==== Blackbody ====
**I. What is a Blackbody?**\\
A blackbody is an idealized physical model under **thermal equilibrium (energy conservation)**, which completely absorbs all wavelengths of incident electromagnetic radiation without reflection or transmission.\\

**Per the law of conservation of energy:**\\
  * **Transparent objects:** Incident radiant energy = Absorption + Reflection + Transmission
  * **Opaque objects:** Incident radiant energy = Absorption + Reflection
When an opaque object has 0 reflectance, its absorptivity and thermal emissivity reach maximum.


An **absolute blackbody** is an idealized physical model that fully converts absorbed energy into thermal radiation. Its** spectral power distribution (SPD)** depends solely on **absolute temperature (unit: Kelvin, K)**, independent of material composition or geometric shape.\\

In the **CIE 1931 xy chromaticity diagram** (see Figure 1), the chromaticity coordinates of a blackbody at different temperatures form a continuous curve known as the Planckian locus. As the temperature increases, the dominant hue of the radiation shifts from warm red (low frequency, long wavelength) to cool blue (high frequency, short wavelength). This locus is an important basis for defining** correlated color temperature (CCT)** and is widely used in light sources, imaging systems, and color calibration.\\
|  {{ :yanding:成像基础知识:光学:通用:1280px-planckianlocus.png?400 |}}  |
^  Figure 1[1]: CIE 1931 xy chromaticity diagram and Planckian locus  ^


**II. Blackbody Simulation Model**\\
In engineering and experimental applications, a** constant-temperature cavity with a small aperture** is commonly used as an approximate blackbody model (Figure 2). When incident electromagnetic waves enter the aperture, they undergo multiple reflections inside the cavity and are fully absorbed, making it difficult to re-emit. This achieves a reflectance approaching 0, allowing the cavity to be regarded as an approximate ideal blackbody.

|  {{ :yanding:成像基础知识:光学:通用:black_body_realization.svg.png?400 |}}  |
^  Figure 2[2]: Schematic of a constant-temperature cavity blackbody simulator  ^
^  Source: https://en.wikipedia.org/wiki/Black_body#/media/File:Black_body_realization.svg  ^

**III. Blackbody Radiation Laws**\\
The thermal radiation characteristics of a blackbody strictly follow three fundamental laws:\\
**(1) Planck's Law of Blackbody Radiation**\\
This law describes the relationship between **the spectral radiant exitance** of a blackbody, wavelength, and absolute temperature. Its core formula is:\\
$$M_\lambda(\lambda, T) = \frac{c_1}{\lambda^5} \frac{1}{e^{\frac{c_2}{\lambda T}} - 1}$$


where:\\
$M_\lambda$: Spectral radiant exitance ($W \cdot m^{-3}$);\\
$\lambda$: Wavelength (m);\\
$T$:Absolute temperature (K);\\
$c_1 = 2\pi h c^2$: First radiation constant ($W \cdot m^2$);\\
$c_2 = \frac{hc}{k_B}$: Second radiation constant ($m \cdot K$);\\
$k_B $: Boltzmann constant ($J \cdot K^{-1}$).\\

| {{ :yanding:成像基础知识:光学:通用:图3普朗克定律.png?600 |}}  |
^  Figure 3[3]:Schematic Diagram of Planck's Law of Blackbody Radiation  ^
^  Source:https://en.wikipedia.org/wiki/Black_body#/media/File:Black_body.svg  ^

As shown in Figure 3, rising blackbody temperature shifts the peak wavelength to shorter wavelengths (Wien's Displacement Law), while total radiant energy increases with the fourth power of temperature (Stefan-Boltzmann Law).\\

**(2) Wien's Displacement Law**\\
This law reveals the inverse relationship between the peak radiation wavelength $\lambda_m$ of a blackbody and its absolute temperature $T$. Its core formula is:\\
$$\lambda_m T = b$$
where:\\
$\lambda_m$: Peak wavelength  ($m$)\\
$T$: Absolute temperature ($K$)\\
$b$: Wien's displacement constant ($m \cdot K$),$b \approx 2.897771955 \times 10^{-3} \, m \cdot K$\\
This law explains that as a blackbody's temperature rises, the peak wavelength of its emission spectrum shifts to shorter wavelengths.\\

**(3) Stefan-Boltzmann Law**\\
This law describes the relationship between the total radiant exitance $M$ of a blackbody (integrated over all wavelengths) and the 4th power of its absolute temperature  $T$.Its core formula is: \\
$$M(T) = \int_0^\infty M_\lambda(\lambda, T) d\lambda = \frac{c_1 \pi^4}{15 c_2^4} T^4 = \sigma T^4$$
where:\\
$M $: Total radiant exitance ($W \cdot m^{-2}$)\\
$T$: Absolute temperature(K)\\
$c_1 = 2\pi h c^2$:  First radiation constant($W \cdot m^2$)\\
$c_2 = \frac{hc}{k_B} $: Second radiation constant  ($m \cdot K$)\\
$\sigma$: Stefan-Boltzmann constant ($W \cdot m^{-2} \cdot K^{-4}$)\\
|  {{ :yanding:成像基础知识:光学:通用:图4斯蒂夫定律.png?400 |}}  |
^  Figure 4[4]: Schematic Diagram of Stefan-Boltzmann Law  ^
^  Source:https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law#/media/File:Stefan_Boltzmann_001.svg  ^


The $T^4$ dependence demonstrates the extreme temperature sensitivity of blackbody radiation: even a small rise in temperature leads to a dramatic increase in total radiant energy, quantitatively validating that an ideal absorber is also an ideal radiator.  
\\


Image sources:\\
[1] https://upload.wikimedia.org/wikipedia/commons/thumb/b/ba/PlanckianLocus.png/960px-PlanckianLocus.png\\
[2]https://en.wikipedia.org/wiki/Black_body#/media/File:Black_body_realization.svg\\
[3] https://commons.wikimedia.org/wiki/File:BlackbodySpectrum_loglog_en.svg\\
[4]https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law#/media/File:Stefan_Boltzmann_001.svg\\