bullet Introduction

The term laser is an anachronism for Light Amplification by Stimulated Emission of Radiation. The use of each one of those words will be explained win the text development below.

Light  is represented by a wave that is the superposition of various (or just one) waves of the form . In the previous expression, is the amplitude of the wave, is the wave vector defined by with where is the wavelength of the perturbation. The parameter represent the angular frequency of the wave. Here,  is the frequency of the oscillation (reference to waves). Finally, the parameter is an arbitrary phase that is determined by the value of the wave at and .

The graph above represents a snap shot of a one-dimensional wave, , let us say, at . In that case, the phase can be written in terms of the coordinate position and, in general,  .

 
bullet Coherent and Incoherent Light


Ordinary light is made out of the superposition of various waves, as the one presented above, with different wavelength and/or frequency and phases. The resulting beam is say to be non-monochromatic incoherent light. If a color filter is placed in the path of such a beam of light, the result is Monochromatic Incoherent Light. A color filter will select a very small range of frequencies (colors) for transmission through the filter; all the other frequencies are blocked by the filter.
Incoherent light refers to the superposition of light with different phases.
The light emitted by a laser is not only monochromatic but also is coherent.
Coherent light refers to the superposition of waves with the same frequency, wave vector, and phase. In three dimensions, the wave vector indicates the direction of propagation of the wave. In the case of coherent light, it is not just required that the wavelength of the different components of the addition are the same (associated to the magnitude of the wave vector) but also the direction of propagation of these components must be the same (non-diverging beams).

The upper graph represents waves with different wavelength (longer for red, smaller for violet). The lower picture has the same characteristic as the upper picture but with additional arbitrary phases among the different rays. The top wave on the graph above represents the addition (superposition) of all the waves below in the graph. The waves below are monochromatic (all have the same frequency) but they have different phases. The addition of the waves has been graphed at the same scale as the other. Notices that rather than a constructive addition of these waves, we have a some how destructive superposition of these waves with a resulting amplitude smaller than the amplitude of the component waves. The top wave on the graph above represents the addition (superposition) of all the waves below in the graph. The waves below are monochromatic (all have the same frequency) but, in this case, they have the same phase. The addition of the waves has been graphed at the same scale as the components of the addition. Notice that now the superposition is constructive and the amplitude of the wave is much bigger than the individual components.

Thus the characteristic of a laser are

  1. light is nearly monochromatic, same ,
  2. light is coherent; that is, waves are exactly in phase, same ,
  3. light does not diverges, same ; and,
  4. light is very intense, the resulting amplitude, , is very high.

The laser effect is produced in a media when more atoms in the media are in an excited state than in the ground state of the atom and the electrons in the excited state jump back to the ground state emitting photons with the characteristics expressed above. This is called population inversion. How this is achieved will be presented below.

bullet Three Levels Laser

The simples operational laser is the three levels laser that involved three atomic levels to produce the laser effect. These levels are the ground state of the atom, an regular excited state, and a meta-stable state of the atom.

Regular excited states are atomic levels where an excited electron can stay after absorbing energy. However, the time that the electron last in such state is of the orders of 10-8 seconds. On the other side, a meta-stable state is an excited state where the electron can last as long as five orders of magnitude longer, 10-3 seconds. Such a long lasting excited state can result in the media reaching the status of population inversion where more atoms are in the excited state than in the ground state.

The sequence of drawings presented at the left corresponds to an atom going through the atomic process involved in the functioning of a laser.  The first drawing of the sequence shows the pumping of energy in the atom by incoherent radiation photons while the electron is in the ground state, . Only one photon from the incoherent radiation is absorbed by the electron pumping it to the regular excited state, . The electron stays on the excited time for a time of the order of 10-8 seconds, . The electron emits a photon with an energy equals to the difference in energy between the regular excited state and the meta-stable state. While emitting, the electron jumps from the regular excited state to the meta-stable state, . Now, the electron stays in the meta-stable state for a time 105 times longer than when it was in the regular excited state. Thus, the electron stays in the meta-state for about 10-3 seconds,   . The last drawing of the sequence shows the emission of the laser photon. This photon has an energy corresponding to the transition energy from the meta-stable state to the ground state of the atom .

