1. Introduction

Guided resonance is a kind of asymmetric Fano resonance which can take place, among others, in photonic crystals (PhC) structures [1]. The periodic index contrast of PhC provides phase matching mechanisms allowing coexistence of resonance and anti-resonance within a narrow frequency range, thus oftentimes leading to an asymmetry resonance line shape in the transmission or reflection spectra [1, 2]. These resonances are very versatile and can be engineered to have tunable Q values and line shapes for different applications [3]. Particularly, Fano resonances with a narrow linewidth and a steep slope, when they are exploited in wavelength interrogation sensing applications, permit that only a small spectral shift is required for detection which, as a consequence, can enhance the sensitivity compared to resonances that have slower slope like Lorentzian line shape resonances.

Guided resonance in silicon, silicon nitride and some other semiconductor materials have been extensively studied. The early studies on GR in 2D PhC structures are presented in ref. [4], where the pioneering experimental results of leaky modes supported by a 2D PhC structure on a suspended AlGaAs slab at optical frequencies are shown. Later, P. Paddon et al. developed 2D vector coupled mode theory to illustrate the resonance modes observed in 2D periodic textured structure [5]. Fan et al. studied these resonances by investigating the dispersion diagrams, the modal patterns, transmission and reflection spectra [1]. In ref. [6] a general scheme to analyze a single optical resonance coupled with multiple input and output ports based on temporal coupled mode theory is developed. A great variety of devices have been explored based on GR and have been used for various applications. For example, by tuning the spectral properties of GR, filtering functions such as all-pass transmission or flattop reflection filters have been achieved [7]. Moreover, a displacement sensor is designed through the investigation of transmission and reflection spectra of a pair closely spaced air bridged PhC slabs by varying the spacing between the slabs [8]. A fiber-tip temperature sensor based on silicon PhC is realized by the coupling of two GRs in two PhC slabs exhibiting a two folds increased temperature sensitivity [9]. GR based silicon membrane reflector is integrated in a VCSEL by transfer-printed technique in order to realize efficient silicon based light sources [10]. In ref. [11] Miroshnichenko et al. proposed a novel interferometer configuration by inserting a cavity exhibiting a Fano resonance into a conventional Mach-Zehnder interferometer. An amplitude dependent bistable resonant transmission or reflection based on a linear chain of particles coupled to a single site defect is presented in ref. [12]. GR have been studied in optical trapping applications and low energy threshold vertical emitting lasers as well via investigation of slow Bloch mode (SBM) in the first Brillouin zone center (Γ point) to generate a high Q Fano resonance [13, 14].

As we mentioned before, most of the devices on GR are exploited in silicon or some other semiconductor materials. There is however very few works concerning the study of Fano resonance on tunable materials like lithium niobate (LN) [15, 16]. LN, a nonlinear optical material with high EO effects [17], is an excellent candidate for EO based E-field sensor applications. With the recent developments of thin film fabrication techniques, high quality single crystalline LN of a few hundred nanometers thickness remains difficult but attainable [18]. When these thin film lithium niobate (TFLN) are combined with a Fano resonance possessing high ER (extinction ratio) and high Q design, tunable active devices with different functionalities can be achieved. Numerical studies of GR properties in square lattice air holes on TFLN with an infinite size PhC structure by varying geometrical parameters such as slab thickness, PhC lattice types and radius of holes have been performed by J. Deng et al. [16], where an experimental reflection spectra of a free standing LN passive device has been shown.

In this paper, we present GR analysis of a rectangular structure on TFLN for E-field sensing applications. The rectangular lattice structure is formed by shifting the neighboring air hole position of two unit cells of a square lattice, thus it is biperiodic. The introduction of biperiodicity enables coupling to highly confined dielectric modes in the IR (infrared) regime with large wavelength separation between modes and it is also beneficial for decreasing the overall PhC size with respect to other PhC geometries [16]. The paper is structured as follows: we first calculate band diagrams of square and rectangular lattices to determine the lattice constant a and filling factor f (i.e. hole radius to period ratio r/a) that is favorable to produce high Q Fano resonances. Afterwards, 3D-FDTD transmission spectra investigations of these resonances are conducted both on infinite and finite size PhC structures. In the last part, optical field factors that is calculated locally in each cell of 3D-FDTD are proposed to accurately predict the sensitivity of GR based E-field sensor.

