1. Introduction

Micro displacement sensor is one of the key elements in Micro-Electro-Mechanical system (MEMS) for its precision measurement of micro-displacement with micro-scale size. It could also be potentially significant for applications such as bio-sensing and atomic force microscopy.

Photonic crystals (PhCs) open new possibilities for ultra-compact, highly wavelength selective optical devices, due to their photonic band gaps (PBGs) [1, 2]. So various designs of micro-sensor based on PhCs have been proposed recently [3–7]. Among these designs, ultrasensitive displacement sensing using photonic crystal waveguides (PCWGs) has been demonstrated [5]. Such device can provide a sensitivity of ~1 [μm ^-1 ] with a light source of 9.02 μm. But 2~3 detectors should be used to obtain the normalized intensity, which complicates the structure of micro-sensor, and the error in any of the detectors will deteriorate the measurement results. Another type of displacement sensing relying on guided resonances in photonic crystal slabs has also been introduced [6, 7]. Its sensing mechanism is based upon photon tunneling and Fano interference, and a 20-dB transmission contrast can be obtained when the distance between two slabs changes by about 1% of operating wavelength. However, since its line shape in the near-field regime is complicated, such device is mainly used as ultra-sensitive displacement switching, rather than as a sensor to measure the exact displacement.

In this paper, we propose a micro displacement sensor and its sensing technique based on line-defect resonant cavity in two-dimensional (2D) PhCs. The line-defect resonant cavity is formed by a fixed PhC segment and a moving one. With a proper operating frequency, the variation of transmitted intensity is a quasi-linear function of the relative displacement between the fixed and mobile PhC segments in a specified range, thus a quasi-linear measurement of micro-displacement is realized with high sensitivity and low Q factor. The sensitivity can be adjusted easily by varying either some relevant radii or operating frequency of the sensing system. Moreover, the sensing range can be broadened by using multiple operating frequencies. The properties of the micro displacement sensor are analyzed theoretically and verified using finite-difference time-domain (FDTD) method.

2. Theoretical analysis

The layout of our micro displacement sensor based on line-defect PhC resonant cavity is shown in Fig. 1. The sensor includes two coplanar PhC segments, one fixed and the other mobile. The two segments are aligned along a common axis and then a line-defect PhC cavity is formed. The light emitted from the light source propagates through the PhC cavity and the transmission intensity is detected by a photo detector. The detected spectrum has a Lorentzian line shape, and the peak occurs when the PhC cavity is at resonance. When the moving PhC segment is shifted along the common axis, the resonant frequency of this PhC cavity will shift accordingly due to the variation of cavity length. In another point of view, for light with a certain frequency incidents upon the PhC, its radiation intensity measured by detector is a function of the relative displacement of moving PhC. Therefore, with a proper operating frequency, this structure can work as a micro displacement sensor with high sensitivity.

 

Fig. 1. Layout of the linear displacement sensor based on 2D PhC with square lattice of dielectric rods. This structure is composed of a fixed PhC segment, a mobile PhC segment, a coherent light source and a photo detector. The radius of blued rods can be tuned to adjust the Q factor of PhC cavity.

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This architecture is somewhat similar to F-P sensor [8, 9], in which the resonance also has a Lorentzian line shape. There are still mainly three differences between them: (1) unlike the traditional F-P cavity, it is a line-defect PhC cavity in our sensor, and the two PhC segments serve as not only mirrors (e.g. the design in Ref [3]), but also PCWGs, which avoids the leakage in the direction perpendicular to the common axis; (2) the transmission properties of these two type of cavities differ a lot, which will be discussed in the following section; (3) the distance between the two PhC segments can be smaller than the operating wavelength owing to the line-defect cavity, thus this sensor can measure the displacement itself is in the nanoscale, however, it is stringent for a conventional F-P sensor to meet this requirement since it no longer functions when the distance between the mirrors is in that order of magnitude.

The transmission coefficient through this symmetric PhC resonant cavity for different frequencies can be expressed approximately as the following Lorentzian function [10]

where ω ₀ is the resonant frequency, and Q is the Q factor of the resonant cavity.

The resonant frequency ω ₀ shifts with the changing of PhC cavity length L ₀, which is define as the separation between the two blued rods in Fig.1. Somewhat like the F-P cavity, in which the shift of resonant frequency satisfies Δω/ω ₀ =-ΔL/L ₀, the shift in sensor’s PhC cavity satisfies Δω = M _L0 ∙ ΔL, where M _L0 decreases with the increasing of L ₀ approximately, but more complicatedly (in the next section, we can see that M _L0 is also a function of resonant frequency). However, for a certain operating frequency, M _L0 can be considered as a constant in a small range of ΔL (e.g. ~20% of operating wavelength).

