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

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References

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  1. 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]
  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 15, 3715-3728 (2007).
    [CrossRef] [PubMed]
  3. 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]
  4. 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]
  5. 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]
  6. A. Ksendzov, Y. Lin, "Integrated optics ring-resonator sensors for protein detection," Opt. Lett. 30, 3344-3346 (2005).
    [CrossRef]
  7. 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]
  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. 24, 242-250 (2006).
    [CrossRef]
  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, B.B. Goldberg, "Optical Sensing of Biomolecules Using Microring Resonators," IEEE J. Sel. Top. Quantum Electron. 12, 148-155 (2006).
    [CrossRef]
  10. R. W. Boyd and J. E. Heebner, "Sensitive disk resonator photonic biosensor," Appl. Opt. 40, 5742-5747 (2001).
    [CrossRef]
  11. 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]
  12. M. Sumetsky, "Optimization of optical ring resonator devices for sensing applications," Opt. Lett. 32, 2577-2579 (2007).
    [CrossRef] [PubMed]
  13. M. Sumetsky and B. Eggleton, "Modeling and optimization of complex photonic resonant cavity circuits," Opt. Express 11, 381-391 (2003).
    [CrossRef] [PubMed]
  14. M. Sumetskii, "Modeling of complicated nanometer resonant tunneling devices with quantum dots," J. Phys.: Condens. Matter,  3, 2651-2664 (1991).
    [CrossRef]
  15. M. Sumetskii, "Resistance resonances for resonant-tunneling structures of quantum dots," Phys. Rev. B,  48, 4586-4591 (1993).
    [CrossRef]
  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]
  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, H. Taniyama, "Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity," Nature Photonics 1, 49-52 (2006).
    [CrossRef]
  19. L. D. Landau and E. M. Lifshitz, Quantum mechanics, (Pergamon Press, 1958).

2007 (2)

2006 (5)

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]

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]

2005 (3)

2004 (1)

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]

2003 (1)

2002 (1)

2001 (1)

1993 (2)

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]

1991 (1)

M. Sumetskii, "Modeling of complicated nanometer resonant tunneling devices with quantum dots," J. Phys.: Condens. Matter,  3, 2651-2664 (1991).
[CrossRef]

1989 (1)

Aldridge, J.C.

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]

Anthes-Washburn, M.

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]

Arnold, S.

Asano, T.

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]

Ashkenazi,

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]

Boyd, R. W.

Chao, C.-Y.

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]

Chao, C-Y.

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]

Chbouki, N.

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]

Chu, S.

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]

Dandliker, R.

De Natale, P.

De Nicola, S.

Desai, T.A

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]

DiGiovanni, D. J.

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]

Driessen, A.

Dulashko, Y.

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]

Eggleton, B.

Fan, X.

Ferraro, P.

Fini, J. M.

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]

Fung, W.

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]

Gagliardi, G.

Gill, D.

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]

Goldberg, B.B.

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]

Greve, J.

Guo, L. J.

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]

Guo, L.J.

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]

Hale, A.

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]

Hanumegowda, N. M.

Heebner, J. E.

Hryniewicz, J.

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]

Keng, D.

King, O.

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]

Klunder, D. J. W.

Krioukov, E.

Ksendzov, A.

Kunishi, W.

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]

Kuramochi, E.

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]

Lin, Y.

Little, B.E.

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]

Maystre, F.

Noda, S.

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]

Noto, M.

Notomi, M.

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]

O’Donnell, M.

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]

Otto, C.

Popat, K.C.

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]

Shinya, A.

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]

Song, B.

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]

Sumetskii, M.

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]

M. Sumetskii, "Resistance resonances for resonant-tunneling structures of quantum dots," Phys. Rev. B,  48, 4586-4591 (1993).
[CrossRef]

M. Sumetskii, "Modeling of complicated nanometer resonant tunneling devices with quantum dots," J. Phys.: Condens. Matter,  3, 2651-2664 (1991).
[CrossRef]

Sumetsky, M.

Tanabe, T.

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]

Taniyama, H.

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]

Teraoka, I.

Ünlü, M.S.

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]

Van, V.

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]

Vollmer, F.

White, I. M.

Yalçin, A.

