Faraday Rotation Using FDTD Technique

Report Submitted By

Mohammad Ismail Hossain

Communications, Systems and Electronics

School of Engineering and Science

Jacobs University Bremen

m.hossain@jacobs-university.de

December 02, 2011

Course: Computational Electromagnetics.

Course Instructor: Prof. Dr. Jon W. Wallace

1

Abstract

In this report, we address to implement Faraday Rotation augment FDTD code tomodel a plasma (anisotropic medium). We show that as a wave propagates throughthe medium, the polarization rotates. We will also observe that for di�erent typeof sampling time we have di�erent status of waves. In order to see the boundarycondition, we use PML boundary condition and observe the e�ect of using boundarycondition. Finally, we visualize these e�ects by using MATLAB simulation.

Contents

1 Introduction 3

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Theoretical Background 4

2.1 Faraday Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Frequency Dependent Permittivity and Susceptibility . . . . . . . . . 4

3 Simulation Results 7

3.1 Demonstrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 MATLAB Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Conclusion 12

2

Chapter 1

Introduction

1.1 Introduction

The �nite-di�erence time-domain (FDTD) technique for the analysis of interactionsof electromagnetic waves with material bodies was formulated for nondispersive by�Yee�. Since that time considerable e�ort has been expended in extending the ap-plication of the method to lossy dielectrics. FDTD may also be applied to computetransient propagation in plasma when the plasma can be characterized by on a com-plex frequency dependent permittivity. Here, we are using pulse normally incidenton an isotropic plasma slab, with the one limitation that the excitation pulse musthave no zero frequency energy components. However, �nite-di�erence time-domain(FDTD) formulation has been developed, which allows explicit calculation of wide-band transient electromagnetic interactions with plasma or anisotropic materials.The method is used to compute wide-band re�ection from an air-water interfaceover a frequency range where the complex permittivity of water varied signi�cantlywith frequency [4]. First, FDTD is applied to a material (plasma) that has a �nitezero frequency conductivity (the polar dielectric model for water used in has zeroconductivity at zero frequency) [4]. Since plasma has the same functional behaviorfor permittivity as good conductors, the results presented here apply to conductorsas well. The second is that the FDTD calculations remain stable over a wide band-width including frequencies where the plasma permittivity is negative. Other FDTDformulations will become Unstable for certain negative values of permittivity thatcause multiplying factors to become singular.

3

Chapter 2

Theoretical Background

2.1 Faraday Rotation

If any transparent solid or liquid is placed in a uniform magnetic �eld, and a beamof plane polarized light is passed through it in the direction parallel to the magneticlines of force (through holes in the pole shoes of a strong electromagnet), it is foundthat the transmitted light is still plane polarized, but that the plane of polarizationis rotated by an angle proportional to the �eld intensity. This "optical rotation"is called the Faraday rotation or Faraday E�ect. The Faraday E�ect or Faradayrotation is a Magneto-optical phenomenon, that is, an interaction between light anda magnetic �eld in a medium. The Faraday E�ect causes a rotation of the plane ofpolarization which is linearly proportional to the component of the magnetic �eld inthe direction of propagation. We will assume that the plasma is linear and isotropicand that its plasma properties (absorbance, refractive index, conductivity, tensilestrength, etc.) can be expressed by a complex frequency dependent permittivityε(ω). The fundamental advantage in dealing with plasma in the frequency domainis that at each single frequency the constitutive parameters are constant. For eachfrequency of interest a separate calculation is made with the appropriate parametervalues.

