Abstract

A novel differential phase detecting surface plasmon resonance (SPR) sensor based on white-light spectral interferometry is presented. Our proposed scheme employs a white-light source for SPR excitation and measures the corresponding SPR phase change at the optimized coupling wavelength with fixed angle of incidence across the visible spectrum. Compared to existing laser based phase detecting schemes, this system offers optimal sensitivity and extended dynamic range of measurement without any compromise in phase detection resolution. Results obtained from sodium chloride solutions indicate that the detection limit is 2.6 × 10−7 RIU over a refractive index range of 10−2 RIU, which is considerably wider than that achievable by existing laser based approach, thus making our scheme very attractive for practical SPR sensing applications.

© 2011 OSA

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References

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  1. J. Homola, Surface Plasmon Resonance Based Sensors, Springer Series on Chemical Sensors and Biosensors (Springer-Verlag, 2006).
  2. M. Piliarik and J. Homola, “Surface plasmon resonance (SPR) sensors: approaching their limits?” Opt. Express 17(19), 16505–16517 (2009).
    [CrossRef] [PubMed]
  3. S. Y. Wu, H. P. Ho, W. C. Law, C. Lin, and S. K. Kong, “Highly sensitive differential phase-sensitive surface plasmon resonance biosensor based on the Mach-Zehnder configuration,” Opt. Lett. 29(20), 2378–2380 (2004).
    [CrossRef] [PubMed]
  4. A. V. Kabashin and P. I. Nikitin, “Surface plasmon resonance interferometer for bio- and chemical-sensors,” Opt. Commun. 150(1-6), 5–8 (1998).
    [CrossRef]
  5. A. V. Kabashin, S. Patskovsky, and A. N. Grigorenko, “Phase and amplitude sensitivities in surface plasmon resonance bio and chemical sensing,” Opt. Express 17(23), 21191–21204 (2009).
    [CrossRef] [PubMed]
  6. W. C. Law, P. Markowicz, K. T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 (2007).
    [CrossRef] [PubMed]
  7. P. P. Markowicz, W. C. Law, A. Baev, P. N. Prasad, S. Patskovsky, and A. V. Kabashin, “Phase-sensitive time-modulated surface plasmon resonance polarimetry for wide dynamic range biosensing,” Opt. Express 15(4), 1745–1754 (2007).
    [CrossRef] [PubMed]
  8. S. P. Ng, C. M. L. Wu, S. Y. Wu, H. P. Ho, and S. K. Kong, “Differential spectral phase interferometry for wide dynamic range surface plasmon resonance biosensing,” Biosens. Bioelectron. 26(4), 1593–1598 (2010).
    [CrossRef] [PubMed]
  9. S. P. Ng, S. Y. Wu, H. P. Ho, and C. M. L. Wu, “A white-light interferometric surface plasmon resonance sensor with wide dynamic range and phase-sensitive response,” IEEE International Conference on Electron Devices and Solid-State Circuits, December 2008, HKSAR.
  10. G. G. Nenninger, M. Piliarik, and J. Homola, “Data analysis for optical sensors based on spectroscopy of surface plasmons,” Meas. Sci. Technol. 13(12), 2038–2046 (2002).
    [CrossRef]
  11. L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, 1995).
  12. P. Hlubina, D. Ciprian, and J. Lunacek, “Spectral interferometric technique to measure the ellipsometric phase of a thin-film structure,” Opt. Lett. 34(17), 2661–2663 (2009).
    [CrossRef] [PubMed]
  13. http://refractiveindex.info/
  14. D. R. Lide ed., CRC Handbook of Chemistry and Physics, 90th ed. (CRC Press, 2010).
  15. W. Yuan, H. P. Ho, C. L. Wong, S. K. Kong, and C. Lin, “Surface plasmon resonance biosensor incorporated in a Michelson interferometer with enhanced sensitivity,” IEEE Sens. J. 7(1), 70–73 (2007).
    [CrossRef]
  16. H. P. Ho, S. Y. Wu, M. Yang, and A. C. Cheung, “Application of white light-emitting diode to surface plasmon resonance sensors,” Sens. Actuators B Chem. 80(2), 89–94 (2001).
    [CrossRef]
  17. P. Hlubina, J. Lunacek, D. Ciprian, and R. Chlebus, “Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals,” Opt. Commun. 281(9), 2349–2354 (2008).
    [CrossRef]

