A method for determining the absorption coefficient of a material from transmittance measurements
By measuring the transmittance of a plane-parallel substrate material (double-sided polished) with a known thickness and calculating the absorption coefficient from the measurement results, it is possible to calculate the internal transmittance for any thickness. This method is explained below.
・Absorption coefficient α
In a material with internal absorption (absorption coefficient α ), the amount of light is attenuated by a factor of exp(-αt) as it propagates from an arbitrary position x in the material to a position a distance t away.
Furthermore, while light propagates through the material over a distance of 1/α , the amount of light decreases by 1/e (approximately 37%). This distance is called the absorption length (or absorption depth) and is used as a guide to the thickness of the material that can be penetrated.
The method for determining the value of this absorption coefficient α from actual measurements of the transmittance of the material will be described below.
・Deriving absorption coefficient α from transmittance measurements
(i) Sample preparation
First, a sample with parallel flat surfaces, both of which are mirror-polished, is prepared, and the plate thickness is measured.
t0 : thickness of sample material
(ii) Derivation of energy reflectance and energy reflectance
Calculate the energy reflectance and energy transmittance shown in the figure below.
R0 : Energy reflectance of the front and back surfaces of the sample material
T0 : Energy transmittance of the front and back surfaces of the sample material
(ii)-1. Derivation of energy reflectance
(ii)-1a. Method of determining from reflectance measurement
By using a back-surface reflection-reducing sheet to suppress multiple reflections, the energy reflectance R 0 can be obtained by actual measurement as follows.
(ii)-1b Method of calculating from refractive index value
The energy reflectance of the front and back surfaces of the measurement medium, taking into account absorption by the medium, is as follows for normal incidence:
*n is the refractive index of the medium, and κ is the extinction coefficient of the medium
Here, when λ is the wavelength, the relationship between κ and α is expressed as follows:
As can be seen from (1)-①a, to calculate the energy reflectance R0 of an absorbing medium, it is necessary to know both the refractive index n and the extinction coefficient κ of that material. However, since we want to determine the absorption coefficient α through measurement, the extinction coefficient κ is an unknown quantity, as can be seen from (1)-①b. Therefore, if left as is, the number of unknown quantities will be greater than the number of equations, making it impossible to solve.
Here, when λ=0.00055 mm, the relationship between the absorption length 1/α and κ can be plotted as follows:
As can be seen from this, unless the absorption length 1/α of the sample medium is as thin as a thin film, κ takes a value that is orders of magnitude smaller than the refractive index, and therefore κ has almost no effect on the value of R0 .
Therefore, if the sample is not a thin film but a bulk with sufficient thickness, and if enough transmitted light can be obtained to measure, then the value of the extinction coefficient κ is sufficiently smaller than the refractive index n, and it can be said that κ can be ignored in the calculation of (1)-①a, and the following formula for a non-absorbing medium can be applied. By making this approximation, R0 can be calculated.
(ii)-2. Derivation of energy transmittance
・The energy transmittance T0 at each surface is expressed by the following formula using the energy reflectance R0 .
(iii) Measurement of total system transmittance
(iii)-1. Measurement of the intensity of incident light
First, the intensity of incident light is measured.
I0 : intensity of incident light
(iii)-2. Measurement of the total system transmittance of the sample material
Next, the intensity of light transmitted through the entire system and the intensity of light reflected through the entire system of the sample material are measured.
*The total intensity of reflected light from the system is not used to derive the absorption coefficient, so it is for reference only.
TM : Total system transmittance of sample material
RM : Total system reflectance of the sample material
The intensity of light transmitted through the entire system and the intensity of light reflected through the entire system are expressed as I0TM and I0RM , respectively, so by dividing these by the I0 actually measured in (iii)-1, the total system transmittance TM and total system reflectance RM can be obtained.
(iv) derivation of Absorption coefficient
As mentioned above, if the absorption coefficient to be sought is α , the amount of light is attenuated by a factor of exp(-αt0) due to absorption while traveling through the thickness t 0 of the sample material.
Taking this into consideration, the branching of light rays due to transmission and reflection and internal absorption can be illustrated as follows:
α : absorption coefficient of the sample material
Here, when performing spectroscopic measurements, all parameters other than t0 are treated as functions of wavelength.
Considering the multiple reflections shown in the figure above, the total transmittance TM and total reflectance RM can be calculated as follows:
Substituting (1)-② into (1)-③a and b and eliminating T0 , we obtain the following equation.
By transforming (1)-③a', the absorption coefficient α can be obtained as follows:
Here, it is possible to similarly modify (1)-③b' and apply the total reflectance RM to calculate α , but in that case the reflected light before it enters the material will be dominant, making it difficult to accurately determine α , so this is not recommended.
In addition, when measuring transmittance, if the absorption by the sample material is extremely high or low, it is not possible to accurately determine the absorption coefficient. In such cases, it is recommended to change the thickness of the sample material to a thickness suitable for measurement, or to apply ellipsometry .