2.2 Frequency Dependent Permittivity and Suscep-

tibility

In the time domain, all of the frequency domain permittivity information is Fouri-ertransformed into a time domain susceptibility function. We have incorporated thenecessary convolution into the FDTD formulation. We assume familiarity with thebasic Yee algorithm [1]. In the time domain we have

D(t) = ε∞ε0E(t) + ε0

ˆ t

0

E(t− τ)χ(τ)dτ (2.1)

Where, ε0 is permittivity of free space, χ(τ) is the electric susceptibility, andwhere ε∞ is the relative permittivity as ω →∞. Actually, ε∞ = 1 for all materials[5] but inclusion of this factor allows for approximate susceptibility functions that

4

are not valid at in�nite frequency, and also allows for comparison with traditionalFDTD. In this project for FDTD we need to use Maxwell curl equations. Which are

∇× E =δB

δt(2.2)

∇×H =δD

δt(2.3)

D = εE = ε0εrE (2.4)

and B = µH (2.5)

Using �Yee� notation, let t=nDt in previous equation and each vector compo-nents of �eld D and E can be expressed by

D(t) ≈ D(n4t) = Dn = ε∞ε0En + ε0

ˆ t

0

E(n4t− τ)χ(τ)dτ (2.6)

Also we can quantize space by x=iDx, y=jDy, z=kDz.

Figure 2.1: �Yee� cell with notation.

From �Yee� cell we can write, for electric �eld Dxε(x)

Dn+1x(i,j,k) −Dn

x(i,j,k)

4t=H

n+ 12

z(i,j+1,k) −Hnz(i,j,k)

4y−H

n+ 12

y(i,j,k+1) −Hny(i,j,k)

4z(2.7)

Similarly we can �nd forDy and Dz.For magnetic �eld Hx

µH

n+ 12

x(i,j,k) −Hn− 1

2

x(i,j,k)

4t=En

y(i,j,k) − Eny(i,j,k−1)

4z−En

y(i,j,k) − Eny(i,j−1,k)

4y(2.8)

Similarly we can �nd forHy and Hz.We spatially quantize the susceptibility in each cell, so that

5

χ(τ, x, y, z) = χ(τ, i, j, k) (2.9)

Plasma Susceptibility [5]: The complex permittivity ε(ω) and susceptibility χ(τ)for anisotropic plasma are given by

ε(ω) = ε0(1 +ω2p

ω(jυc − ω)) = ε0(1 + χ(ω)) (2.10)

where, υc is the collision frequency and ωp is the radiant plasma frequency.

For nonconductive (at zero frequency) materials, the real and imaginary partsof χ(ω) satisfy the Kramers-Kronig relationship and χ(ω) and χ(τ) are Fouriertransform pairs. For conducting materials, including plasmas, due to the pole atω = 0, the Kramers-Kronig relationship must be modi�ed. For the same reason theFourier transform of χ(ω) will yield a non-causal χ(ω) However, a causal expressionfor χ(τ) with Fourier transform equal to χ(ω) except at ω = 0.

χ(τ) =ω2p

υc[1− exp(−υcτ)]U(τ) (2.11)

U(τ) is the unit step function.

6

Chapter 3

Simulation Results

3.1 Demonstrations

To demonstrate the accuracy of FDTD is computing transient electromagnetic in-teractions with plasma in a straight forward manner a one-dimensional problem ofa plane wave incident on a plasma slab will be considered. If we allow only x-direction variation, and consider only Ey and Hz components, then for a plasmawith susceptibility as given by above equations 2.10 and 2.11, ε∞ = 1 and

En+1y (i) =

ε∞(i) +4χ0(i)

ε∞(i) + χ0(i)En

y (i)−1

ε∞(i) + χ∞(i)

n−1∑0

En−my (i)4χm(i)− 1

(ε∞(i) + χ∞(i))ε04x[H

n+ 12

z(i+1)−Hn+ 1

2

z(i) ]

(3.1)

and Hn+ 1

2

z(i+1) ≈ Hn− 1

2

z(i) −4tµ4x

[En+1y (i)− En

y (i)] (3.2)

where, x=iDx as before.For the plasma susceptibility, we readily �nd from above equation of 2.11

