SENSITIVITY AND EVANESCENT FIELD STUDY OF A UNIAXIAL ANISOTROPIC PLANAR WAVEGUIDE BASED OPTICAL SENSOR

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Planar waveguide based optical sensors use the exponential decay of evanescent field in the external medium, to sense alterations occurring in the vicinity of a waveguide surface. This feature makes evanescent wave based optical sensor enable monitoring of chemical and biochemical interactions in real-time with high sensitivity and stability. Furthermore, sensors based on optical LOC systems would sense selective analytes just by choosing the appropriate biological receptors [1]. Recently, it has been reported that plasmonic and silicon photonics-based biosensors are among the most employed evanescent-wave biosensors to analyze nucleic acids with potential applicability in the clinical diagnosis [2]. In addition, Lithium Niobate (LN) is among one of the widely used materials for integrated sensor applications due to its excellent optical, ferroelectric, piezoelectric, and thermoelectric properties [3]. In fact, the fabrication of LiNbO3 thin film crystals exhibiting optical properties comparable to those for bulk crystals along with coupled TE and TM modes [4]. Sensitivity, being most important parameter of an optical sensor has attracted significant research interest [5][6][7]. Thus, planar waveguides comprising left-handed material (LHMs) with different configurations [8], as well as binary and ternary photonic crystal with LHMs layers [9,10]; have been investigated as optical sensors. It has also been observed that the sensitivity of the proposed structures can be significantly improved by the optimization of physical and geometrical parameters of the waveguide layers.
In this paper, the sensitivity and evanescent fields of a slab waveguide sensor constituted of uniaxial anisotropic crystals have been investigated. The influence of light source in addition to the physical and geometrical parameters of the waveguide, on the sensor sensitivity and evanescent fields have been studied in detail for the two kinds of modes propagating simultaneously in a waveguide. In our previous work [11], the influence of physical parameters was briefly studied.

THEORETICAL MODELING
The planar waveguide for an optical sensor was constituted by three layers of uniaxial crystals of lossless and nonmagnetic materials, as presented in Fig. 1. The core refractive index tensor is , whereas, the refractive index tensors of the media above and below the film guide are, respectively, and . Then:  Two waves (TE and TM) are propagated separately in the core of such slab waveguide, considered as an ordinary and extraordinary wave, respectively [12].

Sensitivity of a Birefringent Planar Waveguide Based Sensor
The sensitivity is defined as the rate of change in modal effective refractive index relative to the cover refractive index: The two expressions of the sensor sensitivity for both TE and TM modes are given as [12]: For TE modes For TM modes, N is the effective refractive index, m is the mode order, d is the core thickness and k 0 is the free-space wavenumber. The sensitivity can be expressed as a function of evanescent field intensity [13]: where, P = 0 for TE polarization. However, P = 1 for TM polarization. The intensity of evanescent field can be represented by a confinement factor Γ s , which is the ratio of an electric field intensity in the sensitive region or layer to the entire mode distribution of the guiding mode [14], defined as: Fields Distribution in Each Region for TE Modes Guided TE modes exhibit exponentially decreasing behavior in the cover and substrate, and a sinusoidal behavior in the film region. Afterwards, by using boundary conditions, components of the EM field E y , H x and H z in each region as a function of the amplitude can be extracted as [15]:

Fields Distribution in Each Region for TM Modes
Similarly, different components of the EM field for TM modes can be expressed as: The dielectric constant ε i differs from region to region, i=c, f, s (cover, film, substrate). where

