1. Introduction

The potential importance of integrated optics was not fully realised until 1968. Light propagation in thin films has been proposed and developed extensively since then [1]. The term integrated optics relates to a wide variety of structures where the propagation of light is controlled by a thin dielectric film or by strips of dielectric. The range of laser frequencies available and the types of material used have their limitations. Initially, gas laser and solid-state lasers were used as the light sources in early experiments. There is a possible need for much smaller sources that can be used to achieve the requirement of integrated optics in order to integrate with other applications. For wavelengths smaller than a certain value, say around 0.1 μm, overcoming large absorption and scattering losses becomes a priority since the smaller wavelength range imposes limitations on the practical use of waveguiding.

**88**

*Photonic Crystals - A Glimpse of the Current Research Trends*

metal-organic framework-containing silica-colloidal crystals for vapor sensing. Advanced Materials. 2011;**23**(38):4449-4452

[61] Turduev M, Giden IH, Babayigit C, Hayran Z, Bor E, Boztug C, et al. Midinfrared T-shaped photonic crystal waveg-uide for optical refractive index sensing. Sensors and Actuators B: Chemical. 2017;**245**:765-773

[52] Russell PSJ. Photonic-crystal fibers. Journal of Lightwave Technology.

MacPherson WN, Shephard JD, Maier RRJ, Knight JC, et al. Mid infrared gas sensing using a hollowcore photonic bandgap fibre. In: Optical Fiber Sensors, OSA Technical Digest (CD). Cancun Mexico: Optical Society of America;

[54] Shephard JD, MacPherson WN, Maier RR, Jones JDC, Hand DP, Mohebbi M, et al. Single-mode mid-IR guidance in a hollowcore photonic crystal fiber. Optics Express.

[55] Gayraud N, Kornaszewski LW, Stone JM, Knight JC, Reid DT, Hand DP, et al. Mid-infrared gas sensing using a photonic bandgap fiber. Applied Optics.

[56] Bragg WH, Bragg BWL, et al. The reflection of x-rays by crystals. Proceedings of the Royal Society of London A. 1913;**88**(605):428-438

[57] Asher SA, Holtz J, Liu L, Wu Z. Selfassembly motif for creating submicron periodic materials. Polymerized crystalline colloidal arrays. Journal of the American Chemical Society.

2006;**24**:4729-4749

2006. paper ThA5

2005;**13**:7139-7144

2008;**47**(9):1269-1277

1994;**116**(11):4997-4998

2009;**3**(7):1669-1676

[58] Kobler J, Lotsch BV, Ozin GA, Bein T. Vapor-sensitive bragg mirrors and optical isotherms from mesoporous nanoparticle suspensions. ACS Nano.

[59] Yang H, Jiang P. Macroporous photonic crystal-based vapor detectors created by doctor blade coating. Applied Physics Letters. 2011;**98**(1):011104

[60] Lu G, Farha OK, Kreno LE, Schoenecker PM, Walton KS, Van Duyne RP, et al. Fabrication of

[53] Gayraud N, Stone JM,

In addition, waveguide integrated optics is based on electromagnetic waveguiding at optical frequencies using thin-film optics. In recent years, semiconductor devices have played a major role in the evolution of integrated optics, due to their significant properties relevant to the goal of monolithic integrated optical circuits. In the early 1960s, research on thin-film phenomena became the key route towards developing more complex waveguide properties. The guiding action of planar layers in p-n junctions was observed and reported in 1963 by Yariv and Leite [2] and Bond et al. [3]. Their result has been subsequently used by Nelson and Reinhardt [4] in providing light modulation via the electro-optic effect. Although there was no concern with the optical waveguide circuitry, this work was just the beginning of the new era of planar thin-film waveguides. Light propagation in thin films has been proposed and developed since then [5]. The subject of dielectric periodic microstructures has become a priority ever since the evolution of lasers and integrated optics generally in the early 1960s [5–11]. This great evolution was just the beginning of the new era of development of photonic microstructures on single compact chips. Much research has been carried out with the aim of providing faster optical communication and data processing—whether for entertainment, route switching or computational purposes. In recent years, the motivation towards producing compact and faster communication has become a platform for much research, including switching purposes.

demonstrations have shown that by using an array of holes drilled into the high refractive index material, a stopband is produced where no transmission is allowed over this frequency range [19]. Full PBG structures consist of three-dimensionally periodic structures that inhibit spontaneous emission within the electromagnetic band gap. New design has been developed and innovated based on this concept ever since, although improving the overall performance of this device is still a major concern for full device functionality—and there is also a performance limitation determined by various fabrication processes [19, 20]. 3D photonic crystal structures (PhCs) are one of the possible contenders for the provision of highly compact devices on a single chip that will allow the realisation of complex subsystems. Due to the inherent difficulties of realising and controllably modifying 3D structures, work on 2D and 1D structures has emerged tremendously—which is partly due to the lesser design complexity and the reduction in size. But they can produce some interesting results that have contributed significantly towards the realisation of photonic integrated circuits (PICs). The motivation towards miniaturising PIC devices has expanded the need to put more effort into designing compact photonic crystal-based devices. The massive development of telecommunication infrastructures has created a large demand of multiple applications on a single chip by using a

