Basic Laser Principals

1) Laser Overview
2) Basics of the Atom
3) Absorption And Emission
4) Energizing the Laser Medium
5) Discrete Energy Levels

6) Spontaneous Emission
7) Stimulated Emission
8) Creating a Population Inversion
9) The Optical Resonator
10) Summary of Basic Laser Principals

11) Diffraction
12) Laser Modes
13) Types Of Laser
14) Laser Applications

Laser Overview

Star Wars," "Star Trek," "Battlestar Galactica" -- laser technology plays a pivotal role in science fiction movies and books. It's no doubt thanks to these sorts of stories that we now associate lasers with futuristic warfare and sleek spaceships, but lasers play a pivotal role in our everyday lives, too. The fact is, they show up in an amazing range of products and technologies. You'll find them in everything from CD players to dental drills to high-speed metal cutting machines to measuring systems. Tattoo removal, hair replacement, eye surgery -- they all use lasers. But what is a laser? What makes a laser beam different from the beam of a flashlight? Specifically, what makes a laser light different from other kinds of light? How are lasers classified?
In this tutorial, you'll learn the basics about how lasers work, the physics that enable lasers to emit such special light, the different types of lasers, their different wavelengths and the uses to which we put them.

Lasers come in wide variety of forms, and the processes that go on inside them differ greatly from one type of laser to another. For this reason, it is easy to become distracted by detail that might apply to one type of laser only. In this tutorial, an attempt has been made to concentrate on those features that lasers have in common,however, some of the specific details of various types of lasers will be decribed near the end of the tutorial.
The LASER (Light Amplification by Stimulated Emission of Radiation) is a triumph of modern optics. By exploiting a quantum mechanical effect called stimulated emission, lasers generate a coherent, nearly monochromatic beam of photons. Non-laser light sources typically generate incoherent, unfocused beams of light in which photons, at a multitude of wavelengths and phases, are emitted in random directions, prohibiting certain applications.
To create a laser, two components are necessary - a gain medium and a resonant optical cavity. For a gain medium, certain crystals, glasses, gasses, reactant chemical, semiconductors and even dyed liquids may be used. The gain medium is stimulated by an energy pump source such as an electrical current or another laser. The medium absorbs the energy, exciting the states of the particles in the medium. After a certain threshold, called population inversion, is achieved, shining light through the medium causes more stimulated emission, or release of energy, than absorption.
A resonant optical cavity is a specially sized chamber with a mirror at one end and a semi-silvered mirror at the other. The two reflective surfaces cause light trapped inside to reflect back and forth through the gain medium, acquiring greater energy with each pass. When this effect levels off (reaches equilibrium), the gain is said to be saturated and the light becomes true laser light. Different gain mediums give rise to lasers of different wavelengths.
Two varieties of laser are continuous and pulsed. The continuous laser is more useful for most applications, but the energy in a pulsed laser can be very large. The degree to which the beam diverges over time varies inversely with proportion to its diameter. Small beams diverge rapidly, while larger ones remain coherent.
When the laser was patented by Bell Labs in 1960, it could not immediately be given any applications, although spectrometry, interferometry, radar and nuclear fusion were discussed as potential areas of interest. Today, the laser is among the most versatile of technological wonders, with applications in data storage and retrieval, laser cutting, vision correction, surveying, measurements, holography and displays, and even nuclear fusion. Maximum achievable laser pulse intensity has increased exponentially since the mid-1980s. One day, lasers may be used to generate net energy-producing fusion reactions, providing energy for the entire human race. They also might be used to push solar sails into the depths of outer space.

