Abstract

Optical microcavities are widely used for biological and chemical sensing applications. In these devices, a sensing event is estimated by measuring the shift in the resonant wavelength, or in the quality factor of the microcavity. However, all published works to date only use one of these measures to estimate the sensing event. Here, we show that the estimation accuracy of a sensing event can be improved by employing a combination of both the quality factor and the resonant wavelength measurements in a microcavity sensor. We further demonstrate an experimental application of this model by introducing a refractive index change for a microtoroidal cavity sensor immersed in a liquid. By further using the finite element method simulations in conjunction with the estimator model, we show the existence of three distinct measurement regimes as a function of the quality factor of the microcavity. Finally, the estimator model is extended to develop a sensing metric to compare performance of optical or non-optical sensors.

© 2013 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. K. Hunt and A. M. Armani, “Label-free biological and chemical sensors,” Nanoscale2, 1544–1559 (2010).
    [CrossRef] [PubMed]
  2. F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics267–291 (2012).
  3. L. Maleki and V. S. Ilchenko, “Techniques and devices for sensing a sample by using a whispering gallery mode resonator,” US Patent6490039 (2002).
  4. J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE4629, 172–180 (2002).
    [CrossRef]
  5. A. M. Armani and K. J. Vahala, “Heavy water detection using ultra-high-Q microcavities,” Opt. Lett.31, 1896–1898 (2006).
    [CrossRef] [PubMed]
  6. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421(2003).
    [CrossRef] [PubMed]
  7. J. Barnes, B. Carver, J. M. Fraser, G. Gagliardi, H. P. Loock, Z. Tian, M. W. B. Wilson, S. Yam, and O. Yastrubshak, “Loss determination in microsphere resonators by phase-shift cavity ring-down measurements,” Opt. Express16, 13158–13167 (2008).
    [CrossRef] [PubMed]
  8. M. I. Cheema, S. Mehrabani, A. A. Hayat, Y.-A. Peter, A. M. Armani, and A. G. Kirk, “Simultaneous measurement of quality factor and wavelength shift by phase shift microcavity ring down spectroscopy,” Opt. Express20, 9090–9098 (2012).
    [CrossRef] [PubMed]
  9. L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, 1991).
  10. F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80, 4057–4059 (2002).
    [CrossRef]
  11. A. Armani, D. Armani, B. Min, K. Vahala, and S. Spillane, “Ultra-high-Q microcavity operation in H2O and D2O,” Appl. Phys. Lett.87(2005).
    [CrossRef]
  12. N. Hanumegowda, C. Stica, B. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett.87(2005).
    [CrossRef]
  13. M. I. Cheema, “Towards optimal whispering gallery mode microcavity sensors: Novel techniques and analyses,” Ph.D. thesis, McGill University (2013). (To be published).
  14. M. I. Cheema and A. G. Kirk, “Accurate determination of the quality factor and tunneling distance of axisymmetric resonators for biosensing applications,” Opt. Express21, 8724–8735 (2013).
    [CrossRef] [PubMed]
  15. T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” PNAS108, 5976–5979 (2011).
    [CrossRef] [PubMed]
  16. A. Oraevsky, “Whispering-gallery waves,” Quantum Electron.32, 377–400 (2002).
    [CrossRef]
  17. S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in micro-spheres by protein adsorption,” Opt. Lett.28, 272–274 (2003).
    [CrossRef] [PubMed]
  18. I. M. White and X. Fan, “On the performance quantification of resonant refractive index sensors,” Opt. Express16, 1020–1028 (2008).
    [CrossRef] [PubMed]
  19. J. Hu, X. Sun, A. Agarwal, and L. C. Kimerling, “Design guidelines for optical resonator biochemical sensors,” J. Opt. Soc. Am. B26, 1032–1041 (2009).
    [CrossRef]
  20. J. Homola, I. Koudela, and S. S. Yee, “Surface plasmon resonance sensors based on diffraction gratings and prism couplers: sensitivity comparison,” Sensors and Actuators B: Chemical54, 16–24 (1999).
    [CrossRef]
  21. I. White, H. Oveys, and X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett.31, 1319–1321 (2006).
    [CrossRef] [PubMed]

2013 (1)

2012 (2)

2011 (1)

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” PNAS108, 5976–5979 (2011).
[CrossRef] [PubMed]

2010 (1)

H. K. Hunt and A. M. Armani, “Label-free biological and chemical sensors,” Nanoscale2, 1544–1559 (2010).
[CrossRef] [PubMed]

2009 (1)

2008 (2)

2006 (2)

2005 (2)

A. Armani, D. Armani, B. Min, K. Vahala, and S. Spillane, “Ultra-high-Q microcavity operation in H2O and D2O,” Appl. Phys. Lett.87(2005).
[CrossRef]

N. Hanumegowda, C. Stica, B. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett.87(2005).
[CrossRef]

2003 (2)

2002 (4)

L. Maleki and V. S. Ilchenko, “Techniques and devices for sensing a sample by using a whispering gallery mode resonator,” US Patent6490039 (2002).

