Principles Of Laser Action
Basic principles involved in the generation of lasers
Absorption, spontaneous emission and stimulated emissionAs we all know that atoms and molecules can exist only in certain energy states. The state of lowest energy is called the ground state; all other states have more energy than the ground state and are called excited states. Each excited state, of which there are many, has a fixed amount of energy over and above that of the ground state. Under ordinary conditions, almost all atoms and molecules are in their ground states. Three types of processes are possible for a two-level atomic system. In the first, an incoming photon excites the atomic system from a lower energy state into a higher energy state. This is called absorption or sometimes stimulated absorption. It is called stimulated absorptions because of the fact that the atoms absorb the incident energy at certain frequencies only. Stimulated absorption occurs when a photon strikes an atom with just exactly the proper energy to induce an electronic transition between two energy states. In case a broadband light is incident on a given two level atomic system, we can observe that the complete spectrum is not absorbed but only certain discrete lines are absorbed depending on the difference in their energy levels. This process reduces the lower level population and in the process increases the upper level population. The population or the number of atoms in states E1 and E2 at any time would be N1 and N2 respectively. When radiation passes through a material, it is absorbed according to:
(1) Ix = I0e-αxWhere Ix is the radiance after traveling distance x through the material with absorption coefficient as a and I0 is the initial intensity of light. The absorption depends on the population difference between N1 and N2 and the refractive index of the medium.
Rate of stimulated absorption, R12 (abs), from level 1 to 2 is given as:
(2) R12 (abs) = B12 N1 ρWhere B12 is the Einstein's coefficient for stimulated absorption and has the units as cm3/s2J, N1 is the population in the ground state and ρ is the energy density per unit frequency of the incoming photons.
Once the atom or molecule has been produced in its excited state, there is a probability that it will emit radiation again and return to a lower energy state. This lower energy state may be either the ground state or still one of the excited states but having lower energy level. In the process, a photon is emitted. In this emission process, where the atoms spontaneously goes to a lower energy state through the emission of a photon is called spontaneous emission or fluorescence. This emission process is a random one and the emitted light goes off in all directions, and the wave properties of the light are randomly out of step with each other and thus are incoherent.
The rate of spontaneous emission, R21 (spon), from level 2 to 1 is given as:
(3) R21 (spon) = A21 N2Where A21 is the coefficient of spontaneous emission and has the unit of s-1, N2 is the number of atoms in level 2.
One can observe that this spontaneous decay of the upper level takes place in the absence of an electromagnetic field and the rate is proportional to the population of that level and thus does not depend on the intensity of the excitation source. It is purely a statistical phenomenon related with time and space and is dependent on the lifetime of the excited state. If the transition lifetime is very large, it is considered as a forbidden transition.
Excited atoms can loose their energy not only by spontaneous emission, but also by induced or stimulated emission and therefore the emission output of the system consists of spontaneous and stimulated emissions. The probability of stimulated emission is proportional to the intensity of the energy density of external radiation and the induced emission has a firm phase relationship with it, unlike spontaneous emission. Since the spontaneous photons have no phase relations with each other, the output is incoherent. But stimulated emission has the same phase, direction, spectral and polarization properties as the stimulating field and both are indistinguishable in all aspects. Consequently, the laser output is coherent. In fact it is this stimulated emission, under certain conditions as explained in the earlier section that comes out of the laser device as laser.
Rate of stimulated emission, R21 (stim), from level 2 to 1 is given as:
(4) R21 (stim) = B21 N2 ρWhere B21 is the Einstein's coefficient for stimulated emission and has the dimensions as m3/s2J, N2 is the population in the excited state and ρ is the energy density per unit frequency of the triggering photons.
Considering an ideal material with only two non-degenerate energy levels, where absorption, spontaneous emission and stimulated emission takes place, one can arrive at the following conclusion.
Absorption = spontaneous emission + stimulated emission
(5) i.e. B12N1 r(n) = A21N2 + B21N2 rThis situation is shown in the figure.
