Abstract

A numerical study of second-harmonic generation based on direct solutions of full-wave vector Maxwell equations, conducted by use of a finite-difference time-domain scheme, is reported. Although nonlinear problems have already been solved by finite-difference time-domain schemes, this is the first finite-difference time-domain computation of second-harmonic generation in a nonlinear crystal with a complete description of the electric field. Only spatially plane waves are considered, but the three components of the electric field are taken into account. The advantages and the drawbacks of this new approach are shown: On the one hand, all the spectral components of the waves are computed, but, on the other, the phase mismatch, which is imposed rather than computed, requires the use of a fine temporal mesh. The numerical results obtained for second-harmonic generation of femtosecond pulses in a thin KDP crystal are compared with those obtained by solution of the nonlinear Schrödinger equations. They illustrate the advantages of this method.

© 2000 Optical Society of America

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References

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  1. Y. Combes, “Méthodes numériques pour la résolution des systèmes quasi-linéaires en électromagnétisme,” Ph.D. dissertation (Université Bordeaux 1, Talence, France, 1996).
  2. Ph. Donnat, “Quelques contributions mathématiques en optique nonlinéaire,” Ph.D. dissertation (Ecole Polytechnique, Palaiseau, France, 1994).
  3. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).
  4. P. M. Goorjian and A. Taflove, “Direct time integration of Maxwell’s equations in nonlinear dispersive media for propagation and scattering of femtosecond electromagnetic solitons,” Opt. Lett. 17, 180–182 (1992).
    [CrossRef]
  5. R. M. Joseph and A. Taflove, “FDTD Maxwell’s equations models for nonlinear electrodynamics and optics,” IEEE Trans. Antennas Propag. 45, 364–374 (1997).
    [CrossRef]
  6. C. V. Hile and W. L. Kath, “Numerical solutions of Maxwell’s equations for nonlinear-optical pulse propagation,” J. Opt. Soc. Am. B 13, 1135–1145 (1996).
    [CrossRef]
  7. R. W. Ziolkowski and J. B. Judkins, “Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time,” J. Opt. Soc. Am. B 10, 186–198 (1993).
    [CrossRef]
  8. H. J. Bakker, P. C. M. Planken, and H. G. Muller, “Numerical calculation of optical frequency-conversion processes: a new approach,” J. Opt. Soc. Am. B 6, 1665–1672 (1989).
    [CrossRef]
  9. R. Maleck Rassoul, A. Ivanov, E. Freysz, A. Ducasse, and F. Hache, “Second-harmonic generation under phase-velocity and group-velocity mismatch: influence of cascading self-phase and cross-phase modulations,” Opt. Lett. 22, 268–270 (1997).
    [CrossRef] [PubMed]
  10. N. C. Kothari and X. Carlotti, “Transient second-harmonic generation: influence of effective group-velocity dispersion,” J. Opt. Soc. Am. B 5, 756–764 (1988).
    [CrossRef]
  11. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).
  12. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Heidelberg, 1997).
  13. R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
    [CrossRef]
  14. R. M. Joseph, S. C. Hagness, and A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett. 16, 1412–1414 (1991).
    [CrossRef] [PubMed]
  15. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
    [CrossRef]
  16. F. Zernike, Jr., “Refractive indices of ammonium dihydrogen phosphate and potassium dihydrogen phosphate between 2000 Å and 1.5 μm,” J. Opt. Soc. Am. 54, 1215–1220 (1964).
    [CrossRef]
  17. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode locking pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
    [CrossRef]

1997 (2)

1996 (2)

C. V. Hile and W. L. Kath, “Numerical solutions of Maxwell’s equations for nonlinear-optical pulse propagation,” J. Opt. Soc. Am. B 13, 1135–1145 (1996).
[CrossRef]

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode locking pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

1993 (1)

1992 (1)

1991 (1)

1990 (1)

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[CrossRef]

1989 (1)

1988 (1)

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

1964 (1)

1961 (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Bakker, H. J.

Carlotti, X.

Ducasse, A.

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Freysz, E.

Goorjian, P. M.

Hache, F.

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode locking pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Hagness, S. C.

Hile, C. V.

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Hunsberger, F. P.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[CrossRef]

Ivanov, A.

Joseph, R. M.

Judkins, J. B.

Kath, W. L.

Kothari, N. C.

Kunz, K. S.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[CrossRef]

Luebbers, R.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[CrossRef]

Muller, H. G.

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Planken, P. C. M.

Rassoul, R. Maleck

Schneider, M.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[CrossRef]

Standler, R. B.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode locking pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Taflove, A.