J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE4629, 172–180 (2002).
[CrossRef]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80, 4057–4059 (2002).
[CrossRef]

A. Oraevsky, “Whispering-gallery waves,” Quantum Electron.32, 377–400 (2002).
[CrossRef]

1999 (1)

J. Homola, I. Koudela, and S. S. Yee, “Surface plasmon resonance sensors based on diffraction gratings and prism couplers: sensitivity comparison,” Sensors and Actuators B: Chemical54, 16–24 (1999).
[CrossRef]

Agarwal, A.

Armani, A.

A. Armani, D. Armani, B. Min, K. Vahala, and S. Spillane, “Ultra-high-Q microcavity operation in H2O and D2O,” Appl. Phys. Lett.87(2005).
[CrossRef]

Armani, A. M.

Armani, D.

A. Armani, D. Armani, B. Min, K. Vahala, and S. Spillane, “Ultra-high-Q microcavity operation in H2O and D2O,” Appl. Phys. Lett.87(2005).
[CrossRef]

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421(2003).
[CrossRef] [PubMed]

Arnold, S.

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in micro-spheres by protein adsorption,” Opt. Lett.28, 272–274 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80, 4057–4059 (2002).
[CrossRef]

Barnes, J.

Bearman, G. H.

J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE4629, 172–180 (2002).
[CrossRef]

Braun, D.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80, 4057–4059 (2002).
[CrossRef]

Carver, B.

Cheema, M. I.

Chen, T.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” PNAS108, 5976–5979 (2011).
[CrossRef] [PubMed]

Fan, X.

Flagan, R. C.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” PNAS108, 5976–5979 (2011).
[CrossRef] [PubMed]

Fraser, J. M.

Fraser, S. E.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” PNAS108, 5976–5979 (2011).
[CrossRef] [PubMed]

Gagliardi, G.

Hanumegowda, N.

N. Hanumegowda, C. Stica, B. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett.87(2005).
[CrossRef]

Hayat, A. A.

Herchak, S.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” PNAS108, 5976–5979 (2011).
[CrossRef] [PubMed]

Holler, S.

Homola, J.

J. Homola, I. Koudela, and S. S. Yee, “Surface plasmon resonance sensors based on diffraction gratings and prism couplers: sensitivity comparison,” Sensors and Actuators B: Chemical54, 16–24 (1999).
[CrossRef]

Hu, J.

Hunt, H. K.

H. K. Hunt and A. M. Armani, “Label-free biological and chemical sensors,” Nanoscale2, 1544–1559 (2010).
[CrossRef] [PubMed]

Ilchenko, V. S.

L. Maleki and V. S. Ilchenko, “Techniques and devices for sensing a sample by using a whispering gallery mode resonator,” US Patent6490039 (2002).

J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE4629, 172–180 (2002).
[CrossRef]

Khoshsima, M.

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in micro-spheres by protein adsorption,” Opt. Lett.28, 272–274 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80, 4057–4059 (2002).
[CrossRef]

Kim, J.-H.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” PNAS108, 5976–5979 (2011).
[CrossRef] [PubMed]

Kimerling, L. C.

Kippenberg, T. J.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421(2003).
[CrossRef] [PubMed]

Kirk, A. G.

Kossakovski, D.

J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE4629, 172–180 (2002).
[CrossRef]

Koudela, I.

J. Homola, I. Koudela, and S. S. Yee, “Surface plasmon resonance sensors based on diffraction gratings and prism couplers: sensitivity comparison,” Sensors and Actuators B: Chemical54, 16–24 (1999).
[CrossRef]

Lee, H.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” PNAS108, 5976–5979 (2011).
[CrossRef] [PubMed]

Libchaber, A.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80, 4057–4059 (2002).
[CrossRef]

Loock, H. P.

Lu, T.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” PNAS108, 5976–5979 (2011).
[CrossRef] [PubMed]

Maleki, L.

L. Maleki and V. S. Ilchenko, “Techniques and devices for sensing a sample by using a whispering gallery mode resonator,” US Patent6490039 (2002).

J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE4629, 172–180 (2002).
[CrossRef]

Mehrabani, S.

Min, B.

A. Armani, D. Armani, B. Min, K. Vahala, and S. Spillane, “Ultra-high-Q microcavity operation in H2O and D2O,” Appl. Phys. Lett.87(2005).
[CrossRef]

Nadeau, J. L.

