## Optimization of resonant optical sensors

Optics Express, Vol. 15, Issue 25, pp. 17449-17457 (2007)

http://dx.doi.org/10.1364/OE.15.017449

Acrobat PDF (165 KB)

### Abstract

The sensitivity of the resonant optical sensors, which are based on measurement of the transmission and reflection spectra of optical resonators, is investigated. The following problem is addressed: When the losses of the resonator are known, what is the sharpest possible and the steepest possible shape of the resonant peaks that can be achieved experimentally? This optimization problem is solved for the case of a separated peak, which corresponds to a nondegenerated eigenvalue of the resonator. It is shown that the reflection spectrum possesses better sensitivity than the transmission spectrum. The model of the resonant sensor consisting of two coupled resonators is also considered. This model demonstrates that the sensitivity of transmission spectrum can be significantly increased by modification of the resonator structure. However, for the reflection spectrum, the best sensitivity is still given by a separated resonant peak.

© 2007 Optical Society of America

## 1. Introduction

## 2. Principles of ROS optimization

*λ*, as well as on the other parameters of the ROS,

*γ*=(

*γ*

_{1},

*γ*

_{2},...,

*γ*

*). In practice, not all of these parameters can vary. Some of the parameters are predetermined by the material properties. Other parameters can vary within certain limits determined by geometrical conditions, fabrication quality, etc. Thus, the monitored profile of the resonant spectrum should be optimized by varying the parameters that can be changed in the actual design and fabrication of an ROS. Therefore, the set of ROS parameters,*

_{L}*γ*, should be divided into the subset of variable parameters,

*γ*

*, and the subset of parameters, which should be considered as constants,*

_{var}*γ*

*:*

_{const}*γ*=(

*γ*

*,*

_{var}*γ*

*). In the case, when one is looking for the maximum possible slope,*

_{const}*S*

*, of an ROS spectrum,*

_{max}*P*(

*λ*,

*γ*), the optimization problem is written in the form:

*, is found from the solution of the optimization problem:*

_{max}*γ*

*that can vary with geometric configuration of the ROS and also by parameters*

_{var}*γ*

*that are predetermined by the internal losses of the ROS. The input power and accuracy of measurement of the transmission and reflection powers are assumed to be constant. The transmission and reflection power spectra of an ROS are expressed as a function of these parameters using the generalized Breit-Wigner formula considered in Section 3.*

_{const}## 3. Transmission and reflection of an ROS

*λ*

*. These resonators are coupled to each other, to the input and output waveguide, and to the environment as illustrated in Fig. 1. The coupling coefficient between resonators*

_{n}*n*and

*m*is

*δ*

*and the coupling of a resonator*

_{mn}*n*to an input/output waveguide,

*k*, is defined by the transmission coefficient

*γ*

^{(k)}

*. The internal losses are modeled using the virtual output waveguides, such as the vertically-directed waveguides shown in Fig. 1(a). The resonant transmission and reflection spectrum of such optical resonator can be calculated using the generalized Breit-Wigner formula [13*

_{n}13. M. Sumetsky and B. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express **11**, 381–391 (2003). [CrossRef] [PubMed]

16. M. Sumetskii, “Narrow current dip for the double quantum dot resonant tunneling structure with three leads: Sensitive nanometer Y-branch switch,” Appl. Phys. Lett. , **63**, 3185–3187 (1993). [CrossRef]

*q*is:

*P*

^{(0)}

*is the power input into the waveguide*

_{p}*q*and the sum is taken over all resonators, which couple to the waveguides

*p*and

*q*, and Parameters

*γ*

*in Eq. (5) determine the widths of the uncoupled eigenvalues*

_{m}*λ*

*:*

_{m}*p*,

*P*

*, can be found from the power conservation law:*

_{pp}## 4. Optimization of the transmission and reflection spectrum for a separated resonance

17. T. Asano, W. Kunishi, B. Song, and S. Noda, “Time-domain response of point-defect cavities in two-dimensional photonic crystal slabs using picosecond light pulse,” Appl. Phys. Lett. **88**, 151102 (2006). [CrossRef]

18. T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nature Photonics **1**, 49–52 (2006). [CrossRef]

13. M. Sumetsky and B. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express **11**, 381–391 (2003). [CrossRef] [PubMed]

*γ*=

*γ*

^{(3)}

_{1}is introduced. With Eq. (8) and (7), the reflected power is determined in the form:

12. M. Sumetsky, “Optimization of optical ring resonator devices for sensing applications,” Opt. Lett. **32**, 2577–2579 (2007). [CrossRef] [PubMed]

*γ*

^{(2)}

_{1}=0, and the transmitted light in a ring resonator is conceived as the unfolded reflected light in our resonator.