With a certain operating frequency ω ₁, we can then derive the variation of transmission coefficient (or the output intensity of sensor) when the length of sensor’s PhC cavity (or the micro displacement of the moving PhC) varies from L ₀ to L ₀ + ΔL,

After expanding T(ω ₀ + Δω, ω ₁) into the form of Taylor series, we obtain

wherer $T^{\prime }({\omega }_0,{\omega }_1)=\frac{-2{\left(\frac{{\omega }_0}{2Q}\right)}^2\left({\omega }_0-{\omega }_1\right)}{{\left({\left({\omega }_0-{\omega }_1\right)}^2+{\left(\frac{{\omega }_0}{2Q}\right)}^2\right)}^2}$, and $T^{″}({\omega }_0,{\omega }_1)=\frac{6{\left(\frac{{\omega }_0}{2Q}\right)}^2{\left({\omega }_0-{\omega }_1\right)}^2-2{\left(\frac{{\omega }_0}{2Q}\right)}^4}{{\left({\left({\omega }_0-{\omega }_1\right)}^2+{\left(\frac{{\omega }_0}{2Q}\right)}^2\right)}^3}$.

In practical systems, we should choose a proper operating frequency ω ₁ to make the sensor work in the quasi-linear region [9], in order Eq. (1) to achieve the relatively higher sensitivity, Eq. (2) to reduce the error caused by interpolation, Eq. (3) to simplify the postprocessing of measurement data. When ω ₁ is specified corresponding to T″(ω ₀,ω ₁) ≈0 (or ω ₁ ≈ (1±1/2√3Q ω ₀) can be truncated as a linear function

For|ΔL| < ω ₀/5QM _L0, the relative error caused by this linear approximation is less than 2%. In Eq. (4), the quotient of ΔL is the sensitivity of this micro displacement sensor. It implies that the sensitivity can be enhanced by increasing either the Q factor or M _L0. On the other hand, the linear sensing range is inversely proportional to the sensitivity, so there is a tradeoff between sensing range and sensitivity in this design in response to applications. In order to achieve a proper sensitivity, we can change the radius (or number) of the blued dielectric rods shown in Figure 1 to adjust the Q factor of PhC cavity, and vary operating frequency or the length of PhC cavity to adjust M _L0. Moreover, the sensing range can be broadened by using multiple operating frequencies (e.g. tunable laser source), which is interpreted in the following section.

3. Design and simulation

Based on the theoretical analysis in the preceding section, a micro displacement sensor is designed in a 2D PhC with square lattice of dielectric rods in air. This 2D PhC has a PBG only for TM modes [1], so only the TM modes are concerned in this work. The FDTD code with perfect matched layer (PML) boundary condition is carried out in our simulation.

In the designed structure shown in Fig.1, the radius of rods in PhC bulk is 0.20 a, where a is the lattice constant, and the dielectric constant of these rods is 11.56. This structure has mirror-plane symmetries perpendicular and parallel to the line-defect PhC cavity respectively. For simplicity, we define the position where the distance between the adjacent two columns of rods of fixed and moving segments is a as the original point. The displacement is negative when the moving segments shifts toward the fixed one, and positive in the opposite direction, as indicated in Fig.1. The sensitivity is defined as the ratio between the variation of normalized intensity and the corresponding displacement. The radius of the two rods blued in Fig. 1 is reduced to 0.10 a to obtain a medium sensitivity, and the original length of PhC cavity is 7 a. With these parameters, the cavity has a Q factor of ~40, and M _L0 ≈ 0.021(ω ₀/a) in the range of 0.00~0.20 a, thus the sensor’s sensitivity calculated by Eq. (4) is about 3√3×40×0.021/4 ≈1.09a ^-1.

 

Fig. 2. (a) Lorentzian curves of normalized intensity for different displacements of moving PhC segment, the peak frequency moves from 0.335(2πc/a) to 0.333(2πc/a) when the displacement shifts from 0.00 a to 0.20 a, and the operating frequency is chosen near 0.332(2πc/a); (b) Variation of normalized intensity as a linear function of displacement, the blue square dots present the simulation values, and the red line is the linear regression result.

