Laser Gain

David Sell
December 15, 2014

Submitted as coursework for PH240, Stanford University, Fall 2014


Fig. 1: Illustration of a 3 level system. (Source: Wikimedia Commons

Until the development of the laser in 1960, the only method of generating light was through heat. Prior to the incandescent light bulb, light sources, even for scientific applications, were limited to flames and solar light. A famous example of this is William Herschel's observation of infrared light in 1800 by separating a sun beam with a prism, and observing that the light generated heat even outside of the visible. [1] There is also the Michelson Morley experiment, in which they managed to construct an interferometer, which requires coherent monochromatic light, by filtering flame light. [2] Incandescent sources are no different, and function by heating a filament, and using its thermal emission as an illumination source.

Lasers are an entirely different phenomenon. In fact, L.A.S.E.R. is an acronym which refers to the process by which a laser operates: Light Amplification by Stimulated Emission of Radiation. This article will effectively be an explanation of what it means to generate light in this fashion, and various ways in which it is achieved.

A Little Bit of Quantum Mechanics

Fig. 2: Illustration of a 4 level system. (Source: Wikimedia Commons

As its name suggests, quantum mechanics is heavily focused on the underlying discrete nature of, well, nature. Specifically, every physical system has some sort of discrete ladder of energy levels which is can occupy, and energy can only be added or removed from a system in increments which are compatible with these levels. As systems become more complex, these levels will blur and appear as a continuous space, but at the real nitty-gritty level, the grainy nature will always remain.

For the case of light, it is most important to understand how these quantum mechanical restrictions on energy are present within a simple harmonic oscillator. A simple harmonic oscillator is nothing more than the equivalent of a mass attached to a spring. The special property of such a system is that no matter how far you stretch or compress the spring, the frequency (oscillations per second) at which the mass will spring back and forth will be exactly the same, and is fundamental to this mass-spring system. Quantum mechanically, we find that any system which behaves in this way can only lose and gain energy in increments of h multiplied by its fundamental frequency, where h is Planck's constant (approximately equal to 6.626 × 10-34 joule-seconds, which relates frequency to energy. So this simple harmonic oscillator can only have energies which lie on some 'ladder' of states with equally spaced rungs.

In this way, light is, itself, a simple harmonic oscillator, or generally, a combination of simple harmonic oscillators. Any given light (an electromagnetic field) can contain any number of frequencies, or colors, within. Each color of light corresponds to an electromagnetic field which oscillates at a specific frequency, and just like the case of a simple harmonic oscillator, if you add energy to such a field (effectively stretching the spring further), this frequency will not change. Consequently, an electromagnetic field oscillating at any given frequency can only increase or decrease in energy with multiples of Planck's constant times that frequency. These discrete units of energy change are called photons. This is the reason that light has both particle and wavelike properties, as the electromagnetic field itself still diffracts and interferes in a wavelike fashion, but in its interactions with matter, it only exchanges energy discretely (contrary to popular belief, photons are not flying wiggly lines).

Photons and 2-Level Systems

Employing this concept of discrete energy levels even further, consider a system full of atoms or molecules which are each present in one of two energy levels. Given one of the atoms of molecules in this hypothetical material, it will either have a low energy E1, or a high energy, E2. If we let this system sit on its own, we will find that a majority of the molecules will stay at energy E1, with the ratio of molecules at E1 to those at E2 being determined by the magnitude of their energy difference, and the temperature. The key point is that there will always be more lower energy particles than higher energy particles.

Now introduce some light which is resonant with the energy transition. When I say that it is resonant I mean that its frequency is such that its photon energy is equal to E2-E1. When this light interacts with a particle at energy E1, there is a probability that the particle will absorb a photon, and promote itself to energy E2. Alternatively, when the light interacts with a particle at E2, there is the exact same probability that the particle will add a photon of energy to the light, and drop down to E1. The first process is called stimulated absorption, and the second is called stimulated emission (sound familiar?). There is a third process in which a particle in E2 spontaneously drops to E1, and emits a photon at energy E2-E1. This is called spontaneous emission, and all excited energy states have an associated 'lifetime' which defines the average time it takes for the energy to drop down to a lower level.

Because any system left alone will always have more particles in lower energy states compared to higher energy states, and the probabilities of stimulated absorption from a low energy particle, and stimulated emission from a high energy particle are the same, it follows that if you insert light into any system, there will always be more of it that is absorbed than emitted.

However, consider a special case in which you somehow have created a 2-level system where there are more particles are in the higher energy level than the lower energy level. If you insert resonant light into this system, then you will expect that stimulated emission will occur more often than stimulated absorption, and that, as a result, more light will come out than came in. In fact, this is exactly what would happen. The state in which there are more higher energy particles than lower energy ones is referred to as 'population inversion', and any medium that does this is called a 'gain' medium. Putting a gain medium inside of a cavity tuned to the energy difference of the 2-level system (a cavity is essentially 2 mirrors facing each other, and resonates with light with a wavelength of integer divisions of the mirror separation distance), and keeping that medium in a state of population inversion is called a laser.

