## Micro displacement sensor based on line-defect resonant cavity in photonic crystal

Optics Express, Vol. 14, Issue 1, pp. 298-305 (2006)

http://dx.doi.org/10.1364/OPEX.14.000298

Acrobat PDF (196 KB)

### Abstract

A micro displacement sensor and its sensing technique based on line-defect resonant cavity in photonic crystals (PhCs) are presented. The line-defect resonant cavity is formed by a fixed and a mobile PhC segments. With a proper operating frequency, a quasi-linear measurement of micro-displacement is achieved with sensitivity of 1.15 *a*^{-1} (*a* is the lattice constant) and Q factor of 40. The sensitivity can be adjusted easily by varying either Q factor or operating frequency of the sensing system. In addition, the sensing range can be broadened to -0.55 *a* ~0.60 *a* by using multiple operating frequencies. The properties of the micro displacement sensor are analyzed theoretically and simulated using finite-difference time-domain (FDTD) method.

© 2006 Optical Society of America

## 1. Introduction

2. M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H-Y. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express **12**, 1551–1561 (2004), [CrossRef] [PubMed]

3. E. Chow, A. Grot, L. W. Mirkarimi, M. Sigalas, and G. Girolami, “Ultracompact biochemical sensor built with two-dimensional photonic crystal microcavity,” Opt. Lett. **29**, 1093–1095, (2004). [CrossRef] [PubMed]

5. O. Levy, B. Z. Steinberg, M. Nathan, and A. Boag, “Ultrasensitive displacement sensing using photonic crystal waveguides,” Appl. Phys. Lett. **86**, 104102, (2005). [CrossRef]

*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

6. Wonjoo Suh, M. F. Yanik, Olav Solgaard, and Shanhui Fan, “Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs,” Appl. Phys. Lett. **82**, 1999–2001, (2003). [CrossRef]

7. Wonjoo Suh, Olav Solgaard, and Shanhui Fan, “Displacement sensing using evanescent tunneling between guided resonances in photonic crystal slabs,” J. Appl. Phys. **98**, 033102, (2005). [CrossRef]

## 2. Theoretical analysis

8. Youngmin Kim and Dean P. Neikirk, “Micromachined Fabry-Perot Cavity Pressure Transducer,” IEEE Photonics Technol. Lett. **7**, 1471–1473 (1995). [CrossRef]

9. Jie Zhou, Samhita Dasgupta, Hiroshi Kobayashi, J. Mitch Wolff, Howard E. Jackson, and Joseph T. Boyd, “Optically interrogated MEMS pressure sensors for propulsion applications,” Opt. Eng. **40**, 598–604, (2001). [CrossRef]

3. E. Chow, A. Grot, L. W. Mirkarimi, M. Sigalas, and G. Girolami, “Ultracompact biochemical sensor built with two-dimensional photonic crystal microcavity,” Opt. Lett. **29**, 1093–1095, (2004). [CrossRef] [PubMed]

*ω*

_{0}is the resonant frequency, and

*Q*is the Q factor of the resonant cavity.

*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 Δ

*ω*/

*ω*

_{0}=-Δ

*L*/

*L*

_{0}, the shift in sensor’s PhC cavity satisfies Δ

*ω*=

*M*

_{L0}∙ Δ

*L*, 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 Δ

*L*(e.g. ~20% of operating wavelength).

*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}+ Δ

*L*,

*T*(

*ω*

_{0}+ Δ

*ω*,

*ω*

_{1}) into the form of Taylor series, we obtain

*ω*

_{1}to make the sensor work in the quasi-linear region [9

9. Jie Zhou, Samhita Dasgupta, Hiroshi Kobayashi, J. Mitch Wolff, Howard E. Jackson, and Joseph T. Boyd, “Optically interrogated MEMS pressure sensors for propulsion applications,” Opt. Eng. **40**, 598–604, (2001). [CrossRef]

*ω*

_{1}is specified corresponding to

*T*″(

*ω*

_{0},

*ω*

_{1}) ≈0 (or

*ω*

_{1}≈ (1±1/2√3

*Q*

*ω*

_{0}) can be truncated as a linear function

*L*| <

*ω*

_{0}/5

*QM*

_{L0}, the relative error caused by this linear approximation is less than 2%. In Eq. (4), the quotient of Δ

*L*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

*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 ≈1.09

*a*

^{-1}.

*ω*= 0.333(2

*πc*/

*a*) (

*c*is the velocity of light in free space), a full width at half maximum (FWHM) Δ

*t*= 150

*fs*, 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(2

*πc*/

*a*) to 0.333(2

*πc*/

*a*)). In the other point of view, if a coherent light with the frequency of 0.332(2

*πc*/

*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.

*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 ≈ 0.50

*μm*, 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.

*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(2

*πc*/

*a*) [11

11. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001). [CrossRef] [PubMed]

*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(2

*πc*/

*a*) and 0.305(2

*πc*/

*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.

*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(2

*πc*/

*a*), 0.336(2

*πc*/

*a*), 0.365(2

*πc*/

*a*), 0.402(2

*πc*/

*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(2

*πc*/

*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(2

*πc*/

*a*), which is nearly twice of the sensitivity with operating frequency of 0.332(2

*πc*/

*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.