2. Band diagrams on 2D PhC

Flat and low group velocity bands are features of PhC band structures which can be exploited to design resonances with high Q values. Here two-dimensional (2D) plane wave expansion (PWE) in RSOFT BANDSOLVE are employed to calculate band diagram in order to determine parameters such as lattice constant a and filling factor f. We focus only on in-plane dielectric modes which can be excited by normal incidence illumination. This means that the investigated modes will be located in the Γ-point (the center of the first Brillouin zone) above the light line. The operation wavelength λ_res considered here is in the vicinity of 1.55 μm, which enables the compatibility with most of the telecom devices. Since the lifetime of photons inside the 2D PhC membrane is directly related to the group velocity Vg, a SBM that posses a low Vg would be a good candidate for a high Q cavity design which will then help to enhance the effective EO coefficient and therefore to increase the E-field probe sensitivity [19]. Consequently, dispersion modeling is performed to determine a filling factor f yielding a SBM. Then through the scaling invariant property of Maxwell equations (this is valid when only dealing with non-dispersive material as the case here), a lattice constant a can be determined in order to set the operation wavelength at IR regime.

The studied structure, as shown in Fig. 1(a), is a square lattice of air holes milled in LN material in the xy plane, while it is invariant in z direction. The associated nomenclature of the symmetric points in the irreducible Brillouin zone is shown in Fig. 1(b). The corresponding dispersion curve is shown in Fig. 1(c). The mode that we are looking for is dashed black circled out in Fig. 1(c), which lies in the Γ point above the light line at the normalized frequency a/λ = 0.541. However, there are some disadvantages in applying this mode for designing compact sensor applications. Firstly, a lattice constant of around 800 nm will be needed in order to set the operation wavelength at 1.55 μm as is the case in ref. [16] which is not beneficial to scale down the devices. Secondly, the mode lies near high order modes with small frequency separation, which makes the selected mode more prone to interfere with other nearby modes therefore degrading the resonance visibility in wavelength interrogation based sensing applications. To circumvent these drawbacks, we engineer the band diagram to couple modes in a lower frequency range which is not occupied by any states in the square lattice structure as it is highlighted in the green ellipse region (Fig. 1(c)). To that end, we introduce a shift s to the hole position in every two columns of air holes as schematically shown in Fig. 1(d). As a result, the period in x direction becomes 2a as shown in an orange rectangle unit cell in Fig. 1(d). We refer this lattice as biperiodic rectangular PhC structure. Notice that the method of tailoring structural parameters to enable coupling to preferable modes have been investigated in applications of optical tweezer [13] and photonic refractive index sensor for instance [20]. The principal of this method is to ’band fold’ the preferable mode so that the mode symmetry has a non-zero convolution with the incident beam [21] thus leading to the efficient excitation of the mode. For small values of s (s<0.1a), the band edge mode at symmetry point X (solid black circled out in Fig. 1(c)) is folded to the Γ point above the light line. In this case, mode properties such as group velocity can be assumed to be preserved and be the same as for the square lattice case. Henceforth, we first perform band calculations of square lattice structure. A filling factor of 0.368 is determined which yields a SBM in the symmetry point X. Keeping the same filling factor, we introduce a s = 30 nm to a structure with a = 630 nm as shown in Fig. 1(d) in order to form the biperiodic rectangular structure and and the corresponding dispersion curve is displayed in Fig. 1(e). We can see that two modes (encircled in black) appear at low frequency values in the Γ point above the light line with large frequency separation between other modes (>10% of normalized frequency). The lower frequency mode, which is called first SBM (1st SBM), lies at the normalized frequency a/λ = 0.265, while the second SBM (2nd SBM) lies at the normalized frequency of a/λ = 0.363. By setting λ to 1.55 μm, we obtain the corresponding lattice constant a to be 410 nm and 560 nm respectively. Having determined a and filling factor f, we can proceed to perform 3D-FDTD simulations to fine tune the design parameters in real 3D structures.

