Stimulated Atomic Emissions
This is a continuation from the previous tutorial - impact of profile design on macrobending losses.
Having introduced spontaneous (downward) transitions, we will now look at the stimulated (upward and downward) transitions that are the essential processes in all kinds of laser and maser action.
Atomic Absorption Lines
Suppose we now examine more carefully the absorption of radiation by a collection of atoms as a function of the wavelength of the incident radiation.
Figure 1.16 shows a very elementary example of a grating spectrometer such as might be used for such measurements. (A tunable laser would be a very useful alternative, if one were conveniently available.)
In this spectrometer the radiation from a broadband continuum light source is collected into a roughly parallel beam by a collimating mirror, and is then reflected from a diffraction grating located on a rotatable mount.
At any one orientation of the grating, only one wavelength (rather, a finite but narrow band of wavelengths) is reflected at the correct angle to be collected by another curved mirror, focused down through a narrow slit, and passed through the experimental sample onto a detector.
By rotating the grating, we can tune the wavelength of the radiation that passes through the sample and thereby measure the transmission through the sample as a function of frequency or wavelength. (Figure 1.17 shows a more compact in-line version of such an instrument.)
The result of such an experiment will often appear as shown schematically in Figure 1.18.
The atomic sample will have absorption transitions from the lowest energy level to higher energy levels; so it will exhibit discrete absorption lines—that is, narrow bands of frequency in which the sample exhibits more or less strong absorption—at exactly those wavelengths.
These wavelengths will correspond through Planck's law to the energy gaps between the lowest and higher levels. If there happen to be some atoms already located in higher-lying levels, then absorption lines from those levels to still higher levels may also be seen, as illustrated by transition C in the figure.
These excited-state absorptions, however, will usually appear substantially weaker, simply because there will normally be many fewer atoms in the higher energy levels.
As a specific illustration of atomic absorption, Figure 1.19 shows some of the sharp absorption lines observed when radiation at wavelengths around 540 nm in the visible is transmitted through a crystal of lanthanum fluoride (LaF2) containing a small percentage of the rare-earth ion erbium, or when radiation at wavelengths around 300 nm in the near ultraviolet is transmitted through a crystal of strontium fluoride (SrF2) containing a small percentage of the rare-earth ion gadolinium.
These absorption lines all represent different transitions from the lowest or ground levels of the Er3+ or Gd3+ ions to higher-lying levels, exactly analogous to the terbium levels shown in Figure 1.13. Of course, if a pure lanthanum or strontium fluoride crystal is grown without any erbium or gadolinium present, no such absorption lines are observed.
Absorption Lines in Gases, and Molecular Spectroscopy
Absorption experiments of this sort are, of course, by no means limited to solids or to rare earths. Isolated atoms or ions in gases will exhibit such absorption lines in the visible, and especially the UV.
Molecules in gases, liquids, and solids will exhibit an extremely rich spectrum of absorption lines, notably in the infrared as well as in the visible and ultraviolet.
The absorption lines of atoms and molecules in gases are typically sharper or narrower than those in solids or liquids, since the energy levels in gases are not subject to some of the perturbing influences that tend to broaden or smear out the energy levels in liquids or solids.
As just one more example to illustrate absorption spectroscopy, Figure 1.20 shows a few of the sharp absorption lines characteristic of the formaldehyde molecule H2CO in a narrow range of wavelengths near 3.57 μm.
This particular spectrum was taken by using a continuously tunable laser source (a cw injection diode laser using a lead/cadmium sulfide diode) rather than an incoherent spectrometer.
The dashed envelope in Figure 1.20(a) is the power output of the tunable laser versus wavelength, over a tuning range that is extremely large in absolute terms (~ 3 x 1010 Hz), yet extremely narrow (~ 0.04%) relative to the center frequency. The solid line is the power transmitted through the vapor-filled cell.
Many different molecules exhibit exactly such characteristic sharp lines, specific to the individual molecules, in rich profusion through the near and middle infrared regions.
These sharp lines are extremely useful not only as potential laser lines, but as characteristic signatures of different molecules, for use in chemical diagnostics or in identifying the presence of specific pollutant molecules or hazardous chemicals.
Note that the sensitivity and the laser scanning rate in the experiment allow a small portion of the formaldehyde absorption spectrum to be displayed on an oscilloscope in real time.
