Indian J Phys DOI 10.1007/s12648-017-1130-z ORIGINAL PAPER Binary photonic crystal for refractometric applications (TE case) S A Taya* and S A Shaheen Physics Department, Islamic University of Gaza, P.O.Box 108, Gaza, Palestine Received: 19 November 2016 / Accepted: 24 August 2017 Abstract: In this work, a binary photonic crystal is proposed as a refractometric sensor. The dispersion relation and the sensitivity are derived for transverse electric (TE) mode. In our analysis, the first layer is considered to be the analyte layer and the second layer is assumed to be left-handed material (LHM), dielectric or metal. It is found that the sensitivity of the LHM structure is the highest among other structures. It is possible for LHM photonic crystal to achieve a sensitivity improvement of 412% compared to conventional slab waveguide sensor. Keywords: Photonic crystal; Refractometric application; Sensitivity PACS No.: 42.70.Qs; 87.85.fk; 84.40.Az 1. Introduction In the past few years, extensive research and development activities have been devoted to evanescent- field- based optical waveguide sensors [1–8]. They play an important function in a variety of sensing applications. By means of measuring small changes in optical phase or intensity of the guided light, these sensors present excellent properties such as high sensitivity, fast response, immunity to electromagnetic fields, and safety in the detection of combustible and explosive materials [1]. An optical sensor is a device in which light interacts with the substance to be detected (measurand) and converts light affected by the measurand substance into electrical signal which gives information about the analyte. In slab waveguides, light is confined within the guiding layer, with a small part of the guided mode called the evanescent field that extends to the surrounding media. The evanescent field detects any refractive index variation of the covering medium in homogeneous sensing. The interaction of the evanescent field with the measurand causes a change in the effective refractive index of the guided mode. The change in the effective refractive index due to the change in the analyte index is the sensing criteria. Artificial materials with simultaneously negative electric permittivity (e) and magnetic permeability (l) have received an increasing interest [9–19]. These metamaterials were called left-handed materials (LHMs) or negative-index materials (NIMs). They were named LHMs because the electric field, magnetic field and wave vector of an electromagnetic wave form a left-handed set. Veselago [9] showed that such materials exhibit a number of unusual properties such as negative refractive index, reversal of Snell’s law, and reversal of Doppler shift of radiation. Slab waveguides comprising LHM have been suggested as optical sensors [20–27]. A multilayer waveguide configuration comprising a metal layer and a LHM as a guiding layer was proposed as a metal-clad waveguide sensor [10]. Reflection and transmission coefficients were derived and utilized to investigate the resonance dips at which the reflectance vanishes. The results showed that the proposed configuration has an advantage over the well-known surfaceplasmon resonance (SPR) structure since it gives a much sharper reflectance dip and can result in a considerable sensitivity enhancement [10]. The propagation of transverse magnetic (TM) waves in a four-layer waveguide configuration was investigated for refractometric applications [13]. The waveguide configuration consists of a substrate, a metal layer, a material of LHM as a core layer, and a cladding. The metal layer was very thin so that a peak was obtained in the reflectance profile. The angular position of the reflectance peak was observed to sense any changes in the index of refraction of the cladding. The results showed that for