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Then the sensing characteristic of this sensor is simulated. A Gaussian pulse is excited from light source with a center frequency ω = 0.333(2πc/a) (c is the velocity of light in free space), a full width at half maximum (FWHM) Δt = 150fs, and polarization parallel to the rods. The moving PhC segment moves from 0.00 a to 0.20 a with an interval of 0.05 a, the corresponding transmission spectra normalized with the spectra of light source are plotted in Fig. 2(a) (Since the relative variation of peak transmissions for the five displacements is less than 0.3%, the normalized peak transmissions are all unit roughly). As shown in Fig. 2(a), the Lorentzian curve shifts towards the lower frequency (e.g. its resonant frequency shifts from 0.335(2πc/a) to 0.333(2πc/a)). In the other point of view, if a coherent light with the frequency of 0.332(2πc/a) (we choose this frequency because its five cross points in Fig. 2(a) are all in the quasi-linear regions of Lorentzian curves, and have the maximal linearity) is used as the light source, the variation of normalized transmission coefficient will increase from 0 to 0.237 proportionally when the displacement increases from 0.00 a to 0.20 a, as shown in Fig. 2(b). The sensitivity is about 1.15 a ^-1, which is identical with the theoretical analysis. The regression coefficient is 0.99967, which implies that the sensor is working in a reasonable linear region.

For other displacement regions, our design can also work as a linear sensor with other operating frequencies, which can be specified using the method mentioned above. When the moving PhC segment shifts from -0.55 a to 0.60 a, the operating frequencies, regression coefficients and sensitivities for different displacement regions are listed in Table. 1. The descent of sensitivity is mainly caused by the decline of M _L0 and Q factor of resonant modes. The relative variation of operating frequency is less than 4.6%. Thus we can set up a large range displacement sensor by using a tunable laser as the light source: after the original displacement is calibrated, the sensor knows the original operating frequency and sensitivity, then the relative displacement is measured and accumulated according to the change of light intensity; the operating frequency and sensitivity are altered once the displacement shifts into another region. Therefore, this sensor can measure the displacement in the range of -0.55 a ~0.60 a with sensitivity higher than 0.781 a ^-1. For concreteness, if the lattice constanta ≈ 0.50μm, this sensor will have a sensitivity larger than 1.56 μm ^-1 in the range of -0.28~0.30 μm, with the operating wavelength of 1.49~1.55 μm, and the Q factor of sensor cavity is only ~40.

Table 1. Operating frequency, regression coefficient and sensitivity for different displacement regions

View Table

 

Fig. 3. Electric-field distribution when a PCWG perpendicular to the line-defect cavity is formed: (a) The displacement is -0.60 a, the adjacent two columns of rods of fixed and moving segments form a PCWG; (b) The displacement is 0.70 a, an air PCWG is introduced between these two adjacent two columns of rods.

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If the negative displacement increases from -0.55 a to -0.60 a, the adjacent two columns of rods of fixed and moving segments will form a PCWG perpendicular to the common axis with guided modes near 0.337(2πc/a) [11], as shown in Fig. 3(a), thus the transmission will not exhibit a Lorentzian line shape. On the other hand, if the positive displacement increases from 0.60 a to 0.80 a, an air PCWG will be introduced, as shown in Fig.3(b), which also damages the Lorentzian line shape. Then the light intensities that reach the photo detector are nonlinear except that the operating frequencies are tuned to 0.413(2πc/a) and 0.305(2πc/a) respectively in order to avoid cross coupling, and it is limited by the tunable range of laser source. When the displacement is larger than 0.90 a, all the frequencies of resonant modes can be guided in the cross PCWG, then the modal analyzed in section 2 is no longer valid.

According to Eq. (4), the sensitivity can be adjusted by varying the shift ratio of resonant frequency M _L0. The simulation results show that it is a function of resonant frequency and length of PhC cavity, which differs from the traditional F-P cavity. The structure mentioned above has four resonant frequencies: 0.316(2πc/a), 0.336(2πc/a), 0.365(2πc/a), 0.402(2πc/a). The M _L0 s for each resonant frequency are plotted as red dots in Fig. 4(a). The variation of M _L0 implies that different mode distribution has different sensitivity to the change of cavity structure. Furthermore, the Q values for each frequency have a little difference, which is perhaps caused by the difference of locations of resonant frequencies in PBG Figure 4(a) shows that the sensor has the highest sensitivity with the operating frequency near 0.365(2πc/a). The corresponding transmission spectra are plotted in Fig. 4(b) in the increment of 0.025 a. The sensitivity in the displacement range of 0.00 a ~0.10 a is 2.34 a ^-1 with the operating frequency of 0.361(2πc/a), which is nearly twice of the sensitivity with operating frequency of 0.332(2πc/a). The regression coefficient is 0.99924. The results agree well with the theoretical analysis. On the contrary, with the enhancement of sensitivity, the available linear sensing range for single operating frequency declines inversely, which means that there should be more discrete displacement regions for large range sensing.

 

Fig. 4. (a) M _L0 and theoretical sensitivity for each resonant frequency, the length of PhC cavity is 7 a; (b) Lorentzian curves of normalized intensity for the displacements shifting from 0.00 a to 0.10 a, and the operating frequency is chosen near 0.361(2πc/a).

