Appendix A from the Ph.D. Thesis of Noah Zamdmer

[Ed. note: Noah Zamdmer was kind enough to provide this exerpt from his Ph.D. thesis (N. Zamdmer, "The Design and Testing of Integrated Circuits for Submillimeter-wave Spectroscopy", PhD Thesis, Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 1999). I have reformatted the text for this on-line version. The original appendix and the bibliography are available in gzipped Postscript format.]

Electromagnetic Simulation

We found the finite-difference time-domain (FD-TD) electromagnetic simulator LC [80] to be an extremely useful design tool for laying out coplanar waveguide (CPW) circuits with embedded photoconductors. It convinced us of the importance of designing circuits with evenly symmetric discontinuities to avoid mode mixing, which lead to the mode-preserving circuits discussed in Chapter 3. We can't use it to directly compare the measurements of Chapters 3 and 4 with simulation, because the circuits we designed are too large and finely structured (4.5 mm² chip with 3 m features and 0.3-m-thick metal) to simulate accurately in their entirety. However, it is worthwhile to discuss and provide examples of the use of LC, so that others may take advantage of this useful and free software that can be run easily at M. I. T.

A.1 Simulation of pulse propagation on a CPW

FD-TD simulation is an inherently simple technique¹. Maxwell's curl equations give the time derivatives of the electric and magnetic fields in terms of spatial derivatives of their complementary fields, and are naturally stated as difference equations for a discretized three-dimensional space with discrete time steps. To integrate the fields forward in time, the electric field is used to propagate forward the magnetic field, then the roles are reversed in the next time step to propagate forward the electric field. Various types of materials can be included in the simulation simply by attributing the appropriate values of conductivity, permittivity and permeability to each spatial node.

LC performs its simulation within a regular three-dimensional mesh of points, with a default spacing of 1 mm, and a default time step of 1.668 ps. It allows the user to define three different types of blocks of space within the mesh: materials, sources and probes. The material blocks define the physical properties of the volume enclosed by the block. Source blocks have user-defined electric or magnetic field waveforms on their surfaces, which radiate into the rest of the mesh. Probe blocks are particular regions of the mesh that the user wants to extract information from. This information may be the electric and magnetic field data of the simulation itself, or derived quantities such as the total charge within a probe block, which is derived from the electric field with Gauss's law. LC offers many different types of boundary conditions as well. We usually chose absorbing boundary conditions to minimize the reflections of electromagnetic pulses hitting the boundaries. We found the boundary condition called perfectly matched layers [90] to be the most effective, although it requires the most RAM and computing time.

To verify that LC can accurately simulate pulse propagation on CPW circuits, we simulated a pulse propagating on a straight section of CPW on a GaAs substrate. A top view of the simulation mesh is shown in Figure A-1, and a list of the simulation blocks is in Table A-1. Note that some of the blocks overlap. Since each mesh point can only be of one material type, each material block supersedes the ones listed above it. Some of the probe blocks also overlap, but this simply allows different quantities to be extracted from the same field data. The mesh is basically a 1:1000 replica of the main CPW of the circuit discussed in Chapter 3.

Figure A-1. Top view of the simulated CPW.

Table A-1. Blocks of the pulse propagation simulation, with the coordinates of block limits given in mm.
	x-	x+	y-	y+	z-	z+	waveform symbol
materials
air	-27	27	-25	25	-80	100
GaAs (=13.1)	-27	27	-25	0	-80	100
metal (=3.27x107 /m):
center conductor	-5	5	0	0	-80	100
+x ground plane	10	27	0	0	-80	100
-x ground plane	-27	-10	0	0	-80	100
sources
current source	-6	6	-1	1	-11	-9	i0
probes
center current probe 1	-6	-6	-1	1	9	11	i1
center current probe 2	-6	-6	-1	1	29	31	i2
side current probe 1	9	21	-1	1	9	11	is1
side current probe 2	9	21	-1	1	29	31	is2
voltage probe 1	4	11	-1	1	9	11	v1
voltage probe 2	4	11	-1	1	29	31	v2
front power probe	-25	25	-23	23	30	30	pf
side power probe	25	25	-23	23	-50	30	ps
top power probe	-25	25	23	23	-50	30	pt
bottom power probe	-25	25	-23	-23	-50	30	pb

