The optical mode in the cross-sectional area

There are numerous optical mode solvers that deal with optical eigenvalue problems in a scalar form as shown in equation (2.34), or in a more comprehensive vectorial version, in order to obtain the optical field distribution (i.e., the optical mode) in the cross-sectional area. In contrast to dealing with a 1D slab waveguide problem where an analytical approach (such as a transfer matrix method) exists, a general 2D eigenvalue problem has to be treated numerically. Among many different numerical approaches, the finite difference method (FD) seems to be a popular one for its balance between implementation complexity, computational efficiency and accuracy.

In the 2D domain, equation (2.34) is posed as a boundary value problem of a PDE of elliptical type. Hence, discretization of equation (2.34) under the FD scheme is fairly straightforward since stability is not a concern. For example, we can always stay with the center discretization scheme to gain a second order accuracy. The boundary treatment, however, is crucial especially for those 2D structures with piecewise uniformity, which is most commonly seen in semiconductor optoelectronic devices.

At physical boundaries inside the computation domain, the refractive index is discontinuous, whereas as the solution to equation (2.34), the scalar optical mode must be continuous. Therefore, special treatments are necessary at such boundary points. For example, in many numerical solvers we do not select any mesh grid point at the boundary points to avoid refractive index assignment at these boundaries.

Over the past 30 years, the world has witnessed the rapid development of optoelectronic devices based on III-V compound semiconductors. Past effort has mainly been directed to the theoretical understanding of, and the technology development for, these devices in applications in telecommunication networks and compact disk (CD) data storage. With the growing deployment of such devices in new fields such as illumination, display, fiber sensor, fiber gyro, optical coherent tomography, etc., research on optoelectronic devices, especially on those light emitting components, continues to expand with the pursuit of many experimental explorations on new materials such as group-III nitride alloys and II-VI compounds and novel structures such as quantum wires, dots, and nanostructures.

As the manufacturing technology becomes mature and standardized and few uncertainties are left, design and simulation become the major issue in the performance enhancement of existing devices and in the development of new devices. Recent progress in numerical techniques as well as computing hardware has provided a powerful platform that makes sophisticated computer-aided design, modeling, and simulation possible. So far, the development of optoelectronic devices seems to replicate the history of electronic devices: from discrete to integrated, from technology intensive to design intensive, from trial-and-error experiments to computer-aided simulation and optimization.

The purpose of this book is to bridge the gap between the theoretical framework and the solution to real-world problems, or, more specifically, to bridge the gap between our knowledge acquired on electromagnetic field theory, quantum mechanics, and semiconductor physics and optoelectronic device design and modeling through advanced numerical tools.

Design and modeling of the active region for optical gain

The active region material

Specification of the lasing wavelength and substrate material availability usually dictate the active region material selection. However, in some wavelength bands, the selection of the material is not unique. For example, either one of the following systems: InGaAsP on InP substrate, InAlGaAs on InP substrate, InGaAsN on GaAs substrate, and InAsPN on GaAs substrate, can be chosen as the active region material for semiconductor laser diodes emitting in the 1300 nm band. In this section, however, we will compare only the first two systems, which seem to be more mature in production. Table 10.1 summarizes the features of the two active region structures made of the two different material systems.

Unlike the InGaAsP/InP heterojunction with a smaller conduction band offset (∼36%) compared to the valence band offset (∼64%), the InAlGaAs system has a larger offset (∼71%) on the conduction band side. As such, the InAlGaAs system has a better confining effect on the injected electrons. Hence it has better temperature characteristics as a result of the effective reduction of electron current leakage, which is the dominant form of leakage due to the high electron mobility [2].

In accordance with the active region designs in Table 10.1, the calculated transverse lectric (TE) mode material gain profiles are shown in Fig. 10.1. Since we have used the same well thickness and the same number of QWs in these two material systems, the sheet carrier density in the InAlGaAs and InGaAsP QWs is the same at each carrier injection level.

