## A simple method to measure critical angles for high-sensitivity differential refractometry |

Optics Express, Vol. 20, Issue 2, pp. 1862-1867 (2012)

http://dx.doi.org/10.1364/OE.20.001862

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### Abstract

A total internal reflection-based differencial refractometer, capable of measuring the real and imaginary parts of the complex refractive index in real time, is presented. The device takes advantage of the phase difference acquired by *s*- and *p*-polarized light to generate an easily detectable minimum at the reflected profile. The method allows to sensitively measuring transparent and turbid liquid samples.

© 2012 OSA

## 1. Introduction

*θ*

_{c}) in total internal reflection (TIR) has found increasing use in differential refractometry, particularly for absorbing or turbid samples [1

1. G. H. Meeten and A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol. **6**(2), 214–221 (1995). [CrossRef]

4. W. R. Calhoun, H. Maeta, S. Roy, L. M. Bali, and S. Bali, “Sensitive real-time measurement of the refractive index and attenuation coefficient of milk and milk-cream mixtures,” J. Dairy Sci. **93**(8), 3497–3504 (2010). [CrossRef] [PubMed]

*et al*. [5

5. M. McClimans, C. LaPlante, D. Bonner, and S. Bali, “Real-time differential refractometry without interferometry at a sensitivity level of 10^{(-6)},” Appl. Opt. **45**(25), 6477–6486 (2006). [CrossRef] [PubMed]

^{−6}by using a divergent laser beam, polarized parallel to the plane of incidence (

*p*-polarization), and a linear diode array to precisely find the critical angle. The technique relies on measuring the intensity angular distribution of the divergent laser beam totally reflected by a high-index prism. Normalizing the profile obtained with a particular liquid sample to that obtained with air yields the Fresnel reflectance. At the vicinity of

*θ*

_{c}the reflection changes abruptly, and then, it is possible to find the critical angle with a good sensitivity. Since the absolute value of

*θ*

_{c}is not measured, the results have to be calibrated against a known sample, with an Abbe refractometer or similar, which turn this into a differential technique that needs an origin for refractive indices values.

*p*) and perpendicular (

*s*) to the plane of incidence, to generate a signal that has a minimum at

*θ*

_{c}. In the particular case where the medium is transparent, the value of the minimum is zero. As a consequence, one has a dark stripe on the reflected intensity pattern at the critical angle that makes its determination very easy. For turbid samples, the minimum does not reach zero, but even though is easily determined. This method promptly provides the values of the refractive index and extinction coefficient in real time, without any complicated fitting procedure.

## 2. Experimental section

5. M. McClimans, C. LaPlante, D. Bonner, and S. Bali, “Real-time differential refractometry without interferometry at a sensitivity level of 10^{(-6)},” Appl. Opt. **45**(25), 6477–6486 (2006). [CrossRef] [PubMed]

## 3. Origin of the dark stripe

*θ*

_{c}, we first consider a transparent medium and write the Fresnel refection coefficients for

*s*- and

*p*-polarized light as [6]:

*θ*is the angle of incidence and n = n

_{sample}/n

_{glass}< 1 is the relative refractive index. Above the Brewster’s angle, the

*p*-component acquires a phase π and changes its signal, thus rotating the polarization by 2

*α*, where tan

*α*= |

*r*

_{s}|/|

*r*

_{p}|. Therefore, a fraction of the reflected light passes through the analyzer, as shown in the left part of Fig. 2(a) . As the incident angle approaches

*θ*

_{c}, |

*r*

_{s}| and |

*r*

_{p}| tend to 1, but because the electric field of the

*p*-polarized light gained a π phase change, the polarization of the reflected light is rotated by 90°. As consequence, the light is blocked by the analyzer. Above the critical angle, the amplitudes are always the same (|

*r*

_{s}| = |

*r*

_{p}| = 1), but there is a phase difference occurring during the total internal reflection, given by [6]:

*θ*

_{c}, as seen in the theoretical curve (a) of Fig. 2 and in the experimental pattern of Fig. 3(a) .

7. M. H. Chiu, J. Y. Lee, and D. C. Su, “Complex refractive-index measurement based on Fresnel’s equations and the uses of heterodyne interferometry,” Appl. Opt. **38**(19), 4047–4052 (1999). [CrossRef] [PubMed]

3. W. R. Calhoun, H. Maeta, A. Combs, L. M. Bali, and S. Bali, “Measurement of the refractive index of highly turbid media,” Opt. Lett. **35**(8), 1224–1226 (2010). [CrossRef] [PubMed]

4. W. R. Calhoun, H. Maeta, S. Roy, L. M. Bali, and S. Bali, “Sensitive real-time measurement of the refractive index and attenuation coefficient of milk and milk-cream mixtures,” J. Dairy Sci. **93**(8), 3497–3504 (2010). [CrossRef] [PubMed]