aluminum metal layer, a thickness of about 9 nm represents the optimum metal thickness which corresponds to the highest and sharpest peak. The negative parameters of the LHM were found to have a considerable effect on the performance of the proposed sensor *Corresponding author, E-mail: staya@iugaza.edu.ps Ó 2017 IACS S A Taya and S A Shaheen [13]. A planar waveguide with air core and anisotropic LHM cladding was investigated for refractometric applications [16]. Different from the SPR structures in which the measurand is placed in the evanescent field layer, the proposed sensor contains the measurand in the guiding layer that supports the oscillating field. As a result of the strong concentration of the field in the measurand layer, the proposed structure showed a dramatic sensitivity improvement [16]. Photonic crystals are artificial periodic structures having energy bands for photons which either allow or forbid the propagation of electromagnetic waves of certain frequency ranges in the same manner as the periodic potentials do for electrons in atomic crystals [12]. Depending on the geometry of the structure, photonic crystals can be divided into one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) structures. The most important property which determines practical significance of the photonic crystals is the presence of the photonic band gap. When a radiation with a frequency inside the periodic band gap is incident on the structure, it is completely reflected. Traditional waveguides operate by total internal reflection. In a purely two-dimensional photonic-crystal linear waveguide, a linear one-dimensionally periodic defect is introduced into the crystal, creating a localized band that falls within and is guided by the photonic band gap [15]. Light is therefore prohibited from escaping the waveguide. Photonic crystals were proposed as sensing elements for detection small changes in the refractive index of a medium [2]. The suggested sensor was operated in reflection mode. Ternary photonic crystals are found more sensitive than binary photonic crystal. The fabrication of a refractometric of a quasi-one-dimensional waveguide photonic crystal was reported [3]. The transmission stopband was found to shift by 0.8-nm wavelength for either a cladding refractive index change of 0.05. A novel way to improve the temperature sensitivity in photonic crystals by using a ternary periodic structure was demonstrated [7]. The design of a two-dimensional photonic crystal coupled resonating optical waveguide based integrated-optic sensor platform was proposed [8]. In this work, two-layer slab photonic crystal is investigated for sensing applications. The dispersion relation, sensitivity to any change in an analyte index, and power flow relations are derived, plotted and analyzed for TE mode. 2. Basic equations The geometry of two-layer slab photonic crystal structure is shown in Fig. 1. It consists of two different layers in periodic arrangement which have refractive indices n1 and n2 and thicknesses d1 and d2 , respectively. The refractive index profile of the structure is written as. X n d 1 1 n n 2 d 2 d n n 2 1 1 d 2 d 1 1 n 2 d 2 ... ... 0 n d i ... i z Fig. 1 Two-layer photonic crystal consisting of two different layers in periodic arrangement nð z Þ ¼ n1 n2 0\z\d1 d1 \z\d ð1Þ where nðzÞ ¼ nðz þ dÞ and d ¼ d1 þ d2 . Consider TE polarized light in which the waves are travelling along the x-axis and the electric field is polarized along the y- axis, then there’s a non-zero y-component of the electric field. Helmholtz equation for TE mode can be written as o 2 E y o 2 Ey þ 2 þ k2 Ey ¼ 0; ox2 oz ð2Þ Since the propagation is along x-axis, Eyðx; zÞ can be written as Ey ðx; zÞ ¼ Ey ðzÞeibx : ð3Þ Equation (2) becomes d 2 Ey þ k02 ei li N 2 Ey ðzÞ ¼ 0; 2 dz ð4Þ where b ¼ k0 N and k2 ¼ k02 ei li ; The solution