Abstract

A novel detection method enabled by electro-optically tunable waveguide-coupled surface plasmon resonance sensors is demonstrated. Both the WCSPR response of sensor and the interrogation light are varied simultaneously in this hybrid scheme. Modulation and demodulation of the sensor’s response are achieved by applying a high-frequency AC electrical signal and electrically filtering the detected signal. Scanning the incident angle at a lower speed yields an angular dependent response. Theoretical analyses and experimental results show that the angular-dependent signal is closely related to the derivative of the SPR reflectivity with a sharp, linear jump near the minimum of the SPR peak. Thus, simple linear-fitting and zero-finding algorithms can be used to locate the SPR angle, and sophisticated data processing algorithms and electronic hardware can be avoided.

© 2009 Optical Society of America

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  1. J. Homola, "Surface plasmon resonance sensors for detection of chemical and biological species," Chem. Rev. 108, 462-493 (2008).
    [CrossRef] [PubMed]
  2. K. Kukanskis, J. Elkind, J. Melendez, T. Murphy, and G. Miller, "Detection of DNA hybridization using the TISPR-1 surface plsmon resonance biosensor," Anal. Biochem. 274, 7-17 (1999).
    [CrossRef] [PubMed]
  3. A. A. Kolomenskii, P. D. Gershon, and H. A. Schuessler, "Sensitivity and detection limit of concentration and adsorption measurements by laser-induced surface plasmon resonance," Appl. Opt. 36, 6539-6547 (1997).
    [CrossRef]
  4. K. S. Johnson, S. S. Yee, and K. S. Boolsh, "Calibration of surface plasmon resonance refractometers using locally weighted parametric regression," Anal. Chem. 69, 1844-1851 (1997).
    [CrossRef]
  5. T. M. Chinowsky, L. S. Jung, and S. S. Yee, "Optimal linear data analysis for surface plasmon resonance biosensors," Sens. Actuators B 54, 89-97 (1999).
    [CrossRef]
  6. K. Kurihara, K. Nakamura, and K. Suzuki, "Asymmetric SPR sensor response curve-fitting equation for the accurate determination of SPR resonance angle," Sens. Actuators B 86, 49-57 (2002).
    [CrossRef]
  7. P. Tobiška and J. Homola, "Advanced data processing for SPR biosensors," Sens. Actuators B 107, 162-169 (2005).
    [CrossRef]
  8. C. Thirstrup and W. Zong, "Data analysis for surface plasmon resonance sensors using dynamic baseline algorithm," Sens. Actuators B 106, 796-802 (2005).
    [CrossRef]
  9. T. J. Wang, W. S. Lin, and F. K. Liu, "Integrated-optic biosensor by electro-optically modulated surface plasmon resonance," Bios. Bioelectron. 22, 1441-1446 (2007).
    [CrossRef]
  10. J. N. Yih, F. C. Chien, C. Y. Lin, H. F. Yau, and S. J. Chen, "Angular-interrogation attenuated total reflection metrology system for plasmonic sensors," Appl. Opt. 44, 6155-6162 (2005).
    [CrossRef] [PubMed]
  11. J. J. Chyou, C. S. Chua, Z. H. Shih, C. Y. Lin, K. T. Huang, S. J. Chen, and S. F. Shu, "High efficiency electro-optic polymer light modulator based on waveguide-coupled surface plasmon resonance," Proc. SPIE 5211, 197-206 (2003).
    [CrossRef]
  12. Y. Jiang, Z. Q. Cao, Q. S. Shen, X. M. Dou, and Y. L. Chen, "Improved attenuated-total-reflection technique for measuring the electro-optic coefficients of nonlinear optical polymers," J. Opt. Soc. Am. B 17, 805-808 (2000).
    [CrossRef]
  13. C. C. Teng and H. T. Man, "Simple reflection technique for measuring the electro-optic coefficient of poled polymers," Appl. Phys. Lett. 56, 1734-1736 (1990).
    [CrossRef]
  14. D. R. Lide, "Concentrative properties of aqueous solutions: density, refractive index, freezing point depression, and viscosity" in Handbook of chemistry and physics, (CRC press, Boca Raton, FL, 2005) pp. 8-65.
  15. J. J. Chyou, S. J. Chen, C.-S. Chu, Z. H. Shih, C. Y. Lin, and C. F. Shu, "Fabrication and metrology of E-O polymer light modulator based on waveguide-coupled surface plasmon resonance," Opt. Eng. 44, 034001-034007 (2005).
    [CrossRef]
  16. T. G. Wang and C. W. Hsieh, "Surface Plasmon resonance biosensor based on electro-optically modulated phase detection," Opt. Lett. 32, 2834-2836 (2007).
    [CrossRef] [PubMed]

