# Design of Superlens Using 2D Photonic Crystal with Various Geometries under Polarized Incidence: Design of Superlens Using 2D Photonic Crystal

## INTRODUCTION

Photons have several advantages over electrons. One of the greatest advantages is that the speed of photon much greater than speed of electron in metallic wire. Another advantages is that the band width of dielectric materials is larger than metal. Photons are not directly interacting with electrons, which therefore helps to reduce the energy absorption losses in system design.

Periodic band structures have for years played an important role in many branches of physics. Laudon [1] and Yablonovithch [2] were the first to recommend the idea of designing the material which may affect the properties of photons similar to the way that ordinary semiconductor crystal affects the properties of electrons. In a photonic crystal, the periodic potential (Bragg-like diffraction)occurs due to the lattice mismatch of the periodic potential of different dielectric material constant. It is the combination of dielectric materials that contain a periodicity in dielectric constant. The periodic contrast in static dielectric constant can create a range of forbidden frequency called photonic bandgap [3]. Photonic crystal can be design with any lattice constant and equivalent periodic potential, therefore allowing the gap to be controlled and the existence of photonic bandgap is highly dependent on the lattice geometry and dielectric contrast. Electromagnetic mode with energies resides within a photonic bandgap cannot travel through the photonic crystal and are therefore forbidden.

A photonic crystal might therefore engineered in a different manner to process a photonic bandgap for a specific range of frequency spectrum where which electromagnetic waves are forbidden to exists within the crystal [4-6]. The electromagnetic modes would be prohibited in perfectly periodic photonic crystal. However, if there is defect in the other perfect periodic crystal, localized photonic states could exist[7] within the photonic bandgap. In the recent study, it has been a great interest the photonic band structure, also known as dispersion curve [8—10], for electromagnetic wave travelling in various two-dimensional and three-dimensional photonic crystals. The aim of this study has been to determine the photonic band structure formed by the branches of this dispersion curve that is abstracted by any photonic bandgap. To be useful, the photonic bandgap must exist for all wave vectors values in the Brillouin zone for photonic crystal band structure under study.

**4.1.1 Two-Dimensional Photonic Crystal**

In photonic crystal, the dielectric periodicity creates a photonic bandgap in the electromagnetic dispersion curve for that electromagnetic wave that propagates perpendicular to the layers photonic crystal, therefore, offers the possibility of allowing for unprecedented control and manipulation of light [l l].A two-dimensional photonic crystal is periodic along the two of its axis and homogeneous along the third. Photonic crystal is classified in two categories considering the first photonic crystal is formed by a square lattice of dielectric column of dielectric constant *£•„’ in a different dielectric background * ^{l}e_{b}* The second photonic crystal is made by arranging the dielectric column of a triangular lattice.

**4.1.2 Square Lattice Photonic Crystal**

Figure 4.1 illustrates the first class of two-dimensional photonic crystal to be studied namely: the two-dimensional square lattice of column of high dielectric’ e_{a}’ in the background of low dielectric constant ‘e_{b}’ and the square lattice of column of low- dielectric constant ‘e_{a}’ in the background of high dielectric constant 'e_{b}’

The square lattice consists of a periodic array of parallel dielectric column of circular cross-section and dielectric constant e_{a} w-here the intersection with perpendicular plane form a square lattice. The dielectric column is incorporated in a dielectric material whose dielectric constant is e_{b}.

FIGURE 4.1 (a) The two-dimensional square lattice of column of high dielectric *e _{a}* in the

background of low dielectric constant *e _{a}.* (b) The square lattice of column of low' dielectric constant

*e*in the background of high dielectric constant

_{a}*e*

_{b}.FIGURE 4.2 (a) The two-dimensional triangular lattice of column of high dielectric e_{a} in the

background of low dielectric constant e_{b}. (b) The triangular lattice of column of low dielectric constant e_{a} in the background of high dielectric constant e_{b}.

