# Laser Amplification

This is a continuation from the previous tutorial - ** fiber attenuation loss**.

Using the principles of stimulated emission outlined in the preceding section as a foundation, we next outline briefly how a laser material with an inverted atomic population produces useful laser amplification.

### Signal Absorption and Attenuation

Suppose first that we send a wave of tunable optical radiation through a collection of absorbing atoms, as illustrated in Figure 1.25, with this radiation tuned to a frequency \(\omega\) near the transition frequency \(\omega_{21}\) between two energy levels \(E_1\) and \(E_2\) of the atoms. Let the populations of these energy levels be \(N_1\) and \(N_2\) as shown earlier. (The symbols \(N_1\) and \(N_2\) nearly always mean population densities; i.e., they have dimensions of atoms per unit volume inside the laser medium.)

For an absorbing population difference, we will find that this wave will be absorbed or attenuated with distance in passing through the atoms, in the form

\[\tag{15}\mathcal{E}(z)=\mathcal{E}_0\times\exp[-\alpha(\omega)z]\]

For many atomic transitions the attenuation coefficient \(\alpha(\omega)\) due to the atoms will be given (as we will derive later) by an expression of the general form

\[\tag{16}\alpha(\omega)=\frac{\lambda^2}{4\pi}\frac{\gamma_\text{rad}}{\Delta\omega_a}\frac{N_1-N_2}{1+2[(\omega-\omega_{21})/\Delta\omega_a]^2}\]

This expression contains factors such as the transition wavelength \(\lambda\) (in the laser material); the radiative decay rate \(\gamma_\text{rad}\) of the transition; and the transition linewidth \(\Delta\omega_a\).

Most important, it contains the population difference \(N_1-N_2\), and a lineshape factor (in the final term) giving the frequency lineshape of the transition. This lineshape will in general be a sharp resonance curve, as illustrated in Figure 1.25, with a finite linewidth or bandwidth \(\Delta\omega_a\).

The particular lineshape given by Equation 1.16 is known as a Lorentzian lineshape, and is characteristic of many real atomic transitions. Other transitions, for various reasons, may have somewhat different lineshapes, for example, a doppler-broadened or gaussian lineshape.

The general dependence of the gain coefficient on the important atomic parameters for any real atomic transition will still be very much like Equation 1.16, even though the exact lineshape is somewhat different.

The signal wave passing through such an absorbing laser medium will also experience a small frequency-dependent phase shift due to the atoms, as shown by Figure 1.25(c). This atomic phase shift can have practical implications (such as laser frequency-pulling effects).

### Attenuation Coefficients

Note that the power flow carried by the wave passing through the atoms, or the wave intensity \(I(z)\) (in units of power per unit area), is given by

\[\tag{17}I(z)=|\mathcal{E}(z)|^2=I_0\exp[-2\alpha(\omega)z]\]

Hence the power or intensity attenuates with distance in the form \(\text{d}I(z)/\text{d}z=-2\alpha(\omega)I(z)\). Thus in our notation the power-attenuation coefficient is given by \(2\alpha(\omega)\). We will consistently use \(\alpha\) to represent an amplitude or "voltage" attenuation (or gain) coefficient, and \(2\alpha\) to represent a power or intensity coefficient. In the journal literature, however, \(\alpha\) by itself is often used to represent a power-attenuation or power-gain coefficient.

### Laser Amplification

Suppose now the population difference on an atomic transition can, through some "pumping" process, be made to change sign, creating a population inversion. The same expression for the absorption coefficient \(\alpha(\omega)\) as in Equation 1.16 then remains valid, except that the population difference and absorption coefficient are both reversed in sign. To emphasize this, let us rewrite Equation 1.16 in the form

\[\tag{18}-\alpha(\omega)\equiv\alpha_m(\omega)=\frac{\lambda^2\gamma_\text{rad}}{4\pi\Delta\omega_a}\frac{N_2-N_1}{1+2[(\omega-\omega_{21})/\Delta\omega_a]^2}\]

where \(\alpha_m(\omega)\) means the "molecular" or "maser" or "laser" amplification coefficient. The wave amplitude and power will now grow or amplify with distance in the form

\[\tag{19}\mathcal{E}(z)=\mathcal{E}_0\exp[+\alpha_m(\omega)z]\qquad\text{and}\qquad{I(z)}=I_0\exp[+2\alpha_m(\omega)z]\]

as shown in Figure 1.26(b). The energy for this amplification comes, of course, from the inverted atoms—that is, the upper-level atoms supply energy to the wave, whereas the lower-level atoms still absorb energy. But since there are more upper-level atoms, the net effect is amplification rather than attenuation.

