Professor Mark Csele's Homebuilt Lasers Page

Basic Quantum Physics for Dummies :)

This page is definately not intended to be an all-encompassing discussion of quantum mechanics. Quantum physics is a field in itself and a complex one at that! the purpose of this page is to give the reader (who we assume has at least grade-12 physics) a quick-and-dirty tutorial on how energy levels come about and hence give rise to the many laser lines possible. I hope this page will clarify and demystify the basics and explain why many lasers have discrete output lines. There are other pages on the net as well which describe the key processes in lasing - I encourage you to look at them (see my LINKS page). Quantum is, after all, the key to lasers!

Line Spectra and Photons

When a gas such as hydrogen is put under low pressure and excited electrically it emits light. In analyzing the emitted light using a simple diffraction grating one notices that this light is composed of a series of discrete lines. In the case of hydrogen (chosen because it is the simplest atom) the visible spectrum (between 400nm and 700nm) is actually composed of five discrete lines at 656.3nm in the red, 486.1nm cyan, 434.1nm blue, 410.2nm violet, and 397.0nm in the deep violet. This series of lines is called the Balmer series after the discoverer J.J. Balmer, a Swiss secondary school teacher, who found the relationship between the wavelength of these lines and an integer n. Further investigation showed the reciprocal of the wavelength (1/Lambda) was a function of a constant (the Rydberg constant) and an integer n. The description provided showed that the wavelength emitted corresponds directly to the energy difference between one particular energy level in the hydrogen atom and each of a number of other levels.

To complete the picture consider that emitted energy such as the light emitted from the hydrogen atom is quantized in units called photons. Photons can be thought of as literally a little packet of light and have energy inversely proportional to their wavelength. In other words, E=hc/Lambda where E is the energy of the photon in Joules, h is Planck's constant (6.626*10-34J-s), c is the speed of light (3.00*108m/s) and Lambda is the wavelength of the photon in m.

The energy of a photon of any given wavelength may now be computed in Joules. Often, though, energies this small are expressed in electron-Volts ... the energy a single electron acquires when accelerated across a potential of one volt (abbreviated 'eV'). 1eV = 1.602*10-19J. If we now calculate the energy of 400nm (violet) photons we find an energy of 3.0eV. Red photons at 700nm have an energy of 1.8eV. We can hence say that photons of violet light have more energy that photons of red light.

Einstein introduced this concept of the photon and it has been confirmed by observations of the Photoelectric effect (where incident light can cause electrons to be emitted from a metal surface in a vacuum).

Going back to the Rutherford-Bohr model of the atom ...

Bohr's model postulates that electrons around an atom orbit in a number of possible energy states. In these discrete (allowed) states electrons orbit the nucleus of the atom (according to Newtonian mechanics) but do not radiate energy so long as they are fixed in that orbit. No surprises here, this is analogous to the moon in high orbit around the earth. Bohr further stated that atoms may jump from one state to another (a quantum jump), corresponding to a change in orbit, and in doing so will emit radiation in the form of a photon. The photon will contain the energy difference between the initial higher energy state and the final lower energy state.

Recall that the emission spectra of hydrogen consists of discrete lines. Each of those lines now coresponds with a jump or transition in the energy states of the atom (a change in orbit if you will). In the case of the Balmer series, each line corresponds to an energy-level change which results in the final energy state being n=2 ('second orbit'). Higher-energy photons result from larger jumps (e.g. the violet lines result from transitions from very high orbital states to the final n=2 state while the red line has little energy ... it results from a relatively small jump between the n=3 and n=2 orbital states).

As expected, not all transitions end with n=2 ... some end with n=1 ('first orbit'). These have very high energies being so close to the nucleus (low orbits have higher energies) and so photons emitted in transitions to this final state are in the ultraviolet. This is called the Lyman series of lines. Other transitions end with n=3 and have lower energy changes so photons emitted are in the infrared (the Paschen series).

