Received: 27 April 2016
Revised: 21 September 2016
Accepted: 2 November 2016
DOI 10.1002/jnm.2215
EMF 2016 SYMPOSIUM
Graphics processing unit-based solution of nonlinear Maxwell’s equations for inhomogeneous dispersive media Anton Rudenko
Jean-Philippe Colombier
Univ Lyon, UJM-St-Etienne, Laboratoire Hubert Curien, Saint-Etienne, CNRS UMR 5516, F-42000, France Correspondence Anton Rudenko, Univ Lyon, UJM-St-Etienne, Laboratoire Hubert Curien, CNRS UMR 5516, F-42000, Saint-Etienne, France. Email:
[email protected] and
[email protected] Funding information LABEX MANUTECH SISE French National Research Agency (ANR), Grant/Award Number: ANR-10-LABEX- 0075, ANR-11-IDEX-0007
Tatiana E. Itina
Summary A new approach is developed for fast solution of complex dynamic problems in nonlinear optics. The model consists of the nonlinear Maxwell’s equations coupled with time-dependent electron density equation. The approach is based on the Finite-Difference Time-Domain and the auxiliary differential equation methods for frequency-dependent Drude media with a time-dependent carrier density, changing due to Kerr, photoionization, avalanche, and recombination effects. The system of nonlinear Maxwell-Ampere equations is solved by an iterative fixed-point procedure. The proposed approach is shown to remain stable even for complex nonlinear media and strong gradient fields. Graphics-processing-units technique is implemented by using an efficient algorithm enabling solution of massively 3-dimensional problems within reasonable computation time. KEYWORDS
electron density, femtosecond laser irradiation, GPU-accelerated FDTD, nonlinear Maxwell’s equations, nonlinear optics, photoionization
1
INTRODUCTION
The Finite-Difference Time-Domain (FDTD) method is a powerful technique based on Maxwell’s equations for calculating electric and magnetic fields.1–4 The numerical method was adopted to simulate a complex inhomogeneous dispersive media.5–8 On one hand, up to now, the approach has been mostly used for media with non-transient optical properties. On the other hand, with the recent progress in picosecond and femtosecond high-intensity laser processing, the leading role of nonlinear effects in materials has been underlined.9–11 Thus, an efficient coupling of the electrodynamic approach with the equations describing changes in the local refractive index is of great interest. First attempts were made to consider a complex dynamics of electromagnetic interaction including Kerr effect.12–16 In addition, Newton-iteration and fixed-point iteration methods were proposed to solve the system of nonlinear Maxwell-Ampere equations.16–18 Furthermore, for high intensity laser irradiation, the material characteristics undergo a significant change which is not inherently included in the FDTD solution as it only provides
Int J Numer Model 2016; 1–9
the electromagnetic response of the medium. Thus, a dynamic problem is worth considering. Recently, more sophisticated models have been developed where the Maxwell’s equations are coupled with an electron density evolution equation to investigate radio-frequency wave propagation in a dynamic, magnetized plasma,19 material interaction in vertical-cavity surface-emitting lasers,20 lasing dynamics in 4-level gain systems,21 nonlinear optical phenomena in silicon waveguides22 and transparent dielectrics.11,23–25 In this work, we couple Maxwell’s equations with timedependent electron density equation accounting for Kerr, photoionization, avalanche, and recombination effects. Thus, the model considers the whole variety of nonlinear processes which take place under ultrashort laser irradiation. The proposed procedure can be applied to a wide range of electromagnetic applications. In particular, the method is advantageous in the case of ultrafast laser material processing,26 for modeling ultrashort laser nanoparticle interaction,27,28 and to study the periodic nanostructure formation in dielectric materials by femtosecond laser irradiation.25,29
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Copyright © 2016 John Wiley & Sons, Ltd.
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RUDENKO ET AL.