The laser amplification coefficient \(\alpha_m(\omega)\) thus has exactly the same lineshape and all other properties as the absorption coefficient \(\alpha(\omega)\) for the same transition without inversion. The only difference between stimulated absorption and stimulated emission is in the sign of the population difference. The net atomic phase shift, in fact, also changes sign as the population difference goes from absorbing to amplifying.

### Coherence and "Photons"

We have hardly mentioned photons yet in this book. Many descriptions of laser action use a photon picture like Figure 1.27, in which billiard-ball-like photons travel through the laser medium. Each photon, if it strikes a lower-level atom, is absorbed and causes the atom to make a "jump" upward.

On the other hand, a photon, when it strikes an upper-level atom, causes that atom to drop down to the lower level, releasing another photon in the process. Laser amplification then appears as a kind of photon avalanche process.

Although this picture is not exactly incorrect, we will avoid using it to describe laser amplification and oscillation, in order to focus from the beginning on the coherent nature of stimulated transition processes.

The problem with the simple photon description of Figure 1.27 is that it leaves out and even hides the important wave aspects of the laser interaction process. A photon description leads students to ask questions like, "How do we know that the photon emitted in the stimulated emission process is coherent with the stimulating photon?"

The answer is that the whole stimulated transition process should be treated not as a "photon process" but as a coherent or wave process. These coherence effects are present, and must be considered, in at least two different ways.

First, when an electromagnetic signal wave passes through a collection of atoms, a much more accurate description of the stimulated transition process is that the electromagnetic fields in the wave cause the electronic charges inside the atoms to begin vibrating or oscillating in a coherent relationship to the driving signal fields.

The atoms in fact both respond and reradiate like miniature atomic antennas. The fields reradiated by the individual atoms combine coherently with the incident signal fields to produce absorption or amplification (and also phase shift) in a manner that is both spatially and spectrally coherent, as illustrated in Figure 1.28.

Quantum mechanics tells us in fact that these atoms respond very much like little classical electronic dipole oscillators (as we will discuss in great detail in a later chapter), except that atoms initially in the lower energy level respond in a way that tends to cancel or absorb the incident signal, whereas atoms initially in the upper level respond in exactly opposite phase to the applied signal.

The waves reradiated by the upper-level atoms thus tend to add to the driving signal wave, and amplify it, whereas the wavelets reradiated by lower-level atoms tend to add out of phase to the driving signal and thus attenuate it. Other than this phase difference, the stimulated absorption and emission processes are identical.

### Quantum Description of Stimulated Transitions

A second important aspect of stimulated transitions can also be obscured by the photon picture. In a fully correct quantum description, most atoms are not likely to be exactly "in" one quantum level or another at any given instant of time. Rather, the instantaneous quantum state of any one individual atom is usually a time-varying mixture of quantum states, for example, the upper and lower states of a laser transition.

The populations \(N_1\) and \(N_2\) do not really represent discrete integer numbers of atoms in each level. Rather, each individual atom is partly in the lower level and partly in the upper level (that is, its quantum state is a mixture of the two eigenstates); and the numbers \(N_1\) and \(N_2\) represent averages over all the atoms of the fractional amount that each atom is in the lower or the upper quantum state in its individual state mixture.

Applying an external signal therefore does not cause an individual atom to make a sudden discrete "jump" from one level to the other. Rather, it really causes the quantum-state mixture of each atom to begin to evolve in a continuous fashion. Quantum theory says that an atom initially more in the lower level tends to evolve under the influence of an applied signal toward the upper level, and vice versa.

This changes the state mixture or level occupancy for each atom, and hence the averaged values \(N_1\) and \(N_2\) over all the atoms. Individual atoms do not make sudden jumps; rather, the quantum states of all the atoms change somewhat, but each by a very small amount.

We should emphasize, finally, that laser materials nearly always contain a very large number of atoms per unit volume. Densities of atoms in laser materials typically range from ~ 10^{12} to ~ 10^{19} atoms/cm^{3}.

This density is sufficiently high that laser amplification is an essentially smooth and continuous process, with very little "graininess" or "shot noise" associated with the discrete nature of the atoms involved.

The next tutorial discusses about ** two-lens systems**.