Emission spectroscopy, using the spontaneous emission lines radiated from an excited sample as in Figure 1.5, is thus one way of observing and learning about the discrete transitions and the quantum energy levels of atoms, ions, and molecules.
Absorption spectroscopy, as briefly described here, is another and complementary method of obtaining the same kind of information. These methods are in fact complementary in their utility, since emission spectroscopy tends to give information about downward transitions emanating from high-lying levels, whereas absorption spectroscopy tends to give information about upward transitions from the ground level or low-lying atomic levels.
The formaldehyde example illustrates the possibilities for applying tunable lasers to spectroscopy, to analytical chemistry, and to practical applications such as pollution detection.
Stimulated versus Spontaneous Atomic Transitions
We have now seen that there are two basically different kinds of transition processes that can occur in atoms or molecules.
First, there are spontaneous emission or relaxation transitions, in which atoms spontaneously drop from an upper to a lower level while emitting electromagnetic and/or acoustic radiation at the transition frequency.
Fluorescence, energy decay, and energy relaxation are other names for this process. When atoms emit this kind of fluorescence or spontaneous electromagnetic radiation, each individual atom acts almost exactly like a small randomly oscillating antenna—in most common cases, a small electric dipole antenna—internally driven at the transition frequency.
Each individual atom radiates independently, with a temporal phase angle that is independent of all the other radiating atoms. Thus, the total fluorescent emission from a collection of spontaneously emitting atoms is noise-like in character (Figure 1.21), even though it will be limited in spectral width to the comparatively narrow linewidth of the atomic transition.
Indeed, such spontaneously emitted radiation has all the statistical properties of narrowly bandlimited gaussian noise. We usually refer to it as incoherent emission.
Second, there are the stimulated responses or stimulated transitions—both stimulated absorption and stimulated emission—that occur when an external radiation signal is applied to an atom.
In these transitions each individual atom acts like a miniature passive resonant antenna (again, usually an electric dipole antenna) that is set oscillating by the applied signal itself.
That is, the internal motion or oscillation in the atom is not random, but is driven by and coherent with the applied signal.
Atomic Rate Equations
Suppose we have very many identical atoms, each of which has two just energy levels, \(E_1\) and \(E_2\). (Real atoms will undoubtedly have many other energy levels as well, but we will ignore other levels for the moment.)
Suppose that \(N_1(t)\) of the atoms present are in level \(E_1\) and \(N_2(t)\) atoms are in level \(E_2\). This situation can be illustrated by an energy-level population diagram, as in Figure 1.22.
We have already stated that the spontaneous-relaxation rate down from level \(E_2\) to level \(E_1\) is directly proportional to the upper-level population \(N_2(t)\) and is not influenced at all by the lower-level population \(N_1(t)\).
Hence the spontaneous-emission rate out of level 2 and into level 1 is given by
\[\tag{6}\left.\frac{\text{d}N_2(t)}{\text{d}t}\right|_\text{spontaneous}=\left.-\frac{\text{d}N_1(t)}{\text{d}t}\right|_\text{spontaneous}=-\gamma_{21}N_2(t)\]
where \(\gamma_{21}\) indicates the total spontaneous-transition rate or decay rate (radiative plus nonradiative) from level 2 to level 1.
Suppose now an optical signal is applied to these atoms to cause stimulated transitions, as in the optical pumping or absorption spectroscopy experiments we have just discussed.
This signal must, of course, be tuned in frequency close to the transition frequency of interest, i.e., \(\omega\approx\omega_{21}\pm\Delta\omega_a\), where \(\Delta\omega_a\) is the linewidth of the atomic transition.
We might then characterize the strength of this signal by its intensity \(I\) (dimensions of power per unit area), or by the strength of its \(E\) or \(H\) fields.
In discussions of stimulated transitions, however, the applied signal intensity or energy density is often expressed in units of the number of signal photons \(n(t)\) per unit volume in the applied signal. This does not necessarily imply anything about photons as being billiard-ball-like point particles; it merely means that \(n(t)\) is the electromagnetic energy density of the applied signal divided by the quantum energy unit \(\hbar\omega\).
Such an applied signal will cause atoms initially in the lower energy level to begin making stimulated transitions or "jumps" upward to the upper energy level, at a rate proportional to the applied signal intensity (or power density) times the number of atoms in the starting level.