World's first laser
Theodore Harold Maiman was born in 1927 in Los Angeles, son of an electrical engineer. He studied engineering physics at Colorado University, while repairing electrical appliances to pay for college, and then obtained a Ph.D. from Stanford. Theodore Maiman constructed this first laser in 1960 while working at Hughes Research Laboratories (T.H. Maiman, "Stimulated optical radiation in ruby lasers", Nature, 187, 493, 1960). There is a vertical chromium ion doped ruby rod in the center of a helical xenon flash tube. The ruby rod has mirrored ends. The xenon flash provides optical pumping of the chromium ions in the ruby rod. The output is a pulse of red laser light. (Courtesy of HRL Laboratories, LLC, Malibu, California.) Fusion Ignition:World's largest laser unveiled
By Sharon Gaudin, Computerworld, 05/29/2009
Scientists at the Lawrence Livermore National Laboratory are taking the wraps off the world's largest laser today. The stadium-sized laser, dubbed the National Ignition Facility (NIF), is being housed at the Department of Energy's Lawrence Livermore National Laboratory in Livermore, Calif. The record-breaking laser is made up of 192 individual beams, each about 40 centimeters square. The laser fusion facility is designed to deliver mega amounts of energy with pinpoint precision in billionths of a second, according officials at the scientific research laboratory.

Now, let's start with the fundamentals of laser technology: the basics of what atoms are, how they absorb energy and emit light.
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Basics of the Atom

There are only about 100 different kinds of atoms in the entire universe. Everything we see is made up of these 100 atoms in a nearly unlimited number of combinations. How these atoms are arranged and bonded together determines whether the atoms make up a cup of water, a piece of metal, or the fizz that comes out of your soda can!
Atoms are constantly in motion. They continuously vibrate, move and rotate. Even the atoms that make up the chairs that we sit in are moving around. Solids are actually in motion! Atoms can be in different states of excitation. In other words, they can have different energies. If we apply a lot of energy to an atom, it can leave what is called the ground-state energy level and go to an excited level. The level of excitation depends on the amount of energy that is applied to the atom via heat, light, or electricity and how the atom responds to the particular energy applied. In the simple, classical example, an atom consists of a nucleus (containing the protons and neutrons) and an electron cloud. It’s helpful to think of the electrons in this cloud circling the nucleus in many different orbits, with electrons in higher orbits having more energy than those in lower orbits.

A classic interpretation of what an atom looks like.

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Absorption And Emission

The next few sections will discuss the fundamentals of absorption of energy and emission of photons in an optical medium. A laser is a device that controls the way that energized atoms release photons. "Laser" is an acronym for Light Amplification by Stimulated Emission of Radiation, which describes very succinctly how a laser works.
Although there are many types of lasers, all have certain essential features. In a laser, the lasing medium is “pumped” to get the atoms into an excited state. Typically, very intense flashes of light or electrical discharges pump the lasing medium and create a large collection of excited-state atoms (atoms with higher-energy electrons).

Emission of energy: An atom MAY emit it's photons at random times and in random directions.

It is necessary to have a large collection of atoms in the excited state for the laser to work efficiently. In general, the atoms are excited to a level that is two or three levels above the ground state. This increases the degree of population inversion. The population inversion is the number of atoms in the excited state versus the number in ground state.
Once the lasing medium is pumped, it contains a collection of atoms with some electrons sitting in excited levels. The excited electrons have energies greater than the more relaxed electrons. Just as the electron absorbed some amount of energy to reach this excited level, it can also release this energy. As the figure above illustrates, the electron can simply relax, and in turn rid itself of some energy. This emitted energy comes in the form of photons (light energy). The photon emitted has a very specific wavelength (color) that depends on the state of the electron's energy when the photon is released. Two identical atoms with electrons in identical states will release photons with identical wavelengths.
In 1916, Albert Einstein proposed that there are essentially three processes occurring in the formation of an atomic spectral line. The three processes are referred to as absorption, spontaneous emission, and induced emission, and with each is associated an Einstein coefficient which is a measure of the probability of that particular process occurring. We will discuss each of these processes generally, while avoiding the very complex mathematics that can be involved in quantifying the behavior.