The integrated semiconductor distributed feedback laser and electro-absorption modulator

The device structure

A schematic structure of an integrated semiconductor DFB laser and EAM is shown in Fig. 12.1.

The cross-sectional layer structures in the DFB and EAM sections are shown in Tables 12.1 and 12.2, respectively.

Both sections of this device are made of a ridge waveguide with the same ridge width of 2.0 μm, but with different ridge heights of 2.0 μm in the DFB laser section and 3.0 μm in the EAM section, respectively. Thus, the etched ditch stops on top of the laser active region but goes through the modulator active region. This design makes a better match of the effective indices between the DFB laser and EAM sections and hence the reflection from the interface between these two sections will be effectively reduced. The deep etching in the modulator section also improves the far-field pattern emitted from the front facet. Finally, by removing the EAM active region outside the ridge, the EAM insertion loss will reduce and its extinction ratio will increase as shown in Section 11.1.3.

The laser section and the modulator section are 300 μm and 200 μm long, respectively, the electrodes are isolated by a 10 μm wide and 1.0 μm deep trench etched across the ridge. The DFB laser has a uniform purely index-coupled grating with a normalized coupling coefficient of 4. This design is a balance between the output optical power, immunity to feedback from the modulator, and the lasing wavelength uncertainty in a range roughly equal to the Bragg stop-band width.

A general approach

The material gain treatment

As explained in Section 7.3, it is neither feasible nor necessary to compute the material's optical properties through the physics based gain model in an “online” manner. It is not feasible because of the huge number of times that the gain model has to be invoked. It is not necessary because on many of the times that the model is called up it provides exactly the same results.

For this reason, we take an “offline” approach by calling up the physics based gain model at a coarse mesh grid constructed by multiple variables (i.e., the electron and hole densities, the temperature and the frequency) which covers the entire device operation range, and by establishing a set of analytical formulas which reproduce the required material optical property (i.e., the stimulated and spontaneous emission gains and the refractive index change) in the device operation range. By assuming a universal polynomial, exponential and rational dependence on the carrier density, temperature and frequency, respectively, such formulas are therefore parameterized with the unknown parameters obtained from searching for the best fit between the results from the formulas and from the rigorous calculation on the mesh grid points. Interpolations might be necessary to refine the mesh grid before such a fitting.

Once the analytical formulas are extracted, they will be used as the material model to replace the rigorous model for calculation of gains and refractive index change in an “online” manner.

Figure 9.1 (a) and (b) show comparisons of the stimulated emission gain and refractive index change calculated by the rigorous gain model and analytical formulas.

For a given active region structure with adjacent layers, the rigorous gain is obtained through the following procedure.

Based on the single electron band structures that have been solved in Chapter 3, we are ready to derive models for calculation of material optical properties.

A comprehensive model with many-body effect

Introduction

As summarized in Table 4.1, a real world physics process is described in the time–space domain with time and space vectors as arguments. Through Fourier transform, we can also analyze this process in the frequency (energy) – wave vector (momentum) domain with the frequency and wave vector as arguments. The original arguments and their alternatives form conjugate pairs which satisfy Heisenberg's uncertainty relation.

Table 4.2 shows all possible combinations for describing a physics process in different domains.

Selecting a suitable domain to describe a physics process may bring the following advantages:

• a derivative operation in the original domain (time or space) will become an algebraic operation in the alternative domain (frequency or momentum). Therefore, under an integration transform with the core function obtained from solving an eigenvalue problem, a linear differential equation can be converted to an algebraic equation. In particular, if the linear differential equation contains only invariant parameters, this integration transform is the Fourier transform where the plane wave function serves as the core function, as it is the eigenfunction of any linear and parameter invariant system.

• any periodic function in one domain will be a discrete function after being transformed into its conjugate domain. For example, a periodic time domain function must have a discrete frequency or energy spectrum, a periodic space domain function must have a discrete wave vector or momentum dependence, and vice versa.