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]

Appl. Opt. (2)

Appl. Phys. Lett. (3)

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]

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]

IEEE J. Lightwave Technol. (1)

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]

IEEE J. Sel. Top. Quantum Electron. (2)

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]

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]

J. Phys.: Condens. Matter (1)

M. Sumetskii, "Modeling of complicated nanometer resonant tunneling devices with quantum dots," J. Phys.: Condens. Matter,  3, 2651-2664 (1991).
[CrossRef]

Nature Photonics (1)

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]

Opt. Express (2)

Opt. Lett. (5)

Phys. Rev. B (1)

M. Sumetskii, "Resistance resonances for resonant-tunneling structures of quantum dots," Phys. Rev. B,  48, 4586-4591 (1993).
[CrossRef]

Other (1)

L. D. Landau and E. M. Lifshitz, Quantum mechanics, (Pergamon Press, 1958).

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Figures (3)

Fig. 1.
Fig. 1.

(a) A model of ROS consisting of N elementary optical resonators coupled to each other. (b) A single resonance ROS. (c) A double resonance ROS.

Fig. 2.
Fig. 2.

The plots of resonance peak with maximum possible steepness and sharpness in transmission (a) and reflection (b) spectra.

Fig. 3.
Fig. 3.

Comparison of the optimized transmission spectra of a single and double resonance ROS shown in Fig. 1(b) (curves 1, blue) and in Fig. 1(c) (curves 2, red) having (a)-optimized slope and (b)-optimized sharpness.

Tables (1)

Tables Icon

Table 1. Parameters of (a) transmission and (b) reflection resonance peaks with the maximum possible Q-factor, height, sharpness, and slope.

Equations (20)

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

S max = max λ , γ var ( dP ( λ , γ ) d λ )
Θ max = max λ , γ var ( d 2 P ( λ , γ ) d λ 2 )
P pq = P p ( 0 ) m , n γ m ( p ) γ n ( q ) Q mn 2 .
Q = Λ 1 ,
Λ = ( λ λ 1 + i 2 γ 1 δ 12 δ 1 N δ 12 λ λ 2 + i 2 γ 2 δ 2 N δ N 1 δ N 2 λ λ N + i 2 γ N )
γ m = p γ m ( p )
P pp = P p ( 0 ) q p P pq
P 12 = P 1 ( 0 ) γ 1 ( 1 ) γ 1 ( 2 ) ( λ λ 1 ) 2 + 1 4 ( γ 1 ( 1 ) + γ 1 ( 2 ) + γ ) 2 ,
P 11 = P 1 ( 0 ) P 12 P 13 = P 1 ( 0 ) ( λ λ 1 ) 2 + 1 4 ( γ 1 ( 1 ) γ 1 ( 2 ) γ ) 2 ( λ λ 1 ) 2 + 1 4 ( γ 1 ( 1 ) + γ 1 ( 2 ) + γ ) 2
Q = λ 1 2 ( γ 1 ( 1 ) + γ 1 ( 2 ) + γ )
Q int = λ 1 2 γ .
max ( d P 12 d λ ) = P 1 ( 0 ) 3 3 2 γ = 0.192 γ P 1 ( 0 )
max ( d 2 P 12 d λ 2 ) = P 1 ( 0 ) 2 γ 2 = 0.5 γ 2 P 1 ( 0 )
max ( d P 11 d λ ) = 4 P 1 ( 0 ) 3 3 2 γ = 0.770 γ P 1 ( 0 )
max ( d 2 P 11 d λ 2 ) = 27 P 1 ( 0 ) 8 γ 2 = 3.375 γ 2 P 1 ( 0 )
P 12 = P 1 ( 0 ) [ ( λ λ 2 ) 2 + 1 4 γ 2 ] γ 1 ( 1 ) γ 1 ( 2 ) [ ( λ λ 1 ) ( λ λ 2 ) 1 4 γ 1 γ δ 12 2 ] 2 + 1 4 [ γ 1 ( λ λ 2 ) + γ ( λ λ 1 ) ] 2
P 12 = P 1 ( 0 ) 4 [ 4 ( λ λ 2 ) 2 + γ 2 ] 20 ( λ λ 2 ) 2 8 γ ( λ λ 2 ) + 17 γ 2 ,
max ( d P 12 d λ ) = 0.592 γ P 1 ( 0 ) .
P 12 = P 1 ( 0 ) 4 ( λ λ 2 ) 2 + γ 2 4 ( λ λ 2 ) 2 + ( γ + 2 δ 12 2 γ 1 ( 1 ) ) 2
max ( d 2 P 12 d λ 2 ) = 2 γ 2 P 1 ( 0 )

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