Numerical investigation of nematic liquid crystals in the THz band based on EIT sensor

This paper introduces the concept of electromagnetically induced transparency (EIT) into the permittivity extraction of an anisotropic material—nematic liquid crystal (NLC). A novel two-step strategy is presented to extract the complex permittivity of the NLC at the THz band, which evaluates the relative permittivity tensor from the resonant frequencies and then determines the loss tangent from the quality factor Q of the EIT sensor. The proposed method features high accuracy due to the sharp resonance of the EIT sensor and also high robustness to the thickness of the NLC layer because only amplitude rather than phase information of the transmission coefficients is required. The NLC filled EIT sensor shows a sensitivity of 56.8 m/RIU (the resonance wavelength shift over the refractive index change unit (RIU)) and Figure of Merit (FoM) of 6.92. The uncertainty of the proposed technique in the relative permittivity and loss tangent is 3% and 8.2%, respectively. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (160.3710) Liquid crystals; (130.6010) Sensors; (160.1190) Anisotropic optical materials; (300.6495) Spectroscopy, terahertz. References and links 1. F. Goelden, A. Gaebler, M. Goebel, A. Manabe, S. Mueller, and R. Jakoby, “Tunable liquid crystal phase shifter for microwave frequencies,” Electron. Lett. 45(13), 686–687 (2009). 2. A. L. Franc, O. H. Karabey, G. Rehder, E. Pistono, R. Jakoby, and P. Ferrari, “Compact and Broadband Millimeter-Wave Electrically Tunable Phase Shifter Combining Slow-Wave Effect With Liquid Crystal Technology,” IEEE Trans. Microw. Theory Tech. 61(11), 3905–3915 (2013). 3. C. Weickhmann, N. Nathrath, R. Gehring, A. Gaebler, M. Jost, and R. 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Introduction
Due to its tunable permittivity and relatively low dielectric loss over a wide frequency range from several gigahertzes to optical spectrum, nematic liquid crystals (NLCs) have been widely applied in reconfigurable devices, such as tunable phase shifters [1][2][3], reflectarrays [4][5][6][7], leaky-wave antennas [8,9], frequency selective surfaces (FSSs) [10,11] and liquid crystal display (LCD) screens.In fact, the loss tangent of most NLCs shows a general downward trend as frequency increases [12,13], which is an advantage over other kinds of tunable components such as semiconductor solutions, radio frequency micro-electromechanical systems (MEMS) and ferroelectrics e.g.barium-strontium-titanate (BST) [14].Moreover, NLC-based techniques features low fabrication costs, simple biasing network and ease of integration [4].Therefore, NLC is a kind of promising material in the development of reconfigurable devices for higher frequency, especially for THz band, many investigations have been carried out.Such as the NLC-based tunable THz metamaterials [15] and the advanced modeling techniques of liquid-crystals for terahertz applications [16,17].
In the design and optimization of NLC-based tunable devices, the characterization of NLC is essential [12,18].Free-space methods based on transmission-line (transmission type) or Fabry-Perot (resonant type) theorem are commonly used to extract the complex permittivity at THz band [19][20][21].Directly curve fitting or a closed form relation can be applied in these methods since the E field is with the same direction everywhere and explicit.However, both of the transmission and resonant types are sensitive to the thickness of the NLC layer because phase information of the transmission coefficient is needed in the permittivity extraction procedure.
On the other hand, the concept of electromagnetically induced transparency (EIT) has gained a lot of attention in the free-space material characterization recently because it features sharp transmission window providing an accurate frequency response to the permittivity [23][24][25].In addition, the electromagnetic (EM) field is concentrated around the metal layer making the EIT-based sensor insensitive to the thickness of material under test.However, up to now, the EIT-based sensor is only applied in the characterization of isotropic materials by directly curve-fitting [23][24][25] because the field distribution around the EIT structure is complicated and therefore closed form expressions between the complex permittivity and the frequency response is difficult or even impossible to derive.For the anisotropic case, situation gets more complicated and no relevant reports can be found.
In this paper, we utilize EIT structure to extract the complex permittivity of NLC, which is anisotropic.A novel algorithm is presented to extract the complex permittivity of NLCs at THz band.Firstly, the relative permittivity is extracted regardless of dielectric losses of the NLC because it has little influence on the resonant frequency of the EIT sensor.Secondly, the loss tangent is determined based on the obtained relative permittivity in the first step.Two surface fitting functions are established by changing the polarization configuration of the incident wave.A nonlinear optimization process is applied to find the minimum of the target error functions.The proposed method retains the EIT's properties of high accuracy of frequency response and high robustness to the thickness of the NLC layer, which paves the way for the anisotropic material characterization based on EIT sensor.