of Eq. (4) in the l-th cell can be written as Ey1 ðzÞ ¼ al eiq1 ðzldÞ þ bl eiq1 ðzldÞ ; ðl 1Þd þ d2 \z\ld ð5Þ Ey2 ðzÞ ¼ cl eiq2 ðzldÞ þ fl eiq2 ðzldÞ ; ðl 1Þd\z\ðl 1Þd þ d2 ð6Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where q1 ¼ k0 e1 l1 N 2 and q2 ¼ k0 e2 l2 N 2 . The nonzero components of the magnetic field (Hx and Hz Þ can be calculated using Hx ¼ 1 oEy ; ilx oz from which q1 iq1 ðzldÞ Hx1 ¼ al e bl eiq1 ðzldÞ ; l1 x q2 iq2 ðzldÞ cl e fl eiq2 ðzldÞ ; Hx2 ¼ l2 x and ð7Þ ð8Þ ð9Þ Binary photonic crystal for refractometric applications i oHx ; b q2 oz b iq1 ðzldÞ al e Hz1 ¼ þ bl eiq1 ðzldÞ ; l1 x b iq2 ðzldÞ cl e þ fl eiq2 ðzldÞ : Hz2 ¼ l2 x Hz ¼ ð10Þ a0 b0 ¼ A C B D l al ; bl ð21Þ ð11Þ The electric field vector in a periodic layered medium can be written using Bloch theorem as [17] ð12Þ EKB ðx; zÞ ¼ EKB ðzÞeiKB z eiðxtbxÞ ; The boundary conditions require that the tangential components of E and H to be continuous at z ¼ ðl 1Þd and z ¼ ðl 1Þd þ d2 , yielding a set of homogeneous linear equations for the coefficients ai, bi, ci, and fi. The continuity of Ey and Hx at z ¼ ðl 1Þd gives iq2 d eiq2 d e 1 1 al1 cl ¼ q2 iq2 d q2 iq2 d ; 1 1 bl1 fl qTE e qTE e ð22Þ where EKB ðz þ dÞ ¼ EKB ðzÞ; ð23Þ and KB is the Bloch wave number In terms of column-vector representation, the periodic condition in Eq. (23) for Bloch wave is al a ¼ eikB d l1 : ð24Þ bl bl1 ð13Þ al1 Substituting for from Eq. (16) into (24), we get l2 bl1 where qTE ¼ l q1 . 1 a A B al The continuity of Ey and Hx at z ¼ ½ðl 1Þd þ d2 gives ¼ eikB d l ; ð25Þ b bl C D l iq d iq1 d1 iq1 d1 iq2 d1 2 1 e e cl al e e ¼ qTE iq2 d qTE iq1 d1 : A eikB d al B e e fl b eiq2 d1 eiq2 d1 l q2 q2 ¼ 0: ð26Þ bl C D eikB d ð14Þ By taking the determinant of the matrix equal to zero, then which can be written as iq d iq1 d1 A eikB d B iq1 d1 iq d 2 1 2 1 ð27Þ 1 e e cl al e e ikB d ¼ 0; C De ¼ : qTE iq2 d qTE iq1 d1 iq d iq d 2 1 2 1 e e fl bl e 2 e q2 q2 A eikB d D eikB d BC ¼ 0: ð28Þ ð15Þ Substituting from Eq. (15) into (13), we obtain al1 A B al a ¼ ¼M l ; bl1 bl bl C D But AD BC ¼ 1; which imply that ð16Þ where M is called the transfer matrix with the elements 1 1 qTE 1 e2iq2 d2 A ¼ cosðq2 d2 Þeiq1 d1 eiq2 d1 2 4 q2 1 iq2 1 sinðq2 d2 Þeiq1 d1 þ eiq2 d1 1 þ e2iq2 d2 ; þ 2 qTE 4 ð17Þ 1 qTE q2 B ¼ eiq1 d1 isinðq2 d2 Þ ; ð18Þ 2 q2 qTE 1 1 qTE 1 e2iq2 d2 C ¼ cosðq2 d2 Þeiq1 d1 eiq2 d1 2 4 q2 1 iq2 1 iq2 d1 iq1 d1 sinðq2 d2 Þe e 1 þ e2iq2 d2 ; þ 2 qTE 4 ð19Þ D¼e iq1 d1 1 q2 qTE cosðq2 d2 Þ þ i þ sinðq2 d2 Þ : 2 qTE q2 ð20Þ The transfer matrix of the structure shown in Fig. 1 can be written as e2ikB d ðA þ DÞeikB d þ 1 ¼ 0: ð29Þ The solution of the above equation for eikB d is given by !12 AþD AþD 2 ikB d ¼ 1 : ð30Þ e 2 2 The eigenvectors corresponding to the eigenvalues are obtained from Eq. (26) as a0 B ; ð31Þ ¼ b0 eikB d A Then the corresponding column vectors for the l-th unit cell are given by al B lkB d : ð32Þ ¼e eikB d A bl Multiplying Eq. (29) by eikB d , we get eikB d þ eikB d ¼ A þ D: Using the identity cos x ¼ e ð33Þ ix þe 2 ix , we get S A Taya and S A Shaheen 2 cosðkB d Þ ¼ A þ D: ð34Þ Substituting for A and D from Eqs. (17) and (20) into (34), we get cosðkB dÞ ¼ cosð q2 d2 Þ cosðq1d1 Þ 1 q2 qTE þ sinðq2 d2 Þsinðq1 d1 Þ 2 qTE q2 1 ð35Þ Equation (35) represents the dispersion relation of TE wave. The sensitivity of the effective refractive index to any change in an analyte refractive index is calculated as the change of the effective index (N) with respect to the change in the analyte index (n1); i.e., S¼ oN : on1 sensitivity of the waveguide sensors and the power flowing in the analyte layer. The power is given by 2 Z1 Ey ðzÞ b dz ð40Þ Ptotal ¼ 2x li ðzÞ ð36Þ Differentiating the dispersion relation given by Eq. (35) with respect to N, then calculating S as 1 on1 S¼ ð37Þ oN k2 N 2d2 cosðq1 d1 Þ sinðq2 d2 Þ 0 q2 1 k2 n1 on oN N þ 2d1 cosðq2 