Photonic Crystal for Gas Sensing Application

Abstract

In this paper, 1D Photonic Crystal structure, consisting of alternate layers of silicon and air, is theoretically studied for gas sensing application. With one of the air layers replaced by a different gas layer, the resulting defect mode can be utilized for sensing a particular gas, from the transmittance curve, calculated by Transfer matrix method. Very small refractive index variation, ∆n = 1.5 × 10⁻⁵ and a sensitivity = 979 nm/RIU is found by this sensor.

1 Introduction

Photonic crystals [1] are periodic dielectric structures which are designed to control the propagation of electromagnetic waves. A semiconductor cannot support electrons of energy lying in the electronic bandgap. Similarly, a photonic crystal cannot support photons lying in the photonic bandgap. Those wavelengths that are not able to pass through the structure is called Photonic bandgap. Three types of photonic crystals structures are studied according to their periodicity, those are one-dimensional-, two-dimensional-, and three-dimensional photonic crystal structures [2].

In 1887, the English physicist Lord Rayleigh [3] investigates a periodic system, he found out that there are special frequency ranges, depending on designed structures, that are forbidden in transmission spectra. In 1987, Yablonovitch [4] and John [5] researched on periodic optical structures with more than one dimension, after that this periodic structure is called photonic crystals.

Gas detection [6] is a major part in many areas such as industry, coal mines, environmental studies or medical application for monitoring or controlling respiratory gases. There are various methods for detection of gases, such as, metal oxide semiconductor for gas sensing, where gases are detected by a redox reaction [7], calorimetric methods, in which catalytic oxidation is mainly used to detect the gases [8]. Here we give some recent work on photonic crystal gas sensors. T. Sünner et al. studied a gas sensor that can detect up to 10⁻⁴ RIU refractive index change [9]. An optical-based NO₂ gas monitoring system has been studied by Rshmatet et al. [10]. To sense a gas accurately a very small refractive index change of 1.4 × 10⁻⁵ is achieved by T. Chen et al. using 1D photonic crystal [11]. A 1D photonic crystal with two defects has been designed to sense novel ozone gas [12]. Two alternating layers of MgF₂ and Si with an empty layer in the middle are used to make a sensor which is studied by Bouzidiet et al. [13]. A gas sensor which is studied by S. M. Hamidia et al. where peak frequency measurement is used to sense a gas [14].

In this paper, multilayer stacks of high and low refractive index materials such as Air and Si are chosen to construct 1D photonic crystal gas sensor which is able to sense the presence of a gas, a defect layer, from the change in transmission spectrum, with reference to that of air. Section 2 explains the transfer matrix method [15, 16]. In Sect. 3, the results are discussed and Sect. 4 concludes the result.

2 Theory

2.1 Transmission in the One-Dimensional Photonic Crystal Structure

Figure 1 shows the schematic of 1D photonic crystal. In this structure, alternating periodic layers of silicon and air are used.

Fig. 1

Schematic structure of one-dimensional (1D) photonic crystal structure with defect layer

Consider that first layer is Si (silicon) which has permittivity ε₁, permeability µ₁, thickness d₁, and refractive index n₁ and second layer is air layer which has permittivity ε₂ permeability µ₂, thickness d₂, and refractive index n₂.

The transverse component of E and H field inside the Structure from the Maxwell’s equation, for TE polarization is given by

$$ {\text{E}} = {\text{A}}\, \cdot \,{\text{e}}^{{{\text{i}}\left( {\upomega\,{\text{t}} - {\text{kd}}} \right)}} + {\text{B}}\, \cdot \,{\text{e}}^{{{\text{i}}\left( {\upomega{\text{t}} + {\text{kd}}} \right)}} $$

(1)

$$ {\text{H}} = \frac{1}{\upeta}\left( {{\text{A}}\, \cdot \,{\text{e}}^{{{\text{i}}\left( {\upomega\,{\text{t}} - {\text{kd}}} \right)}} - {\text{B}}\, \cdot \,{\text{e}}^{{{\text{i}}\left( {\upomega\,{\text{t}} + {\text{kd}}} \right)}} } \right), $$

(2)

where k = wave number and η = intrinsic impedance are

$$ k_{L} = \omega \sqrt {(\upvarepsilon_{0}\upmu_{0}\upvarepsilon_{\text{L}}\upmu_{\text{L}} )} $$

(3)

$$ \upeta_{L} = \sqrt {(\upvarepsilon_{0}\upmu_{0} /\upvarepsilon_{\text{L}}\upmu_{\text{L}} )} $$

(4)

L is the total number of periodic layers inside the photonic crystal structure.