Several other experiments were conducted to prove this hypthetical model of photon emission including the Franck-Hertz experiment and an interesting experiment by Gustav Hertz which looked at the spectra of mercury atoms excited directly by electron impact. The electrons used to excite the mercury atoms were given precise energies by exciting them through a known potential. His experiment showed that below 4.9 V no radiation appeared. When electron energy was slowly increased above that point spectral lines began to appear beginning with a UV line at 253.7nm and progressing through a series of spectral lines in no particular order ... until the energy level structure for mercury is considered!

Examining the known energy levels for mercury one finds that the first real energy level occurs at 4.86eV (There is a level at 4.66eV but it is not relevant for our purposes). This implies that in order to emit light from that level by pumping energy into the atom and allowing the atom to emit a photon as the energy state makes the jump back to ground a minimum energy of 4.86eV is required (after all, you can't get more energy out of a system than you put into it). Perfect correspondence with theory!. Consider now the situation where you excite the mercury atom with electrons of energy 8.7eV ... what would the expected spectrum look like? Well, transitions starting from mercury energy levels at 4.86eV, 5.43eV, 6.67eV, 7.69eV, and 7.89eV levels should be allowed but there isn't enough excitation energy to allow the atom to enter other higher energy levels above this, like the level at 8.84eV, so those transitions should not occur. Sure enough, the visible emission spectrum at 8.7eV shows lines at 404.7nm, 435.8nm, 546.1nm, and 407.8nm corresponding to transitions between energy levels 7.69eV & 4.66eV, 7.69eV & 4.86eV, and 7.69eV & 5.43eV respectively). As the excitation energy is increased to 9.7eV, the emission spectrum shown a number of new lines which originate from a number of energy levels at 8.84eV through 9.68eV. Eight new transitions are now possible and hence eight new lines appear, although only five are visible - the other three are UV. Note that the known level at 8.84eV is actually three closely spaced levels which explains the origin of the three UV and two yellow (577nm and 579nm) closely spaced lines.

Problems with the Bohr model

While the Bohr model was a dazzling advent which explained atomic structure well it only works with simple atoms such as hydrogen and hydrogen-like atoms (i.e. single valence or outer-shell electrons). It does not account for the energy-level structures of complex atoms which involve two or more electrons in transitions. One of the major shortcomings of this theory was the angular momentum associated with every Bohr orbit. Orbits in this model are defined by Newtonian mechanics and confined by this. Consider the ground state of hydrogen (n=1) which, according to Bohr theory has orbital angular momentum (after all, even at ground state the electron is orbiting, right?). When quantum states for hydrogen are considered in detail one can prove that the ground state of hydrogen has zero angular momentum. (This is not a trivial bit of math and will be spared here). The same results that Bohr obtained may be arrived at through a different approach which further deviates from classical physics. The new approach, quantum theory, features a wave property for electrons and all other particles. In the (warped) world of quantum theory a particle like an electron can act like a wave and be described by wave mechanics.

Wave properties of Particles ( ... and Quantum Mechanics is born)

In 1924 Louis de Broglie combined the Planck-Einstein relation for photons (that the energy was related to the wavelength) with Einstein's famous statement of mass-to-energy equivalency (E=mc2). He speculated that wave behaviour might be a property of all moving objects. At face value this seems absurd but under the right circumstances wave behaviour is indeed observed!

Central to de Broglie's hypothesis was that any particle with a momentum has an associated wavelength. While the mathematics are somewhat tedious experimental evidence of wave behaviour is more compelling. Consider the traditional optical principle of diffraction. In the time-honoured two-slit experiment a light beam is shone on a card with two parallel slits. A pattern forms on a screen after the slit. The pattern clearly shows diffraction occuring when light passing through each slit interferes to form the fringes. Light does this because it is a wave and waves interfere. Guess what happens when electrons are used instead of light? A similar diffraction pattern appears! Indeed, the electrons behave as waves in this case. But electrons are PARTICLES, right? Guess again ... although we normally consider the electron as a subatomic particle with mass, etc. it clearly exhibits wave behaviour. Furthermore, electrons also refract when passing through, say, the interface to a crystal (air to crystal surface). This is wave behaviour ... from a particle!

Wave-Particle Duality and when to use Quantum

OK, so a particle like an electron can behave as a wave when it has momentum. How do you reconcile this with classical mechanics? Let's draw a few conclusions to keep things straight ...

