Abstract

We have experimentally and theoretically compared intensity and phase measurements of surface plasmon resonance, in order to check the widely claimed superiority of the phase method. Both experiment and theory show a slightly higher sensitivity for intensity detection. Simulations confirm that this result is generally true for a wide range of resonance conditions. The basic reason is that phase measurements must be performed by measuring light intensities in some way, and therefore both modes of operation are limited in similar ways by photon statistics. Sensitivity can only be improved by using larger light intensities.

© 2006 Optical Society of America

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2005

Y. Xinglong, W. Dingxin, W. Xing, D. Xiang, L. Wei and Z. Xinsheng, "A surface plasmon resonance imaging interferometry for protein micro-array detection," Sens. Actuators. B 108, 765- 771 (2005).
[CrossRef]

Y. D. Su, S. J. Chen and T. L. Yen, "Common path phase shift interferometry surface plasmon resonance imaging system," Opt. Lett. 30, 1488-1490 (2005).
[CrossRef] [PubMed]

2004

2003

Y. Xinglong, W. Dingxin, Y. Zibo, "Simulation and analysis of surface plasmon resonance biosensor based on phase detection," Sens. Actuators. B 91, 285-290 (2003).
[CrossRef]

C. M. Wu, Z. C. Jian, S. F. Joe, L. B. Chang, "High-sensitivity sensor based on surface Plasmon resonance and heterodyne interferometry," Sens. Actuators. B 92, 133-136 (2003).
[CrossRef]

2000

A. N. Grigorenko, A. A. Beloglazov, P. I. Nikitin, C. Kuhne, G. Steiner and R. Salzer, "Dark-field surface plasmon resonance microscopy," Opt. Commun. 174, 151-155 (2000).
[CrossRef]

P. I. Nikitin, A. N. Grigorenko, A. A. Beloglazov, M. V. Valeiko, A. I. Savchuk and O. A. Savchuk, "Surface Plasmon Resonance Interferometry for Micro-Array Biosensing," Sens. Actuators. 85, 189-193 (2000).
[CrossRef]

1999

A. N. Grigorenko, P. I. Nikitin and A. V. Kabashin," Phase jumps and interferometric surface plasmon resonance imaging," Appl. Phys. Lett. 75, 3917-3919 (1999).
[CrossRef]

P. I. Nikitin, A. A. Beloglazov, V. E. Kochergin, M. V. Valeiko, T. I. Ksenevich, "Surface plasmon resonance interferometry for biological and chemical sensing," Sens. Actuators. B 54, 43-50 (1999).
[CrossRef]

A. V. Kabashin, V. E. Kochergin and P. I. Nikitin, "Surface plasmon resonance bio- and chemical sensors with phase-polarization contrast," Sens. Actuators. B 54, 51-56 (1999).
[CrossRef]

J. Homola, S. S. Yee, G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Actuators B 54, 3-15 (1999).
[CrossRef]

1998

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

V. E. Kochergin, A. A. Beloglazov, M. V. Valeiko, P. I. Nikitin, "Phase properties of a surface-plasmon resonance from the viewpoint of sensor applications," Quantum. Electron. 28, 444-448 (1998).
[CrossRef]

V. E. Kochergin, M. V. Valeiko, A. A. Beloglazov, T. I. Ksenvich, P. I. Nikitin, "Visualization of the angular dependence of the reflected-radiation phase under conditions of a surface-plasmon resonance and its sensor application," Quantum. Electron. 28, 835-839 (1998).
[CrossRef]

A. V. Kabashin, V. E. Kochergin, A. A. Beloglazov, P. I. Nikitin, "Phase-polarization contrast for surface plasmon resonance biosensors," Biosens. Bioelectron. 13, 1263-1269 (1998).
[CrossRef]

S. Shen, T. Liu and J. Guo, "Optical phase-shift detection of surface plasmon resonance," Appl. Opt. 37, 1747-1751 (1998).
[CrossRef]

1997

A. V. Kabashin, P. I. Nikitin, "Interferometer based on a surface-plasmon resonance for sensor applications," Quantum. Electron. 27, 653-654 (1997).
[CrossRef]

1995

B. Liedberg, C. Nylander, I. Lundstrom, "Biosensing with surface plasmon resonance - how it all started," Biosens. Bioelectron. 10, i-ix (1995).
[CrossRef] [PubMed]

1993

1968

A. Otto, "Excitation of surface plasma waves in silver by the method of frustrated total reflection," Z. Phys. 216, 398-410 (1968).
[CrossRef]

1957

R. H. Ritchie, "Plasma losses by fast electrons in thin films," Phys. Rev. 106, 874−881 (1957).
[CrossRef]

Bartlett, P. N.

