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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 18 — Sep. 4, 2006
  • pp: 8305–8310
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General vector auxiliary differential equation finite-difference time-domain method for nonlinear optics

Jethro H. Greene and Allen Taflove  »View Author Affiliations


Optics Express, Vol. 14, Issue 18, pp. 8305-8310 (2006)
http://dx.doi.org/10.1364/OE.14.008305


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Abstract

The auxiliary differential equation finite-difference time-domain method for modeling electromagnetic wave propagation in dispersive nonlinear materials is applied to problems where the electric field is not constrained to a single vector component. A full-vector Maxwell’s equations solution incorporating multiple-pole linear Lorentz, nonlinear Kerr, and nonlinear Raman polarizations is presented. The application is illustrated by modeling a spatial soliton having two orthogonal electric field components. To the best of our knowledge, the numerical technique presented here is the first to model electromagnetic wave propagation with two or three orthogonal vector components in dispersive nonlinear materials. This technique offers the possibility of modeling sub-wavelength interactions of vector spatial solitons.

© 2006 Optical Society of America

1. Introduction

Emerging applications in nanophotonics involve electromagnetic wave interactions with materials having frequency-dependent and intensity-dependent polarizations. Auxiliary differential equation (ADE) methods extend the finite-difference time-domain (FDTD) method to incorporate polarization by time-stepping auxiliary differential equations with Maxwell’s curl equations [1

1 . T. Kashiwa and I. Fukai , “ A treatment by FDTD method of dispersive characteristics associated with electronic polarization ,” Microwave Optics Tech. Lett. 3 , 203 – 205 ( 1990 ). [CrossRef]

, 2

2 . 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]

]. Nakamura et al. validated experimentally an ADE FDTD model of nonlinear femtosecond ultrabroadband-pulse propagation in silica fiber [3

3 . S. Nakamura , Y. Koyamada , N. Yoshida , N. Karasawa , H. Sone , M. Ohtani , Y. Mizuta , R. Morita , H. Shigekawa , and M. Yamashita , “ Finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation for 12 fs laser pulse propagation in a silica fiber ,” IEEE Photonics Technol. Lett. 14 , 480 – 482 ( 2002 ). [CrossRef]

, 4

4 . S. Nakamura , N. Takasawa , and Y. Koyamada , “ Comparison between finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation and experimental results for slightly chirped 12 fs laser pulse propagation in a silica fiber ,” IEEE J. of Lightwave Technol. 23 , 855 – 863 ( 2005 ). [CrossRef]

]. The Fujii ADE FDTD method [5

5 . M. Fujii , M. Tahara , I. Sakagami , W. Freude , and P. Russer , “ High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media ,” IEEE J. Quantum Electron. 40 , 175 – 182 ( 2004 ). [CrossRef]

] is an efficient reformulation which eliminates the need to solve a system of N equations at each time-step, where N is the number of poles of the chromatic dispersion.

In this paper, we provide details of how to apply Fujii’s method to nonlinear optics problems where the electric field has two or three orthogonal vector components. We designate this technique the general vector auxiliary differential equation (GVADE) FDTD method. This method reproduces published results for temporal soliton propagation in a dispersive nonlinear material [5

5 . M. Fujii , M. Tahara , I. Sakagami , W. Freude , and P. Russer , “ High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media ,” IEEE J. Quantum Electron. 40 , 175 – 182 ( 2004 ). [CrossRef]

]. The GVADE FDTD method is then used to model spatial soliton propagation with two orthogonal electric field components in fused silica. We use the same chromatic dispersion, nonlinear Kerr polarization, and nonlinear Raman polarization used by Nakamura et al., which models realistic physical effects in silica.