 

Fig. 1 (a) Sketch of the studied 2D infinite square lattice of air holes in LN where the orange square displays the unit cell with lattice constant a. (b) Reciprocal space associated to the square lattice (Γ, M, X) and rectangular lattice (Γ, M₁, X₁). (c) Band diagram for square lattice structure in (a) with electric field lies in the xy plane and a filling factor f of 0.368. The studied SBM in Γ point is black dashed circled out at normalized frequency of 0.545. The band edge black circled mode at symmetry point X lies at normalized frequency of 0.261. (d) Sketch of rectangular lattice air holes in LN formed by shifting holes center positions of s value in every two columns. The orange rectangle displays the unit cell with period of 2a in x direction. The dashed circles nearby solid line circles are square lattice hole positions. (e) Band diagram for rectangular structure in (d) with parameters values a = 630 nm, s = 30 nm, electric field in xy plane, filling factor f of 0.368. The two SBMs in Γ point are black circled out and lie at normalized frequency of 0.265 and 0.363 respectively.

Download Full Size | PDF

3. 3D-FDTD transmission simulations

In this section, we present a systematic 3D-FDTD (home-made codes) study by investigating zero order normalized transmission spectra of infinite and finite size PhC structures with the parameters previously determined by PWE calculations.

In the case of structure with infinite PhC size, the structure consists of a TFLN slab milled with a rectangular lattice air holes structure suspended in air as shown in Fig. 2(b). The calculation with parameters determined from PWE for both square lattice (Fig. 2(a)) and rectangular lattice structure is first conducted. Then the rectangular PhC is studied by varying the hole conicity (angle θ) and the shift value s in order to see the influence of these geometrical parameters on the resonance properties such as Q and ER.

 

Fig. 2 (a) Unit cell of square lattice air hole on TFLN of lattice constant a, radius r, TFLN slab thickness t. (b) Unit cell of rectangular lattice structure on TFLN by shifting hole position s, the black arrows at the bottom show the illumination direction. (c) Sketch of conical air hole with θ as the conicity angle. (d) Sketch of finite size air membrane type PhC structure, the coordinate (not relevant to lithium niobate crystalline orientation) shows the direction definition in agreement with that in Fig. 1(a). Period along x direction is 2a, while along y direction is a. (e) N ×N holes of finite size air bridged type PhC structure, with N = 6 shown in the sketch.

Download Full Size | PDF

In the case of structure with finite PhC size, both the air membrane structure (Fig. 2(d), vertically has three layers i.e. 700 nm TFLN/1.2 μm SiO₂/LN) and the air bridged structure (Fig. 2(e), vertically has only a layer of thin film lithium niobate self-suspended in the air) are studied. These finite structure studies are conducted by varying the shift value s and the PhC size N (number of rows) in order to see the influence on resonance properties such as Q and ER.