• Classical Physics such as Newtonian mechanics provides a VERY clear view of how macroscopic particles and object behave.
• Depending on the circumstances, Quantum Physics may best describe the properties and behaviours of particles like electrons
Of course defining those circumstances is the key. The problem now becomes one of statistics and probabilities.

Hydrogen Revisited

Before we really confuse the issue it might be best to come back down to the practical level and apply quantum theory to practical things like hydrogen energy levels and emission spectra. Using wave mechanics (featuring the scary-sounding Schroedinger equation) one can compute the permitted energy states for hydrogen or any other atom (hydrogen being the simplest). For given values of n (just like in the Bohr model) one ends up with radial functions which describe the PROBABILITY of finding an electron there. Put a different way, Bohr has electrons in circular orbits where the energy depends solely on distance from the nucleus. In quantum an electron having a particluar energy can be described as having a good 'probability' of being in a certain defined area.

NOTE: If you're thinking that WE took a quantum leap you're right. There is enough math between de Broglie's hypothesis and the new quantum model to bury a Physics student for a term. If you're interested in the math I suggest An Introduction to Quantum Physics by French and Taylor (Publ: Norton)

Now, the atom is in 3-D and so can be described by various parameters. First off we've got the principle quantum number n which is familiar from the Bohr model. As well we have an orbital angular momentum number l which describes the number of possible angular momentum states. A third parameter m describes azimuthal quantum states. Suffice it to say the most important number here is n and for each value of n there are n-1 values for l

Now, if wave mechanics are applied to each possible state of n we can plot the probability distributions for each value ... in other words areas where we are most likely to find an electron of that energy. Each state is described by not only n but also by the other parameters to give the total energy of that particular quantum state. There are hence many physical areas in which one may find an electron which has the same total energy. In some areas it will have more angular momentum but the total energy remains constant. For n=1 the probability 'shell' is what is termed an 's' orbital and resembles a 3-D sphere around the nucleus (just like the Bohr model) as shown on the diagram. In fact, when the radial probability density for the ground-state hydrogen atom is plotted as a function of x, the most probable value for the radius is a0, the Bohr radius! For n=2 the shell resembles either a larger sphere (another 's' orbital, but larger) or a set of dumbells (a 'p' orbital) about the centre. For n=3 even more states are possible - these orbitals can be of either the 's', 'p', or 'd' type.

Now just to complicate matters add a fourth parameter, spin, to describe the state it is in and you can see where we deviate from the Bohr model. For any quantum value n there are a total of 2n2 states of essentially the same energy possible! Why does this matter when all states with a given number n have the same energy anyway? The answer lies in magnetic interactions due to the spin of the electrons. Each described state takes on a slightly different energy which gives rise to multiple energy levels (like the closely spaced ones we saw in the mercury spectrum).

More On Spin

Electrons spin like tiny planets - this gives them an intrinsic angular momentum. Evidence of this is found in the sodium D spectral lines. When a low-pressure sodium lamp (like a yellow streetlight) is examined spectroscopically a bright yellow spectral line is seen. If this line is examined closer (i.e. with a decent spectrograph) the line is actually seen to be TWO lines at 589.593nm and 588.996nm. The energy difference between these two lines in only 1/1000th of the total photon energy! This split is brought about by the spin of the electrons which lead to this spectral emission. Spinning electrons may be modelled as tiny dipoles and exhibit a magnetic effect.

As well as everything else (like principle quantum state), the angular momentum is also quantized and may only be in allowed states. This is not a trivial result but was indeed proven to be true in a famous experiment by Stern and Gerlach. In the experiment a beam of cesium atoms was deflected magnetically. The deflection always occured the same way for a range of applied magnetic fields proving that angular momentum was indeed quantized to finite, discrete values.