2010

S. P. Ng, C. M. L. Wu, S. Y. Wu, H. P. Ho, and S. K. Kong, “Differential spectral phase interferometry for wide dynamic range surface plasmon resonance biosensing,” Biosens. Bioelectron. 26(4), 1593–1598 (2010).
[CrossRef] [PubMed]

2009

2008

P. Hlubina, J. Lunacek, D. Ciprian, and R. Chlebus, “Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals,” Opt. Commun. 281(9), 2349–2354 (2008).
[CrossRef]

2007

W. C. Law, P. Markowicz, K. T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 (2007).
[CrossRef] [PubMed]

P. P. Markowicz, W. C. Law, A. Baev, P. N. Prasad, S. Patskovsky, and A. V. Kabashin, “Phase-sensitive time-modulated surface plasmon resonance polarimetry for wide dynamic range biosensing,” Opt. Express 15(4), 1745–1754 (2007).
[CrossRef] [PubMed]

W. Yuan, H. P. Ho, C. L. Wong, S. K. Kong, and C. Lin, “Surface plasmon resonance biosensor incorporated in a Michelson interferometer with enhanced sensitivity,” IEEE Sens. J. 7(1), 70–73 (2007).
[CrossRef]

2004

2002

G. G. Nenninger, M. Piliarik, and J. Homola, “Data analysis for optical sensors based on spectroscopy of surface plasmons,” Meas. Sci. Technol. 13(12), 2038–2046 (2002).
[CrossRef]

2001

H. P. Ho, S. Y. Wu, M. Yang, and A. C. Cheung, “Application of white light-emitting diode to surface plasmon resonance sensors,” Sens. Actuators B Chem. 80(2), 89–94 (2001).
[CrossRef]

1998

A. V. Kabashin and P. I. Nikitin, “Surface plasmon resonance interferometer for bio- and chemical-sensors,” Opt. Commun. 150(1-6), 5–8 (1998).
[CrossRef]

Baev, A.

W. C. Law, P. Markowicz, K. T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 (2007).
[CrossRef] [PubMed]

P. P. Markowicz, W. C. Law, A. Baev, P. N. Prasad, S. Patskovsky, and A. V. Kabashin, “Phase-sensitive time-modulated surface plasmon resonance polarimetry for wide dynamic range biosensing,” Opt. Express 15(4), 1745–1754 (2007).
[CrossRef] [PubMed]

Cheung, A. C.

H. P. Ho, S. Y. Wu, M. Yang, and A. C. Cheung, “Application of white light-emitting diode to surface plasmon resonance sensors,” Sens. Actuators B Chem. 80(2), 89–94 (2001).
[CrossRef]

Chlebus, R.

P. Hlubina, J. Lunacek, D. Ciprian, and R. Chlebus, “Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals,” Opt. Commun. 281(9), 2349–2354 (2008).
[CrossRef]

Ciprian, D.

P. Hlubina, D. Ciprian, and J. Lunacek, “Spectral interferometric technique to measure the ellipsometric phase of a thin-film structure,” Opt. Lett. 34(17), 2661–2663 (2009).
[CrossRef] [PubMed]

P. Hlubina, J. Lunacek, D. Ciprian, and R. Chlebus, “Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals,” Opt. Commun. 281(9), 2349–2354 (2008).
[CrossRef]

Grigorenko, A. N.

Hlubina, P.

P. Hlubina, D. Ciprian, and J. Lunacek, “Spectral interferometric technique to measure the ellipsometric phase of a thin-film structure,” Opt. Lett. 34(17), 2661–2663 (2009).
[CrossRef] [PubMed]

P. Hlubina, J. Lunacek, D. Ciprian, and R. Chlebus, “Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals,” Opt. Commun. 281(9), 2349–2354 (2008).
[CrossRef]

Ho, H. P.

S. P. Ng, C. M. L. Wu, S. Y. Wu, H. P. Ho, and S. K. Kong, “Differential spectral phase interferometry for wide dynamic range surface plasmon resonance biosensing,” Biosens. Bioelectron. 26(4), 1593–1598 (2010).
[CrossRef] [PubMed]

W. C. Law, P. Markowicz, K. T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 (2007).
[CrossRef] [PubMed]

W. Yuan, H. P. Ho, C. L. Wong, S. K. Kong, and C. Lin, “Surface plasmon resonance biosensor incorporated in a Michelson interferometer with enhanced sensitivity,” IEEE Sens. J. 7(1), 70–73 (2007).
[CrossRef]

S. Y. Wu, H. P. Ho, W. C. Law, C. Lin, and S. K. Kong, “Highly sensitive differential phase-sensitive surface plasmon resonance biosensor based on the Mach-Zehnder configuration,” Opt. Lett. 29(20), 2378–2380 (2004).
[CrossRef] [PubMed]

H. P. Ho, S. Y. Wu, M. Yang, and A. C. Cheung, “Application of white light-emitting diode to surface plasmon resonance sensors,” Sens. Actuators B Chem. 80(2), 89–94 (2001).
[CrossRef]

Homola, J.

