© Huw Owens - University of Manchester : Interactions of Light and Matter Refraction Reflection Absorption Dr Huw Owens

© Huw Owens - University of Manchester : Introduction Refraction of Light – Snell’s Law Surface Reflection of Light – Fresnel’s Law Absorption of light – Beer-Lambert Law

© Huw Owens - University of Manchester : Refraction of Light – Snell’s Law Snell’s Law – When light travels through a medium of refractive index n 1 Encounters and enters a medium of refractive index n 2 then the light beam is bent through an angle according to the following equation :- Where i is the angle of incidence and r is the angle of refraction. An Example:- A typical paint resin has a refractive index similar to that of ordinary glass (n=1.5) and so a beam of radiation incident on the surface at 45 degrees will be bent towards towards the normal by 17 degrees to a refraction angle of approximately 28 degrees.

© Huw Owens - University of Manchester : Refraction of Light – Snell’s Law continued Pigment particles Embedded in polymer Air(refractive index n i ) Paint medium (refractive index n r ) i r i = angle of incidence r = angle of refraction

© Huw Owens - University of Manchester : Diffuse Reflection

© Huw Owens - University of Manchester : Effect of angle of incidence

© Huw Owens - University of Manchester : Surface Reflection of Light – Fresnel’s Law A light beam incident normally (vertically) on a surface or any boundary between two phases of differing refractive index will suffer partial back-reflection according to Fresnel’s law: If the incident light beam is white then the light reflected from the surface will also be white. The small percentage of the white light reflected from the surface affects the visually perceived colour, and instrumentally measured reflectance values should indicate whether specular reflection is included (SPIN) or excluded (SPEC). Where ρ is the reflection factor for unpolarised light and n is n 2 /n 1

© Huw Owens - University of Manchester : Surface Reflection of Light

© Huw Owens - University of Manchester : Colour Models A colour model is a mathematical relationship between the input to a coloration system and the output, expressed as colorimetric coordinates. The physics and chemistry of a coloration system are more appropriately expressed in spectral terms. Mixing laws that are spectrally based are always more accurate. A colour mixing law defines a relationship between the colour components and the spectral data such that the relationship is linear.

© Huw Owens - University of Manchester : Linear Models A linear model has two requirements: Scalability – The spectral data should be scalable. That is, changes in the amount of colorant should only affect the level or amplitude of the spectral curve, not its shape. Additivity – The spectral data of mixtures should be an additive combination of the measurement of the individual components. (For example, the measurements of a display’s white should be equal to the sum of the measurements of the individual red, green and blue channels)

© Huw Owens - University of Manchester : Linear Models - Scalability

© Huw Owens - University of Manchester : Linear Models - continued The main advantages of using linear systems is the use of algebra. Scalability and additivity are always assumed when performing algebra (e.g. solving simultaneous equations) In some cases the relationship between the scalars of the linear system and the controls of the coloration system is nonlinear. The nonlinearity follows and exponential function that is called gamma. In textile dyeing, increased concentration starts to have a reduced effect in colouring the substrates the amount of dye approaches this maximum that can be applied. This also results in a nonlinearity. We us the term effective concentration to represent the linear system scalars.

© Huw Owens - University of Manchester : CRT Display Gamma

© Huw Owens - University of Manchester : Dyebath concentration

© Huw Owens - University of Manchester : Simple-Subtractive Mixing Subtractive mixing – refers to the removal of light coming from a light source by an object. The ways in which light can be removed include absorption and scattering. We call the case that involves only absorption and not scattering simple-subtractive mixing. We call the more complex situation, where there is scattering and absorption, complex-subtractive mixing. Examples of materials that do not scatter light (ignoring first surface reflections) include coloured filters (glass, gelatin and plastic) and coloured liquids

© Huw Owens - University of Manchester : Simple-Subtractive Mixing The appropriate colour-mixing law results in a linear system. The spectral properties of each colorant are scalable, and the mixtures are and additive combination of the individual components. Evaluation of the mixing law – Produce a series of specimens by varying the amount of a single dye (known as a tint ladder, a concentration or thickness series, or a colour ramp) Produce a set of mixtures using several colorants of known amounts.

© Huw Owens - University of Manchester : Simple-Subtractive Mixing Evaluation of the mixing law continued– Rather than varying the amount of colorant in a fixed amount of material (e.g. Varying the concentration of a dye in a liquid cell) we can fix the amount of colorant while varying the amount of material (e.g. producing a series of coloured glasses of different thickness) Mixtures can be made by gluing different coloured glasses into “sandwiches”.

© Huw Owens - University of Manchester : Simple-Subtractive Mixing Linear System In 1729, Bouguer discovered spectral transmittance could be transformed using logarithms to achieve a linear system. His experiments used coloured glass of different thicknesses. Suppose a 1-cm thick glass transmitted one-half of the light incident upon it at some arbitrary wavelength. It may seem that if the glass were 2-cm thick then it would transmit no light. BUT, he found that transmittance was ¼ of the light. If the glass thickness was 3-cm thick then the transmittance was 1/8 of the light.