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2
NUMERICAL MODEL
The model of ultrafast light propagation in inhomogeneous dispersive media consists of non-magnetic nonlinear Maxwell’s curl equations, which are written as follows ⎧ ⎪ ⎨ ⎪ ⎩
⃗ 𝜕E 𝜕t ⃗ 𝜕H 𝜕t
= =
⃗ ∇×H − 𝜖1 (J⃗D 𝜖0 0 ⃗ E − ∇× , 𝜇0
+ J⃗Kerr + J⃗pi )
Then, we couple Equation 1 with electron density equation. The time-dependent conduction-band carrier density is described with a rate equation taking into account multiphoton ionization, avalanche ionization and recombination9 as 𝜕𝜌 𝜌 , = (𝜌0 − 𝜌)wpi (I) + Wav (I, 𝜌) − 𝜕t 𝜏rec
(1)
⃗ is the electric field, H ⃗ is the magnetic field and where E ⃗JD is current derived from the Drude model for the dispersive media, J⃗Kerr is Kerr polarization current, and J⃗pi is the photoionization term.23,24 To avoid nonphysical reflections, we complete the FDTD grid at the edges by uniaxial perfect matched layers (UPML).30 The optical properties of the medium change as a result of several nonlinear physical processes. The first one is the heating of the conduction band electrons, which is modeled by the Drude model with a time-dependent carrier density as follows e2 𝜌(t) ⃗ 𝜕 J⃗D = −𝜈e J⃗D + E, (2) 𝜕t me where e is the elementary charge; me is the electron mass; 𝜌 is the free carrier density and 𝜈 e is the electron collision frequency. The second contribution is a nonlinearity because of the Kerr effect. The third nonlinear electric polarization field 3 3 3 ∑ ∑ ∑ (3) (3) is defined as Pi = 𝜒ijkl Ej Ek El , where 𝜒ijkl is the
(5)
where 𝜏 rec is the electron recombination time. It is worth noting that a more general approach can be used to estimate accurately the contribution of the avalanche ionization W av (I,𝜌) based on the multiple rate equation.10 However, as the discretization of the system of Rethfeld’s equations is similar to the discretization of the single rate equation, we will treat here only the Equation 5 with W av = wav (I)𝜌, where wav (I) is considered to be dependent only on the laser intensity.
3
TEMPORAL DISCRETIZATION
3.1
Nonlinear currents
The system of Maxwell’s equations 1 is discretized temporally as follows ⎧ ⎪ ⎨ ⎪ ⎩
⃗ t−1∕2 ⃗ t ⃗ t+1∕2 −H E) H = − (∇× Δt 𝜇0 ⃗t ⃗ t+1∕2 ⃗ t+1 −E E = (∇×H) − 𝜖1 Δt 𝜖0 0
( ) t+1∕2 t+1∕2 t+1∕2 J⃗D + J⃗Kerr + J⃗pi .
(6)
j=1 k=1 l=1
4th order component of the electrical susceptibility.31 For isotropic materials, the components of the polarization field | ⃗ |2 are defined as Pi = 𝜒3 |E | E , where 𝜒 3 is the third-order sus| | i ceptibility equal for all polarization components. Substituting ⃗ this expression to J⃗Kerr = 𝜖0 𝜕P results in the following Kerr polarization current
𝜕t
J⃗Kerr = 𝜖0 𝜒3
) ( | ⃗ |2 ⃗ 𝜕 |E |E | | 𝜕t
.
(3)
In addition, we consider a complete Keldysh photoionization rate wpi (I),32 including multiphoton and tunneling ionization. The corresponding current is written as follows ⃗ 𝜌0 − 𝜌 wpi (I)E . (4) 𝜌0 | ⃗ |2 |E | | | Here, Eg is the electron band gap in the absence of the electric | ⃗ |2 field; 𝜌0 is the saturation density; I = 𝛼 |E | is the intensity, | | √𝜖 n 0 𝛼 = 2 𝜇 is the normalizing constant, and n is the refracJ⃗pi = Eg
0
tive index of unexcited material with 𝜌 = 0. The rate wpi (I) depends on the laser intensity. For a fixed laser wavelength, the value of the photoionization rate can be found separately from the complete Keldysh formalism32 and included to the calculation problem in the form of data array.