A. K. Sheridan, R. D. Harris, P. N. Bartlett and J. S. Wilkinson, "Phase interrogation of an integrated optical SPR sensor," Sens. Actuators. B 97, 114-121 (2004).
[CrossRef]

Beloglazov, A. A.

A. N. Grigorenko, A. A. Beloglazov, P. I. Nikitin, C. Kuhne, G. Steiner and R. Salzer, "Dark-field surface plasmon resonance microscopy," Opt. Commun. 174, 151-155 (2000).
[CrossRef]

P. I. Nikitin, A. N. Grigorenko, A. A. Beloglazov, M. V. Valeiko, A. I. Savchuk and O. A. Savchuk, "Surface Plasmon Resonance Interferometry for Micro-Array Biosensing," Sens. Actuators. 85, 189-193 (2000).
[CrossRef]

P. I. Nikitin, A. A. Beloglazov, V. E. Kochergin, M. V. Valeiko, T. I. Ksenevich, "Surface plasmon resonance interferometry for biological and chemical sensing," Sens. Actuators. B 54, 43-50 (1999).
[CrossRef]

V. E. Kochergin, M. V. Valeiko, A. A. Beloglazov, T. I. Ksenvich, P. I. Nikitin, "Visualization of the angular dependence of the reflected-radiation phase under conditions of a surface-plasmon resonance and its sensor application," Quantum. Electron. 28, 835-839 (1998).
[CrossRef]

A. V. Kabashin, V. E. Kochergin, A. A. Beloglazov, P. I. Nikitin, "Phase-polarization contrast for surface plasmon resonance biosensors," Biosens. Bioelectron. 13, 1263-1269 (1998).
[CrossRef]

V. E. Kochergin, A. A. Beloglazov, M. V. Valeiko, P. I. Nikitin, "Phase properties of a surface-plasmon resonance from the viewpoint of sensor applications," Quantum. Electron. 28, 444-448 (1998).
[CrossRef]

Chang, L. B.

C. M. Wu, Z. C. Jian, S. F. Joe, L. B. Chang, "High-sensitivity sensor based on surface Plasmon resonance and heterodyne interferometry," Sens. Actuators. B 92, 133-136 (2003).
[CrossRef]

Chen, S. J.

Dingxin, W.

Y. Xinglong, W. Dingxin, W. Xing, D. Xiang, L. Wei and Z. Xinsheng, "A surface plasmon resonance imaging interferometry for protein micro-array detection," Sens. Actuators. B 108, 765- 771 (2005).
[CrossRef]

Y. Xinglong, W. Dingxin, Y. Zibo, "Simulation and analysis of surface plasmon resonance biosensor based on phase detection," Sens. Actuators. B 91, 285-290 (2003).
[CrossRef]

Gauglitz, G.

J. Homola, S. S. Yee, G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Actuators B 54, 3-15 (1999).
[CrossRef]

Grigorenko, A. N.

A. N. Grigorenko, A. A. Beloglazov, P. I. Nikitin, C. Kuhne, G. Steiner and R. Salzer, "Dark-field surface plasmon resonance microscopy," Opt. Commun. 174, 151-155 (2000).
[CrossRef]

P. I. Nikitin, A. N. Grigorenko, A. A. Beloglazov, M. V. Valeiko, A. I. Savchuk and O. A. Savchuk, "Surface Plasmon Resonance Interferometry for Micro-Array Biosensing," Sens. Actuators. 85, 189-193 (2000).
[CrossRef]

A. N. Grigorenko, P. I. Nikitin and A. V. Kabashin," Phase jumps and interferometric surface plasmon resonance imaging," Appl. Phys. Lett. 75, 3917-3919 (1999).
[CrossRef]

Guo, J.

Harris, R. D.

A. K. Sheridan, R. D. Harris, P. N. Bartlett and J. S. Wilkinson, "Phase interrogation of an integrated optical SPR sensor," Sens. Actuators. B 97, 114-121 (2004).
[CrossRef]

Ho, H. P.

