I.  Requirements for a laser

A.  Laser gain medium

Nearly all lasers are produced as a result of electrons jumping from an excited energy level within a radiating species to a lower-lying energy level and, in the process, radiating light that contributes to the laser beam. Those radiating species can include:

atoms such as in the red helium-neon (HeNe) laser, the visible and ultraviolet argon ion and helium-cadmium (HeCd) lasers, and the green and yellow copper vapor lasers (CVL)

molecules such as in the infrared carbon dioxide (CO₂) laser, the ultraviolet excimer lasers such as ArF and KrF, and the pulsed N₂ laser

liquids such as those involving various organic dye molecules dilutely dissolved in various solvent solutions

dielectric solids such as those involving neodymium atoms doped in YAG or glass to make the crystalline Nd:YAG or Nd:glass lasers

semiconductor materials such as gallium arsenide or indium phosphide crystals or various mixtures of impurities blended with  those and other semiconductor species

Each of the above species contains a lowest energy level referred to as the ground state in which the electrons predominantly reside at room temperature, as indicated by level 0 in Figure 5-3.

Figure 5-3  Simplified energy diagram of an atom showing excitation and emission processes

The electrons are moved to higher-lying (excited) levels such as 1 and 2 by means of various pumping processes that will be described in the next section. They then decay back to lower-lying levels within a period of time called the lifetime of the level, and eventually find their way back to the ground state when the pumping source is removed. There are three types of processes involving the interaction of light beams with atoms that have electrons residing in various energy levels. Examples of those are depicted in Figure 5-4.

Figure 5-4  The three radiation processes that can occur when light interacts with matter (atoms)

First an electron residing in level 2 can spontaneously jump to level 1, radiating a photon of light when it does so. That process is known as spontaneous emission as indicated in Figure 5-4a. Most excited energy levels undergo spontaneous emission. Each level has a specific lifetime t over which it will remain in that level before decaying to a lower-lying level. That lifetime is determined by the interactions of the electron with the other electrons and nuclei of that atom. Typical lifetimes of electrons residing in specific levels that decay by radiating in the visible portion of the spectrum are of the order of 10–100 nsec. Of course the ground state cannot decay further and thus has infinite lifetime. The photon radiated during spontaneous emission has the exact wavelength l₂₁ and frequency n₂₁ corresponding to the difference in energy DE₂₁ of the two involved energy levels (1 and 2 in this case) according to the relationship

DE21  =  hn21  =

(5-3)

in which h is Planck’s constant such that h = 6.63 × 10^–34 joule-sec and c is the speed of light, c = 3 × 10⁸ m/sec. Also the wavelength l₂₁ is generally given in meters (often expressed in micrometers (mm) or nanometers (nm). Because different materials have different energy-level arrangements, they radiate at different wavelengths and thus emit different colors or frequencies of light that are specific to the material. Nearly all the light we see originates from such transitions between energy levels of various kinds of matter.

The second process is absorption, shown in Figure 5-4b, which occurs if the atom has its electron in level 1 of Figure 5-3 and a photon of light of wavelength l₂₁ collides with the atom. During the collision, the photon is absorbed by the atom and the electron is moved up to the higher energy level 2. This process is the way light interacts with practically all of matter. It can happen from any energy level that is occupied (generally the ground state) and always boosts the atom to a higher-lying level while eliminating the photon. This often results in heating of the absorbing material.

The third process, shown in Figure 5-4c, is referred to as stimulated emission. It results when an electron is in a higher-lying level, such as level 2 in Figure 5-3, and a photon of light of wavelength l₂₁ collides with the atom. During the collision the photon stimulates the atom to radiate a second photon having exactly the same energy DE₂₁ (and wavelength according to Equation 5-3) as that of the incident photon and traveling in exactly the same direction in order to satisfy the laws of conservation of energy and momentum. Hence, one photon leads to two identical photons, which, in effect, leads to an amplification process. A photon has been gained at the expense of the loss of energy stored within the atom.

