Abstract

We investigate the effect of the divergence of a Gaussian laser beam on the resonance curve and the sensitivity of optical sensors based on surface-plasmon resonance (SPR). For He–Ne laser beams it is found that, for beams with a waist radius of less than 300 µm, the SPR-curve characteristics differ appreciably from the case in which a plane wave is considered. Simple expressions for the sensitivity of (bio)chemical sensors are given. A simple Lorentzian model is used to estimate the maximum possible sensitivity when a multilayer system is used to enhance the resonance peak. It was found that the sensitivity can reach its highest value when the width of the SPR curve is equal to the laser-beam divergence. The results could be particularly important when a SPR curve is used to measure the absolute value of the refractive index of a sample or the dielectric constant and the thickness of a metal layer.

© 1999 Optical Society of America

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References

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  1. J. Homola, S. S. Yee, G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B 54, 3–15 (1999).
    [CrossRef]
  2. C. Nylander, B. Liedberg, T. Lind, “Gas detection by means of surface plasmon resonance,” Sens. Actuators B 3, 79–88 (1982).
    [CrossRef]
  3. K. Matsubara, S. Kawata, S. Minami, “Optical chemical sensor based on surface plasmon resonance measurement,” Appl. Opt. 27, 1160–1163 (1988).
    [CrossRef] [PubMed]
  4. G. Margheri, A. Mannoni, F. Quercioli, “High-resolution angular and displacement sensing based on the excitation of surface plasma waves,” Appl. Opt. 36, 4521–4525 (1997).
    [CrossRef] [PubMed]
  5. J. Meléndez, R. Carr, D. Bartholomew, H. Taneja, S. S. Yee, C. Jung, C. Furlonf, “Development of a surface plasmon resonance sensor for commercial applications,” Sens. Actuators B 38–39, 375–379 (1997).
  6. V. I. Chegel, Y. M. Shirshov, E. V. Piletskaya, S. A. Piletsky, “Surface plasmon resonance sensor for pesticide detection,” Sens. Actuators B 48, 456–460 (1998).
    [CrossRef]
  7. G. Sakai, K. Ogata, T. Uda, N. Miura, N. Yamazoe, “A surface plasmon resonance-based immunosensor for highly sensitive detection of morphine,” Sens. Actuators B 49, 5–12 (1998).
    [CrossRef]
  8. E. Kretschmann, “The determination of the optical constants of metals by excitation of surface plasmon resonance,” Z. Phys. 241, 313–324 (1971).
    [CrossRef]
  9. W. M. Robertson, Fullerton, “Reexamination of the surface-plasma-wave technique for determining the dielectric constant and thickness of metal films,” J. Opt. Soc. Am. B 6, 1584–1589 (1989).
  10. R. P. H. Kooyman, H. Kolkman, J. Van Gent, J. Greeve, “Determination of dielectric permittivity and thickness of a metal layer from a surface plasmon resonance experiment,” Appl. Opt. 29, 1974–1978 (1990).
    [CrossRef] [PubMed]
  11. W. Lukosz, “Principles and sensitivities of integrated and surface plasmon resonance sensors for direct affinity sensing and immunosensing,” Biosens. Bioelectron. 6, 215–225 (1991).
    [CrossRef]
  12. A. A. Kolomenskii, P. D. Gershon, 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]
  13. P. A. Gass, S. Schalk, J. R. Sambles, “Highly sensitive optical measurement techniques based on acoustic-optic devices,” Appl. Opt. 33, 7501–7510 (1994).
    [CrossRef] [PubMed]
  14. M. J. Jory, G. W. Bradberry, P. S. Cann, J. R. Sambles, “A surface-plasmon-resonance optical sensor using acousto-optics,” Meas. Sci. Technol. 6, 1193–1200 (1995).
    [CrossRef]
  15. F. Caruso, M. J. Jory, G. W. Bradberry, J. R. Sambles, D. N. Furlong, “Acousto-optic surface plasmon resonance measurements on thin films on gold,” J. Appl. Phys. 83, 1023–1028 (1998).
    [CrossRef]
  16. S. Shen, T. Liu, J. Guo, “Optical phase-shift detection on surface plasmon resonance,” Appl. Opt. 37, 1747–1751 (1998).
    [CrossRef]
  17. H. M. Lai, F. C. Cheng, W. K. Tang, “Goos–Hänchen effect around and off the critical angle,” J. Opt. Soc. Am. A 3, 550–557 (1986).
    [CrossRef]
  18. A. García-Valenzuela, R. Diaz-Uribe, “Detection limits of an internal-reflection sensor for the optical beam deflection method,” Appl. Opt. 36, 4456–4462 (1997).
    [CrossRef] [PubMed]
  19. R. E. Collin, Antennas and Radiowave Propagation (McGraw-Hill, New York, 1985), Appendix to Chap. 4, pp. 284–286.
  20. J. Villatoro, A. García-Valenzuela, “Measuring optical power transmission near the critical angle for sensing beam deflection,” Appl. Opt. 37, 6648–6653 (1998).
    [CrossRef]
  21. A. García-Valenzuela, M. P. Gomar, C. G. Segundo, V. F. Aburto, “Dynamic reflectometry near the critical angle for high resolution sensing of the index of refraction,” Sens. Actuators B 52, 236–242 (1998).
    [CrossRef]
  22. I. Stemmler, A. Brecht, G. Gauglitz, “Compact surface plasmon resonance-transducers with spectral readout for biosensing applications,” Sens. Actuators B 54, 98–105 (1999).
    [CrossRef]
  23. K. Matsubara, S. Kawata, S. Minami, “Multilayer system for high-precision surface plasmon resonance sensor,” Opt. Lett. 15, 75–77 (1990).
    [CrossRef] [PubMed]
  24. J. van Gent, P. V. Lambeck, H. J. M. Kreuwel, G. J. Gerritsma, E. J. R. Sudhölter, D. N. Reinhoudt, T. J. A. Popma, “Optimization of a chemooptical surface plasmon resonance based sensor,” Appl. Opt. 29, 2843–2849 (1990).
    [CrossRef] [PubMed]

