# Laser Output-Beam Properties

This is a continuation from the previous tutorial - ** materials and fabrication technologies in optical fiber manufacturing**.

The output beam from a laser oscillator thus basically consists of electromagnetic radiation, or light, that is not fundamentally different in kind from the radiation emitted by any other source of electromagnetic radiation.

There are several important and fundamental differences in detail, however, between the "incoherent" light emitted by any thermal light source, such as the flashlight in Figure 1.42, and the "coherent" light emitted by a laser oscillator.

The output beams produced by laser oscillators in fact have much more in common with the outputs of conventional low-frequency electronic oscillators, such as transistors or vacuum tubes, than they do with any kind of thermal light sources.

Laser beams are often described as being different from ordinary light sources in being both spatially coherent and temporally or spectrally coherent. These rather vague phrases refer to some characteristic laser output-beam properties that we will review briefly in this tutorial.

An important point to keep in mind is that all these coherence properties arise primarily from the classical resonant-cavity properties of the laser resonator, as we described in the preceding section, rather than from any of the quantum transition properties of the laser atoms.

## Ideal Laser Monochromaticity and Frequency Stability

The flashlight shown in Figure 1.42, like any other thermal light source, emits a generally broadband continuum of light at many different wavelengths. There are light sources, such as discharge lamps, that emit only comparatively few spectral lines or narrow bands of wavelengths, but the spectral widths of the light emitted by even the best such sources are still limited by the linewidths of the atomic transitions in the discharge atoms.

The output beams from most lasers can be, by contrast, highly monochromatic, and in ideal lasers can consist almost entirely of a single frequency. That is, the output signal from a near-ideal laser will be a nearly pure, constant-amplitude, highly stable, single-frequency sine wave, exactly like the signal generated by a highly stable electronic oscillator in any other frequency range.

Atomic transitions typically have fractional atomic linewidths \(\Delta\omega_a/\omega\) ranging from 1 part in 100 (broadband dye or semiconductor materials) to narrower than 1 part in 10^{6} (narrow-line atomic transitions in gases); and it is this linewidth that characterizes the spontaneous or fluorescent emission from such atoms.

In absolute terms such linewidths range from a few GHz (as in typical doppler-broadened gas lasers in the visible) to a few tens or hundreds of GHz (as in typical solid-state lasers). The short-term spectral purity of a good-quality single-frequency laser oscillator, by contrast, can range from a few tens of MHz (in a moderately well-stabilized gas laser) down to only a few Hz in a very highly stabilized system.

As we have said, it is the laser cavity and not the laser atomic transition that is primarily responsible for these spectral properties. Continuous oscillations can be sustained in a laser resonator only at those discrete axial-mode frequencies where the round-trip phase shift inside the laser cavity is an integer \(q\) times \(2\pi\).

The laser atomic transition then serves primarily to provide gain at these cavity resonance frequencies, not to determine the oscillation frequency (except for small, second-order frequency-pulling effects that we have not yet discussed).

## Spectral Purity in Practical Lasers

Both the short-term frequency jitter and the long-term frequency drift of a laser oscillator usually result primarily from mechanical vibrations and noise, thermal expansion, and other effects that tend to change the length \(L\) of the laser cavity.

Very highly stabilized laser oscillators can nonetheless have long-term absolute frequency stabilities better than 1 part in 10^{10}, and short-term spectral purities as high as 1 part in 10^{13}, making them equal to or better than the best atomic clocks available in any frequency range.

The ultimate limit on laser spectral purity is finally set by quantum noise fluctuations caused by the spontaneous emission from the atoms inside the laser cavity.

These quantum noise effects, which are described by the so-called "Schawlow-Townes formula," can be observed with great difficulty only on the very best and most highly stabilized laser oscillators.

## Laser Statistical Characteristics

In addition to being highly frequency-stable, a good-quality laser oscillator will generally have all the other statistical and amplitude-stabilization properties associated with a coherent electronic oscillator in any frequency range.

The most basic of these properties is that the instantaneous optical field in a single oscillating cavity mode will be essentially a pure optical-frequency sine wave, whose amplitude remains closely stabilized to the steady-state value at which the saturated laser gain just equals the net mode losses.

This is usually a self-stabilizing situation: if the gain increases slightly above the loss because of some random fluctuation, the oscillation amplitude begins to grow slightly, and the slightly increased signal amplitude pulls the gain back down.

Conversely, if the amplitude fluctuates slightly above its average value, this pushes the gain down below the loss, and pulls the oscillation amplitude back down.