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

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

In general, photonic crystal device structures exhibit a strong optical confinement covering a fairly large frequency spectrum. Strong optical confinement is needed in a small volume to provide a suitable platform in the optical emission properties—thus creating enhancement of the luminescent 'atoms' through spontaneous emission. By creating a 'defect' or a small region surrounded by the photonic crystal arrangement, the basic properties of that photonic crystal lattice are significantly changed. In other words, the photonic crystal has the capability to localise light when a 'defect' is introduced within the periodicity of the crystal arrangement —thus forming a micro-cavity that is surrounded by a highly reflective mirror region. For example, in 2D photonic crystal structures, a 'defect' or micro-cavity can be formed by simply removing one or more holes [21–23]—or by changing the surrounding hole sizes [24–26]. Light that is strongly confined within the channel waveguide formed by the photonic crystal arrangement (square [27] or hexagonal lattice [28, 29] is directly coupled into the micro-cavity region. In this design arrangement, light may be guided through the structure by removing a single row of holes to form a channel waveguide—and in this way, light can propagate at the characteristic frequency of the cavity, within the band gap. Channel waveguides may be designed to have different widths, W, such as W1 [30]—where a single row of holes is removed to provide a channel waveguide. In other examples, W3 [31] consists of three hole removed and W0.7 [32]—i.e. a situation where the spacing between two blocks of photonic crystal is additionally increased by 0.7 of a lattice spacing. Recently, low propagation losses, 4.1 dB/cm, have been obtained in a single-line defect W1 PhC channel waveguide [33] which shows that PhCs can provide a suitable platform for designing low loss devices. In 1D photonic crystal structures, the micro-cavity has great potential for producing a high-quality factor in a small volume—thus providing a suitable platform to design a wavelength selective device, for example, for WDM applications using passive components such as multiplexers/de-multiplexers, optical switching, sensors and optical filters. On the other hand, in 1D photonic crystals, micro-cavities may be formed by creating a defect and using a smaller hole in the middle of a single-row crystal, as

In photonic crystal (PhC) micro-cavity structures, the optical properties may be characterised by the Q/V ratio (often called the Purcell factor [12, 13]), where Q is the quality factor and V is the modal volume corresponding to the particular

combination of several optical subsystems.

shown in Figure 1.

91

In addition, the advance in photonic technology for many applications has emerged on a large scale, whether using active devices such as III–V semiconductor materials or even silicon and silica passive devices. But the latter two materials still work as a separate system, although the main aim is still to achieve a monolithic photonic integration that is capable of handling any application in a single chip. The developments based on the concepts formulated by Purcell [12, 13] regarding the effect of radiation properties due to the presence of mirrors have been discussed extensively. These ideas led to the new concept of photonic crystals (PhCs) [14, 15]. Instead of manipulating the electrons that are involved in the use of the conventional electronic properties of solids, where they can produce an electronic band gap, photons are manipulated in periodic structures (photonic crystals)—and can exhibit stopband and photonic band-gap behaviour. In other words, photons are not allowed to propagate through the 'crystal'structures at all—and there can be a forbidden gap or band gap.

Much of the attraction in the research areas of the micro- and nanophotonic structures comes from the use of high refractive index contrast materials such as silicon-on-insulator (SOI) that have been increasingly used in recent years. This development is due to the ability of silicon technology to support monolithic integration of optical interconnects and form fully functional photonic devices that can be incorporated into CMOS chips. Soref and Lorenzo [16] have demonstrated the possibilities of passive and active silicon waveguides as long ago as 1985, with single-crystal silicon grown epitaxially on a heavily doped silicon substrate. The advances of silicon-based and silicon-on-insulator optoelectronics have also been noted by Jalali et al. [17] and Masini et al. [18].

### 2. Photonic crystal and photonic wire waveguides

The concept of photonic band-gap structures was independently proposed by John [14] and by Yablonovitch [15]. PBG structures create the condition where over a certain photon energy range, light can travel through the periodic structure—and is reflected back when impinging onto the crystal and is not allowed to propagate thus creating a so-called forbidden zone. In 1991, the first experimental

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

demonstrations have shown that by using an array of holes drilled into the high refractive index material, a stopband is produced where no transmission is allowed over this frequency range [19]. Full PBG structures consist of three-dimensionally periodic structures that inhibit spontaneous emission within the electromagnetic band gap. New design has been developed and innovated based on this concept ever since, although improving the overall performance of this device is still a major concern for full device functionality—and there is also a performance limitation determined by various fabrication processes [19, 20]. 3D photonic crystal structures (PhCs) are one of the possible contenders for the provision of highly compact devices on a single chip that will allow the realisation of complex subsystems. Due to the inherent difficulties of realising and controllably modifying 3D structures, work on 2D and 1D structures has emerged tremendously—which is partly due to the lesser design complexity and the reduction in size. But they can produce some interesting results that have contributed significantly towards the realisation of photonic integrated circuits (PICs). The motivation towards miniaturising PIC devices has expanded the need to put more effort into designing compact photonic crystal-based devices. The massive development of telecommunication infrastructures has created a large demand of multiple applications on a single chip by using a combination of several optical subsystems.