Homola, J.

J. Homola, S. S. Yee, G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Actuators B 54, 3-15 (1999).
[CrossRef]

Jian, Z. C.

C. M. Wu, Z. C. Jian, S. F. Joe, L. B. Chang, "High-sensitivity sensor based on surface Plasmon resonance and heterodyne interferometry," Sens. Actuators. B 92, 133-136 (2003).
[CrossRef]

Joe, S. F.

C. M. Wu, Z. C. Jian, S. F. Joe, L. B. Chang, "High-sensitivity sensor based on surface Plasmon resonance and heterodyne interferometry," Sens. Actuators. B 92, 133-136 (2003).
[CrossRef]

Kabashin, A. V.

A. N. Grigorenko, P. I. Nikitin and A. V. Kabashin," Phase jumps and interferometric surface plasmon resonance imaging," Appl. Phys. Lett. 75, 3917-3919 (1999).
[CrossRef]

A. V. Kabashin, V. E. Kochergin and P. I. Nikitin, "Surface plasmon resonance bio- and chemical sensors with phase-polarization contrast," Sens. Actuators. B 54, 51-56 (1999).
[CrossRef]

A. V. Kabashin, V. E. Kochergin, A. A. Beloglazov, P. I. Nikitin, "Phase-polarization contrast for surface plasmon resonance biosensors," Biosens. Bioelectron. 13, 1263-1269 (1998).
[CrossRef]

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

A. V. Kabashin, P. I. Nikitin, "Interferometer based on a surface-plasmon resonance for sensor applications," Quantum. Electron. 27, 653-654 (1997).
[CrossRef]

Kochergin, V. E.

P. I. Nikitin, A. A. Beloglazov, V. E. Kochergin, M. V. Valeiko, T. I. Ksenevich, "Surface plasmon resonance interferometry for biological and chemical sensing," Sens. Actuators. B 54, 43-50 (1999).
[CrossRef]

A. V. Kabashin, V. E. Kochergin and P. I. Nikitin, "Surface plasmon resonance bio- and chemical sensors with phase-polarization contrast," Sens. Actuators. B 54, 51-56 (1999).
[CrossRef]

A. V. Kabashin, V. E. Kochergin, A. A. Beloglazov, P. I. Nikitin, "Phase-polarization contrast for surface plasmon resonance biosensors," Biosens. Bioelectron. 13, 1263-1269 (1998).
[CrossRef]

V. E. Kochergin, A. A. Beloglazov, M. V. Valeiko, P. I. Nikitin, "Phase properties of a surface-plasmon resonance from the viewpoint of sensor applications," Quantum. Electron. 28, 444-448 (1998).
[CrossRef]

V. E. Kochergin, M. V. Valeiko, A. A. Beloglazov, T. I. Ksenvich, P. I. Nikitin, "Visualization of the angular dependence of the reflected-radiation phase under conditions of a surface-plasmon resonance and its sensor application," Quantum. Electron. 28, 835-839 (1998).
[CrossRef]

Kong, S. K.

Kostianovski, S.

Kretschmann, E.

E. Kretschmann, H. Raether, "Radiative decay of nonradiative surface plasmons excited by light," Z. Naturforsch. A. 23, 2135-2136 (1968).

Ksenevich, T. I.

P. I. Nikitin, A. A. Beloglazov, V. E. Kochergin, M. V. Valeiko, T. I. Ksenevich, "Surface plasmon resonance interferometry for biological and chemical sensing," Sens. Actuators. B 54, 43-50 (1999).
[CrossRef]

Ksenvich, T. I.

V. E. Kochergin, M. V. Valeiko, A. A. Beloglazov, T. I. Ksenvich, P. I. Nikitin, "Visualization of the angular dependence of the reflected-radiation phase under conditions of a surface-plasmon resonance and its sensor application," Quantum. Electron. 28, 835-839 (1998).
[CrossRef]

Kuhne, C.

A. N. Grigorenko, A. A. Beloglazov, P. I. Nikitin, C. Kuhne, G. Steiner and R. Salzer, "Dark-field surface plasmon resonance microscopy," Opt. Commun. 174, 151-155 (2000).
[CrossRef]

Law, W. C.

Liedberg, B.

B. Liedberg, C. Nylander, I. Lundstrom, "Biosensing with surface plasmon resonance - how it all started," Biosens. Bioelectron. 10, i-ix (1995).
[CrossRef] [PubMed]

Lin, C.

