Nonlinear Drude Model

Transcription

1 Nonlinear Drude Model Jeremiah Birrell July 8, Perturbative Study of Nonlinear Drude Model In this section we compare the 3rd order susceptibility of the Nonlinear Drude Model to that of the Kerr effect. The equations of motion in the nonlinear Drude model are the Lorentz force law with a linear damping term. v(t) = q [E(t) + v(t) B(t)] v(t)/τ (1) m In the following we assume a plane wave with E in the x direction and B in the y direction: E(t) = Ee i(kz ωt) B(t) = E(t)/c Since this will produce no force in the y direction we can ignore the y component of the Lorentz force. Equation (1) then reduces to the following two equations for v x and v z : v x (t) = q m [E(t) v zb(t)] v x (t)/τ () v z (t) = q m v x(t)b v z (t)/τ (3) Expanding x(t) and z(t) as Fourier series and inserting these into () and (3) gives: x n ( inω) E n e inωt =qee ikz iωt 1 x n ( inω)e n e inωt τ qe z n ( inω)e n e ikz i(n+1)ωt mc z n ( inω) E n e inωt = 1 x n ( inω)e n e inωt τ + qe mc x n ( inω)e n e ikz i(n+1)ωt 1

2 Equating terms gives: qeikz x 1 = mω(ω + i τ ) qiωe ikz z = mcω(4ω + i τ )x 1 qiωe ikz x 3 = mcω(9ω + 3i τ )z Substituting z and x 1 into the expression for x 3 gives: q 3 e 3ikz x 3 = c m 3 ω(9ω + 3i i τ )(4ω + τ )(ω + i τ ) (4) The polarization associated with the motion x(t) is P (t) = qρx(t) where ρ is the number density of electrons. The third harmonic component of the polarization in the nonlinear Drude model is therefore P 3 (t) = qρx 3 E(t) 3. The polarization associated with the Kerr effect is P Kerr (t) = ɛ 0n ce(t) 3 Using the value of x 3 from (4) the ratio of there two becomes: P 3 P Kerr = ρq 4 e 3ikz ɛ 0 n c 3 m 3 ω(9ω + 3i i τ )(4ω + τ )(ω + i τ ) (5) We let λ = 800nm, τ = s, n = 10 3 m /W, q be the electron charge, m the electron mass, and assume fully ionized air ρ /m 3. Substituting in these values along with the values of c and ɛ 0 gives: P P Kerr Derivation of Numerical Algorithms The model we are using for the current as a function of time is: J(t) = q ρ(t )V (t, t )dt + qρ( )V (t, ) (6) Here ρ is the number density of electrons and V (t, t ) is the velocity at time t computed via the nonlinear Drude model (1) with the initial condition V (t, t ) = 0. The second term is required to account for a nonzero initial charge density. The original plan for computing J(t) was to solve for V (t, t ) on a triangular array {(t, t ) : t t f, t t} and then do the integral in (6) explicitly. We compare results from several numerical ode algorithms for solving for J(t), some of which follow the method outlined here and one method that explicitly steps forward in J without first solving for V (t, ) on the triangular grid. The

3 simplest method considered here for integrating the nonlinear Drude model is Heun s method. Given an ode of the form ẏ(t) = F (t, y(t)) y : R R n, F : R R n R n (7) Heun s method proceeds as follows. Given a uniform grid of points t n with spacing t we make the integral trapezoidal approximation of to write the following: y(t n+1 ) y(t n ) = n+1 t n ẏ(t)dt = n+1 t n F (t, y(t))dt (8) t (F (t n+1, y(t n+1 )) + F (t n, y(t n ))) (9) Next we use the Euler approximation to replace y(t n+1 ) in (9) with y(t n ) + tf (t, y(t n )) Solving (8) and (9) for y(t n+1 ) with this substitution gives: y(t n+1 ) y(t n ) + t [F (t n, y(t n )) + F (t n+1, y(t n ) + tf (t n, y(t n ))] (10) The right hand side of (10) only depends on t and y n, allowing us to iteratively solve for y(t). The nonlinear Drude model is an ode of the form (7) with F (t, V ) = q [E(t) + V B(t)] V/τ (11) m Therefore we can use this algorithm to solve for V (t, t ) and then compute J(t). The above method, though functional, requires us to compute and store a great deal of extra data, in the form of V (t, t ), than we are really interested in. Using Heun s method we can construct an algorithm that skips the above step of solving for V (t, t ) and allows us to iteratively compute J n+1 in terms of J n, ostensibly reducing the number computing cycles and memory used. The starting point for this algorithm is the following: + t ρ(t )V (t + t, t )dt = ρ(t )V (t + t, t )dt + ρ(t )V (t + t, t )dt + t + t t ρ(t )V (t + t, t )dt ρ(t)v (t + t, t) where we have used the trapezoidal approximation as well as the fact that V (t + t, t + t) = 0 due to initial conditions. This implies J(t + t) ρ(t )V (t + t, t )dt (1) + t ρ(t)v (t + t, t) (13) + ρ( )V (t + t, ) (14) 3