The number of stimulated upward transitions per unit time caused by the applied signal can then be written as
\[\tag{7}\left.\frac{\text{d}N_2(t)}{\text{d}t}\right|_\text{stimulated upward}=Kn(t)N_1(t)\]
That is, the stimulated upward transition rate is directly proportional to the photon density \(n\) of the applied signal.
Each such upward transition absorbs one quantum of energy from the applied signal and—at least in an elementary description—transfers it to one of the atoms which is lifted upward. This is the process of stimulated absorption.
But the essential point is that the same applied signal will also cause any atoms initially in the upper energy level to begin making similar stimulated transitions or jumps downward in energy, at a rate which is again proportional to the applied signal intensity times the number of atoms in the initial (i.e., upper) level.
The number of stimulated downward transitions per unit time can thus similarly be written as
\[\tag{8}\left.\frac{\text{d}N_2(t)}{\text{d}t}\right|_\text{stimulated downward}=-Kn(t)N_2(t)\]
This is the process of stimulated emission. The atoms in this case jump downward, giving up energy. This energy must go into the stimulating optical signal, which is therefore strengthened or amplified.
The constant \(K\) in each of these equations is just a proportionality constant that measures the absolute strength of the stimulated response on the particular atomic transition.
A fundamental and essential point, however, is that this proportionality constant necessarily has exactly the same value for transitions in either direction. This constant \(K\) will also be largest for an applied signal tuned exactly to the atomic transition frequency, and will rapidly become small to negligible as the signal frequency \(\omega\) is tuned away from the transition frequency \(\omega_{21}\) by more than a linewidth or so.
The total rate equation for the atomic populations in this simple example, including stimulated plus spontaneous transitions, is thus given by
\[\tag{9}\begin{align}\left.\frac{\text{d}N_2(t)}{\text{d}t}\right|_\text{total}&=\left.\frac{\text{d}N_2(t)}{\text{d}t}\right|_\text{stimulated upward}+\left.\frac{\text{d}N_2(t)}{\text{d}t}\right|_\text{stimulated downward}+\left.\frac{\text{d}N_2(t)}{\text{d}t}\right|_\text{spontaneous}\\&=Kn(t)\times[N_1(t)-N_2(t)]-\gamma_{21}N_2(t)=-\left.\frac{\text{d}N_1(t)}{\text{d}t}\right|_\text{total}\end{align}\]
where \(n(t)\) is directly proportional to the applied signal intensity or power density.
Stimulated Transitions and Laser Amplification
The total rate at which atoms make signal-stimulated transitions between two energy levels (i.e., "up" minus "down") is thus given by \(Kn(t)\times[N_1(t)-N_2(t)]\). Each upward transition transfers \(\hbar\omega\) of energy from the signal to the atoms; each downward transition does the reverse.
But this implies that the net rate at which energy per unit volume is absorbed from the signal by the atoms is then given by this net flow rate times the energy \(\hbar\omega\) per jump. That is, the net energy transfer rate to the atoms is
\[\tag{10}\frac{\text{d}U_a}{\text{d}t}=Kn(t)\times[N_1(t)-N_2(t)]\times\hbar\omega\]
where \(U_a\) is the energy density in the forced internal oscillation of the atoms.
This same energy must at the same time be coming out of the signal. Hence the energy density \(U_\text{sig}(t)=n(t)\times\hbar\omega\) in the applied signal must be decreasing with time according to the reverse expression
\[\tag{11}\frac{\text{d}U_\text{sig}}{\text{d}t}=-K[N_1(t)-N_2(t)]\times{n(t)}\times\hbar\omega=-K[N_1(t)-N_2(t)]\times{U}_\text{sig}(t)\]
or, in terms of photon density,
\[\tag{12}\frac{\text{d}n}{\text{d}t}=-K[N_1(t)-N_2(t)]n(t)\]
The signal energy density \(U_\text{sig}(t)\), or the photon density \(n(t)\), may thus either decay or grow with time, depending on the sign of the population difference \(\Delta{N}(t)=N_1(t)-N_2(t)\) in the square brackets.
The signal growth rate described by Equation 1.12 leads to the essential concept of laser amplification. This equation says that if an external signal is applied to a collection of atoms where there are more atoms in the lower energy level than in the upper, or where \(N_1(t)\gt{N_2}(t)\), then the net transition rate or net flow of atoms between the levels will be upward. In this case net energy is being supplied to the atoms by the applied signal; so the applied signal must become absorbed or attenuated.
