Towards an Integrated Multi-Physics Component & Circuit Solver#
It is important to understand the limitations of multi-physical solvers in order to know exactly how inaccurate our results are, and what we are really simulating.
Optical#
This references are based on Chrostowski’s Silicon Photonic Design Chapter 2:
If we observe into the cross section of a waveguide towards the transverse propagation of light, we can observe that the optical wave modes are propagating invariantly in time. Maxwell’s equvations time-harmonic solutions are solved into the frequency domain by these solvers. this solution is for a specific frequency. To analyse a broadband pulse at multiple wavelength modes, the mode solver has to run a frequency sweep.
Full solutions include Finite Element Method and Finite Difference algorithms. Approximated solutions include the Effective Index Method.
Finite Element Method solvers are good for unstructured multi-geometrical shapes or 3D modelling.
Finite Difference solvers are good for high-index contrast structures.
3D FDTD#
Three-dimensional Maxwell’s equations are solved by this tool. We explore light interaction with sub-wavelength features such as those that can be lithographically fabricated in a silicon chip. Accuracy converges to an exact solution as the mesh size is reduced, hence this is the most accurate solver. Simulation time step can be sub-femtosecond so large processing units are required for large scale use. It accurately models a wide-optical-bandwidth response, and can generate scattering parameters for its simulations.
Used to calculate:
Waveguide bends
Coupling coefficients between waveguides
Bragg reflections and transmission
Field profiles and reflections
Procedure:
Generate material definitions
Generate optical structure
Simulation volume is defined. Mesh \(\Delta x\) >14-18 points per wavelength. Simulation time \(\propto \frac{1}{\Delta x^4}\).
Define matching boundary layers.
Optical sources are injected. Because FDTD is a time-to-frequency simulation, a short pulse length translates to a broad optical spectrum.
Monitors are used to measure electronic and optical field quantities. Fundamental monitor is time-domain monitor, and this is wavelength dependent.
Perform convergence simulations to make sure there are no numerical errors.
Perform analysis on the signal profiles.
Common solvers that implement this:
- Tidy3D
- Lumerical FDTD.
2.5D with Effective Index FDTD#
Can be similar to 3D FDTD when there is no coupling between TE and TM modes.
Used to calculate:
Planar photonic integrated circuits
Waveguide systems
Ring resonators
Planar omi-directional propagation without assumptions like an optical -axis
Common solvers that implement this:
- Tidy3D
- Lumerical MODE.
Eigenmode Expansion Method EME#
The propagation of light in local fields is decomposed into modes at that position. Uses the scattering-parameters to connect the next section of the device which makes it inherently bi-directional as it includes both forward and backwards propagation for an infinite number of modes.
Used to calculate:
Waveguide sections
Directional couplers
Interferometer region in a MMI
Electronic & Optical#
This is the type of modelling that we particularly care about. However, it is important to note the importance and effect of each individual simulation implementation accordingly. This is where the multi-physical component-level domain simulations are integrated.
Mixed-Signal Electronic-Photonic Co-Simulation#
Time Synchronisation Complexities#
One of the main complexities of this type of simulation software is the mapping of different dimensions of time. For example, the time of the laser wave is has its electromagnetic components changing in the femtosecond regime, whereas digital signals may be changing in the nanosecond regime. This inherently creates a level of complexity and computational optimisation required to simulate photonic and electronic networks that standard electronics does not have to solve.
Electronic and optical simulations need to be synchronised into a single time domain to have a continuous signal. This means that it is necessary to integrate electrical and optical solvers in some form and to some level of reasonable translation. It is not necessary to simulate a whole system at picosecond resolution to observe photonic transient effects, but to observe transient effects, picosecond resolution might be desired - whilst steady state might not.
This leads to a complex proposition: how to integrate transient and steady state time-domain solvers to simulate both electronics and photonics systems?
Tools Integration#
The implementation mechanism followed is to create parametric SPICE components that can be closely integrated with our existing device models and have a mapping to the electronic and photonic simulation software. The way this is implemented also follows microservices architectures as to minimise the computational architecture required for these systems.
However, one main aspect of complexity is the event-driven nature of multi-physical simulations. For example, a photonic pulse detection event creates an analog stimuli in time which triggers a digital simulation event. As such, simulating accurately, multi-physical triggers is a tricky implementation problem due to the linking events between the domain solvers.
Why is this analysis important? Photonic operations and electronic operations may not be occurring at the same rate. The optical signals may be changing when the electronics is steady, the electronic signals may be changing when the photonics is steady, or the electronics and the photonics are changing both at the same time. These are the potential scenarios that time-dependent simulations must account for.
Integration Schemes Discussion & Philosophy#
To try to build a monolithic simulator that can solve for all of these domains at the same time limits the potential further multi-physical modelling of more complex systems, such as quantum states. As such, we want to be as open and as modular as possible in order to make this simulation tool as useful as possible.
