This chapter describes the process of optoelectronic parameter extraction starting from the geometric parameters in components, followed by their use in circuit simulations. The techniques described can be used for components such as waveguides, couplers, Y-junctions, grating couplers, edge couplers, waveguide couplers, ring resonators, modulators, and filters and photo detectors, as described in Parts II and III. For fixed-cell designs, the components can be simulated once using the appropriate physical solvers and the optoelectronic parameters can be extracted for later use. For parameterized-cell designs, it is necessary to perform parameter sweeps over a range of possible geometries to create lookup tables or validated phenomenological models. We then show examples of how these extracted optoelectronic parameters can be used in photonic circuit simulations.

Need for photonic circuit modelling

Silicon photonics is a technology enabling large-scale integration of photonic components into photonic circuits. There is increasing interest in photonic integrated circuits in silicon systems for a variety of applications including on-chip and inter-chip communication systems. Broad adoption of silicon photonics technology for circuits and systems requires standardization in the design flow that is similar to what is available for electrical circuit design. This chapter describes methods for modelling optical circuits. The methods presented here are integrated as part of a design methodology that takes advantage of commercial electronic-design automation (EDA) tools for circuit design and layout, as described in Chapter 10.

Numerical methods such as finite-difference time domain (FDTD) and eigen-mode solvers coupled with solutions to optoelectronic equations are workhorses in silicon photonic component-level design. These methods unfortunately do not scale well as the number of components in the photonic circuit increases. Thus, modelling approaches that are computationally efficient yet accurately represent complex nanophotonic devices need to be employed in circuit simulations. In addition, the physical layout can affect the circuit response and needs to be considered. These issues have been addressed by the CMOS electronics industry, and are beginning to be incorporated into silicon photonic design.

This chapter describes the tools used by photonic integrated circuit designers, in particular for those focusing on the physical implementation of the design. We begin with a discussion of process design kits (PDKs) typically provided by the fabrication foundry. This is followed by a discussion of what electronic design automation (EDA) tools offer, including a library of components, schematic capture, (schematic-driven) layout, and design rule checking. We also provide suggestions for space-efficient photonic mask layout.

Process design kit (PDK)

A process design kit is a set of documentation and data files that describe a fabrication process at a semiconductor foundry and enable the user to complete a design. A typical PDK contains: documentation including technology details, mask layout instructions, and design rules; a library of cells such as modulators, detectors, etc.; component models and/or experimental data; and design verification tools. PDKs usually contain the foundry's proprietary information and trade secrets, thus are not always openly available.

In this section, we describe a Generic Silicon Photonics (GSiP) PDK, which is implemented in Mentor Graphics (Pyxis and Calibre) and Lumerical INTERCONNECT tools and is available for download. The purpose of this kit is to demonstrate the functionality of a silicon photonics design flow implementation, with no restrictions to its distribution. This kit can be adapted for different fabrication processes, and also provides insight into what PDK and libraries available today offer [1,2].

The components of the GSiP PDK include the following.

• Fabrication process parameters, mask layer table.

• Library: a small example library of components, including fibre grating couplers; waveguides, waveguide bends, and a splitter; a ring modulator; and an electrical bond pad.

1. – Component symbols for schematic capture: using the provided components and/or user-provided components, circuits can be designed at the schematic level.

2. – Component models: the library components include circuit models implemented in Lumerical INTERCONNECT.

3. – Component physical layout: mask layout for components is implemented in fixed-layout cells (i.e. GDS, e.g. Y-branch splitter) and parameterized (i.e. PCells, e.g. ring modulator).

• Schematic capture: this functionality allows the designer to create a schematic of their system.

In this chapter, we present an overview of the simulation and design tools useful for silicon photonics component and circuit design.

The design methodology for silicon photonic systems is illustrated in Figure 2.1. The order of the material presented in this book follows this illustration from top down. The design of passive silicon photonic components is considered in Part II, Chapters 3–5, while active component design is considered in Part III, Chapters 6–7. Model synthesis is described throughout these sections, and in more detail in Chapter 9, in which we describe optical circuit modelling techniques. Circuit modelling initially is focused on predicting the system behaviour in the presence of external stimulus, namely electrical and optical signals. Once a circuit is designed, the designer uses the schematic to lay out the components in a physical mask layout using a variety of design aids, as described in Chapter 10. This is followed by verification, including manufacturing design rule checking (DRC, DFM), layout versus schematic checking (LVS), test considerations, lithography simulation, and parasitic extraction. The results of the verification are fed back into the circuit simulations to predict the system response including effects due to the physical implementation (e.g. lithography effects, fabrication non-uniformity, temperature, waveguide lengths, and component placement). In this step, the circuit simulation takes into account not only the external stimulus but also the fabrication process (Chapter 11) and environmental variations. The design methodology was presented in Reference [1].

