Optical absorption, loss and gain

OPTICAL ABSORPTION, LOSS AND GAIN

The photon flux associated with an electromagnetic wave traveling through a semiconductor is denoted by

I_ph = I⁰_ph exp ( - ɑx) …(1)

where ɑ (the absorption coefficient) is usually positive and

I⁰_ph  is the incident light intensity at x = 0.

The optical intensity (which is the photon flux multiplied by the photon energy h v) falls as the wave travels. The electrons are pumped in the conduction band and holes in the valence band. The electron-hole recombination process (photon emission) can be stronger than electron-hole generation (photon absorption).

In general, the gain coefficient is defined by

gain coefficient = emission coefficient - absorption coefficient.

Let f^e (E^e) and f^h (E^h) the electron and hole occupation. The emission coefficient depends upon the product of ƒ^e (E^e) and f^h (E^h).

Similarly, the absorption coefficient depends upon the product of (1 - f^e (E^e)) and (1 - f^h (E^h)). Here the energies E^e and E^h are related to the photon energy by the condition of vertical k-transitions.

For these transitions we have

The occupation probabilities f^e and f^h are found by the quasi-Fermi levels for electrons and holes.

The gain is the difference of the emission and absorption coefficient. It is now proportional to

The optical wave has a general spatial intensity dependence:

I_ph = I⁰_ph exp (hv) x)  … (4)

and if g (h v) is positive, the intensity grows because additional photos are added by emission. The condition for positive gain requires "inversion" of the semiconductor system, eqn (3).

f^e (E^e) + f^h (E^h) > 1

The quasi-Fermi levels must penetrate their respective bands for this condition to be satisfied.