When a large group of atoms is assembled and irradiated with light, most of those atoms are in the ground-state energy level (see Figure 5-3). If the photons of the impinging light have the appropriate energy DE₂₀ for example, as indicated in Figure 5-3, the light will be absorbed according to the following expression for the variation of intensity I with the distance L into the material

(5-4a)

in which I₀ is the intensity of the beam when it first reaches the atoms, s₂₀ is referred to as the cross section for absorption or emission of those two levels, and N₀ is the population density of atoms residing in level 0 (number of atoms per unit volume). If N₀ is in atoms/cm³ and L is in cm, the absorption cross section s₂₀ must be expressed in units of area or cm² (hence the name cross section). Equation 5-4a indicates that the amount of beam absorption depends on both the number density of atoms residing in level 0 and the length L or thickness of the medium comprising those atoms as indicated in Figure 5-5. Also, the exponential factor suggests quite rapid absorption if the exponent is large. For example, e^–2 = 0.135 and e^–4 = 0.018. Hence, if either the length of the medium or the population is doubled, the beam intensity drops by nearly a factor of 8!  Chemists have used this effect for many decades to measure the concentration of a material dissolved into a solvent. Equation 5-4a is known as Beer’s law, and the product s₂₀ N₀ is referred to as the absorption coefficient. This absorption process is also one of the techniques used in pumping lasers in order to transfer population to level 2 from level 0, as will be discussed later.

Figure 5-5  Intensity variation versus depth z into an absorbing sample

The absorption described above could have been equally applied if population initially existed in level 1, and light of energy DE₂₁ and wavelength l₂₁ would be absorbed by the medium according to the following equation

(5-4b)

An alternative situation will now be considered. Suppose that we were able to “pump” (excite) a significant amount of population of the medium from level 0 to level 2 according to Equation 5-4a. Also, for the time being let us assume that there is no population in level 1. (This is an unlikely scenario but we will do this as a “thought” experiment for illustrative purposes.)  Then again, let us consider having a beam of photons of energy DE₂₁ and wavelength l₂₁ enter the medium. According to the earlier discussion, and considering the process described in Figure 5-4c, the only process that can occur is stimulated emission, and we would expect more photons to be generated as the beam progresses. That is exactly what happens!  Since the absorption indicated in Figure 5-4b and also described in Equation 5-4a is a symmetrical process with the stimulated emission process of Figure 5-4c, it is not surprising that the beam evolves in a similar way to that of Equation 5-4a except that a sign reversal must be made in the exponent to reflect the process of photon production instead of photon absorption. This can be described mathematically in the equation below

(5-5)

in which we now have the population density N₂ in the expression along with the appropriate cross section s₂₁.

Now, if population is allowed to be in both level 1 and level 2, both absorption and stimulated emission will occur within the medium and therefore Equations 5-4 and 5-5 must be combined to give

(5-6)

as indicated in Figure 5-6. Hence, if more population exists in level 2 than in level 1, N₂ will be greater than N₁ and the exponent of Equation 5-6 will be positive. The beam will grow and emerge from the medium with a greater intensity than when it entered. In other words, for amplification or gain to occur, the condition must be

(5-7)

Having N₂ be larger than N₁ is known as having a population inversion, which is not a normal, naturally occurring relationship. This would be the equivalent of having a mountain in which there is more dirt at higher levels than at lower levels. The mountain would taper inward toward the bottom rather than outward, which is generally an unstable situation. The only way to maintain such an “inversion” is to continually transfer or “pump” more dirt up to higher levels by a conveyor belt or some other process. The equivalent transfer to higher levels, or “pumping” is also required in lasers to maintain the population inversion of level 2 with respect to level 1 such that amplification can be produced.

Figure 5-6  Absorption and stimulated emission effects combined in a laser gain medium

Population inversions in gases—Inversions in gases are generally produced by applying a voltage across a gas discharge tube that consists of a long, narrow glass or ceramic tube serving to confine the gain medium, with electrodes installed at each end of the tube. In its simplified form the electrodes, which are essentially electrical feedthroughs, are attached to each end of the tube to allow a voltage to be applied across the length of the tube. The tube is then filled with a low-pressure gas or gas mixture that includes the species that will serve as the gain medium. The applied voltage produces an electric field within the laser tube that accelerates the electrons within the gas. Those electrons collide with the gas atoms and excite the atoms to excited energy levels, some of which serve as upper laser levels. Lower-lying levels, those to which higher-lying levels can transition, typically decay to the ground state faster than the higher-lying levels, thereby establishing a population inversion between some of the higher and lower levels as indicated in Figure 5-7. This inversion can be envisioned by considering that, if the lower levels drain out faster than the upper levels, there will be less population left in those lower levels than in the higher-lying levels. The laser light then occurs when the higher-lying levels decay to the lower levels while radiating photons at the wavelengths corresponding to the energy separation between the levels. In many instances the excitation is a two-step process in which the electrons first excite a long-lived or metastable (storage) level or they ionize the atom, leaving an ion of that species and another electron. In either case, that level then transfers its stored energy to the upper laser level via a subsequent collision with the laser species. The laser transitions in gaseous laser media typically occur at relatively precise, discrete wavelengths that correspond to the energy difference of inherently narrow energy levels.