We can found a relation between the electromagnetic field components E_L and H_L inside the structure and the electromagnetic field components E₁ and H₁ in the air in matrix form using the transfer matrix method.

$$ \left| {\begin{array}{*{20}c} {{\text{E}}_{1} } \\ {{\text{H}}_{1} } \\ \end{array} } \right| = M_{1} M_{2} \ldots M_{L - 1} M_{L} \left| {\begin{array}{*{20}c} {{\text{E}}_{\text{L}} } \\ {{\text{H}}_{\text{L}} } \\ \end{array} } \right| $$

(5)

M_Lis the characteristic matrix of Lth layer

$$ M_{L} = \left| {\begin{array}{*{20}c} {\cos \left( {{\text{k}}_{{\text{L}}} {\text{d}}_{{\text{L}}} } \right)} & {{\text{j}}\eta _{1} \sin \left( {{\text{k}}_{{\text{L}}} {\text{d}}_{{\text{L}}} } \right)} \\ {\frac{{\text{j}}}{{\eta _{{\text{L}}} }}\sin \left( {{\text{k}}_{{\text{L}}} {\text{d}}_{{\text{L}}} } \right)} & {\cos \left( {{\text{k}}_{{\text{L}}} {\text{d}}_{{\text{L}}} } \right)} \\ \end{array} } \right| $$

(6)

If we consider the angle of incident light (θ_L) then the characteristic matrix of Lth layer can be written as

$$ M_{L} = \left| {\begin{array}{*{20}c} {{ \cos }\left( {\updelta_{\text{L}} } \right)} & {{\text{j}}\upgamma_{\text{L}} { \sin }\left( {\updelta_{\text{L}} } \right)} \\ {\frac{\text{j}}{{\upgamma_{\text{L}} }}{ \sin }\left( {\updelta_{\text{L}} } \right)} & {{ \cos }\left( {\updelta_{\text{L}} } \right)} \\ \end{array} } \right| $$

(7)

δ_L and γ_L being the matrix parameters and depending on the incident angle θ_L of light, the optical constants and the layer thickness are expressed as,

$$ \updelta_{\text{L}} = {\text{k}}_{\text{L}}\, {\text{d}}_{\text{L}}\, \cos \left( {\uptheta_{\text{L}} } \right) $$

(8)

$$ \upgamma_{\text{L}} =\upeta_{\text{L}} /\cos\uptheta_{\text{L}} \left( {{\text{TE}}\,{\text{mode}}} \right) $$

(9)

By considering the characteristic matrix of each layer, we can obtain the transmission matrix of the whole structure, where p is the number of periods,

$$ \mathop \prod \limits_{1}^{\text{p}} ({\text{M}}_{\text{p}} ) = \left[ {\begin{array}{*{20}c} {{\text{m}}_{11} } & {{\text{m}}_{12} } \\ {{\text{m}}_{21} } & {{\text{m}}_{22} } \\ \end{array} } \right] $$

(10)

$$ {\text{Mp}} = {\text{M}}_{1} \times {\text{ M}}_{2} $$

(11)

The transmittance t is defined as the ratio of the fluxes of the transmitted wave to the flux of the incident wave.

$$ t = \frac{{2\upeta_{\text{i}}^{ - 1} }}{{({\text{m}}_{11} + {\text{m}}_{12}\upeta_{\text{s}}^{ - 1} )\upeta_{\text{i}}^{ - 1} + ({\text{m}}_{21} +\upeta_{s}^{ - 1} {\text{m}}_{22} )}} $$

(12)

Here η_i and η_s are the intrinsic impedance of the first and last medium of the structure which is given as,

$$ \upeta_{\text{s}} = \left\{ {\begin{array}{*{20}c} {\frac{{\upeta_{\text{s}} \cos \,\uptheta_{\text{s}} }}{{{\text{Z}}_{0} }}\,{\text{TE}}\,{\text{mode}}} \\ {\frac{{\cos \,\uptheta_{\text{s}} }}{{\upeta_{\text{s}}\, {\text{Z}}_{0} }}{\text{TM}}\,{\text{mode}}} \\ \end{array} } \right. $$