A well-stabilized single-frequency laser can in fact have almost negligible amplitude fluctuations, limited mostly by random fluctuations in the pumping rate and the cavity parameters.

The output signal from a well-stabilized high-quality single-frequency laser can thus be best described as an optical sine wave with a highly stabilized amplitude and frequency, whose amplitude changes very little, but whose absolute phase drifts randomly and slowly through all possible values, because of small random environmental fluctuations and ultimately because of quantum noise.

## Laser Signals Versus Narrowband Incoherent (Thermal) Signals

The output signal from such a high-quality laser will also differ in another quite fundamental way from the spontaneous emission emitted by any thermal or "incoherent" light source.

Suppose that the output signal from some very bright thermal light source could be first filtered through some extraordinarily narrowband optical filter, and then amplified through some very high-gain linear optical amplifier (perhaps a laser amplifier), so that the resulting signal was both as narrowband and as powerful as a typical high-quality laser beam. (Though this conceptual experiment would be extremely difficult in practice, there is no fundamental barrier to it in principle.)

This output will then also look like an optical-frequency sine wave, but this sine wave will not have constant amplitude or phase, no matter how narrowly filtered it may be. Rather, it will always look something like the incoherent narrowband noise wave in the lower part of Figure 1.43.

Suppose that we write the instantaneous electric field for both the signals in Figure 1.43 in the form \(\mathcal{E}(t)=E(t)\cos[\omega_0t+\phi(t)]\) where \(\omega_0\) is the midband or carrier frequency, and \(E(t)\) and \(\phi(t)\) are the slowly varying amplitude and phase of the signals.

We can then represent each signal during any short interval of time by its instantaneous phasor amplitude \(E(t)e^{j\phi(t)}\) where this phasor amplitude moves around in time in the complex plane as shown in Figure 1.44(a).

For a thermal noise source, the instantaneous phasor amplitude will then move slowly but randomly through many different phase angles and amplitudes, tracing out a two-dimensional random walk as shown in Figure 1.44(b).

The bandwidth of the noise signal, no matter how narrow, will determine only how rapidly the phasor moves around within this region—not how far it moves, or with what probability distribution.

This noise signal, though having the same power and bandwidth (and hence the same power spectral density) as the laser, will still have the statistical character of narrowband gaussian noise.

That is, both the phase and the amplitude of this thermal signal will fluctuate slowly with time, at a rate given essentially by the inverse bandwidth of the signal.

The probability distribution for the instantaneous phasor amplitude of the thermal signal will be a "gaussian molehill" (Figure 1.45), with the \(x\) and \(y\) axes corresponding to the amplitudes of the \(\sin(\omega_0t)\) and \(\cos(\omega_0t)\) components of the signal, and the height of the molehill corresponding to the probability of the signal having these \(\sin\) and \(\cos\) components at any instant.

However, the laser oscillator signal, like any other conventional oscillator, will fluctuate primarily only in phase, with only small fluctuations in amplitude about its steady-state value.

Its phase angle will wander slowly but randomly through all possible phase angles, in a manner corresponding to its small residual frequency uncertainty; but its amplitude will not. Its probability distribution will thus be a "gaussian molerun" (Figure 1.46) rather than a molehill.

## Laser Temporal Coherence

The preceding descriptions make more precise what is generally meant by the "temporal coherence" of a laser output signal. However, the term "coherence" is often used carelessly, both in discussions of lasers and in other situations, and this has led to some confusion.

The term coherence necessarily refers not to one property of a signal at a single point in space and time, but to a relationship, or a family of relationships, between one signal at one point in space and time, and the same or another signal at other points in space and time.

There are, for example, certain precise mathematical definitions of coherence functions as used in coherence theory. These functions give the degree of correlation, described in a specific mathematical fashion, between two signals observed at different points in space and/or time.

More colloquially, a signal is called "temporally coherent" if there is strong correlation in some sense between the amplitude and/or phase of the signal at any one time and at earlier or later times.

Both the amplitude and the phase of a good-quality laser oscillator will in fact change only slowly with time, so that the amplitude and phase of the output sine wave from the laser at any one time will be strongly correlated with the amplitudes and phases at considerably earlier or later times.

A good laser beam might thus be said to be temporally coherent because of this strong correlation between the amplitudes and phases of the signal at not very different points in time.

Much the same might be said, however, of the narrowband noise signal described earlier, since there is considerable coherence between the signals at any two times that are less than one reciprocal bandwidth apart.

In fact, a high degree of coherence in the formal mathematical sense does not by itself imply that signals are the kind of "clean" and amplitude-stable sinusoidal oscillation signal generated by a good laser oscillator. Two highly disorderly or irregular signals can still have a very high degree of coherence between themselves.