*k*by k(θ) = k[4π

^{−1}, where L = n

^{2}- k

^{2}-sin

^{2}θ and M

^{2}= P

^{2}- 2L sin

^{2}θ - sin

^{4}θ, with P

^{2}= n

^{2}+ k

^{2}. After some algebra, the above set of equations provides the amplitudes and phase of the reflectivity coefficients. The intensity reaching the CCD is proportional to |

*r*

_{s}|

^{2}+ |

*r*

_{p}|

^{2}+ 2 |

*r*

_{s}| |

*r*

_{p}| cos(δ

_{s}- δ

_{p}) and is shown in Fig. 2 for two values of attenuation coefficients, defined as α = (4π/λ) k n

_{glass}. Therefore, by measuring the values of the minimum position and its amplitude one can find the complex refractive index. For practical purposes, we analyzed several curves like those of Fig. 2 and found that for

*α*< 200 cm

^{−1}, the minimum can be characterized approximately by:

*θ*

_{min}and

*V*

_{min}have to be calibrated against some reference sample with known values of

*n*and

*k*. The calibration for

*n*may be achieved by measuring two transparent solutions in a commercial refractometer and the calibration for

*k*may be obtained by analyzing the transmission of a collimated beam through a thin cell containing some phantom material, according to the method described in [8

8. S. T. Flock, B. C. Wilson, and M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. **14**(5), 835–841 (1987). [CrossRef] [PubMed]

## 4. Results and discussion

*θ*

_{c}can be found with a good sensitivity, even though the minimum value is not zero.

5. M. McClimans, C. LaPlante, D. Bonner, and S. Bali, “Real-time differential refractometry without interferometry at a sensitivity level of 10^{(-6)},” Appl. Opt. **45**(25), 6477–6486 (2006). [CrossRef] [PubMed]

*θ*

_{c}( = sin

^{−1}n

_{water}/n

_{SF10}) for the several spectral lines of the laser and plot it against the pixel position where the dark line appears. For a given wavelength

*λ*

_{0}, the critical angle measured is denoted as

*θ*

_{c}(λ

_{0}). For a nearby wavelength

*λ*, the new angle is

*θ*

_{c}(λ) =

*θ*

_{c}(λ

_{0}) + Δ

*θ*, such that n(λ) = sin

*θ*

_{c}(λ) ≈n(λ

_{0}) + cos

*θ*

_{c}(λ

_{0}) Δ

*θ*, for small Δ

*θ*. Therefore, a linear behavior between

*n*and Δ

*θ*is expected for sufficiently small variations of

*n*. Indeed, such dependence was observed for the orange/red lines (594, 604, 612 and 633 nm) but the green line (543 nm) deviates somewhat from the linear behavior. Furthermore, the chromatic focal shift should also be considered in the case of large spectral variation. In this experiment n

_{water}/n

_{SF10}changed by more than 10

^{−3}, but in measurements where the relative indices change less than 10

^{−4}, and the wavelength is fixed, the linear behavior can be safely assumed.

*θ*

_{c}and the pixel number, we used this refractometer to demonstrate changes in the refractive index that are smaller than 10

^{−5}. Solutions of different volume concentrations of ethanol (n = 1.36371 @ 532 nm, 20 °C) in water (n = 1.33524 @ 532 nm, 21.5 °C), ranging from 0 to 1%, were used. The values of refractive indices of ethanol, water and the SF10 glass were taken from [9]. For calibration, we used a commercial Pulfrich refractometer with a resolution better than 10

^{−5}. The indices were measured at 546 nm (Hg, spectral line

*e*) and then corrected to 532 nm. The temperature was kept constant in ±0.2 °C around 22 °C. Figure 4 depicts the results obtained with the Pulfrich refractometer and with the proposed technique, proving that it is capable of providing sensitivity better than 10

^{−5}. The refractive indices obtained with the Pulfrich refractometer can be used to calibrate the pixel number scale. Note that after a calibration, prior to carry out any further measurements of unknown samples, the position of the CCD relative to the lens cannot be changed.