M. Piliarik and J. Homola, “Surface plasmon resonance (SPR) sensors: approaching their limits?” Opt. Express 17(19), 16505–16517 (2009).
[CrossRef] [PubMed]

G. G. Nenninger, M. Piliarik, and J. Homola, “Data analysis for optical sensors based on spectroscopy of surface plasmons,” Meas. Sci. Technol. 13(12), 2038–2046 (2002).
[CrossRef]

Kabashin, A. V.

A. V. Kabashin, S. Patskovsky, and A. N. Grigorenko, “Phase and amplitude sensitivities in surface plasmon resonance bio and chemical sensing,” Opt. Express 17(23), 21191–21204 (2009).
[CrossRef] [PubMed]

W. C. Law, P. Markowicz, K. T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 (2007).
[CrossRef] [PubMed]

P. P. Markowicz, W. C. Law, A. Baev, P. N. Prasad, S. Patskovsky, and A. V. Kabashin, “Phase-sensitive time-modulated surface plasmon resonance polarimetry for wide dynamic range biosensing,” Opt. Express 15(4), 1745–1754 (2007).
[CrossRef] [PubMed]

A. V. Kabashin and P. I. Nikitin, “Surface plasmon resonance interferometer for bio- and chemical-sensors,” Opt. Commun. 150(1-6), 5–8 (1998).
[CrossRef]

Kong, S. K.

S. P. Ng, C. M. L. Wu, S. Y. Wu, H. P. Ho, and S. K. Kong, “Differential spectral phase interferometry for wide dynamic range surface plasmon resonance biosensing,” Biosens. Bioelectron. 26(4), 1593–1598 (2010).
[CrossRef] [PubMed]

W. Yuan, H. P. Ho, C. L. Wong, S. K. Kong, and C. Lin, “Surface plasmon resonance biosensor incorporated in a Michelson interferometer with enhanced sensitivity,” IEEE Sens. J. 7(1), 70–73 (2007).
[CrossRef]

S. Y. Wu, H. P. Ho, W. C. Law, C. Lin, and S. K. Kong, “Highly sensitive differential phase-sensitive surface plasmon resonance biosensor based on the Mach-Zehnder configuration,” Opt. Lett. 29(20), 2378–2380 (2004).
[CrossRef] [PubMed]

Law, W. C.

Lin, C.

W. Yuan, H. P. Ho, C. L. Wong, S. K. Kong, and C. Lin, “Surface plasmon resonance biosensor incorporated in a Michelson interferometer with enhanced sensitivity,” IEEE Sens. J. 7(1), 70–73 (2007).
[CrossRef]

S. Y. Wu, H. P. Ho, W. C. Law, C. Lin, and S. K. Kong, “Highly sensitive differential phase-sensitive surface plasmon resonance biosensor based on the Mach-Zehnder configuration,” Opt. Lett. 29(20), 2378–2380 (2004).
[CrossRef] [PubMed]

Lunacek, J.

P. Hlubina, D. Ciprian, and J. Lunacek, “Spectral interferometric technique to measure the ellipsometric phase of a thin-film structure,” Opt. Lett. 34(17), 2661–2663 (2009).
[CrossRef] [PubMed]

P. Hlubina, J. Lunacek, D. Ciprian, and R. Chlebus, “Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals,” Opt. Commun. 281(9), 2349–2354 (2008).
[CrossRef]

Markowicz, P.

W. C. Law, P. Markowicz, K. T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 (2007).
[CrossRef] [PubMed]

Markowicz, P. P.

Nenninger, G. G.

G. G. Nenninger, M. Piliarik, and J. Homola, “Data analysis for optical sensors based on spectroscopy of surface plasmons,” Meas. Sci. Technol. 13(12), 2038–2046 (2002).
[CrossRef]

Ng, S. P.

S. P. Ng, C. M. L. Wu, S. Y. Wu, H. P. Ho, and S. K. Kong, “Differential spectral phase interferometry for wide dynamic range surface plasmon resonance biosensing,” Biosens. Bioelectron. 26(4), 1593–1598 (2010).
[CrossRef] [PubMed]

Nikitin, P. I.

