Introduction

Spectral measurement is an important means of analyzing the optical properties of materials, and is widely used in chemical analysis, material characterization, biomedicine and other fields. Transmittance, reflectance and absorbance are the three most fundamental measurement modes; accurately understanding and applying their calculation formulas is crucial for obtaining reliable measurement results. This article systematically explains the physical principles, mathematical definitions and calculation formulas of these three measurement modes.

Transmittance Measurement

Physical Principle

Transmittance describes the ability of light to pass through a sample, defined as the ratio of the transmitted light intensity to the incident light intensity. According to the Beer-Lambert law, the attenuation of light in a medium is related to the optical path length and the concentration in the medium.

Mathematical Definition

Basic Transmittance Formula

Transmittance $T$ is defined as the ratio of the transmitted light intensity $I$ to the incident light intensity $I_0$:

$$T = \frac{I}{I_0}$$

Where:
- $I_0$ is the incident light intensity (unit: W/m²)
- $I$ is the transmitted light intensity (unit: W/m²)
- $T$ is the transmittance (dimensionless, ranging $0 \leq T \leq 1$)

Percentage Transmittance

In practical applications, percentage transmittance is often used:

$$T(\%) = \frac{I}{I_0} \times 100\%$$

Beer-Lambert Law

For a homogeneous medium, the relationship between transmittance, the concentration of the absorbing substance and the optical path length is described by the Beer-Lambert law:

$$T = 10^{-\varepsilon \cdot c \cdot l} = e^{-\alpha \cdot l}$$

Where:
- $\varepsilon$ is the molar absorption coefficient (unit: L·mol⁻¹·cm⁻¹)
- $c$ is the concentration of the absorbing substance (unit: mol/L)
- $l$ is the optical path length (unit: cm)
- $\alpha$ is the absorption coefficient (unit: cm⁻¹)

Internal and External Transmittance

Two definitions of transmittance must be distinguished:

External transmittance (including surface reflection loss):

$$T_{external} = \frac{I_{transmitted}}{I_{incident}}$$

Internal transmittance (excluding surface reflection loss):

$$T_{internal} = \frac{I_{transmitted}}{I_{incident} - I_{reflected}} = \frac{T_{external}}{1 - R}$$

where $R$ is the surface reflectance.

Reflectance Measurement

Physical Principle

Reflectance characterizes the ability of a sample surface or interior to reflect incident light. Depending on the mechanism of reflection, it can be divided into specular reflection and diffuse reflection.

Mathematical Definition

Basic Reflectance Formula

Reflectance $R$ is defined as the ratio of the reflected light intensity $I_r$ to the incident light intensity $I_0$:

$$R = \frac{I_r}{I_0}$$

Specular Reflectance

For a smooth surface, the Fresnel reflection law applies. The reflectance at normal incidence is:

$$R_{normal} = \left(\frac{n_1 - n_2}{n_1 + n_2}\right)^2$$

Where:
- $n_1$ is the refractive index of the incident medium
- $n_2$ is the refractive index of the sample

Diffuse Reflectance

For rough surfaces, a diffuse-reflection model is used. The diffuse reflectance is described by the Kirchhoff approximation:

$$R_{diffuse} = \frac{\int_{2\pi} I_r(\theta, \phi) \cos\theta \, d\Omega}{I_0}$$

Where:
- $\theta$ is the reflection angle
- $\phi$ is the azimuth angle
- $d\Omega = \sin\theta \, d\theta \, d\phi$ is the solid-angle element

Bidirectional Reflectance Distribution Function

A more precise description uses the bidirectional reflectance distribution function (BRDF):

$$f_r(\theta_i, \phi_i; \theta_r, \phi_r) = \frac{dL_r(\theta_r, \phi_r)}{L_i(\theta_i, \phi_i) \cos\theta_i \, d\Omega_i}$$

Where:
- $L_i$ is the incident radiance
- $L_r$ is the reflected radiance
- $\theta_i, \phi_i$ is the incident direction
- $\theta_r, \phi_r$ is the reflection direction

Integrated Reflectance

The hemispherical integrated reflectance (total reflectance):

$$R_{total} = \int_{2\pi} f_r(\theta_i, \phi_i; \theta_r, \phi_r) \cos\theta_r \, d\Omega_r$$

Absorbance Measurement

Physical Principle

Absorbance, also called optical density, is a physical quantity characterizing the ability of a substance to absorb light. Absorbance has a logarithmic relationship with transmittance and is additive, which facilitates quantitative analysis.

Mathematical Definition

Basic Absorbance Formula

Absorbance $A$ is defined as the common logarithm of the reciprocal of transmittance:

$$A = -\log_{10}(T) = \log_{10}\left(\frac{I_0}{I}\right)$$

Or expressed with the natural logarithm:

$$A = \frac{1}{\ln(10)} \ln\left(\frac{I_0}{I}\right) \approx 0.4343 \ln\left(\frac{I_0}{I}\right)$$

Conversion Relationship with Transmittance

$$A = -\log_{10}(T) = 2 - \log_{10}[T(\%)]$$

$$T = 10^{-A}$$

Absorbance Form of the Beer-Lambert Law

$$A = \varepsilon \cdot c \cdot l$$

This is the fundamental formula for quantitative spectral analysis, showing that:
- Absorbance is proportional to the concentration of the absorbing substance
- Absorbance is proportional to the optical path length
- The proportionality coefficient is the molar absorption coefficient $\varepsilon$