Lipson, S. G.

Liu, T.

Lundstrom, I.

B. Liedberg, C. Nylander, I. Lundstrom, "Biosensing with surface plasmon resonance - how it all started," Biosens. Bioelectron. 10, i-ix (1995).
[CrossRef] [PubMed]

Nikitin, P. I.

P. I. Nikitin, A. N. Grigorenko, A. A. Beloglazov, M. V. Valeiko, A. I. Savchuk and O. A. Savchuk, "Surface Plasmon Resonance Interferometry for Micro-Array Biosensing," Sens. Actuators. 85, 189-193 (2000).
[CrossRef]

A. N. Grigorenko, A. A. Beloglazov, P. I. Nikitin, C. Kuhne, G. Steiner and R. Salzer, "Dark-field surface plasmon resonance microscopy," Opt. Commun. 174, 151-155 (2000).
[CrossRef]

P. I. Nikitin, A. A. Beloglazov, V. E. Kochergin, M. V. Valeiko, T. I. Ksenevich, "Surface plasmon resonance interferometry for biological and chemical sensing," Sens. Actuators. B 54, 43-50 (1999).
[CrossRef]

A. V. Kabashin, V. E. Kochergin and P. I. Nikitin, "Surface plasmon resonance bio- and chemical sensors with phase-polarization contrast," Sens. Actuators. B 54, 51-56 (1999).
[CrossRef]

A. N. Grigorenko, P. I. Nikitin and A. V. Kabashin," Phase jumps and interferometric surface plasmon resonance imaging," Appl. Phys. Lett. 75, 3917-3919 (1999).
[CrossRef]

A. V. Kabashin, V. E. Kochergin, A. A. Beloglazov, P. I. Nikitin, "Phase-polarization contrast for surface plasmon resonance biosensors," Biosens. Bioelectron. 13, 1263-1269 (1998).
[CrossRef]

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

V. E. Kochergin, A. A. Beloglazov, M. V. Valeiko, P. I. Nikitin, "Phase properties of a surface-plasmon resonance from the viewpoint of sensor applications," Quantum. Electron. 28, 444-448 (1998).
[CrossRef]

V. E. Kochergin, M. V. Valeiko, A. A. Beloglazov, T. I. Ksenvich, P. I. Nikitin, "Visualization of the angular dependence of the reflected-radiation phase under conditions of a surface-plasmon resonance and its sensor application," Quantum. Electron. 28, 835-839 (1998).
[CrossRef]

A. V. Kabashin, P. I. Nikitin, "Interferometer based on a surface-plasmon resonance for sensor applications," Quantum. Electron. 27, 653-654 (1997).
[CrossRef]

Nylander, C.

B. Liedberg, C. Nylander, I. Lundstrom, "Biosensing with surface plasmon resonance - how it all started," Biosens. Bioelectron. 10, i-ix (1995).
[CrossRef] [PubMed]

Otto, A.

A. Otto, "Excitation of surface plasma waves in silver by the method of frustrated total reflection," Z. Phys. 216, 398-410 (1968).
[CrossRef]

Pao, M. C.

Raether, H.

E. Kretschmann, H. Raether, "Radiative decay of nonradiative surface plasmons excited by light," Z. Naturforsch. A. 23, 2135-2136 (1968).

Ribak, E. N.

Ritchie, R. H.

R. H. Ritchie, "Plasma losses by fast electrons in thin films," Phys. Rev. 106, 874−881 (1957).
[CrossRef]

Salzer, R.

A. N. Grigorenko, A. A. Beloglazov, P. I. Nikitin, C. Kuhne, G. Steiner and R. Salzer, "Dark-field surface plasmon resonance microscopy," Opt. Commun. 174, 151-155 (2000).
[CrossRef]

Savchuk, A. I.

P. I. Nikitin, A. N. Grigorenko, A. A. Beloglazov, M. V. Valeiko, A. I. Savchuk and O. A. Savchuk, "Surface Plasmon Resonance Interferometry for Micro-Array Biosensing," Sens. Actuators. 85, 189-193 (2000).
[CrossRef]

Savchuk, O. A.