1999

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

I. Stemmler, A. Brecht, G. Gauglitz, “Compact surface plasmon resonance-transducers with spectral readout for biosensing applications,” Sens. Actuators B 54, 98–105 (1999).
[CrossRef]

1998

V. I. Chegel, Y. M. Shirshov, E. V. Piletskaya, S. A. Piletsky, “Surface plasmon resonance sensor for pesticide detection,” Sens. Actuators B 48, 456–460 (1998).
[CrossRef]

G. Sakai, K. Ogata, T. Uda, N. Miura, N. Yamazoe, “A surface plasmon resonance-based immunosensor for highly sensitive detection of morphine,” Sens. Actuators B 49, 5–12 (1998).
[CrossRef]

F. Caruso, M. J. Jory, G. W. Bradberry, J. R. Sambles, D. N. Furlong, “Acousto-optic surface plasmon resonance measurements on thin films on gold,” J. Appl. Phys. 83, 1023–1028 (1998).
[CrossRef]

A. García-Valenzuela, M. P. Gomar, C. G. Segundo, V. F. Aburto, “Dynamic reflectometry near the critical angle for high resolution sensing of the index of refraction,” Sens. Actuators B 52, 236–242 (1998).
[CrossRef]

S. Shen, T. Liu, J. Guo, “Optical phase-shift detection on surface plasmon resonance,” Appl. Opt. 37, 1747–1751 (1998).
[CrossRef]

J. Villatoro, A. García-Valenzuela, “Measuring optical power transmission near the critical angle for sensing beam deflection,” Appl. Opt. 37, 6648–6653 (1998).
[CrossRef]

1997

1995

M. J. Jory, G. W. Bradberry, P. S. Cann, J. R. Sambles, “A surface-plasmon-resonance optical sensor using acousto-optics,” Meas. Sci. Technol. 6, 1193–1200 (1995).
[CrossRef]

1994

1991

W. Lukosz, “Principles and sensitivities of integrated and surface plasmon resonance sensors for direct affinity sensing and immunosensing,” Biosens. Bioelectron. 6, 215–225 (1991).
[CrossRef]

1990

1989

1988

1986

1982

C. Nylander, B. Liedberg, T. Lind, “Gas detection by means of surface plasmon resonance,” Sens. Actuators B 3, 79–88 (1982).
[CrossRef]

1971

E. Kretschmann, “The determination of the optical constants of metals by excitation of surface plasmon resonance,” Z. Phys. 241, 313–324 (1971).
[CrossRef]

Aburto, V. F.

A. García-Valenzuela, M. P. Gomar, C. G. Segundo, V. F. Aburto, “Dynamic reflectometry near the critical angle for high resolution sensing of the index of refraction,” Sens. Actuators B 52, 236–242 (1998).
[CrossRef]

Bartholomew, D.

