5. The finite-difference time-domain (FDTD) approach

There have been several methods used for computational purposes, especially for modelling photonic crystal structures and photonic wire waveguides. The finitedifference time-domain (FDTD) approach is a commonly used technique because it provides both the spatial and temporal properties of the structure with a single calculation, making it suitable for the analysis of many structures. However it requires a lot of time to compute a single run. This technique uses the famous Maxwell's equations based on the Yee mesh [56], published in 1966. Yee has proposed this technique in order to derive a numerical scattering problem and electromagnetic absorption on the basis of Maxwell's equations. The computational domain was first established, in order to determine the physical region within which the calculation will be performed. The electric field, E, and the magnetic field, H, are distinguished at every point within the domain by specifying the material used at each domain point (in xyz directions). The materials involved could be free space (air), metal or dielectric material. A light source in the form of a plane wave is then impinging on the chosen material. Later in 1994, the technique called the perfectly matched layer (PML) boundary condition was introduced [57]. It was

Modelling of Photonic Crystal (PhC) Cavities: Theory and Applications DOI: http://dx.doi.org/10.5772/intechopen.84961

used as an absorption mechanism for electromagnetic wave incident on the edge of the computational domain in space. The FDTD method can be implemented in either 2D or 3D computations—but it requires a lot of memory and power consumption for a single computational run, especially for a large device in 3-D. 2D FDTD reduces time and memory requirement significantly. It employs a refractive index approximation or average refractive index of the slab—called effective index method (EIM). By using this method, the cross-sectional index profile is usually transformed to the one-dimensional index profile by using EIM [58, 59]. In the EIM approach, the eigenvalue of the equivalent slab waveguide is an approximate index value of the original waveguide. Although the EIM approach provides a good approximation, it still suffers from errors in the vicinity of the cut-off [60–63]. At the beginning of this present work, this method is used to investigate the preliminary behaviour of the device with the assumption that losses are negligible. In order to reduce simulation time and power consumption, 2D FDTD approach was initially used throughout the course to analyse the general optical behaviour of the device structures—implementing EIM. Since EIM method is only an approximation of the actual refractive index obtained by taking into account the whole ridge waveguide structures, at least a small discrepancy between the simulations measured results is very much to be expected. On the other hand, the 3D FDTD method can give a better estimate of the properties, although it is time- and power-consuming, which is still a major concern.

During this present work, different types of commercial software have been used. The Fullwave RSoft computational software has been used at the beginning of this work, where only 2D computation was deeply explored due to the longer time and high power consumption for 3D FDTD. Based on the concept proposed by Yee [56], several key pieces of information are needed to solve the basic propagation problem in optical waveguide which comprised of:


For 2D FDTD computation, the average refractive index, n, or effective index, neff, of the slab waveguide of a material is used rather than the actual refractive of that particular material. This can be obtained using mode-matching method available in the Fimm-wave® commercial software by Photon Design®. This method includes the approximation of refractive index in both propagation direction of vertical and horizontal confinement of the slab waveguide. The transverse section of the device is first simulated using Fimm-wave® simulation tools. It shows the intensity of light in guiding mode, confinement of light inside the slab and the effective index, neff. It also shows the leaky region where light is not confined inside the slab. Figure 4 shows the contour plot of the TE fundamental mode of the waveguide. It shows the intensity of light confinement along the core at 1.52 μm wavelength at different etching depths. It is suggested that the different etching depths will give rise to the abrupt change of the effective index, neff, at the

approximately 180 nm, which is useful for some filter designs and some optical communications applications. This wide stopband may be compared with the limited bandwidth of the stopband or may be contrasted with the significantly smaller

In addition, for this grating condition, the total length, L, of the waveguide Bragg grating of 11 μm is longer by a factor of four in order to achieve a practical stopband spectrum, as compared with the hole grating structures (3 μm). The present work will demonstrate the design, fabrication and characterisation of the 1D PhC-based micro-cavity, which is potentially useful for wavelength division multiplexing (WDM) in PhC devices. A single row of PhC holes is embedded in a narrow photonic wire waveguide to allow sufficient optical coupling for integration with other photonic devices. This thesis will address the importance of using a combination of hole tapering with a different hole diameter at the interface between the un-patterned wire and the cavity mirror, as well into the micro-cavity region in order to achieve large optical transmission together with a high-resonance Qfactor value. Achieving high Q-factor together with large optical transmission remains a significant challenge. The key points towards designing an ultrahigh Qfactor device that confines light in such a small volume lie in reducing the modal mismatch between the un-patterned wire and the PhC or grating sections. Therefore, designing a tapered structure to reduce the modal mismatch at the interfaces between the mirror region and the PhW waveguide sections is necessary. One of the

approaches used to overcoming this situations is the use of a taper structure

add-drop filter switching experiments, slow light and non-linear optics.