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Energizing the Active Medium - Absorption Of Energy

Consider the illustration above. Although more modern views of the atom do not depict discrete orbits for the electrons, it can be useful to think of these orbits as the different energy levels of the atom. In other words, if we apply some heat or light energy to an atom, we might expect that some of the electrons in the lower-energy orbits would jump (transition) to higher-energy orbits farther away from the nucleus.
This is a highly simplified view of things, but it actually reflects the core idea of how atoms work in terms of lasers.

Absorption of energy: An atom absorbs energy in the form of heat, light, electricity. Electrons may move from a lower-energy orbit to a higher-energy orbit.
Any material that will serve as the amplifying medium for the lasing process must have a large population of these excited electrons. Increasing the intensity of a light beam that passes through an amplifying medium requires putting additional energy into the light beam. This energy comes from the amplifying medium which must in turn have energy fed into it in some way. In laser terminology, the process of energizing the amplifying medium is known as "pumping".
There are many configurations and methods for pumping an amplifying medium. In chemical lasers, a hyperactive chemical reaction is used to excite the electrons. In some gas lasers, a high intensity radio frequency (RF) field is passed through the gas. When the lasing medium is a solid, pumping is usually achieved by irradiating it with intense light. This light is absorbed by atoms or ions within the medium raising them into higher energy states.
In older lasers, Xenon-filled flashtubes positioned as shown at right are sometimes used as a simple source of pumping light. Passing a high voltage electric discharge through the flashtubes causes them to emit an intense flash of white light, some of which is absorbed by the amplifying medium.
The assembly of flashtubes is enclosed within a polished metal reflector (not shown in the diagram above) to concentrate as much light as possible on the amplifying medium. A laser that is pumped in this way will have a pulsed output. Pumping an amplifying medium by irradiating it with intense light is referred to as optical pumping. The source of pumping light can be another laser. Some types of laser that were originally pumped using xenon-filled flashtubes are now, interestingly, pumped by laser diodes.
This is known as a Diode Pumped Solid State Laser or DPSSL) and can be a compact source of intense light. The laser diodes themselves use electricity to emit their laser light. One of the advantages of using diodes is that the wavelength of their output can be closely matched with the absorption bands of the lasing medium, thus improving the efficiency of the laser. We will discuss this laser configuration in some detail in "Types Of Lasers".

Gaseous amplifying media have to be contained in some form of enclosure or tube and are often pumped by passing an electric discharge through the medium itself. The mechanism by which this elevates atoms or molecules in the gas to higher energy states depends upon the gas that is being excited and is often complex.

In many gas lasers, the end windows of the laser tube are inclined at an angle and they are referred to as Brewster windows. Brewster windows are able to transmit a beam that is polarized in the plane of the diagram without losses due to reflection. Such a laser would have an output beam that is polarized. Polarization of light is discussed in another tutorial.

The diagram illustrates pumping by passing a discharge longitudinally through the gaseous amplifying medium but, in some cases, the discharge takes place transversely from one side of the medium to the other. Many lasers that are pumped by an electric discharge can produce either a pulsed output or a continuous output depending upon whether the discharge is pulsed or continuous.
Various other methods of pumping the amplifying medium in a laser are used. For example, laser diodes are pumped by passing an electric current across the junction where the two types of semiconductor within the diode come together.

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Discrete Quantum Levels

It must be understood that the excited energy states can only occur at discrete (or quantum) levels. The "orbits" of the electrons can occur only at defined "distances" and the electrons jump precisely from one state up to a higher one, and down again. Each atom or ion has it's own discrete energy transition levels. Because each energy level can contain only a fixed number of electrons, each orbit is associated with a particular range of electron energy, and thus each orbit must fill completely before electrons can be added to a higher orbit (these levels are also known as "shells"). The electrons in the outermost shell determine the light emission properties of the atom. Some examples of the energy states and how they are used in the laser absorption and emission process are given in the diagrams below.

Historical framework
The primary indication for the existence of discrete energy levels came from the study of the spectrum of emissions of energetically excited atomic systems. Historically, the most important such spectrum is that of the simplest atom, hydrogen, a system of one proton and one electron bound together by their electromagnetic attraction.