2008

J. Homola, "Surface plasmon resonance sensors for detection of chemical and biological species," Chem. Rev. 108, 462-493 (2008).
[CrossRef] [PubMed]

2007

T. J. Wang, W. S. Lin, and F. K. Liu, "Integrated-optic biosensor by electro-optically modulated surface plasmon resonance," Bios. Bioelectron. 22, 1441-1446 (2007).
[CrossRef]

T. G. Wang and C. W. Hsieh, "Surface Plasmon resonance biosensor based on electro-optically modulated phase detection," Opt. Lett. 32, 2834-2836 (2007).
[CrossRef] [PubMed]

2005

J. N. Yih, F. C. Chien, C. Y. Lin, H. F. Yau, and S. J. Chen, "Angular-interrogation attenuated total reflection metrology system for plasmonic sensors," Appl. Opt. 44, 6155-6162 (2005).
[CrossRef] [PubMed]

J. J. Chyou, S. J. Chen, C.-S. Chu, Z. H. Shih, C. Y. Lin, and C. F. Shu, "Fabrication and metrology of E-O polymer light modulator based on waveguide-coupled surface plasmon resonance," Opt. Eng. 44, 034001-034007 (2005).
[CrossRef]

P. Tobiška and J. Homola, "Advanced data processing for SPR biosensors," Sens. Actuators B 107, 162-169 (2005).
[CrossRef]

C. Thirstrup and W. Zong, "Data analysis for surface plasmon resonance sensors using dynamic baseline algorithm," Sens. Actuators B 106, 796-802 (2005).
[CrossRef]

2003

J. J. Chyou, C. S. Chua, Z. H. Shih, C. Y. Lin, K. T. Huang, S. J. Chen, and S. F. Shu, "High efficiency electro-optic polymer light modulator based on waveguide-coupled surface plasmon resonance," Proc. SPIE 5211, 197-206 (2003).
[CrossRef]

2002

K. Kurihara, K. Nakamura, and K. Suzuki, "Asymmetric SPR sensor response curve-fitting equation for the accurate determination of SPR resonance angle," Sens. Actuators B 86, 49-57 (2002).
[CrossRef]

2000

1999

K. Kukanskis, J. Elkind, J. Melendez, T. Murphy, and G. Miller, "Detection of DNA hybridization using the TISPR-1 surface plsmon resonance biosensor," Anal. Biochem. 274, 7-17 (1999).
[CrossRef] [PubMed]

T. M. Chinowsky, L. S. Jung, and S. S. Yee, "Optimal linear data analysis for surface plasmon resonance biosensors," Sens. Actuators B 54, 89-97 (1999).
[CrossRef]

1997

K. S. Johnson, S. S. Yee, and K. S. Boolsh, "Calibration of surface plasmon resonance refractometers using locally weighted parametric regression," Anal. Chem. 69, 1844-1851 (1997).
[CrossRef]

A. A. Kolomenskii, P. D. Gershon, and H. A. Schuessler, "Sensitivity and detection limit of concentration and adsorption measurements by laser-induced surface plasmon resonance," Appl. Opt. 36, 6539-6547 (1997).
[CrossRef]

1990

C. C. Teng and H. T. Man, "Simple reflection technique for measuring the electro-optic coefficient of poled polymers," Appl. Phys. Lett. 56, 1734-1736 (1990).
[CrossRef]

Boolsh, K. S.

K. S. Johnson, S. S. Yee, and K. S. Boolsh, "Calibration of surface plasmon resonance refractometers using locally weighted parametric regression," Anal. Chem. 69, 1844-1851 (1997).
[CrossRef]

Cao, Z. Q.

Chen, S. J.

