In order to analyze the experimental data and understand the physics involved in the 2D structure, we have to do the calculation based on modal expansion. The unit system we adapt here is SI units.
First of all, we divide the whole system into three regions which are I, II, and III respectively as shown in Fig. 4.1(b). RegionI is the region of reflection where the EM fields can be expanded by the eigenmodes of Helmholtz’s equations in free space and the incident light is given in this region. RegionII is the structure region where the EM fields inside the hole can be expanded by rectangular waveguide modes of perfect electrical conductor (PEC). RegionIII is the substrate region which can be seen as another kind of free space except the light velocity there has to be divided by refraction index of the substrate material. The 2D structure under study is an infinite array of holes drilled periodically in a metal film of thicknessh. Fig. 4.1(a) depicts the definition of the primitive unit cell. The primitive unit cell is defined as a rectangular with length and width being A, B respectively, while the length and width of the rectangular hole inside the unit cell is denoted by a, b respectively.
We start from the Maxwell’s equations for complex time-harmonic fields in source free case:
( ) iωμ0 ( ) 0
∇×H r − E r =
( ) iωε ε0 ( ) ( ) 0
∇×E r + r H r =
(4.1) (4.2)
0 ( ) ( ) 0 ε ε
∇⋅ r E r = ,
where the related coefficients are the same as the standard notations in any electromagnetic textbook. In the following derivation, the only approximation we make is that the metal is considered to be perfect electrical conductor (PEC). This is a good approximation because our system is operated in THz region, where the skin depth of the metal with good conductivity is about only several tens of nanometer.
Combine (4.1) and (4.2) , we obtain
After further manipulation we can obtain two Helmholtz’s equations,
2
Both electric and magnetic fields satisfy the above two Helmholtz’s equations in each region. At any boundary the EM fields in between the regions satisfy the following boundary conditions,
where the numbers of subscripts mean different regions, n is the unitary vector normal to the surface, and Js and ρs are surface current and surface charge respectively. In the following, we will first attack this problem at each region individually and then match the boundary conditions at the interface of each region.
All EM fields at each region are governed by (4.7) and (4.8) . To simplify the derivation, we learned from Garcia’s paper in 2008 [16] to use Dirac’s notation for representation of each field, or eigenfunction. For example, the electric field can be written as E r( )= r E// E z( ) . The reason why we deliberately separate the z-dependent function from Dirac’s notation will be apparent in our derivation soon. To further simplify the derivation, according to Garcia, for the same mode E-field and H-field have the following relation
mode Ymode mode
− ×z H = ± E ,
where Ymode is the modal admittance. The choice of + or – depends on the propagation direction +z or –z respectively. Consequently we can consider only the eigenmodes of electric field.
Basically, we can categorize the system into two types, the free space type and the inside hole type. In free space, the eigenmodes of (4.7) are plane waves obeying Bragg diffraction law, i.e., k=k0+G, where k0 and k are incident and reflected wavevectors respectively, and 2 (m n )
A B
π
= +
G x y . In the previous expression ( , )m n denoting the diffraction order is a pair of arbitrary integers. Moreover, the plane wave eigenmodes can be further decomposed to two orthogonal functions based on the directions of polarization, p and s. The definitions of p- and s-polarizations are shown
(4.13)
Fig. 4.1(a) Top view of unit cell of the rectangular lattice.
Fig. 4.1(b) Schematics of the system under study.
Fig. 4.1(c) Schematics of the system with incident light.
which can be denoted by TE,pq and TM,pq , where ( , )p q represents a certain order of waveguide mode.
After choosing the expansion basis with consideration of Bloch’s theorem, we can write down the EM fields in different regions:
z Region I, (z<0)
Given a normal incident wave, the EM fields can be expressed as follows,
{ }
(1),00 1mnp mnp mns mns
mn
( )
coefficients to be calculated.z Region III, (z>z2)
Now we write down the real space expression for each eigenmode explicitly:
(4.20)
2 2
There is one important thing that has to be mentioned. For TE waveguide modes,
// TE,pq
surface have to be continuous everywhere on the surface and magnetic fields parallel to the surface have to be continuous on the holes area. Therefore, as Garcia did, we project the matching equations onto plane wave eigenmodes for electric fields and rectangular waveguide eigenmodes for magnetic fields. Besides, we can match the boundary only on one Wigner-Seitz unit cell, i.e., R=0, because our structure has perfect periodicity and thus satisfies Bloch’s theorem. Therefore we can abbreviate TM,pq,R=0 and TE,pq,R=0 to TM,pq and TE,pq respectively. The boundary conditions are matched at two interfaces:
At z=z1 (z1− = −z2 h): area of unit cell to obtain
{ }
an area of a hole to obtain of unit cell to obtain
( )
With the four simultaneous equations (4.36), (4.40), (4.43), and (4.46) , we can (4.47)
[ ]
Thus (4.36), (4.40), (4.43), and (4.46) can be expressed as(4.58)
[ ] [ ] [ ][ ] [ ][ ][ ]
a1 + b1 = M a2 + M E b2[ ] [ ] [ ] [ ]
M † Y1{
a1 − b1}
=[ ] [ ] [ ][ ]
Y2{
a2 − E b2} [ ] [ ] [ ][ ] [ ]
a3 = M { E a2 + b2 }[ ] [ ][ ] [ ] [ ][ ] [ ]
M † Y3 a3 = Y2{
E a2 − b2}
.With (4.53) being given and the four matrix equations (4.59), (4.60), (4.61) and (4.62) , we can determine the four column matrices (4.54), (4.56), (4.57) and (4.58) . The last step is to calculate the transmittance of this system. Because the system is considered to be lossless, the transmittance in each region must be equal. We use Poynting’s theorem to calculate the energy flux through a unit cell at a given z,
( ) ( )
*( )
unit cell
1Re ˆ , , , , d d
J z 2 ⎧ x y z x y z x y⎫
⎡ ⎤
= ⎨ ⋅⎣ × ⎦ ⎬
⎩
∫∫
z Ev Hv ⎭.Finally, the transmittance can be obtained by dividing ( )J z by incoming energy flux
J0.
(4.63) (4.62) (4.61) (4.60) (4.59)