Within the framework of classical physics, the structure of the hydrogen atom poses fundamental problems. The first is the existence of a stable ground state: An electron in orbit around a proton is in constant acceleration, and therefore, according to Maxwell's classical electromagnetic theory, should continuously radiate away energy. Furthermore, the radiation emitted as the atom decays to a lower energy state should form a continuous spectrum of frequencies. However, the hydrogen atom both possesses a stable ground state and emits radiation at only a discrete set of frequencies.
In 1913 N. Bohr made a fundamental advance by postulating that the angular momentum of the electron-proton system could take on only a discrete set of values. The angular momentum is said to be quantized. A consequence is that the excitation energies of the hydrogen atom also have a discrete spectrum. Bohr made the further postulate that the atom decays from an excited level, Ek, only by making a transition to a lower energy level, Ej, emitting a single light quantum (photon) in the process. The energy, E?, of this photon is given by the conservation of energy, Ey = Ek - Ej. Although Bohr's postulates are in many ways without real foundation, they were later justified and extended by the development of quantum mechanics.

Three Level Laser
Here's what happens in a three level laser.

The diagram at right shows the transitions for a three level laser. The pump causes an excitation from the ground state to the second excited state. This state is a rather short-lived state, so that the atom quickly decays into the first excited level. Decays back to the ground state also occur, but these atoms can be pumped back to the second excited state again. The first excited state is a long-lived (i.e. metastable) state which allows the atom to "wait" for the "passer-by" photon, (we will talk about the significance of this in the Stimulated Emission section), while building up a large population of atoms in this state. The lasing transition, in this laser, is due to the decay of the atom from this first excited metastable state to the ground state.
There are also Four Level Lasers in which the lasing transition is from the second highest state to the third highest state.

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Spontaneous Emission

Once an electron moves to a higher-energy orbit, it eventually wants to return to the ground state. When it does, it releases its energy as a photon -- a particle of light. You see atoms releasing energy as photons all the time. For example, when the heating element in a toaster turns bright red, the red color is caused by atoms, excited by heat, releasing red photons. When you see a picture on a TV screen, what you are seeing is phosphor atoms, excited by high-speed electrons, emitting different colors of light. Anything that produces light -- fluorescent lights, gas lanterns, incandescent bulbs -- does it through the action of electrons changing orbits and releasing photons.

The processes of the absorption and spontaneous emission of light are illustrated at right. A photon of light is absorbed by an atom in which one of the outer electrons is initially in a low energy state denoted by 0. The energy of the atom is raised to the upper energy level, 1, and remains in this excited state for a period of time that is typically less than 10-6 second. It then spontaneously returns to the lower state, 0, with the emission of a photon of light. Absorption is referred to as a resonant process because the energy of the absorbed photon must be equal to the difference in energy between the levels 0 and 1. This means that only photons of a particular frequency (or wavelength) will be absorbed. Similarly, the photon emitted will have energy equal to the difference in energy between the two energy levels.

These common processes of absorption and spontaneous emission cannot give rise to the amplification of light. The best that can be achieved is that for every photon absorbed, another is emitted.
More technically speaking, spontaneous emission is the radiation emitted when a quantum mechanical system drops spontaneously from an excited level to a lower level. This radiation is emitted according to the laws of probability without regard to the simultaneous presence of similar radiation. The rate of spontaneous emission is proportional to the Einstein "A" coefficient and is inversely proportional to the radiative lifetime.
The Einstein Coefficients are a set of probability coefficients that express the probabilities of stimulated and spontaneous radiative transitions between stationary energy levels.
The radiative lifetime of an excited electronic state e.g. in a laser gain medium is the lifetime which would be obtained if radiative decay via the unavoidable spontaneous emission were the only mechanism for depopulating this state. It is given by the equation:

which shows that high emission cross sections and a large emission bandwidth inevitably lead to a low radiative lifetime. This is because the cross sections describe not only the strength of stimulated emission but also that of spontaneous emission. The derivation of this equation is based on an equation for the mode density of free space, as is also used e.g. for the derivation of Planck's law for the power spectral density of thermal radiation. This means that the equation does not hold in microcavities (as often used in experiments on quantum electrodynamics), because such cavities can substantially modify the mode density.
Note also the influence of the refractive index via the mode density. If fluorescence lifetime measurements are done using a powder with a grain size well below the wavelength of light, the refractive index of the ambient medium (rather than that of the powder grains) becomes relevant. For example, the upper-state lifetime measured for powder in air can be longer compared with that for solid crystals. Such observations should not be misinterpreted as evidence for quenching effects in crystals.
Another important aspect is that a shorter mean wavelength of the emission implies a shorter radiative lifetime. This results from the increased mode density of the radiation field. A consequence is that ultraviolet lasers tend to have a higher threshold pump power than e.g. infrared lasers.
As the gain efficiency of a laser medium is (in simple cases) proportional to the product of the maximum emission cross section and the upper-state lifetime (the s– product), lasers based on broadband gain media have a higher threshold pump power.
The actual lifetime of an electronic level can be lower than the radiative lifetime, if non-radiative quenching processes also significantly depopulate the level. This means that the quantum efficiency of the transition is below unity.
If the quantum efficiency is known to be close to unity, the above equation can be used for obtaining the absolute scaling of emission cross sections, the wavelength dependence of which is already known from the shape of the emission spectrum (? Füchtbauer–Ladenburg equation). In other cases, where the scaling of emission cross sections is known (e.g. obtained from absorption cross sections via the reciprocity method), the quantum efficiency of the fluorescence can be obtained by comparing the calculated radiative lifetime with the upper-state lifetime

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Stimulated Emission

Stimulated emission is a very uncommon process in nature but it is central to the operation of lasers. Above it was stated that an atom in a high energy, or excited, state can return to the lower state spontaneously. However, if a photon of light interacts with the excited atom, it can stimulate a return to the lower state. One photon interacting with an excited atom results in two photons being emitted. Furthermore, the two emitted photons are said to be in phase, i.e. thinking of them as waves, the crest of the wave associated with one photon occurs at the same time as on the wave associated with the other. This feature ensures that there is a fixed phase relationship between light radiated from different atoms in the amplifying medium and results in the laser beam produced having the property of coherence.

The MAGIC That Enables L.A.S.E.R.

The "magic" of lasers is that if a photon of the same "color" (wavelength) passes in close proximity to an excited atom, it will stimulate that atom to emit it's photon at that instant in precisely the same direction and with the same phase as the stimulating photon. Now, we have two photons that are propagating in lock step! The mirrors cause the effect to be Amplified many fold and lasing action begins. If the medium is being pumped at a rate that equals the number of photons being emitted as laser light, the laser is in equilibrium and can emit a continuous average level of laser energy.

Stimulated emission is the process that can give rise to the amplification of light. As with absorption, it is a resonant process; the energy of the incoming photon of light must match the difference in energy between the two energy levels. Furthermore, if we consider a photon of light interacting with a single atom, stimulated emission is just as likely as absorption; which process occurs depends upon whether the atom is initially in the lower or the upper energy level. However, under most conditions, stimulated emission does not occur to a significant extent. The reason is that, under most conditions, that is, under conditions of thermal equilibrium, there will be far more atoms in the lower energy level, 0, than in the upper level, 1, so that absorption will be much more common than stimulated emission. If stimulated emission is to predominate, we must have more atoms in the higher energy state than in the lower one. This unusual condition is referred to as a population inversion and it is necessary to create a population inversion for laser action to occur.