2. Electromagnetic wave propagation in dispersive nonlinear materials

To derive the GVADE method, we formulate Maxwell’s equations using polarization current:

×E=μ0Ht,
(1)
×H=ε0Et+J,
(2)

where E and H are the electric and magnetic field vectors and J is the polarization current, Pt.. In materials with multiple-pole linear Lorentz polarization, instantaneous Kerr nonlinear polarization, and Raman nonlinear polarization, the polarization current is J = J Lorentz +J Kerr + J Raman. The linear Lorentz polarization models the chromatic dispersion and contains a contribution from multiple resonances:

JLorentz=p=13JLorentzp,
(3)

where each J Lorentzp corresponds to the polarization current due to a single pole of the Sellmeier expansion where the phasor polarization is given by

P˜Lorentzp=ε0χ(1)E˜=ε0βpωp2ωp2ω2E˜,
(4)

where βp and ωp are the strength and frequency, respectively, of the pth resonance [6

6 . G. P. Agrawal , Nonlinear Fiber Optics, 3rd ed . Academic Press, San Diego, CA , 2001 .

].

In general, the third-order nonlinear polarization is

PNLrt=ε0χ(3)tt1tt2tt3Ert1Ert2Ert3dt1dt2dt3,
(5)

where χ (3) is the third-order susceptibility tensor [7

7 . R. W. Boyd , Nonlinear Optics, 2nd ed . Academic Press, San Diego, CA , 2003 .

]. For a simple model of the electron response accounting for nonresonant incoherent (intensity-dependent) nonlinear effects, the third-order nonlinear polarization can be described by the Born-Oppenheimer approximation,

PNL(t)=ε0χ0(3)Eg(tt)E(t)2dt,
(6)

where g(t) is the causal response function and the induced polarization is assumed to lie in the same direction as the electric field [6

6 . G. P. Agrawal , Nonlinear Fiber Optics, 3rd ed . Academic Press, San Diego, CA , 2001 .

]. To model the Kerr and the Raman polarizations,

g(t)=αδ(t)+(1α)gRaman(t),
(7)

where α represents the relative strengths of the Kerr and Raman polarizations, δ(t) is a Dirac delta function that models the instantaneous Kerr nonresonant virtual transitions, and

gRaman(t)=(τ12+τ22τ1τ22)exp(tτ2)sin(tτ1)U(t),
(8)

where g Raman(t) is an approximation of the Raman response function with parameters τ 1 and τ 2 chosen to fit the Raman-gain spectrum [6

6 . G. P. Agrawal , Nonlinear Fiber Optics, 3rd ed . Academic Press, San Diego, CA , 2001 .

], and U(t) is the Heaviside step function. From Eq. (6) and Eq. (7), the polarization and polarization current from the Kerr nonlinearity are

PKerr(t)=ε0χ0(3)Eαδ(tt)E(t)2dt=αε0χ0(3)E2E,
(9)
JKerr(t)=PKerrt=tαε0χ0(3)E2E.
(10)

Writing Eq. (6) as a convolution, the polarization from the Raman effect is

PRaman(t)=ε0E[χRaman(3)(t)*E2],
(11)

where

χRaman(3)(t)=(1α)χRaman(3)(t)gRaman(t).
(12)

3. General vector auxiliary differential equation FDTD method

To implement the Fujii ADE method for the general electric field vector case, we use the formulation of Maxwell’s equations which emphasizes the polarization current, J, instead of the polarization, P, because this eliminates the need to store and update the electric displacement field. The GVADE method can be used for any electric field having two or three orthogonal vector components. However, for a concise derivation, we consider the case where derivatives with respect to the z-coordinate are zero and the electromagnetic field is composed of Ex , Ey , and Hz . Eq. (2) will be solved using FDTD combined with a semi-implicit method in which E n+1 will be updated in terms of H n+1/2 and J n+1/2, where the superscript indicates the time-step.

3.1. Linear Lorentz polarization

Expression of Eq. (4) in the form of an equivalent polarization current phasor gives:

J˜Lorentzp=ε0βpωp2(ωp2ω2)E˜.
(13)

Multiplying both sides of Eq. (13) by (ωp2 -ω 2), and transforming to the time domain, yields

ωp2JLorentzp+2JLorentzpt2=ε0βpωp2Et.
(14)

Finite-differencing Eq. (14) centered at time-step n and solving for J n+1 Lorentzp yields

JLorentzpn+1=αpJLorentzpnJLorentzpn1+γpEn+1En12Δt
(15)

where

αp=2ωp2(Δt);γp=ε0βpωp2(Δt)2.
(16)