4 where the factor of q was absorbed into the definition of ρ(t), making it the charge density. We now proceed to use Heun s method to approximate V (t 1 + t, t ) in the above. First we make the following definitions: K 1 (t 1, t ) F (t 1, V (t 1, t )) K (t 1, t ) F (t 1 + t, V (t 1, t ) + tk 1 (t 1, t )) Using these definitions along with (10) we can write: V (t 1 + t, t ) V (t 1, t ) + t (K 1(t 1, t ) + K (t 1, t )) (15) We can now substitute this approximation into each of (1), (13), and (14). First consider (1): = ρ(t )V (t + t, t )dt ρ(t )V (t, t )dt + = J 1 (t) + t ρ(t )[V (t, t ) + t (K 1(t, t ) + K (t, t ))]dt ρ(t ) t (K 1(t, t ) + K (t, t ))dt ρ(t )(K 1 (t, t ) + K (t, t ))dt where we have defined J 1 (t) J(t) ρ( )V (t, ). We now break up the last integral: ρ(t )K1(t, t )dt = = q m ([ρ(t) ρ()]e(t) + ρ(t )( q m [E(t) + V (t, t ) B(t)] V (t, t )/τ)dt ρ(t )V (t, t )dt B(t)) = q m ([ρ(t) ρ()]e(t) + J 1 (t) B(t)) J 1 (t)/τ ρ(t )V (t, t )dt /τ = ρ(t )K(t, t )dt = ρ(t )F (t + t, V (t, t ) + tk 1 (t, t ))dt ρ(t )( q m [E(t + t) + (V (t, t ) + tk 1 (t, t )) B(t + t)] [V (t, t ) + tk 1 (t, t )]/τ)dt = q m ([ρ(t) ρ()]e(t + t) + [J 1 (t) + t 1/τ(J 1 (t) + t ρ(t )K 1 (t, t )dt ] B(t + t)) ρ(t )K 1 (t, t )dt ) 4

5 Now consider (13): t ρ(t)v (t + t, t) t = t 4 ρ(t)(v (t, t) + t [K 1(t, t) + K(t, t)]) t ρ(t)(f (t, 0) + F (t + t, tf (t, 0))) = 4 ρ(t)( q (E(t) + E(t + t) m + q t q t E(t) B(t + t)) m mτ E(t)) Finally (14): ρ( )V (t + t, ) ρ( )(V (t, ) + t [K 1(t, ) + K(t, )]) = ρ( )(V (t, ) + t [K 1(t, ) + K(t, )]) K 1 (t, ) = q m (E(t) + V (t, ) B(t)) V (t, )/τ K (t, ) = q m (E(t + t) + (V (t, ) + tk 1 (t, )) B(t + t)) (V (t, ) + tk 1 (t, ))/τ With these expressions substituted in for (1), (13), and (14) we obtain an expression for J(t + t) in terms of J(t), E(t), B(t), ρ(t), and V (t, ). E(t) and B(t) are givens and ρ(t) is either given or is calculated from E(t) so we now have an algorithm that will allow us to iteratively calculate J(t). The only difference from a standard iteration algorithm is that we also need to know V (t, ), which is not given. This means that we must have additional memory allocated to store a single value of V (t, ). Then at each step we can calculate both J(t + t) and V (t + t, ) (the latter using the Heun s method algorithm) and overwrite V (t, ) with V (t + t, ) as in the following pseudo code: for i = 0 to N do K1=F(i,V) K=F(i+1,V+dt*K1) J(i+1)=G(J(i),E(i),B(i),ρ(i),V) V=V+dt/*(K1+K) end for where G denotes the iterative function we just derived and F as in (11). This algorithm will be referred to as the direct algorithm in the remained of the paper. 3 Comparison of Numerical Algorithms Here we compare the solutions calculated by various algorithms to exact or approximately exact solutions. In the following tables the normalized L 1 error 5

6 of an approximation y to an exact solution y exact is given by: y(t) y exact (t) dt/ The normalized L error is given by: ( y(t) y exact (t) dt) 1/ /( The maximum relative error is given by: y exact (t) dt max((y(t i ) y exact (t i ))/y exact (t i )) y exact (t) dt) 1/ The equation of motion has an analytic solution for E(t) and B(t) constants. Below are tables comparing the exact solution to V (t) as calculated by various algorithms. Heun s Method N Maximum Relative Error Time (s) Matlab ode Matlab ode Matlab ode Matlab ode15s

7 Matlab ode3s We now take E(t) to be a Gaussian pulse multiplied by a sine wave. Below are tables comparing the V (t) as calculated by various algorithms to the output of ode45 with relative error tolerance of Heun s Method N 3rd Harmonic Relative Error Time (s) Matlab ode Matlab ode Matlab ode Matlab ode15s Matlab ode3s

8 These results show that Heun s method and the Matlab algorithms perform similarly in terms of accuracy and time to compute. Since we require a method that works on a uniform grid and none of the Matlab algorithms do, we will no longer consider any of the Matlab algorithms. Heun s method will now be the algorithm we evaluate the direct algorithm against. Next we compare J(t) as calculated by Heun s method and integrating to that calculated by the direct algorithm for constant E, B. Heun s Method N Normalized L 1 Error Normalized L Error Maximum Relative Error Time (s) Direct Algorithm N Normalized L 1 Error Normalized L Error Maximum Relative Error Time (s) We now take E(t) to be a Gaussian pulse multiplied by a sine wave. The outputs of the direct algorithm and Heun s method and integrating are compared to the output of the direct algorithm with a grid size of N = 3. Heun s Method N Normalized L 1 Error Normalized L Error 3rd Harmonic Relative Error Time (s) Direct Algorithm N Normalized L 1 Error Normalized L Error 3rd Harmonic Relative Error Time (s) Note that we have no data on calculating J(t) via Heun s method and integrating for N greater than 1 due to ram limitations. 4 Conclusions From these results we can see that both the direct algorithm and Heun s method perform almost identically in terms of accuracy. This is not unexpected since the same approximations were made in deriving both algorithms. Additionally, the normalized L 1 error of both algorithms reduces by a factor of 4 for each factor of increase in N. This is consistent with an algorithm of order accuracy. Finally, the direct algorithm is approximately a factor of N times faster than Heun s method and integration. This is also not unexpected, as Heun s method takes on the order of N iterations to compute J(t) as opposed to N iterations for the direct algorithm. 8