Abstract

We present in this work the scalar potential formulation of second harmonic generation process in χ(2) nonlinear analysis. This approach is intrinsically well suited to the applications of the concept of circuit analysis and synthesis to nonlinear optical problems, and represents a novel alternative method in the analysis of nonlinear optical waveguide, by providing a good convergent numerical solution. The time domain modeling is applied to nonlinear GaAs asymmetrical waveguide with dielectric discontinuities in the hypothesis of quasi phase matching condition in order to evaluate the efficiency conversion of the second harmonic signal. The accuracy of the modeling is validated by the good agreement with the published experimental results. The effective dielectric constant method allows to extend the analysis also to 3D optical waveguides.

© 2008 Optical Society of America

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References

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  1. T. Rozzi and M. Farina, Advanced electromagnetic analysis of passive and active planar structures, (IEE Electromagnetic wave series 46, London. 1999), Chap. 2.
  2. C. G. Someda, Onde elettromagnetiche, (UTET Ed., Torino. 1996), Chap. I.
  3. M. A. Alsunaidi, H. M. Masoudi, and J. M. Arnold, "A time-domain for the analysis of second harmonic generation in nonlinear optical structures," IEEE Photon. Technol. Lett. 12, 395-397 (2000).
    [CrossRef]
  4. A. Massaro and T. Rozzi, "Rigorous time-domain analysis of dielectric optical waveguides using Hertzian potentials formulation," Opt. Express 14, 2027-2036 (2006).
    [CrossRef] [PubMed]
  5. A. Massaro, M. Grande, R. Cingolani, A. Passaseo, and M. De Vittorio, "Design and modeling of tapered waveguide for photonic crystal slab by using time-domain Hertzian potentials formulation," Opt. Express 15, 16484-16489 (2007).
    [CrossRef] [PubMed]
  6. N. Marcuvitz, and J. Schwinger, "On the representation of the electric and magnetic field produced by currents and discontinuities in wave guides," J. Appl. Phys. 22, 806-820 (1951).
    [CrossRef]
  7. N. C. Frateschi, A. Rubens, and B. De Castro, " Perturbation theory for the wave equation and the effective refractive index approach," IEEE J. Quantum Electron. QE-22, 12-15 (1986).
    [CrossRef]
  8. A. Yariv, Quantum Electronics, (John Wiley & Sons, 3rd ed., Canada.1989), Chap. 22.
  9. L. Striscione, M. Centini, C. Sibilia, and M. Bertolotti, "Entangled guided photon generation in (1+1)-dimensional photonic crystals," Phys. Rev. A 74, (2006).
  10. A. Taflove and S. C. Hagness, Computational Electrodynamic: the Finite-difference Time-domain Method, (Arthec House Publishers, sec. ed., London 2000), Chaps. 2, 3, 4, 7.
  11. G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
    [CrossRef]
  12. G. Khanarian, "Theory of design parameters for quasi-phase-matched waveguides and application to frequency doubling in polymer waveguides," IEEE J. Sel. Top. Quantum Electron. 7, 793-805 (2001).
    [CrossRef]
  13. E. U. Rafailov, P. L. Alvarez, C. T. A. Brown, W. Sibbett, R. M. De la Rue, P. Millar, D. A. Yanson, J. S. Roberts, and P. A. Houston, "Second-harmonic generation from a first-order quasi-phase-matched GaAs/AlGaAs waveguide crystal," Opt. Lett. 26, 1984-1986 (2001).
    [CrossRef]
  14. A. Massaro, L. Pierantoni, and T. Rozzi, "Accurate analysis and modelling of laminated multilayered 3-D optical waveguides," IEEE J. Quantum Electron. 40, 1478-1489 (2004).
    [CrossRef]
  15. T. Suhara, and H. Nishihara, "Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped grating," IEEE J. Quantum Electron. 26, 1265-1276 (1990).
    [CrossRef]
  16. A. Yariv, "Coupled-mode theory for guided-wave optics," IEEE J. Quantum Electron. 9, 919-933 (1973).
    [CrossRef]

2007 (1)

2006 (2)

A. Massaro and T. Rozzi, "Rigorous time-domain analysis of dielectric optical waveguides using Hertzian potentials formulation," Opt. Express 14, 2027-2036 (2006).
[CrossRef] [PubMed]

L. Striscione, M. Centini, C. Sibilia, and M. Bertolotti, "Entangled guided photon generation in (1+1)-dimensional photonic crystals," Phys. Rev. A 74, (2006).