d2 Þ sinðq1 d1 Þ 0 q1 2 2 l K0 N q2 k0 on1 N sinðq1 d1 Þ sinðq2 d2 Þ þ 1 3 n1 l2 q1 q2 oN q1 l1 q2 k02 on1 N cosðq1 d1 Þ sinðq2 d2 Þ n1 þ d1 l2 q1 q1 oN l q2 k2 N d2 1 cosðq2 d2 Þ sinðq1 d1 Þ 0 q2 l2 q1 ! on l2 K02 n1 oN1 N q1 2 þ þ 3 k0 N sinðq1 d1 Þ sinðq2 d2 Þ q1 d2 l1 q2 l q1 k2 on1 N þ d1 2 cosðq1 d1 Þ sinðq2 d2 Þ 0 n1 l1 q2 q1 oN 2 l q1 k N d2 2 cosðq2 d2 Þ sinðq1 d1 Þ 0 ¼ 0 q2 l1 q2 ð38Þ After some arrangement, we obtain h i n1 2 dq11 u dq11 sTE v þ q12 rTE w 1 2 2 i; S¼ h q2 q1 d1 d2 N 2 q1 u þ q2 v þ q2 q2 rTE w dq11 v þ dq22 u sTE 1 2 ð39Þ where u = cosðq2 d2 Þ sinðq1 d1Þ; w ¼ sinðq1 d1 Þ sinðq2 d2 Þ; sTE ¼ ll1 qq21 þ ll2 qq12 ; and rTE ¼ ll1 qq21 ll1 qq21 :It is useful to 2 1 2 2 find the total time-average power carried by the photonic crystal layers. There is a close connection between the For l th cell, the power for the first layer can be written as 2 b al 2iq1 d2 pl ¼ e e2iq1 d 2xll 2iq1 ð41Þ b2l 2iq1 d2 2iq1 d e e þ 2al bl d1 : 2iq1 and for the second layer as 2 2iq2 d2 b cl 0 2iq2 d2 2 1e pl ¼ 1e þ 2cl fl d2 : fl 2xl2 2iq2 2iq2 ð42Þ 3. Results and discussion The dispersion relation given by Eq. (35) was solved numerically using Maple 17 software and the sensitivity was calculated using Eq. (39). In the following computations, a two layer photonic crystal was assumed in which the first layer was considered to be the analyte layer [2, 3] and the second layer was assumed to be LHM, dielectric or metal. This section was divided into three subsections. In the first subsection, the second layer was assumed to be LHM. In the second and third subsections, the second layer was considered metal and dielectric, respectively. In all subsections, the sensitivity was plotted with different parameters of the structure such as layer thickness and wave frequency. 3.1. Left-handed material (LHM) The LHM is characterized by e2 and l2 which can be written as [20] eeff ðxÞ ¼ 1 x2p ; x2 þ icx ð43Þ leff ðxÞ ¼ 1 Fx2 ; x2 x2 ixc ð44Þ where x° is the resonance frequency,c is the electron scattering rate, xp is the plasma frequency and F is the fractional area occupied by the split ring. We consider x0 = 4.0 GHz, xp = 10.0 GHz, F = 0.56 and c = 0.012 xp. These values are experimental values as given in many references [26]. The frequency range was taken from 4.0 to 6.0 GHz in which e2 and l2 are Binary photonic crystal for refractometric applications 1.4 1.0 0.8 0.6 0.4 0.2 8 10 12 14 16 18 20 22 24 d2(mm) Fig. 3 Sensitivity of the proposed sensor versus the thickness of LHM layer for different values of the analyte thickness for e1 = 1.77, n1 = 1.33, F = 0.56, c = 0.012xp, x = 5.0 GHz, x0 = 4.0 GHz, xp = 10.0 GHz and kB = p/(5d) sensitivity reaches a maximum value of 1.28 at d1 = 7.2 mm and d2 = 13.36 mm, whereas it has a value of 1.24 at d2 = 12.3 mm for d1 = 7.1 mm. But at d1 = 7.0 mm, the sensitivity reaches 1.18 at an optimum LHM thickness of 12.12 mm. It can be noted from Fig. 3 that when d1 increases from 7.0 to 7.1 mm, the maximum sensitivity enhances by 3.2%. On the other hand, the maximum sensitivity improves by 5.1% as d1 increases from 7.1 to 7.2 mm. The dependence of the sensitivity of the proposed sensor on the thickness of the LHM layer for different values of electron scattering rate (c) is plotted in Fig. 4. The sensitivity can be enhanced with decreasing the electron scattering rate (c). As can be seen from the figure, as the electron scattering rate changes, the maximum sensitivity occurs at the same value of an optimum LHM thickness. The maximum sensitivity obtained has a value of 1.03 at 1.1 1.4 =5.0 GHz 1.2 1.0 =5.1 GHz 0.9 =5.2 GHz 1.0 Sensitivity Sensitivity d1=7.0mm d1=7.1mm d1=7.2mm 1.2 Sensitivity simultaneously negative according to Eqs. (43) and (44), the analyte layer is assumed to be water with n1 = 1.33. Figure 2 shows the sensitivity of the proposed photonic crystal sensor