Now that angular momentum is involved it may be seen that energy alone cannot fully describe a quantum state. In most cases both energy and angular momentum are required to specify a quantum state. Consider a simple 2-D example first in which a quantum state can be described by the principle quantum number, n, and the index of orbital angular momentum, m. In this system the values for orbital momentum can take on only certain allowed values. With n=0 m can only equal 0. With n=1 the allowed values of m are -1 or +1. With n=2 the allowed values of m are 0, -2 and +2. So, for each principle quantum state (i.e. 'Bohr orbit') there are a number or angular momentum values possible. For a real atom in our 3-D world the situation becomes increasingly complex and four quantum numbers are usually required to fully describe a state. By this point we shall abandon vector notation to describe states (in which we describe the quantum numbers as a \ grouping of four) and instead use spectroscopic notation which is common in scientific circles and will be found in almost all literature describing specific laser transitions.

Spectroscopic Notation

Quantum states for a single electron are described by four parameters n, l, j , and m. In quantum physics they would be written as |n,l,j,m> (this is vector notation) however it is customary in spectroscopy to use a different notation. First, the l term is converted into a letter designation, so for l=0,1,2,3,4 we now write the designation as s,p,d,f,g,h instead. For an electron in state n=3 and l=1 we would write it as 3p. Two other terms are added to describe the state fully: the spin-orbit interaction parameter j (which is a fraction) is added as a subscripted suffix and finally a multiplicity number which denotes that each level for a given l is a doublet is written as a superscript after the value of n. As an example one might see a particular level written as 3p3/2 which indicates we are in the n=3 state, the p shell (l=1), and the 3/2 denotes the contribution from spin-orbit momentum to this level. Often another number is added to denote the multiplicity of a level of given l. In this case the level becomes 32p3/2 where the '2' shows that the fine structure turns each level (except l=0) into a doublet: two lines with slightly different energy brought on by differences in angular momentum. If the 'p' were written as a capital letter such as 32P3/2 this would denote that orbital angular momentum is involved.

As a practical example consider first the alpha line for the hydrogen atom (n=3 to n=2 transition). The upper level has n=3 but l=2 (and 2l+1 levels exist for m) so that possible states for this level are 3S1/2,3P1/2, 3S3/2,3D3/2, and 3D5/2. In all the upper 'level' is really FIVE separate levels!. The lower level with n=2 is also composed of multiple hyperfine levels since l=1 and so m had 3 allowed levels. That 'level' is actually 2S1/2, 2P1/2, and 2P3/2. In all, 15 hyperfine transitions are possible. In reality, about 7 can be resolved by high resolution spectroscopy. The situation is complicated again when a larger atom is considered which has multiple electrons in it's outer shell (which, of course, interact). In the case of argon the inner 1S shell has 2 electrons, the next shells are 2S with 2 electrons and 2P with 6. Finally the highest shell has 2 electrons in the 3S and 6 electrons in the 3P shells. When argon becomes an ion, as it does in the argon ion laser, all sorts of interactions occur in the outer shell!

Consider now the common helium-neon gas laser. The actual laser transition happens in the neon gas from from an upper 3S2 level to a lower 2P4 level (in a simpler notation ... the actual notation of these levels is more complex). From a Bohr perspective we have a transition from the third to the second level (one laser line should hence result) but there are at least eight possible such transitions all leading to lasing action (indeed, green, orange, and yellow HeNe lasers use those transitions). Using our new notation (brought to us by quantum theory) we can easily discern all possible transitions.

Laser Transitions In a Gas Laser

The helium-neon (HeNe) laser contains helium and neon gases at a ratio of about 10:1. Helium serves as a 'pump' gas allowing energy to be coupled into neon atoms at the correct energy level. In order for lasing action to occur a population inversion must be maintained - that is we must ensure there is a higher population of atoms at the upper energy state than the lower state. In the HeNe laser a direct electrical discharge pumps energy into helium atoms bringing them up to high energy 2S states (about 20eV above ground level). Note that all energy levels are is Paschen notation in which '1s' referes to the first excited state, not the actual electron configuration of the atom. The abundant helium atoms then transfer their energy to neon atoms by collision. Of course they transfer this energy to neon atoms' energy levels which are almost identical in energy to their own 2S state ... in this case neon's 3S levels. Neon atoms then lose energy in transitions between the 3S and 2P states and in doing so emit photons. Depending on which lower level the transition ends on, a photon of red, yellow, orange, or green light is emitted. If the transition is from 3S2 (one of the four levels in the '3s' group) to 2P4 (one of the ten levels in the '2p' group), the photon is red (633nm). If the transition is from 3S2 to 2P10, the photon is green (543nm). Red transitions (at 633nm) are preferred and have a high gain although this does not preclude lasing on the green transition - the laser's optics and gases simply must be adjusted to preferentially laser the green transition over the red. This is possible of course by ensuring the mirrors reflect only green light ... stimulated emission and hence amplification will hence occur only on that wavelength. All possible (allowed) laser transition complete for gain in the laser media but selection of one wavelength over another via the mirror's coating ensures only one wavelength is produced. It might be noted as well that there is an infrared transition at 1152nm which is actually the strongest. This transition originates from a level lower than neon's 3s level and is not shown here. This infrared transition is so strong it must be supressed (by ensuring the mirrors do not reflect IR back into the tube) in order to ensure the red or any other visible transition will lase!