2004 (1)

A. Massaro, L. Pierantoni, and T. Rozzi, "Accurate analysis and modelling of laminated multilayered 3-D optical waveguides," IEEE J. Quantum Electron. 40, 1478-1489 (2004).
[CrossRef]

2001 (2)

G. Khanarian, "Theory of design parameters for quasi-phase-matched waveguides and application to frequency doubling in polymer waveguides," IEEE J. Sel. Top. Quantum Electron. 7, 793-805 (2001).
[CrossRef]

E. U. Rafailov, P. L. Alvarez, C. T. A. Brown, W. Sibbett, R. M. De la Rue, P. Millar, D. A. Yanson, J. S. Roberts, and P. A. Houston, "Second-harmonic generation from a first-order quasi-phase-matched GaAs/AlGaAs waveguide crystal," Opt. Lett. 26, 1984-1986 (2001).
[CrossRef]

2000 (1)

M. A. Alsunaidi, H. M. Masoudi, and J. M. Arnold, "A time-domain for the analysis of second harmonic generation in nonlinear optical structures," IEEE Photon. Technol. Lett. 12, 395-397 (2000).
[CrossRef]

1990 (1)

T. Suhara, and H. Nishihara, "Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped grating," IEEE J. Quantum Electron. 26, 1265-1276 (1990).
[CrossRef]

1986 (1)

N. C. Frateschi, A. Rubens, and B. De Castro, " Perturbation theory for the wave equation and the effective refractive index approach," IEEE J. Quantum Electron. QE-22, 12-15 (1986).
[CrossRef]

1981 (1)

G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

1973 (1)

A. Yariv, "Coupled-mode theory for guided-wave optics," IEEE J. Quantum Electron. 9, 919-933 (1973).
[CrossRef]

1951 (1)

N. Marcuvitz, and J. Schwinger, "On the representation of the electric and magnetic field produced by currents and discontinuities in wave guides," J. Appl. Phys. 22, 806-820 (1951).
[CrossRef]

Alsunaidi, M. A.

M. A. Alsunaidi, H. M. Masoudi, and J. M. Arnold, "A time-domain for the analysis of second harmonic generation in nonlinear optical structures," IEEE Photon. Technol. Lett. 12, 395-397 (2000).
[CrossRef]

Alvarez, P. L.

Arnold, J. M.

M. A. Alsunaidi, H. M. Masoudi, and J. M. Arnold, "A time-domain for the analysis of second harmonic generation in nonlinear optical structures," IEEE Photon. Technol. Lett. 12, 395-397 (2000).
[CrossRef]

Bertolotti, M.

L. Striscione, M. Centini, C. Sibilia, and M. Bertolotti, "Entangled guided photon generation in (1+1)-dimensional photonic crystals," Phys. Rev. A 74, (2006).

Brown, C. T. A.

Centini, M.

L. Striscione, M. Centini, C. Sibilia, and M. Bertolotti, "Entangled guided photon generation in (1+1)-dimensional photonic crystals," Phys. Rev. A 74, (2006).

Cingolani, R.

De Castro, B.

N. C. Frateschi, A. Rubens, and B. De Castro, " Perturbation theory for the wave equation and the effective refractive index approach," IEEE J. Quantum Electron. QE-22, 12-15 (1986).
[CrossRef]

De la Rue, R. M.

De Vittorio, M.

Frateschi, N. C.

N. C. Frateschi, A. Rubens, and B. De Castro, " Perturbation theory for the wave equation and the effective refractive index approach," IEEE J. Quantum Electron. QE-22, 12-15 (1986).
[CrossRef]

Grande, M.

Houston, P. A.

Khanarian, G.

G. Khanarian, "Theory of design parameters for quasi-phase-matched waveguides and application to frequency doubling in polymer waveguides," IEEE J. Sel. Top. Quantum Electron. 7, 793-805 (2001).
[CrossRef]

Marcuvitz, N.

N. Marcuvitz, and J. Schwinger, "On the representation of the electric and magnetic field produced by currents and discontinuities in wave guides," J. Appl. Phys. 22, 806-820 (1951).
[CrossRef]

Masoudi, H. M.