Using Eq. (15), and averaging across step n and n+1, gives

JLorentzpn+12=12[(1+αp)JLorentzpnJLorentzpn1+γp2Δt(En+1En1)].
(17)

3.2. Nonlinear Kerr polarization

The nonlinear Kerr polarization at step n+12 is obtained from Eq. (10). The finite-difference expression for the time derivative of the Kerr polarization centered at step n+12 is

JKerrn+12=αε0χ0(3)Δt{(En+1)2En+1(En)2En}.
(18)

3.3. Nonlinear Raman polarization

Eq. (11) is solved by introducing a scalar auxiliary variable for the convolution,

S(t)χRaman(3)(t)*E(t)2
(19)

with Fourier transform

S(ω)χRaman(3)(ω){E(t)2},
(20)

where

χRaman(3)(ω)=(1α)χ0(3)ωRaman2ωRaman2+2δRamanω2,
(21)
ωRamanτ12+τ22τ12τ22;δRaman=1τ2.
(22)

Inserting Eq. (21) into Eq. (20), multiplying by (ωRaman2 + 2 δ Raman -ω 2), and transforming to the time domain yields the auxiliary differential equation,

ωRaman2+2δRamanSt+2St2=(1α)χ0(3)ωRaman2E2.
(23)

Because Eq. (23) contains a second derivative, it is finite-differenced, centered at step n:

Sn+1=[2ωRaman2(Δt)2δRamanΔt+1]Sn+[δRamanΔt1δRamanΔt+1]Sn1
+[(1α)χ0(3)ωRaman2(Δt)2δRamanΔt+1](En)2.
(24)

Using Eq. (11) and Eq. (19), the finite-difference expression for the time derivative of the Raman polarization centered at step n+12 is

JRamann+12=ε0Δt(En+1Sn+1EnSn).
(25)

3.4. Solution for electric field

Now that all terms in Eq. (2) are known at step n+12,En+1, E n+1 is determined from

×Hn+12=ε0Δt(En+1En)+p=13JLorentzpn+12+JKerrn+12+JRamann+12.
(26)

Because E is a vector quantity, solving Eq. (26) for E n+1 requires solving a nonlinear system of coupled equations. This is efficiently solved using a multi-dimensional Newton’s method, where zeroes of objective functions are determined by iteration. For the field components {Ex ,Ey ,Hz } considered here, Eq. (26) is solved for Exn+1 and Eyn+1 by substituting Eq. (17), Eq. (18), and Eq. (25). We then define the objective function vector [X Y]T as

[XY]=×Hn+12+ε0Δt(En+1En)
+12p=13[(1+αp)JLorentzpnJLorentzpn1+γp2Δt(En+1En1)]
+αε0χ0(3)Δt{(En+1)2En+1(En)2En}+ε0Δt(En+1Sn+1EnSn)
(27)

Next, we define Gxg and Gyg to be the gth guesses for Exn+1 and Eyn+1. Newton’s method updates the guesses until the objective functions X and Y are sufficiently close to zero. Each subsequent guess is made from the current guess by:

[Gxg+1Gyg+1]=[GxgGyg](J1[XY])g,
(28)

where J is the Jacobian matrix, ∂(X,Y)/∂(Gx ,Gy ), with elements

J11=ε0Δt+1t(γ1+γ2+γ3)+ε0Δt[αχ0(3)(3Gx2+Gy2)+Sn+1],
J12=2ε0Δtαχ0(3)GxGy,J21=2ε0Δtαχ0(3)GxGy,
J22=ε0Δt+1t(γ1+γ2+γ3)+ε0Δt[αχ0(3)(Gx2+3Gy2)+Sn+1],
(29)

and ∣g indicates evaluation using the values from the gth guess.

4. GVADE FDTD simulation of temporal and spatial solitons

We applied the GVADE method to model temporal and spatial solitons in dispersive nonlinear materials. First, we modeled temporal soliton propagation in a material with single pole linear Lorentz dispersion and Kerr and Raman nonlinear polarizations. The same pulse, grid, and material from [5

5 . M. Fujii , M. Tahara , I. Sakagami , W. Freude , and P. Russer , “ High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media ,” IEEE J. Quantum Electron. 40 , 175 – 182 ( 2004 ). [CrossRef]

] were used; the results, shown in Fig. 1, reproduce those of [5

5 . M. Fujii , M. Tahara , I. Sakagami , W. Freude , and P. Russer , “ High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media ,” IEEE J. Quantum Electron. 40 , 175 – 182 ( 2004 ). [CrossRef]

].