3.1. 3D-FDTD transmission simulations of infinite PhC structure

The infinite rectangular structure (Fig. 2(b)) in the 3D-FDTD calculations consists of a computational window of 2a×a in the xy plane. The structure is illuminated by a normally incident pulsed plane wave. Bloch periodic boundary condition is applied in x and y direction to realize the infinite structure. In the z direction, PML (perfectly matched layer) is applied to inhibit parasitical reflections on the computational window borders. Specular transmission response in the spectral domain is then calculated by making Fourier transform of the time varying electric field on the detector plane which is placed above the PhC top surface. The normalization is performed with respect to the incident signal. The spatial step size along x and y directions is uniform and it is fixed to Δx = Δy = 30 nm. Along the z direction, nonuniform meshing is employed to better describe the membrane thickness and to reduce the calculation time and computing memory. In order to faithfully describe the cylindrical air holes, an averaging technique is applied to define the refractive index distribution by considering the mean value of neighboring cells. This generates a gradual change in the refractive index distribution therefore helping to avoid spurious numerical dispersion due to abrupt changes in the index distribution staircase effect. The simulations start with the parameters determined from PWE calculations on square and rectangular lattice structure. The results are shown in Fig. 3(a). Let us first have a look at the transmission curves corresponding to the square lattice structure (the dashed green, dashed blue and solid red curves in Fig. 3(a)). As expected, there is no resonance appearing in the IR wavelength regime in the dashed green and the dashed blue curves cases (corresponding to parameters of a = 410 nm, r = 206 nm, t = 700 nm and a = 560 nm, r = 151 nm, t = 700 nm respectively). When a increases to 900 nm (solid red curve), resonances appear in the vicinity of 1.55 μm. Nevertheless, these resonances are close to each other with small frequency separation (for instance, there is only 21 nm separation in the resonance dip wavelength of the lowest two modes) between each other which is in good agreement with PWE calculations (Fig. 1(c)). Now let us investigate the rectangular lattice with s = 30 nm. The results are shown in the solid green and solid blue curves in Fig. 3(a), where two resonance dips (highlighted in orange rectangles) with zero to unity transmission and with large inter mode frequency separation appear in the IR zone which is not occupied by any mode in the square lattice case. The electric field distributions of the two blue encircled SBMs (Fig. 3(a)) are shown in Fig. 3(b). They are obtained by two continuous wave calculations at the transmission dip wavelength of 1446 nm and 1867 nm respectively. The top figure corresponds to the 1st SBM mode at 1867 nm, while bottom figure corresponds to the 2nd SBM mode at 1446 nm. We can clearly see that these two SBMs are dielectric modes for which electric fields are strongly confined in the LN material with a minimum electric field intensity centered on the air holes.

 

Fig. 3 (a) Normalized transmission spectra for square and rectangular infinite structures. Dashed blue, green curves and solid red curve correspond to square lattice structure. Solid blue and green curves correspond to rectangular structure with parameters calculated from PWE prediction to set the two SBMs operating at 1.55 μm. The modes in solid blue curve under highlighted orange rectangles are the folded two SBMs due to super lattice. (b) Normalized electric field amplitude distribution over one unit cell calculated at 100 nm below the PhC top surface. The upper one corresponds to 1st SBM while the bottom one corresponds to 2nd SBM as indicated in Fig. 3(a). (c) Normalized transmission spectra for a = 630 nm, r = 230 nm, s = 30 nm, t = 700 nm, varying θ. (d) Normalized transmission spectra for a = 630 nm, r = 230 nm, t = 700 nm, varying s.

Download Full Size | PDF

Simulations varying lattice constant a are conducted in order to set λ_res into the vicinity of 1.55 μm for a slab thickness of t = 700 nm. The results show that structure parameters of a = 450 nm and r = 165 nm yield λ_res at 1533 nm which corresponds to the 1st SBM in Fig. 1(e), while parameter of a = 630 nm, r = 230 nm yield λ_res at 1557 nm which corresponds to the 2nd SBM in Fig. 1(e).

Studies of the robustness of SBMs with respect to fabrication imperfections are also conducted by varying the hole conicity θ. Fig. 3(c) shows the normalized transmission study as a function of the hole angle, where θ is varied from 0° to 2.4°. Conical angle definition is schematically shown in Fig. 2(c). The θ is calculated by arctan(0.5(d₁ − d₂)/t), where the fixed parameters are t = 700 nm and d₁ = 460 nm. d₂ is varied to define different angle values. Hole conicity introduces a red shift of the resonance wavelength (1° induces approximately a 10 nm shift) and a slight broadening of the resonance linewidth. Nevertheless, we can conclude that the existence of the resonance is robust under variation of hole conicity which are often unavoidable in fabrication processes.

Studies varying the shift value s were also conducted as shown in Fig. 3(d). For s=0, the structure corresponds to that of a square lattice (black curve in Fig. 3(d)), no resonance appears in the transmission spectra. The parameter s has negligible effect on the resonance wavelength λ_res and ER. As s decreases, the resonance linewidth gets narrower, yielding a higher Q value of the resonance.