Types Of Energy Levels in Lasers

Most gas lasers have very well defined wavelengths which correspond to precise transitions of the type discussed above. These are Electronic Transitions which involve only electrons of the species. Precise and well-defined states characterize these transitions which are responsible for most visible gas laser transitions. Other energy levels possible are Vibrational levels brought on, in molecules, by various supported modes of vibrations. In the case of a molecule like nitrogen (N2), the two nitrogen atoms are free to vibrate only in certain allowed ways. Think of the molecule as two weights attached by a spring (the bond between the two atoms). Carbon-dioxide molecules have similar vibrational levels. Finally Rotational levels involve the rotation of molecules. Energies for these transitions are low and hence correspond to the far-infrared region (e.g. the carbon-dioxide laser's output is brought about by rotational energy levels and occurs in the far infrared at 10.6 microns).

Energy levels for a laser might involve more than one mechanism. In the nitrogen laser, for example, the lasing transition is a vibronic one involving both electronic and vibrational levels. This leads to a gas laser transition with relatively broadband output (0.1nm) - many times wider than 'traditional' gas lasers. The energy levels defining this gas laser are really 'bands' where transitions can occur between these bands giving a range of possible wavelengths rather than a sharply defined line. Dye lasers usually involve all three mechanisms which leads to a continuum output over a given range. By tuning the feedback mechanism one can selectively amplify a given wavelength - these lasers are truly tunable.

Remember, too, that quantum mechanics can give rise to amazingly large numbers of closely-spaced levels. This happens in ion lasers (see below) where there are a number of laser transitions closely spaced (in the case of an argon ion laser, ten lasing lines appear in the blue-green-violet region).

To complicate matters there are also four-level lasers like the YAG. In the YAG laser energy absorbed by the laser rod pumps an upper energy level (actually two levels but who's counting :) which decays quickly (without radiating energy) to an upper laser level. From there neodymium atoms emit a photon at 1064nm (IR) and fall in energy to a lower laser state. Finally atoms at the lower state fall to ground state and the cycle begins again. Incidentally, the lasing transition in the YAG laser is from an 'F' level to an 'I' level (n=6 to n=3). Four levels are involved here: the upper pumped level, two lasing levels, and the ground level. Four level lasers are notoriously efficient.

One problem that might be encountered along the way is the lifetime of lasing levels. Some levels have very long lifetimes (metastable). If the lower level of the lasing transition has a longer lifetime than the upper level then lasing action may not be possible at all or might be possible only in pulsed mode. Many practical lasers such as the nitrogen and copper-vapour laser have this situation. In both these cases, the upper-lasing levels can be preferentially filled, quickly, by a pulsed discharge of 100's of amperes or more! If the upper level fills quickly and population inversion can be achieved, lasing action will begin. Soon, though, the lower lasing level fills, the population at that level exceeding the upper level (and staying that way since it's lifetime is longer) and lasing ceases. In the case of the nitrogen laser, lasing ceases after 10-20nS. In the copper-vapour laser, about 150nS. Both of these lasers operate strictly in pulsed mode and CW laser action cannot be achieved.