Fig. 1: Spontaneous and simulated processes in a two-level systemAt any given instance, under normal circumstances, both stimulated and spontaneous emissions may occur, but the probability of stimulated emission is pretty low. One can find out this ratio of spontaneous to stimulated emission using one of the following equations:
(7)where ρ is the radiation energy density and is equal to Nhn, N being the number of photons of frequency n per unit volume and k is Boltzmann's constant. Considering a case of ordinary bulb having a filament temperature of about 5000K and emitting radiation in the wavelength range of 0.6 micron corresponding to frequency of 5 x 10 14 Hz, the probability of stimulated emission is approximately one hundredth of that of the spontaneous emission. At lower temperatures, it would even be orders less than this.
The ratio of the probability of spontaneous to stimulated light emission depends directly on the frequency of emission or inversely to the wavelength. Thus in the microwave region, stimulated emission is more probable than spontaneous, hence the early production of the maser. In the optical region, spontaneous emission is more likely than stimulated emission and this gets worse as we go into the UV and X-ray regions of the spectrum.
Under thermal equilibrium, the population N2 and N1 of levels E2 and E1 respectively governed by the fact that the rate of upward transitions should be equal to rate of downward transitions.
The population density of atoms N1 and N2 in ground level E1 and excited state E2 can be estimated using Boltzmann's relationship as follows:
(8)Since, (E2 - E1) / kT is always positive, irrespective of the value of temperature T, N2 must be less than N1 if the system is remain at thermal equilibrium. At the most the excited state population N2 (t) reaches a steady state at t → ∞, and the highest proportion of atoms that can exist in the excited state N2/Ntotal<1/2. Under these conditions the material always acts as an absorber of incident photons.
The above discussion implies that in a two level system the number of atoms in the excited state can never exceed the number in the ground state and hence can never work as a laser. If the system is to act as a laser, an incident photon must have a higher probability of causing stimulated emission than of being absorbed i.e. the rate of stimulated emission must exceed that of absorption. In other words, the laser action is possible only when N2 > N1. This non-equilibrium condition is known as called population inversion.
Before we discuss about the techniques of population inversion and laser action, these are some additional important points related to Absorption, spontaneous emission and stimulated emission:
- In case of spontaneous emission of a photon, the probability of its emission is inversely related to the average length of time that an atom can reside in the upper level of the transition before it relaxes. This time is known as the SPONTANEOUS LIFETIME. Typically, the spontaneous lifetime is of the order of 10-8 - 10-9 sec. The shorter the spontaneous lifetime, the greater is the probability that spontaneous emission will occur.
- In certain materials, there are energy levels, which has the spontaneous lifetime of the order of microseconds to a few milliseconds. These levels are known as METASTABLE levels. The probability of transitions involving metastable levels is relatively low.
- As the likelihood of spontaneous emission decreases the conditions that favor stimulated emission are enhanced. If an atom is excited into a metastable state it can stay there long enough for a photon of the correct frequency to arrive. Such a situation promotes stimulated emission at the expense of spontaneous emission.
- In case of stimulated emission, atoms in an upper energy level can be triggered or stimulated in phase by an incoming photon of a specific energy. The incident photon must have an energy corresponding to the energy difference between the upper and lower states. The emitted photons have the same energy as incident photon. These photons are in phase with the triggering photon and also travel in its direction.
- Stimulated processes like stimulated absorption, or stimulated emission require incoming photons of the right frequency, whereas spontaneous emission can take place in the absence of incoming photon also.
- Spontaneous emission is completely isotropic. Stimulated processes, on the other hand, have a built-in preference for emission in the direction of the incident flux of photons.
Population Inversion and Laser OperationAs discussed above, whenever light is incident on the material, there is competition between absorption, spontaneous emission and stimulated emission processes. Under normal equilibrium conditions, the population of various levels is given by Boltzmann's relationship and thus N2 will always be less than N1. Further, stimulated photon emission is much less than the spontaneous photon emission and the absorption. For a system to work as a laser one requires that stimulated emission should exceed photon absorption; it leads us to the following two conditions:
- N2 > N1: i.e. Population Inversion
- As per equation (6) or (7), the value of ρ (the radiation energy density which is equal to Nhn) should be as large as possible.
