# Two Prism Critical Angle Refractometry with Attenuating Media

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background Theory

## 3. Method’s Description

`NSolve`, to conclude that the real roots are always unique, at least within the physically meaningful range $0<{R}_{c},{R}_{c}^{\prime}<1$.

## 4. Uncertainty Assessment

## 5. Limitations and Further Insights

## 6. Conclusions

- 1.
- A pair of angles (${\theta}_{c},{\theta}_{c}^{\prime}$) is all that the method needs as input. To the benefit of precision, reflectance values are not required in the computational process.
- 2.
- The routine is general, since it applies with s and p polarised light, facilitating the characterisation of optically isotropic and anisotropic samples.
- 3.
- Uncertainty in the determination of the output quantities (${n}_{r},{n}_{i}$) decreases as attenuation grows from zero, which is ideally suited for analysing media that attenuate light as much as, for example, most forms of biological matter.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) A typical prism-coupling refractometer. The sample is attached to the base of a transparent reference prism, which sits on a rotary table. Incoming through the prism’s front facet, a collimated linearly polarized laser beam hits the interface at a variable incidence angle $\theta $. The reflected light exits the prism through its rear facet, heading towards a photodiode (PD), where the reflectance profile $R\left(\theta \right)$ is monitored. (

**b**) Fresnel reflectance profiles $R\left(\theta \right)$ at two interfaces between a sample and two different prisms. Bell-like curves are corresponding derivatives $dR/d\theta $, peaking at ${\theta}_{c}$ and ${\theta}_{c}^{\prime}$. Calculations assume s polarisation and $n=0.75$, $k=5\xb7{10}^{-3}$, $\mathrm{\Lambda}=1.2$.

**Figure 2.**(

**a**) Iso-${\theta}_{c}$ curves in the ($n,k$) plane (${\theta}_{c}=48.{5}^{\circ}$) arrows mark the direction of increasing $\mathrm{\Delta}$ in steps of $0.{001}^{\circ}$, starting from a base $\mathrm{\Delta}=9.{882}^{\circ}\phantom{\rule{0.222222em}{0ex}}(9.{845}^{\circ})$ for s (p) polarisation. (

**b**) Iso-n and iso-k curves in the (${\theta}_{c},\mathrm{\Delta}$) plane. Parameter n assumes constant values $0.749,\phantom{\rule{0.222222em}{0ex}}0.7495,\phantom{\rule{0.222222em}{0ex}}0.75,\phantom{\rule{0.222222em}{0ex}}0.7505$ and $0.751$ for both polarisations. Parameter k assumes constant values $0,\phantom{\rule{0.222222em}{0ex}}0.0025,\phantom{\rule{0.222222em}{0ex}}0.005,\phantom{\rule{0.222222em}{0ex}}0.0075$ and $0.001$ for s polarisation, as well as 0 and $0.01$ for p polarisation. The common transparency line ($k=0$) separates the s polarisation grid on the left, from the p polarisation grid on the right.

**Figure 3.**Normalized relative error ${U}_{real}$ (

**a**) and ${U}_{im}$ (

**b**) as a function of k, for both polarisation states. Throughout, we assume constant values for $n=0.75$ and $\mathrm{\Lambda}=1.2$.

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**MDPI and ACS Style**

Koutsoumpos, S.; Giannios, P.; Moutzouris, K. Two Prism Critical Angle Refractometry with Attenuating Media. *Instruments* **2022**, *6*, 21.
https://doi.org/10.3390/instruments6030021

**AMA Style**

Koutsoumpos S, Giannios P, Moutzouris K. Two Prism Critical Angle Refractometry with Attenuating Media. *Instruments*. 2022; 6(3):21.
https://doi.org/10.3390/instruments6030021

**Chicago/Turabian Style**

Koutsoumpos, Spyridon, Panagiotis Giannios, and Konstantinos Moutzouris. 2022. "Two Prism Critical Angle Refractometry with Attenuating Media" *Instruments* 6, no. 3: 21.
https://doi.org/10.3390/instruments6030021