versus the thickness of the LHM layer for different values of the wave frequency. The sensitivity can be dramatically enhanced with decreasing the guided wave frequency. As the frequency increases, the wavelength decreases and the wave confinement in the LHM layer increases. Consequently the evanescent field in the analyte medium decreases and as a result the sensitivity also decreases. It is also clear that as the wave frequency increases, the sensitivity peak shifts toward higher value of optimum thickness of the LHM layer. For x = 5.0 GHz, the maximum sensitivity is 1.28 obtained at d2 = 13.4 mm whereas it is 1.057 at d2 = 14.4 mm for x = 5.1 GHz. On the other hand, for x = 5.2 GHz, the sensitivity reaches a peak of 0.87 at an optimum thickness of 16.7 mm. It is worth comparing our results for the sensitivity with those of the conventional slab waveguide sensor comprising lossless dielectric media which was proposed by Tiefenthaler [24]. The maximum sensitivity obtained by Tiefenthaler and his co-workers was 0.25. Our results show that it is possible for LHM photonic crystal to achieve a sensitivity improvement of 412% compared to conventional slab waveguide sensor. This improvement was calculated as ð1:280:25Þ100% . 0:25 Figure 3 shows the sensitivity versus the LHM layer thickness for some different values of the analyte thickness. It is clear that the maximum sensitivity can be enhanced with increasing the analyte layer thickness. As the analyte layer thickness increases, the evanescent field in the analyte medium is enhanced and as a result the maximum sensitivity also increases. It is also observed that as the analyte layer thickness increases, the sensitivity shifts toward higher value of optimum LHM thickness. The 0.8 0.6 0.8 0.7 0.6 0.5 0.4 0.4 0.2 8 10 12 14 16 18 20 22 24 d 2 (mm) Fig. 2 Sensitivity of the proposed sensor versus the thickness of LHM layer for different values of frequency for e1 = 1.77, n1 = 1.33, d1 = 7.13 mm, x0 = 4.0 GHz, F = 0.56, c = 0.012xp, xp = 10.0 GHz and kB = p/(5d) 14 15 16 17 18 19 20 21 22 23 d2(mm) Fig. 4 Sensitivity of the proposed sensor versus the thickness of LHM layer for different values of electron scattering rate for e1 = 1.77, n1 = 1.33, F = 0.56, kB = p/(3d), x = 5.0 GHz, x0 = 4.0 GHz, xp = 10.0 GHz and d1 = 7.1 mm S A Taya and S A Shaheen c = 0.012 xp. For c = 0.013 xp, the sensitivity has a peak value of 0.99 whereas it is 0.95 at c = 0.014 xp. As c reduces from 0.014 xp to 0.013 xp, the maximum sensitivity can be improved by 4.2%, whereas when c reduces from 0.013 xp to 0.012 xp, the maximum sensitivity can be improved by 4.0%. In Figs. 2, 3 and 4, the sensitivity increases with increasing the LHM layer thickness and peaks at an optimum value of the thickness. At this value of the thickness, the evanescent field in the analyte is maximum. For thicknesses beyond the optimum one, the sensitivity decreases towards extremely low values and this can be attributed to the high confinement of the electromagnetic wave in the LHM layer. Increasing the LHM layer thickness beyond the optimum value enhances the electric field in the LHM layer and the sensitivity decreases. The variation of the sensitivity of the proposed sensor with the analyte thickness for different values of the thickness of LHM layer is illustrated in Fig. 5. As is clearly seen from the figure, the sensitivity increases slowly with d1 until it reaches 0.87 at d1 = 6.7 mm, then it shows a sharp increase with further increasing of d1. This behavior can be attributed to the part of the propagating field in the analyte layer. As the thickness of the analyte layer increases, this part increases. One interesting feature can be seen from the figure, the sensitivity can reach 2.17 for d1 = 8.82 mm and d2 = 14.1 mm which is extremely high value. The sensitivity exhibits a very slight