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Creating a Population Inversion

All lasers contain an energized substance that can increase the intensity of light passing through it by virtue of a pumping function that continuously replenishes the population of excited energy levels. This substance is called the amplifying medium or, sometimes, the gain medium, and it can be a solid, a liquid or a gas. Whatever its physical form, the amplifying medium must contain atoms, molecules or ions, a high proportion of which can store energy that is subsequently released as light. The achievement of a significant population inversion in atomic or molecular energy states is a precondition for laser action. When the rate of repopulation exceeds the rate at which photons leave the cavity (and other losses), then a Population Inversion has occurred and lasing will reach equilibrium.
In a neodymium YAG (Nd:YAG) laser, the amplifying medium is a rod of yttrium aluminium garnate (YAG) containing ions of the lanthanide metal neodymium (Nd). In a dye laser, it is a solution of a fluorescent dye in a solvent such as methanol. In a helium-neon laser, it is a mixture of the gases helium and neon. In a laser diode, it is a thin layer of semiconductor material sandwiched between other semiconductor layers. The factor by which the intensity of the light is increased by the amplifying medium is known as the gain. The gain is not a constant for a particular type of medium. It's magnitude depends upon the wavelength of the incoming light, the intensity of the incoming light, the length of the amplifying medium and also upon the extent to which the amplifying medium has been energized.
Once the medium is pumped and contains a sufficient population of excited electrons, a random process of excitation, relaxation and emission of photons will be occurring. What is needed is to use the Stimulated Emission effect to get many of these processes to emit photons that are in phase and in a single direction. This is done by using reflecting mirrors at each end of the resonant Laser Cavity. In most cases, one end of the cavity will use a totally reflecting mirror and the other will use a mirror that allows just enough photons to pass through it (partially reflecting) so that the population of excited electrons is not depleted. Rather, the choice of reflectivity for this end mirror is selected so that photons leave the cavity at the same rate that the electrons are being re-energized. In this way, the cavity can operate at equilibrium.
A population inversion cannot be achieved with just two levels (one level and ground state) because the probabability for absorption and for spontaneous emission is exactly the same, as shown by Einstein and expressed in the Einstein A and B coefficients. The lifetime of a typical excited state is about 10-8 seconds, so in practical terms, the electrons drop back down by photon emission about as fast as you can pump them up to the upper level.

Finding substances in which a population inversion can be set up is central to the develpment of new kinds of laser. The case of Ruby laser illustrates one of the ways of achieving the necessary population inversion. The first material used was synthetic ruby. Ruby is crystalline alumina (Al2O3) in which a small fraction of the Al3+ ions have been replaced by chromium ions, Cr3+. It is the chromium ions that give rise to the characteristic pink or red colour of ruby and it is in these ions that a population inversion is set up in a ruby laser.
In a ruby laser, a rod of ruby is irradiated with the intense flash of light from xenon-filled flashtubes. Light in the green and blue regions of the spectrum is absorbed by chromium ions, raising the energy of electrons of the ions from the ground state level to the broad F bands of levels. Electrons in the F bands rapidly undergo non-radiative transitions to the two metastable E levels. A non-radiative transition does not result in the emission of light; the energy released in the transition is dissipated as heat in the ruby crystal.

The metastable levels are unusual in that they have a relatively long lifetime of about 4 milliseconds (4 x 10-3 s), the major decay process being a transition from the lower level to the ground state. This long lifetime allows a high proportion (more than a half) of the chromium ions to build up in the metastable levels so that a population inversion is set up between these levels and the ground state level. This population inversion is the condition required for stimulated emission to overcome absorption and so give rise to the amplification of light. In an assembly of chromium ions in which a population inversion has been set up, some will decay spontaneously to the ground state level emitting red light of wavelength 694.3 nm in the process. This light can then interact with other chromium ions that are in the metastable levels causing them to emit light of the same wavelength by stimulated emission. As each stimulating photon leads to the emission of two photons, the intensity of the light emitted will build up quickly. This cascade process in which photons emitted from excited chromium ions cause stimulated emission from other excited ions is indicated below:

The ruby laser is often referred to as an example of a three-level system. More than three energy levels are actually involved but they can be put into three categories.These are; the lower level form which pumping takes place, the F levels into which the chromium ions are pumped, and the metastable levels from which stimulated emission occurs. Other types of laser operate on a four level system and , in general, the mechanism of amplification differs for different lasing materials. However, in all cases, it is necessary to set up a population inversion so that stimulated emission occurs more often than absorption.