J. N. Yih, F. C. Chien, C. Y. Lin, H. F. Yau, and S. J. Chen, "Angular-interrogation attenuated total reflection metrology system for plasmonic sensors," Appl. Opt. 44, 6155-6162 (2005).
[CrossRef] [PubMed]

J. J. Chyou, S. J. Chen, C.-S. Chu, Z. H. Shih, C. Y. Lin, and C. F. Shu, "Fabrication and metrology of E-O polymer light modulator based on waveguide-coupled surface plasmon resonance," Opt. Eng. 44, 034001-034007 (2005).
[CrossRef]

J. J. Chyou, C. S. Chua, Z. H. Shih, C. Y. Lin, K. T. Huang, S. J. Chen, and S. F. Shu, "High efficiency electro-optic polymer light modulator based on waveguide-coupled surface plasmon resonance," Proc. SPIE 5211, 197-206 (2003).
[CrossRef]

Chen, Y. L.

Chien, F. C.

Chinowsky, T. M.

T. M. Chinowsky, L. S. Jung, and S. S. Yee, "Optimal linear data analysis for surface plasmon resonance biosensors," Sens. Actuators B 54, 89-97 (1999).
[CrossRef]

Chu, C.-S.

J. J. Chyou, S. J. Chen, C.-S. Chu, Z. H. Shih, C. Y. Lin, and C. F. Shu, "Fabrication and metrology of E-O polymer light modulator based on waveguide-coupled surface plasmon resonance," Opt. Eng. 44, 034001-034007 (2005).
[CrossRef]

Chua, C. S.

J. J. Chyou, C. S. Chua, Z. H. Shih, C. Y. Lin, K. T. Huang, S. J. Chen, and S. F. Shu, "High efficiency electro-optic polymer light modulator based on waveguide-coupled surface plasmon resonance," Proc. SPIE 5211, 197-206 (2003).
[CrossRef]

Chyou, J. J.

J. J. Chyou, S. J. Chen, C.-S. Chu, Z. H. Shih, C. Y. Lin, and C. F. Shu, "Fabrication and metrology of E-O polymer light modulator based on waveguide-coupled surface plasmon resonance," Opt. Eng. 44, 034001-034007 (2005).
[CrossRef]

J. J. Chyou, C. S. Chua, Z. H. Shih, C. Y. Lin, K. T. Huang, S. J. Chen, and S. F. Shu, "High efficiency electro-optic polymer light modulator based on waveguide-coupled surface plasmon resonance," Proc. SPIE 5211, 197-206 (2003).
[CrossRef]

Dou, X. M.

Elkind, J.

K. Kukanskis, J. Elkind, J. Melendez, T. Murphy, and G. Miller, "Detection of DNA hybridization using the TISPR-1 surface plsmon resonance biosensor," Anal. Biochem. 274, 7-17 (1999).
[CrossRef] [PubMed]

Gershon, P. D.

Homola, J.

J. Homola, "Surface plasmon resonance sensors for detection of chemical and biological species," Chem. Rev. 108, 462-493 (2008).
[CrossRef] [PubMed]

P. Tobiška and J. Homola, "Advanced data processing for SPR biosensors," Sens. Actuators B 107, 162-169 (2005).
[CrossRef]

Hsieh, C. W.

Huang, K. T.

J. J. Chyou, C. S. Chua, Z. H. Shih, C. Y. Lin, K. T. Huang, S. J. Chen, and S. F. Shu, "High efficiency electro-optic polymer light modulator based on waveguide-coupled surface plasmon resonance," Proc. SPIE 5211, 197-206 (2003).
[CrossRef]

Jiang, Y.

Johnson, K. S.

K. S. Johnson, S. S. Yee, and K. S. Boolsh, "Calibration of surface plasmon resonance refractometers using locally weighted parametric regression," Anal. Chem. 69, 1844-1851 (1997).
[CrossRef]

Jung, L. S.

T. M. Chinowsky, L. S. Jung, and S. S. Yee, "Optimal linear data analysis for surface plasmon resonance biosensors," Sens. Actuators B 54, 89-97 (1999).
[CrossRef]

Kolomenskii, A. A.

Kukanskis, K.

K. Kukanskis, J. Elkind, J. Melendez, T. Murphy, and G. Miller, "Detection of DNA hybridization using the TISPR-1 surface plsmon resonance biosensor," Anal. Biochem. 274, 7-17 (1999).
[CrossRef] [PubMed]

Kurihara, K.

K. Kurihara, K. Nakamura, and K. Suzuki, "Asymmetric SPR sensor response curve-fitting equation for the accurate determination of SPR resonance angle," Sens. Actuators B 86, 49-57 (2002).
[CrossRef]

Lin, C. Y.