The second condition that requires higher value of r necessitates the use of an additional supply of large amount of energy of correct wavelength to excite the desired transition. The process is known as pumping. Various techniques include optical, electrical, chemical, gas dynamic etc.
Population inversion though is the primary condition, but in itself is not sufficient for producing a laser. As there are certain losses of the emitted photons within the material itself in addition to spontaneous emission, one has to think about the geometry that can overcome these losses and there is overall gain. This requires an optical cavity or resonator.
The principle behind the laser is like this. Suppose we can produce a large number of atoms all in excited states. If one of the atoms emitted spontaneously, then the emitted photon would stimulate other atoms to emit. These emitted photons would, in turn, stimulate further emission. The result would be an intense burst of coherent radiation.
These issues have been discussed below:
Fig. 2: Basic Laser systemA representative laser system is shown in Figure (2). It consists of three basic parts.
- An active medium with a suitable set of energy levels to support laser action.
- A source of pumping energy in order to establish a population inversion.
- An optical cavity or resonator to introduce optical feedback and so maintain the gain of the system overcoming all losses.
Active laser medium or gain medium: Laser medium is the heart of the laser system and is responsible for producing gain and subsequent generation of laser. It can be a crystal, solid, liquid, semiconductor or gas medium and can be pumped to a higher energy state. The material should be of controlled purity, size and shape and should have the suitable energy levels to support population inversion. In other words, it must have a metastable state to support stimulated emission. Most lasers are based on 3 or 4 level energy level systems, which depends on the lasing medium. These systems are shown in figs 3a and 3b. In case of a three-level laser, the material is pumped from level 1 to level 3, which decays rapidly to level 2 through spontaneous emission. Level 2 is a metastable level and promotes stimulated emission from level 2 to level 1.
On the other hand in a four level laser, the material is pumped to level 4, which is a fast decaying level, and the atoms decay rapidly to level 3, which is a metastable level. The stimulated emission takes place from level 3 to level 2 from where the atoms decay back to level 1. Four level lasers is an improvement on a system based on three level systems. In this case, the laser transition takes place between the third and second excited states. Since lower laser level 2 is a fast decaying level which ensures that it rapidly gets empty and as such always supports the population inversion condition.
- Excitation or pumping mechanism: Absorption of the energy by the atoms, electrons, ions or molecules as the case may be, of the active medium is a primary requisite in the generation of laser. In order to excite these elements to higher energy levels, an excitation or pumping mechanism is necessary. It is well known that under the equilibrium state, as per Boltzman?s conditions, higher energy levels are much less populated than the lower energy levels. One of the requirements of laser action is population inversion in the levels concerned. i.e. to have larger population in the upper levels than in the lower ones. Otherwise absorption will dominate at the cost of stimulated emission. There are various types of excitation or pumping mechanisms available, the most commonly used ones are optical, electrical, thermal or chemical techniques, which depends on the type of the laser gain medium employed. For example, Solid state lasers usually employ optical pumping from high energy xenon flash lamps (e.g., ruby, Nd:YAG) or from a second pump laser or laser diode array (e.g., DPSS frequency doubled green lasers). Gas lasers use an AC or DC electrical discharge through the gas medium, or external RF excitation, electron beam bombardment, or a chemical reaction. The DC electrical discharge is most common for 'small' gas lasers (e.g., helium-neon, argon ion, etc.). DC most often pumps semiconductor lasers current. Liquid (dye) lasers are usually pumped optically.
- Optical resonator: Optical resonator plays a very important role in the generation of the laser output, in providing high directionality to the laser beam as well as producing gain in the active medium to overcome the losses due to, straying away of photons from the laser medium, diffraction losses due to definite sizes of the mirrors, radiation losses inside the active medium due to absorption and scattering etc. In order to sustain laser action, one has to confine the laser medium and the pumping mechanism in a special way that should promote stimulated emission rather than spontaneous emission. In practice, photons need to be confined in the system to allow the number of photons created by stimulated emission to exceed all other mechanisms. This is achieved by bounding the laser medium between two mirrors as shown in figure 2. On one end of the active medium is the high reflectance mirror (100% reflecting) or the rear mirror and on the other end is the partially reflecting or transmissive mirror or the output coupler. The laser emanates from the output coupler, as it is partially transmissive. Stimulated photons can bounce back and forward along the cavity, creating more stimulated emission as they go. In the process, any photons which are either not of the correct frequency or do not travel along the optical axis are lost.