Figure 2 illustrates the full-vector capability of the GVADE method. Here, we model the +x-directed propagation of a higher-order spatial soliton having the electromagnetic field vector components {Ex ,Ey ,Hz }. We assume for the vector electric field the three-pole set of linear Sellmeier dispersions and the instantaneous Kerr and dispersive Raman nonlinearities published in [3

3 . S. Nakamura , Y. Koyamada , N. Yoshida , N. Karasawa , H. Sone , M. Ohtani , Y. Mizuta , R. Morita , H. Shigekawa , and M. Yamashita , “ Finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation for 12 fs laser pulse propagation in a silica fiber ,” IEEE Photonics Technol. Lett. 14 , 480 – 482 ( 2002 ). [CrossRef]

, 4

4 . S. Nakamura , N. Takasawa , and Y. Koyamada , “ Comparison between finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation and experimental results for slightly chirped 12 fs laser pulse propagation in a silica fiber ,” IEEE J. of Lightwave Technol. 23 , 855 – 863 ( 2005 ). [CrossRef]

]. The strengths and resonant frequencies of the linear dispersions from Sellmeier’s equation are given by: β 1 = 0.69617, β 2 = 0.40794, β 3 = 0.89748, ω 1 = 2.7537 × 1016 rad/s, ω 2 = 1.6205 × 1016 rad/s, and ω 3 = 1.90342 × 1014 rad/s. The third-order electric susceptibility scalar is χ0(3) = 1.89 × 10-22 m2/V2. The relative strengths of the Kerr and Raman polarizations are given by the parameter α = 0.7. The Raman polarization parameters are τ 1 = 12.2 fs and τ 2 = 32 fs. The soliton was modeled on a modified Yee grid with collocated electric field components [8

8 . A. Taflove and S. C. Hagness , Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed . Artech House, Norwood, MA , 2005 .

]. The grid was 6000 by 2615 cells with a spatial resolution of Δx = Δy = 8nm and a temporal resolution of Δt = 3.34 × 10-18s. The spatial and temporal resolutions were chosen empirically based on convergence tests. The chosen time-step was one quarter of the time-step required by the Courant limit for standard FDTD [8

8 . A. Taflove and S. C. Hagness , Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed . Artech House, Norwood, MA , 2005 .

]. The magnetic field was excited by a hard source at the far-left side of the grid (x = 0) having an initial transverse profile

Fig. 1. GVADE simulation results for temporal soliton propagation in a dispersive nonlinear material. These results reproduce those of [5].
Hz(t)=H0sin(ωct)sech(yw),
(30)

where H 0 = 4.77×107 A/m, ωc = 4.35×1015 rad/s (λ 0 = 433nm), and w = 667 nm. From Fig. 2, we see that the calculated solution exhibits a periodic expansion and contraction that is characteristic of higher-order solitons [9

9 . A T. Ryan and G. P. Agrawal , “ Spatiotemporal coupling in dispersive nonlinear planar waveguides ,” J. Opt. Soc. Am. B 12 , 2382 – 2389 ( 1995 ). [CrossRef]

].

Fig. 2. GVADE simulation results of a +x-directed higher-order spatial soliton with field components {Ex ,Ey ,Hz } in a material with a three-pole Sellmeier linear dispersion, an instantaneous Kerr nonlinearity, and a dispersive Raman nonlinearity: magnitude of Hz .

5. Conclusion

The importance of modeling electromagnetic wave propagation in dispersive nonlinear materials will increase with modern engineering of nanophotonic devices. In this paper, we showed how the Fujii ADE FDTD method can be applied to nonlinear optics problems where the electric field has two or three orthogonal vector components. The GVADE method is, to the best of our knowledge, the first numerical technique to model electromagnetic wave propagation with two or three orthogonal electric field vector components in dispersive nonlinear materials.