Here, each nonlinear process is introduced as a current. The terms are added to the Maxwell-Ampere electric field update equation and defined on t + 12 timestep. In what follows, we show how each current is discretized and how the system of nonlinear Equation 6 is solved numerically. The Drude current, responsible for the heating of the conduction band electrons, is defined from the Equation 2. It is solved by the auxiliary differential equation (ADE) technique.2 The basis of the method is to express the relation⃗ and the electric ship between the electric displacement field D ⃗ field E with a differential equation rather than with a convolutional integral which is used in recursive convolutional method.33 The major advantage of the ADE is the simplicity in modeling complex materials. In this approach, the dispersive complex media is modeled simply by adding the current directly to the Maxwell-Ampere differential Equation 6. It does not complicate the method if the other nonlinear effects are considered.15,34 The Drude current 2 is discretized as ⃗ t+1 + 𝜌t E ⃗t 1 − 𝜈e Δt ∕2 𝜌t+1 E e2 Δt J⃗Dt+1 = J⃗Dt + . 1 + 𝜈e Δt ∕2 me (1 + 𝜈e Δt ∕2) 2 (7) t+1∕2 Applying the approximation J⃗D = (J⃗Dt + J⃗Dt+1 )∕2, we find the Drude current’s discretization at time t + 1/2 to introduce directly in Maxwell-Ampere Equation 6 as follows
RUDENKO ET AL.
t+1∕2 J⃗D =
3
J⃗Dt 1+
𝜈e Δt 2
⃗ t+1 + 𝜌t E ⃗t 𝜌t+1 E e2 Δt + . me (2 + 𝜈e Δt ) 2
(8)
Kerr contribution 3 is introduced by using the following equation: ( ) ⃗ t+1 − I t E ⃗t I t+1 E t+1∕2 J⃗Kerr = 𝜖0 𝜒3 . (9) 𝛼Δt
( ) t ⃗ t+1 = C1 I t , I t+1 E ⃗ E k k +
(
C2 Ikt , Ikt+1
)
⃗ t+1 (𝜌0 − 𝜌t+1 ) wt E ⃗ t (𝜌0 − 𝜌t ) ⎞ ⎛ wt+1 E pi pi t+1∕2 ⎟ . (10) = 𝛼Eg ⎜ + J⃗pi ⎜ ⎟ 2𝜌0 I t 2𝜌0 I t+1 ⎝ ⎠
⃗ (∇ × H)
t+1∕2
t Δt J⃗D − 𝜖0 1 + 𝜈 e Δ t
Fixed-point iteration algorithm
By substituting equations 8–10 to 6, we obtain a system of nonlinear Maxwell’s equations. One can note, that all nonlinear processes here 2–4 are intensity-dependent. Thus, the equations for each electric field could not be solved separately as it was possible for nonlinear Kerr and Raman media in 2D-TE case by recursive convolutional dispersive method with iterative and non-iterative schemes12,14,35 or by finding the analytical solution for cubic equation.13 The first algorithm for nonlinear dispersive media which could be implemented for 3D-FDTD and could handle any kind of nonlinearities was proposed by Greene and Taflove.18 The algorithm was based on the multi-dimensional Newton’s iteration for 3 electric field components. The method requires the calculation of the Jacobian matrix at each iteration which is cumbersome from a view of computational speed17 and is delicate in the case when the nonlinear process is not defined by analytical formula as Keldysh photoionization in our model. Furthermore, the nonlinear equation is of high order with electric field as a variable which results in high number of required iterations. Recently, a new method for solving