In general, photonic crystal device structures exhibit a strong optical confinement covering a fairly large frequency spectrum. Strong optical confinement is needed in a small volume to provide a suitable platform in the optical emission properties—thus creating enhancement of the luminescent 'atoms' through spontaneous emission. By creating a 'defect' or a small region surrounded by the photonic crystal arrangement, the basic properties of that photonic crystal lattice are significantly changed. In other words, the photonic crystal has the capability to localise light when a 'defect' is introduced within the periodicity of the crystal arrangement —thus forming a micro-cavity that is surrounded by a highly reflective mirror region. For example, in 2D photonic crystal structures, a 'defect' or micro-cavity can be formed by simply removing one or more holes [21–23]—or by changing the surrounding hole sizes [24–26]. Light that is strongly confined within the channel waveguide formed by the photonic crystal arrangement (square [27] or hexagonal lattice [28, 29] is directly coupled into the micro-cavity region. In this design arrangement, light may be guided through the structure by removing a single row of holes to form a channel waveguide—and in this way, light can propagate at the characteristic frequency of the cavity, within the band gap. Channel waveguides may be designed to have different widths, W, such as W1 [30]—where a single row of holes is removed to provide a channel waveguide. In other examples, W3 [31] consists of three hole removed and W0.7 [32]—i.e. a situation where the spacing between two blocks of photonic crystal is additionally increased by 0.7 of a lattice spacing. Recently, low propagation losses, 4.1 dB/cm, have been obtained in a single-line defect W1 PhC channel waveguide [33] which shows that PhCs can provide a suitable platform for designing low loss devices. In 1D photonic crystal structures, the micro-cavity has great potential for producing a high-quality factor in a small volume—thus providing a suitable platform to design a wavelength selective device, for example, for WDM applications using passive components such as multiplexers/de-multiplexers, optical switching, sensors and optical filters. On the other hand, in 1D photonic crystals, micro-cavities may be formed by creating a defect and using a smaller hole in the middle of a single-row crystal, as shown in Figure 1.

In photonic crystal (PhC) micro-cavity structures, the optical properties may be characterised by the Q/V ratio (often called the Purcell factor [12, 13]), where Q is the quality factor and V is the modal volume corresponding to the particular

In addition, waveguide integrated optics is based on electromagnetic waveguiding at optical frequencies using thin-film optics. In recent years, semiconductor devices have played a major role in the evolution of integrated optics, due to their significant properties relevant to the goal of monolithic integrated optical circuits. In the early 1960s, research on thin-film phenomena became the key route towards developing more complex waveguide properties. The guiding action of planar layers in p-n junctions was observed and reported in 1963 by Yariv and Leite [2] and Bond et al. [3]. Their result has been subsequently used by Nelson and Reinhardt [4] in providing light modulation via the electro-optic effect. Although there was no concern with the optical waveguide circuitry, this work was just the beginning of the new era of planar thin-film waveguides. Light propagation in thin films has been proposed and developed since then [5]. The subject of dielectric periodic microstructures has become a priority ever since the evolution of lasers and integrated optics generally in the early 1960s [5–11]. This great evolution was just the beginning of the new era of development of photonic microstructures on single compact chips. Much research has been carried out with the aim of providing faster optical communication and data processing—whether for entertainment, route switching or computational purposes. In recent years, the motivation towards producing compact and faster communication has become a platform for much

Photonic Crystals - A Glimpse of the Current Research Trends

In addition, the advance in photonic technology for many applications has emerged on a large scale, whether using active devices such as III–V semiconductor materials or even silicon and silica passive devices. But the latter two materials still work as a separate system, although the main aim is still to achieve a monolithic photonic integration that is capable of handling any application in a single chip. The developments based on the concepts formulated by Purcell [12, 13] regarding the effect of radiation properties due to the presence of mirrors have been discussed extensively. These ideas led to the new concept of photonic crystals (PhCs) [14, 15]. Instead of manipulating the electrons that are involved in the use of the conventional electronic properties of solids, where they can produce an electronic band gap, photons are manipulated in periodic structures (photonic crystals)—and can exhibit stopband and photonic band-gap behaviour. In other words, photons are not allowed to propagate through the 'crystal'structures at all—and there can be a

Much of the attraction in the research areas of the micro- and nanophotonic structures comes from the use of high refractive index contrast materials such as silicon-on-insulator (SOI) that have been increasingly used in recent years. This development is due to the ability of silicon technology to support monolithic integration of optical interconnects and form fully functional photonic devices that can be incorporated into CMOS chips. Soref and Lorenzo [16] have demonstrated the possibilities of passive and active silicon waveguides as long ago as 1985, with single-crystal silicon grown epitaxially on a heavily doped silicon substrate. The advances of silicon-based and silicon-on-insulator optoelectronics have also been

The concept of photonic band-gap structures was independently proposed by John [14] and by Yablonovitch [15]. PBG structures create the condition where over a certain photon energy range, light can travel through the periodic structure—and is reflected back when impinging onto the crystal and is not allowed to propagate—

thus creating a so-called forbidden zone. In 1991, the first experimental

research, including switching purposes.

forbidden gap or band gap.

90

noted by Jalali et al. [17] and Masini et al. [18].