We are currently exploring the relationships of predictions made by nonlinear Schrödinger (NLS) equation theory and the GVADE technique for a variety of scalar and vector spatial solitons. Our initial aim is to determine the transition between the the two approaches, i.e., where a computation based upon the full-vector Maxwell’s equations is required to properly model the vector wave physics. Furthermore, we intend to explore a regime where the NLS equation is arguably inapplicable – the scattering of a spatial soliton by a sub-wavelength particle.

References and links

1 .

T. Kashiwa and I. Fukai , “ A treatment by FDTD method of dispersive characteristics associated with electronic polarization ,” Microwave Optics Tech. Lett. 3 , 203 – 205 ( 1990 ). [CrossRef]

2 .

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]

3 .

S. Nakamura , Y. Koyamada , N. Yoshida , N. Karasawa , H. Sone , M. Ohtani , Y. Mizuta , R. Morita , H. Shigekawa , and M. Yamashita , “ Finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation for 12 fs laser pulse propagation in a silica fiber ,” IEEE Photonics Technol. Lett. 14 , 480 – 482 ( 2002 ). [CrossRef]

4 .

S. Nakamura , N. Takasawa , and Y. Koyamada , “ Comparison between finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation and experimental results for slightly chirped 12 fs laser pulse propagation in a silica fiber ,” IEEE J. of Lightwave Technol. 23 , 855 – 863 ( 2005 ). [CrossRef]

5 .

M. Fujii , M. Tahara , I. Sakagami , W. Freude , and P. Russer , “ High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media ,” IEEE J. Quantum Electron. 40 , 175 – 182 ( 2004 ). [CrossRef]

6 .

G. P. Agrawal , Nonlinear Fiber Optics, 3rd ed . Academic Press, San Diego, CA , 2001 .

7 .

R. W. Boyd , Nonlinear Optics, 2nd ed . Academic Press, San Diego, CA , 2003 .

8 .

A. Taflove and S. C. Hagness , Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed . Artech House, Norwood, MA , 2005 .

9 .

A T. Ryan and G. P. Agrawal , “ Spatiotemporal coupling in dispersive nonlinear planar waveguides ,” J. Opt. Soc. Am. B 12 , 2382 – 2389 ( 1995 ). [CrossRef]

OCIS Codes
(190.3270) Nonlinear optics : Kerr effect
(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(190.5650) Nonlinear optics : Raman effect

ToC Category:
Nonlinear Optics

History
Original Manuscript: June 26, 2006
Revised Manuscript: August 22, 2006
Manuscript Accepted: August 24, 2006
Published: September 1, 2006

Citation
Jethro H. Greene and Allen Taflove, "General vector auxiliary differential equation finite-difference time-domain method for nonlinear optics," Opt. Express 14, 8305-8310 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-18-8305


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References

  1. T. Kashiwa and I. Fukai, "A treatment by FDTD method of dispersive characteristics associated with electronic polarization," Microwave Opt. Tech. Lett. 3, 203-205 (1990). [CrossRef]
  2. 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]
  3. S. Nakamura, Y. Koyamada, N. Yoshida, N. Karasawa, H. Sone, M. Ohtani, Y. Mizuta, R. Morita, H. Shigekawa, and M. Yamashita, "Finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation for 12 fs laser pulse propagation in a silica fiber," IEEE Photonics Technol. Lett. 14, 480-482 (2002). [CrossRef]
  4. S. Nakamura, N. Takasawa, and Y. Koyamada, "Comparison between finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation and experimental results for slightly chirped 12 fs laser pulse propagation in a silica fiber," J. Lightwave Technol. 23, 855-863 (2005). [CrossRef]
  5. M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, "High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media," IEEE J. Quantum Electron. 40, 175-182 (2004). [CrossRef]
  6. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, San Diego, CA, 2001).
  7. R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, San Diego, CA, 2003).
  8. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, Norwood, MA, 2005).
  9. A. T. Ryan and G. P. Agrawal, "Spatiotemporal coupling in dispersive nonlinear planar waveguides," J. Opt. Soc. Am. B 12, 2382-2389 (1995). [CrossRef]

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