A method for determining the absorption coefficient of a material from transmittance measurements
The transmittance of a plane-parallel substrate material (double-sided polished) with a known thickness is measured, the absorption coefficient is calculated from the measurement results, and a formula for calculating the transmittance for any thickness is derived.
・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
First, let us define the parameters. In the actual measurement of the transmittance of a parallel plane substrate material (double-sided polished) with a known thickness ,
I0 : Amount of incident light
T0 : Energy transmittance of the front and back surfaces of the measurement material
R0 : Energy reflectance of the front and back surfaces of the measurement material
t0 : thickness of the material to be measured
TM : Total transmittance of the measuring material (measured quantity)
RM : Total reflectance of the measurement material
α : Absorption coefficient of the measurement material (unknown quantity)
Here, spectroscopic measurement may be performed as a function of wavelength other than t0. By deriving α from the measurement results , the following relational expression is finally obtained.
t: material thickness (variable)
T: Total transmittance of the material (variable)
At this point, the following can be said:
・The energy transmittance T0 at each surface is expressed by the following formula using the energy reflectance R0 .
*How to assign R0 will be explained later.
・While traveling through the thickness t0 of the measurement material, the amount of light is attenuated by a factor of exp(-αt0 ).
Based on the above, the branching of light rays due to transmission and reflection and internal absorption can be illustrated as follows:
Considering the multiple reflections shown in the figure above, the total transmittance TM and total reflectance RM can be expressed as follows:
Substituting ① for ②a,b and eliminating T0 , we obtain the following equation.
By transforming ②a', the absorption coefficient α can be obtained as follows:
Here, it is also possible to calculate α by similarly transforming ②b' and applying the total reflectance RM, but in that case the reflected light before it enters the material becomes dominant, making it difficult to accurately determine α, so this is not recommended.
To use the absorption coefficient α thus determined to find the total transmittance T as a function of an arbitrary thickness t, the following equation can be obtained by replacing TM with T and t0 with t in ②a'.
・How to assign the energy reflectance R0 for each surface
(i) Method of determining from reflectance measurement
By using a backside anti-reflection sheet to suppress multiple reflections, the energy reflectance R0 can be obtained by actual measurement as follows.
(ii) Calculation method based on refractive index
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:
Here, when λ is the wavelength, the relationship between κ and α is expressed as follows:
As can be seen from ④a, in order 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 in the first place, as can be seen from ④b.
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 sufficient transmitted light can be obtained to measure, the value of the extinction coefficient κ is sufficiently smaller than the refractive index n at that point and can be ignored, and the following calculation formula for a non-absorbing medium can be applied.