Optical waveguide mode solver

An eigenmode solver (or mode solver) solves for optical modes in a cross-section of an arbitrary waveguide geometry (or a 3D geometry) at a particular frequency. A waveguide mode is a transverse field distribution that propagates along the waveguide without changing shape; the solution is time-invariant. An example mode profile is shown in Figure 2.2a.

Eigenmode solvers determine time-harmonic solutions to Maxwell's equations in the frequency domain. Since they provide a solution for a single optical frequency, numerous simulations are required to obtain wavelength sweeps needed to study waveguide dispersion.

This chapter discusses one of the most challenging aspects of silicon photonics, namely the laser. It is highly desirable to have a silicon-compatible material that can provide optical emission and optical gain, for light sources (lasers, LEDs) and for on-chip optical amplifiers. Silicon is an indirect-band semiconductor, hence very inefficient at light generation. The common semiconductors used to make lasers at wavelengths where silicon is transparent (> 1.1 µm), such as In P-based compounds, have a crystal-lattice constant that is much larger than silicon, hence are difficult to grow on silicon. Germanium is the closest-matched material and has a smaller band gap than silicon; however, it is also an indirect-band semiconductor.

Present-day multi-project wafer (MPW) foundries (see Section 1.5.5) do not provide monolithic or hybrid-integrated lasers, and users rely on external lasers. While edge and grating couplers have both seen improvements in coupling efficiency, the lack of an on-chip source limits the potential applications of these chips. Laser integration is not widely available, and in turn the design of lasers for silicon photonics is an evolving research area. While there are design methodologies for the various laser-integration approaches described below, these depend on the type of approach, hence silicon photonic laser design is not a typical “text-book” topic.

This chapter describes the challenges associated with integrating lasers and optical amplifiers on the silicon photonics platform. It begins with the easiest method of getting light on the chip – namely using external lasers. We then discuss approaches with increasing level of difficulty: co-packaging, epitaxial bonding (hybrid lasers), monolithic growth, and germanium lasers.

External lasers

One approach for silicon photonic systems is to consider the laser as an optical power supply [1], similar to electrical power supplies, both delivering a constant amplitude.

There are several advantages for keeping the laser off-chip.

1. (1) Thermal management is simplified by keeping the laser off-chip. Lasers are typically 10%–30% efficient, hence an on-chip laser would be a contributing heat source; the heat output is 3–10 times higher than the optical power of the laser itself.

2. […]

While the stationary behavior of the laser was described in Part I, now we focus on its dynamical aspects. Some of them have been illustrated already in Chapter 19, but only in the single-mode regime and for class-A and class-B lasers, in which the only instability which arises concerns the trivial stationary solution and leads to the transition from the non lasing to the lasing state. Chapter 20, on the other hand, provided a general picture of single-mode and multimode instabilities in active as well as in passive systems, and of the structural relations which link single-mode and multimode instabilities. In this chapter we focus on the instabilities which arise in the laser and lead to spontaneous temporal oscillations and chaos.

We start in Section 22.1 with the linear-stability analysis of the trivial stationary solution in the general multimode case for the standard laser. The results of this analysis have been anticipated in previous chapters, but here you will find their derivation. The same problem is considered in Section 22.2 for the laser without inversion, but limited to the single-mode case under fully resonant conditions as in Section 17.5.

Of fundamental importance is the analogy between the single-mode laser model and the Lorenz model, which is prototypical for the general field of chaos. This matter is discussed in Section 22.3, and is immediately followed by the treatment of the resonant single-mode laser instability in Section 22.4.

On the other hand, multimodal instabilities in the ring laser are discussed in Sections 22.5 and 22.6. The first is devoted to the amplitude instability, completing the picture given in Section 20.2, and the second to the phase instability. The multimode amplitude instability is related to a classic dynamical phenomenon in lasers, mode-locking, which topic is illustrated in Chapter 23.

Finally, Section 22.7 is devoted to the matter of amplitude instabilities again, but in the case of a Fabry–Perot laser containing an ultrathin medium discussed in Section 14.5.

This chapter concludes the discussion about temporal instabilities and self-oscillations, considering the instabilities that arise in the systems analyzed in Chapter 13, with the exception of optical bistability, which has been discussed separately in Chapter 11.