M. A. Alsunaidi, H. M. Masoudi, and J. M. Arnold, "A time-domain for the analysis of second harmonic generation in nonlinear optical structures," IEEE Photon. Technol. Lett. 12, 395-397 (2000).
[CrossRef]

Massaro, A.

Millar, P.

Mur, G.

G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

Nishihara, H.

T. Suhara, and H. Nishihara, "Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped grating," IEEE J. Quantum Electron. 26, 1265-1276 (1990).
[CrossRef]

Passaseo, A.

Pierantoni, L.

A. Massaro, L. Pierantoni, and T. Rozzi, "Accurate analysis and modelling of laminated multilayered 3-D optical waveguides," IEEE J. Quantum Electron. 40, 1478-1489 (2004).
[CrossRef]

Rafailov, E. U.

Roberts, J. S.

Rozzi, T.

A. Massaro and T. Rozzi, "Rigorous time-domain analysis of dielectric optical waveguides using Hertzian potentials formulation," Opt. Express 14, 2027-2036 (2006).
[CrossRef] [PubMed]

A. Massaro, L. Pierantoni, and T. Rozzi, "Accurate analysis and modelling of laminated multilayered 3-D optical waveguides," IEEE J. Quantum Electron. 40, 1478-1489 (2004).
[CrossRef]

Rubens, A.

N. C. Frateschi, A. Rubens, and B. De Castro, " Perturbation theory for the wave equation and the effective refractive index approach," IEEE J. Quantum Electron. QE-22, 12-15 (1986).
[CrossRef]

Schwinger, J.

N. Marcuvitz, and J. Schwinger, "On the representation of the electric and magnetic field produced by currents and discontinuities in wave guides," J. Appl. Phys. 22, 806-820 (1951).
[CrossRef]

Sibbett, W.

Sibilia, C.

L. Striscione, M. Centini, C. Sibilia, and M. Bertolotti, "Entangled guided photon generation in (1+1)-dimensional photonic crystals," Phys. Rev. A 74, (2006).

Striscione, L.

L. Striscione, M. Centini, C. Sibilia, and M. Bertolotti, "Entangled guided photon generation in (1+1)-dimensional photonic crystals," Phys. Rev. A 74, (2006).

Suhara, T.

T. Suhara, and H. Nishihara, "Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped grating," IEEE J. Quantum Electron. 26, 1265-1276 (1990).
[CrossRef]

Yanson, D. A.

Yariv, A.

A. Yariv, "Coupled-mode theory for guided-wave optics," IEEE J. Quantum Electron. 9, 919-933 (1973).
[CrossRef]

IEEE J. Quantum Electron. (4)

N. C. Frateschi, A. Rubens, and B. De Castro, " Perturbation theory for the wave equation and the effective refractive index approach," IEEE J. Quantum Electron. QE-22, 12-15 (1986).
[CrossRef]

A. Massaro, L. Pierantoni, and T. Rozzi, "Accurate analysis and modelling of laminated multilayered 3-D optical waveguides," IEEE J. Quantum Electron. 40, 1478-1489 (2004).
[CrossRef]

T. Suhara, and H. Nishihara, "Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped grating," IEEE J. Quantum Electron. 26, 1265-1276 (1990).
[CrossRef]

A. Yariv, "Coupled-mode theory for guided-wave optics," IEEE J. Quantum Electron. 9, 919-933 (1973).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

G. Khanarian, "Theory of design parameters for quasi-phase-matched waveguides and application to frequency doubling in polymer waveguides," IEEE J. Sel. Top. Quantum Electron. 7, 793-805 (2001).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

M. A. Alsunaidi, H. M. Masoudi, and J. M. Arnold, "A time-domain for the analysis of second harmonic generation in nonlinear optical structures," IEEE Photon. Technol. Lett. 12, 395-397 (2000).
[CrossRef]

IEEE Trans. Electromagn. Compat. (1)

G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

J. Appl. Phys. (1)

N. Marcuvitz, and J. Schwinger, "On the representation of the electric and magnetic field produced by currents and discontinuities in wave guides," J. Appl. Phys. 22, 806-820 (1951).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (1)

L. Striscione, M. Centini, C. Sibilia, and M. Bertolotti, "Entangled guided photon generation in (1+1)-dimensional photonic crystals," Phys. Rev. A 74, (2006).