2. Photonic crystal and photonic wire waveguides

work, light confined within a PhC/PhW structure is directly coupled into the micro-

These kinds of device structure also have the capability of providing compact structures in small device volumes, as compared to other more complex structures such as ring resonators—which occupy larger device volumes. Furthermore, 1D PhC/PhW structures may also be preferred, since they can exhibit large band gaps as compared to what 3D photonic crystal structures can offer—thus making the PhC/PhW approach a contender for filter devices that can be integrated with other photonic devices. The fact that they share similar concepts with grating filters helps in understanding how these devices operate. With the material properties of SOI, an extremely small waveguide working in single-mode operation can be realised, with a reduction in propagation losses from 50 dB in 1996 [44] to 1.7 dB/cm in late 2006 [45], and most recently a propagation loss value of 0.91 dB/cm [46] has been achieved. The other features that have led to increasing attention to this area of research are the ability to provide a platform for the confinement of light within a small volume—for example, when a defect or a spacer is introduced between periodic mirrors. With this condition, light can be trapped within a small cavity, thus producing resonances that occur at certain frequencies within the stopband. These structures have been characterised by their high Q-factor value, adequate

Maxwell's equations are important for an understanding of light propagation in photonic crystals. They are central for the solution of electromagnetic problems in dielectric media—for a variety of different lengths and dielectric scales, which are

In photonic crystals, the famous Maxwell's equations are used to study light propagation in photonic crystal structure. The propagation of light in a medium is governed by the four well-known microscopic Maxwell's equations, written here in cgs units [21, 47, 48]. The microscopic forms of the Maxwell equation are given by

∇ � E þ

<sup>∇</sup> � <sup>H</sup> � <sup>1</sup>

or in mks/SI unit they can be written as

93

1 c

c

∂B ∂t 

∂D ∂t 

<sup>¼</sup> <sup>4</sup><sup>π</sup> c

∇ � B ¼ 0 (1)

∇ � D ¼ 4πρ (2)

∇ � B ¼ 0 (5)

∇ � D ¼ ρ (6)

¼ 0 (3)

J (4)

cavity by using a tapered hole arrangement, as shown in Figure 1.

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

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

normalised optical transmission and small modal volume.

related to each other.

3. Photonic crystals: the theory

#### Figure 1.

1D PhC/PhW waveguide structures with a series of PhC hole of periodic spacing, a, and hole diameter, d, embedded in 500 nm wire. The tapered hole introduced has a number of hole tapered outside cavity, NTO, and the number of tapered hole within cavity, NTI, with cavity length, c. A periodic mirror has N number of equally spaced hole.

micro-cavity and its characteristic electromagnetic resonant modes. Thus designing high Q-factor optical micro-cavities confined in a small volume, V, may be useful for high-speed optical processing—where light is confined within a small volume on the order of (λ/2n)<sup>3</sup> —and λ is the emission wavelength and n is the refractive index of the given material. Recently, designing ultrasmall micro-cavity devices based on 1D PhC/PhW dimensions has been of major interest because of their capability to provide extremely high Q/V values, close to the theoretical values of the modal volume, V = (λ/2n)<sup>3</sup> . Q-factor values as large as 10<sup>8</sup> have been achieved, but this experimental value was based on silica toroids [34]—but this design has a relatively large modal volume, corresponding to a Q/V value of approximately 5 <sup>10</sup><sup>4</sup> (λ/2n).

Therefore, in most optical telecommunication applications, there is a need to have 1D PhC/PhW device structures that are necessary for manipulation of light at the infrared wavelength (around 1550 nm)—ruled by its capability of confined light within a small volume, V. Due to the fabrication challenges and the capability of designing structures that occupy very small areas, one-dimensional PhC structures have been preferred, although there are practical performance limitations. The devices typically consist of a single row of holes embedded in a narrow single-mode photonic wire waveguide. On the other hand, photonic wire (PhW) device structures based on total internal reflection (TIR) concepts have shown a capability for reduced loss, together with less complexity. They can also provide strong optical confinement due to the large refractive index contrast between the waveguide core and its surrounding cladding, leading also to small device volumes and compact structures [35]. In addition the photonic wire approach also gives great flexibility for the design of structures such as sharp bends, abrupt Y-junctions, small device volumes, micro-cavities and Mach-Zehnder (MZ) structures [36–42]. In other words, this concept is based on high refractive index contrast where light is confined in such a narrow ridge waveguide. The combination of one-dimensional photonic crystal (PhC) structures and photonic wire (PhW) waveguides in high refractive index materials such as silicon-on-insulator (SOI) became increasingly important in a number of research areas. In order to obtain a wide range of device functionality, the reduction of propagation losses in narrow photonic wires is equally as important as enhancing the performance of the device structures.

On the other hand, 1D PhC/PhW device structures have increasingly became a topic of interest—and use a mirror design that is based on a single periodic row of holes embedded in a narrow ridge waveguide, as shown in Figure 1. This approach was first introduced by Foresi et al. [43]. The periodic hole mirror characteristics can be varied by changing several parameters—such as hole diameter, cavity spacing and hole spacing—as will be described in detail in Chapter 4. In the present

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

work, light confined within a PhC/PhW structure is directly coupled into the microcavity by using a tapered hole arrangement, as shown in Figure 1.

These kinds of device structure also have the capability of providing compact structures in small device volumes, as compared to other more complex structures such as ring resonators—which occupy larger device volumes. Furthermore, 1D PhC/PhW structures may also be preferred, since they can exhibit large band gaps as compared to what 3D photonic crystal structures can offer—thus making the PhC/PhW approach a contender for filter devices that can be integrated with other photonic devices. The fact that they share similar concepts with grating filters helps in understanding how these devices operate. With the material properties of SOI, an extremely small waveguide working in single-mode operation can be realised, with a reduction in propagation losses from 50 dB in 1996 [44] to 1.7 dB/cm in late 2006 [45], and most recently a propagation loss value of 0.91 dB/cm [46] has been achieved. The other features that have led to increasing attention to this area of research are the ability to provide a platform for the confinement of light within a small volume—for example, when a defect or a spacer is introduced between periodic mirrors. With this condition, light can be trapped within a small cavity, thus producing resonances that occur at certain frequencies within the stopband. These structures have been characterised by their high Q-factor value, adequate normalised optical transmission and small modal volume.