dependence on d2 as it changes from 13.8 to 14.1 mm. 3.2. Metal material In this subsection, we assume the layer of thickness d2 to be metal. The proposed structure can be operated using many 2.5 metals such as Au, Ag, Al, Ni, Cr, etc. We considered two metals: nickel (Ni) and chromium (Cr). The thickness of the metal is taken in the nanometer range since the wavelength of the propagating light and thickness of layers should be comparable. Figures 6 and 7 show the sensitivity of the proposed sensor versus the thickness of metals (Ni) and (Cr) layer, respectively, for different values of the wavelength (k). The sensitivity increases with increasing d2 until it reaches a maximum value at optimum d2, then decreases very slightly with further increasing of d2 until it has a fixed value. Increasing the metal layer thickness beyond the optimum value does not show a significant effect on the sensitivity. As can be seen from Figs. 6 to 7, increasing the wavelength decreases the sensitivity of the proposed sensor. It is also obvious that as the wavelength decreases, the sensitivity peak shifts toward lower value of optimum thickness of the metal layer. We mean that the sensitivity peak occurs at lower values of the metal layer thickness as the wavelength decreases. For k = 632.8 nm, the sensitivity has a maximum value of 0.385 at Ni thickness of 148.7 nm whereas it has a maximum value of 1.27 at Cr thickness of 156.49 nm. For k = 650 nm, the sensitivity has a peak of 0.375 at Ni thickness of 150.2 nm, whereas it has a peak of 1.15 at Cr thickness of 161.95 nm. On the other hand, the sensitivity of Ni metal structure has a peak of 0.348 at a thickness of 170 nm for k = 700 nm, but for Cr metal structure it has a peak of 0.86 at a thickness of 200 nm for the same wavelength. It can be seen from Fig. 6 that as k decreases from 700 to 650 nm, the maximum sensitivity improves by 7.75% for Ni metal whereas it enhances by 33.7% for Cr metal as shown in Fig. 7. As k decreases from 650 to 632.8 nm, the maximum sensitivity enhances by 2.7% for 0.40 d2=13.8 mm d2=13.9 mm 2.0 0.38 d2=14.1 mm 0.34 1.5 Sensitivity Sensitivty 0.36 1.0 0.5 0.32 0.30 0.28 0.26 0.24 0.0 0 2 4 6 8 10 d1(mm) Fig. 5 Sensitivity of the proposed sensor versus analyte thickness for different values of the thickness of LHM layer for e1 = 1.77, n1 = 1.33, F = 0.56, c = 0.012xp, x = 5.0 GHz, x0 = 4.0 GHz, xp = 10.0 GHz and kB = p/(5d) 0.22 50 100 150 200 250 300 350 400 450 d2(nm) Fig. 6 Sensitivity of the proposed sensor versus the thickness of metal (Ni) layer for different values of wavelength (k) for e1 = 1.77, l2 = 1, kB = p/(5d) and l1 = 1, e2 = - 9.96 ? 14.66i, d1 = 550 nm Binary photonic crystal for refractometric applications 1.3 1.3 1.2 1.1 Sensitivity Sensitivity 1.2 1.0 0.9 0.8 1.1 1.0 0.9 0.7 d1=650nm d1=660nm d1=670nm 0.8 0.6 50 100 150 200 250 0.7 300 50 100 150 d2 200 250 300 d2(nm) Fig. 7 Sensitivity of the proposed sensor versus the thickness of metal (Cr) layer for different values of wavelength (k) for e1 = 1.77, l2 = 1, kB = p/(5d) and d1 = l1 = 1, e2 = - 1.1 ? 20.79i, 670 nm Fig. 9 Sensitivity of the proposed sensor versus the thickness of metal layer (Cr) for different values of the analyte thickness for k = 632.8 nm, e1 = 1.77, l1 = 1, e2 = - 1.1 ? 20.79i, l2 = 1 and kB = p/(5d) Ni metal as shown in Fig. 6 whereas it improves by 10.4% for Cr metal as shown in Fig. 7. It is worth mentioning that d1 = 550 nm in case of using Ni metal and d1 = 670 nm in case of using Cr metal are found numerically to correspond to the maximum sensitivity and the following figures will confirm that. The sensitivity of the proposed sensor as a function of the metal layer thickness is plotted in Fig. 8 for (Ni) and in Fig. 9 for (Cr) for different values of the analyte thicknesses. Figures 8 and 9 show that the sensitivity can be increased by