P. I. Nikitin, A. N. Grigorenko, A. A. Beloglazov, M. V. Valeiko, A. I. Savchuk and O. A. Savchuk, "Surface Plasmon Resonance Interferometry for Micro-Array Biosensing," Sens. Actuators. 85, 189-193 (2000).
[CrossRef]

Shen, S.

Sheridan, A. K.

A. K. Sheridan, R. D. Harris, P. N. Bartlett and J. S. Wilkinson, "Phase interrogation of an integrated optical SPR sensor," Sens. Actuators. B 97, 114-121 (2004).
[CrossRef]

Steiner, G.

A. N. Grigorenko, A. A. Beloglazov, P. I. Nikitin, C. Kuhne, G. Steiner and R. Salzer, "Dark-field surface plasmon resonance microscopy," Opt. Commun. 174, 151-155 (2000).
[CrossRef]

Su, Y. D.

Valeiko, M. V.

P. I. Nikitin, A. N. Grigorenko, A. A. Beloglazov, M. V. Valeiko, A. I. Savchuk and O. A. Savchuk, "Surface Plasmon Resonance Interferometry for Micro-Array Biosensing," Sens. Actuators. 85, 189-193 (2000).
[CrossRef]

P. I. Nikitin, A. A. Beloglazov, V. E. Kochergin, M. V. Valeiko, T. I. Ksenevich, "Surface plasmon resonance interferometry for biological and chemical sensing," Sens. Actuators. B 54, 43-50 (1999).
[CrossRef]

V. E. Kochergin, M. V. Valeiko, A. A. Beloglazov, T. I. Ksenvich, P. I. Nikitin, "Visualization of the angular dependence of the reflected-radiation phase under conditions of a surface-plasmon resonance and its sensor application," Quantum. Electron. 28, 835-839 (1998).
[CrossRef]

V. E. Kochergin, A. A. Beloglazov, M. V. Valeiko, P. I. Nikitin, "Phase properties of a surface-plasmon resonance from the viewpoint of sensor applications," Quantum. Electron. 28, 444-448 (1998).
[CrossRef]

Wei, L.

Y. Xinglong, W. Dingxin, W. Xing, D. Xiang, L. Wei and Z. Xinsheng, "A surface plasmon resonance imaging interferometry for protein micro-array detection," Sens. Actuators. B 108, 765- 771 (2005).
[CrossRef]

Wilkinson, J. S.

A. K. Sheridan, R. D. Harris, P. N. Bartlett and J. S. Wilkinson, "Phase interrogation of an integrated optical SPR sensor," Sens. Actuators. B 97, 114-121 (2004).
[CrossRef]

Wu, C. M.

C. M. Wu and M. C. Pao, "Sensitivity-tunable optical sensors based on surface Plasmon resonance and phase detection," Opt. Express 12, 3509-3514 (2004).
[CrossRef] [PubMed]

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Y. Xinglong, W. Dingxin, W. Xing, D. Xiang, L. Wei and Z. Xinsheng, "A surface plasmon resonance imaging interferometry for protein micro-array detection," Sens. Actuators. B 108, 765- 771 (2005).
[CrossRef]

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Y. Xinglong, W. Dingxin, W. Xing, D. Xiang, L. Wei and Z. Xinsheng, "A surface plasmon resonance imaging interferometry for protein micro-array detection," Sens. Actuators. B 108, 765- 771 (2005).
[CrossRef]

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Y. Xinglong, W. Dingxin, W. Xing, D. Xiang, L. Wei and Z. Xinsheng, "A surface plasmon resonance imaging interferometry for protein micro-array detection," Sens. Actuators. B 108, 765- 771 (2005).
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Y. Xinglong, W. Dingxin, W. Xing, D. Xiang, L. Wei and Z. Xinsheng, "A surface plasmon resonance imaging interferometry for protein micro-array detection," Sens. Actuators. B 108, 765- 771 (2005).
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Figures (7)

Fig. 1.
Fig. 1.

The Kretschman configuration for measuring surface plasmon resonance.

Fig. 2.
Fig. 2.

Calculated behavior of the phase and intensity of light (λ=692nm) reflected from a prism of SF11 glass with gold films of thicknesses between 30 and 80nm covered by water. (a) shows the intensity reflected, (b) the phase anomaly between 50 and 55nm thickness, and (c) the interference intensity when a coherent reference wave of constant amplitude and phase is added to the reflected light. The interferogram shows no signs of anomaly at any gold layer thickness.