© Huw Owens - University of Manchester : Bouguer – Thickness and Spectral Transmittance 1cm 2cm 3cm 1cm 100% 50% 25% 12.5%, where T λ,i is the internal spectral transmittance, t λ is the internal spectral transmittance at unit thickness (i.e. 1cm) and b is the thickness. NB, The “internal transmittance” refers to the transmittance “inside” of the material - that is, the transmittance taking into account specular reflections at each surface - due to changes in refractive index.

© Huw Owens - University of Manchester : Absorption and Thickness Each centimetre of material has an exponential effect rather than a subtractive effect. This led to the logarithmic transformation of transmission known as absorption. If the absorption is 0.3 for a 1-cm thick piece of glass, it will be 0.6 for a 2-cm thick glass and 0.9 for a 3-cm thick glass. Thus the scalability requirement is satisfied when we describe the glass by its spectral absorption rather than its spectral transmittance. In 1760 Lambert rediscovered Bouguer’s law and often “Lambert’s law” is used to describe the relationship between absorption and thickness.

© Huw Owens - University of Manchester : Absorption and Thickness About 100 years later Beer found the same principles that describe the relationship between transmittance and thickness for transparent materials applied to liquids of varying concentration (Beer, 1852; 1854). Thus the linear mixing law for coloured materials that do not scatter light is known as the Bouguer-Beer law or the Lambert-Beer law. The Bouguer-Beer law states that absorbance, A λ, is equal to the product of the colorant’s absorptivity, a λ, the thickness of the sample b and the concentration of the colorant, c: For colour mixtures, the absorbances of each individual colorant add together:

© Huw Owens - University of Manchester : Beer’s law Beer found that changes in concentration also had an exponential effect on transmittance 1 unit of colorant 100% 50% 2 units of colorant 100% 25%, where T λ,i is the internal spectral transmittance, t λ is the internal spectral transmittance at unit thickness (I.e. 1cm) and b is the Thickness.

© Huw Owens - University of Manchester : Testing the mixing law We use a colour ramp to test the validity of the selected mixing law. The spectral transmittances are transformed into spectral absorbances after first accounting for surface reflections and any inherent colour on the substrate (Bouguer’s law) or solvent and cell (Beer’s law). When the spectral curves are normalised and plotted, they should lie nearly on top of one another. Thus we have scalability.scalability

© Huw Owens - University of Manchester : Additivity The additivity requirement is verified by measuring the spectral transmittances of several mixtures and transforming into spectral absorbances. This is done by solving simultaneous equations or least squares. The measured and estimated curves should nearly lie on top of one another. The final step is to define scalars for each colorant and relating them to thickness or concentration.

© Huw Owens - University of Manchester : Numerical Example Finding Dye concentrations in a transparent sample by the use of Bouguer-Beer’s law Sample Y contains 1.38 units of yellow dye Sample B contains 1.07 units of blue dye Sample G contains unknown amounts of the yellow and the blue dyes How do we find the unknown dye concentrations?

© Huw Owens - University of Manchester : Numerical Example (Continued) Because there are two unknown concentrations to be determined, we need to consider what happens at two wavelengths. These should be chosen at relatively flat parts of the absorbance curves – one where the yellow dye absorbs strongly (high absorbance) and the blue dye absorbs very little. The other where the reverse is true. A look at the figure shows wavelengths 415nm, and 610nm, 630nm or 670nm could be used. The first step is to read the absorbance A for each sample at each wavelength from the curves.

© Huw Owens - University of Manchester : Numerical Example (continued) Wavelength Sample415nm630nm Y B G Wavelength Sample415nm630nm Y B Calculate the absorptivities a for the yellow and blue dyes at each wavelength. Bouger-Beer law states that A = abc, but because the sample thickness b is the same for all three samples it cancels out in this calculation.

© Huw Owens - University of Manchester : Numerical Example (continued) Now use the mixing law, which states that A G = A Y +A B In which these absorbances all refer to the green sample. Using the equivalent equation A G =a Y c Y +a B c B Where c Y and c B are the concentrations we wish to find we can write - WavelengthMixing Equation 415 nm0.610 = 0.741cy+0.075cB(1) 630 nm0.488 = 0.006cY cB(2)

© Huw Owens - University of Manchester : Numerical Example (continued) These two equations with two unknowns (c Y and c B ), can be solved as follows 0.488=0.006c Y c B (2) 0.005=0.006c Y c B (3) 0.483=0.616c B (4) 0.784=cBcB Finally, this value of c B can be substituted into Eq (1) to obtain c Y : 0.610=0.741c Y c B (1) 0.610=0.741c Y * =0.741c Y 0.743=cYcY

© Huw Owens - University of Manchester : Numerical Example (Continued) The values c B = units and c Y =0.743 units are the desired concentrations. Mixtures of three dyes can be treated similarly, but the equations are more complicated. Simple-subtractive colorant mixing is widely used in colour photography and in the dyeing of transparent plastics.