## Laser Spatial Coherence

We have already noted that a good-quality laser oscillator can also oscillate in a single transverse-mode pattern, which has a definite and specific amplitude and phase pattern across any transverse plane inside the laser, and particularly across the output mirror.

In this situation there is a very high degree of correlation between the instantaneous amplitudes, and especially between the instantaneous phase angles, of the wavefront at any two points across the output beam.

We can then also say that the output beam possesses a very high degree of "spatial coherence" (in the transverse direction) as well as the temporal coherence discussed above.

Often this lowest-order output-beam pattern will vary reasonably smoothly in amplitude, and its phase variation will approximate reasonably closely either a plane wave or a spherical wave (which can be converted into a plane wave with a simple lens).

In contrast, if there are badly distorted optical elements inside the laser cavity, the amplitude and especially the phase profile across the beam may be badly distorted.

But if this pattern still represents a single transverse cavity mode, however badly distorted, then there will still be a high degree of coherence between the wavefront phasor at different transverse points; i.e., this beam will still be "spatially coherent" in some sense.

In principle, we could therefore design a complex "deaberrating lens" or deaberrating spatial filter that can convert this distorted but stationary wavefront into a smooth and uniphase wavefront of the type that is desirable in a laser output beam.

## Laser Beam Collimation

Thermal light sources not only usually emit many wavelengths, but also emit them quite randomly, in essentially all directions. Even if we capture some fraction of this radiation and collimate it with a lens or mirror, as in a searchlight or in the flashlight in Figure 1.42, the resulting degree of collimation, or the amount of radiation emitted per unit solid angle, is still much smaller than in even a very poor quality laser oscillator.

A single-transverse-mode laser oscillator can produce (usually in practice, and always in principle) an output beam that is more or less uniform in amplitude and constant in phase ("uniphase") across its full output aperture of width or diameter \(d\).

Such a beam can propagate for a sizable distance with very little diffraction spread; will have a very small far-field angle at still larger distances; and can be focused into a spot only a few wavelengths in diameter.

Elementary diffraction theory says, for example, that a uniphase plane wave coming from an aperture of diameter \(d\) will have a minimum angular diffraction spread \(\Delta\theta\) in the far field (Figure 1.47) given by

\[\tag{32}\Delta\theta\approx\frac{\lambda}{d}\qquad(\text{in radians})\]

For a visible laser with \(\lambda\) = 0.5 μm and an output aperture of, say, \(d\) = 0.5 cm, this gives an angular spread of \(\Delta\theta\approx10^{-4}\) radians, which we might alternatively express as 0.1 milliradians or 100 μrad.

The axial distance over which this same beam will stay approximately parallel and collimated before diffraction spreading begins to significantly increase the beam size—sometimes called the Rayleigh range—is then given (see Figure 1.47) by \(d/z_R\approx\lambda/d\), or

\[\tag{33}z_R\approx{d^2}/\lambda\]

A visible beam with a diameter of 5 mm thus has a Rayleigh range of \(z_R\approx50\) meters.

Suppose this same Uniphase beam is magnified by a 20-power telescope attached to the laser output and focused to infinity, as in Figure 1.48. Then the source aperture diameter is increased to \(d\) = 10 cm, and these results change to \(\Delta\theta\approx5\) μrad and \(z_R\approx\) 20 km. Uniphase laser beams can be propagated for very large distances with very small diffraction spreads.

## Laser Beam Focusing

Suppose this same uniphase laser beam with initial diameter d is focused down to a spot of diameter \(d_0\) by means of a simple lens of focal length \(f\). The diameter \(d_0\) of the focused spot can then be calculated by applying the same angular spread condition in reverse, to obtain

\[\tag{34}\Delta\theta\approx\frac{\lambda}{d_0}\approx\frac{d}{f}\]

or

\[\tag{35}d_0d\approx{f\lambda}\]

since the focal point will occur essentially one focal length \(f\) beyond the lens.

Suppose we follow the common practice in optics of defining the "\(f\)-number" or "\(f\)-stop" of the focusing lens by \(f^{\#}\equiv{f/d}\), i.e., focal length over diameter. (We are really defining this quantity in terms of the input beam diameter rather than of the lens diameter, but this of course determines the minimum lens diameter that can be employed.)

The approximate diameter of the focused spot can then be written as simply

\[\tag{36}d_0\approx{f^{\#}\lambda}\]

Photography buffs will know that lenses with \(f^{\#}\ge10\) are fairly easy to obtain; lenses with \(f^{\#}\) less than about 2 become expensive; and lenses with \(f^{\#}\) approaching unity become very expensive.