Beyond 2 Levels

Unfortunately, it is fundamentally impossible to create population inversion within a system whose components only have 2 energy levels. The reason for this is that anything that is capable of promoting an atom to the higher energy state is equally capable of pulling an atom down to the lower energy state. To solve this issue, it is necessary to take advantage of systems with 3 or more energy levels.

In general, laser gain systems can be described in terms of either a 3 or 4-level system (see Figs. 1 and 2). Laser gain in a 3-level system is achieved through the following sequence:

  1. A 'pump' which is resonant with the E1 to E3 transition is used to promote ground state atoms into state 3.

  2. Atoms at E3 quickly (microsecond scale) decay to E2 in a manner which does not emit a photon (generally, the energy is released as heat into the ambient medium).

  3. E2 is 'metastable', meaning that an atom in E2 will generally take on the order of 10s of milliseconds to spontaneously decay to E1.

  4. Assuming the rate at which the pump excites atoms is greater than the rate of spontaneous decay, the system will maintain population inversion, amplifying any incident light which is resonant with the E2-E1 transition. In most cases, the light used to start this amplification process is provided by spontaneous emission (an exception being something like a fiber amplifier in which the input signal is what starts the process).

In 1960, the first ever laser was created by Theodore Maiman using ruby (Cr3+:Al2O3) as its gain medium, which operates as a 3-level system. [3]

Generating gain with a 3-level system is not particularly efficient, and is generally not used in more modern laser systems. The reason for this is that the resting state of the particles prior to absorbing energy from a pump is within a ground state that can also be excited by the laser you are trying to amplify. As a result, your amplified beam will often encounter atoms which are sitting in the ground state, and it will lose energy by inducing stimulated absorption. The solution to this is to use a quasi-3-level system, or a 4-level system. In a 4-level system, gain is achieved through this sequence (see Fig. 2):

  1. A pump resonant with the E1 to E4 transition excites atoms through stimulated absorption to the E4 state.

  2. Atoms at E4 quickly decay to E3 without radiating.

  3. E3 is metastable, so the pumped system generates population inversion.

  4. When stimulated emission occurs, pulling an atom from E3 to E2, it will again quickly decay back to E1, preventing unwanted stimulated absorption from occurring.

The only difference between a quasi-3-level system and a 4-level system is that E1 and E2 are sufficiently far apart in a 4-level system that the ratio of atoms in E2 to those in E1 when the system is in thermal equilibrium is very small. A quasi-3-level system is one where E1 and E2 are generally part of the same energy band, but that band has been broadened by some mechanism, so there is a non-negligible ratio of atoms in E2 versus E1 in thermal equilibrium. Nd:YAG lasers are a very commonly used 4-level systems which operate at near infrared wavelengths. [4] There are, in fact, a large variety of YAG lasers, operating in the near to mid infrared, which use 4-level gain, or at least a quasi-3-level gain. 4-level systems tend to have very high potential gain, and are often used in applications such as laser cutting or welding. [5]

An example of a quasi-3-level system is the gain medium of an erbium doped fiber amplifier (EDFA). EDFAs are used to amplify signals in long distance optical fibers, and play a large part in modern network infrastructure. They amplify at roughly 1550nm, and can be pumped with either 980nm or 1480nm light, the former creating gain as a quasi-3-level system (in which the E1 band is spread), and the latter a quasi-2-level system (where E2 is spread). [6]

© David Sell. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.


[1] W. Herschel, "Experiments on the Refrangibility of the Invisible Rays of the Sun," Phil. Trans. Roy. Soc. Lond. 90, 284 (1800).

[2] A. A. Michelson and E. W. Morley. "On the Relative Motion of the Earth and the Luminiferous Ether," Am. J. Sci. 34, 333 (1887).

[3] T. H. Maiman, "Stimulated Optical Radiation in Ruby," Nature 187, 493 (1960).

[4] J. E. Geusic, H. M. Marcos, and L. G. Van Uitert. "Laser Oscillations in Nd-Doped Yttrium Aluminum, Yttrium Gallium and Gadolinium Garnets," Appl. Phys. Lett.4, 182 (1964).

[5] E. C. Honea et al., "115-W Tm: YAG Diode-Pumped Solid-State Laser," IEEE J. Quantum Elect. 33, 1592 (1997).

[6] P. C. Becker. A. A. Olsson and J. R. Simpson, Erbium-Doped Fiber Amplifiers: Fundamentals and Technology (Academic press, 1999).