Absorbance of a Multi-Component System

For a system containing $n$ absorbing components, the total absorbance is the sum of the absorbances of the components:

$$A_{total} = \sum_{i=1}^{n} A_i = \sum_{i=1}^{n} \varepsilon_i \cdot c_i \cdot l$$

Verification of Absorbance Additivity

Suppose two absorbing substances are mixed, with absorbances:

$$A_1 = \varepsilon_1 c_1 l, \quad A_2 = \varepsilon_2 c_2 l$$

The total absorbance after mixing:

$$A_{mix} = -\log_{10}(T_{mix}) = -\log_{10}(T_1 \cdot T_2) = A_1 + A_2$$

This verifies the additivity principle of absorbance.

Relationships Among the Three Measurement Modes

Energy Conservation Relationship

For a non-luminous sample, the distribution of the incident light energy satisfies:

$$I_0 = I_T + I_R + I_A$$

Where:
- $I_T$ is the transmitted light intensity
- $I_R$ is the reflected light intensity (including specular and diffuse reflection)
- $I_A$ is the absorbed light intensity

Normalized form:

$$T + R + A_{abs} = 1$$

where $A_{abs} = I_A/I_0$ is the absorptance (note the distinction from absorbance $A$).

Relationship Between Absorbance and Absorptance

$$A_{abs} = 1 - T - R = 1 - 10^{-A} - R$$

For low-absorbance samples ($A \ll 1$):

$$A_{abs} \approx A \cdot \ln(10) - R \approx 2.303A - R$$

Correction Formulas in Practical Measurement

Baseline Correction

Baseline correction is required in practical measurement to eliminate systematic errors:

Transmittance Baseline Correction

$$T_{corrected} = \frac{T_{sample} - T_{dark}}{T_{reference} - T_{dark}}$$

Where:
- $T_{sample}$ is the sample measurement value
- $T_{reference}$ is the reference measurement value
- $T_{dark}$ is the dark-current measurement value

Reflectance Baseline Correction

$$R_{corrected} = \frac{R_{sample} - R_{dark}}{R_{reference} - R_{dark}} \times R_{std}$$

where $R_{std}$ is the calibrated reflectance of the standard reference plate.

Stray-Light Correction

The effect of stray light $S$ on the measurement:

$$T_{measured} = T_{true}(1 - S) + S$$

Correction formula:

$$T_{true} = \frac{T_{measured} - S}{1 - S}$$

Nonlinear-Response Correction

Correction of detector nonlinear response (polynomial fitting):

$$I_{true} = a_0 + a_1 I_{raw} + a_2 I_{raw}^2 + a_3 I_{raw}^3$$

The coefficients $a_0, a_1, a_2, a_3$ are determined by calibration with a standard light source.

Error Analysis

Transmittance Measurement Error

The relative error propagation of transmittance:

$$\frac{\Delta T}{T} = \sqrt{\left(\frac{\Delta I}{I}\right)^2 + \left(\frac{\Delta I_0}{I_0}\right)^2}$$

For a photon-noise-limited system:

$$\frac{\Delta T}{T} = \sqrt{\frac{1}{N_I} + \frac{1}{N_{I_0}}}$$

where $N_I$ and $N_{I_0}$ are the photon counts at the corresponding light intensities.

Absorbance Measurement Error

The error propagation of absorbance:

$$\Delta A = \frac{0.4343}{T} \Delta T$$

When $T = 0.368$ (i.e. $A = 0.434$), the absorbance measurement error is minimized.

Concentration Measurement Error

The concentration error based on the Beer-Lambert law:

$$\frac{\Delta c}{c} = \frac{\Delta A}{A} = \frac{0.4343}{T \cdot \ln(1/T)} \Delta T$$

Optimal measurement range: $0.2 < A < 0.8$ (corresponding to $15\% < T < 63\%$)

Application Examples

Example 1: Solution Concentration Determination

For a certain substance, $\varepsilon = 1.2 \times 10^4$ L·mol⁻¹·cm⁻¹, the optical path length is $l = 1$ cm, and the measured $T = 0.316$.

Calculation steps:

  1. Calculate the absorbance:
    $$A = -\log_{10}(0.316) = 0.500$$

  2. Calculate the concentration:
    $$c = \frac{A}{\varepsilon \cdot l} = \frac{0.500}{1.2 \times 10^4 \times 1} = 4.17 \times 10^{-5} \text{ mol/L}$$

Example 2: Thin-Film Thickness Determination

For a thin film with absorption coefficient $\alpha = 500$ cm⁻¹, the measured $T = 0.606$.

Calculate the thickness:

$$l = -\frac{\ln(T)}{\alpha} = -\frac{\ln(0.606)}{500} = 1.0 \times 10^{-3} \text{ cm} = 10 \text{ μm}$$

Conclusion

Transmittance, reflectance and absorbance are the three fundamental physical quantities of spectral measurement, and their calculation formulas constitute the theoretical basis of quantitative spectral analysis. Accurately understanding and applying these formulas, combined with appropriate error analysis and correction methods, is the key to obtaining reliable measurement results. In practical applications, one should choose the appropriate measurement mode according to the sample characteristics and measurement requirements, and strictly control the measurement conditions to ensure data accuracy and repeatability.