J. Meléndez, R. Carr, D. Bartholomew, H. Taneja, S. S. Yee, C. Jung, C. Furlonf, “Development of a surface plasmon resonance sensor for commercial applications,” Sens. Actuators B 38–39, 375–379 (1997).

Bradberry, G. W.

F. Caruso, M. J. Jory, G. W. Bradberry, J. R. Sambles, D. N. Furlong, “Acousto-optic surface plasmon resonance measurements on thin films on gold,” J. Appl. Phys. 83, 1023–1028 (1998).
[CrossRef]

M. J. Jory, G. W. Bradberry, P. S. Cann, J. R. Sambles, “A surface-plasmon-resonance optical sensor using acousto-optics,” Meas. Sci. Technol. 6, 1193–1200 (1995).
[CrossRef]

Brecht, A.

I. Stemmler, A. Brecht, G. Gauglitz, “Compact surface plasmon resonance-transducers with spectral readout for biosensing applications,” Sens. Actuators B 54, 98–105 (1999).
[CrossRef]

Cann, P. S.

M. J. Jory, G. W. Bradberry, P. S. Cann, J. R. Sambles, “A surface-plasmon-resonance optical sensor using acousto-optics,” Meas. Sci. Technol. 6, 1193–1200 (1995).
[CrossRef]

Carr, R.

J. Meléndez, R. Carr, D. Bartholomew, H. Taneja, S. S. Yee, C. Jung, C. Furlonf, “Development of a surface plasmon resonance sensor for commercial applications,” Sens. Actuators B 38–39, 375–379 (1997).

Caruso, F.

F. Caruso, M. J. Jory, G. W. Bradberry, J. R. Sambles, D. N. Furlong, “Acousto-optic surface plasmon resonance measurements on thin films on gold,” J. Appl. Phys. 83, 1023–1028 (1998).
[CrossRef]

Chegel, V. I.

V. I. Chegel, Y. M. Shirshov, E. V. Piletskaya, S. A. Piletsky, “Surface plasmon resonance sensor for pesticide detection,” Sens. Actuators B 48, 456–460 (1998).
[CrossRef]

Cheng, F. C.

Collin, R. E.

R. E. Collin, Antennas and Radiowave Propagation (McGraw-Hill, New York, 1985), Appendix to Chap. 4, pp. 284–286.

Diaz-Uribe, R.

Fullerton,

Furlonf, C.

J. Meléndez, R. Carr, D. Bartholomew, H. Taneja, S. S. Yee, C. Jung, C. Furlonf, “Development of a surface plasmon resonance sensor for commercial applications,” Sens. Actuators B 38–39, 375–379 (1997).

Furlong, D. N.

F. Caruso, M. J. Jory, G. W. Bradberry, J. R. Sambles, D. N. Furlong, “Acousto-optic surface plasmon resonance measurements on thin films on gold,” J. Appl. Phys. 83, 1023–1028 (1998).
[CrossRef]

García-Valenzuela, A.

Gass, P. A.

Gauglitz, G.

I. Stemmler, A. Brecht, G. Gauglitz, “Compact surface plasmon resonance-transducers with spectral readout for biosensing applications,” Sens. Actuators B 54, 98–105 (1999).
[CrossRef]

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

Gerritsma, G. J.

Gershon, P. D.

Gomar, M. P.

A. García-Valenzuela, M. P. Gomar, C. G. Segundo, V. F. Aburto, “Dynamic reflectometry near the critical angle for high resolution sensing of the index of refraction,” Sens. Actuators B 52, 236–242 (1998).
[CrossRef]

Greeve, J.

Guo, J.

Homola, J.

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

Jory, M. J.

F. Caruso, M. J. Jory, G. W. Bradberry, J. R. Sambles, D. N. Furlong, “Acousto-optic surface plasmon resonance measurements on thin films on gold,” J. Appl. Phys. 83, 1023–1028 (1998).
[CrossRef]

M. J. Jory, G. W. Bradberry, P. S. Cann, J. R. Sambles, “A surface-plasmon-resonance optical sensor using acousto-optics,” Meas. Sci. Technol. 6, 1193–1200 (1995).
[CrossRef]

Jung, C.