A. V. Kabashin and P. I. Nikitin, “Surface plasmon resonance interferometer for bio- and chemical-sensors,” Opt. Commun. 150(1-6), 5–8 (1998).
[CrossRef]

Patskovsky, S.

Piliarik, M.

M. Piliarik and J. Homola, “Surface plasmon resonance (SPR) sensors: approaching their limits?” Opt. Express 17(19), 16505–16517 (2009).
[CrossRef] [PubMed]

G. G. Nenninger, M. Piliarik, and J. Homola, “Data analysis for optical sensors based on spectroscopy of surface plasmons,” Meas. Sci. Technol. 13(12), 2038–2046 (2002).
[CrossRef]

Prasad, P. N.

P. P. Markowicz, W. C. Law, A. Baev, P. N. Prasad, S. Patskovsky, and A. V. Kabashin, “Phase-sensitive time-modulated surface plasmon resonance polarimetry for wide dynamic range biosensing,” Opt. Express 15(4), 1745–1754 (2007).
[CrossRef] [PubMed]

W. C. Law, P. Markowicz, K. T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 (2007).
[CrossRef] [PubMed]

Roy, I.

W. C. Law, P. Markowicz, K. T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 (2007).
[CrossRef] [PubMed]

Wong, C. L.

W. Yuan, H. P. Ho, C. L. Wong, S. K. Kong, and C. Lin, “Surface plasmon resonance biosensor incorporated in a Michelson interferometer with enhanced sensitivity,” IEEE Sens. J. 7(1), 70–73 (2007).
[CrossRef]

Wu, C. M. L.

S. P. Ng, C. M. L. Wu, S. Y. Wu, H. P. Ho, and S. K. Kong, “Differential spectral phase interferometry for wide dynamic range surface plasmon resonance biosensing,” Biosens. Bioelectron. 26(4), 1593–1598 (2010).
[CrossRef] [PubMed]

Wu, S. Y.

S. P. Ng, C. M. L. Wu, S. Y. Wu, H. P. Ho, and S. K. Kong, “Differential spectral phase interferometry for wide dynamic range surface plasmon resonance biosensing,” Biosens. Bioelectron. 26(4), 1593–1598 (2010).
[CrossRef] [PubMed]

S. Y. Wu, H. P. Ho, W. C. Law, C. Lin, and S. K. Kong, “Highly sensitive differential phase-sensitive surface plasmon resonance biosensor based on the Mach-Zehnder configuration,” Opt. Lett. 29(20), 2378–2380 (2004).
[CrossRef] [PubMed]

H. P. Ho, S. Y. Wu, M. Yang, and A. C. Cheung, “Application of white light-emitting diode to surface plasmon resonance sensors,” Sens. Actuators B Chem. 80(2), 89–94 (2001).
[CrossRef]

Yang, M.

H. P. Ho, S. Y. Wu, M. Yang, and A. C. Cheung, “Application of white light-emitting diode to surface plasmon resonance sensors,” Sens. Actuators B Chem. 80(2), 89–94 (2001).
[CrossRef]

Yong, K. T.

W. C. Law, P. Markowicz, K. T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 (2007).
[CrossRef] [PubMed]

Yuan, W.

W. Yuan, H. P. Ho, C. L. Wong, S. K. Kong, and C. Lin, “Surface plasmon resonance biosensor incorporated in a Michelson interferometer with enhanced sensitivity,” IEEE Sens. J. 7(1), 70–73 (2007).
[CrossRef]

Biosens. Bioelectron.

W. C. Law, P. Markowicz, K. T. Yong, I. Roy, A. Baev, S. Patskovsky, A. V. Kabashin, H. P. Ho, and P. N. Prasad, “Wide dynamic range phase-sensitive surface plasmon resonance biosensor based on measuring the modulation harmonics,” Biosens. Bioelectron. 23(5), 627–632 (2007).
[CrossRef] [PubMed]

S. P. Ng, C. M. L. Wu, S. Y. Wu, H. P. Ho, and S. K. Kong, “Differential spectral phase interferometry for wide dynamic range surface plasmon resonance biosensing,” Biosens. Bioelectron. 26(4), 1593–1598 (2010).
[CrossRef] [PubMed]

IEEE Sens. J.

W. Yuan, H. P. Ho, C. L. Wong, S. K. Kong, and C. Lin, “Surface plasmon resonance biosensor incorporated in a Michelson interferometer with enhanced sensitivity,” IEEE Sens. J. 7(1), 70–73 (2007).
[CrossRef]

Meas. Sci. Technol.

