**2.2 Analysis of FEM with anisotropic PML**

Technically, a PML is not a boundary condition but an additional domain that absorbs the incident radiation without producing reflections. It can have arbitrary thickness and is specified to be made of an artificial absorbing material. The material has anisotropic

A Novel Compact Photonic Crystal Fibre Surface

equal to d2 throughout this study.

propagating in the z – direction.

**4. Simulation results** 

*n*

using data from Johnson and Christy(Johnson and Christy 1972).

(Sellmeier 1871);

Plasmon Resonance Biosensor for an Aqueous Environment 85

Extra air holes of diameter d01 are inserted between the main air holes as a means of reducing the propagation losses whilst ensuring efficient coupling between the core guided and plasma modes. A small air hole of diameter d1 is introduced into the core to facilitate phase matching with a plasmon by lowering the refractive index of the core-guided mode. The first layer of holes work as a low refractive index cladding, enabling mode guidance in the fiber core. The second ring has two slots of uniform thickness d3, which houses the analyte. Theses slots are coated with gold of thickness tAu. The gap between slots is set to be

The background material is made of silica which is modeled using the Sellmeier equation

() 1 *B B <sup>B</sup>*

where n is the refractive index, λ , the wavelength in µm, B(i = 1,2,3) and C(i = 1,2,3) are Sellmeier coefficients. The Sellmeier coefficients used for the background material are B1=0.696166300, B2=0.407942600, B3=0.897479400, C1=4.67914826×10-3μm2, C2=1.35120631×10-2 μm2 and C3= 97.9340025 μm2 (Sellmeier 1871). The permittivity of gold and silver is modeled

Simulations were carried out using a full-vectorial finite element method (FEM) with perfectly matched layers (PMLs). The cross section of the proposed PCF SPR biosensor is divided into many sub-domains with triangular shaped elements in such a way that the step index profiles can be exactly represented. Due to the symmetrical nature of the PCF structure, only one-quarter of the sensor cross section is divided into curvilinear hybrid elements. This results in a computational window area of (5 μm × 5 μm) terminated by a PML of width = 1 μm. The mesh size is 19,062 elements. Modal analysis of the fundamental mode has been performed on the cross section in the x-y plane of the PCF as the wave is

We begin our analysis by investigating the potential of the proposed PCF for sensing. The structural parameters used are Λ = 1.5 μm, d1/Λ= 0.2, d2/Λ= 0.35, d01/Λ = 0.15, d3= 1.5 μm and tAu = 40 nm. The slots in the second ring are first filled with an analyte whose refractive index, na = 1.33 (water) after which the confinement loss of the fundamental mode is calculated. The process is repeated with an analyte of refractive index of 1.34 and the calculated loss spectra for both cases are plotted in Fig 2. It can be observed from Fig. 2 that there are two major attenuation peaks which correspond to the excitation of plasmonic modes on the surface of the metalized channels filled with aqueous analyte , na=1.33. It is important to note that the shape of a metallized surface can have a significant effect on the plasmonic excitation spectrum. Hence, a planar metallized surface supports only one

plasmonic peak, while a cylindrical metal layer can support several plasmonic peaks.

By changing the analyte refractive index from 1.33 to 1.34, it is observed in Fig. 2 that there is a corresponding shift (dashed curves) in the resonant attenuation peaks. This transduction mechanism is commonly used for detecting the bulk analyte refractive index changes, as

2 2 2 1 2 3 222

*CCC*

123

(8)

permittivity and permeability that matches that of the physical medium outside the PML in such a way that there are no reflections regardless of the angle of incidence, polarisation and frequency of the incoming electromagnetic radiation (Koshiba and Saitoh 2001; Buksas 2001). The PML formulation can be deduced from Maxwell's equations by introducing a complex-valued coordinate transformation under the additional requirement that the wave impedance should remain unaffected (Buksas 2001).

If one considers a PML which is parallel to one of the Cartesian coordinate planes, an s matrix of the form (Koshiba 1992; Koshiba and Saitoh 2001):

$$\begin{aligned} \begin{bmatrix} \mathbf{s} \end{bmatrix} &= \begin{bmatrix} \frac{\mathbf{s}\_y \mathbf{s}\_z}{\mathbf{s}\_x} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{0} & \frac{\mathbf{s}\_x \mathbf{s}\_z}{\mathbf{s}\_y} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & \frac{\mathbf{s}\_x \mathbf{s}\_y}{\mathbf{s}\_z} \end{bmatrix} \\\\ \begin{bmatrix} \mathbf{s} \end{bmatrix}^{-1} &= \begin{bmatrix} \frac{\mathbf{s}\_x}{\mathbf{s}\_y \mathbf{s}\_z} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{0} & \frac{\mathbf{s}\_y}{\mathbf{s}\_x \mathbf{s}\_z} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & \frac{\mathbf{s}\_z}{\mathbf{s}\_x \mathbf{s}\_y} \end{bmatrix} \end{aligned} \tag{5}$$

can be substituted into Eqn. (1) to permit the use of anisotropic PML. The modified equation thus becomes;

$$\nabla \times \left( [\boldsymbol{\nu}] [\boldsymbol{s}]^{-1} \nabla \times \boldsymbol{\Phi} \right) - k\_0^2 [\boldsymbol{q}] [\boldsymbol{s}] \boldsymbol{\Phi} = \mathbf{0} \tag{6}$$