3.2. 3D-FDTD transmission simulations of finite size PhC structure

In order to realize the proposed structure, the actual size of the PhC has to be determined (which will be quantified by the number of air hole rows N). Simulations are performed by replacing the Bloch periodic boundary condition in xy plane in infinite structure case by PML as absorbing boundary conditions. The structure is illuminated by a Gaussian beam with beam waist of half the size of the PhC. The near field time varying electromagnetic (EM) fields are recorded by putting a detector above the PhC top surface. The EM fields are then processed through Fourier transformation to calculate the specular transmission where the normalization is done by the corresponding incident signal. Finite size PhC on both air membrane (Fig. 2(d)) and air bridged (Fig. 2(e)) configurations are studied with 3D-FDTD. The air membrane structure consists of a 700 nm thickness TFLN on 1.2 μm thickness silica buffer layer bonded on LN substrate. The air membrane is formed by opening a square region in silica layer under the PhC region with a side length of 600 nm larger than the size of PhC.

Air membrane structure calculations are conducted by varying the hole position shift s and the number of rows N. The thickness of TFLN and buffer layer silica are fixed at 700 nm and 1.2 μm respectively according to the wafer specifications provided by NANOLN [18]. The 3D infinite structure calculations show that two couple of parameters ((a = 450 nm, r = 165 nm) and (a = 630 nm, r = 230 nm)) can set the 1st SBM and 2nd SBM to operate at the vicinity of 1.55 μm respectively. Considering a PhC size N of 30 (empirically large enough to excite the resonance modes), then cross section areas of 13 μm × 13 μm and 19 μm × 19 μm are required to excite the 1st SBM and the 2nd SBM respectively. Nevertheless, a larger lattice constant a yields a larger air hole space which is easier for fabrication. Henceforth, the structure parameters in finite size PhC simulations are mainly performed for lattice constant of a = 630 nm and r = 230 nm configuration.

In Fig. 4(a), the finite air membrane PhC structure with parameters t = 700 nm, N = 30, a = 630 nm, r = 230 nm is studied by varying s from 0 (square lattice) to 30 nm by a step size of 5 nm. The λ_res corresponding to the 2nd SBM occurs at the same value (around 1560 nm) as for the infinite PhC case. The parameter s has a negligible effect on the position of λ_res similar to that in infinite structure studies (Fig. 3(d)). Nevertheless the shift parameter s can tune the ER and the Q value of the resonance. The ER of the resonance degrades as s decreases while the resonance linewidth gets narrower, implying a higher Q value.

 

Fig. 4 (a) Normalized transmission spectra of a finite PhC size air membrane rectangular lattice structure with N = 30, a = 630 nm, r = 230 nm, t = 700 nm as fixed parameters and s being the parameter that varies. (b) Normalized transmission spectra of air membrane rectangular structure of a = 630 nm, r = 230 nm, t = 700 nm, s = 15 nm and varying N. (c) Normalized transmission spectra of air membrane and air bridged structure of N = 30, t = 700 nm for three different configurations (see inset for geometrical parameters). (d) Normalized electric field amplitude distribution of purple circled resonance in Fig. 4(c) which corresponds to air bridged structure with a = 630 nm, r = 230 nm, s = 15 nm, N = 30 and t = 700 nm.

Download Full Size | PDF

Figure 4(b) depicts the normalized transmission of the air membrane structure for the parameters t = 700 nm, a = 630 nm, r = 230 nm, s = 15 nm while varying the PhC size (N) as 16, 20, 26, 30, 36 and infinite. The smaller the N, the smaller the ER and the broader the resonance linewidth. Henceforth, PhC size N can also be used to tune the ER and the Q value of the resonance. For applications requiring high ER and Q as well as a compact size, N = 30 and s = 15 nm could be a good compromise. Whereas, a smaller s yields a higher Q but compromise have to make between ER and PhC size N.

