Abstract

One can use the angle-modulated reflectance of a Gaussian beam near the critical angle to sense with high resolution the index of refraction of the external medium. We analyze in detail the reflectivity of a Gaussian beam near the critical angle and its dependence on the optical absorption of the external medium. The given formulation is relatively simple and is useful in discerning the effects of the various parameters involved on the reflectivity and its differentials with respect to the angle of incidence. The results presented can be readily used for the quantitative design of novel sensors based on modulated reflectance near the critical angle. We provide a simple algebraic expression for the loss of sensitivity of modulated reflectance near the critical angle as the sample’s absorption coefficient increases. We find that, in a typical case, the sensitivity has decreased to approximately half its value for transparent samples when the absorption coefficient has increased to 25 cm-1. We conclude that modulated reflectance near the critical angle remains a competitive technique for monitoring the index of refraction of an external medium with an absorption coefficient of as much as 120 cm-1. We compared experimentally obtained curves of the first differential of the reflectivity with respect to the angle of incidence with theory and found good agreement.

© 2000 Optical Society of America

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References

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  1. T. Kohno, N. Ozawa, K. Miyamoto, T. Musha, “High-precision optical surface sensors,” Appl. Opt. 27, 103–108 (1988).
    [CrossRef]
  2. P. S. Huang, S. Kiyono, O. Kamada, “Angle measurement based on the internal reflection effect: a new method,” Appl. Opt. 31, 6047–6055 (1992).
    [CrossRef] [PubMed]
  3. P. S. Huang, J. Ni, “Angle measurement based on the internal reflection effect using elongated critical-angle prisms,” Appl. Opt. 35, 2239–2241 (1996).
    [CrossRef] [PubMed]
  4. 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]
  5. J. Villatoro, A. García-Valenzuela, “Measuring optical power transmission near the critical angle for sensing beam deflection,” Appl. Opt. 37, 6648–6653 (1998).
    [CrossRef]
  6. P. S. Huang, “Use of thin films for high-sensitivity angle measurement,” Appl. Opt. 38, 4831–4836 (1999).
    [CrossRef]
  7. P. R. Jarvis, G. H. Meeten, “Critical-angle measurement of refractive index of absorbing materials: an experimental study,” J. Phys. E 19, 296–298 (1986).
    [CrossRef]
  8. J. Rheims, J. Köser, T. Wriedt, “Refractive index measurements in the near-IR using an Abbe refractometer,” Meas. Sci. Technol. 8, 601–605 (1997).
    [CrossRef]
  9. P. A. Gass, S. Schalk, J. R. Sambles, “Highly sensitive optical measurement techniques based on acousto-optic devices,” Appl. Opt. 33, 7501–7510 (1994).
    [CrossRef] [PubMed]
  10. L. Hui, X. Shusen, “Measurement method of the refractive index of biotissue by total internal reflection,” Appl. Opt. 35, 1793–1795 (1996).
    [CrossRef]
  11. A. García-Valenzuela, M. Peña-Gomar, C. García-Segundo, V. Flandes-Aburto, “Dynamic reflectometry near the critical angle for high-resolution sensing of the index of refraction,” Sensors Actuators B 52, 236–242 (1998).
    [CrossRef]
  12. K. J. Kasunic, “Comparison of Kretschmann–Raether angular regimes for measuring changes in bulk refractive index,” Appl. Opt. 39, 61–64 (2000).
    [CrossRef]
  13. S. Nemoto, “Measurement of the refractive index of liquid using laser beam displacement,” Appl. Opt. 31, 6690–6694 (1992).
    [CrossRef] [PubMed]
  14. E. Moreels, C. de Greef, R. Finsy, “Laser light refractometer,” Appl. Opt. 23, 3010–3013 (1984).
    [CrossRef] [PubMed]
  15. T. Fukano, I. Yamaguchi, “Separation of measurement of the refractive index and the geometrical thickness by use of a wavelength-scanning interferometer with a confocal microscope,” Appl. Opt. 38, 4065–4073 (1999).
    [CrossRef]
  16. J. Villatoro, A. García-Valenzuela, “Sensitivity of optical sensors based on laser-excited surface-plasmon waves,” Appl. Opt. 38, 4837–4844 (1999).
    [CrossRef]
  17. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).
  18. G. H. Meeten, A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol. 6, 214–225 (1995).
    [CrossRef]
  19. G. H. Meeten, A. N. North, “Refractive index measurement of turbid colloid fluids by transmission near the critical angle,” Meas. Sci. Technol. 2, 441–447 (1991).
    [CrossRef]
  20. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 3, p. 39.

