Reflectivity of a Gaussian beam near the critical angle with external optically absorbing media

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.

Design of an optical probe based on angle scanning reflectometry near the critical angle

“…1. 9,10 The internal reflection takes place at the flat side of the lens, which is in contact with the sample ͑interface plane͒. To scan the angle of incidence, either the laser or the lens L must be rotated about the reflection point C. As the angle of incidence changes, the reflected beam rotates and the detector must be displaced to a new position.…”

“…7,9,10 It turns out that the shape of the reflectivity curve and its angle differential around the critical angle depend strongly on the beam's waist radius for typical laser beams. 9 In general terms, the larger the beam's waist radius ͑i.e., the more collimated the beam is͒, the larger the sensitivity around the critical angle. In the laboratory setup, one may use a collimating lens (L 1 in Fig.…”

“…In previous works, [7][8][9][10] we have studied techniques based on angle scanning reflectometry ͑ASR͒ near the critical angle for measuring the real and imaginary parts of the RI of transparent, absorbing, and turbid liquids, as well as for sensing small variations of the refractive index with high resolution. The angle differential profile near the critical angle is measured by using what we could call angle scanning dynamic reflectometry ͑ASDR͒.…”

“…The angle differential profile near the critical angle is measured by using what we could call angle scanning dynamic reflectometry ͑ASDR͒. [7][8][9][10] The technique consists of harmonically modulating the angle of incidence of a laser beam and measuring the amplitude of the ac signal at the photodetector. The mean angle of incidence can be scanned about the critical angle by mechanical rotation of the laser or of the optical lens used as the internal medium in the internal reflection with the sample.…”

<title>Optical device for sensing the index of refraction of liquids with high turbidity</title>

“…The case of using modulated PRCA with a gaussian beam for sensing highly absorbing media was recently analyzed in Ref. [7], where simple expressions of the reflectance and its first and second derivatives near the critical angle were derived. E-mail: garciaa(a1eph.cinstrum.unam.mx; Fax: (525) 622 8620…”