The underlying physics in device operation

Figure 1.1 shows the major physical processes and their linkages in the operation of optoelectronic devices.

To capture these physical processes, we need the following models and knowledge:

1. a model that describes wave propagation along the device waveguide (electromagnetic wave theory);

2. a model that describes the optical properties of the device material platform (semiconductor physics);

3. a model that describes carrier transport inside the device (quasi-electrostatic theory);

4. a model that describes thermal diffusion inside the device (thermal diffusion theory).

Therefore, the above four aspects should be included in any model established for simulation of optoelectronic devices.

Modeling and simulation methodologies

There are two major approaches in device modeling and simulation.

(1) Physics modeling: a direct approach based on the first principle physics-based model.

The required governing equations in the preceding four aspects are all derived from first principles, such as the Maxwell equations (including electromagnetic wave theory for the optical field distribution and quasi-electrostatic theory for the carrier transport), the Schrödinger equation (for the semiconductor band structure), the Heisenberg equation (for the gain and refractive index change), and the thermal diffusion equation (for the temperature distribution).

This model gives the physical description of what exactly happens inside the device and is capable of providing predictions on device performance in every aspect, once the device building material constants, the structural geometrical sizes, and the operating conditions are all given.

This approach is usually adopted by device designers who work on developing devices themselves.

In contrast to the conventional holographic process, in which intensity variations in an interference pattern between an object beam and a reference beam are recorded, polarization holography employs beams with two different polarizations for recording information. In this case, the polarization state of the resultant beam is recorded on a suitable medium. This process was discovered by Sh. D. Kakicheshvili, from Georgia (which was then part of the USSR). Currently there is only one monograph on polarization holography, namely that written in Russian by Kakicheshvili (Polyarizatsionnaya golografiya, Nauka, 1989). However, because of its complex presentation, this monograph is not easily amenable for application to practical work.

Polarization holographic storage has several unique properties: (1) it is possible to achieve theoretically 100% diffraction efficiency even in thin films; (2) the diffracted beams have unique polarization properties, depending on the polarization of the recording and read-out beams; (3) it is possible to fabricate polarization-sensitive optical elements; and (4) the optical elements fabricated with polarization holography are achromatic, allowing their use at all wavelengths.

Our book intends to fill a gap in the area of holography, and documents research done during the last two decades. High-capacity holographic storage remains a hot topic. This book is intended for scientists as well as students at graduate and postgraduate level. A basic undergraduate optics background is assumed, and a well-prepared undergraduate physics major will be able to appreciate the subtleties of polarization holography.

Polarization holograms have been shown to possess some extraordinary properties: they are capable of achieving a 100% diffraction efficiency; and they are able to reconstruct the polarization properties of the object beam in addition to providing intensity and wavelength information. Since the reconstructed images have different polarization from that of the undiffracted light, their signal-to-noise ratio is higher than in conventional holography. Polarization holograms also exhibit achromaticity. It is possible to fabricate a half-wave plate, or a polarization beamsplitter for linearly and circularly polarized light, independently of the wavelength of operation.

Polarization holograms in materials such as alkali halides, arsenic trisulfide and bacteriorhodopsin have been used to demonstrate several interesting properties. The most efficient material available today for polarization holography is based on azobenzene. Azobenzene-containing polymers have fast response, and the polarization holograms based on azobenzene polymers are stable at room temperature. However, the fast response depends on the intensity of the interfering beams. There are other drawbacks with this material. Azobenzene absorbs blue and green light. Thus any optical element based on azobenzene can be used only in the red and infrared. While polarization holograms fabricated in azobenzene polymers have been stable over many years under ambient conditions, they are not stable under high-temperature treatment. Amorphous polymers with high glass-transition temperatures have been shown to retain light-induced anisotropy until approximately 200°C. Liquid-crystalline polymers that depend on the reorientation of entire domains are more susceptible to degradation at temperatures around 100°C.