If, however, we can somehow produce a condition of population inversion, in which there are more atoms in the upper level than in the lower, or \(N_2\gt{N_1}\), then both the quantity \(N_1\)—\(N_2\) and hence the net energy flow between signal and atoms will change sign.
The net stimulated-transition rate for the atoms will now be in the downward direction. Net energy will then be given up by the atoms, and taken up by the applied signal.
This energy flow will in fact produce a net amplification of that signal, at a rate proportional to the population difference and to the strength of the external signal.
Boltzmann's Principle
One of the fundamental laws of thermodynamics, Boltzmann's Principle, states that when a collection of atoms is in thermal equilibrium at a positive temperature \(T\), the relative populations of any two energy levels \(E_1\) and \(E_2\) are given by
\[\tag{13}\frac{N_2}{N_1}=\exp\left(-\frac{E_2-E_1}{kT}\right)\]
which of course means that
\[\tag{14}\Delta{N}\equiv{N_1}-N_2=\left(1-e^{-\hbar\omega/kT}\right)N_1\]
Thus for a collection of atoms in equilibrium at a normal positive temperature \(T\), an upper-level population is always smaller than a lower-level population (much smaller if the energy gap \(E_2-E_1\) is an optical-frequency gap).
The total stimulated-transition rate on such an equilibrium transition is thus always absorptive or attenuating rather than amplifying.
To create laser amplification, we must find some pumping process which will put more atoms into an upper level than into a lower level, and thus create a nonequilibrium condition of population inversion.
Coherence in Stimulated Transitions
If we want, we can think of the basic stimulated transition process as the sum of two separate processes: in one, atoms initially in the lower energy level are stimulated by the applied signal to make transitions upward; in the other, atoms initially in the upper energy level are stimulated by the applied signal to make transitions downward.
It is vital to understand, however, that the stimulated-transition probability produced by an applied signal (probability of transition per atom and per second) is always exactly the same in both directions. The net flow of atoms is thus always from whichever level has the larger population at the moment, to whichever level has the smaller population.
There is also no conceivable way to "turn off" one or the other of the stimulated absorption or emission processes separately. If the lower level is more heavily populated, the signal is attenuated. If the upper level is more heavily populated, the signal is amplified. This is the essential amplification process in all lasers and other stimulated-emission devices.
It is also essential to keep in mind that the stimulated transition process we have been introducing here results from a resonant response of the atomic wave function, or of the atomic charge cloud in each individual atom, to the applied signal. That is, the internal induced oscillation or dipole response that is produced in each atom is stimulated by and thus fully coherent with the applied signal.
The net amplification (or attenuation) process is thus a fully coherent one, in which the atomic oscillations follow the driving optical signal coherently in amplitude and phase.
The output signal from an amplifying laser medium is a linear reproduction of the input signal, and of any amplitude modulation or phase modulation that may be on the input signal, except that (i) the output signal is amplified or increased in magnitude; (ii) the signal modulation may be decreased somewhat in bandwidth because of the finite bandwidth of the atomic response; and (iii) the signal in general has a small amount of spontaneous emission noise added to it.
Spontaneous Versus Stimulated Transitions
Note also that in a collection of laser atoms with a population inversion, and with an applied signal present, both the spontaneous transitions and the stimulated transitions will occur simultaneously and essentially independently.
The stimulated-transition rates and the spontaneous-relaxation rate can be simply added together. The spontaneous emission, however, will emerge in all directions, as in Figure 1.24, and will have the spectral and statistical character of narrowband random noise; whereas the stimulated emission (and absorption) will all be in the same direction and at the same frequency as the applied signal.
In a laser amplifier the input signal will thus be amplified by the stimulated transitions. At the same time, a small amount of the spontaneous emission (in essence, that portion traveling exactly parallel to the applied signal) will be added to the output signal by the spontaneous emission process.
The spontaneous emission in this situation thus acts essentially like a small additive amplifier noise source insofar as the stimulated amplification process is concerned.
Unless the applied signal is very small, approaching the noise limit of the laser amplifier, the added spontaneous-emission noise can normally be ignored in discussions of the basic stimulated amplification process.
The next tutorial discusses about landscape lenses and the influence of stop position.