Air membrane structures, compared to air bridged structures, are more amenable for fabrication and manipulation since they only require PhC milling and wet etching processes. On the other hand, TFLN air bridged structures are still difficult but feasible if the LN thin film is bonded on materials such as silicon that can be easily etched. Henceforth, we conduct simulations on finite size of air bridged PhC and compare them to air membrane structures as shown in Fig. 4(c). The dashed lines correspond to air membrane structures while solid lines correspond to air bridged structures. Black lines correspond to structures with lattice constant a = 450 nm and s = 15 nm, red lines correspond to structures with lattice constant a = 630 nm and s = 30 nm, while green curves correspond to structure with lattice constant a = 630 nm and s = 15 nm. All the curves in Fig. 4(c) have been obtained by setting N = 30, t = 700 nm. As far as the properties of the resonance such as ER and Q value are concerned, air bridged structures show superior performance than the same configuration but in air membrane. Concerning the fabrication difficulties, the air membrane structure is more preferable and mechanically more stable with the structure bonded on a substrate. Figure 4(d) shows the electric field amplitude distributions for the purple circled resonance in Fig. 4(c) which is recorded 100 nm below the PhC top surface (inside LN material). The black circles correspond to the air hole positions. We can see nodal planes (zero electric field) formed in the middle of the air holes expelling light out towards the LN material, thus enhancing the light confinement in LN.

4. Sensitivity analysis

The main objective of our analysis is to find a suitable LN PhC configuration for E-field sensing applications. Unlike centrosymmetric crystals such as silicon for which linear EO effect is absent, LN belongs to the class of trigonal crystal symmetry and exhibits large EO effect. Consequently, its optical index variation with respect to the external applied electric field $\overrightarrow{E}=(E_x,E_y,E_z)$ is linked to the EO tensor as follows:

In order to exploit the largest EO effect, one must overlap the largest E-field component to be measured with r₃₃ which is along the z crystalline direction thus making the sensor exhibiting anisotropic characteristics. By neglecting other small EO element contributions, the EO effect can be simplified as:

4.1. Sensitivity analyzed with $\overline{f_{opt}}$

When LN has been designed with a slow light PhC structure that enables E-field confinement of the optical mode in the material, the induced EO effect (and therefore the sensitivity) can be enhanced. A generalized local field theory from nonlinear homogeneous media [23] to the case of PhC [24] had been employed to show that nonlinear susceptibilities of the PhC media are enhanced by local field factors which was supported by good agreements between experiments and simulations [25–29]. In ref. [27–29], an optical field factor is employed to analyze the enhanced EO effect in LN. It is estimated as a single averaged value to represent the ensemble enhancement effect, denoted here as $\overline{f_{opt}}$, which is the ratio of electric field amplitude integration over all the LN material region in the PhC structure to the same quantity obtained for bulk LN as follows:

The effective EO coefficient $r_{33}^{eff}$ can be quantified with a multiplication by ${\overline{f_{opt}}}^2$ in Eq. 2 in order to take into account the effect of the light enhancement induced EO effect exaltation. The square in $\overline{f_{opt}}$ originates from that r₃₃ is proportional to the second order susceptibility χ⁽²⁾, thus having the same proportionality with respect to the electric field amplitude. In order to obtain a large $r_{33}^{eff}$, one needs to orient the largest optical E-field component along the z crystalline axis. In rectangular GR structure, E_y ((see Fig. 4(d)) is the major electric field of the excited mode, thus the z crystalline direction should be oriented along y direction in order to exploit the largest EO effect. Consequently, X or Y-cut wafers of LN can be considered for the experimental realization.

For structures where light is confined in a defect region and the PhC serves as mirrors, this $\overline{f_{opt}}$ quantification is valid and well agreements between experiments and simulations had been demonstrated [27–29]. However, in the case of GR where there is no defect region, the resonance wavelength are dependent on the each local refractive index variation along the PhC structure. Therefore, it is not appropriate to extend the $\overline{f_{opt}}$ quantification method to GR structure analysis. In fact, this $\overline{f_{opt}}$ quantification had already been employed in sensitivity analysis of our early studies on Suzuki Phase Lattice (SPL) based temperature sensor [30] and it yielded a higher value than that of the experimental measurements. In that paper [30], the SPL based temperature sensor exhibited an experimental sensitivity of about 0.77 nm/°C, while the estimated sensitivity reached 2.3 nm/°C through $\overline{f_{opt}}$ method. In the same paper [30], the simulated reflection spectra at increased temperature estimating by $\overline{f_{opt}}$ yields a similar line shape with that of room temperature one. Whereas the line shape of the reflectivity was degraded with the increasing of temperature in experiments (Fig. 3 (a) in ref. [30]). Nevertheless, the field factor $\overline{f_{opt}}$ can give a correct variation trend of light confinement in GR structure with respect to geometrical parameters.

