Determining Material Absorption Coefficients from Transmittance Measurements: Theory and Methods
If you want to know the internal transmittance of a certain medium, you can calculate the internal transmittance for any thickness if you know the absorption coefficient. Therefore, this page shows how to derive the absorption coefficient of an unknown medium from the results of transmittance measurement.
table of contents:
・Absorption coefficient α
In a material with internal absorption (absorption coefficient α ), the intensity of light is attenuated exp(-αx) times as it propagates from the incident surface of the material to a position at a distance x.
Furthermore, while light propagates through the material over a distance of 1/α, the intensity 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.
・Derivation of absorption coefficient α from transmittance measurements - For solids -
(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
Here, surface reflection can be taken into account through calculations or actual measurements, so we do not intentionally use an anti-reflection coating (because if an anti-reflection coating were used, it would make it more complicated to take into account surface reflection and absorption by the film).
(ii) Derivation of energy transmittance and energy reflectance for transmission and reflection at the boundary surface
The energy transmittance and energy reflectance of transmission and reflection at the interface between air and the sample material are derived as shown in the figure below.
R0 : Energy reflectance at the interface between air and sample material
T0 : Energy transmittance at the interface between air and sample material
(ii)-1. Derivation of energy reflectance
(ii)-1a. Method of determining from total system reflectance measurement
By using a back-surface reflection eliminating sheet as described below, multiple reflections can be suppressed.
As a result, the total reflected light itself becomes reflected light from the interface (single surface) between the air and the sample material, and by measuring the intensity of this reflected light, the energy reflectance can be calculated.
Specifically, the energy reflectance R0 can be calculated by dividing the total reflected light intensity I0R0 of the sample material by the incident light intensity I0 .
(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, the refractive index n is a known quantity.
Furthermore, 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
Without a coating, it can be assumed that there is no absorption at the interface between the air and the sample material, so the energy transmittance T0 at each surface can be expressed by the following equation using the energy reflectance R0 .
Furthermore, by applying (1)-①a', it can also be expressed in terms of the refractive index of the sample material as follows:
(iii) Derivation of the absorption coefficient from the measured total transmittance of the sample material.
The intensity of incident light and the total transmitted light intensity for the sample material will be measured.
*Here, the total reflected light intensity, which is not necessary when deriving the absorption coefficient, is also treated as a reference value.
I0 : intensity of incident light
IT : Total transmitted light intensity of the sample material
TM : Total transmittance of sample material
( IR : Total reflected light intensity of the sample material)
( RM : Total reflectance of the sample material)
Dividing the total transmitted light intensity IT and the total reflected light intensity IR of the sample material by the incident light intensity I0 yields the total transmittance TM and the total reflectance RM of the sample material.
(iii)-1 Derivation of absorption coefficient from measured results
As mentioned above, if the absorption coefficient to be sought is α , the intensity of light is attenuated by a factor of exp(-αt0) due to absorption while traveling through the thickness t0 of the sample material.
Taking this into consideration, the absorption coefficient α can be calculated as follows.
(iii)-1a. Simple calculation method
The branching of light rays due to transmission and reflection at the boundary surface, internal absorption, and the intensity of light transmitted through the entire system can be illustrated as follows (however, multiple reflections are not taken into account here).
From the above figure, the absorption coefficient α is calculated as follows:
From (1)-② and (1)-③,
Furthermore, by applying (1)-①a', it can also be expressed as follows.
(iii)-1b. Method taking multiple reflections into account
The diagram below shows the branching of light rays due to transmission and reflection at the boundary surface, internal absorption, and the total transmitted and reflected light, taking into account multiple reflections.
Taking into account the multiple reflections shown in the figure above, the total system transmittance TM and total system reflectance RM of the sample material are calculated as follows:
Substituting (1)-② into (1)-④a,b and eliminating T0 , we obtain the following equation.
By transforming (1)-④a', the absorption coefficient α can be obtained as follows:
Furthermore, by substituting (1)-①a', α can also be expressed as follows using the refractive index n of the medium:
Here, it is possible to similarly transform (1)-④b' and apply the total reflectance RM to calculate α , 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. Therefore, as mentioned above, the absorption coefficient can actually be found by measuring only the intensity of light transmitted through the entire system.
(iii)-2. Comparison between simple calculation and calculation taking multiple reflections into account
The relationship between TM and t0α at each refractive index value is plotted as follows, both in the simplified calculation and when multiple reflections are taken into consideration.
However, the dashed line shows the result of the simple calculation, and the solid line shows the result when multiple reflections are taken into account.
As can be seen from this figure, the larger the refractive index n of the medium and the smaller the value of t0α , the larger the difference between the calculated value when the simplified calculation and the calculated value when multiple reflections are taken into account.