The current source excites a gaussian pulse of current in the center conductor of the CPW. The transient current i₀(t) reaches a maximum of 1 A 0.956 ns after the start of the simulation, and has a full width at half maximum of 0.527 ns. Our simulation lasted 3000 times steps, which is about 5 ns. While LC has a few different types of waveforms it can generate itself, we supplied LC with an input file with the current waveform because it seems to have trouble padding a self-defined gaussian pulse with zero current following the main peak.

We used three different types of probes to characterize the pulse propagation. Current probes give the line integral of the magnetic field around the probe block. We defined the orientation of our four current probes to give the current in the z direction. Note that two current probes define loops that wrap around the center conductor, and the other two wrap around the portion of the ground planes that carry the most transient current. The voltage probes give the electric field integral in a given direction, which we defined to be the x direction. We positioned two voltage probes to give the transient voltage between the center conductor and one ground plane. The power probes are planes which give the surface integral of the Poynting vector. We placed the power probes around the current source to compare the power radiated and guided away from it.

The simulated waveforms of all the probed quantities are shown in Figure A-2.

Figure A-2. Simulated waveforms of all the probed quantities.

There are many signs that the simulation is accurate and physical. For example, the pulse's group velocity, as derived from the temporal difference between the peaks in i₀(t) and i₂(t), is 1.04x10⁸ m/s, which is close to expected value of 1.1x10⁸ m/s The ratio of the voltage peaks to the current peaks is 34.5 , which is close to the expected characteristic impedance of 44 [49]. We don't expect the derived characteristic impedance to be very accurate, because the mesh spacing of 1 mm is not much smaller then the 5 mm distance between each ground plane and the center conductor. The current pulse i₂(t) is slightly dispersed, with a faster trailing edge than leading edge. This agrees with theory, which predicts that the effective dielectric constant of a CPW is an increasing function of frequency [72] and agrees with previous experiments as well [73]. The guided power p_f is about 20 times greater than p_f, the power radiated into the GaAs substrate, and much greater still than the power radiated into air and to the side. This is as expected, because the pulse width of about 55 mm is greater than the 20 mm distance between ground planes, thus the ground planes effectively shield the substrate an air from radiation. That more power is radiated into the GaAs substrate than into air is also as expected, since GaAs presents a lower impedance to the radiating current than air does [44]. Also as expected, the currents i_s1 and i_s2 in the ground plane are very similar to the currents i₁ and i₂ in the center conductor, but are about half the magnitude and have the opposite sign.

There are a few non-physical results that must be noted. The boundaries at z=-80 mm and z=100 mm both produce reflections that are evident in the current and voltage waveforms near a time of 3 ns. Fortunately, these features are of low amplitude, and have substantial temporal separation from the main peaks. The difference in amplitude between the excitation current i₀ and the propagating currents i₁ and i₂ is also non-physical. The amplitude of i₂ is slightly less than that of i₁, due to both resistive and radiative attenuation, but both are much less than the amplitude of i₀. The power probes show that radiation into the substrate does not account for the difference. Perhaps the difference arises because an isolated current source is intrinsically non-physical.

Since the absolute magnitude of the waveforms calculated by LC are slightly dubious, the conclusion we draw from the above investigation is that LC is very good for comparing the relative transmission properties of different discontinuities in a transmission line circuit. To make the main features of waveforms distinguishable from those features due to reflections at simulation boundaries, we need simply position our probes far from the boundaries.