2000 (1)

1999 (3)

1998 (2)

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

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

1997 (2)

J. Rheims, J. Köser, T. Wriedt, “Refractive index measurements in the near-IR using an Abbe refractometer,” Meas. Sci. Technol. 8, 601–605 (1997).
[CrossRef]

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]

1996 (2)

1995 (1)

G. H. Meeten, A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol. 6, 214–225 (1995).
[CrossRef]

1994 (1)

1992 (2)

1991 (1)

G. H. Meeten, A. N. North, “Refractive index measurement of turbid colloid fluids by transmission near the critical angle,” Meas. Sci. Technol. 2, 441–447 (1991).
[CrossRef]

1988 (1)

1986 (1)

P. R. Jarvis, G. H. Meeten, “Critical-angle measurement of refractive index of absorbing materials: an experimental study,” J. Phys. E 19, 296–298 (1986).
[CrossRef]

1984 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

de Greef, C.

Diaz-Uribe, R.

Finsy, R.

Flandes-Aburto, V.

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

Fukano, T.

García-Segundo, C.

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

García-Valenzuela, A.

Gass, P. A.

Huang, P. S.

Hui, L.

Jarvis, P. R.

P. R. Jarvis, G. H. Meeten, “Critical-angle measurement of refractive index of absorbing materials: an experimental study,” J. Phys. E 19, 296–298 (1986).
[CrossRef]

Kamada, O.

Kasunic, K. J.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 3, p. 39.

Kiyono, S.

Kohno, T.

Köser, J.

J. Rheims, J. Köser, T. Wriedt, “Refractive index measurements in the near-IR using an Abbe refractometer,” Meas. Sci. Technol. 8, 601–605 (1997).
[CrossRef]

Meeten, G. H.

G. H. Meeten, A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol. 6, 214–225 (1995).
[CrossRef]

G. H. Meeten, A. N. North, “Refractive index measurement of turbid colloid fluids by transmission near the critical angle,” Meas. Sci. Technol. 2, 441–447 (1991).
[CrossRef]

P. R. Jarvis, G. H. Meeten, “Critical-angle measurement of refractive index of absorbing materials: an experimental study,” J. Phys. E 19, 296–298 (1986).
[CrossRef]

Miyamoto, K.

Moreels, E.

Musha, T.

Nemoto, S.

Ni, J.

North, A. N.

G. H. Meeten, A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol. 6, 214–225 (1995).
[CrossRef]

G. H. Meeten, A. N. North, “Refractive index measurement of turbid colloid fluids by transmission near the critical angle,” Meas. Sci. Technol. 2, 441–447 (1991).
[CrossRef]

Ozawa, N.

Peña-Gomar, M.

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

Rheims, J.

J. Rheims, J. Köser, T. Wriedt, “Refractive index measurements in the near-IR using an Abbe refractometer,” Meas. Sci. Technol. 8, 601–605 (1997).
[CrossRef]

Sambles, J. R.

Schalk, S.

Shusen, X.

Villatoro, J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Wriedt, T.

J. Rheims, J. Köser, T. Wriedt, “Refractive index measurements in the near-IR using an Abbe refractometer,” Meas. Sci. Technol. 8, 601–605 (1997).
[CrossRef]

Yamaguchi, I.

Appl. Opt. (13)

T. Kohno, N. Ozawa, K. Miyamoto, T. Musha, “High-precision optical surface sensors,” Appl. Opt. 27, 103–108 (1988).
[CrossRef]

P. S. Huang, S. Kiyono, O. Kamada, “Angle measurement based on the internal reflection effect: a new method,” Appl. Opt. 31, 6047–6055 (1992).
[CrossRef] [PubMed]

P. S. Huang, J. Ni, “Angle measurement based on the internal reflection effect using elongated critical-angle prisms,” Appl. Opt. 35, 2239–2241 (1996).
[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]

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

P. S. Huang, “Use of thin films for high-sensitivity angle measurement,” Appl. Opt. 38, 4831–4836 (1999).
[CrossRef]

K. J. Kasunic, “Comparison of Kretschmann–Raether angular regimes for measuring changes in bulk refractive index,” Appl. Opt. 39, 61–64 (2000).
[CrossRef]

S. Nemoto, “Measurement of the refractive index of liquid using laser beam displacement,” Appl. Opt. 31, 6690–6694 (1992).
[CrossRef] [PubMed]

E. Moreels, C. de Greef, R. Finsy, “Laser light refractometer,” Appl. Opt. 23, 3010–3013 (1984).
[CrossRef] [PubMed]

T. Fukano, I. Yamaguchi, “Separation of measurement of the refractive index and the geometrical thickness by use of a wavelength-scanning interferometer with a confocal microscope,” Appl. Opt. 38, 4065–4073 (1999).
[CrossRef]

J. Villatoro, A. García-Valenzuela, “Sensitivity of optical sensors based on laser-excited surface-plasmon waves,” Appl. Opt. 38, 4837–4844 (1999).
[CrossRef]