Maxwell's equations are important for an understanding of light propagation in photonic crystals. They are central for the solution of electromagnetic problems in dielectric media—for a variety of different lengths and dielectric scales, which are related to each other.

### 3. Photonic crystals: the theory

In photonic crystals, the famous Maxwell's equations are used to study light propagation in photonic crystal structure. The propagation of light in a medium is governed by the four well-known microscopic Maxwell's equations, written here in cgs units [21, 47, 48]. The microscopic forms of the Maxwell equation are given by

$$\nabla \cdot B = \mathbf{0} \tag{1}$$

$$\nabla \cdot \mathbf{D} = 4\pi\rho \tag{2}$$

$$\nabla \times E + \frac{1}{c} \left( \frac{\partial B}{\partial t} \right) = \mathbf{0} \tag{3}$$

$$\nabla \times H - \frac{1}{c} \left( \frac{\partial D}{\partial t} \right) = \frac{4\pi}{c} f \tag{4}$$

or in mks/SI unit they can be written as

$$\nabla \cdot B = \mathbf{0} \tag{5}$$

$$
\nabla \cdot \mathbf{D} = \rho \tag{6}
$$

micro-cavity and its characteristic electromagnetic resonant modes. Thus designing high Q-factor optical micro-cavities confined in a small volume, V, may be useful for high-speed optical processing—where light is confined within a small volume on

1D PhC/PhW waveguide structures with a series of PhC hole of periodic spacing, a, and hole diameter, d, embedded in 500 nm wire. The tapered hole introduced has a number of hole tapered outside cavity, NTO, and the number of tapered hole within cavity, NTI, with cavity length, c. A periodic mirror has N number of equally

Photonic Crystals - A Glimpse of the Current Research Trends

of the given material. Recently, designing ultrasmall micro-cavity devices based on 1D PhC/PhW dimensions has been of major interest because of their capability to provide extremely high Q/V values, close to the theoretical values of the modal

experimental value was based on silica toroids [34]—but this design has a relatively large modal volume, corresponding to a Q/V value of approximately 5 <sup>10</sup><sup>4</sup> (λ/2n). Therefore, in most optical telecommunication applications, there is a need to have 1D PhC/PhW device structures that are necessary for manipulation of light at the infrared wavelength (around 1550 nm)—ruled by its capability of confined light within a small volume, V. Due to the fabrication challenges and the capability of designing structures that occupy very small areas, one-dimensional PhC structures have been preferred, although there are practical performance limitations. The devices typically consist of a single row of holes embedded in a narrow single-mode photonic wire waveguide. On the other hand, photonic wire (PhW) device structures based on total internal reflection (TIR) concepts have shown a capability for reduced loss, together with less complexity. They can also provide strong optical confinement due to the large refractive index contrast between the waveguide core and its surrounding cladding, leading also to small device volumes and compact structures [35]. In addition the photonic wire approach also gives great flexibility for the design of structures such as sharp bends, abrupt Y-junctions, small device volumes, micro-cavities and Mach-Zehnder (MZ) structures [36–42]. In other words, this concept is based on high refractive index contrast where light is confined in such a narrow ridge waveguide. The combination of one-dimensional photonic crystal (PhC) structures and photonic wire (PhW) waveguides in high refractive index materials such as silicon-on-insulator (SOI) became increasingly important in a number of research areas. In order to obtain a wide range of device functionality, the reduction of propagation losses in narrow photonic wires is equally as important as enhancing the performance of the device structures.

On the other hand, 1D PhC/PhW device structures have increasingly became a topic of interest—and use a mirror design that is based on a single periodic row of holes embedded in a narrow ridge waveguide, as shown in Figure 1. This approach was first introduced by Foresi et al. [43]. The periodic hole mirror characteristics can be varied by changing several parameters—such as hole diameter, cavity spacing and hole spacing—as will be described in detail in Chapter 4. In the present

—and λ is the emission wavelength and n is the refractive index

. Q-factor values as large as 10<sup>8</sup> have been achieved, but this

the order of (λ/2n)<sup>3</sup>

Figure 1.

spaced hole.

92

volume, V = (λ/2n)<sup>3</sup>

$$\nabla \times E + \left(\frac{\partial B}{\partial t}\right) = \mathbf{0} \tag{7}$$

and

become

95

E rð Þ¼ ;<sup>t</sup> E rð Þeiω<sup>t</sup> <sup>¼</sup> <sup>0</sup> (16)

∇ � H rð Þ¼ ∇ � D rð Þ¼ 0 (17)

∇xH rð Þ (19)

<sup>P</sup> (20)

H rð Þ (18)

c <sup>2</sup>

By substituting Eqs. (15) and (16) into the Maxwell equations (11)–(14), the equation is deduced to a simple condition (two divergence) as shown below:

where H(r) and E(r) and the field components at t = 0. By deriving Eqs. (15) and

(16) and substituting them into Eqs. (13) and (14), the Maxwell equation will

<sup>∇</sup> � H rð Þ <sup>¼</sup> <sup>ω</sup>

E rð Þ¼ �ic

also be used for more complex structures such as 2D and 3D PhCs.