A.2 Pulse propagation through a discontinuity

One structure we would like to know the transmission properties of is the double photoconductor described in Chapter 3. The data presented in the chapter suggest that this structure transmits a large portion of an electromagnetic pulse incident from the main CPW of the experimental circuit. To verify this, we created the simulation mesh shown in Figure A-3, which is basically a 1:1000 replica of a section of the experimental circuit with a double photoconductor. A list of the simulation blocks is in Table A-2. We created another simulation for comparison with the same size and probes as the one shown in Figure A-3, but with a straight CPW and no discontinuity.

Figure A-3. Top view of the simulated CPW with a double photoconductor. The colors and symbols for materials, sources and probes are the same as in Figure A-1.

In Figure A-4 we compare the transmitted current i₁ and the radiated and guided power of the two simulations. Note that i₁ for the simulation with no discontinuity is mainly a delayed version of i₁ and i₂ of the simulation discussed in the previous section, and shown in Figure A-2. This is to be expected, because of the greater distance between source and probe for the later simulations. Note also that the discontinuity causes only a 39% reduction in the transmitted current. We see that more power is transmitted through the double photoconductor than diverted to the parasitic CPWs in the x direction, which are used to bias the double photoconductor.

Figure A-4. Comparison of current and power waveforms for simulations with and without a double-photoconductor discontinuity. p_f is the power in the probe plane at extreme +z, p_s is the power in the plane at extreme +x, p_b is the power in the plane at extreme -y, and p_b is the power in the plane at extreme +y (see Figure A-3 and Table A-2).

With the simulator we can see in detail how the pulse on the main CPW couples to the CPWs orthogonal to it. Figure A-5 (a) shows the currents i_sn, i_sn andi_sn in the CPW orthogonal to the main CPW in the z direction. The waveforms show that two pulses of opposite sign are launched onto the parasitic CPW, with a small time delay between them. These pulses in i_sn and i_sn are induced by the currents in the ground planes of the main CPW, as shown in Figure A-5 (b). i_sc is bipolar becuase it is composed of separate pulses of opposite sign induced by the currents in the nearby ground planes. The odd propagating mode of the main CPW (see FIgure 2-19) does not couple to the odd propagating mode of the parasitic CPW, but rather to the slotline-like modes localized on either side of the parasitic CPW. This must be kept in mind when considering what impedance the parasitic CPWs present to the main CPW.

Figure A-5. (a): Currents in the CPW used to bias the double photoconductor. (b): Circuit diagram showing the currents induced by the source pulse, which explains why i_sn and i_sn are of opposite sign.

The simulation reveals that the double-photoconductor structure is fairly transmissive, and gives some insight into how it interferes with pulse propagation. To learn even more from simulation, we could vary the simulated discontinuity to maximize the transmitted power, so as to avoid diverting power to the parasitic CPWs or reflecting it back to the pump photoconductor. As a first step in this direction, we created a simulated circuit identical to the one in Figure A-3, except that its CPW in the z direction at high-z has a lower characteristic impedance Z₀ than the same CPW of the original simulation. The dimensions of the original simulation's high-z CPW are 10 mm-10 mm-10 mm, while those of the new simulation are 5 mm-10 mm-5 mm, the same dimensions as those of the low-z CPW of both simulations. In Figure A-6 we compare the transmitted power and current waveforms of the two simulations. The original simulation, with the high-z CPW with the higher Z₀, has less transmitted current but more transmitted power. Apparently increasing the load seen by a pulse incident on the discontinuity makes the load a better match to the CPW that the pulse is incident from (the main CPW at low z).

Figure A-6. Comparison of transmitted current i₁ and transmitted power p_f of simulations with CPWs of different characteristic impedance at high-z.

As stated in Sec. 3.4, the best way to experimentally verify FD-TD simulations of discontinuities is to use a mobile electric field probe to compare the electric field waveforms in different regions of a circuit, much as we use multiple probes in our simulation. While we did not perform such experiments, most of LC's calculations are physically reasonable, and give us confidence that LC is a useful tool for the design of CPW circuits. To future users of LC at M. I. T., we must add that LC only runs on SGI computers, the fastest of which (as of May 1999) are the O2 machines in the Athena cluster in room 4-035.

LC Model Files

Model A-1	Model A-2