J. J. Chyou, S. J. Chen, C.-S. Chu, Z. H. Shih, C. Y. Lin, and C. F. Shu, "Fabrication and metrology of E-O polymer light modulator based on waveguide-coupled surface plasmon resonance," Opt. Eng. 44, 034001-034007 (2005).
[CrossRef]

J. N. Yih, F. C. Chien, C. Y. Lin, H. F. Yau, and S. J. Chen, "Angular-interrogation attenuated total reflection metrology system for plasmonic sensors," Appl. Opt. 44, 6155-6162 (2005).
[CrossRef] [PubMed]

J. J. Chyou, C. S. Chua, Z. H. Shih, C. Y. Lin, K. T. Huang, S. J. Chen, and S. F. Shu, "High efficiency electro-optic polymer light modulator based on waveguide-coupled surface plasmon resonance," Proc. SPIE 5211, 197-206 (2003).
[CrossRef]

Lin, W. S.

T. J. Wang, W. S. Lin, and F. K. Liu, "Integrated-optic biosensor by electro-optically modulated surface plasmon resonance," Bios. Bioelectron. 22, 1441-1446 (2007).
[CrossRef]

Liu, F. K.

T. J. Wang, W. S. Lin, and F. K. Liu, "Integrated-optic biosensor by electro-optically modulated surface plasmon resonance," Bios. Bioelectron. 22, 1441-1446 (2007).
[CrossRef]

Man, H. T.

C. C. Teng and H. T. Man, "Simple reflection technique for measuring the electro-optic coefficient of poled polymers," Appl. Phys. Lett. 56, 1734-1736 (1990).
[CrossRef]

Melendez, J.

K. Kukanskis, J. Elkind, J. Melendez, T. Murphy, and G. Miller, "Detection of DNA hybridization using the TISPR-1 surface plsmon resonance biosensor," Anal. Biochem. 274, 7-17 (1999).
[CrossRef] [PubMed]

Miller, G.

K. Kukanskis, J. Elkind, J. Melendez, T. Murphy, and G. Miller, "Detection of DNA hybridization using the TISPR-1 surface plsmon resonance biosensor," Anal. Biochem. 274, 7-17 (1999).
[CrossRef] [PubMed]

Murphy, T.

K. Kukanskis, J. Elkind, J. Melendez, T. Murphy, and G. Miller, "Detection of DNA hybridization using the TISPR-1 surface plsmon resonance biosensor," Anal. Biochem. 274, 7-17 (1999).
[CrossRef] [PubMed]

Nakamura, K.

K. Kurihara, K. Nakamura, and K. Suzuki, "Asymmetric SPR sensor response curve-fitting equation for the accurate determination of SPR resonance angle," Sens. Actuators B 86, 49-57 (2002).
[CrossRef]

Schuessler, H. A.

Shen, Q. S.

Shih, Z. H.

J. J. Chyou, S. J. Chen, C.-S. Chu, Z. H. Shih, C. Y. Lin, and C. F. Shu, "Fabrication and metrology of E-O polymer light modulator based on waveguide-coupled surface plasmon resonance," Opt. Eng. 44, 034001-034007 (2005).
[CrossRef]

J. J. Chyou, C. S. Chua, Z. H. Shih, C. Y. Lin, K. T. Huang, S. J. Chen, and S. F. Shu, "High efficiency electro-optic polymer light modulator based on waveguide-coupled surface plasmon resonance," Proc. SPIE 5211, 197-206 (2003).
[CrossRef]

Shu, C. F.

J. J. Chyou, S. J. Chen, C.-S. Chu, Z. H. Shih, C. Y. Lin, and C. F. Shu, "Fabrication and metrology of E-O polymer light modulator based on waveguide-coupled surface plasmon resonance," Opt. Eng. 44, 034001-034007 (2005).
[CrossRef]

Shu, S. F.

J. J. Chyou, C. S. Chua, Z. H. Shih, C. Y. Lin, K. T. Huang, S. J. Chen, and S. F. Shu, "High efficiency electro-optic polymer light modulator based on waveguide-coupled surface plasmon resonance," Proc. SPIE 5211, 197-206 (2003).
[CrossRef]

Suzuki, K.

