff2n1c E2*E3 exp(iΔkz),
z+1v2 tE2=jωdeff2n2c E1*E3 exp(iΔkz),
z+1v3 tE3=j2ωdeff2n3c E1E2 exp(-iΔkz),
deff=12 |χ(2)(2ω; ω, ω)|.
vi(ωi)=dωidki=cni+ωidωidni.
Ii(z, t)=12 0μ01/2ni|Ei(z, t)|2=I0|Ai(z, t)|2(i=1, 2, 3),
Ai=Eini2I0 0μ01/21/2(i=1, 2, 3),
z+1v1 tA1=jΓA2*A3 exp(jΔkz),
z+1v2 tA2=jΓA1*A3 exp(jΔkz),
z+1v3 tA3=j2ΓA1A2 exp(-jΔkz),
Γ=ωc deff(n1n2n3)1/2 2I0μ001/21/2,
Ai(0, t)=Ai0 exp-(1+jbi)tΔt2=Ii0I01/2 exp-(1+jbi)tΔt2(i=1, 2),
A3(0, t)=0,
n2=A+B1-Cλ2-Dλ2,
k2ω(ϕ)=kωϕ+kω,z,
2n2ω(ϕ)=nω(ϕ)+nω,z,
n(ϕ)=cos2(ϕ)ny2+sin2(ϕ)nx21/2.

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