4.2. Sensitivity analyzed with f_opt (x, y, z)

Having realized the importance of each local refractive index variations to the λ_res of GR, we proposed a more accurate Δλ_res sensitivity analysis by calculating a local f_opt (x, y, z). It is calculated locally cell by cell in 3D-FDTD algorithm enabling better description of inhomogeneous field enhancement distribution along the structure.

In order to theoretically analyze the Δλ_res sensitivity with respect to the measurand, $f_{opt}^2(x,y,z)$ is integrated in Eq. 2 (similar to the $\overline{f_{opt}}$) in order to quantify the light enhancement induced $r_{33}^{eff}$. Consequently, the corresponding induced refractive index local variation in PhC structure can be expressed as:

This sensitivity analysis employing f_opt (x, y, z) method had been utilized in sensitivity analysis of SPL guided resonance based temperature sensor [30] and well agreement between simulations and experiments was achieved. Therefore, we will employ here the same method for accurate sensitivity analysis of rectangular lattice GR based E-field sensor.

4.3. Sensitivity analysis of GR based E-field sensors

The E-field sensor device sensitivity (i.e. the minimum detectable E-field) depends on the Δλ_res sensitivity with respect to the E-field, the resolution of the OSA (Optical Spectrum Analyzer) and the E-field enhancement from antenna effect (if electrodes are designed) [31, 32]. In the following we theoretically analyze the Δλ_res sensitivity and through some assumptions of other factors, redthe minimum detectable E-field employing the GR structure as E-field sensor are deduced. The λ_res of the structure is first determined by 3D-FDTD transmission investigation. Then the Δλ_res sensitivity is analyzed by conducting 3D-FDTD CW numerical calculations at resonance dip wavelengths to obtain f_opt (x, y, z) distributions over all the PhC volume. Next, different E_z values are assumed and substituted into Eq. 4 in order to obtain Δn(x, y, z). Then 3D-FDTD transmission calculations are performed by including this local refractive index variation in the description of the structure in order to estimate the Δλ_res.

Notice that the refractive index variations Δn(x, y, z) are wavelength dependent since f_opt (x, y, z) are wavelength dependent. In general, the f_opt (x, y, z, λ) reaches its maxima at the λ_res. Henceforth, the above methods of investigating the Δλ_res sensitivity is in principal only correct for the wavelength of λ_res since the Δn(x, y, z, λ) is modified according to the f_opt (x, y, z, λ_res). However, the Δλ_res sensitivity obtained by choosing λ_res as operating wavelength yields the highest sensitivity compared to otherwise operating at off resonance wavelength. Therefore, the Δλ_res sensitivity can represent the sensitivity of the device. The negative effect of modifying the Δn(x, y, z) by only considering the f_opt (x, y, z, λ_res) is that it leads to the distortion of transmission spectra line shape compared to the experimental measurements as in ref. [30](see Fig. 3 of ref. [30]). This distortion originates from that the Δn(λ) at any wavelength is fixed at Δn(λ_res) so that the dispersive property of the index variation is not taken into account.