Other (4)

A. Taflove and S. C. Hagness, Computational Electrodynamic: the Finite-difference Time-domain Method, (Arthec House Publishers, sec. ed., London 2000), Chaps. 2, 3, 4, 7.

A. Yariv, Quantum Electronics, (John Wiley & Sons, 3rd ed., Canada.1989), Chap. 22.

T. Rozzi and M. Farina, Advanced electromagnetic analysis of passive and active planar structures, (IEE Electromagnetic wave series 46, London. 1999), Chap. 2.

C. G. Someda, Onde elettromagnetiche, (UTET Ed., Torino. 1996), Chap. I.

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Figures (11)

Fig. 1.
Fig. 1.

(a) Asymmetrical slab waveguide with nonlinear GaAs core, ns1 and nf1 are the effective refractive indexes (SH index and fundamental index respectively) of the region characterized by the thickness D, ns2 and nf2 are the effective refractive indexes (SH index and fundamental index respectively) of the region characterized by the thickness d; (b) and related transmission line model with generators at discontinuous interfaces.

Fig. 2.
Fig. 2.

Block diagram: analysis of SH generation process in discontinuous GaAs waveguide. The testing is related to experimental results of [13].

Fig. 3.
Fig. 3.

Measured and scalar time domain numerical results of a periodic GaAs/Al0 ,4Ga0,6As waveguide. Inset: simulated structure with Lc=2.97 µm, Ls=4.73 µm. The effective indexes of the region I are: ns1=3.321, nf1=3.29; the effective indexes of the region II are ns2=3.1651, nf2=3.161. The time step used in the simulation is Δt=1.6679*10-16 sec. and the spatial step is Δz=5*10-8 m. PSH refers to the SH output power.

Fig. 4.
Fig. 4.

Normalized discrete Fourier transform DFT at the end of the simulated waveguide of: a) fundamental mode Ψf, and b) SH mode Ψs, respectively.

Fig. 5.
Fig. 5.

Time evolution of the normalized fundamental (λ f 0 =1.55 µm) and SH (λ S 0 =0.775 µm) signals generated at the end of grating region in an asymmetrical slab waveguide with nonlinear GaAs core: χ(2)=200 pm/V, d=0.22 µm, D=0.36 µm, Λ=6.021 µm (QPM condition), ns1=3.2313, nf1=3.1052, ns2=3.3187, nf2=3.1895. The total grating length is 5Λ.

Fig. 6.
Fig. 6.

(a) Pump source signals for different T0 values. (b) Fundamental and SH spectrum for different T 0 values at the end of the grating (section a of the inset). The time step used in the simulation is Δt=1.6679*10-16 sec. and the spatial step is Δz=5*10-8 m.

Fig. 7.
Fig. 7.

SH conversion efficiency in the case of uniform GaAs slab waveguide and of QPM GaAs grating and sinusoidal source as pump signal.

Fig. 8.
Fig. 8.

(a) 3D ridge waveguide with nonlinear GaAs core; (b) EDC approach.

Fig. 9.
Fig. 9.

Coupling coefficients C f,S (z) for different values of Lc and LS. C f,S (z)=C1(z) refers to a QPM grating with LC=LS=0.2275µm, C f,S (z)=C2(z) refers to a grating with LC=0.2275µm and LS=0.5 µm, and finally C f,S (z)=C3(z) refers to a grating with LC=0.2275µm and LS=0.1 µm.

Fig. 10.
Fig. 10.

Ridge waveguide (QPM condition): |Af(z)|2,|BS(z)|2, and Cf,S(z) are the fundamental, the SH and the coupling coefficient amplitudes, respectively. The initial value |Af(z=0)|2=1 represents the pump signal.

Fig. 11.
Fig. 11.

SH mode peak development and fundamental mode dip development. The minimum length of coupling efficiency is at z=1,25 µm. The coupling efficiency is obtained with the ideal assumption of no mode-losses.