J. Meléndez, R. Carr, D. Bartholomew, H. Taneja, S. S. Yee, C. Jung, C. Furlonf, “Development of a surface plasmon resonance sensor for commercial applications,” Sens. Actuators B 38–39, 375–379 (1997).

Kawata, S.

Kolkman, H.

Kolomenskii, A. A.

Kooyman, R. P. H.

Kretschmann, E.

E. Kretschmann, “The determination of the optical constants of metals by excitation of surface plasmon resonance,” Z. Phys. 241, 313–324 (1971).
[CrossRef]

Kreuwel, H. J. M.

Lai, H. M.

Lambeck, P. V.

Liedberg, B.

C. Nylander, B. Liedberg, T. Lind, “Gas detection by means of surface plasmon resonance,” Sens. Actuators B 3, 79–88 (1982).
[CrossRef]

Lind, T.

C. Nylander, B. Liedberg, T. Lind, “Gas detection by means of surface plasmon resonance,” Sens. Actuators B 3, 79–88 (1982).
[CrossRef]

Liu, T.

Lukosz, W.

W. Lukosz, “Principles and sensitivities of integrated and surface plasmon resonance sensors for direct affinity sensing and immunosensing,” Biosens. Bioelectron. 6, 215–225 (1991).
[CrossRef]

Mannoni, A.

Margheri, G.

Matsubara, K.

Meléndez, J.

J. Meléndez, R. Carr, D. Bartholomew, H. Taneja, S. S. Yee, C. Jung, C. Furlonf, “Development of a surface plasmon resonance sensor for commercial applications,” Sens. Actuators B 38–39, 375–379 (1997).

Minami, S.

Miura, N.

G. Sakai, K. Ogata, T. Uda, N. Miura, N. Yamazoe, “A surface plasmon resonance-based immunosensor for highly sensitive detection of morphine,” Sens. Actuators B 49, 5–12 (1998).
[CrossRef]

Nylander, C.

C. Nylander, B. Liedberg, T. Lind, “Gas detection by means of surface plasmon resonance,” Sens. Actuators B 3, 79–88 (1982).
[CrossRef]

Ogata, K.

G. Sakai, K. Ogata, T. Uda, N. Miura, N. Yamazoe, “A surface plasmon resonance-based immunosensor for highly sensitive detection of morphine,” Sens. Actuators B 49, 5–12 (1998).
[CrossRef]

Piletskaya, E. V.

V. I. Chegel, Y. M. Shirshov, E. V. Piletskaya, S. A. Piletsky, “Surface plasmon resonance sensor for pesticide detection,” Sens. Actuators B 48, 456–460 (1998).
[CrossRef]

Piletsky, S. A.

V. I. Chegel, Y. M. Shirshov, E. V. Piletskaya, S. A. Piletsky, “Surface plasmon resonance sensor for pesticide detection,” Sens. Actuators B 48, 456–460 (1998).
[CrossRef]

Popma, T. J. A.

Quercioli, F.

Reinhoudt, D. N.

Robertson, W. M.

Sakai, G.

G. Sakai, K. Ogata, T. Uda, N. Miura, N. Yamazoe, “A surface plasmon resonance-based immunosensor for highly sensitive detection of morphine,” Sens. Actuators B 49, 5–12 (1998).
[CrossRef]

Sambles, J. R.

F. Caruso, M. J. Jory, G. W. Bradberry, J. R. Sambles, D. N. Furlong, “Acousto-optic surface plasmon resonance measurements on thin films on gold,” J. Appl. Phys. 83, 1023–1028 (1998).
[CrossRef]

M. J. Jory, G. W. Bradberry, P. S. Cann, J. R. Sambles, “A surface-plasmon-resonance optical sensor using acousto-optics,” Meas. Sci. Technol. 6, 1193–1200 (1995).
[CrossRef]

P. A. Gass, S. Schalk, J. R. Sambles, “Highly sensitive optical measurement techniques based on acoustic-optic devices,” Appl. Opt. 33, 7501–7510 (1994).
[CrossRef] [PubMed]

Schalk, S.

Schuessler, H. A.

Segundo, C. G.

A. García-Valenzuela, M. P. Gomar, C. G. Segundo, V. F. Aburto, “Dynamic reflectometry near the critical angle for high resolution sensing of the index of refraction,” Sens. Actuators B 52, 236–242 (1998).
[CrossRef]

Shen, S.

Shirshov, Y. M.