The parameters sx, sy, and sz are complex valued scaling parameters. These parameters are set to α (α = 1 – αj for leaky mode analysis), when we want to absorb the travelling wave in that direction and unity when no absorption is need. Thus, the absorption by the PML can be controlled by appropriate choice of αj. A parabolic profile is assumed for αj such that:

$$
\alpha\_j = \alpha\_{j\max} \left(\frac{\rho}{\alpha}\right)^2 \tag{7}
$$

where ρ is the distance from the beginning of the PML and ω, the thickness of the PML.

#### **3. Sensor design and numerical modelling**

The proposed PCF SPR biosensor (Fig. 1) consists of circular air holes arranged in a hexagonal lattice, with a small circular air hole at the center. The air hole to air hole spacing is denoted by Λ, whilst d2 represents the diameters of the circular air holes in the first ring.

permittivity and permeability that matches that of the physical medium outside the PML in such a way that there are no reflections regardless of the angle of incidence, polarisation and frequency of the incoming electromagnetic radiation (Koshiba and Saitoh 2001; Buksas 2001). The PML formulation can be deduced from Maxwell's equations by introducing a complex-valued coordinate transformation under the additional requirement that the wave

If one considers a PML which is parallel to one of the Cartesian coordinate planes, an s

0 0

*x y z*

> *z x y*

<sup>0</sup> *ps k qs* <sup>0</sup> (6)

*s s s*

2

(4)

(5)

(7)

*s s s*

0 0

0 0

*s*

*s s*

*y x z*

0 0

can be substituted into Eqn. (1) to permit the use of anisotropic PML. The modified equation

<sup>1</sup> <sup>2</sup>

The parameters sx, sy, and sz are complex valued scaling parameters. These parameters are set to α (α = 1 – αj for leaky mode analysis), when we want to absorb the travelling wave in that direction and unity when no absorption is need. Thus, the absorption by the PML can be controlled by appropriate choice of αj. A parabolic profile is assumed for αj such that:

> *j j*max

where ρ is the distance from the beginning of the PML and ω, the thickness of the PML.

The proposed PCF SPR biosensor (Fig. 1) consists of circular air holes arranged in a hexagonal lattice, with a small circular air hole at the center. The air hole to air hole spacing is denoted by Λ, whilst d2 represents the diameters of the circular air holes in the first ring.

**3. Sensor design and numerical modelling** 

0 0

*s s*

*s*

*x z y*

0 0

*x y z*

*s s s*

*y z x*

*s s s*

impedance should remain unaffected (Buksas 2001).

matrix of the form (Koshiba 1992; Koshiba and Saitoh 2001):

<sup>1</sup>

*s*

thus becomes;

*s*

Extra air holes of diameter d01 are inserted between the main air holes as a means of reducing the propagation losses whilst ensuring efficient coupling between the core guided and plasma modes. A small air hole of diameter d1 is introduced into the core to facilitate phase matching with a plasmon by lowering the refractive index of the core-guided mode. The first layer of holes work as a low refractive index cladding, enabling mode guidance in the fiber core. The second ring has two slots of uniform thickness d3, which houses the analyte. Theses slots are coated with gold of thickness tAu. The gap between slots is set to be equal to d2 throughout this study.

The background material is made of silica which is modeled using the Sellmeier equation (Sellmeier 1871);

$$m(\lambda) = \sqrt{1 + \frac{B\_1 \lambda^2}{\lambda^2 - C\_1} + \frac{B\_2 \lambda^2}{\lambda^2 - C\_2} + \frac{B\_3 \lambda^2}{\lambda^2 - C\_3}}\tag{8}$$

where n is the refractive index, λ , the wavelength in µm, B(i = 1,2,3) and C(i = 1,2,3) are Sellmeier coefficients. The Sellmeier coefficients used for the background material are B1=0.696166300, B2=0.407942600, B3=0.897479400, C1=4.67914826×10-3μm2, C2=1.35120631×10-2 μm2 and C3= 97.9340025 μm2 (Sellmeier 1871). The permittivity of gold and silver is modeled using data from Johnson and Christy(Johnson and Christy 1972).

Simulations were carried out using a full-vectorial finite element method (FEM) with perfectly matched layers (PMLs). The cross section of the proposed PCF SPR biosensor is divided into many sub-domains with triangular shaped elements in such a way that the step index profiles can be exactly represented. Due to the symmetrical nature of the PCF structure, only one-quarter of the sensor cross section is divided into curvilinear hybrid elements. This results in a computational window area of (5 μm × 5 μm) terminated by a PML of width = 1 μm. The mesh size is 19,062 elements. Modal analysis of the fundamental mode has been performed on the cross section in the x-y plane of the PCF as the wave is propagating in the z – direction.
