emittance | Strolls with my Dog

Imagine a bucolic scene on a clear sunny day at the equator, sand warmed by the tropical sun with a typical irradiance ( $E$ ) of about 1000 watts per square meter. As discussed earlier we could express this quantity as illuminance in lumens per square meter ( $lx$ ) – or as a certain number of photons per second ( $\Phi$ ) over an area of interest ( $\mathcal{A}$ ).

(1) $\begin{equation*} E = \frac{\Phi}{\mathcal{A}} \; (W, lm, photons/s) / m^2 \end{equation*}$

How many $photons/s$ per unit area can we expect on the camera’s image plane (irradiance $E_i$ )?

Figure 1. Irradiation transfer from scene to sensor.

In answering this question we will discover the Camera Equation as a function of opening angles – and set the stage for the next article on lens pupils. By the way, all quantities in this article depend on wavelength, which will be assumed in the formulas to make them more readable.

Continue reading Angles and the Camera Equation →

How many photons are emitted by a light source? To answer this question we need to evaluate the following simple formula at every wavelength in the spectral range of interest and add the values up:

(1) $\begin{equation*} \frac{\text{Power of Light in }W/m^2}{\text{Energy of Average Photon in }J/photon} \end{equation*}$

The Power of Light emitted in $W/m^2$ is called Spectral Exitance, with the symbol $M_e(\lambda)$ when referred to units of energy. The energy of one photon at a given wavelength is

(2) $\begin{equation*} e_{ph}(\lambda) = \frac{hc}{\lambda}\text{ joules/photon} \end{equation*}$

with $\lambda$ the wavelength of light in meters and $h$ and $c$ Planck’s constant and the speed of light in the chosen medium respectively. Since Watts are joules per second the units of (1) are therefore $photons/m^2/s$ . Writing it more formally:

(3) $\begin{equation*} M_{ph} = \int\limits_{\lambda_1}^{\lambda_2} \frac{M_e(\lambda)\cdot \lambda \cdot d\lambda}{hc} \text{ $\frac{photons}{m^2\cdot s}$} \end{equation*}$

Continue reading Photons Emitted by Light Source →

Musings about Photography