Wave Manipulation by Topology Optimization - Solid Mechanics

2.3 Helmholtz’ equation 5
Wave-type u A B k0
E-polarized Ez μ −1
r ɛr ω √ μ0ɛ0
H-polarized Hz ɛ −1
r μr ω √ μ0ɛ0
Acoustic p ρ −1
r
κ −1
r
ω ρ0κ −1
0
Table 2.1 Parameter relations for general notation.
a physical field. The physical field at time t = 0 is easily obtained as the real part
of the complex phasor. Multiplying an obtained phasor solution with e jφ1 results in
the phasor field for the phase, φ1.
2.3 Helmholtz’ equation
Throughout the thesis we will assume wave propagation in structures with infinite
extension in one dimension. In-plane modes describe wave propagation in a plane
perpendicular to the infinite extension. Traditionally in a Cartesian coordinate
system in-plane modes are given in the (x, y)-plane with the z-axis being out-ofplane.
With the E =(0, 0,Ez) T field being out-of-plane and H =(Hx,Hy,0) T field
being in-plane the wave equation in (2.5) simplifies to the scalar Helmholtz’ equation
(here presented in a general form)
∇· A∇u + k 2 0Bu = 0 (2.6)
where u = Ez, A = μ −1
r and B = ɛc for the Ez polarization. In case of the magnetic
field being out-of-plane we instead use u = Hz, A = ɛ −1
c
and B = μr for the Hz
polarization. The big advantage is of course that the out-of plane scalar field only
needs to be computed in the (2D) plane thereby easing the computational efforts.
Having obtained the solution for the out-of-plane field the in-plane field can easily
be derived from Maxwell-Amperes or Faradays laws in equations (2.1) and (2.2),
respectively.
The scalar Helmholtz’ equation does not only govern the in-plane wave propagation
for an electromagnetic wave. It also covers acoustic wave propagation[6] as
well as certain types of elastic wave propagation[7], with appropriate choice of the
coefficients A and B, c.f. table 2.1. In case of acoustic wave propagation u describes
the pressure field, p, with A being the inverse of the mass density, ρr, andB being
the inverse of the bulk modulus, κr. Here we have related the acoustic material
properties to that of air at room temperature to cast the acoustic wave propagation
in the same form as the electromagnetic case. Thus k0 = ω/c is the wave number
in air with c =1/ ρ0κ −1
0 being the speed of sound in air, where ρ0 =1.204 kg m −3
and κ0 =1.42 · 10 5 Pa.
Appropriate boundary conditions along with the stated Helmholtz equation complete
the boundary value problem which governs the physics for the problems considered
in this thesis. In order to solve the boundary value problem numerically