All the power in a truly uniphase laser beam can thus be focused into a spot a few laser wavelengths in diameter, if we use a powerful lens. (Microscope objectives are usually used for this purpose, at least for laser beams that are not too high in power. A focusing lens for single-wavelength laser radiation of course requires no correction for chromatic aberration, which helps.)

## Nonideal Laser Oscillators: Multimode and Multifrequency Oscillation

Many real lasers can produce output beams which come very close to the ideal temporal and spatial behavior described in the preceding paragraphs. Other lasers, however—especially including some of the higher-power laser systems—are more likely to oscillate in both multiple axial and multiple transverse cavity modes.

The coherence properties, both temporal and spatial, of such lasers then necessarily deteriorate relative to more ideal single-mode lasers; and the effort to obtain both single-axial-mode (or single frequency) oscillation, and single-transverse-mode (or "diffraction limited") beam quality, provides a continuing struggle for those who design and construct lasers.

Forcing a practical laser to oscillate in only a single centermost axial mode within the atomic linewidth is most easily accomplished if the laser cavity is made short in order to increase the \(c/2L\) axial mode spacing, and if the atomic linewidth is narrow.

The laser transition should also preferably be "homogeneously" rather than "inhomogeneously" broadened (we will define these specialized terms later). Special mode-selection techniques employing intracavity etalons and other special filters can also be used to reinforce one selected axial mode and suppress others.

Many practical lasers, however, actually oscillate in several axial modes simultaneously, usually in only a few, but perhaps in a few hundred in extreme cases. The outputs from such lasers, though no longer single-frequency, can still be quite narrowband compared to incoherent light sources; and multi-axial-mode oscillation is not a serious defect for many practical laser applications.

In such multi-axial-mode lasers there are more likely to be large random fluctuations of individual mode amplitudes, as individual mode frequencies drift across the gain profile because of thermal cavity expansion, and as individual modes compete with each other.

The total intensity in all the axial modes is, however, somewhat more likely to remain constant. Real laser devices can also be operated in various internally modulated and pulsed forms, and may be subject to various kinds of instabilities and relaxation oscillations, such as "spiking," which we will discuss in more detail in later tutorials.

The output signals from such less-than-perfect lasers may thus usually be described as the summation of several simultaneous and independent oscillation frequencies, and may have substantial random variations in amplitude and frequency for each separate oscillation.

Such a rather random multifrequency output, though not really the same as a gaussian random noise signal, may appear much like random noise according to various statistical and spectral measures.

## Real Laser Oscillators: Multiple-Transverse-Mode Oscillation

Many real lasers produce output beams which also approach the desirable single-transverse-mode character. A laser beam having the necessary single-mode and uniphase character is often said to be "diffraction limited," since its far-field diffraction angle and focal spot size will approach the ideal limits given just above; whereas beams whose far-field angular spread or focused spot size are \(k\) times larger than this are said to be \(k\) times diffraction-limited in performance.

More detailed diffraction calculations show that the far-field beam spread of a nonideal beam from an aperture of diameter d is not greatly affected by the exact amplitude pattern of the beam across the aperture; that is, it does not matter greatly whether the amplitude pattern is uniform, gaussian, cosine, or Bessel function, nor do moderate amplitude ripples on the beam lead to serious far-field beam spreading.

However, phase variations across the beam wavefront, whether random or regular in character, do begin to substantially increase the far-field beam spread or the focal spot size as soon as they approach the order of 90° phase shift—a distortion of more than a quarter of an optical wavelength—anywhere across the beam width.

A rough argument for the deterioration in beam quality that results from multiple-transverse-mode operation can be developed as follows. Let us call the number of simultaneously oscillating transverse modes in some real laser \(N_\text{tm}\).

Then the far-field angular spread of the output beam from that laser will usually be ~ \(N_\text{tm}^{1/2}\) times larger than the ideal value for a uniphase beam coming from an aperture of the same size, and the focused spot diameter will be ~ \(N_\text{tm}^{1/2}\) times larger than for an ideal beam. (The spot area will, of course, be ~ \(N_\text{tm}\) times larger.)

The ratio \(N_\text{tm}^{1/2}\) is sometimes referred as the "times diffraction limited" or "TDL" ratio of the real laser oscillator. This TDL ratio may range from about 1 up to a few factors of ten in real lasers. (In practice, a designer can often insert some suitable aperture inside a real laser cavity to improve the transverse beam quality, at the price of a corresponding reduction in total output power.)

The next tutorial introduces ** a few practical laser examples**.