K. Kurihara, K. Nakamura, and K. Suzuki, "Asymmetric SPR sensor response curve-fitting equation for the accurate determination of SPR resonance angle," Sens. Actuators B 86, 49-57 (2002).
[CrossRef]

Teng, C. C.

C. C. Teng and H. T. Man, "Simple reflection technique for measuring the electro-optic coefficient of poled polymers," Appl. Phys. Lett. 56, 1734-1736 (1990).
[CrossRef]

Thirstrup, C.

C. Thirstrup and W. Zong, "Data analysis for surface plasmon resonance sensors using dynamic baseline algorithm," Sens. Actuators B 106, 796-802 (2005).
[CrossRef]

Tobiška, P.

P. Tobiška and J. Homola, "Advanced data processing for SPR biosensors," Sens. Actuators B 107, 162-169 (2005).
[CrossRef]

Wang, T. G.

Wang, T. J.

T. J. Wang, W. S. Lin, and F. K. Liu, "Integrated-optic biosensor by electro-optically modulated surface plasmon resonance," Bios. Bioelectron. 22, 1441-1446 (2007).
[CrossRef]

Yau, H. F.

Yee, S. S.

T. M. Chinowsky, L. S. Jung, and S. S. Yee, "Optimal linear data analysis for surface plasmon resonance biosensors," Sens. Actuators B 54, 89-97 (1999).
[CrossRef]

K. S. Johnson, S. S. Yee, and K. S. Boolsh, "Calibration of surface plasmon resonance refractometers using locally weighted parametric regression," Anal. Chem. 69, 1844-1851 (1997).
[CrossRef]

Yih, J. N.

Zong, W.

C. Thirstrup and W. Zong, "Data analysis for surface plasmon resonance sensors using dynamic baseline algorithm," Sens. Actuators B 106, 796-802 (2005).
[CrossRef]

Anal. Biochem.

K. Kukanskis, J. Elkind, J. Melendez, T. Murphy, and G. Miller, "Detection of DNA hybridization using the TISPR-1 surface plsmon resonance biosensor," Anal. Biochem. 274, 7-17 (1999).
[CrossRef] [PubMed]

Anal. Chem.

K. S. Johnson, S. S. Yee, and K. S. Boolsh, "Calibration of surface plasmon resonance refractometers using locally weighted parametric regression," Anal. Chem. 69, 1844-1851 (1997).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

C. C. Teng and H. T. Man, "Simple reflection technique for measuring the electro-optic coefficient of poled polymers," Appl. Phys. Lett. 56, 1734-1736 (1990).
[CrossRef]

Bios. Bioelectron.

T. J. Wang, W. S. Lin, and F. K. Liu, "Integrated-optic biosensor by electro-optically modulated surface plasmon resonance," Bios. Bioelectron. 22, 1441-1446 (2007).
[CrossRef]

Chem. Rev.

J. Homola, "Surface plasmon resonance sensors for detection of chemical and biological species," Chem. Rev. 108, 462-493 (2008).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B

Opt. Eng.

J. J. Chyou, S. J. Chen, C.-S. Chu, Z. H. Shih, C. Y. Lin, and C. F. Shu, "Fabrication and metrology of E-O polymer light modulator based on waveguide-coupled surface plasmon resonance," Opt. Eng. 44, 034001-034007 (2005).
[CrossRef]

Opt. Lett.

Proc. SPIE

J. J. Chyou, C. S. Chua, Z. H. Shih, C. Y. Lin, K. T. Huang, S. J. Chen, and S. F. Shu, "High efficiency electro-optic polymer light modulator based on waveguide-coupled surface plasmon resonance," Proc. SPIE 5211, 197-206 (2003).
[CrossRef]

Sens. Actuators B

T. M. Chinowsky, L. S. Jung, and S. S. Yee, "Optimal linear data analysis for surface plasmon resonance biosensors," Sens. Actuators B 54, 89-97 (1999).
[CrossRef]

K. Kurihara, K. Nakamura, and K. Suzuki, "Asymmetric SPR sensor response curve-fitting equation for the accurate determination of SPR resonance angle," Sens. Actuators B 86, 49-57 (2002).
[CrossRef]

P. Tobiška and J. Homola, "Advanced data processing for SPR biosensors," Sens. Actuators B 107, 162-169 (2005).
[CrossRef]

C. Thirstrup and W. Zong, "Data analysis for surface plasmon resonance sensors using dynamic baseline algorithm," Sens. Actuators B 106, 796-802 (2005).
[CrossRef]

Other

D. R. Lide, "Concentrative properties of aqueous solutions: density, refractive index, freezing point depression, and viscosity" in Handbook of chemistry and physics, (CRC press, Boca Raton, FL, 2005) pp. 8-65.