have a low one. In optics, the Q-factor is generally given by [49, 50]:

<sup>Q</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> <sup>f</sup> <sup>0</sup><sup>E</sup>

Thus Eq. (18) derived has H components which become a master equation for dielectric medium, in particular photonic crystal with only magnetic field, H(r) component. This can also be used to recover an electric field component, E(r), of

ωεð Þr

The final equation given above (18) and (19) is only used primarily to understand the basic concepts of photonic crystal (PhC) structures. These concepts can

One-dimensional (1D)-PhC micro-cavities embedded in narrow photonic wire have been widely studied. A small shift in the periodic mirrors—in particular one situated in the middle of the periodic mirror—will produce a sharp resonance peak in the middle of stopband. This resonant condition oscillates naturally at certain frequencies with greater amplitudes than others within the system. The Q-factor is particularly useful in determining the qualitative behaviour of a system. For some telecom applications, such as dense wavelength division multiplexing (DWDM), the performance of those resonances is determined by their quality factors and optical transmission at a certain resonance frequency. The quality factor of a system is a dimensionless parameter that defines the first-order behaviour, for the decay, of an oscillating frequency within a micro-cavity. It is characterised by the ratio of the resonant frequency to the bandwidth of the resonance or by the decrease in the amplitude of the wave propagating through a system, within an oscillation period. Equivalently, it compares the frequency at which the system oscillates to the rate of energy dissipated by the system. A higher Q-factor value indicates a lower rate of energy dissipation relative to the oscillation frequency, so the oscillations die out more slowly. For example, a pendulum suspended from a high-quality bearing, oscillating in air, would have a high Q, while a pendulum immersed in oil would

∇ �

the Maxwell equation given by

4. PhC/PhW micro-cavities

1 εð Þr

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

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

$$\nabla \times H - \left(\frac{\partial D}{\partial t}\right) = f \tag{8}$$

Based on Joannopoulos and Jackson [21, 47], Eqs. (1)–(4) are given in cgs units, whereas Eqs. (5)–(8) are given in mks/SI units, where the physical quantities are given as

B magnetic flux density in Tesla, T; D electric flux density in Coulombs per square m, C/m<sup>2</sup> ; E electric field strength in Volt per metre, V/m; H magnetic field strength in Ampere per metre, A/m; ρ electric charge density in Coulombs per cubic metre, C/m<sup>3</sup> ; J electric current density in Ampere per square metre, A/m2 .

The detailed derivation of each counterpart of Maxwell's equations is given by Jackson in Ref. [47]. For propagation in mixed dielectric medium, ρ and J are set to zero, since there are no free charges or currents in the homogeneous dielectric material. By assuming that the applied field strength is small and behave linearly, the dielectric flux density, D, can be related to the electric field density by the power series of

$$D\_i = \sum\_j \varepsilon\_{ij} E\_j + \sum\_j k \chi\_{ijk} E\_j E\_k + O\left(E^3\right) \tag{9}$$

Since the electric field strength E (r,ω) and displacement field D (r,ω) are related to the scalar dielectric constant of the microscopic and isotropic material, ε (r,ω)�χ and the higher order term can be neglected. In low loss dielectric materials, ε(r) can be treated as purely real, thus producing the electric field density written as

$$\mathbf{D(r) = e \ (r) \to (r)}\tag{10}$$

In addition, for most dielectric material, the magnetic permeability, μr, is approximately equal to 1, giving the magnetic flux density, B, equal to the magnetic field strength, H. The flux density of the dielectric material can be written as D = ε.E where the permittivity, ε, is real. Therefore the Maxwell equation can be rewritten as already illustrated in [21, 47] as

$$\nabla \cdot H(r, t) = \mathbf{0} \tag{11}$$

$$\nabla \cdot \varepsilon(r)E(r,t) = 0 \tag{12}$$

$$\nabla \times E(r, t) + \frac{1}{c} \left( \frac{\partial H(r, t)}{\partial t} \right) = \mathbf{0} \tag{13}$$

$$\nabla \times H(r, t) - \frac{\varepsilon(r)}{c} \left( \frac{\partial E(r, t)}{\partial t} \right) = \mathbf{0} \tag{14}$$

Then the harmonic mode of the E and H field components propagating in the dielectric medium is considered as

$$\mathbf{H}(\mathbf{r}, \mathbf{t}) = \mathbf{H}(\mathbf{r})\mathbf{e}^{i\alpha t} = \mathbf{0} \tag{15}$$

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

and

∇ � E þ

Photonic Crystals - A Glimpse of the Current Research Trends

given as

square m, C/m2

metre, C/m<sup>3</sup>

power series of

as already illustrated in [21, 47] as

dielectric medium is considered as

94

<sup>∇</sup> � <sup>H</sup> � <sup>∂</sup><sup>D</sup>

∂B ∂t 

∂t 

; E electric field strength in Volt per metre, V/m; H magnetic field

Based on Joannopoulos and Jackson [21, 47], Eqs. (1)–(4) are given in cgs units, whereas Eqs. (5)–(8) are given in mks/SI units, where the physical quantities are

B magnetic flux density in Tesla, T; D electric flux density in Coulombs per

strength in Ampere per metre, A/m; ρ electric charge density in Coulombs per cubic