Quantum Mechanics of Ions: The Argon Laser

Unlike the ubiquitous helium-neon laser the argon laser is an ion laser in which lasing transitions occur on energy levels in ionized argon (an argon atom with an electron removed). Looking back at the HeNe laser's energy structure you'll see that the lasing transitions in neon occur at an energy of about 20eV above ground. Neon, however, requires about 22eV to ionize so these energy levels exist in non-ionized (neutral) neon. This is not all that strange since no one ever claimed that ionization is required to excite an atom to higher energy: in the Bohr model an electron simply moves to a higher orbit (in quantum, the explanation is a lot more complex but still energy can increase in a neutral atom). In the case of the ion laser the gas is first ionized - for argon we're looking at pumping 20eV into the atom just to get it ionized - then energy levels involved in lasing transitions occur above that level. The diagram to the left depicts the 'big picture' of the argon ion's energy levels. The upper laser levels (there are several, clumped tightly together) are about 20 eV above the ground state of the argon ion or nearly 36 eV above the ground state of the argon atom. Obviously, this is a very high energy which will require a large pump energy to build-up a high population of ions in that high-energy state. The dynamics of the argon ion are good for CW laser action in that the lifetime of the lower level is very short compared to the upper level. This allows population inversion to be maintained so long as a large pump energy is available (and all argon lasers need large pump energies - most have between 10 to 70A continuously through the discharge!). The short lifetime of the lower lasing level lead to another problem though in that ions in that energy state (i.e. having just emmitted a coherent photon) drop rapidly to the ground state of argon ion. This is a large jump and results in the spontaneous emission of a 74-nm extreme-ultraviolet photon (the energy had to go somewhere, right?). From an efficiency standpoint, this means an excited ion at the upper lasing level loses about 2 eV of energy in producing the coherent photon then loses 18 eV in spontaneous emission of that UV photon. These dynamics limit efficiency severely. As well the extreme UV light from that emission can damage many optical materials so mirrors and windows in an argon laser must be built to withstand such punishment!

The excitation to the upper-laser levels is inefficient at best. Bridges, the 'father' of the argon laser, describes three major mechanisms which raise the argon ion to the upper laser levels. One is energy transfer from an electron to a ground-state argon ion. Another is collisional transfer of energy from an electron to the argon ion in an excited metastable state similar to the way in which helium pumps neon's levels in the HeNe laser. The third is decay of higher levels produced by electron excitation. All three effects combine to pump the argon ions to the upper levels for lasing. Looking at the fine structure of the argon ion's energy structures you see nine upper levels terminating at two lower levels. This give a possible eighteen laser lines for the argon laser. Not all lase as only ten known lasing transitions are shown on the diagram. Many of these energy states are actually the same major energy state split by spin. Since the argon atom (normally stable with the outer shell filled - it is, after all, an inert gas) is now an ion there is an imbalance in spin. In the inner shells electrons are paired so that total spin and total orbital momentum are zero but in an ion there is an imbalance which results in an angular momentum of the atom. This imbalance gives rise to the hyperfine structure you see here which in turn leads to a plethora of possible laser transitions between each of these hyperfine levels. Were it not for these effects, we would expect the argon laser to have two upper and one lower level so that only two laser transitions would result. If you've ever seen an argon running you realize there are a multitude of lines which appear - the diagram above shows ten. Quantum is a beautiful thing which leads to the rich emission spectrum of this laser which might otherwise be a single or two line spectrum.

The two most powerful lines of the argon laser are in the blue at 488 nm and the green at 514.5 nm. These transitions originate from the upper 4p4D0 level although they have different spin. Both terminate at the same lower level (which rapidly decays to ion ground state anyway so there are no issues of population build-up in that level which might otherwise stop the lasing process).

These two diagrams show the VISIBLE laser lines which are produced from singly-ionized argon. Singly-ionized krypton gas also produces a rich spectra of laser lines (although it is even less efficient than argon). Argon can be doubly-ionized in which case ultraviolet laser lines are produced. As you might imagine UV argon lasers require even more pump energy which shows up as higher excitation currents in the tube! Fundamentals of Light Sources and Lasers by Csele, 2004, John Wiley & Sons, ISBN 0-471-47660-9 An Introduction to Quantum Physics by A.P. French and Edwin F. Taylor, 1978, Norton and Co., ISBN 0-393-09106-0