V. I. Chegel, Y. M. Shirshov, E. V. Piletskaya, S. A. Piletsky, “Surface plasmon resonance sensor for pesticide detection,” Sens. Actuators B 48, 456–460 (1998).
[CrossRef]

Stemmler, I.

I. Stemmler, A. Brecht, G. Gauglitz, “Compact surface plasmon resonance-transducers with spectral readout for biosensing applications,” Sens. Actuators B 54, 98–105 (1999).
[CrossRef]

Sudhölter, E. J. R.

Taneja, H.

J. Meléndez, R. Carr, D. Bartholomew, H. Taneja, S. S. Yee, C. Jung, C. Furlonf, “Development of a surface plasmon resonance sensor for commercial applications,” Sens. Actuators B 38–39, 375–379 (1997).

Tang, W. K.

Uda, T.

G. Sakai, K. Ogata, T. Uda, N. Miura, N. Yamazoe, “A surface plasmon resonance-based immunosensor for highly sensitive detection of morphine,” Sens. Actuators B 49, 5–12 (1998).
[CrossRef]

van Gent, J.

Villatoro, J.

Yamazoe, N.

G. Sakai, K. Ogata, T. Uda, N. Miura, N. Yamazoe, “A surface plasmon resonance-based immunosensor for highly sensitive detection of morphine,” Sens. Actuators B 49, 5–12 (1998).
[CrossRef]

Yee, S. S.

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

J. Meléndez, R. Carr, D. Bartholomew, H. Taneja, S. S. Yee, C. Jung, C. Furlonf, “Development of a surface plasmon resonance sensor for commercial applications,” Sens. Actuators B 38–39, 375–379 (1997).

Appl. Opt.

K. Matsubara, S. Kawata, S. Minami, “Optical chemical sensor based on surface plasmon resonance measurement,” Appl. Opt. 27, 1160–1163 (1988).
[CrossRef] [PubMed]

R. P. H. Kooyman, H. Kolkman, J. Van Gent, J. Greeve, “Determination of dielectric permittivity and thickness of a metal layer from a surface plasmon resonance experiment,” Appl. Opt. 29, 1974–1978 (1990).
[CrossRef] [PubMed]

P. A. Gass, S. Schalk, J. R. Sambles, “Highly sensitive optical measurement techniques based on acoustic-optic devices,” Appl. Opt. 33, 7501–7510 (1994).
[CrossRef] [PubMed]

A. García-Valenzuela, R. Diaz-Uribe, “Detection limits of an internal-reflection sensor for the optical beam deflection method,” Appl. Opt. 36, 4456–4462 (1997).
[CrossRef] [PubMed]

G. Margheri, A. Mannoni, F. Quercioli, “High-resolution angular and displacement sensing based on the excitation of surface plasma waves,” Appl. Opt. 36, 4521–4525 (1997).
[CrossRef] [PubMed]

S. Shen, T. Liu, J. Guo, “Optical phase-shift detection on surface plasmon resonance,” Appl. Opt. 37, 1747–1751 (1998).
[CrossRef]

J. van Gent, P. V. Lambeck, H. J. M. Kreuwel, G. J. Gerritsma, E. J. R. Sudhölter, D. N. Reinhoudt, T. J. A. Popma, “Optimization of a chemooptical surface plasmon resonance based sensor,” Appl. Opt. 29, 2843–2849 (1990).
[CrossRef] [PubMed]

A. A. Kolomenskii, P. D. Gershon, 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]

J. Villatoro, A. García-Valenzuela, “Measuring optical power transmission near the critical angle for sensing beam deflection,” Appl. Opt. 37, 6648–6653 (1998).
[CrossRef]

Biosens. Bioelectron.

W. Lukosz, “Principles and sensitivities of integrated and surface plasmon resonance sensors for direct affinity sensing and immunosensing,” Biosens. Bioelectron. 6, 215–225 (1991).
[CrossRef]

J. Appl. Phys.

F. Caruso, M. J. Jory, G. W. Bradberry, J. R. Sambles, D. N. Furlong, “Acousto-optic surface plasmon resonance measurements on thin films on gold,” J. Appl. Phys. 83, 1023–1028 (1998).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Meas. Sci. Technol.

M. J. Jory, G. W. Bradberry, P. S. Cann, J. R. Sambles, “A surface-plasmon-resonance optical sensor using acousto-optics,” Meas. Sci. Technol. 6, 1193–1200 (1995).
[CrossRef]

Opt. Lett.