; J electric current density in Ampere per square metre, A/m2

Since the electric field strength E (r,ω) and displacement field D (r,ω) are related to the scalar dielectric constant of the microscopic and isotropic material, ε (r,ω)�χ and the higher order term can be neglected. In low loss dielectric materials, ε(r) can be treated as purely real, thus producing the electric field density written as

In addition, for most dielectric material, the magnetic permeability, μr, is approximately equal to 1, giving the magnetic flux density, B, equal to the magnetic field strength, H. The flux density of the dielectric material can be written as D = ε.E where the permittivity, ε, is real. Therefore the Maxwell equation can be rewritten

> 1 c

> > c

Then the harmonic mode of the E and H field components propagating in the

<sup>∂</sup>H rð Þ ; <sup>t</sup> ∂t 

> <sup>∂</sup>E rð Þ ; <sup>t</sup> ∂t

∇ � E rð Þþ ; t

<sup>∇</sup> � H rð Þ� ; <sup>t</sup> <sup>ε</sup>ð Þ<sup>r</sup>

The detailed derivation of each counterpart of Maxwell's equations is given by Jackson in Ref. [47]. For propagation in mixed dielectric medium, ρ and J are set to zero, since there are no free charges or currents in the homogeneous dielectric material. By assuming that the applied field strength is small and behave linearly, the dielectric flux density, D, can be related to the electric field density by the

> Di ¼ ∑ j

εijEj þ ∑ j

¼ 0 (7)

¼ J (8)

<sup>k</sup>χijkEjEk <sup>þ</sup> O E<sup>3</sup> (9)

D(r) = ε (r) E (r) (10)

∇ � H rð Þ¼ ; t 0 (11)

∇ � εð Þr E rð Þ¼ ; t 0 (12)

H rð Þ¼ ;<sup>t</sup> H rð Þe<sup>i</sup>ω<sup>t</sup> <sup>¼</sup> <sup>0</sup> (15)

¼ 0 (13)

¼ 0 (14)

.

$$\mathbf{E}(\mathbf{r}, \mathbf{t}) = \mathbf{E}(\mathbf{r})\mathbf{e}^{i\alpha\mathbf{t}} = \mathbf{0} \tag{16}$$

By substituting Eqs. (15) and (16) into the Maxwell equations (11)–(14), the equation is deduced to a simple condition (two divergence) as shown below:

$$\nabla \cdot H(r) = \nabla \cdot D(r) = \mathbf{0} \tag{17}$$

where H(r) and E(r) and the field components at t = 0. By deriving Eqs. (15) and (16) and substituting them into Eqs. (13) and (14), the Maxwell equation will become

$$\nabla \times \left(\frac{1}{\varepsilon(r)} \nabla \times H(r)\right) = \left(\frac{\alpha}{c}\right)^2 H(r) \tag{18}$$

Thus Eq. (18) derived has H components which become a master equation for dielectric medium, in particular photonic crystal with only magnetic field, H(r) component. This can also be used to recover an electric field component, E(r), of the Maxwell equation given by

$$E(r) = \left[\frac{-ic}{a\varepsilon(r)}\right] \nabla \mathbf{x} H(r) \tag{19}$$

The final equation given above (18) and (19) is only used primarily to understand the basic concepts of photonic crystal (PhC) structures. These concepts can also be used for more complex structures such as 2D and 3D PhCs.

### 4. PhC/PhW micro-cavities

One-dimensional (1D)-PhC micro-cavities embedded in narrow photonic wire have been widely studied. A small shift in the periodic mirrors—in particular one situated in the middle of the periodic mirror—will produce a sharp resonance peak in the middle of stopband. This resonant condition oscillates naturally at certain frequencies with greater amplitudes than others within the system. The Q-factor is particularly useful in determining the qualitative behaviour of a system. For some telecom applications, such as dense wavelength division multiplexing (DWDM), the performance of those resonances is determined by their quality factors and optical transmission at a certain resonance frequency. The quality factor of a system is a dimensionless parameter that defines the first-order behaviour, for the decay, of an oscillating frequency within a micro-cavity. It is characterised by the ratio of the resonant frequency to the bandwidth of the resonance or by the decrease in the amplitude of the wave propagating through a system, within an oscillation period. Equivalently, it compares the frequency at which the system oscillates to the rate of energy dissipated by the system. A higher Q-factor value indicates a lower rate of energy dissipation relative to the oscillation frequency, so the oscillations die out more slowly. For example, a pendulum suspended from a high-quality bearing, oscillating in air, would have a high Q, while a pendulum immersed in oil would have a low one. In optics, the Q-factor is generally given by [49, 50]:

$$Q = \frac{2\pi f\_0 E}{P} \tag{20}$$

where ε is the stored energy in the cavity and P is the power dissipated within the cavity, given by

$$P = -\frac{dE}{dt} \tag{21}$$

The Q-factor is equal to the ratio of the resonant frequency to the bandwidth of the cavity resonance shown in Figure 2.

Ideally, the average lifetime of a resonant photon in the cavity is proportional to the cavity's Q. Resonant systems respond to frequencies close to their natural frequency much more strongly than they respond to other frequencies. A system with a high Q resonates with greater amplitude (at the resonant frequency) than one with a low Q-factor, and its response falls off more rapidly as the frequency moves away from resonance. Therefore the physical interpretation of resonance is given by its general equation:

$$Q = \frac{f\_0}{\Delta f} \tag{22}$$

gratings where 82% light was reflected back. Hole gratings show more pronounce stopbands compared to their counterpart. As mentioned before, stronger reflection was observed for the hole grating. The hole gratings have a bigger stopband of

Different types of PhW waveguides micro-cavity. (a) 1D PhC/PhW waveguides with cavity length (distance between two hole edges of the hole spacer) c, hole periodic spacing (distance between centre-to-centre hole), a and number of periodic holes, N (b) PhW Bragg gratings waveguides with cavity length, c, period, Λ, and number of recess period, N, and (c) transmission spectra of Bragg grating waveguides and 1D PhC/PhW.