G. G. Nenninger, M. Piliarik, and J. Homola, “Data analysis for optical sensors based on spectroscopy of surface plasmons,” Meas. Sci. Technol. 13(12), 2038–2046 (2002).
[CrossRef]

Opt. Commun.

P. Hlubina, J. Lunacek, D. Ciprian, and R. Chlebus, “Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals,” Opt. Commun. 281(9), 2349–2354 (2008).
[CrossRef]

A. V. Kabashin and P. I. Nikitin, “Surface plasmon resonance interferometer for bio- and chemical-sensors,” Opt. Commun. 150(1-6), 5–8 (1998).
[CrossRef]

Opt. Express

Opt. Lett.

Sens. Actuators B Chem.

H. P. Ho, S. Y. Wu, M. Yang, and A. C. Cheung, “Application of white light-emitting diode to surface plasmon resonance sensors,” Sens. Actuators B Chem. 80(2), 89–94 (2001).
[CrossRef]

Other

J. Homola, Surface Plasmon Resonance Based Sensors, Springer Series on Chemical Sensors and Biosensors (Springer-Verlag, 2006).

http://refractiveindex.info/

D. R. Lide ed., CRC Handbook of Chemistry and Physics, 90th ed. (CRC Press, 2010).

L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, 1995).

S. P. Ng, S. Y. Wu, H. P. Ho, and C. M. L. Wu, “A white-light interferometric surface plasmon resonance sensor with wide dynamic range and phase-sensitive response,” IEEE International Conference on Electron Devices and Solid-State Circuits, December 2008, HKSAR.

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Figures (6)

Fig. 1
Fig. 1

(a) (Color online) Spectral reflectance and phase of p-polarization obtained from Fresnel equations; 1(b) simulated zero mean spectral interferogram based on optical path difference of 140 microns and resonance data of Fig. 1(a).

Fig. 2
Fig. 2

(a) (Color online) Spectral phase response of selected wavelength with increment NaCl concentration; 2(b) estimated limit of detection of our white-light spectral interferometer based on simulated data from Fresnel equations [9] with 0.01° stability and reference [14].

Fig. 3
Fig. 3

(Color online) Setup of our white-light SPR spectral interferometer for differential phase measurement: 1, 40W warm-white light emitting diode (LED); 2, collimator; 3, broadband linear polarizer oriented 45° to the optical axis; 4, broadband non-polarizing beamsplitter; 5, SPR probe cell which is attached to the peristaltic pump; 6 and 8, λ/20 high precision mirrors; 7, SPR reference cell which is filled with distilled water; 9, broadband polarizing beamsplitter which divides the two polarizations into different path; 10 and 11, collimators that collect the interference fringes into corresponding channel; 12, dual channel spectrometer; 13, personal computer for signal processing and data storage.

Fig. 4
Fig. 4

(Color online) Experimental spectral interference fringes acquired at (a) the presence of distilled water and (b) after injection of 10% NaCl solution into the probe cell, the dip in TM (p-polarized) spectral fringe amplitude and corresponding phase shift between TM and TE (s-polarized) fringes imply the optimized SPR occurs at about 646nm with water and it was shifted to about 675nm with 10% NaCl solution.

Fig. 5
Fig. 5

(a) (Color online) shows the differential phases of A: distilled water, B: 2% NaCl, C: 4% NaCl, D: 6% NaCl, E: 8% NaCl and F: 10% NaCl solutions; (b) is the corresponding derivatives of A to F with respect to wavelength which shows the maximum differential phase occurs at different wavelengths as A’ to F’.

Fig. 6
Fig. 6

(a) (Color online) illustrates the nonlinear differential phase response of G: 645.15nm, H: 648.35nm, I: 653.75nm, J: 659.05nm, K: 663.65nm, L: 667.95nm; (b) shows the calculated Limit of Detection (LOD) of our system within the region of linear phase response of each wavelength over the 10% NaCl concentration (10−2 RIU).

Equations (3)

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

r p,s ( λ ) = R p,s ( λ ) exp [ i φ p,s ( λ ) ] ,
{ S p ( λ ) = V p ( λ ) cos [ φ air + φ p ( λ ) + φ noise ] , S s ( λ ) = V s ( λ ) cos [ φ air + φ s ( λ ) + φ noise ] ,
σ R I = δ n δ Y     σ S D ,

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