Torner, L.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode locking pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Weinreich, G.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Zernike Jr., F.

Ziolkowski, R. W.

IEEE Trans. Antennas Propag. (2)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

R. M. Joseph and A. Taflove, “FDTD Maxwell’s equations models for nonlinear electrodynamics and optics,” IEEE Trans. Antennas Propag. 45, 364–374 (1997).
[CrossRef]

IEEE Trans. Electromagn. Compat. (1)

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (4)

Opt. Lett. (3)

Opt. Quantum Electron. (1)

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode locking pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Phys. Rev. Lett. (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Other (4)

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Heidelberg, 1997).

Y. Combes, “Méthodes numériques pour la résolution des systèmes quasi-linéaires en électromagnétisme,” Ph.D. dissertation (Université Bordeaux 1, Talence, France, 1996).

Ph. Donnat, “Quelques contributions mathématiques en optique nonlinéaire,” Ph.D. dissertation (Ecole Polytechnique, Palaiseau, France, 1994).

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Figures (5)

Fig. 1
Fig. 1

Electric-field intensity for the second-harmonic pulse inside the 50-µm thick type I KDP crystal in a (cw, E=1×105 V/m) configuration without phase matching.

Fig. 2
Fig. 2

(a)–(c) Three components of the electric field generated by a 10-fs and 4×1010 V/m fundamental pulse in a 25-µm thick type I KDP crystal. (d) Zoom of the temporal evolution of the three spatial pulse components on a small time scale.

Fig. 3
Fig. 3

Comparison of the Maxwell and the NLS solutions for the spatial Ex and Ey harmonic electric-field components generated by a 10-fs and 4×1010 V/m fundamental pulse propagating along the x axis in a 25-µm-thick type I KDP crystal. Maxw2 (solid curve): solution of the Maxwell equation along the Y axis; Maxw1 (long–short dashed curve): solution of the Maxwell equation along the x axis; SNL1 (short-dashed curve): solution of the NLS equation for the pulse at the fundamental frequency ω; SNL2 (long-dashed curve): solution of the NLS equation for the pulse at the second-harmonic frequency 2ω.

Fig. 4
Fig. 4

Second-harmonic [(a), (c)] and fundamental [(b), (d)] fields generated in a 100-µm-thick type I KDP crystal by a 10 fs and 4×1010 V/m fundamental pulse after propagation in a 100-µm-thick type I KDP crystal. Solutions labeled Maxwell and NLS were computed by the full-wave Maxwell equations and the NLS equations, respectively.

Fig. 5
Fig. 5

(a) Second-harmonic pulse generated in a 100-µm-thick type I KDP crystal by 3 fs and 7.5×108 V/m fundamental pulses and computed by solution of the full-wave Maxwell equations (dashed curve) and by the NLS (solid curve) equations. (b) Fourier transforms of the second-harmonic pulses, computed by the full-wave Maxwell equations (solid curve) and by the NLS (dashed curve) equations.

Tables (1)

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Table 1 Computed Coherence Lengths

Equations (65)