*, is achieved for the largest possible Q-factor,*

_{FWHM}*Q*=

*Q*

*, which corresponds to*

_{int}*γ*

^{(1)}

_{1}=

*γ*

^{(2)}

_{1}=0. In this case,

*P*

_{12}=0 and

*R*

_{11}=1, i.e. the peak is absent and no sensing is possible (column 1 in Table 1 (a) and (b)). Alternatively, the relative height of the resonance achieves its maximum equal to 1 when max(

*γ*

^{(1)}

_{1},

*γ*

^{(2)}

_{1})=∞ and

*Q*=0. Then

*P*

_{12}=1 and

*P*

_{11}=0, i.e. the peak becomes infinitely broad and, again, no sensing is possible (see column 2 in Table 1 (a) and (b)). Thus, the maximum slope and sharpness of the resonance is achieved at an intermediate value of Q-factor, 0<

*Q*<

*Q*

*. Eq. (8) and (9) allow simple solution of the optimization problem. The parameters λ*

_{int}_{1},

*λ*

^{(1)}

_{1},

*γ*

^{(2)}

_{1}, and are variable because they can be modified by changing the geometry of the resonator and, in particular, by changing the distance between the input/output waveguides and the cavity. The parameter

*γ*is a constant because it determines the internal loss of the ROS. The following calculations allow to find the optimized transmission and reflection resonant peaks depicted in Fig. 2(a) and (b), respectively. The sharpest peak in transmission is achieved at

## 5. Optimization of transmission spectrum for two coupled resonators of equal loss

*γ*. An example of this type of resonator, which can be created in photonic crystals, is illustrated in the inset of Fig. 1(c). There are five variable parameters of this ROS: the wavelength eigenvalues

*λ*

_{1}and

*λ*

_{2}, coupling between resonators,

*δ*

_{12}, and also transmission coefficients between resonators 1 and waveguides 1and 2:

*γ*

^{(1)}

_{1}and

*γ*

^{(2)}

_{1}. From Eq.(3)–(6), the transmission spectrum of the ROS shown in Fig. 1(c) is found in the form:

*λ*

_{1},

*λ*

_{2},

*δ*

_{12},

*γ*

^{(1)}

_{1}, and

*γ*

^{(2)}

_{1}. It was found that the transmission slope reaches the maximum for very large

*γ*

^{(1)}

_{1}=

*γ*

^{(2)}

_{1}>>

*γ*. The latter condition means that the resonance 1 should be very broad. In other words, the waveguides 1 and 2 should be very strongly coupled to each other and should practically compose a single waveguide coupled to Resonator 2. The whole device becomes a single resonance ROS consisting of Resonator 2 side-coupled to the waveguide. In contrast to this device, the ROS in Fig. 1(b) consists of the resonator positioned inline with the waveguides. Numerical simulation showed that the slope can achieve maximum only if

*δ*

^{2}

_{12}=

*γγ*

^{(1)}

_{1}/2 and

*λ*

_{2}-

*λ*

_{1}=

*γ*

^{(1)}

_{1}/2. With these relations, Eq. (16) yields the optimized transmission spectrum:

*γ*

^{(1)}

_{1}=

*γ*

^{(2)}

_{1}>>

*γ*.and

*γ*

^{(1)}

_{1}=

*γ*

^{(2)}

_{1}>>|

*λ*-

*λ*

_{1}|. Thus, again, the optimized ROS is a side-coupled single resonance device. In contrast to Eq. (17), in this case

*γ*

^{(1)}

_{1}=

*γ*

^{(2)}

_{1}>>|

*λ*-

*λ*

_{1}| and the transmission spectrum is symmetric with respect to the position λ

_{2}of the second resonance:

*δ*

_{12}=0. Then the performance of the double resonance structure is equivalent to that of the inline single resonance structure and the optimum parameters of the resonator 1 should be equal to the corresponding parameters determined in Section 4.