increasing the analyte thickness. It is observed from Fig. 8 that the sensitivity has maximum values of 0.370, 0.378, and 0.385 at Ni thickness of 145.54 nm for d1 = 530 nm, 540 and 550 nm, respectively, but Fig. 9 shows that it has maximum values of 1.14, 1.22 and 1.27 at Cr thickness of 163.37 nm for d1 = 650, 660 and 670 nm, respectively. As d1 increases from 530 to 540 nm, the sensitivity improves by 2.16% whereas it enhances by 1.85% as d1 increases from 540 to 550 nm for Ni metal structure. On the other hand, the sensitivity enhances by 7.0% when d1 increases from 650 to 660 nm, but it improves by 5.0% as d1 increases from 660 to 670 nm for Cr metal structure. The sensitivity of the proposed sensor as a function of the analyte thickness is illustrated in Fig. 10 for (Ni) and in Fig. 11 for (Cr) for different values of the metal thicknesses. It can be shown from Fig. 10 that the sensitivity of the proposed sensor with (Ni) metal can be increased by increasing the thickness of analyte layer whereas Fig. 11 shows that the sensitivity of the proposed sensor with (Cr) metal can be increased by increasing the thickness of analyte layer until it reaches a maximum value at an optimum d1 = 670 nm, then decreases with further increasing of d1. In a similar to Fig. 5, this behavior can be 0.39 0.40 0.38 0.38 0.36 0.36 Sensitivity Sensitivity 0.37 0.35 0.34 0.33 d1=530nm 0.32 d1=550nm d1=540nm 50 100 150 200 250 300 350 400 d2(nm) Fig. 8 Sensitivity of the proposed sensor versus the thickness of metal layer (Ni) for different values of the analyte thickness for k = 632.8 nm, e1 = 1.77, l1 = 1, e2 = - 9.96 ? 14.66i, l2 = 1 and kB = p/(5d) d2=180nm d2=200nm d2=220nm 0.34 0.32 0.30 0.28 0.26 400 420 440 460 480 500 520 540 560 d1(nm) Fig. 10 Sensitivity of the proposed sensor versus the analyte thickness for different values of thickness of the metal layer (Ni) for k = 632.8 nm, e1 = 1.77, l1 = 1, e2 = - 9.96 ? 14.66i, l2 = 1 and kB = p/(5d) S A Taya and S A Shaheen 1.4 1.2 1.2 1.0 Sensitivity 1.0 Sensitivity 1.4 d2=150nm d2=170nm d2=190nm 0.8 0.6 0.8 0.6 0.4 0.2 0.4 d1=100nm d1=105nm 0.0 0.2 d1=110nm -0.2 210 600 650 700 750 215 220 800 3.3. Dielectric material In this subsection, we assume the layer of thickness d2 to be dielectric, silicon dioxide, of parameters e2 = 4.0 and l2 = 1. The dependence of sensitivity on the thickness of the dielectric layer for various values of the analyte layer is shown in Fig. 12. It is found that the sensitivity increases very sharply with increasing the thickness of dielectric layer and reaches a maximum value at an optimum thickness, then decreases very slowly with further increasing of the thickness of dielectric layer beyond the optimum value. The behavior of the sensitivity with the analyte thickness is also illustrated in Fig. 12. It is obvious from the figure that as d1 decreases, the maximum sensitivity shifts toward higher values of dielectric layer. For d1 = 100 nm, the sensitivity has a maximum value of 1.31 at d2 = 218.77 nm whereas it has a peak of 1.23 at d2 = 215.82 nm for d1 = 105 nm. The sensitivity has a peak of 1.16 at d2 = 212.72 nm for d1 = 110 nm. An enhancement of 6.5% can be obtained when the analyte thickness is reduced from 105 to 100 nm whereas the sensitivity increases by 6.0% as the analyte thickness is reduced from 110 to 105 nm. 235 240 Fig. 12 Sensitivity of the proposed sensor versus the thickness of dielectric layer for different values of thickness of the analyte layer for k = 632.8 nm, e1 = 1.77, l1 = 1, e2 = 4, l2 = 1 and kB = p/ (4d) Figure 13 is plotted to study the variation of the sensitivity with the thickness of dielectric layer for different wavelengths. The Figure shows that the maximum sensitivity increases with increasing the wavelength. It is also clear that the maximum sensitivity at the optimal thickness of dielectric