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Pi1=ijEj,
Pi1=iiEi
Pi1=_ijEj,
¯xx=XX(cos θ)2+ZZ(sin θ)2,¯xy=0,
¯xz=(XX-ZZ)sin θ cos θ,¯yx=0,
¯yy=YY,¯yz=0,¯zx=(XX-ZZ)sin θ cos θ,
¯zy=0,¯zz=XX(sin θ)2+ZZ(cos θ)2.
Dx=¯xxEx+¯xzEz=¯xx-¯xz¯zx¯zzEx.
XX=YY=1.479715+0.7795605611-ω20.27452372×1033+13.005221-ω20.8882644×1028,
ZZ=1.42934875+0.7033192541-ω20.28931235×1033+3.22799241-ω20.8882644×1028,
P=E+F+G,
H+1ωH2δ2Hδ2t=αHE,
H(t)=αHωH-tE(τ)sin[ωH(t-τ)]δτ.
H(t+Δt)=cos(ωHΔt)H(t)+sin(ωHΔt)HC(t)+αHωHtt+ΔtE(τ)sin[ωH(t+Δt-τ)]δτ,
Pi(2)=χijkEjEk.
χijk=χikj.
PX(2)=2χXYZEYEZ,
PY(2)=2χXYZEXEZ,
PZ(2)=2χZXYEXEY,
χijk=2dijk.
χXYZ=χZXY.
ξxxx=-(4χXYZ+2χZXY)sin θ(cos θ)2 sin ϕ cos ϕ,
ξxxy=-(χXYZ+χZXY)sin θ cos θ[(cos ϕ)2-(sin ϕ)2],
ξxxz=[2χXYZ(cos θ)3-2(χXYZ+χZXY)×(sin θ)2 cos θ]sin ϕ cos ϕ,
ξxyy=2χZXY sin θ sin ϕ cos ϕ,
ξxyz=[χXYZ(cos θ)2-χZXY(sin θ)2][(cos ϕ)2-(sin ϕ)2],
ξxzz=[4χXYZ sin θ(cos θ)2-2χZXY(sin θ)3]sin ϕ cos ϕ,
ξyxx=-2χXYZ sin θ cos θ[(cos ϕ)2-(sin ϕ)2],
ξyxy=2χXYZ sin θ sin ϕ cos ϕ,
ξyxz=χXYZ[(cos θ)2-(sin θ)2][(cos ϕ)2-(sin ϕ)2],
ξyyy=0,
ξyyz=-2χXYZ cos θ sin ϕ cos ϕ,
ξyzz=2χXYZ sin θ cos θ[(cos ϕ)2-(sin ϕ)2],
ξzxx=-[4χXYZ(sin θ)2 cos θ-2χZXY(cos θ)3]sin ϕ cos ϕ,
ξzxy=[χZXY(cos θ)2-χXYZ(sin θ)2][(cos ϕ)2-(sin ϕ)2],
ξzxz=[2(χXYZ+χZXY)sin θ(cos θ)2-2χXYZ(sin θ)3]sin ϕ cos ϕ,
ξzyy=-2χZXY cos θ sin ϕ cos ϕ,
ξzyz=(χXYZ+χZXY)sin θ cos θ[(cos ϕ)2-(sin ϕ)2],
ξzzz=(4χXYZ+2χZXY)(sin θ)2 cos θ sin ϕ cos ϕ.
Ez=-¯zx¯zzEx-ξzyy¯zzEy2(typeI),
Ez=ξzxyEy-¯zx¯zz-ξzyzEyEx(typeII).
Dy=YYEy+2ξyxyExEy+2ξyyzEzEy,
χeff2=χZXY sin θ sin(2ϕ) ¯xxZZ,
Dy=YYEy+2ξyxxExEx+2ξyxzExEz+2ξyzzEzEz,
χeff2=χZXY sin 2θ cos(2ϕ) ¯xx2XXZZ.
Dy=YYEy+2χeff2ExEy-2χZXYχXYZ(sin 2ϕ cos θ)2Ey3 ¯xxXXZZ.
δBδt=-rot E,δDδt=rot H.
Hyi+(1/2)n+(1/2)=Hyi+(1/2)n-(1/2)-ΔtµΔz(Exi+1n-Exin),
Dxin+1=Dxin-ΔtΔz[Hyi+(1/2)n+(1/2)-Hyi-(1/2)n+(1/2)],
D=Dlinear+Dnonlinear=P(1)(Ex, Ey, Ez)+P(2)(Ex, Ey, Ez).
DXin+1=XXEXin+1+FXin+1+GXin+1+2χXYZEYin+1EZin+1,
FXin+1=[FXin(2-ωFX2Δt2)-FXin-1+Δt2αFXEXin],
GXin+1=[GXin(2-ωGX2Δt2)-GXin-1+Δt2αGXEXin]
FXin+1=cos(ωFXΔt)FXin+sin(ωFXΔt)[(FXiC n+(Δt/2)αFXωFXEXin)],
GXin+1=cos(ωGXΔt)GXin+sin(ωGXΔt)[GXiC n+(Δt/2)αGXωGXEXin],
FXiC n+1=cos(ωFXΔt)FXiC n-sin(ωFXΔt)FXin+(Δt/2)αFXωFX[EXin cos(ωFXΔt)+EXin+1],
GXiC n+1=cos(ωGXΔt)GXiC n-sin(ωGXΔt)GXin+(Δt/2)αGXωGX[EXin cos(ωGXΔt)+EXin+1]
Pi2(Ein+1,k+δE)=Pi2(Ein+1,k)+ΛikδE,
Ein+1,k+1=Ein+1,k+(+Λik)-1Din+1-Pi2(Ein+1,k)+Ein+1,k+iFin+1+iGin+1,
Step1:EnHn+1/2Step2:Hn+1/2Dn+1Maxwellequations;
Step3:En,FnFn+1En,GnGn+1ODEortimeintegral;
Step4:Dn+1,Fn+1,Gn+1En+1
Constitutiverelation.
δA1δz+1ug1δA1δt=j ω2n1c2χeff2A2A1* exp(-jΔkz),
δA2δz+1ug2δA2δt=j ω2n2c2χeff2A22 exp(jΔkz),

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