Section 25.1 focuses on the laser with an injected signal. As already mentioned in Section 13.1, in this system self-pulsing instabilities are rather ubiquitous, and on exploring the parameter space one meets an extremely rich variety of oscillatory behaviors. For example, under extremely high-gain conditions, breathing, spiking and chaos have been identified over various ranges of the driving field strength [294]. In this section we show some results that arise under parametric conditions that are more accessible experimentally. We illustrate especially the coexistence, for the same parametric values, of different oscillatory behaviors that are reached by starting from different initial conditions.

On the other hand, Section 25.2 deals with a laser with a saturable absorber. In this system even the full scenario of stationary intensity solutions in the single-mode model is very complex. For the sake of simplicity, in this section we focus on a simpler rate-equation model that is obtained from the single-mode model of Eqs. (13.13)–(13.17) by adiabatically eliminating the atomic polarization of both active and passive medium. As in Chapter 13, we assume that the atomic transition frequencies are equal and also coincide with a cavity frequency. In this way all the variables in the model can be assumed real, and the linearization around the stationary solution described in Section 13.2 leads to a cubic eigenvalue equation. The spontaneous oscillations which arise from this instability are called repetitive passive Q-switching in the literature.

Section 25.3 considers the stability of the stationary solutions of the degenerate optical parametric oscillator described in Section 13.4 and outlines the scenario of period doubling and chaos that arises under appropriate parametric conditions.

In the first section we discuss the stationary solutions for a laser with an injected signal, the active counterpart of the optical bistability analyzed in Chapter 11.

Next, in Section 13.2 we consider another very interesting system, namely a laser with a saturable absorber, a free-running laser in which the optical cavity contains a passive medium in addition to the active medium. The single-mode model for this system can be constructed by combining the single-mode models for the free-running laser and for optical bistability. The discussion is then focussed on its stationary states.

In Section 13.3 we return to the single-mode model for optical bistability and, by introducing an appropriate “cubic limit” on the parameters in play, we derive a cubic model for dispersive optical bistability with two-level atoms. The stationary equation for such a model formally coincides with that derived and discussed for a Kerr medium in Section 11.2.

In the last section we start from the equations for a quadratic nonlinear medium in the degenerate configuration derived in Chapter 6 and combine them with the appropriate boundary conditions for the two field envelopes in play, assuming that the quadratic medium is contained in a ring cavity as happens in an optical parametric oscillator or in second-harmonic generation in a cavity. Next, by introducing the low-transmission limit and the single-mode limit we derive a two-mode model that describes the dynamics of a degenerate optical parametric oscillator or of second-harmonic generation in a cavity. Finally, for the case of a parametric oscillator we illustrate extensively the stationary solutions of the equations.

A laser with an injected signal

From a technical point of view, this system is interesting because of the possibility of stabilizing the frequency and the phase of separate lasers by taking advantage of the known affinity of nonlinear oscillators to lock onto a periodic forcing element.

In 1972 Spencer and Lamb [125] predicted the possibility of bistable behavior in a laser with an injected signal before the surge of interest in optical bistability in the passive configuration.

In Chapters 27 and 28 we discussed the topic of optical spatial pattern formation in planar cavities. In this chapter, instead, we consider the case of ring cavities with spherical mirrors, on the basis of the discussion of transverse modes in Chapter 26.

In the first section we derive, first of all, a set of modal equations to be coupled with the atomic equations, with a configuration similar to that of the modal equations of Chapter 12, with the difference that now transverse modes are included in addition to longitudinal modes. This provides an extremely complex context, and therefore in the following we focus on the case of quasi-planar cavities, in which it is possible, by introducing the low-transmission and single-longitudinal-mode approximations, to derive a single-longitudinal-mode model that represents, in the framework of cavities with spherical mirrors, the counterpart of the single-longitudinal-mode model for planar cavities discussed in Chapter 27.

In Section 29.2 we consider the simplest configuration, in which only the fundamental Gaussian transverse mode is assumed to have negligible losses, so that it is possible to consider a single-mode model that includes only the Gaussian mode. This is precisely the model considered in Section 24.3.1 for the comparison with experimental data concerning the single-mode instability of optical bistability. We calculate the stationary curve, and in the laser case we demonstrate the validity of the mode-pulling formula also in this configuration.

In the following section we consider, instead, the general case in which several transverse modes of a laser are involved and demonstrate that the modal equations admit multimodal stationary intensity states, a configuration that is not possible in the case of longitudinal modes, at least in the low-transmission approximation. In this phenomenon, which we call cooperative frequency locking, the modes agree a common oscillation frequency that is given by the average of the (mode-pulled) modal frequencies, with weights given by the modal intensities. Numerical and experimental evidence of cooperative frequency locking is provided.

Three elements, introduced in Part I of the book, play a key role in the book itself.