J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE4629, 172–180 (2002).
[CrossRef]

Oraevsky, A.

A. Oraevsky, “Whispering-gallery waves,” Quantum Electron.32, 377–400 (2002).
[CrossRef]

Oveys, H.

Patel, B.

N. Hanumegowda, C. Stica, B. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett.87(2005).
[CrossRef]

Peter, Y.-A.

Scharf, L. L.

L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, 1991).

Spillane, S.

A. Armani, D. Armani, B. Min, K. Vahala, and S. Spillane, “Ultra-high-Q microcavity operation in H2O and D2O,” Appl. Phys. Lett.87(2005).
[CrossRef]

Spillane, S. M.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421(2003).
[CrossRef] [PubMed]

Stica, C.

N. Hanumegowda, C. Stica, B. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett.87(2005).
[CrossRef]

Sun, X.

Teraoka, I.

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in micro-spheres by protein adsorption,” Opt. Lett.28, 272–274 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80, 4057–4059 (2002).
[CrossRef]

Tian, Z.

Vahala, K.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” PNAS108, 5976–5979 (2011).
[CrossRef] [PubMed]

A. Armani, D. Armani, B. Min, K. Vahala, and S. Spillane, “Ultra-high-Q microcavity operation in H2O and D2O,” Appl. Phys. Lett.87(2005).
[CrossRef]

Vahala, K. J.

A. M. Armani and K. J. Vahala, “Heavy water detection using ultra-high-Q microcavities,” Opt. Lett.31, 1896–1898 (2006).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421(2003).
[CrossRef] [PubMed]

Vollmer, F.

F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics267–291 (2012).

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in micro-spheres by protein adsorption,” Opt. Lett.28, 272–274 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80, 4057–4059 (2002).
[CrossRef]

White, I.

I. White, H. Oveys, and X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett.31, 1319–1321 (2006).
[CrossRef] [PubMed]

N. Hanumegowda, C. Stica, B. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett.87(2005).
[CrossRef]

White, I. M.

Wilson, M. W. B.

Yam, S.

Yang, L.

F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics267–291 (2012).

Yastrubshak, O.

Yee, S. S.

J. Homola, I. Koudela, and S. S. Yee, “Surface plasmon resonance sensors based on diffraction gratings and prism couplers: sensitivity comparison,” Sensors and Actuators B: Chemical54, 16–24 (1999).
[CrossRef]

Appl. Phys. Lett. (3)

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80, 4057–4059 (2002).
[CrossRef]

A. Armani, D. Armani, B. Min, K. Vahala, and S. Spillane, “Ultra-high-Q microcavity operation in H2O and D2O,” Appl. Phys. Lett.87(2005).
[CrossRef]

N. Hanumegowda, C. Stica, B. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett.87(2005).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nanophotonics (1)

F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics267–291 (2012).

Nanoscale (1)

H. K. Hunt and A. M. Armani, “Label-free biological and chemical sensors,” Nanoscale2, 1544–1559 (2010).
[CrossRef] [PubMed]

Nature (1)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421(2003).
[CrossRef] [PubMed]

Opt. Express (4)

Opt. Lett. (3)

PNAS (1)

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” PNAS108, 5976–5979 (2011).
[CrossRef] [PubMed]

Proc. SPIE (1)

J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE4629, 172–180 (2002).
[CrossRef]

Quantum Electron. (1)

A. Oraevsky, “Whispering-gallery waves,” Quantum Electron.32, 377–400 (2002).
[CrossRef]

Sensors and Actuators B: Chemical (1)

J. Homola, I. Koudela, and S. S. Yee, “Surface plasmon resonance sensors based on diffraction gratings and prism couplers: sensitivity comparison,” Sensors and Actuators B: Chemical54, 16–24 (1999).
[CrossRef]

US Patent (1)

L. Maleki and V. S. Ilchenko, “Techniques and devices for sensing a sample by using a whispering gallery mode resonator,” US Patent6490039 (2002).

Other (2)

L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, 1991).

M. I. Cheema, “Towards optimal whispering gallery mode microcavity sensors: Novel techniques and analyses,” Ph.D. thesis, McGill University (2013). (To be published).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

The change in wavelength (Δλ) and quality factor (ΔQ) is induced by a sensing event. Each measurement has an experimental noise, i.e., Nλ, and NQ. The estimator estimates ΔX from the outputs (measurements) of the system.

Fig. 2
Fig. 2

(a) Experimental setup. FG-Function Generation, Det-Detector. Flow cell is same as the one used in [8] (b)Scanning electron microscope image of a typical microtoroidal cavity (c) Cross section of the microtoroidal cavity showing its dimensions: Major diameter, D, and minor diameter, d.