layer shifts toward larger values of dielectric thickness as the wavelength increases. For k = 632.8 nm, the sensitivity has a maximum value of 1.16 at d2 = 212.81 nm and d1 = 110 nm whereas it has a maximum value of 1.30 at d2 = 218.67 nm and d1 = 100 nm. On the other hand, for k = 650 nm, the sensitivity reaches a peak of 1.22 at an optimum dielectric thickness of 219.68 nm and d1 = 110 nm whereas it has a peak of 1.29 at an optimum thickness 227.74 nm and d1 = 100 nm. As can be seen from Fig. 13 when k increases from 632.8 to 650 nm, the sensitivity enhances by 0.8% for d1 = 100 nm whereas it enhances by 5.2% for d1 = 110 nm. 1.4 1.2 1.0 Sensitivity attributed to the part of the propagating field in the analyte layer. As the thickness of the analyte layer increases, this part increases until it reaches a maximum value. After that it decays again towards lower values. For 180 B d2 B 220 nm, changing the thickness of metal layer has an ignorable effect on the sensitivity of Ni metal structure as noted from Fig. 10. Also changing the thickness of Cr layer has a slight effect on the sensitivity in the range of 150 B d2 B 190 nm. 230 d2(nm) d1(nm) Fig. 11 Sensitivity of the proposed sensor versus the analyte thickness for different values of the thickness of metal layer (Cr) for k = 632.8 nm, e1 = 1.77, l1 = 1, e2 = - 1.1 ? 20.79i, l2 = 1 and kB = p/(5d) 225 d1=110nm 0.8 d1=100nm 0.6 0.4 0.2 0.0 -0.2 210 215 220 225 230 235 240 d2(nm) Fig. 13 Sensitivity of the proposed sensor versus the thickness of dielectric layer for different values of wavelength (k), for d1 = (100,110)nm, e1 = 1.77, l1 = 1, e2 = 4, l2 = 1 and kB = p/ (4d) Binary photonic crystal for refractometric applications d2=218nm d2=220nm 1.40 1.35 d2=225nm Sensitivity 1.30 1.25 1.20 sensor. When the results of the structures comprising LHM, metal, and dielectric are compared, it can be concluded that the highest sensitivity was achieved with LHM structures. It is worth mentioning that the proposed sensor in the current work can be fabricated in a similar manner to that proposed in reference 3. 1.15 1.10 References 1.05 96 98 100 102 104 106 108 110 112 114 d1(nm) Fig. 14 Sensitivity of the proposed sensor versus the thickness of analyte layer for different values of thickness of the dielectric layer for k = 632.8 nm, e1 = 1.77, l1 = 1, e2 = 4, l2 = 1 and kB = p/ (4d) Figure 14 is plotted to illustrate the effect of the thickness of the analyte layer on the sensitivity at some values of the thickness of the dielectric layer. It is clear that the sensitivity of the proposed sensor increases with decreasing the thickness of the analyte. The sensitivity has the range of 1.06 B S B 1.33 at d2 = 220 nm. For d2 = 225 nm, the sensitivity has the range of 1.05 B S B 1.22 whereas it has the range of 1.06 B S B 1.39 for d2 = 218 nm. It is also obvious that the maximum sensitivity enhances by 4.5% as d2 decreases from 225 to 220 nm, but it improves by 9.0% as d2 decreases from 220 to 218 nm. 4. Conclusion In this work, a two-layer photonic crystal was assumed in which the first layer was considered to be the analyte layer and the second layer was considered to be LHM, dielectric or metal. It is possible for LHM photonic crystal structure to achieve a sensitivity of 2.17 which is extremely high value whereas a metal photonic crystal structure can achieve a sensitivity of 1.27. The dielectric photonic crystal structure can reach a sensitivity of 1.39. The sensitivity improvement with LHM photonic crystal was 412% for TE mode when compared to conventional slab waveguide [1] J Lou, L Tong and Z Ye Opt. Exp. 13 2135 (2005) [2] A Banerjee Prog. Electromagn. Res. PIER 89 11 (2009) [3] W Hopman, P Pottier, D Yudistira, J Lith, P Lambeck, R Rue, A Driessen, H Hoekstra and R Ridder IEEE J. Sel. Top. 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