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Radiometry, Photometry & Optical Calibration
1. Definitions and governing physics
Spectral radiance $L_\lambda$ has units W m⁻² sr⁻¹ m⁻¹, and spectral irradiance $E_\lambda$ is W m⁻² m⁻¹. For an ideal Lambertian scene under appropriate geometry, $E_\lambda=\int_\Omega L_\lambda\cos\theta\,d\Omega$.
Planck’s blackbody spectral radiance is
\[B_\lambda(T)=\frac{2hc^2}{\lambda^5}\left[\exp\left(\frac{hc}{\lambda k_BT}\right)-1\right]^{-1}.\]Brightness temperature is the temperature of an ideal blackbody producing the measured spectral radiance at a stated wavelength; it need not equal physical temperature for a non-black surface.
Optical étendue is $G=\int n^2\cos\theta\,dA\,d\Omega$. It limits how much radiance a passive optical system can couple and is central to imaging-system throughput.
2. Detector response and uncertainty
For incident spectral power $P_\lambda$ and total throughput $T(\lambda)$, photon count expectation in integration time $t$ is
\[N_e=t\int\frac{P_\lambda T(\lambda)\lambda}{hc}\eta(\lambda)\,d\lambda,\]where $\eta$ is detector quantum efficiency. With shot noise, dark signal $N_d$ and read noise $\sigma_r$ in electrons rms, a simplified SNR is
\[\mathrm{SNR}=\frac{N_e}{\sqrt{N_e+N_d+\sigma_r^2}},\]assuming independent Poisson photoelectron/dark counts and Gaussian read noise. Add background, spatial nonuniformity and correlated noise for a real camera.
NEP in W/√Hz is the input optical power amplitude spectral density corresponding to unit output SNR per √Hz under a specified convention. Specific detectivity $D^*$ must state detector area and bandwidth conventions; it is not a stand-alone detector ranking across unlike measurement conditions.
3. Calibration plan for a flight camera
| Calibration | Question answered |
|---|---|
| Dark and bias frames | How much signal exists without illumination? |
| Flat field | How does pixel response vary spatially? |
| Wavelength/wavenumber calibration | Where is each spectral channel centered? |
| Linearity and saturation | How faithfully do counts track incident flux? |
| Absolute radiometry | How do digital numbers convert to physical radiance? |
| Stray-light/PSF/MTF | How does out-of-field or off-axis illumination contaminate the observation? |
| Thermal/radiation repeat | Does performance drift under mission conditions? |
Worked estimate: a 1 pW monochromatic signal at 1 µm corresponds to approximately $5.03\times10^6$ incident photons/s. At 60% quantum efficiency and perfect other throughput, the expected photoelectron rate is $3.02\times10^6$ electrons/s.
Modeling exercise: compute blackbody spectral radiance for 250–350 K and integrate through a realistic detector filter curve. Propagate uncertainty in filter transmission and detector responsivity separately.