χ0(i) =ω2p

υc4t−

ω2p

υc[1− exp(−υcτ)] (3.3)

and 4χm(i) = −(ω2p

υc)exp(−mυc4t)[1− exp(−υcτ)]2 (3.4)

At �rst glance, it appears that evaluation of the summation (convolution) termof will require storing a large number of past times values of En

y . But since thesusceptibility function is an exponential, the summation can be updated recursively,and only one additional number need be stored for each electric �eld component ateach spatial index [2]. Thus we de�ne

ψny (i) =

n−1∑m=1

En−my (i)4χm(i) (3.5)

Where ψny (i) is a single real variable. The value of ψ

ny (i) at the present time step

is related to that at the previous time step by

7

ψny (i) = En−1

y (i)4χ1(i) + [exp(−υc4t)]ψn−1y (i) (3.6)

Using this recursive approach to evaluate the convolution summation, calcula-tions were made for a plasma slab 1.5 cm thick. The one-dimensional problem spaceconsists of 800 spatial cells each 75 µm thick, with the plasma slab occupying cells300 through 500. The time step is 0.125 ps. The plasma considered has a plasmafrequency of 28.7 GHz and a collision frequency υc of 2.0 × 1010 Hz. In order toeliminate zero frequency incident energy, calculations were made for a normally in-cident plane wave with a time behavior given by the derivative of a Gaussian pulse.Absorbing boundaries [3] were used at the terminations of the problem space toeliminate unwanted re�ections.

3.2 Results

In order to establish the accuracy we took the number of 04 snapshots for di�erentvalue of time steps such as for 400, 800, 1100 and 1600. Since we can observe that in�gure 3.1 for free space propagation and when it turns to enter plasma (anisotropic)medium it polarization automatically changes after that in �gure 3.4 it will continueswith changed polarization. If we did not use absorb boundary condition we will getunwanted re�ection. We can see the code and �gures below.

0 100 200 300 400 500 600 700 800−1.5

−1

−0.5

0

0.5

1

1.5 Plasma (Anisotropic) Material Free Space Free Space

Time Step = 400

Cell Index

Ele

ctric

Fie

ld (

v/m

)

Figure 3.1: Wave propagation for time step 400

8

0 100 200 300 400 500 600 700 800−1.5

−1

−0.5

0

0.5

1

1.5 Plasma (Anisotropic) Material Free Space Free Space

Time Step = 800

Cell Index

Ele

ctric

Fie

ld (

v/m

)

Figure 3.2: Wave propagation for time step 800

0 100 200 300 400 500 600 700 800−1.5

−1

−0.5

0

0.5

1

1.5 Plasma (Anisotropic) Material Free Space Free Space

Time Step = 1100

Cell Index

Ele

ctric

Fie

ld (

v/m

)

Figure 3.3: Wave propagation for time step 1100

9

0 100 200 300 400 500 600 700 800−1.5

−1

−0.5

0

0.5

1

1.5 Plasma (Anisotropic) Material Free Space Free Space

Time Step = 1600

Cell Index

Ele

ctric

Fie

ld (

v/m

)

Figure 3.4: Wave propagation for time step 1600

3.3 MATLAB Code

Faraday Rotation with Plasma (anisotropic) E�ect and augment 1D FDTDclear all; %Initialization of valuesSC=800; % Number of spatial cells%kc=�x(SC/2);Eo=8.854e-12; % Value of epsilondx=75e-6; % Thickness of each cell % Total thickness 800x75e-6 = 6 cmdt=0.125e-12;% Time step PcS=300;% Plasma cell e�ect start from 300PcE=500; % Plasma cell e�ect end at 500% Total plasma thickness 200x75e-6 = 1.5 cmfp=28.7e9; % Plasma FrequencyWp=2*pi*fp; % Radiant Plasma Frequency Vc=2e10; % Collision FrequencyNS=1600; % Number of time steps% Generation of zero matrixes for �eldEy=zeros(1,SC);Dy=zeros(1,SC);Hz=zeros(1,SC); % Generation of zero matrixes for single real variable "Shi"Sy=zeros(1,SC); Sy1=zeros(1,SC); Sy2=zeros(1,SC); % Initial conditiont0=100.0;