*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 2.75

*a*

^{-1}correspondingly. Therefore the sensitivity can be enhanced by optimizing the length of sensor cavity.

*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(2

*πc*/

*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.

*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(2

*πc*/

*a*) to 0.331(2

*πc*/

*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.

## 4. Conclusion

*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.

*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(2

*πc*/

*a*)~0.343(2

*πc*/

*a*). The method for displacement sensing with high sensitivity and large sensing range was demonstrated.

12. Steven G. Johnson, Pierre. R. Villeneuve, Shanhui Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B **62**, 8212–8222 (2000). [CrossRef]

## Acknowledgments

## References and Links

1. | J. Joannopoulos, R, Meade, and J. Winn, |

2. | M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H-Y. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express |

3. | E. Chow, A. Grot, L. W. Mirkarimi, M. Sigalas, and G. Girolami, “Ultracompact biochemical sensor built with two-dimensional photonic crystal microcavity,” Opt. Lett. |

4. | J. Topolancik, P. Bhattacharya, J. Sabarinathan, and P.-C. Yu, “Fluid detection with photonic crystal-based multichannel waveguides,” Appl. Phys. Lett. |

5. | O. Levy, B. Z. Steinberg, M. Nathan, and A. Boag, “Ultrasensitive displacement sensing using photonic crystal waveguides,” Appl. Phys. Lett. |

6. | Wonjoo Suh, M. F. Yanik, Olav Solgaard, and Shanhui Fan, “Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs,” Appl. Phys. Lett. |

7. | Wonjoo Suh, Olav Solgaard, and Shanhui Fan, “Displacement sensing using evanescent tunneling between guided resonances in photonic crystal slabs,” J. Appl. Phys. |

8. | Youngmin Kim and Dean P. Neikirk, “Micromachined Fabry-Perot Cavity Pressure Transducer,” IEEE Photonics Technol. Lett. |

9. | Jie Zhou, Samhita Dasgupta, Hiroshi Kobayashi, J. Mitch Wolff, Howard E. Jackson, and Joseph T. Boyd, “Optically interrogated MEMS pressure sensors for propulsion applications,” Opt. Eng. |

10. | H. A. Haus, |

11. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

12. | Steven G. Johnson, Pierre. R. Villeneuve, Shanhui Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B |

**OCIS Codes**

(230.3990) Optical devices : Micro-optical devices

(250.5300) Optoelectronics : Photonic integrated circuits

**ToC Category:**

Optical Devices

**Citation**

Zhenfeng Xu, Liangcai Cao, Claire Gu, Qingsheng He, and Guofan Jin, "Micro displacement sensor based on line-defect resonant cavity in photonic crystal," Opt. Express **14**, 298-305 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-1-298

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### References

- J. Joannopoulos, R, Meade, and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
- M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H-Y. Ryu, "Waveguides, resonators and their coupled elements in photonic crystal slabs," Opt. Express 12, 1551-1561 (2004). [CrossRef] [PubMed]
- E. Chow, A. Grot, L. W. Mirkarimi, M. Sigalas, and G. Girolami, "Ultracompact biochemical sensor built with two-dimensional photonic crystal microcavity," Opt. Lett. 29, 1093-1095, (2004). [CrossRef] [PubMed]
- J. Topolancik, P. Bhattacharya, J. Sabarinathan, and P.-C. Yu, "Fluid detection with photonic crystal-based multichannel waveguides," Appl. Phys. Lett. 82, 1143-1145, (2003). [CrossRef]
- O. Levy, B. Z. Steinberg, M. Nathan, and A. Boag, "Ultrasensitive displacement sensing using photonic crystal waveguides," Appl. Phys. Lett. 86, 104102, (2005). [CrossRef]
- Wonjoo Suh, M. F. Yanik, Olav Solgaard, and Shanhui Fan, "Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs," Appl. Phys. Lett. 82, 1999-2001, (2003). [CrossRef]
- Wonjoo Suh, Olav Solgaard, and Shanhui Fan, "Displacement sensing using evanescent tunneling between guided resonances in photonic crystal slabs," J. Appl. Phys. 98, 033102, (2005). [CrossRef]
- Youngmin Kim, and Dean P. Neikirk, "Micromachined Fabry-Perot Cavity Pressure Transducer," IEEE Photonics Technol. Lett. 7, 1471-1473 (1995). [CrossRef]
- J. Zhou, S. Dasgupta, H. Kobayashi, J. M. Wolff, H. E. Jackson, and J. T. Boyd, "Optically interrogated MEMS pressure sensors for propulsion applications," Opt. Eng. 40, 598-604, (2001). [CrossRef]
- H. A. Haus, Waves and Felds in Optoelectronics (Prentice-Hall, Englewood Cliffs, USA, 1985).
- S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173-190 (2001). [CrossRef] [PubMed]
- S. G. Johnson, P.. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Linear waveguides in photonic-crystal slabs," Phys. Rev. B 62, 8212-8222 (2000). [CrossRef]

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