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The Optical Resonator

Pumped amplifying media as described in Section 4 could be used to increase the intensity of light at particular wavelengths and such amplifiers are often incorporated into laser systems. However, except in a few exceptional cases, light amplifiers would not be regarded as lasers. A laser consists of a pumped amplifying medium positioned between two mirrors as indicated below. The purpose of the mirrors is to provide what is described as 'positive feedback'. This means simply that some of the light that emerges from the amplifying medium is reflected back into it for further amplification. An amplifier with positive feedback is known as an oscillator.

The space between the two mirrors is known as the laser cavity. The beam within the cavity undergoes multiple reflections between the mirrors and is amplified each time it passes through the amplifying medium. One of the mirrors reflects almost all of the light that falls upon it (total reflector in the above diagram). The other mirror reflects between 20% and 98% of the incident light depending upon the type of laser, the light that is not reflected being transmitted through the mirror. This transmitted portion constitutes the output beam of the laser.
The laser cavity has several important functions. Following pumping, spontaneous emission of light from excited atoms within the amplifying medium initiates the emission of low intensity light into the laser cavity. This light is increased in intensity by multiple passes through the amplifying medium so that it rapidly builds up into an intense beam. In the absence of cavity mirrors, this self-starting process, or oscillation, would not occur.
The cavity ensures that the divergence of the beam is small. Only light that travels in a direction closely parallel to the axis of the cavity can undergo multiple reflections at the mirrors and make multiple passes through the amplifying medium. More divergent rays execute a zig-zag path within the cavity and wander out of it.
The laser cavity also improves the spectral purity of the laser beam. Usually, the amplifying medium will amplify light within a narrow range of wavelengths. However, within this narrow range, only light of particular wavelengths can undergo repeated reflection up and down the cavity. The characteristics that a light beam within the cavity must possess in order to undergo repeated reflections define what is referred to as a cavity mode. Light which may still be amplified by the amplifying medium but which does not belong to one of these special modes of oscillation is rapidly attenuated and will not be present in the output beam. This behaviour is similar to that of a vibrating guitar string in that a particular string will only vibrate at certain frequencies. In a similar way, an optical cavity will only sustain repeated reflections for particular well-defined wavelengths of light.

Various resonant cavity configurations.

Resonator Stability

Only certain ranges of values for R1, R2, and L produce stable resonators in which periodic refocussing of the intracavity beam is produced. If the cavity is unstable, the beam size will grow without limit, eventually growing larger than the size of the cavity mirrors and being lost. By using methods such as ray transfer matrix analysis, it is possible to calculate a stability criterion:

Values which satisfy the inequality correspond to stable resonators.
The stability can be shown graphically by defining a stability parameter, g for each mirror:

and plotting g1 against g2 as shown. Areas bounded by the line g1 g2 = 1 and the axes are stable. Cavities at points exactly on the line are marginally stable; small variations in cavity length can cause the resonator to become unstable, and so lasers using these cavities are in practice often operated just inside the stability line.

Graph showing stable resonator solution space.

A simple geometric statement describes the regions of stability: A cavity is stable if the line segments between the mirrors and their centers of curvature overlap, but one does not lie entirely within the other.
In the confocal cavity a ray, which is deviated from its original direction in the middle between the cavity, is maximally (compared to other cavities) displaced on the return to the middle. This prevents amplified spontaneous emission and is important for a good beam quality and high power amplifiers. In wave optics this is expressed by the eigenvalue degeneration of the modes. On every turn to the left, the 0,0 mode and the 1,0 mode are 90° out of phase, but on the turn back, they are 180° out of phase. Interference of the modes then leads to a displacement.