# e-book Quantum Electronics for Atomic Physics (Oxford Graduate Texts) Contents:

The fairly subtle distinction between the photon statistics of a coherent laser beam and that of thermal radiation will be treated in some detail. The internal intensity transmission is t2 for a traveling-wave cavity and t4 for a standingwave cavity. Although most lasers using either type of cavity have only one partially transmitting mirror, we will continue to allow for the possibility of two partially transmitting mirrors.

The fractional roundtrip loss is 6. We thus have the condition for laser oscillation: 2 6. As the gain constant is increased via an external pumping mechanism , the laser will reach threshold when the gain is exactly equal to the losses. Of course, by increasing the pumping, the power output will increase, but the gain will remain the same. This seeming paradox will be resolved in the next section.

Therefore, the output power will increase with increased pumping. This equation also explains how laser oscillation is initiated. Thus, the clamping mechanism of the gain is saturation. This will yield the net Pumping in three-level and four-level laser systems power per unit volume, Pe , provided by the amplifying medium. The internal power is obtained by multiplying this by the mode volume, Vm , which is the volume overlap between the amplifying medium and the radiation.

The only conveniently available parameter which can be adjusted to maximize the power output is the output coupler transmission, T. The result is: 6. A plot of the output power versus T appears in Fig. Figure 6. For simplicity, we assume that all of the levels have a multiplicity of 1. The excitation rate of level 2 due to the laser radiation is given by eqn 5. These equations can be solved in the steady-state by setting the derivatives equal to zero and solving the resulting algebraic equations.

Noting that the equations for N1 and N2 are the same, we have only two independent equations which can be solved for the inversion. If we add a level somewhat below level 1, we can avoid this and greatly increase the ease of obtaining an inversion. This is the four-level laser scheme. We also found that the real part of the susceptibility produced a phase shift. This phase shift will alter the cavity resonances from their values in the absence of the amplifying medium. To summarize the results from Chapter 5, a wave propagating in the amplifying medium will be described by!

This expression is valid for both standing wave and traveling wave cavities. This change in the resonant frequencies from those of the empty cavity is called frequency pulling. This relation is a general result but can be derived from eqns 5. The actual cavity mode s at Laser oscillation and pumping mechanisms which oscillation takes place is determined by the requirement that the gain is equal to the loss. The question arises: is it possible for a laser to oscillate simultaneously at more than one cavity mode frequency? The answer depends upon the nature of the broadening in the amplifying medium.

A set of gain versus frequency curves appears in Fig. As the pumping increases, the gain curve will increase in both amplitude and width until threshold is reached where the peak gain just equals the loss. The peak gain is clamped at this level and the laser will oscillate at the frequency corresponding to the peak. All of this is a consequence of the behavior of the homogeneous gain curve under various levels of saturation, as discussed in Chapter 5. One can therefore state the general principle that at any particular time, a homogeneously broadened laser is inherently single-mode.

Again, at exact threshold, the laser will oscillate at the peak of the gain curve. Since the model of the inhomogeneously broadened medium is a series of homogeneously broadened packets distributed in frequency over the inhomogeneous linewidth, each cavity mode Spatial hole burning can oscillate independently of the others provided that the gain at the mode frequency is greater than or equal to the loss.

The plot shows a number of holes at several cavity modes with the bottom of the hole clamped to the loss as required by the condition for laser oscillation. For all frequencies between cavity modes, the gain can be greater than the loss since no clamping can occur at frequencies not corresponding to a cavity mode possible laser oscillation can only occur at a cavity mode.

It is as though a large number of independent lasers exist whose peak gain curve frequencies are distributed over the inhomogeneous linewidth. Thus, the general principle is that an inhomogeneously broadened medium can sustain laser oscillation at a number of modes within the inhomogeneous linewidth. It is due to the spatial inhomegeniety which is caused by the standing wave intensity distribution inside the gain medium. The phenomenon is called spatial hole burning and is illustrated in Fig.

Possible other modes due to spatial hole burning Gain Available gain 0 z Fig.

The presence of a spatially periodic gain is clearly depicted in Fig. The hole burning formerly discussed is called spectral hole burning. In a standing wave, the clamping of the gain to the loss at each point is no longer a reasonable condition for laser operation. The power output in a standing wave laser can be calculated by equating the intensity increase to the integrated loss and performing a fair bit of algebra. The possibility of multimode oscillation in a standing wave laser containing a homogeneously broadened medium can be explained by Fig. Thus, we have possible multimode behavior due to spatial inhomogenieties as well as spectral inhomogenieties.