(iv) Summary
As mentioned above, we have shown how to determine the absorption coefficient from transmittance measurements of a sample material. However, if the sample material has extremely high absorption and almost no transmitted light is obtained, or if the sample material has almost no absorption and only the effects of surface reflection appear in the measurement results, it is difficult to determine the absorption coefficient from the measurement results. In such cases, it may be possible to measure the absorption coefficient by adjusting the thickness of the sample material to a thickness suitable for measurement, or by applying ellipsometry .
・Derivation of absorption coefficient α from transmittance measurement - For liquids -
(i) Preparation of the container for the sample
When measuring the transmittance of a liquid, a container for the sample is necessary. This container must have a mirror-like and flat incident and exit surface with respect to the measurement light. While the effect of absorption by the container itself can be removed by calculation, from the perspective of achieving a good signal-to-noise ratio (SNR), it is desirable to have as little absorption as possible.
t01 : Thickness of the container (inlet side)
ts : thickness of the sample layer
t02 : Thickness of the container (exit side)
n0 : Refractive index of the container
α0 : absorption coefficient of the container
(ii) Transmission measurement of the container only
First, measure the transmittance of only the measuring container.
I0 : Incident light intensity
Iref : Intensity of light transmitted through the entire system, including the container
Tref : Total system transmittance of the container only
T0 : Energy transmittance at the interface between the container and air
From the figure, the following can be said:
Here, since there are many layers, if multiple reflections are taken into consideration, the calculations become quite complicated, so multiple reflections are not taken into consideration here.
Since Tref can be obtained from both (2)-①a' (measurement of I0 and Iref ) and (2)-①b' (calculation), it is a good idea to confirm that they match.
Here, similar to (1)-②', the energy transmittance of the interface can be calculated using the following formula.
(iii) Measurement of the transmittance of the sample and derivation of the absorption coefficient
(iii)-1. Measurement of the transmittance of the sample
Once the transmittance measurement of the measuring container is complete, the sample is placed into the container, and the transmittance of the sample is then measured.
IM : Intensity of light transmitted through the entire sample container
TM : Total system transmittance of the sample container
Ts : Energy transmittance at the interface between the container and the sample
ns : refractive index of the sample
αs : Absorption coefficient of the sample (unknown quantity)
From the figure, the following can be said:
Here, since there are many layers, if multiple reflections are taken into consideration, the calculations become quite complicated, so multiple reflections are not taken into consideration here.
Also, similar to (1)-②', the energy transmittance of the interface can be calculated using the following formula.
(iii)-2. Derivation of the absorption coefficient of the sample
From (2)-①b and (2)-③b,
Substituting (2)-② and (2)-④,
(iv) Measurement of sample transmittance and derivation of absorption coefficient – when the container is wet with the sample solution –
If the outside of the container is wetted with the sample solution, light loss due to scattering and refraction will occur, and if the wettability is high, the reflectivity of the interface may change due to the liquid film, so caution is necessary. The following model illustrates this situation. Here, the sample solution film on the outside of the container is assumed to be sufficiently thin, and its absorption is ignored.
(iv)-1. Measurement of the transmittance of the sample
IM' : Total transmitted light intensity in the sample container (if the container is wet with the sample solution)
TM' : Total system transmittance of the sample container (when the container is wet with the sample solution)
Ts : Energy transmittance at the interface between the container and the sample.
Ts0 : Energy transmittance at the air-sample interface
ns : Refractive index of the sample
αs' : Absorption coefficient of the sample (when the container is wet with the sample solution) (unknown quantity)
From the figure, the following can be said:
Here, since there are many layers, if multiple reflections are taken into consideration, the calculations become quite complicated, so multiple reflections are not taken into consideration here.
Also, similar to (1)-②', the energy transmittance of the interface can be calculated using the following formula.
(iv)-2. Derivation of the absorption coefficient of the sample
From (2)-①b and (2)-⑥b,
Substitute (2)-②, (2)-④, and (2)-⑦,
Thus, since the liquid film formed on the outer surface alters the transmittance, it is important to keep the cell surface clean and free of liquid film in order to ensure the reproducibility and reliability of the measurement results.
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Related Topics
The derivation of the energy reflectance (normal incidence) in a non-absorbent medium is performed in (6)-⑥⑨ below.
The derivation of the absorption coefficient is shown in (10)-⑤ below, and the derivation of the energy reflectance in the absorbing medium (normal incidence) is shown in (12)-①-2 below.
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References
[1] Swanepoel, R., “Determination of the thickness and optical constants of amorphous silicon” (J. Phys. E, 1983)
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Related literature
[A] Born, M. & Wolf, E., Principles of Optics, Cambridge University Press.
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Update History
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2026-05 Added "Measurement of sample transmittance and derivation of absorption coefficient – when the container is wet with the sample solution –"
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2026-03 Added "Derivation of absorption coefficient α from transmittance measurement - For liquids -"
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2026-02 Completely revised
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2025-11 Newly released