**4.1.3 Triangular Lattice Photonic Crystal**

The triangular lattice structure consists of periodic array of parallel dielectric column circular cross-section and dielectric constant e_{a} where intersection with a perpendicular plane form a triangular lattice. The dielectric columns are incorporated in a dielectric material in which constant is e_{b}

The square lattice and triangular lattice photonic crystals are the subjects of interest because these are now being used for current experimental purposes.

## MATHEMATICAL FORMULATION

Analogy of the photonic crystal structure is found with semiconductors. Without the presence of external currents and sources, the Maxwell’s equation can be modified as the following that is very similar to the Schrodinger equation. The equation of photonic crystal is given by

where symbols have usual significances. In this context, we will discuss about the solution of the equation with the approach already carried out, and consecutive inferences can be derived from it.

### Photonic Band Structure

Equation 4.1 represents a linear Hermitian eigenvalue problem. The eigenvalues or eigen-frequencies are determined entirely by the microscopic dielectric function e(r). If e(r) is completely periodic as in an ideal photonic crystal, then results are denoted by a wave vector *к* and a band index *n.* The collection of eigen-frequency in the dispersion curve as a function of wave vector is usually taken under consideration of the first Brillouin zone and is called as photonic band structure.

The photonic band structure depends on lattice constant, the radius of the dielectric column in two dimensions and the dielectric contrast between the cylinders relative to the material making up the background of the photonic crystal. Photonic bandgap may exist for a variety of photonic crystal. The gap is a region of frequency where no electromagnetic modes may subsist within the photonic crystal for any value of wave vector *к* within the Brillouin zone. The band above the photonic bandgap is commonly known as *a* air band and the band below the photonic bandgap is commonly known as a dielectric band. Since electromagnetic modes are prohibited, spontaneous emission will be prohibited in cases in which photonic bandgap overlays the electronic bandgap. It is desirable to obtain a photonic band structure with a photonic bandgap, within which the travelling of electromagnetic wave is prohibited for all wave vector *к* within the Brillouin zone.

Previously, theoretical study of electromagnetic wave in photonic crystal is based on scalar approximation in w'hich vector nature of the electromagnetic field is ignored. These calculations did not give correct result for the band structure. Nowadays, theoretical studies of photonic crystal rely on fully vectored electromagnetic field.

The translation symmetry of photonic crystal can be broken in addition of a dielectric effect. This is essentially a change in the dielectric constant of a specific region within the otherwise perfectly periodic photonic crystal. These dielectric defects may result in defect states within the photonic bandgap, permitting a localized mode to exist about the defect within the crystal. Dielectric defects result in defect states whining the photonic bandgap just below the air band, while air defects result in defect states just above the dielectric band of photonic band structure.

### Two-Dimensional Photonic Crystal

In two dimension, owing to the presence of minor symmetry in the plane perpendicular to the dielectric column, it becomes possible to decouple the electromagnetic modes into ТЕ and TM modes with reference to the plane normal to the dielectric column.

The general rule presented by Joannoploulos [ 12] is that TM bandgaps are recommended in the lattice of outlying region of high dielectric constant and that ТЕ bandgaps are recommended in a connected lattice of high dielectric constant. The design of photonic bandgap of both ТЕ and TM modes’ designs are required a photonic crystal that attempts to combine the lattice which favored both ТЕ and TM modes bandgap. Such crystal will have high dielectric constant region that are almost entirely isolated but which are still connected by narrow high veins. The triangular lattice of column of low dielectric constant e_{a} in the background of high dielectric constant e_{b} is an example of such photonic crystal. The remnants of high dielectric between the columns can be considered to be the isolated region of high dielectric constant required for TM bandgap. If the column is slightly smaller than the maximum packing would allow, then narrow veins of high dielectric constant will connect the mostly isolated region and allow ТЕ bandgap to exits as well.