Let us first investigate the Δλ_res sensitivity of infinite PhC structures. Infinite structure sensitivity results for two sets of configurations that yield λ_res near the vicinity of 1.55 μm are shown in Fig. 5 where different curves correspond to different s values. We can see that λ_res changes quasi linearly with respect to E-field. The Δλ_res sensitivity are analyzed by calculating Δλ_res/E_z between each neighboring data points, then the mean value is taken to represent the sensitivity. The smaller the s, the higher the sensitivity as can be seen from the steeper slope of the curves. Notice that the variation trend of resonance Q versus s (Fig. 3(d)) is the same with sensitivity versus s which indicates that a higher Q can yield a higher sensitivity in GR structures. In Fig. 5(a), corresponding to a = 630 nm configuration, an s=5 nm (black dashed curve) yields a sensitivity around 0.4 pm/V, while it decreases down to 0.017 pm/V (red dashed curve) for an s=30 nm. The same s value with configuration of a = 450 nm (corresponds to the 1st SBM) yields a better sensitivity, with 1.2 pm/V for s=5 nm and 0.023 pm/V for s=30 nm as shown in Fig. 5(b). The higher sensitivity achieved with the configuration of the 1st SBM (a= 450 nm) than the 2nd SBM is due to its slower group velocity as can be deduced from the flatter curvatures of Fig. 1(e) indicated by ellipses.

 

Fig. 5 Numerically calculated plot showing the λ_res as a function of the E-field (the insets show the zoom view of the first few data points) for infinite PhC air bridged structure with parameters of (a) a = 630 nm, r = 230 nm, t = 700 nm and varying s, (b) a = 450 nm, r = 165 nm, t = 700 nm and varying s.

Download Full Size | PDF

Now let us investigate the Δλ_res sensitivity of finite PhC structures. Since the heavy computational burden of the finite structure calculations, sensitivity study with respect to different E-field values is only carried out for structures with parameters of a = 630 nm, r = 230 nm, s= 30 nm, t = 700 nm and N = 30 (about 19 μm × 19 μm cross section) corresponding to 2nd SBM. The counterpart of 1st SBM yields a higher GR quality factor Q, therefore simulations take more computational time to obtain converged results. The corresponding Δλ_res sensitivity results are tabulated in Table 1. The Δλ_res sensitivity satisfies a quasi-linear relationship with a mean value of sensitivity of 0.0028 pm/V. Studies for E_z = 5 × 10³ V/m are carried out for the same structure parameters with two different s values of 20 nm and 25 nm. The estimated sensitivities are 0.007 and 0.0045 pm/V respectively. These results show that a smaller s in finite structure case also yields a higher sensitivity. The Δλ_res sensitivity in finite structure with size of N=30 drops at around 16% of that in infinite case.

Table 1. Resonance dip wavelengths shift with respect to different E_z

View Table

Let us estimate the minimum detectable E-field with the structure parameters of a = 450 nm, r = 165 nm, t = 700 nm, s = 5 nm and N=30. The infinite structure Δλ_res sensitivity is 1.2 pm/V. Considering the above 16% drop of sensitivity in finite structure case, a 0.19 pm/V Δλ_res sensitivity is expected. The minimum detectable E-field of the sensor also depends on other factors such as OSA resolution that is assumed to be 0.1 pm (this is feasible in experiments such as stimulated Brillouin scattering based high resolution OSA). If the E-field sensor is designed with electrodes (small enough so that they do not disturb the E-field to be measured), an E-field enhancement from the antenna effect will further enhance the devices’ sensitivity. For example, assuming a bowtie antenna design as that in ref. [32], a 10⁴ field enhancement can be obtained. With this field enhancement, an E-field of about 50 μV/m can generate 0.5 V/m E-field inside the feed gap of the antenna which will induce around 0.1 pm shift in λ_res that can be detected by a high resolution OSA. Consequently, our designed Fano based PhC structure is estimated able to detect a minimum E-field as small as 50 μV/m.

5. Conclusion

This study demonstrated that GR are favorable for highly sensitive sensing applications due to its merits such as high ER and tunable Q of resonance. The rectangular lattice PhC with a shift value s of the neighboring air hole position had been demonstrated to add more freedom to tune the resonance Q value, ER and to enable the coupling of high light confinement dielectric modes which is then exploited as E-field sensors. A local field factor that is calculated locally in each FDTD cell is proposed to accurately estimate the sensitivity of GR based E-field sensors and a sensitivity of 50 μV/m could be achieved with the proposed structure. These systematical studies of designing high Q, ER and steep slope GR and accurate sensitivity analysis will pave the way to improve the bulk LN based devices such as filters, optical switches, sensors and modulators etc.