Figure 5-7  Inversion processes in gases, liquids, solids, and semiconductors

Population inversions in liquids—Most excited energy levels in liquids decay so rapidly due to collisions with the surrounding nearby atoms or molecules that they can’t stay around long enough to participate in a lasing process. There are some molecules however, namely organic dye molecules, that do have a sufficiently long lifetime in an upper energy level (of the order of 1–5 nsec) so they can participate in the laser process by being excited to an upper laser level. These molecules also have the ability to radiate the energy from that level rather than lose the energy due to decay by collisions. Those molecules are the dyes that are used to color cloth and other objects that we use in our everyday life. When dissolved in a solvent such as alcohol or water, they can be concentrated in sufficient quantity to be used as a laser gain medium. In these dissolved dye solutions, electrons cannot be made to flow in the form of an electrical current within the liquid as they can in gases. Therefore the pumping of the upper laser levels must be carried out by optical means such as a flashlamp or another laser as shown in Figure 5-7. When the light is applied to the dye solution, it is absorbed at certain wavelengths by the dye as described by Equation 5-4a, placing the dye molecules in highly excited upper laser levels. A population inversion is then produced between those levels and a very broad range of lower-lying energy levels, thereby allowing the possibility for a wide range of laser wavelengths to be produced within the gain medium. Those lower levels are not initially pumped by the light and therefore are sufficiently empty to produce the inversion. Dye lasers thus allow the possibility of wide wavelength tunability and have been used extensively in doing a variety of spectroscopic studies in which very specific laser wavelengths are desired.

Population inversions in crystalline solids and glasses—As in the case of liquids, when energy levels in solids are excited, typically by irradiating those solids with light, the levels tend to decay much more rapidly via collisions with their surrounding neighbors rather than by radiating their energy in the form of light. In a few cases, however, specific types of atoms are embedded into a transparent host material (such as a specific crystalline solid or a glass) at concentrations of up to 1 part in 100, and the atoms radiate their energy rather than decay by collisions. These specific types of atoms, such as chromium or neodymium, consist of a radiating electron surrounded by a “screen” of other electrons that protect that radiating electron from being bombarded by collisions from neighboring atoms. The consequence is that the atoms can absorb pump light that passes through the transparent host medium and can then subsequently radiate that energy. Gemstones such as rubies fall into that category. Ruby, a desired gemstone and also the material that comprised the gain medium for the first laser, consists of chromium atoms doped into a transparent sapphire (Al₂O₃) host crystal. The color of the ruby crystal is determined by the chromium atoms, which absorb light in the blue and green regions of the spectrum and radiate in the red.

When these types of laser crystals absorb light, the energy ends up in excited energy levels that serve as the upper laser level. These crystals have the property that the upper laser level has a very long lifetime before it decays by radiating when compared to all other types of laser gain media. The population inversion in most of these lasers occurs by the lower laser levels being rapidly depleted by collisions with the neighboring atoms (see Figure 5-7) since these levels are not screened or protected as are the upper laser levels. An exception to this is the ruby laser in which the lower laser level is the ground state. In this case the pumping power must be excessively high in order to pump more than half of the chromium atoms into the upper laser level to produce an inversion.

In these solid-state laser gain media, some of the doping atoms produce very broad excited energy levels and others have very narrow energy levels. The broad energy levels allow a broad wavelength region over which gain or amplification occurs and thus allow broad wavelength tunability of the lasers. The narrow energy levels produce lasers operating over a very narrow wavelength region or narrow bandwidth.