The dielectric column of dielectric constant * ‘e_{a}’* is assumed to be parallel to the z-axis. The convergence of the axis of these columns with the

*X-Y*plane from a two-dimensional lattice (square or triangular) is given by the vector

where vector a,and *a _{2}* are the two primitive translation vector of the lattice and / and

*m*are any two integers. For the square lattice of lattice constant. the primitive lattice vector

*a,*and

*a*are given by

_{2}

For the triangular lattice constant *a,* the primitive lattice vector *a*, and *a _{2}* are

The dielectric constant of the this composite system is therefore position dependent, and referred by tfrj.The area of the primitive unit cell of this lattice is equal to Au = lal x a2l.

**4.2.2.1 Real-Space Representation of Square Lattice**

The following Figure 4.3(a) is the real-space depiction of the two-dimensional square lattice column of high dielectric column e_{a} in the background of low dielectric constant e_{b}.

**4.2.2.2 Reciprocal Space Representation of Square Lattice**

Figure 4.3(b) is the reciprocal space depiction of two-dimensional square lattice of column. The reciprocal space representation of two-dimensional square lattice is shown in Figure 4.4 with the square Brillouin zone. The triangular irreducible portion of the Brillouin zone is shown in the upper right corner of Brillouin zone.

Figure (4.4) is shown as an expanded view of the Brillouin zone for two-dimensional square lattice. The *k-space* depiction of high symmetry *X, f. M* defines the irreducible Brillouin zone and is given by

FIGURE 4.3 (a) The real-space depiction of the two-dimensional square lattice of column high dielectric constant in a background of low dielectric constant, (b) The reciprocal depiction of the two-dimensional square lattice column high dielectric constant in a background of low dielectric constant, (c) The reciprocal space depiction of two-dimensional square lattice shown in (b) with the square Brillouin zone. The triangular irreducible portion of the Brillouin zone is shown in the upper right corner of Brillouin zone.

FIGURE 4.4 An expanded view of the Brillouin zone for the two-dimensional square lattice. The triangular irreducible portion of the Brillouin zone is shown in the upper right corner of the Brillouin zone. The *к-space* defines of high symmetry *Х.Г. M.*

The irreducible portion of the Brillouin zone defines the k-space trajectory to be followed when calculating the photonic band structure. The band structure for all other *к* - space points outside the irreducible portion of the Brillouin zone can be found by taking advantage of the symmetry of the Brillouin zone.

FIGURE 4.5 Schematic illustration of the unit cell construction for 2D square lattice of dielectric column. Shown in the figure are the inverse dielectric function for the unit cell dielectric background (f_{b}“ = —) and the dielectric column (f”= —).

Within each unit cell of the square lattice, we have a single dielectric column (Figure 4.5).

### Real-Space Representation of Triangular Lattice

Figure 4.6 shows the real-space representation of two-dimensional triangular lattice of column of high dielectric constant *e _{a}* in a background of low dielectric constant

*e*

_{b}.Figure 4.6b shows the reciprocal space depiction of the two-dimensional triangular lattice of column of high dielectric constant *e _{a}* in a background of low dielectric constant

*t*with the reciprocal space depiction of the two-dimensional triangular lattice with the hexagonal irreducible Brillouin zone.

_{h}Figure4.7 shows an expanded view of Brillouin zone. The *к-space* points high symmetry *f. M, ->* defines the irreducible Brillouin zone and is given as

FIGURE 4.6 (a) Real-space depiction of two-dimensional triangular lattice of column of high dielectric constant *e„* in a background of low' dielectric constant *s _{h}.* (b) Reciprocal space depiction of the two-dimensional triangular lattice of column of high dielectric constant

*e*in a background of low dielectric constant

_{a}*e*(c) Reciprocal space depiction of the two-dimensional triangular lattice with the hexagonal irreducible Brillouin zone.

_{b}.FIGURE 4.7 An expanded view of the Brillouin zone for the two-dimensional triangular lattice. The triangular irreducible portion of the Brillouin zone is shown. The *к-space* points high symmetry *f*, *M,* defines the irreducible Brillouin zone.

The irreducible portion of the Brillouin zone defines *к-space* trajectory to be followed when calculating the photonic band structure. The band structure of all other *k-* space points outside the irreducible portion of the Brillouin zone can be found by taking advantages of the symmetry of the Brillouin zone.