## 6. Summary

## References and links

1. | F. Maystre and R. Dandliker, “Polarimetric fiber optical sensor with high sensitivity using a Fabry-Perot structure,” Appl. Opt. |

2. | G. Gagliardi, S. De Nicola, P. Ferraro, and P. De Natale, “Interrogation of fiber Bragg-grating resonators by polarization-spectroscopy laser-frequency locking,” Opt. Express |

3. | M. Noto, F. Vollmer, D. Keng, I. Teraoka, and S. Arnold, “Nanolayer characterization through wavelength multiplexing of a microsphere resonator,” Opt. Lett. |

4. | I. M. White, N. M. Hanumegowda, and X. Fan, “Subfemtomole detection of small molecules with microsphere sensors,” Opt. Lett. |

5. | Ashkenazi, C.-Y. Chao, L. J. Guo, and M. O’Donnell“Ultrasound detection using polymer microring optical resonator,” Appl. Phys. Lett. |

6. | A. Ksendzov and Y. Lin, “Integrated optics ring-resonator sensors for protein detection,” Opt. Lett. |

7. | C-Y. Chao, W. Fung, and L.J. Guo, “Polymer Microring Resonators for Biochemical Sensing Applications,” IEEE J. Sel. Top. Quantum Electron. |

8. | M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and D. J. DiGiovanni, “The Microfiber Loop Resonator: Theory, Experiment, and Application,” IEEE J. Lightwave Technol. |

9. | A. Yalçin, K.C. Popat, J.C. Aldridge, T.A Desai, J. Hryniewicz, N. Chbouki, B.E. Little, O. King, V. Van, S. Chu, D. Gill, M. Anthes-Washburn, M.S. Ünlü, and B.B. Goldberg, “Optical Sensing of Biomolecules Using Microring Resonators,” IEEE J. Sel. Top. Quantum Electron. |

10. | R. W. Boyd and J. E. Heebner, “Sensitive disk resonator photonic biosensor,” Appl. Opt. |

11. | E. Krioukov, D. J. W. Klunder, A. Driessen, J. Greve, and C. Otto, “Integrated optical microcavities for enhanced evanescent-wave spectroscopy,” Opt. Lett. , |

12. | M. Sumetsky, “Optimization of optical ring resonator devices for sensing applications,” Opt. Lett. |

13. | M. Sumetsky and B. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express |

14. | M. Sumetskii, “Modeling of complicated nanometer resonant tunneling devices with quantum dots,” J. Phys.: Condens. Matter , |

15. | M. Sumetskii, “Resistance resonances for resonant-tunneling structures of quantum dots,” Phys. Rev. B , |

16. | M. Sumetskii, “Narrow current dip for the double quantum dot resonant tunneling structure with three leads: Sensitive nanometer Y-branch switch,” Appl. Phys. Lett. , |

17. | T. Asano, W. Kunishi, B. Song, and S. Noda, “Time-domain response of point-defect cavities in two-dimensional photonic crystal slabs using picosecond light pulse,” Appl. Phys. Lett. |

18. | T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nature Photonics |

19. | L. D. Landau and E. M. Lifshitz, |

**OCIS Codes**

(060.2370) Fiber optics and optical communications : Fiber optics sensors

(130.6010) Integrated optics : Sensors

(230.3990) Optical devices : Micro-optical devices

(230.5750) Optical devices : Resonators

(230.4555) Optical devices : Coupled resonators

**ToC Category:**

Sensors

**History**

Original Manuscript: September 24, 2007

Revised Manuscript: October 24, 2007

Manuscript Accepted: October 24, 2007

Published: December 10, 2007

**Virtual Issues**

Vol. 3, Iss. 1 *Virtual Journal for Biomedical Optics*

Physics and Applications of Microresonators (2007) *Optics Express*

**Citation**

M. Sumetsky, "Optimization of resonant optical sensors," Opt. Express **15**, 17449-17457 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-25-17449