Population inversions in semiconductors—Inversions in semiconductors are produced when joining a p-doped semiconductor material with an n-doped semiconductor material in a similar way to that of producing a transistor to create a pn junction. The n-doped material contains an excess of electrons and the p-doped material has an excess of holes (a material with excess positive charge). When a voltage is applied across the junction, with the positive voltage on the p side, the electrons are pulled through the junction toward the positive electrode and the holes are attracted to the negative side, producing an electrical current flow across the junction. The electrons and holes meet within the junction and are attracted to each other because of opposite charges. When they meet, they recombine and emit radiation and also can produce a population inversion. This inversion occurs between energy levels located above and below the semiconductor bandgap (see Figure 5-7), the gap in energy below which the material is transparent. This energy typically corresponds to a wavelength in the infrared, and hence most semiconductors radiate in the infrared and are not transparent in the visible spectral region like glass is. However, semiconductor lasers are under development to operate in the green and blue regions of the spectrum. At very low currents, a population inversion does not occur even though recombination radiation is emitted. In fact, such nonlaser-like emission is the source of radiation from a light-emitting diode (LED). In comparision, to produce a population inversion, a very high current density is applied within the junction region. However, this high current density leads to excessive heat deposition in the material; therefore a significant part of the development of semiconductor lasers involves how to remove the heat, or to make smaller junctions so that less current is required. The material and its corresponding energy bandgap determine the laser wavelength.

Equation 5-6 describes the way in which a beam is amplified if a population inversion exists between two energy levels such as 1 and 2, as described above. An inversion is a necessary condition for making a laser but not a sufficient condition. The exponential factor in Equation 5-6 must be high enough for the beam to grow and develop into the kind of narrow beam that is expected from a laser. For example, if the exponent turns out to have a value of only 0.00001, there will be no noticeable beam growth even though Equation 5-7 might be satisfied. The exponent of Equation 5-6 consists of a cross section s₂₁ that is characteristic of a specific material and also a specific radiative transition in that material. It is referred to as a cross section because it has dimensions of length² or area, as we mentioned earlier. Table 5-1 lists cross sections for some of the laser transitions described in this course. The population difference N₂ – N₁, which is sometimes expressed as DN₂₁, is a value determined by the power available from the pumping source. Values of DN₂₁ are also shown in Table 5-1. Such pumping processes are described in the next section. The other factor that affects gain is the length of the gain medium. If s₂₁ and DN₂₁ are not quite sufficient, the length L of the gain medium can be increased to increase the exponent of Equation 5-6. We will show later that the amplifier length L can be effectively increased by putting mirrors around it such that the beam will pass back and forth through it many times during the beam’s growth process.

Table 5-1

It is useful to describe the product of s₂₁ and DN₂₁ as the small-signal-gain coefficient g₂₁ or

Hence, Equation 5-6 can be rewritten as

(5-9)

By considering the units of both s₂₁ (length²) and DN₂₁ (l/length³) we can see that g₂₁ has the units of 1/length. Hence, if s₂₁ is given in units of cm² and DN₂₁ is given in units of (1/cm³), g₂₁ will be given in (1/cm), more commonly expressed as cm^–1. Values of the cross sections s₂₁ and DN₂₁, and the small-signal gain g₂₁, are listed in Table 5-1 for some of the lasers described in this module.

Bandwidth of laser gain medium—The bandwidth of the laser gain medium determines the range of wavelengths over which amplification can occur for any specific laser. This bandwidth is expressed in either a wavelength range Dl_G or a frequency range Dl_G. These two expressions are related by

(5-10)

in which l is the laser wavelength and c is the speed of light. The bandwidth of the gain medium is usually determined by the bandwidth over which the spontaneous emission occurs for a given laser transition. This bandwidth is determined by specific properties of the energy levels involved in the transitions, such as their lifetimes, how the atoms interact with other atoms, how closely the atoms are arranged, etc. Typically, atomic gas lasers have bandwidths of the order of 1 GHz (10⁹ Hz). Molecular lasers have bandwidths that are sometimes a factor of 10 to 10,000 larger than that due to the closeness in wavelength of several molecular transitions that overlap in frequency. Solid-state lasers can have relatively narrow bandwidths of the order of 100 GHz in cases such as the Nd:YAG laser, or very wide bandwidths, of the order of 100 THz (10¹⁴ Hz) in the case of the titanium sapphire laser. Semiconductor lasers have bandwidths typically of 10¹³ Hz. Comparisons of the laser gain bandwidths for the HeNe, Nd:YAG, and Ti:Al₂O₃ lasers are shown in Figure 5-8. These various bandwidths are not the bandwidths of the laser beam that emerges from the amplifier but do indicate the range over which amplification can occur. Laser mirror cavity properties primarily determine the bandwidth of the emerging laser beam, as will be described later under laser beam properties.

Figure 5-8  Laser gain bandwidths for the HeNe, Nd:YAG, and Ti:Al₂O₃ lasers