10

spread=25.0;% pulse width% Initial values for Absorbing boundary conditions to avoid re�ectionEy_low_m1=0.0;Ey_low_m2=0.0;Ey_high_m1=0.0;Ey_high_m2=0.0; T=0;%%%%%%Main Loop %%%%%%%%%%%%for n=1:NST=T+1; % Loop for free spacefor i=2:SCDy(i)=Dy(i)+0.5*(Hz(i-1)-Hz(i));end% Source Pulse %pulse=-2.0*((t0-T)./spread).*exp(-1.*((t0-T)./spread)^2);Dy(5)=Dy(5)+pulse; % loop for plasma e�ect %for i=2:SCif (i >= PcS & i <= PcE)Ey(i)=(Dy(i)-Sy(i));Sy(i)=(1+exp(-Vc.*dt)).*Sy1(i)-exp(-Vc.*dt).*Sy2(i)+((Wp^2).*dt/Vc).*(1-exp(-Vc.*dt)).*Ey(i);Sy2(i)=Sy1(i);Sy1(i)=Sy(i);else Ey(i)=Dy(i);end ; end% Absorbing boundary conditions to avoid re�ectionEy(1)=Ey_low_m2;Ey_low_m2=Ey_low_m1;Ey_low_m1=Ey(2);Ey(SC)=Ey_high_m2;Ey_high_m2=Ey_high_m1;Ey_high_m1=Ey(SC-1);for i=1:SC-1Hz(i)=Hz(i)+0.5*(Ey(i)-Ey(i+1));endplot(Ey,'b','LineWidth',1.5)%plot(Hz,'b','LineWidth',1.5) rectangle('Position',[300, -1.5, 200, 3],'LineWidth',1.5)grid on text((301),1.4,' Plasma (Anisotropic) Material');text((101),1.4,' Free Space');text((601),1.4,' Free Space');title(['Time Step = ',num2str(n)]);xlabel('Cell Index');ylabel('Electric Field (v/m)');pause(0.002) ; end

11

Chapter 4

Conclusion

Transient electromagnetic interactions with isotropic plasma have been explicitlycalculated in the time domain using a FDTD formulation. This technique involvesconvolution of the time domain plasma susceptibility with the electric �eld, updatedat each time step. Since the susceptibility function is exponential, the convolutionsummation can be evaluated recursively, with only one additional real variable storedfor each electric �eld component. In a plasma or any other anisotropic medium,the permittivity is a tensor (a matrix, not a scalar), which causes the di�erentcomponents of the E �eld to be coupled.

12

Bibliography

[1] K. S . Yee, �Numerical solution of initial boundary value problems involvingMaxwell's equations in isotropic media,� IEEE Trans. Antennas Propagat., vol.AP-14, pp. 302-307, May 1966.

[2] A. Bayliss and E. Turkel, �Radiation boundary conditions for vol. AP-22, pp.819- 821, NOV. 1974. wave-like equations,� Comm. Pure. Appl. Maih., vol. AP-33, pp. 707-725, 1980.

[3] G. Mur, �Absorbing boundary conditions for �nite-di�erence approximation ofthe time-domain electromagnetic-�eld equations,� IEEE Trans. Electromagn.Compat., vol. EMC-23,pp. 1073-1077, NOV. 1981.

[4] R. Luebbers, F. Hunsberger, K. Kunz, R. Standler, and M. Schneider, �Afrequency- dependent �nite di�erence time domain formulation for dispersivematerials,� IEEE Trans. Electromagn. Compat., vol. 32, pp. 222-227, Aug. 1990.

[5] L. D. Landau et al., Electrodynamics of Continuous Media, 2nd ed. Elmsford,NY:Pergamon, 1984.

13