Coplanar waveguide theory

For ideal, translationally invariant transmission lines with perfect conductors, that are in addition completely surrounded by a uniform dielectric medium, the principal wave that can exist on the transmission line is a TEM wave [51], where the electric and magnetic fields for an electromagnetic wave traveling

g Gold g w s w Polyimide high-resistivity Si ( >10'000ρ Ωcm) x y z ε =4.2r ε =11.9r tan δ =0.016 σ σ t tpoly tSi Currents: -Imw/2 Imw -Imw/2

Voltages: Ground Vmw Ground

t

20 mμ

Figure 2.1.1: (Left) Schematic drawing of a CPW structure that is inte-

grated on our atom chip. The thicknesses aret_{poly ≈}6µmand t_Si= 525µm. The conductivity of gold isσ= 4.5_×107_Ω−1_m−1 _{at room temperature. The}

voltages and currents indicated correspond to that of an ideal CPW mode. (Right) Exemplary microfabricated CPW structure (here, t= 800 nm). along y are given by

E(r, t) = Re [E(x, z) exp(iωt₋γy)], (2.1.1) B(r, t) = Re [B(x, z) exp(iωt₋γy)]. (2.1.2)

TEM waves are convenient because the distribution of the transverse fields can be calculated by combining computationally cheap 2D electro- and mag- netostatic analyses.

Conductor loss In micrometer sized transmission lines, conductor loss

plays an important role [109]. A finite conductivity σ not only leads to

attenuation but also changes the current distribution within the wire via the skin effect [104], which affects the electromagnetic field within and around the wire. For gold andω= 2π_×6.8GHz, the skin depthδskin =

p

2/ωµ0σ =

0.9µmis comparable to the dimensions of the micron sized structures on our chip (see Section 2.1.3).

Planar transmission lines, like a CPW, where the dielectric does not fill the complete surrounding, and/or where the conductors have only finite con- ductivity (see Figure 2.1.1) do not support true TEM waves. There are longitudinal field components at the dielectric-air interface as well as inside the conductors of finite conductivity σ. In the low-frequency limit, where λ_mw/(s+ 2w) 1 and for weak conductor losses, the dominant mode of propagation is very similar to the true TEM case, i.e. the longitudinal field components are much smaller than the transverse [110]. These are called

quasi-TEM modes [51]. Typically the condition

Figure 2.1.2: Visualization of the electric field distribution E(x, z) of the two modes supported by CPW structures. The modes are called (a) theCPW mode and (b) the slotline or odd mode. In our experiment, any coupling of

power from the CPW mode to the slotline mode is undesired. The potentials of the CPW wires are indicated on the bottom. The figure has been adapted from [111].

is sufficient to guarantee quasi-TEM behavior [109].

Since a CPW consists of three wires, it has two eigenmodes, the CPW

and theslotline mode (see Figure2.1.2). We excite only the CPW mode (by

pulling both grounds to the same potential), but discontinuities in the CPW geometry can couple power from the CPW mode to the slotline mode. This could be avoided by equilibrating the potential between both grounds by properly spaced conductive bridges [112, 113], but this is incompatible with the DC currents on the CPW grounds as required on our chip. We carefully designed and tested the structures on our chip to minimize such coupling between modes.

The electric and magnetic field distribution E(r) and B(r) around a waveguide can in general be computed with a computationally very expen- sive full-wave 3D simulation, using software packages likeHFSS1 _orYatpac.2 If the condition for quasi-TEM behavior is satisfied (Eq. (2.1.3)), then for a piece of transmission line with constant cross section, the fields E(x, z) and B(x, z) can be calculated using a 2D quasistatic simulation that takes conductor loss into account, as described in [109, 69, 114].3 Both methods yield a field distribution which takes the current distribution in the wires into account, that is strongly influenced by the skin effect [104].

At distances d from the waveguide, with ds, w, and t, the simulation

1_{From Ansoft, www.ansoft.com} 2_{www.yatpac.org}

R L

C G

Figure 2.1.3: Equivalent circuit for a piece of transmission line with con-

stant cross section. The parameters R, L, C and G (all defined per unit

length) are defined as integrals of E(x, z) and B(x, z). From these param- eters, the complex propagation constant γ and the complex characteristic

impedance Zc can be calculated. While for typical waveguides GωC, for

micron sized structuresR and ωLcan be of comparable magnitude.

can be simplified even further to a static simulation, where the waveguide currents are approximated by homogeneous currents that flow on infinitely thin waveguide wires, and B(x, z)is calculated via the law of Biot-Savart.

In many cases, where the structure is not translational invariant (therefore a 2D quasistatic or static analysis is inappropriate) and full-wave 3D simu- lations are too expensive, one can use 3D planar simulations using software packages likeSonnet4 _{orMicrowave Office.}5 _{Their outputs areS-parameters} [115] and current distributions on infinitely thin layers. The correspond- ing microwave magnetic near-field distribution can be calculated from the current distribution again using the law of Biot-Savart.

Equivalent circuit model A piece of a transmission line with constant

cross-section, where the (quasi)-TEM condition is fulfilled (Eq. (2.1.3)), can be related to an equivalent circuit model [109] (see Figure 2.1.3), which is very helpful for intuition. Such a circuit contains the series resistance R,

inductanceL, capacitanceCand shunt conductanceG, which are all defined

per unit length of the transmission line. These parameters are defined as integrals of E(x, z) and B(x, z) [51], and we calculate them using E(x, z) andB(x, z)from a quasi-static simulation. From these, the parameters which describe the wave propagation on the transmission line can be derived:

Zc=

r

R+iωL

G+iωC, (2.1.4)

γ =α+iβ =p(R+iωL)(G+iωC). (2.1.5) Zc = V_Imwmw is called the characteristic impedance, and is in general complex.

A complex-valued Zc leads to a phase shift between electric and magnetic

4_{www.sonnetsoftware.com} 5_{From AWR, www.awrcorp.com}

fields, but this does not influence the microwave potential for the atoms. Changes ofZcalong the transmission line lead to mismatch loss (reflections).

The quantity γ is the propagation constant, which contains α quantifying

microwave attenuation and β = 2π

λmw with λmw the wavelength of the guided wave. In a lot of cases, for the calculation of Zc and γ it is not necessary to

perform a numerical simulation. For most regimes, approximate analytical formulas exist [109], which yield useful results. A helpful tool for estimates is the program TX-Line.6

Scaling For an ideal CPW with perfect conductors,Zcis only a function

of s

s+2w [69, 108] and therefore the idealized CPW can be tapered without

change in Zc by scalings and w by a common factor. For a real CPW, the

same is approximately true.