Fig. 3.
Fig. 3.

Four interferograms simulated as input to the phase-step algorithm at the phase anomaly. The interferograms are the result of interference between the p- polarized wave which interacts with the surface plasmons and the s- polarized wave which serves as a reference wave. Although the phase of the wave shows a very steep slope at the resonance angle (shown by the arrow), none of the four interferograms (which the phase is calculated from) show any unusual behavior.

Fig. 4.
Fig. 4.

The optical system used for the comparison measurements. (1) LED source, (2) collimation lens, (3) rotatable polarizer, (4) Kretschman prism, (5) objective lens, (6) phaseshifter, (7) rotatable analyzer, (8) imaging lens, (9) CCD. Angle scanning was provided by a linear motion of the LED source.

Fig. 5.
Fig. 5.

The refractive index measurement accuracy Δn as a function of the input photon flux, plotted to show dependence on N . The effect of using averages over different numbers of CCD pixels is shown. In each case the theoretical curves (full lines) include correction for readout noise, R N , with broken lines for R N =0. (a) refers to the intensity mode and (b) to the interferometric mode of measurement.

Fig. 6.
Fig. 6.

The error in the refractive index measurement in the interferometric mode, when the phase difference between the S- and P-polarizations was varied with the phase shifter. The agreement between the simulated and observed values indicates that phase-shifter errors could not account for the poorer performance of the interferometric mode. The optimum phase difference was the value used in the experiments.

Fig. 7.
Fig. 7.

Simulated experiments and theoretical model for sensitivity as a function of Au film thickness. The error bars shown indicate numerical uncertainties, arising from finite-number simulations. The poor value of the phase mode at 50nm arises from resonant absorption.

Equations (14)

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I ( n , θ ) = I 0 2 [ R ( n , θ ) 2 cos 2 ( α ) cos 2 ( β ) + R ( n , θ ) P sin 2 ( α ) sin 2 ( β ) +
1 2 R ( n , θ ) S R ( n , θ ) P sin ( 2 α ) sin ( 2 β ) cos ( φ ( n , θ ) P φ ( n , θ ) S + Δ ψ ) ]
I ( n , θ ) = I 0 2 R ( n , θ ) P
I ( n , θ ) I 0 M ( n , θ ) = I 0 2 [ R ( n , θ ) S 4 + R ( n , θ ) P 4 + 1 2 R ( n , θ ) S R ( n , θ ) P cos ( φ ( n , θ ) P φ ( n , θ ) S + Δ ψ ) ]
φ ( n , θ ) = arctan ( I ( n , θ , 4 ) I ( n , θ , 2 ) I ( n , θ , 1 ) I ( n , θ , 3 ) )
Δ n ( n , θ ) = ( N 0 R ( n , θ ) + R N 2 b ) 1 2 ( N 0 R ( n , θ ) P n ) 1
Δ n ( n , θ ) = R N 0 R ( n , θ ) ( N 0 R ( n , θ ) n ) 1
Δ n ( n , θ ) = ( N 0 M ( n , θ ) + R N 2 b ) 1 2 ( N 0 M ( n , θ ) n ) 1
Δ n ( n , θ ) = ( A 2 ( Δ I ( n , θ , 1 ) 2 + Δ I ( n , θ , 3 ) 2 ) + Δ I ( n , θ , 2 ) 2 + Δ I ( n , θ , 4 ) 2 ) 1 2 N 0 φ ( n , θ ) n ( 1 + A 2 ) M ( n , θ , 1 ) M ( n , θ , 3 )
A M ( n , θ , 4 ) M ( n , θ , 2 ) M ( n , θ , 1 ) M ( n , θ , 3 ) Δ I j ( N 0 M ( n , θ , j ) + R N 2 b ) 1 2
Δ n ( n , θ ) ( N 0 φ ( n , θ ) n R ( n , θ ) S R ( n , θ ) P ) 1
n wav = [ k w k ] 1 k w k n ( θ k )
Δ n wav = [ k w k ] 1 = [ k Δ n ( θ k ) 2 ] 1
M ( n ) n = 1 2 [ 1 4 R ( n ) P n + R ( n ) S 2 ( 1 2 1 R ( n ) P cos ( φ ( n ) P φ ( n ) S ) R ( n ) P n R ( n ) P sin ( φ ( n ) P φ ( n ) S ) φ ( n ) P n ) ]

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