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

L. Hui, X. Shusen, “Measurement method of the refractive index of biotissue by total internal reflection,” Appl. Opt. 35, 1793–1795 (1996).
[CrossRef]

J. Phys. E (1)

P. R. Jarvis, G. H. Meeten, “Critical-angle measurement of refractive index of absorbing materials: an experimental study,” J. Phys. E 19, 296–298 (1986).
[CrossRef]

Meas. Sci. Technol. (3)

J. Rheims, J. Köser, T. Wriedt, “Refractive index measurements in the near-IR using an Abbe refractometer,” Meas. Sci. Technol. 8, 601–605 (1997).
[CrossRef]

G. H. Meeten, A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol. 6, 214–225 (1995).
[CrossRef]

G. H. Meeten, A. N. North, “Refractive index measurement of turbid colloid fluids by transmission near the critical angle,” Meas. Sci. Technol. 2, 441–447 (1991).
[CrossRef]

Sensors Actuators B (1)

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

Other (2)

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 3, p. 39.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

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Figures (8)

Fig. 1
Fig. 1

Plot of J(η) as a function of η for b = 0.0, 0.2, 0.4, … , 4. The curves appear in successive order, with the highest curve corresponding to b = 0. The vertical dashed line corresponds to the critical angle. We generated the plot by numerically integrating the integral defined in Eq. (6).

Fig. 2
Fig. 2

Plot of K(η) as a function of η for b = 0.0, 0.2, 0.4, … , 4. The curves appear in successive order, with the highest curve corresponding to b = 0. The vertical dashed line corresponds to the critical angle. We generated the plot by numerically integrating the integral defined in Eq. (6).

Fig. 3
Fig. 3

Dependence of the position of the maximum, η m , of J(η) and its value at maximum, J m ).

Fig. 4
Fig. 4

Decrease of sensitivity as parameter b increases. Solid curve, exact numerical calculations with the integral definition of K(η) in Eq. (6). Open circles, 1/(1 + αb)2, with α = 0.56811. Dashed lines localize the points where S n = 1/2 and S n = 1/10.

Fig. 5
Fig. 5

Fig. 5. (k 1ω0)3/2 times normalized sensitivity S n as a function of the radius of beam waist ω0 for n 1 = 1.51, 2 ≈ 1.33, and λ = 0.63 µm. We calculated the curves by using Eq. (10). Curves correspond to absorption coefficients a from 10 to 120 cm-1 in steps of 10 cm-1. The first two are labeled; the rest are in successive order.

Fig. 6
Fig. 6

Schematic of the experimental arrangement that we used to measure the first differential of the reflectivity. PZT, for a bimorph piezoelectric cell used to vibrate the mirror.

Fig. 7
Fig. 7

First differential of the sensitivity as a function of the angle of incidence. Continuous curves, theory, squares, experimental values. (a) a = 0 (pure water), (b) a = 0.85 cm-1, (c) a = 8.1 cm-1, (d) a = 13 cm-1. The experimental uncertainties in the measurements are contained within the filled squares.

Fig. 8
Fig. 8

Signal response after injection of 0.1 mL of EtOH into 60 mL of a blue ink–water solution with an absorption coefficient of a = 13 cm-1.

Tables (1)

Tables Icon

Table 1 Numerical Examples of Maximum of First and Second Differentials and Angular Locations of Maximum of the First Differential and of the Normalized Sensitivity for Several Values of the Absorption Coefficienta

Equations (12)

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

Rθi=ω0k12π- |fθ|2 exp-ω02k122θ-θi2dθ,
fθ=ky1-αky2ky1+αky2,
fθ=1-2α ky2ky1.
|fθ|2=1-4αn12-ε2hγ+h2γ2+ε221/221/2,
Rθi=1+DIη2πω0k11/2,
Iη=-1/2u+u2+b21/2 exp-1/2×u-η2du, D=4α4ε2/n12-ε21/4, b=ω0k1ε2/h.
dRdθi=Dω0k11/22π Jη, d2Rdθi2=Dω0k13/22π Kη,
Jη=-1/2u+u2+b21/2u-ηexp-1/2×u-η2du, Kη=-1/2u+u2+b21/2u-η2-1exp-1/2u-η2du.
a=bh/n1nˆ2ω0.
dRdθimax=1.28 Dω0k11/22π, d2Rdθi2max=0.669 Dω0k13/22π.
S=ΔirmsΔnˆ2=κP0δθi2n12-nˆ22d2Rdθi2=κP0δθiD4πn12-nˆ221/2ω0k13/2Kη,
S=0.699×κP0Dδθi4πn12-nˆ22ω0k13/21+0.56811n1nˆ2ω0a/h2.

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