In a laser there are three basic processes, stimulated absorption, spontaneous emission, and stimulated emission. The drawings below shows these processes.

Stimulated Absorption: In this process a photon with an energy equals to the difference in energy between two of the atomic levels interacts with the atom. The electron of the atom is in the lower energy state, absorbs the photon, and jumps to the higher energy state (excited state). The photon is completely absorbed by the atom. The final state of the atom is excited.
Spontaneous emission: In this case, one of the electron is not in the lowest possible energy state; thus, the atom is excited. Spontaneously, the atom emits a photon with an energy equals to the difference in energy between the energy when the electron is in the excite state and the energy of the level when the electron is in the relaxed state (lower energy state). During this emission the electron jumps down to the lower energy state. The atoms give up energy which is carried by the emitted photon and it is left unexcited. On the other side, the emitted photon has an energy equals to the difference in energy between the two atomic states involved. However, there is not a prefer direction of motion for the emitted photon; in fact, the direction of emission is completely random.
Stimulated emission: When a photon with the same energy as the difference between two atomic levels reaches the atom and the corresponding electron to those levels is in the excited state, the electron jumps down to the lower energy state emitting an additional photon. After this process, two photons are emerging from the atom for every incident photon that interacts with the atom inducing the emission.

 

bullet Emission and Absorption of Photons

In this section, the relations between the atomic parameters and the incident electromagnetic radiation are study. Relations between the parameters associated with the stimulated absorption, spontaneous emission, and stimulated emission are presented.

The drawing on the left represents the two energy levels involved in the emission and absorption of photons. With the lower energy and the higher energy level. The sample to be study has many atoms with exactly the same characteristics. In addition, the sample has many atoms with the electron on the higher energy state and many atoms with the electron in the lower energy state. All these atoms are in thermal equilibrium at the absolute temperature with the electromagnetic radiation of frequency and energy density .
Let us call the number of atoms in the state and the number of atoms in the state .
Thus, the number of atoms that absorb a photon and jump from the state with energy to the state with energy can be written as

where is a constant that depends on the properties of the states and .

In the same form, the number of atoms that can spontaneously drop from the state to the state by emitting a photon is where is a constant that represents the probability for this atomic transition and the corresponding photon emission.

At the same time, if is the corresponding constant representing the probability for stimulated emission, the number of atoms that emit photons by either spontaneous or stimulated emission can be written as .

For the atomic system media at thermal equilibrium with the radiation, the number of transitions from  to is the same as the number of transitions from to ,

At the previous equality,  the radiation energy density can be obtained,

Applying, Maxwell-Boltzmann distribution to the number of atoms in a particular state, and where is Boltzmann constant. The ratio between the two number of atoms in the two states is . Therefore,

The previous result can be compared with Planck radiation formula, , and the following equalities among the various quantities are deduced,

and

These result imply the following conclusions:

:

  1. Stimulated emission DOES occur.

  2. The stimulated emission and the stimulated absorption have the same weight in the transition numbers. That is, if the number of atoms in the two atomic states are about the same, the probability for an atom of the simple to undergo stimulate d emission is the same as the probability of an atom in the simple to undergo and stimulated absorption.

  3. An incident photon of energy has the same probability of causing an atom to emit a photon of this energy as of being absorbed by an atom of the simple when the radiation and the simple are at thermal equilibrium.

:

  1. The spontaneous emission occurrence in the media increases rapidly with the energy difference between the two atomic states when compared to the stimulated emission.

 

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by Luis F. Sez, Ph. D.    Comments and Suggestions: LSaez@dallaswinwin.com