The first is the derivation of the Maxwell–Bloch equations, which is carried out in Chapters 2–4. In particular the derivation of the field-envelope equation, discussed in Section 3.1, is explained with special care. The derivation is based, of course, on the slowly varying envelope approximation and on the paraxial approximation. In textbooks these two approximations are usually discussed separately and rather briefly. We show that the two approximations are intimately connected, and only in this way it is possible to demonstrate that, while the second-order derivatives with respect to time and to the longitudinal coordinate z can be neglected, the second-order derivatives with respect to the transverse coordinates x and y must be kept. It is also possible to demonstrate that the term ∇(∇ · E) in the Maxwell wave equation for the electric field can indeed be neglected. An additional bonus of this approach is that, once one has shown in detail the derivation of the envelope equation in the simplest context, it becomes straightforward to generalize it to the more complex cases of frequency dispersion in the background refractive index (Section 5.4) and of quadratic and cubic nonlinearities (Chapters 6 and 7).

The second key element is the use of a multiple limit in the parameters, which in the literature is called the mean-field limit or uniform-field limit, whereas in this book we prefer to name it more precisely the low-transmission limit. This plays a central role when we derive modal equations and single-mode models from sets of equations with propagation. In this way one avoids, for example, steps like the phenomenological introduction of a damping term in the field equation(s) to describe the escape of photons from the cavity, and derives the damping from the boundary conditions of the cavity in a transparent way (Chapters 12 and 14).

In Chapter 9 we discussed the case of a nonlinear cavity containing an active medium. In this framework, the outstanding phenomenon which arises is laser emission, and we calculated the stationary solutions for the ring laser. In this chapter, instead, we turn our attention to the passive nonlinear cavity and in this framework the outstanding phenomenon is optical bistability, i.e. the existence of two distinct stable stationary states that coexist for the same fixed values of the parameters in play. This phenomenon can arise also in the case of an active system contained in a cavity, as we have already noted in Section 9.1 and as we will discuss further in connection with the laser with injected signal, but it became popular from the studies of nonlinear passive optical cavities in the 1970s and 1980s.

In contrast to the laser case, in the passive configuration we use an input coherent field, which is injected into the cavity. We discussed this case in Chapter 8 and derived a general equation for the transmission of the cavity (Eq. (8.23)), which we used for an empty cavity and for a linear cavity in Sections 8.4 and 8.5, respectively. In this chapter we will discuss the full nonlinear configuration.

It is customary to identify two extreme cases of optical bistability called absorptive optical bistability and dispersive or refractive optical bistability. In both cases an essential ingredient to obtain the bistable behavior is the feedback action exerted by the mirrors of the cavity. In the purely absorptive configuration, in which there is exact resonance among the frequencies of the input field, of the atoms and of the cavity, the other essential ingredient is the saturable nonlinearity of the absorption. In the dispersive case, instead, the main feature is the nonlinearity of the refractive index.

These two cases will be illustrated in Sections 11.1 and 11.2, respectively, both “exactly” and in the low-transmission limit.

Although the discovery of oscillatory instabilities (both single-mode and multimode) in free-running lasers was achieved in the 1960s, investigations on oscillatory instabilities in optical bistability did not take place until the 1970s and 1980s.

The first section of this chapter does not deal with oscillatory instabilities, but instead concerns an interesting dynamical phenomenon in optical bistability that arises when the input field approaches a boundary of the bistable domain, and is called critical slowing down. This has been investigated both theoretically and experimentally.

The remainder of this chapter is entirely devoted to oscillatory instabilities, following a historical order. With an inverse approach with respect to the laser case, we discuss multimode instabilities first. In Section 24.2.1 we start with the description of the instability which arises under exactly resonant conditions. While this is the counterpart in optical bistability of the resonant multimode laser instability discussed in Section 22.5, a striking difference from the laser case is that in the passive configuration there is no corresponding single-mode instability. This difference disappears, however, when we consider the generalization of the multimode instability of optical bistability to the detuned configuration. Finally we show that in the detuned case it has been possible to achieve a convincing experimental observation of the multimode instability in a long microwave cavity.

After some relevant miscellaneous considerations in Section 24.2.2, in Section 24.2.3 we address the subject of the multimode Ikeda instability which leads to period doubling and chaos and aroused a lot of interest in optical chaos. We emphasize also the intrinsic connections with the standard multimode instability discussed in Section 24.2.1, showing, in particular, that the Ikeda instability requires conditions far from the low-transmission limit, i.e. that ∝L is not small.

In Section 24.3 we discuss single-mode instabilities in optical bistability. In Section 24.3.1 we illustrate a configuration for the single-mode instability that includes also the Gaussian shape of the electric field in the beam section, and leads to periodic, close-to-sinusoidal self-pulsations.