5. The finite-difference time-domain (FDTD) approach

consisting of holes with different sizes through progressive increase of the hole size into the mirror region [52]. On the other hand, the same model has also been used with short taper sections incorporated into a 1D micro-cavity-based system [53]. Using these concepts, the impact of progressive tapering using different hole diameters has shown a huge improvement in enhancing the quality factor of the micro-

Therefore, 1D PhC/PhW micro-cavities can provide higher optical confinement in smaller volumes that are closer to the theoretical value of 0.055 (λ/n)<sup>3</sup> [54] which has a great potential in high-index contrast materials such as silicon-oninsulator (SOI) to be used in some telecommunication applications such as DWDM,

There have been several methods used for computational purposes, especially for modelling photonic crystal structures and photonic wire waveguides. The finitedifference time-domain (FDTD) approach is a commonly used technique because it provides both the spatial and temporal properties of the structure with a single calculation, making it suitable for the analysis of many structures. However it requires a lot of time to compute a single run. This technique uses the famous Maxwell's equations based on the Yee mesh [56], published in 1966. Yee has proposed this technique in order to derive a numerical scattering problem and electromagnetic absorption on the basis of Maxwell's equations. The computational domain was first established, in order to determine the physical region within which the calculation will be performed. The electric field, E, and the magnetic field, H, are distinguished at every point within the domain by specifying the material used at each domain point (in xyz directions). The materials involved could be free space (air), metal or dielectric material. A light source in the form of a plane wave is then impinging on the chosen material. Later in 1994, the technique called the perfectly matched layer (PML) boundary condition was introduced [57]. It was

stopband of the rectangular recess grating.

Photonic Crystals - A Glimpse of the Current Research Trends

cavity [54, 55].

98

silicon where symmetric field distribution is obtained (see Figure 4(a)) as

Modelling of Photonic Crystal (PhC) Cavities: Theory and Applications

based on this design is 2.97, which will be used for 2D FDTD computation.

Depositing silica on top of the photonic wire can also improve the confinement of TE fundamental mode of the photonic wire significantly. From Figure 5, 100, 200 and 400 nm SiO2 have been deposited on the photonic wire. But to reduce the device preparation complexity and process development, the slab waveguide design based on fully etched silicon is considered throughout this work. The calculated neff

The value of neff is fed into the full-wave simulation tool by using either pulsed or continuous Gaussian source for slab waveguide. The finite computational domain is optimised in space covering the area between 10 and 20 μm in length and 2 μm in

The effective index, neff, at different etching depths for symmetric (silica deposition on top) and asymmetric

waveguide (no silica deposition). (a) Shallow etched. (b) Deep etched.

compared to shallow- and deep-etched silicon.

DOI: http://dx.doi.org/10.5772/intechopen.84961

Figure 5.

101

Figure 4. Contour plot of the TE fundamental mode intensity. (a) Fully etched. (b) Shallow etched. (c) Deep etched.

boundary of silicon core and silica cladding (lower cladding) where some of the light are reflected back into the cladding (backscattering). This can be improved by etching slightly deeper into the lower cladding by around 20–40 nm, thus reducing scattering losses. The effective index calculated using the Fimm-wave™ simulation tool for 500 nm wide ridge waveguides at different etching depths is given in Figure 4(a) and (b). More profound field intensity is obtained for fully etched

Modelling of Photonic Crystal (PhC) Cavities: Theory and Applications DOI: http://dx.doi.org/10.5772/intechopen.84961

silicon where symmetric field distribution is obtained (see Figure 4(a)) as compared to shallow- and deep-etched silicon.

Depositing silica on top of the photonic wire can also improve the confinement of TE fundamental mode of the photonic wire significantly. From Figure 5, 100, 200 and 400 nm SiO2 have been deposited on the photonic wire. But to reduce the device preparation complexity and process development, the slab waveguide design based on fully etched silicon is considered throughout this work. The calculated neff based on this design is 2.97, which will be used for 2D FDTD computation.