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Meanwhile, M _L0 declines roughly with the increasing of cavity length, but the Q factor of resonant mode increases at the same time as shown in Fig. 5(a). Consequently the sensitivity oscillates with the cavity length as indicated in Fig. 5(b). For instance, when the cavity length is altered from 5 a to 15 a, the sensitivity increases from 1.85 a ^-1 to 2.75a ^-1 correspondingly. Therefore the sensitivity can be enhanced by optimizing the length of sensor cavity.

 

Fig.5. (a)M _L0 and Q factor for different lengths of PhC cavity, the length increases from 7 a to 17 a, and the operating frequency is nearly 0.361(2πc/a); (b) Theoretical sensitivity as a function of the cavity length.

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The sensitivity of sensor can also be adjusted by varying the Q factor of PhC cavity. In our design, Q factor can be changed by changing the radius (or number) of the blued dielectric rods in Fig. 1. For instance, we tune the radius from 0.10 a to 0.15 a. The Q factor correspondingly increases from ~40 to ~200, so the sensitivity will be four times higher theoretically. The transmission spectra of sensor are shown in Fig. 6. The sensitivity in the displacement range of 0.00 a ~ 0.075 a is 5.56 a ^-1 with the operating frequency of 0.3302(2πc/a), and 5.56/1.18≈4.71. The regression coefficient is 0.99996. The results prove the validity of our theoretical modal. Higher sensitivity can be achieved with larger Q factor in this structure, but simultaneously narrower sensing range.

 

Fig. 6. (a) Lorentzian curves of normalized intensity for the displacements shifting from 0.00 a to 0.075 a, the Q factor of sensor’s PhC cavity is enhanced to ~200 by increasing the radius of blued rods from 0.10 a to 0.15 a; (b) Variation of normalized intensity as a linear function of displacement, and the operating frequency is chosen near 0.332(2πc/a).

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In actual implementations, the robustness of the sensor against dimensional errors is a significant factor. The effects of structural disorder on the sensor’s performances are checked succinctly in the following two cases: (1) for the shift of moving PhC segment in the perpendicular direction, 0.05 a is introduced; and (2) for random errors involved in the radius and locations of dielectric rods, 5% and 2% relative error respectively. In case (1), the proper operating frequency almost retains invariable, the sensitivity is 1.18, and the regression coefficient is 0.99965. In case (2), the operating frequency moves slightly from 0.332(2πc/a) to 0.331(2πc/a), the sensitivity is 1.20, and the regression coefficient is 0.99997. Figure 7 shows the performances of the error-introduced and error-free structures, and we note that they have the similar behavior.

 

Fig. 7. Sensing performances of the micro displacement sensor in the cases of (1) the moving PhC segment shifts 0.05 a in the perpendicular direction; (2) random errors are introduced in the radius and locations of dielectric rods; (3) error-free structure. The initial radius of blued rods in Fig.2 is 0.20 a, and L ₀ =7a.

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4. Conclusion

In summary, we have presented a micro displacement sensor based on line-defect resonant cavity in 2D PhCs with square lattice of dielectric rods in air, and its sensing technique. The sensor is formed by a fixed and a mobile PhC segments. With a proper operating frequency, the variation of radiation in the output port is a quasi-linear function of the relative displacement of the mobile PhC segment in a specified range. For the sensor with 7 a long line-defect resonant cavity, nearly 1.15 a ^-1 sensitivity is obtain in the range of 0.00 a ~0.20 a, and the Q factor of resonant cavity is only ~40; the sensitivity reaches 2.75 a ^-1 after the operating frequency and cavity length are optimized. We also tuned the Q factor of cavity from ~40 to ~200, the sensitivity is enhanced to 5.56 a ^-1. Since the product of sensitivity and sensing range is nearly constant, the sensing range with single operating frequency decreases inversely. All these results agree well with the theoretical analysis.

Furthermore, the sensing range can be broadened by using multiple operating frequencies. The micro displacement sensor can measure the displacement in the range of -0.55 a ~0.60 a, which is divided into 6 sub-ranges, with sensitivity higher than 0.781 a ^-1 if the light source can be tuned in the range of 0.328(2πc/a)~0.343(2πc/a). The method for displacement sensing with high sensitivity and large sensing range was demonstrated.

We notice that our FDTD simulation is based on 2D model and vertical confinement of light is not considered. For practical sensors, the proposed structure should be realized in different forms, such as PhC slab [12], in which vertical confinement is provided by index guiding. In such practical structures, performance of the sensor will be degraded by imperfect vertical confinement, and the effect of vertical unwanted displacement (e.g., polarization mixing) should also be considered. However, the theoretical analysis presented in this paper can still be used after including the out-of-plane loss of the sensor’s cavity. This subject is under investigation.