The proposed EIT based sensor
The schematic layout of the EIT based sensor, including all geometrical parameters, is shown in Fig. 1(a).The gold layer is a Jerusalem cross structure with four gaps in the four quadrants, which is deposited on a fused silica substrate and then is spin-coated with rubbed polyimide thin layer for alignment purpose [4].The proposed EIT sensor features a 90°rotation symmetry.Therefore, before loaded with anisotropic NLCs, its electromagnetic response is identical in both x and y directions, which is important for the following extraction procedure.To contain the NLC, another fused silica with same surface treatment without metal layer is stacked on the top of the EIT sensor supported by spherical spacers forming an EIT sensor based NLC cell [refer to Fig. 1(b)].The schematic layout of the EIT based sensor, including all geometrical parameters, is shown in Fig. 1(a).The gold layer is a Jerusalem cross structure with four gaps in the four quadrants, which is deposited on a fused silica substrate and then is spin-coated with rubbed polyimide thin layer for alignment purpose [4].The proposed EIT sensor features a 90°rotation symmetry.Therefore, before loaded with anisotropic NLCs, its electromagnetic response is identical in both x and y directions, which is important for the following extraction procedure.To contain the NLC, another fused silica with same surface treatment without metal layer is stacked on the top of the EIT sensor supported by spherical spacers forming an EIT sensor based NLC cell [refer to Fig. 1(b)].
Numerical simulation of the proposed EIT based NLC cell is performed by the frequency domain solver in CST Microwave Studio (MWS) [26] to demonstrate the EIT effect.The geometry of the unit cell is designed as w = 55 m, l = 420 m, a = 700 m and s = 30 m.The thickness of the NLC layer is 120 m.The fused silica has a permittivity of 3.78, a loss tangent of 0.0015, and a thickness of 300 m.Boundary condition of "unit cell" is applied in x and y direction and a Gaussian pulse with a spectrum ranging from 250 GHz to 350 GHz is chosen as the excitation signal.The dielectric constants r⊥ and r|| of NLC is assumed as 2 and 4, respectively, regardless of dielectric losses for time-saving purpose.r = 1 is always valid for NLCs as it is a non-ferromagnetism material [18].The polarization of the normally incident plane wave and the orientation of NLC molecules n are in the same direction along y-axis.The corresponding transmission coefficient of the sensor is depicted in Fig. 2. a sharp transmission peak (i.e. a transparent window) can be observed between two stop-bands, which can be regarded as the EIT-like effect brought by the excitation of the trapped mode, a high-Q mode formed by counter-propagating currents [27].The sharp resonant of the EIT sensor provides an accurate strategy for the characterization of NLCs.To visualize the physical mechanism of the transparency window, the current distributions induced by the incident plane wave at f 1 = 304.50GHz, f 2 = 309.46GHz and f 3 = 312.35GHz, respectively, are simulated and compared as shown in Fig. 3. Remarkable resonances of dipole mode [27] can be observed in Figs.3(a) and 3(c) (i.e.current oscillations symmetric with respect to y-axis in outer and inner strips, respectively).The reradiated electric field by the resonant strip(s) is considerably enhanced and has opposite phase with respect to the incident electric field, which leads to the two stop-bands in the transmission spectrum of Fig. 2. In Fig. 3(b), the induced currents of the inner and outer strips appear to be similar magnitude with opposite phase.The re-radiated electric field is very weak and hence has little influence on the incident electric field.The transparent window is therefore formed.