Laser actionInteraction of electromagnetic radiation with matter produces absorption and spontaneous emission. Absorption and spontaneous emission are natural processes. For the generation of laser, stimulated emission is essential. Stimulated emission has to be induced or stimulated and is generated under special conditions as stated by Einstein in his famous paper of 1917. i.e. ?when the population inversion exists between upper and lower levels among atomic systems, it is possible to realize amplified stimulated emission and the stimulated emission has the same frequency and phase as the incident radiation?. Einstein combined Plank? law with Boltzmann?s statistics in formulating the concept of stimulated emission. In electronic, atomic, molecular or ionic systems the upper energy levels are less populated than the lower energy levels under equilibrium conditions. Pumping mechanism excites say, atoms to a higher energy level by absorption (Figs.3a and 3b).
The atom stays at the higher level for a certain duration and decays to the lower stable ground level spontaneously, emitting a photon, with a wavelength decided by the difference between the upper and the lower energy levels. This is referred to as natural or spontaneous emission and the photon is called spontaneous photon. The spontaneous emission or fluorescence has no preferred direction and the photons emitted have no phase relations with each other, thus generating an incoherent light output (Fig.4). But it is not necessary that the atom is always de-excited to ground state. It can go to an intermediate state, called metastable state with a radiation less transition, where it stays for a much longer period than the upper level and comes down to lower level or to the ground state. Since period of stay of atoms in the metastable state is large, it is possible to have a much larger number of atoms in metastable level in comparison to the lower level so that the population of metastable state and the lower or ground state is reversed. i.e. there are more atoms in the upper metastable level than the lower level. This condition is referred to as population inversion. Once this is achieved, laser action is initiated in the following fashion. The atom in the metastable state comes down to the ground state emitting a photon. This photon can stimulate an atom in the metastable state to release its photon in phase with it. The photon thus released is called stimulated photon. It moves in the same direction as the initiating photon, has the same wavelength and polarization and is in phase with it, thus producing amplification. Since there are a large number of initiating photons, it forms an initiating electromagnetic radiation field. An avalanche of stimulated photons is generated, as the photons traveling along the length of the active medium stimulates a number of excited atoms in the metastable state to release their photons. This is referred to as the stimulated emission. These photons are fully reflected by the rear reflector (100% reflective) and the number and consequently the intensity of stimulated photons increases as they traverse through the active medium, thus increasing the intensity of radiation field of stimulated emission. At the output coupler, a part of these photons are reflected and the rest is transmitted as the laser output. This action is repeated and the reflected photons after striking the rear mirror, reach the output coupler in the return path. The intensity of the laser output increases as the pumping continues. When the input pumping energy reduces, the available initiating and subsequently the stimulated photons decrease considerably and the gain of the system is not able to overcome the losses, thus laser output ceases. Since the stimulation process was started by the initiating photons, the emitted photons can combine coherently, as all of them are in phase with each other, unlike in the case of spontaneous emission and coherent laser light is emitted (Fig.5). Though the laser action will continue as long as the energy is given to the active medium, it may be stated that pulsed laser is obtained if the population inversion is available in a transient fashion and continuous wave (CW) laser is possible if the population inversion is maintained in a steady-state basis. If the input energy is given by say a flash lamp, the output will be a pulsed output and the laser is called a pulsed laser. If equilibrium can be achieved between the number of photons emitted and the number of atoms in the metastable level by pumping with a continuous arc lamp instead of a flash lamp, then it is possible to achieve a continuous laser output, which is called continuous wave laser.
This web site does not intend to provide complete rate equations related to laser generation; only the salient features of the same have been given above.
- W. K. Koechner, Solid State Laser Engineering, Spriger-Verlag, London
- Lasers principles
- Laser fundamentals
- Sam's laser FAQ
Updated: 6 April, 2015