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

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

Figure 3.

97

where f0 is the central frequency of the resonance and Δf is the frequency difference within at 3 dB points or ½ of the total energy stored in the micro-cavity system. In this present work, several different types of resonator have been studied, namely, waveguide Bragg gratings and 1D PhC/PhW waveguides—as shown in Figure 3. Unlike the Bragg grating waveguide [51], which has a rectangular recess embedded on a photonic wire waveguide, a single row of holes is used as a set of mirrors.

This structure consists of a single row of holes drilled in the 500 nm width of wire waveguides. Those holes acted as a periodic mirror where light impinging on the PhC bounced back provide a band gap where light is forbidden to propagate at certain frequency. A spacer was introduced symmetrically between the periodic mirrors—thus producing a narrow resonance in the transmission. The concepts for this kind of structure were proposed by Krauss and Foresi [5, 20]. But the Q-factor at this [43] resonance condition obtained was small (�500). The PhC hole mirrors resulted in a wide stopband (approximately 182 nm), using eight PhC mirrors holes, whereas 32 period waveguide Bragg gratings were used and showed a narrower stopband of approximately 88 nm. This difference is due to the fact that the light was coupled more strongly in the periodic hole mirrors—where 95% of the light was reflected with the periodic hole arrangement, as compared to the waveguide Bragg

#### Figure 2.

A typical resonance frequency resulted from micro-cavity structures defined by the central resonance frequency, f0, and the bandwidth of the frequency at 3 dB points (energy at the steady state condition).

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

#### Figure 3.

where ε is the stored energy in the cavity and P is the power dissipated within

<sup>P</sup> ¼ � dE

The Q-factor is equal to the ratio of the resonant frequency to the bandwidth of

Ideally, the average lifetime of a resonant photon in the cavity is proportional to the cavity's Q. Resonant systems respond to frequencies close to their natural frequency much more strongly than they respond to other frequencies. A system with a high Q resonates with greater amplitude (at the resonant frequency) than one with a low Q-factor, and its response falls off more rapidly as the frequency moves away from resonance. Therefore the physical interpretation of resonance is given by

<sup>Q</sup> <sup>¼</sup> <sup>f</sup> <sup>0</sup>

where f0 is the central frequency of the resonance and Δf is the frequency difference within at 3 dB points or ½ of the total energy stored in the micro-cavity system. In this present work, several different types of resonator have been studied, namely, waveguide Bragg gratings and 1D PhC/PhW waveguides—as shown in Figure 3. Unlike the Bragg grating waveguide [51], which has a rectangular recess embedded on a photonic wire waveguide, a single row of holes is used as a set of

This structure consists of a single row of holes drilled in the 500 nm width of wire waveguides. Those holes acted as a periodic mirror where light impinging on the PhC bounced back provide a band gap where light is forbidden to propagate at certain frequency. A spacer was introduced symmetrically between the periodic mirrors—thus producing a narrow resonance in the transmission. The concepts for this kind of structure were proposed by Krauss and Foresi [5, 20]. But the Q-factor at this [43] resonance condition obtained was small (�500). The PhC hole mirrors resulted in a wide stopband (approximately 182 nm), using eight PhC mirrors holes, whereas 32 period waveguide Bragg gratings were used and showed a narrower stopband of approximately 88 nm. This difference is due to the fact that the light was coupled more strongly in the periodic hole mirrors—where 95% of the light was reflected with the periodic hole arrangement, as compared to the waveguide Bragg

A typical resonance frequency resulted from micro-cavity structures defined by the central resonance frequency,

f0, and the bandwidth of the frequency at 3 dB points (energy at the steady state condition).

dt (21)

<sup>Δ</sup><sup>f</sup> (22)

the cavity, given by

its general equation:

mirrors.

Figure 2.

96

the cavity resonance shown in Figure 2.

Photonic Crystals - A Glimpse of the Current Research Trends

Different types of PhW waveguides micro-cavity. (a) 1D PhC/PhW waveguides with cavity length (distance between two hole edges of the hole spacer) c, hole periodic spacing (distance between centre-to-centre hole), a and number of periodic holes, N (b) PhW Bragg gratings waveguides with cavity length, c, period, Λ, and number of recess period, N, and (c) transmission spectra of Bragg grating waveguides and 1D PhC/PhW.

gratings where 82% light was reflected back. Hole gratings show more pronounce stopbands compared to their counterpart. As mentioned before, stronger reflection was observed for the hole grating. The hole gratings have a bigger stopband of

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 stopband of the rectangular recess grating.

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

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

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

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

is still a major concern.

problem in optical waveguide which comprised of:

• The refractive index distribution, n(x,y,z)

• The boundary of PML layer

• Spatial grid size, Δx and Δy

99

• Electromagnetic field excitation (plane wave or Gaussian)

• Time step, Δt, and the total length of the simulation time

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

• Finite computational domain in x, y and z direction

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 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 microcavity [54, 55].

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