Any object surface can radiate heat to and receive heat from outside, if an object can absorb all the incident radiation, regardless of the frequencies and directions, this object is called Black Body. A ball cavity with a small hole can be regarded as a black body, since any radiation entering the ball cavity can only reflect inside it, thus totally absorbed.

A mode is a possible standing electromagnetic wave inside the black body. At thermal equilibrium, the space inside the black body has an average energy per mode <E>. Scientists found if we compute this <E> supposing that energy were continuous from zero frequency to infinity frequency, the total energy inside the limited black body would be infinity! This is impossible.

Planck solved this problem. His fundamental hypothesis was that energy of a given mode could not have arbitrary continuous values from 0 to infinity, on the other hand, this energy is discrete. The permitted values of this energy are integral multiplies of a fundamental quantity which is proportional to the frequency of this mode, that is, Planck assumed that the energy of the mode is:

E = nhn

Where n is a positive integer and h is the Planck constant.

This implies that energy exchange involves a discrete amount of energy hn , this minimum quantity that can be changed is called a light quantum or photon. The average energy of the mode is:

<E>=hn /[exp(hn /kT)-1]

Where k is the Boltzmann constant, T is the Kelvin temperature.

Is this too abstract for you? Let’s see the following phenomena.

The discharge helium lamp appears to emit pinkish-white light when viewed directly, but when viewed through the diffraction grating, each wavelength in the light is diffracted at a different angle, we can thus observe that the light is made up of different discrete wavelengths or colors. There is a strong yellow line at 588 nm, there are also other violet, green, red and deep red lines. The following figure plots the distribution of the lines with their intensities.

What does this tell us? The radiation of the lamp is not continuous in frequency! It can be readily explained that these discrete lines originate from the fact that the energy distributions of particles are discrete, i.e., the particles (atoms, molecules, ions) have energy levels, and the radiation is from the difference of these energy levels. The actual energy levels of an atom are very complex even for elements like helium.

We usually simplify the actual complex energy levels into the interested energy levels, like the following four level model below. We use E0,E1, E2, E3 to represent different energy levels with E0<E1<E2<E3. We use N0, N1, N2 and N3 to represent the populations of atoms or particles at this energy level. Sometimes we draw the energy levels with certain width, some are wider, some are narrower, the width of the energy levels represent the range that the absorbed incident radiation frequency can be varied.

When we incident the laser medium with a wide bandwidth EM radiation, which is called pumping, the atoms will be raised from lower energy levels to higher energy levels. Usually we assume that pumping process only pumps atoms at ground level to a certain high level, for example, from E0 to E3. This does not mean that pumping doesn’t happen between other levels. We can simplify our model in this way because the width of E3 is much wider than the other levels, so that the numbers of atoms pumped to E3 level is much greater than that to the other levels, pumping between other levels can be neglected.

Also worth attention is that the atoms at certain energy levels have different lifetimes, that is, the atoms at E_j after an average time T_ji will change to E_i, E_j>E_i, T_ji is called the radiative decay time between E_j and E_i.