Equations (46)

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2 Ψ e , h ( x , y , z , t ) μ ε eff 2 Ψ e , h ( x , y , z , t ) t 2 = 0
Z e , h = 1 ε eff e , h μ 0 ε 0
2 Ψ e , h ( x , y , z , t ) μ ε eff 2 Ψ e , h ( x , y , z , t ) t 2 μ 2 P e , h pert ( x , y , z , t ) t 2 = 0
P e , h pert ( x , y , z , t ) = Δ ε ( x , y , z , t ) Ψ e, h ( x , y , z , t )
Δ ε = ε i + 1 ε i i = z longitudinal position
Ψ e , h n + 1 ( i ) = Ψ e , h n ( i + 1 ) ( b a ) + Ψ e , h n ( i ) ( 2 a 2 b a ) + Ψ e , h n 1 ( i ) ( 1 ) + Ψ e , h n ( i 1 ) ( b a )
Ψ e , h n + 1 ( i ) = Ψ e , h n ( i + 1 ) ( b a ) + Ψ e , h n ( i ) ( 2 a 2 b a ) + Ψ e , h n 1 ( i ) ( 1 ) + Ψ e , h n ( i 1 ) ( b a )
a = μ ε ( Δ t ) 2
a = a + μ Δ ε ( Δ t ) 2
b = 1 ( Δ z ) 2
2 Ψ e , h ( x , y , z , t ) μ 0 ε 0 n 2 2 Ψ e , h ( x , y , z , t ) t 2 μ 0 ε 0 2 P e , h NL ( x , y , z , t ) t 2 = 0
2 Ψ e = μ 0 ε 0 n a 2 2 Ψ e t 2 + μ 0 ε 0 χ ( 2 ) ( ω 1 ) 2 ( Ψ h Ψ g ) t 2
2 Ψ h = μ 0 ε 0 n b 2 2 Ψ h t 2 + μ 0 ε 0 χ ( 2 ) ( ω 2 ) 2 ( Ψ e Ψ g ) t 2
2 Ψ g = μ 0 ε 0 n c 2 2 Ψ g t 2 + μ 0 ε 0 χ ( 2 ) ( ω 3 ) 2 ( Ψ e Ψ h ) t 2
2 Ψ e = μ 0 ε 0 n a 2 2 Ψ e t 2 + μ 0 ε 0 χ ( 2 ) ( ω 1 ) ·
· ( Ψ h 2 Ψ g t 2 + Ψ h 2 Ψ h t 2 + 2 Ψ h t Ψ g t )
2 Ψ h = μ 0 ε 0 n b 2 2 Ψ h t 2 + μ 0 ε 0 χ ( 2 ) ( ω 2 ) ·
· ( Ψ e 2 Ψ g t 2 + Ψ h ' 2 Ψ e t 2 + 2 Ψ e t Ψ g t )
2 Ψ g = μ 0 ε 0 n c 2 2 Ψ g t 2 + μ 0 ε 0 χ ( 2 ) ( ω 3 ) ·
· ( Ψ e 2 Ψ h t 2 + Ψ h 2 Ψ e t 2 + 2 Ψ e t Ψ h t )
2 Ψ f = μ 0 ε 0 n f 2 2 Ψ f t 2 + 2 μ 0 ε 0 χ ( 2 ) ·
· ( Ψ f 2 Ψ s t 2 + Ψ s 2 Ψ f t 2 + 2 Ψ f t Ψ s t )
2 Ψ s = μ 0 ε 0 n s 2 2 Ψ s t 2 + 2 μ 0 ε 0 χ ( 2 ) ·
· ( Ψ f 2 Ψ f t 2 + Ψ f t Ψ f t )
Z f = 1 n f μ 0 ε 0 , Z s = 1 n s μ 0 ε 0
Ψ f n ( i + 1 ) 2 Ψ f n ( i ) + Ψ f n ( i 1 ) Δ 2 z = μ 0 ε 0 n f 2 Ψ f n + 1 ( i ) 2 Ψ f n ( i ) + Ψ f n 1 ( i ) Δ 2 t + 2 μ 0 ε 0 χ ( 2 ) ·
· ( Ψ f n ( i ) Ψ s n + 1 ( i ) 2 Ψ s n ( i ) + Ψ s n 1 ( i ) Δ 2 t + Ψ s n ( i ) Ψ f n + 1 ( i ) 2 Ψ f n ( i ) + Ψ f n 1 ( i ) Δ 2 t +