Theory description
There are four parameters: r⊥ , r|| , tan ⊥ and tan || to be estimated in the characterization of NLCs.In conventional technique of the free-space permittivity extraction of NLCs, electric field E in the NLC layer is with the same direction everywhere, such as parallel slabs based transmission type [19,20] and Fabry-Perot cavity based resonant type [21,22] structures.Therefore, the parallel components: r|| , tan || and the vertical components: r⊥ , tan ⊥ can be independently derived by setting the incident electric field E parallel [refer to the case 1 shown in Fig. 4(a)] and perpendicular [refer to Fig. 4(b), namely, case 2] to the orientation of NLC molecules n , respectively.However, the distribution of electric field E is complicated and multi-directional in the NLC region for the proposed EIT sensor as depicted in Fig. 4(c).The transmission coefficients of case 1 and case 2 are no longer independently determined by parallel and vertical components of the NLC.In other words, the existing algorithm is invalid for the EIT-based permittivity extraction of NLCs.The NLC is low-loss medium with the typical loss tangent in the order of 10 −2 .The influence of tan on the frequency shift can be therefore neglected [28].For further explanation, two types of NLCs with different loss-tangent combinations are filled into the EIT cell.Simulation results are shown in Fig. 5.The variation of tan ||,⊥ has little influence on the resonant frequency in both cases, which means that the resonant frequency of the EITbased sensor is approximately determined by the real part of the complex permittivity.In addition, the distribution of electric field E is also primarily sensitive to the r|| , r⊥ , which is closely related to the power dissipation level of the sensor.Therefore, the extraction of the dielectric properties of the NLCs is performed by two separate steps.Firstly, extract the r|| , r⊥ from the resonant frequencies.Then determine the tan || , tan ⊥ from the quality factor Q based on the extracted r|| , r⊥ .

Relative permittivity extraction
Due to the complicated field distribution in the EIT sensor, closed form expression of the relation between the relative permittivity r|| , r⊥ and the corresponding resonant frequency f C1 , f C2 (resonant frequency in Case 1 and Case 2) is difficult or even impossible to derive.It is worth noting that in the millimeter and sub-millimeter wave band, according to [10,12,[19][20][21][22]29], the relative permittivity of NLCs satisfies the follow relation without exception as In this case, numerical simulations are carried out by calculating each point in the permittivity plane [ r|| (2:4), r⊥ (2:4)] with a fixed step length of 0.01.Configurations of the simulation model are shown in Fig. 4 (a) and (b).The results of f C1 , f C2 versus r|| , r⊥ are plotted in Fig. 6.It is obvious that the resonant frequency decreases with the r|| , r⊥ in both cases.The three points f C1 , f C2 and the corresponding ( r|| , r⊥ ) are collinear and parallel to z-axial [red dots in Fig. 6].f C1 is equal to f C2 at the points meeting r|| = r⊥ (isotropic case).Referring to the diagram, the r|| and r⊥ can be extracted according to the obtained f C1 and f C2 .However, in practice, directly checking diagram is not an artful method for the parameters extraction.In order to conduct the extraction procedure automatically through using computers, the plotted surfaces for the Cases 1 and 2 can be fitted by two polynomial functions.Since a full surface, namely, Surf_1 [refer to Fig. 6] can be modeled by mirroring the surface of Case 1 from variable area of r|| > r⊥ to r|| < r⊥ along the diagonal line r|| = r⊥ , the two fitting functions will therefore feature identical polynomial coefficient P mn (listed in Table 1) with transposed r|| and r⊥ as defines a position in the plane ( r|| , r⊥ ).The chosen order of the polynomial is 3, 5 with a root-mean-square error (RMSE) of 0.42 GHz because no remarkable improvements of fitting level can be achieved for higher order indexes.Deviations in the extracted permittivity brought by this discrepancy will be discussed in following segment.The next step is to search for the zeros of f C1f C1m , and f C2f C2m simultaneously, where f C1m and f C2m are the measured resonant frequencies in Case 1 and Case 2, respectively.Due to the nonlinearity of these Eqs, numerical solution is necessary.One of the methods to solve such a root finding problem is the Newton-Raphson iteration.However, this method is applicable only if the initial values of r|| and r⊥ are very closed to the final solution, otherwise it could be diverged [30].In this case, we use the nonlinear optimization method to search the minimum of the error function as where .
This error function is similar to a paraboloid as shown in Fig. 7, which is quite simple and low dimension.Therefore, the calculation of the inverse Hessian matrix error function can be made directly without requiring complex procedures [30].Then the solution can be obtained by the iteration procedure as where the subscript k is the number of iteration steps.Assuming the case of f C1m = 306.1 GHz, f C2m = 310.3GHz (corresponding to r⊥ = 2.5, r|| = 3.3 in the simulation results), the solution of the proposed extraction method converges to the point r⊥ = 2.48, r|| = 3.27 as depicted in the inset of Fig. 7.
where C1 , C2 is the resonant wavelength of Case 1 and Case 2, respectively.This definition is different from that proposed in [24], which is only suitable for the isotropic rather than anisotropic case.The average FWHM (Full Width Half Maximum) of the sensor in the two cases is 8.We have compared the corresponding FoM values of sensing refractive index (permittivity) based on EIT-like sensor with coupled split-ring and spiral structure [23], Jerusalem cross in ring type [24], detuned split ring [31], planar EIT-like metamaterial [32], and our proposed EIT-like sensor.The sensing performance of our EIT-like sensor is comparable with other works as listed in Table 2.It is worth noting that only this work is applicable for the characterization of anisotropic material whereas the others are for the isotropic case.