Influence of Waveguide and Source Parameters on the Sensor Sensitivity and Evanescent Fields
As, the sensitivity expressions and evanescent field components can be expressed as functions of waveguide and the light source parameters. We have developed a program in Matlab tool to observe the influence of physical, geometrical and the source parameters on the sensitivities of TE and TM modes as well as their influence on the evanescent fields in Effect of the Source's Parameters Effect of the Field Amplitude. Equation (6) describes that the sensor becomes more sensitive as the intensity of evanescent field increases, where the intensities of the electric and magnetic evanescent fields are function of their amplitudes. Fig. 2 illustrates the sensitivity as a function of the frequency for TE 0 and TM 0 modes for different fraction of modal power located in the cover. The observations revealed that, the sensitivity increases with the increase fraction of modal power located in the cover. Though, for TE 0 , the sensitivity observed to be maximum for lower frequencies however, decreases for higher frequencies. Whereas, for TM 0 , the sensitivity is minimal at lower frequencies while improves at higher frequencies. The improved sensitivities at higher modal power located in the cover resulted from enhanced amplitudes of evanescent field in the region, as illustrated in Fig. 3 and 4. It has also been observed that, amplitudes of evanescent field components of TM modes are more important than those of TE modes, hence, the sensitivity of TM modes are always greater than that of TE modes.   Effect of the Source's Wavelength. Sensitivities as a function of the core thickness for TE 0 and TM 0 modes, are presented in Fig. 5 observed at different source's wavelengths. The curves show that, sensitivity of the planar waveguidebased sensor increases by increasing the wavelength of source, however, the curves are shifted towards higher core thicknesses. This implies that, at higher wavelengths, maximum sensitivities can be obtained for higher core thicknesses and vice versa. The increase of the sensitivity for higher source's wavelengths can be justified by an increase in the intensities of evanescent field, as illustrated in Fig. 6 and 7. Effect of the Geometrical Parameter Fig. 8, represents the sensitivity as a function of the frequency for TE 0 and TM 0 modes at different core thickness. The observations revealed that, by increasing the core thickness, sensitivity of the sensor decreases and curves shifted towards the lower frequencies. The reduced sensitivity is resulted from the decrease in sensing area limited by the evanescent fields, normal to the waveguide at z = 0 and the super surface of the waveguide (x = 0). Thickness S e n s itiv ity   In fact, at greater core thickness, maximum part of an optical signal enters into the core of waveguide as energy and ultimately reduces the evanescent field intensities, as illustrated in Fig. 9 and 10. Effect of the Physical Parameters Physical parameters studied are the refractive indices of cover, substrate and the core of waveguide. Expressions (2) and (4) demonstrated the influence of physical parameters of waveguide on the sensitivity of the sensor. However, their influence on the evanescent fields are expressed by the electromagnetic field components in the cover.
Effect of the Cover Refractive Index. Fig. 11 illustrated the sensitivities as a function of the frequency of TE0 and TM 0 modes when the refractive index of cover was taken as 1.48 and 1.70. For TE 0 , the observed maximal sensitivities are 0.15 and 0.37, corresponding to the frequencies 3×10 14 Hz and 1.25×10 14 Hz, respectively. However, for TM 0 , the observed maximal sensitivities are 0.23 and 0.47, corresponding to the frequencies 6.25×10 14 Hz and 3.25×10 14 Hz, respectively. E v a n e s c e n t fie ld s Exc, d=100 nm Exc, d=150 nm Exc, d=200 nm Ezc, d=100 nm Ezc, d=150 nm Ezc, d=200 nm Figure 11. Sensitivity as a function of the frequency for TE0 and TM0 modes for different refractive index of the cover material nc, (ns=1.72, d=100 nm, LiNbO3 as guiding film) The curves show that, sensitivity of slab waveguide-based sensor increases by increasing the cover refractive index for both TE and TM modes. In addition to that, the position of maximal sensitivities exhibit shift toward lower frequencies.
As the refractive index of cover must be always lesser than that for the substrate, one can conclude that, for both TE and TM modes, the sensitivity of a sensor increases by making the cover refractive index as close as possible to that for the substrate.
Nevertheless, for different cover refractive indices, small changes in the evanescent field components are evident, near to the cut-off frequencies, Fig. 12 and 13. These changes are caused by the variation in effective refractive indices of TE0 and TM 0 modes induced by fluctuations in core refractive index, Fig. 14. These changes in evanescent field components in the cover, close to cut-off frequencies, have minimal influence on sensitivity of the slab waveguide-based sensor. Effect of the Substrate Refractive Index. Fig. 15 represents the sensitivities of TE 0 and TM 0 modes as a function of the frequency, for various refractive index of substrate. Graphs depicted the maximal values of sensitivities and their corresponding frequencies. For TE 0 , maximal sensitivities are observed to be 0.245 and 0.11, corresponding to 2 x 10 14 Hz and 4 10 14 Hz frequency, respectively. For TM 0 , maximal sensitivities are 0.34 and 0.175, corresponding to the frequencies as 5 10 14 Hz and 6 10 14 Hz, respectively. The sensitivity of sensor decreases as a function of substrate refractive index for both TE and TM modes. Furthermore, the position of maximal sensitivities exhibit shifts toward higher frequencies.  Curves show a small change in the evanescent field components caused by of the variation in refractive index of substrate; and, these small changes occur near the cut-off frequencies of each mode. Far from these frequencies, there has been no changes in evanescent field components. However, these changes are caused by small variation in effective refractive indices of substrate, Fig. 18. One can conclude that, change in evanescent field have negligible influence on the sensitivity of a slab waveguide sensor Fig. 15.    For TE 0 , maximal sensitivities are observed to be 0.125 and 0.24 corresponding to 6 x 10 14 Hz and 2 x 10 14 Hz, respectively. However, for TM 0 , maximal sensitivities are 0.15 and 0.34 corresponding to 8 x 10 14 Hz and 5 x 10 14 Hz, respectively. The observations revealed that, sensitivity of the planar waveguide-based sensor increases as a function of core refractive index for both TE and TM modes. In addition to that, the position of the maximal sensitivities exhibit shifts toward lower values of frequency.
On the other hand, evanescent field components as illustrated in Fig. 20 and 21 for different core refractive indices, exhibit moderate changes caused by variation in core refractive index; in fact, these changes occur far from the cut-off frequencies of each mode. These changes are caused by the variation in effective refractive indices of TE 0 and TM 0 modes induced at different core refractive index, Fig. 22. These changes in evanescent field, far from the cut-off frequencies, have negligible influence on the sensitivity of a slab waveguide sensor, since the sensitivity peaks appeared near the cutoff frequencies. Exc, n f =1.80 Exc, n f =2.2880 Ezc, n f =1.80 Ezc, n f =2.2880 CONCLUSIONS In this work, sensitivity and evanescent field of a planar waveguide-based optical sensor prepared of birefringent materials have been investigated. The influence of source, geometrical and physical parameters of the planar waveguide on the sensitivity of sensor and evanescent field are studied. The observations in all cases revealed that, the maximal sensitivities of TM modes and their corresponding frequencies are greater than those for TE modes. Furthermore, improved sensitivity and evanescent field in the cover has been observed at in higher amplitude and wavelength of the light source. An increase in the core thickness however, reduces the sensitivity of sensor owing to the decrement of evanescent field in the cover. Although, any increment in refractive indices of cover and the core resulted in an improved sensor sensitivity. On the contrary, higher refractive index induces of substrate reduces the sensor sensitivity. The variation in physical parameters of the waveguide sensor have no or negligible influence on the evanescent fields.