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### References

- F. Maystre and R. Dandliker, "Polarimetric fiber optical sensor with high sensitivity using a Fabry-Perot structure," Appl. Opt. 28, 1995-2000 (1989). [CrossRef] [PubMed]
- G. Gagliardi, S. De Nicola, P. Ferraro, and P. De Natale, "Interrogation of fiber Bragg-grating resonators by polarization-spectroscopy laser-frequency locking," Opt. Express 15, 3715-3728 (2007). [CrossRef] [PubMed]
- M. Noto, F. Vollmer, D. Keng, I. Teraoka, and S. Arnold, "Nanolayer characterization through wavelength multiplexing of a microsphere resonator," Opt. Lett. 30, 510-512 (2005). [CrossRef] [PubMed]
- I. M. White, N. M. Hanumegowda, and X. Fan, "Subfemtomole detection of small molecules with microsphere sensors," Opt. Lett. 30, 3189-3191 (2005). [CrossRef] [PubMed]
- Ashkenazi, C.-Y. Chao, L. J. Guo, and M. O’Donnell, "Ultrasound detection using polymer microring optical resonator," Appl. Phys. Lett. 85, 5418-5420 (2004). [CrossRef]
- A. Ksendzov, Y. Lin, "Integrated optics ring-resonator sensors for protein detection," Opt. Lett. 30, 3344-3346 (2005). [CrossRef]
- C-Y. Chao, W. Fung, L.J. Guo, "Polymer Microring Resonators for Biochemical Sensing Applications," IEEE J. Sel. Top. Quantum Electron. 12, 134-142 (2006). [CrossRef]
- M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, and D. J. DiGiovanni, "The Microfiber Loop Resonator: Theory, Experiment, and Application," IEEE J. Lightwave Technol. 24, 242-250 (2006). [CrossRef]
- A. Yalçin, K.C. Popat, J.C. Aldridge, T.A Desai, J. Hryniewicz, N. Chbouki, B.E. Little, O. King, V. Van, S. Chu, D. Gill, M. Anthes-Washburn, M.S. Ünlü, B.B. Goldberg, "Optical Sensing of Biomolecules Using Microring Resonators," IEEE J. Sel. Top. Quantum Electron. 12, 148-155 (2006). [CrossRef]
- R. W. Boyd and J. E. Heebner, "Sensitive disk resonator photonic biosensor," Appl. Opt. 40, 5742-5747 (2001). [CrossRef]
- E. Krioukov, D. J. W. Klunder, A. Driessen, J. Greve, and C. Otto, "Integrated optical microcavities for enhanced evanescent-wave spectroscopy," Opt. Lett., 27, 1504-1506 (2002). [CrossRef]
- M. Sumetsky, "Optimization of optical ring resonator devices for sensing applications," Opt. Lett. 32, 2577-2579 (2007). [CrossRef] [PubMed]
- M. Sumetsky and B. Eggleton, "Modeling and optimization of complex photonic resonant cavity circuits," Opt. Express 11, 381-391 (2003). [CrossRef] [PubMed]
- M. Sumetskii, "Modeling of complicated nanometer resonant tunneling devices with quantum dots," J. Phys.: Condens. Matter, 3, 2651-2664 (1991). [CrossRef]
- M. Sumetskii, "Resistance resonances for resonant-tunneling structures of quantum dots," Phys. Rev. B, 48, 4586-4591 (1993). [CrossRef]
- M. Sumetskii, "Narrow current dip for the double quantum dot resonant tunneling structure with three leads: Sensitive nanometer Y-branch switch," Appl. Phys. Lett., 63, 3185-3187 (1993). [CrossRef]
- T. Asano, W. Kunishi, B. Song, and S. Noda, "Time-domain response of point-defect cavities in two-dimensional photonic crystal slabs using picosecond light pulse," Appl. Phys. Lett. 88, 151102 (2006). [CrossRef]
- T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, H. Taniyama, "Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity," Nature Photonics 1, 49-52 (2006). [CrossRef]
- L. D. Landau and E. M. Lifshitz, Quantum mechanics, (Pergamon Press, 1958).

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