+ 2 · Ψ f n + 1 / 2 ( i ) Ψ f n 1 / 2 ( i ) Δ t · Ψ s n + 1 / 2 ( i ) Ψ s n 1 / 2 ( i ) Δ t )
Ψ s n ( i + 1 ) 2 Ψ s n ( i ) + Ψ s n ( i 1 ) Δ 2 z = μ 0 ε 0 n s 2 Ψ s n + 1 ( i ) 2 Ψ s n ( i ) + Ψ s n 1 ( i ) Δ 2 t + 2 μ 0 ε 0 χ ( 2 ) ·
· ( Ψ f n ( i ) Ψ f n + 1 ( i ) 2 Ψ f n ( i ) + Ψ f n 1 ( i ) Δ 2 t + Ψ f n + 1 / 2 ( i ) Ψ f n 1 / 2 ( i ) Δ t · Ψ f n + 1 / 2 ( i ) Ψ f n 1 / 2 ( i ) Δ t )
Ψ f n + 1 ( i ) · ( μ 0 ε 0 n f 2 Δ 2 z + 2 μ 0 ε 0 χ ( 2 ) Δ 2 z Ψ s n ( i ) ) = Ψ f n ( i ) · [ ( 2 Δ 2 t + 2 μ 0 ε 0 n f 2 Δ 2 z
2 μ 0 ε 0 χ ( 2 ) Δ 2 z · ( Ψ s n + 1 ( i ) 2 Ψ s n ( i ) + Ψ s n 1 ( i ) ) 2 μ 0 ε 0 χ ( 2 ) Δ 2 z Ψ s n ( i ) ] +
+ 4 μ 0 ε 0 χ ( 2 ) Δ 2 z Ψ f n + 1 / 2 ( i ) · ( Ψ s n + 1 / 2 ( i ) + Ψ s n 1 / 2 ( i ) ) +
+ 4 μ 0 ε 0 χ ( 2 ) Δ 2 z Ψ f n 1 / 2 ( i ) · ( Ψ s n + 1 / 2 ( i ) Ψ s n - 1 / 2 ( i ) ) )
Ψ s n + 1 ( i ) · ( μ 0 ε 0 n s 2 Δ 2 z ) = Ψ s n ( i + 1 ) Δ 2 t + Ψ s n ( i ) ( 2 μ 0 ε 0 n s 2 Δ 2 z 2 Δ 2 t ) +
+ Ψ s n ( i 1 ) Δ 2 t Ψ s n 1 ( i ) μ 0 ε 0 n s 2 Δ 2 z Ψ f n ( i ) Ψ f n + 1 ( i ) 2 μ 0 ε 0 χ ( 2 ) Δ 2 z +
+ Ψ f n ( i ) Ψ f n ( i ) 4 μ 0 ε 0 χ ( 2 ) Δ 2 z Ψ f n ( i ) Ψ f n 1 ( i ) 2 μ 0 ε 0 χ ( 2 ) Δ 2 z
Ψ f n + 1 / 2 ( i ) Ψ f n + 1 / 2 ( i ) 2 μ 0 ε 0 χ ( 2 ) Δ 2 z + Ψ f n + 1 / 2 ( i ) Ψ f n 1 / 2 ( i ) 2 μ 0 ε 0 χ ( 2 ) Δ 2 z +
+ Ψ f n 1 / 2 ( i ) Ψ f n + 1 / 2 ( i ) 2 μ 0 ε 0 χ ( 2 ) Δ 2 z Ψ f n 1 / 2 ( i ) Ψ f n 1 / 2 ( i ) 2 μ 0 ε 0 χ ( 2 ) Δ 2 z
L c = π 2 β f β SH = λ 0 f 4 n f n s
2 π Λ = Δ k
Ψ f = exp ( ( t · Δ t / T 0 ) 2 ) · cos ( ω 0 f · t · Δ t )
d A f ( z ) d z = j C f , s ( z ) B s ( z ) A f ( z ) * exp ( j 2 δ z )
d B S ( z ) d z = j C s , f ( z ) [ A f ( z ) ] 2 exp ( j 2 δ z )
C f , s ( z ) = ω 0 f + + Δ ε ( z ) E t f E s t d x d y
Δ ε ( z ) = 1 π l = 1 ( n 1 2 n 3 2 l ) · sin ( 2 l π z L c + L s )

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