This form of multimode behavior is most easily cured by using a traveling wave cavity instead of a standing wave cavity: the resulting laser is usually called a ring laser. Another treatment is to place the gain medium very close to one of the standing wave cavity mirrors. We will postulate the existence of photons in a somewhat ad hoc manner Some consequences of the photon model for laser radiation and derive a number of important laser beam properties from this model.

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The determination of the occupation number requires the steady-state solution of two rate equations: one for the atoms and one for the photons. The remarkable jump in n above threshold is akin to a phase transition. Since photons are bosons, it is a Bose—Einstein condensation of photons: the sudden appearance of an enormous number of particles in the same quantum state. This is one of the several remarkable features of the radiation from a laser; we will discuss others in the following sections. Its magnitude can be obtained from a Michelson interferometer Fig.

## Quantum Electronics for Atomic Physics and Telecommunication

Light sources can be divided into two distinct classes, depending upon the photon statistics of the radiation they emit. Conventional light sources based upon a thermal emitter or a discharge lamp are said to emit chaotic light. The other type of source is a laser above threshold whose radiation is said to be coherent.

The usual practice is to average a large number of measurements over time. In order to calculate the average over t as required by eqn 6. From eqn 6. For the particular case of a thermal blackbody source, it is very straightforward to obtain the same result.

Most chaotic sources have bandwidths of many MHz and would require counting times of less than a microsecond to observe their departure from Poisson statistics. Let us assume that we have a fourlevel laser and that the lower laser level has negligible population.

At the other extreme, a conventional laboratory continuous wave dye or Ti-sapphire laser can have an ultimate linewidth of less than one Hz. The increased linewidth from the latter factors is called technical noise.

## ECE 677 - Winter 2017

When we discuss laser frequency stabilization, we will show how one can remove the technical frequency noise and even reduce the laser linewidth to below the Shawlow—Townes limit. The discussions of the laser oscillation condition and pulling comes largely from Yariv and the discussions of the Schawlow—Townes limit and photon statistics are similar to those in Milonni and Eberly For simplicity, assume that the laser is a ring laser with very low losses. Estimate the pulling when the cavity resonance is detuned 50 MHz from the gain curve center.

Due to its dominance, the ubiquitous diode laser warrants at least one chapter of its own. Several of the laser systems described in this chapter are currently being replaced by simpler and less expensive diode and diode-pumped solid state systems. In the category of lasers whose days are probably numbered in the atomic physics laboratory are argon ion lasers and dye lasers.

Some commercial realizations of these lasers will be described.

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The diode-pumped frequency-doubled Nd:YVO4 laser and the Ti-sapphire laser pumped by the former will be discussed. Finally, the formerly ubiquitous He-Ne laser currently supplanted by red diode lasers will be described, in part for historical reasons and also because it employs an interesting and unique excitation transfer mechanism for pumping. We will start with the He-Ne laser. A DC glow discharge is maintained by a high voltage power supply connected to an anode and cathode placed inside the tube.

Thus, for a typical laser with a 2 mm diameter capillary, about 2 Torr of helium and 0. The laser emits radiation at a number of wavelengths and the optimum parameters have a slight wavelength dependence.

The pumping mechanism can be understood by reference to Fig. The principal pumping 21 S 5s 4p Collision 23 S 4s nm The s-states have four levels and the p-states have ten.

These lifetimes are much shorter than expected for the 23 S and 21 S states since they are fairly rapidly quenched by the discharge. Once the helium atoms are excited to the metastable states, they transfer their excitation, via collisions, to the 4s and 5s states of neon. This process is greatly enhanced by the accidental near resonance between these levels and the metastable levels in helium.

This step pumps the neon atoms into their upper lasing level. The reason that there is an optimum discharge current is the following. Let the ground state population in helium be N1 , the mestatable population be N2 and the discharge current be I. Lower state pop. Gain Populations in neon We assume that, via collisions, the excited state neon populations will be proportional to the helium metastable populations.

Thus, the upper laser level in neon will saturate with increasing current. However, the lower lasing level is increasingly populated as the current increases, due to electron bombardment.