Fig. 3
Fig. 3

Experimental results. Δλ is positive (i.e. λ shifts towards red) and ΔQ is negative (i.e. Q decreases) as ΔX increases. This behavior is consistent with the fundamental micro-cavity theory and is verified by simulations.

Fig. 4
Fig. 4

A normal probability distribution fit (pdf) to the experimental noise data (σθ = 4.10 × 10−4 deg.) for phase shift (θ) of the waveguide signal w.r.t to the reference signal (FG2).The sum of the areas of the rectangles is equal to one. From the noise data, the noise (σQ) in ΔQ calculation is 30 for the quality factor of 5 × 104. The noise (σλ) in Δλ measurement is 46 fm. See Section 3 for further details and discussion.

Fig. 5
Fig. 5

Modeling Results. D−Δλ: Δλ is dominant contributor in the estimator. D−ΔQ: ΔQ is dominant contributor in the estimator. AEC: Approximately equal contribution from the two measurements. For details of the simulation and other symbols, see Section 2 and Section 4.

Tables (1)

Tables Icon

Table 1 Estimation Results. From the commercial SPR system (GenOptics, France), ΔX = 1.95 × 10−3 RIU.

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

Δ λ = f λ ( Δ X ) + N λ ,
Δ Q = f Q ( Δ X ) + N Q ,
Δ λ = m λ Δ X + N λ ,
Δ Q = m Q Δ X + N Q ,
y = Δ X m + N ,
y = [ Δ λ Δ Q ] , m = [ m λ m Q ] , N = [ N λ N Q ] .
N λ ~ 𝒩 [ μ λ , σ λ 2 ] = 1 σ λ 2 π exp ( 1 2 ( Δ λ μ λ σ λ ) 2 ) ,
N Q ~ 𝒩 [ μ Q , σ Q 2 ] = 1 σ Q 2 π exp ( 1 2 ( Δ Q μ Q σ Q ) 2 ) ,
P ( Δ λ ) = 𝒩 [ Δ X m λ + μ λ , σ λ ] 2 ,
P ( Δ Q ) = 𝒩 [ Δ X m Q + μ Q , σ Q 2 ] .
f Δ X ( y ) = 1 2 π det ( R ) 1 2 exp ( 1 2 ( y m y ) R 1 ( y m y ) ) ,
m y = [ Δ X m λ + μ λ Δ X m Q + μ Q ] , R = [ σ λ 2 0 0 σ Q 2 ] .
f Δ X ( y ) = 𝒩 [ Δ X m λ + μ λ , σ λ 2 ] 𝒩 [ Δ X m Q + μ Q , σ Q 2 ] ,
ln f Δ X ( y ) = ln 1 σ λ σ Q ( 2 π ) ( Δ λ Δ X m λ μ λ ) 2 2 σ λ 2 ( Δ Q Δ X m Q μ Q ) 2 2 σ Q 2 .
Δ X ln f Δ X ( y ) = 0.
Δ X ^ = ( σ Q m Q ) 2 ( Δ λ μ λ m λ ) + ( σ λ m λ ) 2 ( Δ Q μ Q m Q ) ( σ λ m λ ) 2 + ( σ Q m Q ) 2 .
R = [ σ λ 2 σ λ Q σ λ Q σ Q 2 ] .
Δ X ^ = ( σ Q m Q ) 2 ( Δ λ μ λ m λ ) + ( σ λ m λ ) 2 ( Δ Q μ Q m Q ) + σ λ Q m λ m Q ( Δ λ μ λ m λ + Δ Q μ Q m Q ) ( σ λ m λ ) 2 + ( σ Q m Q ) 2 + σ λ Q m λ m Q .
( Δ λ f λ ( Δ X ) μ λ ) f λ ( Δ X ) σ λ 2 + ( Δ Q f Q ( Δ X ) μ Q ) f Q ( Δ X ) σ Q 2 = 0 ,
Δ λ = m λ N Δ X 2 + m λ L Δ X + N λ = f λ ( Δ X ) + N λ ,
Δ Q = m Q N Δ X 2 + m Q L Δ X + N Q = f Q ( Δ X ) + N Q ,
S M = m ( Δ A μ ) σ 2 ,
S M c = m ( Δ A μ ) σ 2 γ ,
Δ A 1 = m 1 Δ X + N 1 ,
Δ A n = m n Δ X + N n ,
Δ X ^ = 1 Z ( m 1 ( Δ A 1 μ 1 ) σ 1 2 Sensor 1 + m 2 ( Δ A 2 μ 2 ) σ 2 2 Sensor 2 + + m n ( Δ A n μ n ) σ n 2 Sensor n ) ,
Z = i = 1 i = n m i 2 σ i 2 .

Metrics