(13)

where

$$ Z_{0} = \sqrt {\frac{{\mu_{0} }}{{\varepsilon_{0} }}} $$

(14)

Hence the transmittance T spectrums can be obtained by using the expressions

$$ {\text{T}} = \left| {\text{t}} \right|^{2} $$

(15)

The sensitivity of this sensor is given by the following expression:

$$ {\text{S}} = \Delta\uplambda/\Delta {\text{n}}\,\left( {{\text{nm}}/{\text{RIU}}} \right) $$

(16)

3 Results and Discussion

The analysis using the transfer matrix approach uses four periodic layers of silicon and air on each side of the defect layer.

Different lengths of defect layer are studied to get defect mode within 1.55 μm range, assuming a thickness of Si layer of d_Si = 540 nm and thickness of Air layer of d_Air = 440 nm, as shown in Fig. 2. For a normal incident of light, defect mode appears within the photonic bandgap, depending on the length of the defect layer. The wavelength of defect mode shifts when cavity length increases from 700 to 900 nm.

Fig. 2

Transmission spectrum of 1D photonic crystal for different defect layers thickness. Si layer thickness d_Si = 540 nm and air layer thickness d_Air = 440 nm

The wavelength shifting is linear with the length of the defect layer as shown in Fig. 3. This shift of the wavelength within the photonic bandgap also depends on the thickness of the Si layer (d_Si).

Fig. 3

Wavelength of defect layer as a function of defect layers thickness for d_Si = 540 nm, d_Air = 440 nm where cavity length changes from 700 to 900 nm

When light incident on the structure at an angle other than normal incidence, defect mode shifts in lower wavelength value as shown in Fig. 4. It may also be noted from Fig. 4 that the photonic bandgap increases with increase in incident angle of light which is clear from Fig. 5.

Fig. 4

Transmission spectrum for different incident angle of light for Si layer thickness d_Si = 540 nm and air layer thickness d_Air = 440 nm

Fig. 5

Photonic bandgap for different incident angle of light for Si layer thickness = 540 nm and air layer thickness = 440 nm

Figure 6a, b shows the transmission characteristics of 1D photonic crystal in the presence of different gases such as Helium (He), Hydrogen (H₂), Carbon dioxide (CO₂), Methane (CH₄), Argon (Ar), Neon (Ne), and Sulfur dioxide (SO₂) in the defect layer. The shift in wavelength peak-position is measured taking air as reference. It may be seen from the figure that the peak-position shifts to lower wavelengths for gases having refractive indices less than air, while it shifts to higher wavelengths for gases having higher refractive indices. Thus, the result can be utilized for sensing of different gases.

Fig. 6

Transmission spectrum for different gases with air as the reference medium d_defect = 860 nm

To study the sensitivity of the device, different thicknesses of silicon and air layer are studied. Curve A, B, C, and D in Fig. 7 denote the corresponding results. Sensitivity, calculated from (16), for those designed structures are 440 nm/RIU, 550 nm/RIU, 654 nm/RIU, and 979 nm/RIU respectively.

Fig. 7

Wavelength shift as a function of the refractive index change. A (d_Si = 540 nm, d_Air = 440 nm, d_defect = 800 nm), B (d_Si = 380 nm, d_Air = 280 nm, d_defect = 700 nm), C (d_Si = 340 nm, d_Air = 620 nm, d_defect = 700 nm), D (d_Si = 160 nm, d_Air = 240 nm, d_defect = 700 nm)

4 Conclusion

In this paper, a 1D photonic crystal structure as a gas sensor is studied. The sensor can sense the shift in wavelength of transmission spectra, using Transfer matrix method. The study shows that the defect mode inside the bandgap can shift in between the gap of the photonic bandgap if the length of the Si layer or the length of the defect layer or the angle of incidence of the light on the structure is changed. As a result, this structure is very sensitive to the refractive index change of gases which are present in the dry air. The sensitivity of this sensor will increase if we consider different designs of the device structure.