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Figures (8)

Fig. 1.
Fig. 1.

Experimental setup?D, photo detector; P, ZF7 prism; S, ZF7 substrate glass; U, upper gold layer; L, lower gold layer; W, EO waveguide layer)

Fig. 2.
Fig. 2.

Angular dependent response of the WCSPR sensor for water.

Fig. 3.
Fig. 3.

Reflectivity measurements under different glucose solutions using the conventional angular interrogation method

Fig. 4.
Fig. 4.

(a) Angle shift based conventional angular interrogation for different voltages ; (b) SPR Angle shift with voltage variation

Fig. 5.
Fig. 5.

The voltage modulation curve at the fixed angle 34.478°

Fig. 6.
Fig. 6.

Experimental results of R and the modulated power I and the differential from R to θ versus incident angle for 1% solution

Fig. 7.
Fig. 7.

The angle shift versus the refractive index of the analytes

Fig. 8.
Fig. 8.

Measurement of the WCSPR angle based on different interrogation schemes and data processing algorithms.

Equations (18)

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

R = r 1,5 2 r i , 5 = r i , i + 1 + r i + 1,5 e 2 j d i + 1 k z , i + 1 1 + r i , i + 1 r i + 1,5 e 2 j d i + 1 k z , i + 1 ( i = 3,2,1 ; j = 1 ) r i , i + 1 = n i + 1 2 k z , i + 1 n i 2 k z , i n i + 1 2 k z , i + 1 + n i 2 k z , i ( i = 1,2,3,4 ) k zi = ( 2 π λ ) 2 n i 2 k 0 x 2 ( i = 1,2,3,4,5 ) k 0 x = 2 π λ n 1 sin θ }
R = R ( θ , V )
V = V 0 sin ( ω 0 t )
g ( V 0 sin ω 0 t ) = n = 0 V 0 2 n g ( 2 n ) ( 0 ) 4 n ( n ! ) 2
+ k = 1 n ( 1 ) k cos ( 2 k ω 0 t ) 2 n = k V 0 2 n g ( 2 n ) ( 0 ) 4 n ( n + k ) ! ( n k ) !
+ k = 0 n ( 1 ) k sin ( ( 2 k + 1 ) ω 0 t ) 2 n = k V 0 2 n + 1 g ( 2 n + 1 ) ( 0 ) 4 n ( n + 1 + k ) ! ( n k ) !
R ( θ , V 0 sin ω 0 t ) = n = 0 V 0 2 n R ( 2 n ) ( θ , 0 ) 4 n ( n ! ) 2
+ k = 1 n ( 1 ) k cos ( 2 k ω 0 t ) 2 n = k V 0 2 n R ( 2 n ) ( θ , 0 ) 4 n ( n + k ) ! ( n k ) !
+ k = 0 n ( 1 ) k sin ( ( 2 k + 1 ) ω 0 t ) 2 n = k V 0 2 n + 1 R ( 2 n + 1 ) ( θ , 0 ) 4 n ( n + 1 + k ) ! ( n k ) !
F ( θ , V 0 , ω 0 ) = [ R ( θ , 0 ) + V 0 2 4 2 R V 2 V = 0 + V 0 4 64 4 R V 4 V = 0 + ]
+ [ V 0 1 ! R V V = 0 + V 0 3 8 3 R V 3 V = 0 + V 0 5 192 5 R V 5 V = 0 + ] × πi 2 δ ( ω ω 0 )
+ j = 2 A j δ ( ω j ω 0 ) +
I = n = 0 V 0 2 n + 1 4 n n ! ( n + 1 ) ! 2 n + 1 R V 2 n + 1 V = 0 = V 0 R V V = 0 + V 0 3 8 3 R V 3 V = 0 +
f = V 0 2 8 r
V 0 = 8 f r
V 0 = 458.83 f
R ' ( θ , δV ) = R ( θ δθ , 0 ) = R ( θ βδV , 0 )
R ' ( θ 0 , V ) V V = 0 = β R ( θ , 0 ) θ θ = θ 0

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