References on mixed-signal simulation with an open-source focus:
Of all the open-source mixed-signal simulator software, ngspice is in strong active development by a large amount of organisations. It makes sense to bootstrap on top of it for these simulations. However, we need to implement event-driven simulators accordingly with both photonic-event driven inputs.
Implementation Scheme#
ngspice can be treated as a dominant simualator. As of writing, this is based on the latest version 40 manual . It is capable of implementing event-driven mixed-mode models as described in Chapter 12. This is fundamentally based on building .model directives in the SPICE netlist, and compiling ngspice with the --enable-xspice in our simulation .configure command.
Legacy - Considered Potential Implementation Schemes#
Discretisation of the Simulation#
This potential implementation consists of:
Modularise the time-domain operations
Compute photonic and electronic transients only when they are changing.
Append the corresponding data into a total global time with various levels of resolution depending on the reference signal
The goal is to enable to minimize computational cost of simulating such a system, whilst being flexible and simple enough for any potential system simulation configuration. This means that we are not simulating an electronic and photonic system at a femto-second resolution, but we still resolve the time-scales throughout signal transitions if we desire.
The electronic simulation implementation is what could be called micro-SPICE in the sense that it is a minimal implementation of transition simulation for a circuit. The ideal is that transients are computed in high resolution and steady-state is computed operationally.
However, this scheme is invalid when considering feedback signals.
Assuming Independent Solvers#
One potential implementation of operation specific electronic transient modelling is fundamental. Say, an electronic circuit is idle. This means for the period of time that this circuit is idle and the environment is constant, it can be approximated that the circuit is in a very similar constant state with some level of variation dependent only on external factors eg. temperature or so on. We can also say that the initial conditions of the circuit, say some DC biasing, voltage-controlled-current-sources or current-controlled-voltage-sources DC, or similar are controlled by signals that may be external to the circuit in question. Now, this is valid whenever there is no remaining transient memory effects on the initial condition of the circuit, in this case you need to model full transient effects as in normal SPICE.
Say, our electronic circuit is initially in idle state (which we control the external parameters that affect this state), and we can control initial conditions such as the DC biasing and other time-independent states of our circuit. An opto-electronic signal creates a time-dependent transient input signal, and we begin our analogue simulation of our circuit, the pulse has an end width, and the time-dependent input signal reduces to back to a zero or steady noisy time-independent input. The internal time-dependent properties of the circuit reset to their time-independent steady-state.
Then there is the other aspect of complexity, which is how to enable the multi-domains to access the other domain simulation data state.
I know some experienced people who are reading this are probably
thinking: why bother even doing this, surely tools such as Cadence
already solve this through their mixed-signals AMS simulators. However,
any extension of this is strongly limited by their often incomplete
documentation and closed-source software. It is also limited by
excluding people who do not know Verilog-A (even if I do), such as many
photonics engineers who design their chips in Python. This is some
motivation behind a piel design flow.
One way is simply not allow them to access any other domain than their own.
Motivation for Integrated Solvers#
At the digital time-step, there is a possibility that an analogue signal contains memory or previous states from the transitions. This creates an aspect of complexity in modelling these systems. It means that the time-synchronisation between the digital and analogue system must be possible. The aspect of complexity is the multi-domain interaction between the solvers, for example, if the digital solver is triggered by a photonic-analogue signal or related.
Co-Routine Dominant Solver Limits Feedback#
However, this is possible by implementing subroutines. In cocotb,
the simulations are run as asynchronous coroutines with an analog python model. We can follow this
exact principle in terms of implementing a multi-domain simulatior. Each
cocotb simulation has a Timer, and it is possible that for every
step in time that we desire, we can implement another subroutine. This
would be easy to do if the photonic and analogue simulations were
time-independent because then the digital signals control the
operation flow, and it can just be considered a functional-based system
dominated by the digital Timer. it is not particularly complicated
either to have feedback in between digital-photonic logic as through
that subroutine, we have full control over the triggering of digital
signals and related. However, my question of complexity really returns
to the analogue interaction of the signals. If it is digitally-directed,
and there is no analogue feedback interaction onto the digital logic,
then the co-routine would not have to change and is not affected other
than driving the analogue signals.
Part of the question becomes on having to run an analogue simulation at
the same time as the digital one in order to verify that there would not
be remnant analogue memory in between the driving pulses. This makes
sense in particular when the digital Timer clock overlaps in between
the analogue signal as this changes the initial state of the analogue
simulation. We could keep a track of the initial state, but it might be
a bit complex even when considering a standard RC network and driving
pulses accordingly. The other way to do so, is to perform the SPICE
simulation, and discretize at particular points in time as is the way
with AMS.
Photonic Time Delay Synchronisation#
Another complexity of simulating these systems is that photonic pulses
might also be propagating in time alongside the electronic signals.
sax already implements some functionality to analyse the time-delay
of such photonic circuits, but a resolution mechanism is required to
interconnect these circuits together and a corresponding time-delay
model needs to be provided to the components.