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# Blackbody Radiation and Planck's Law

Excerpt from Optical Design Fundamentals for Infrared Systems, Second Edition

A blackbody is defined as a perfect radiator which absorbs all radiation incident upon it.

In his investigation, to find a relation between the radiation emitted by a blackbody as a function of temperature and wavelength, Max Planck (1858–1947) developed the now famous equation named after him. His efforts laid the foundation of the quantum theory, for which he received the Nobel prize in 1918.^{1}

On October 19, 1900, he first reported his findings, which were based on his experimental work.^{1} Only two months later, on December 14, he presented the theoretical derivation of the equation3 that described the blackbody radiation curve:

where *W*λ = spectral radiant emittance (W cm^{−2} μm^{−1}), λ = wavelength (μm), *T* = blackbody temperature (K), *C*_{1} = 37,418, and *C*_{2} = 14,388, when the area is in square centimeters. *e* = base of the natural logarithm (2.718.....).

In Eq. (1.8) notice there is a strong dependence on the wavelength and that *W*λ goes to zero when λ = 0 and ∞. Planck's curve for a 500 K blackbody is shown in Fig. 1.

**Figure 1 ***Planck's curve for a blackbody source with T = 500 K.*

To determine the radiant emittance over a spectral band, we integrate under the Planck curve between the limiting wavelengths λ1 and λ2 . An example of a selected spectral band is indicated in Fig. 2.

**Figure 2 ***Radiant emittance within a selected spectral bandwidth.*

To perform the integration, one can use published tables or special slide rules.^{2} A very convenient way is to apply Simpson's rule with the aid of a calculator. For that purpose we state that

Using a Δλ of 0.05 or even 0.1 μm is sufficient for summations over the midwavelength (MWIR) and long-wavelength (LWIR) infrared regions (3–5, and 8–12 μm). It is also good to remember that if the spectral band is relatively narrow, *W* ≈ *W*λ Δλ .

### References

- E. Hecht,
*Optics*,*Second Edition*, Addison-Wesley (1990). *Infrared Radiation Calculator*, Infrared Information Analysis Center, Ann Arbor, MI, and EG&G Judson, Montgomeryville, PA and others.

**Citation:**

M. Riedl, *Optical Design Fundamentals for Infrared Systems, Second Edition*, SPIE Press, Bellingham, WA (2001).

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