*α*< 200 cm

^{−1}) [4

4. W. R. Calhoun, H. Maeta, S. Roy, L. M. Bali, and S. Bali, “Sensitive real-time measurement of the refractive index and attenuation coefficient of milk and milk-cream mixtures,” J. Dairy Sci. **93**(8), 3497–3504 (2010). [CrossRef] [PubMed]

*V*

_{min}. Three types of milk of the same brand were used, with nominal fat concentration of ~0%, 1% and 3%. We don’t have any information on the accuracy of these values. Figure 5(a)-(c) shows the the dark line pattern observed for each concentration, while Fig. 5(d) gives the refractive index, calibrated with the Pulfrich refractometer using water/ethanol solutions, and the attenuation coefficient, calibrated in transmission measurements of the 532 nm collimated laser beam through a 200 μm-thick quartz cell containing either our milk samples or the water/ethanol solution as reference [8

8. S. T. Flock, B. C. Wilson, and M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. **14**(5), 835–841 (1987). [CrossRef] [PubMed]

2. A. J. Jääskeläinen, K. E. Peiponen, and J. A. Räty, “On reflectometric measurement of a refractive index of milk,” J. Dairy Sci. **84**(1), 38–43 (2001). [CrossRef] [PubMed]

**93**(8), 3497–3504 (2010). [CrossRef] [PubMed]

## 4. Conclusions

*s*- and

*p*-polarizations acquire upon internal reflection. The light reflected by the interface between the base of the semi-cylindrical lens and the sample presents a dark line that can be used to find the real (position of the line) and imaginary (value of the minimum) parts of the complex refractive index in real time. This method allowed the buildup of a sensitive, rugged and user friendly differential refractometer for transparent and turbid samples [10].

## Acknowledgments:

## References and links

1. | G. H. Meeten and A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol. |

2. | A. J. Jääskeläinen, K. E. Peiponen, and J. A. Räty, “On reflectometric measurement of a refractive index of milk,” J. Dairy Sci. |

3. | W. R. Calhoun, H. Maeta, A. Combs, L. M. Bali, and S. Bali, “Measurement of the refractive index of highly turbid media,” Opt. Lett. |

4. | W. R. Calhoun, H. Maeta, S. Roy, L. M. Bali, and S. Bali, “Sensitive real-time measurement of the refractive index and attenuation coefficient of milk and milk-cream mixtures,” J. Dairy Sci. |

5. | M. McClimans, C. LaPlante, D. Bonner, and S. Bali, “Real-time differential refractometry without interferometry at a sensitivity level of 10 |

6. | G. R. Fowles, Introduction to Modern Optics, Holt, Rinehart and Winston, Inc., New York, 1968. |

7. | M. H. Chiu, J. Y. Lee, and D. C. Su, “Complex refractive-index measurement based on Fresnel’s equations and the uses of heterodyne interferometry,” Appl. Opt. |

8. | S. T. Flock, B. C. Wilson, and M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. |

9. | |

10. | S. C. Zilio, “Refratômetro diferencial para medir o índice de refração e coeficiente de atenuação de um líquido em tempo real”, Patent pending. |

**OCIS Codes**

(230.0230) Optical devices : Optical devices

(260.6970) Physical optics : Total internal reflection

(290.7050) Scattering : Turbid media

(280.4788) Remote sensing and sensors : Optical sensing and sensors

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: November 21, 2011

Revised Manuscript: December 27, 2011

Manuscript Accepted: December 27, 2011

Published: January 12, 2012

**Virtual Issues**

Vol. 7, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

S. C. Zilio, "A simple method to measure critical angles for high-sensitivity differential refractometry," Opt. Express **20**, 1862-1867 (2012)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-2-1862

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### References

- G. H. Meeten and A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol.6(2), 214–221 (1995). [CrossRef]
- A. J. Jääskeläinen, K. E. Peiponen, and J. A. Räty, “On reflectometric measurement of a refractive index of milk,” J. Dairy Sci.84(1), 38–43 (2001). [CrossRef] [PubMed]
- W. R. Calhoun, H. Maeta, A. Combs, L. M. Bali, and S. Bali, “Measurement of the refractive index of highly turbid media,” Opt. Lett.35(8), 1224–1226 (2010). [CrossRef] [PubMed]
- W. R. Calhoun, H. Maeta, S. Roy, L. M. Bali, and S. Bali, “Sensitive real-time measurement of the refractive index and attenuation coefficient of milk and milk-cream mixtures,” J. Dairy Sci.93(8), 3497–3504 (2010). [CrossRef] [PubMed]
- M. McClimans, C. LaPlante, D. Bonner, and S. Bali, “Real-time differential refractometry without interferometry at a sensitivity level of 10(-6),” Appl. Opt.45(25), 6477–6486 (2006). [CrossRef] [PubMed]
- G. R. Fowles, Introduction to Modern Optics, Holt, Rinehart and Winston, Inc., New York, 1968.
- M. H. Chiu, J. Y. Lee, and D. C. Su, “Complex refractive-index measurement based on Fresnel’s equations and the uses of heterodyne interferometry,” Appl. Opt.38(19), 4047–4052 (1999). [CrossRef] [PubMed]
- S. T. Flock, B. C. Wilson, and M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys.14(5), 835–841 (1987). [CrossRef] [PubMed]
- http://refractiveindex.inf
- S. C. Zilio, “Refratômetro diferencial para medir o índice de refração e coeficiente de atenuação de um líquido em tempo real”, Patent pending.

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