Sens. Actuators B

A. García-Valenzuela, M. P. Gomar, C. G. Segundo, V. F. Aburto, “Dynamic reflectometry near the critical angle for high resolution sensing of the index of refraction,” Sens. Actuators B 52, 236–242 (1998).
[CrossRef]

I. Stemmler, A. Brecht, G. Gauglitz, “Compact surface plasmon resonance-transducers with spectral readout for biosensing applications,” Sens. Actuators B 54, 98–105 (1999).
[CrossRef]

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

C. Nylander, B. Liedberg, T. Lind, “Gas detection by means of surface plasmon resonance,” Sens. Actuators B 3, 79–88 (1982).
[CrossRef]

J. Meléndez, R. Carr, D. Bartholomew, H. Taneja, S. S. Yee, C. Jung, C. Furlonf, “Development of a surface plasmon resonance sensor for commercial applications,” Sens. Actuators B 38–39, 375–379 (1997).

V. I. Chegel, Y. M. Shirshov, E. V. Piletskaya, S. A. Piletsky, “Surface plasmon resonance sensor for pesticide detection,” Sens. Actuators B 48, 456–460 (1998).
[CrossRef]

G. Sakai, K. Ogata, T. Uda, N. Miura, N. Yamazoe, “A surface plasmon resonance-based immunosensor for highly sensitive detection of morphine,” Sens. Actuators B 49, 5–12 (1998).
[CrossRef]

Z. Phys.

E. Kretschmann, “The determination of the optical constants of metals by excitation of surface plasmon resonance,” Z. Phys. 241, 313–324 (1971).
[CrossRef]

Other

R. E. Collin, Antennas and Radiowave Propagation (McGraw-Hill, New York, 1985), Appendix to Chap. 4, pp. 284–286.

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Figures (5)

Fig. 1
Fig. 1

Schematic illustration of the Kretschmann geometry. The coordinate systems used in the analysis are shown. The subscript 1 represents the prism; the subscript 2, the Ag layer; and the subscript 3, the sample medium. The thickness of the Ag film is d.

Fig. 2
Fig. 2

SPR curve for three values of w 0 obtained under the assumption of a BK7 glass prism, a 56-nm-thick Ag film, air as the sample medium, and a wavelength of 632.8 nm [by use of Eq. (9)].

Fig. 3
Fig. 3

Main characteristics of a SPR curve as a function of w 0: (a) the position of the minimum versus w 0, (b) the value at the minimum versus w 0, (c) the curve width versus w 0. The filled circles shown in all three cases represent a typical value of a He–Ne laser beam-waist radius.

Fig. 4
Fig. 4

First differential versus the angle of incidence for three values of w 0. For the plot Eq. (10) was solved numerically.

Fig. 5
Fig. 5

Maximum of the integral of Eq. (15) versus the reflectivity width to the laser-beam (half-angle) divergence ratio Q.

Equations (16)

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

Hr=A -- Qˆkexp-w024ky2+kz2×expjk·rdkzdky,
Fk=r12+r23 expj2dky21+r12r23 expj2dky2,
rlm=mkyl-lkymmkyl+lkym,
kyl,m=k0l,m-n12 sin2 θ1/2,  l, m=1, 2, 3,
Hrx, y, z=A -- QˆkFkexp-w024kz2+ky sin θi-kx cos θi2×expjkxx-kyy+kzz+kx sin θi+ky cos θiLdkydkzsin θi,
Hrr, θ, ϕ=2Aπkxj expjkrr sin θiexp-w024kz2+ky sin θi-kx cos θi2×exp[jkx sin θi+ky cos θiLFkQˆk,
Pr=limrcμ02n10π/202π r2 sin θ|Hrr|2dϕdθ.
Pr=Bk120π/202πsin3 θ cos2 ϕ |Fk|2sin2 θi×exp-w02k122sin2 θ sin2 ϕ+sin2θi-θdϕdθ,
Rθi=w0k12π- |Fk|2 exp- w02k122θi-θ2dθ.
Rθi=-w0k132π-θi-θ|Fk|2×exp-w02k122θi-θ2dθ.
Ry3=Rθiθi=θsθiy3,
y3Rθi=2Rθi2θi=θspθspy3,
|Faθ|2=R0-Cθ-θsp2+W2,
Rθi=4α2π IQ,
IQ=-2Q2x-x02+2Q2x exp-x2dx
Rθimax1.28α2π.

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