the system of nonlinear equations has been proposed by Ammann17 and then implemented by Francés.16,36 In this method, the Newton’s iteration for Ex t+1 , Ey t+1 and Ez t+1 is replaced by fixed-point iteration for I t + 1 . The intensity has to be introduced in the equations and is calculated as I t+1 = 𝛼(Ex t+1 Ex t+1 + Ey t+1 Ey t+1 + Ez t+1 Ez t+1 ), where Ex t+1 , Ey t+1 and Ez t+1 are taken from the ADE method for nonlinear medium. In our case, the algorithm is especially advantageous as all the nonlinear processes are intensity-dependent. At the same time, computational speed is increased and the accuracy is maintained. The schematics of the calculation procedure is shown in Figure 1. Firstly, the magnetic fields are updated according to simple Yee’s discretization.1 Then, we start the iterations and calculate the electric fields according to nonlinear ADE method as
(11)
,
with the coefficients that include the Kerr effect and the photoionization’s contributions ( ) wtpi Eg Δt 𝜌t t 1 − + 2𝜒I − 𝛼 t k ( ) 𝜈e Δt+2 Ik 𝜖0 𝜌0 C1 Ikt , Ikt+1 = ( ) t+1 w 𝜔t+1 t+1 pi Eg Δt pl 2𝜖∞ + 2𝜒Ikt+1 + 𝛼 I t+1 1 − 𝜌𝜌 + 𝜈 Δt+2 𝜖 (
C2 Ikt , Ikt+1
)
𝜔tpl
0
k
=
2Δt wt+1 Eg Δt
pi 2𝜖∞ + 2𝜒Ikt+1 + 𝛼 I t+1 k
𝜖0
0
( 1−
𝜌t+1 𝜌0
e
) +
𝜔t+1 pl
,
𝜈e Δt+2
(12) 𝜒
where the plasma frequency is 𝜔tpl = e mΔt𝜖 𝜌 , 𝜒 = 𝛼3 , and 𝜖 ∞ e 0 is the medium permeability. After that, we check if the resulting intensity as the sum of the squared electric fields satisfies the Maxwell-Ampere’s nonlinear equation with possible deviation not more than a tolerance 𝜖 (Figure 1). If not, we update again the C1 and C2 coefficients and electric fields taking into account the new value of the calculated intensity. If yes, we finish the iteration procedure, calculate the electron density defined by 5 and update the polarization currents according to 7 which are used at the next timestep for calculation of C1 and C2 defined in equation 12. 2
3.2
]
2
2𝜖∞ −
The finite-difference equation for the ionization current 4 is calculated by
[
3.3
2 t
Convergence of the iteration procedure
In what follows, we investigate the convergence of the proposed iteration algorithm. According to Banach fixedpoint theorem,37 fixed-point iteration method f (I t+1 ) = ⃗ t+1 E ⃗ t+1 ) = I t+1 converges if the derivation of the intensity 𝛼(E | t+1 | function | 𝜕f𝜕I(It+1 ) | < 1 is limited ∀I t + 1 from the applied | | intensity interval. This condition yields [ ( )] ⃗t J Δt D ⃗ t + C2 (I t+1 ) (∇ × H) ⃗ t+1∕2 − 𝛼 C1 (I t+1 )E · 𝜖0 1 + 𝜈 e Δ t 2 )] [ ( t Δt J⃗D 1 ′ t+1 ⃗ t ′ t+1 t+1∕2 ⃗ C1 (I )E + C2 (I ) (∇× H) − < . 𝜖0 1 + 𝜈 e Δ t 2 2
(13)
Because of the continuity of the equations, and because temporal and spatial grid have accurately high resolutions respectively, the terms containing spatial and temporal derivations ⃗ t+1∕2 and J⃗t can be considered small compared as (∇ × H) D with the terms proportional to the electric field and hence can be neglected. Therefore, the condition yields C1 (I t+1 )C1 ′ (I t+1 )I t
