Precise determination of the optical properties of turbid media using an optimized integrating sphere and advanced Monte Carlo simulations. Part 2: experiments

Abstract

Based on theoretical investigations of the light propagation within an integrating sphere, we developed an accurate method to determine the optical properties of scattering media using an integrating sphere-based setup. The method takes into account the exact sphere geometry as well as the different angular distributions of the reflected and transmitted light from the sample and the calibration standard. We tested our novelties successfully in theory with Monte Carlo simulations and in practice using a 3D printed and professionally coated integrating sphere. As a result, we were able to determine precisely the effective scattering coefficient, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msubsup> <mml:mi>μ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> </mml:mrow> <mml:mi class="MJX-variant" mathvariant="normal">′</mml:mi> </mml:msubsup> </mml:math> , and the absorption coefficient, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>a</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , between 400 nm and 1500 nm in a range of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>a</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>e</mml:mi> <mml:mo>−</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> <mml:mspace width="thickmathspace"/> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">m</mml:mi> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>10</mml:mn> <mml:mspace width="thickmathspace"/> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">m</mml:mi> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msubsup> <mml:mi>μ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> </mml:mrow> <mml:mi class="MJX-variant" mathvariant="normal">′</mml:mi> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mn>0.2</mml:mn> <mml:mspace width="thickmathspace"/> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">m</mml:mi> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>100</mml:mn> <mml:mspace width="thickmathspace"/> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">m</mml:mi> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . Usually, the accuracy was around 1% for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msubsup> <mml:mi>μ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">s</mml:mi> </mml:mrow> <mml:mi class="MJX-variant" mathvariant="normal">′</mml:mi> </mml:msubsup> </mml:math> and around 3% for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>a</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> for turbid phantom media with an optical thickness <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>τ</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>μ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> </mml:mrow> <mml:mi class="MJX-variant" mathvariant="normal">′</mml:mi> </mml:msubsup> <mml:mi>d</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:math> and a transmittance signal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mo>&gt;</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> <mml:mo>.</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi mathvariant="normal">%</mml:mi> </mml:math> .

Topics & Concepts