The value of neff is fed into the full-wave simulation tool by using either pulsed or continuous Gaussian source for slab waveguide. The finite computational domain is optimised in space covering the area between 10 and 20 μm in length and 2 μm in

#### Figure 5.

The effective index, neff, at different etching depths for symmetric (silica deposition on top) and asymmetric waveguide (no silica deposition). (a) Shallow etched. (b) Deep etched.

boundary of silicon core and silica cladding (lower cladding) where some of the light are reflected back into the cladding (backscattering). This can be improved by etching slightly deeper into the lower cladding by around 20–40 nm, thus reducing scattering losses. The effective index calculated using the Fimm-wave™ simulation tool for 500 nm wide ridge waveguides at different etching depths is given in Figure 4(a) and (b). More profound field intensity is obtained for fully etched

Contour plot of the TE fundamental mode intensity. (a) Fully etched. (b) Shallow etched. (c) Deep etched.

Photonic Crystals - A Glimpse of the Current Research Trends

Figure 4.

100

width since large computational area will contribute to a longer simulation time and also consume more memory and power.

6. Conclusion

Author details

103

In conclusion, the 2D simulation approach is used to produce preliminary designs for the device, together with the employment of an effective index approximation based on the waveguide properties of the base material structure discussed earlier. 2D simulation helps with obtaining a better understanding of the general behaviour of the device—but 3D simulation gives more accurate prediction of the results, at the expense of much greater time and energy consumption. Both 2D and 3D simulations were carried out using the commercial software using finitedifference time-domain (FDTD) approach. This chapter in particular demonstrated the detailed theoretical models of single-row PhC cavities embedded in narrow (typically 500 nm wide) photonic wire waveguides based on silicon-on-insulator (SOI). The device structures have been designed to operate in TE polarisation at wavelengths around 1550 nm. The compactness together with high reflectivity and possibilities for an active tuning capability make the device suitable as a basic building block for incorporation into integrated circuits where several functions are realised on a single chip—i.e. what are commonly known as high-density photonic integrated circuits (PICs). On the other hand, it may also be useful in providing one of the solutions for the design of compact filters for either coarse or dense wavelength division multiplexing situations, for high-speed switching and non-linear optics. For instance, FDTD approach used in this chapter has shown a significantly good agreement with the measured result—thus it can be used as a method to obtain

Modelling of Photonic Crystal (PhC) Cavities: Theory and Applications

DOI: http://dx.doi.org/10.5772/intechopen.84961

a preliminary result before the actual design for fabrication is proposed.

1 Institute of Microengineering and Nanoelectronics, University Kebangsaan

3 Schoolf of Engineering, University of Glasgow, Glasgow, United Kingdom

2 John Mc Kay Lab, School of Engineering and Applied Science (SEAS), Harvard

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Ahmad Rifqi Md Zain1,2\* and Richard M. De La Rue3

\*Address all correspondence to: rifqi@ukm.edu.my

Malaysia (UKM), Bangi, Selangor, Malaysia

University, Cambridge, MA, United States

provided the original work is properly cited.

The space must be exactly proportionate to the size of the optical waveguide. The thickness of PML required for the device operating at around 1.52 μm wavelength is 0.5 μm to provide better electromagnetic wave absorption at the boundary. Other parameters that contribute to the accuracy of the simulation are determined by the choice of the spatial grid or mesh size, where smaller grid spacing gave more accurate computation. In other words, the closer the resolution in simulation to the actual device, the more accurate simulation will be established.

During the second half of the computational process, the author has used other commercially available software to compute all the device structures. This software is found to be more accurate than the previous simulation tool. On the other hand, by using the same parameters in the crystal wave tool as previously used in fullwave, the simulation time has been reduced by a factor of five, and the result obtained is closer to the measured result as shown in one of the example in Figure 6. The comparison is made by using 12-period 1D PhC mirrors with diameters of 350 nm and periodic spacing of 360 nm. By looking at the band edge location of the measured result in Figure 6, the 2D FDTD crystal wave shows closer result (band edge) than the one computed using RSoft tools where the deviation of 83 nm is observed between the simulation and the measured result. But this is understood to be due to small deviation in the dimension of the structures produced after fabrication process as real devices.

No further investigation is made in reference to the discrepancy between the different examples of commercial software, but the problem has been addressed to the relevant personnel. As a result, based on further tests carried out using different measurements run to compare the results with the simulation, the CrystalWave software have been chosen as the relevant tools that are well-suited to the design structures used throughout this present work.

#### Figure 6.

An example showing a comparison between 2D FDTD computed using different simulations tools (RSoft and crystal wave) with the measured result.