Loss tangent extraction
Once the relative permittivity is obtained, we can extract the loss tangent tan || , tan ⊥ of the NLC using same procedure aforementioned.Firstly, numerical simulations are carried out and the results of Q C1 , Q C2 versus tan || , tan ⊥ in both cases are plotted in Fig. 8.The half-power frequencies f 1 and f 2 are used to determine the resonant frequency and the quality factor is defined as Using (11) instead of the absolute peak in transmission magnitude provides a more accurate estimate of the center frequency in the presence of noise [33].
The different field distribution of two cases making the Q-factors in different level, which are not continuous at tan || = tan ⊥ .Therefore, the polynomial coefficients of the fitting functions are different in the two cases, which is denoted as C1 mn P and C2 mn P , respectively.
The polynomial functions are established as where tan tan defines a position in the plane (tan || , tan ⊥ ).The chosen order of the polynomial is 3, 3 with a RMSE of 0.13.Deviations in the extracted loss tangent brought by this discrepancy will also be discussed in following segment.Then the error function is defined as where ( ) and ( ) This error function is also similar to a paraboloid as shown in Fig. 9 and similar iteration procedure is carried out as .

Uncertainty analysis
The inherent uncertainty in the extracted complex permittivity is caused by the discrepancies between the fitting function and the numerical data.The maximum root-mean-square error (RMSE) of the Eqs.( 4) and ( 5) is 0.42 GHz.To evaluate the uncertainty of the calculated relative permittivity brought by this frequency discrepancy, all of eight frequency combinations are considered with a variation step of 0.42 GHz, which is denoted as (f C1 ± 0.42, f C2 ), (f C1 , f C2 ± 0.42), (f C1 ± 0.42, f C2 ± 0.42).The corresponding results of the relative permittivity are calculated and shown in Fig. 10(a) indicating that the uncertainty of r|| and r⊥ is 3% and 2.8%, respectively.As the maximum RMSE of the Eqs.( 15) and ( 16) is 0.13, same procedure is applied to determine the uncertainty of the loss tangent.The results calculated from eight Q-factor combinations (Q C1 ± 0.13, Q C2 ), (Q C1 , Q C2 ± 0.13), (Q C1 ± 0.13, Q C2 ± 0.13) are shown in Fig. 10(b).The uncertainty of tan ⊥ and tan || is 7.6% and 8.2%, respectively.Additional error is originated from the fabrication tolerance.In fact, the discrepancies of geometry dimensions can be neglected for the existing fabrication technology.For instance, with the typical dimensional tolerance of 2-m [5], the shift of resonant frequency of numerical results is less than 0.17 GHz.To determine the uncertainty in the frequency deviation to the thickness variation of the NLC layer, a numerical sensitivity analysis is applied as shown in Fig. 11.The shift of resonant frequency is less than 0.1 GHz with the thickness variation from 110 m to 130 m implying the high robustness of the proposed method compared to the conventional design.

Fig. 1 .
Fig. 1.Schematic layout of (a) the EIT sensor and (b) the EIT sensor based NLC cell.

Fig. 4 .
Fig. 4. Schematics of the two configurations with the orientation of the NLC molecules parallel (a) and perpendicular (b) to the incident (E) field denoted as Case 1 (extraordinary transmission) and Case 2 (ordinary transmission), respectively.The two cases can be switched by rotating the sample with 90 degree; (c) (E)-field distribution in the NLC layer.

Fig. 7 .
Fig. 7. Error function versus r|| , r⊥ with assumption: f C1m = 306.1 GHz and f C2m = 310.3GHz.The relative permeability of NLC is r = 1.Then the sensitivity m NLC [the resonance wavelength shift over the refractive index change unit (RIU)] of the EIT sensor can be defined and calculated as 21 m.Then the Figure of Merit (FoM) is defined and calculated as LC

Fig. 10 .
Fig. 10.Deviations in the relative permittivity (a) and loss tangent (b) brought by the inherent error of the proposed technique (the results with and without deviations are marked as stars and crosses, respectively).

Fig. 11 .
Fig. 11.Transmission coefficients of the proposed EIT sensor with different thickness of the NLC layer.