Laser Oscillation and Laser Cavity Modes
This is a continuation from the previous tutorial - laser pumping and population inversion.
Adding laser mirrors and hence signal feedback, as we will do in this section, is then the final step necessary to produce coherent laser oscillation and thus to obtain a working laser oscillator.
Condition for Build-Up of Laser Oscillation
Suppose in fact that we have a laser rod or a laser tube containing atoms that are properly pumped so as to produce population inversion and amplification on a certain laser transition. To make a coherent oscillator using this medium, we must then add partially reflecting, carefully aligned end mirrors to the laser medium, as shown in Figures 1.1 or 1.4.
Suppose that we do this, and that a small amount of spontaneous emission at the laser transition frequency starts out along the axis of this device, being amplified as it goes.
This radiation will reflect off one end mirror and then be reamplified as it passes back though the laser medium to the other end mirror, where it will of course again be sent back though the laser medium (Figure 1.31).
If the round-trip laser gain minus mirror losses is less than unity, this radiation will decrease in intensity on each pass, and will die away after a few bounces. But, if the total round-trip gain, including laser gain and mirror losses, is greater than unity, this noise radiation will build up in amplitude exponentially on each successive round trip; and will eventually grow into a coherent self-sustained oscillation inside the laser cavity formed by the two end mirrors.
The threshold condition for the build-up of laser oscillation is thus that the total round-trip gain—that is, net laser gain minus net cavity and coupling losses—must have a magnitude greater than unity.
Steady-State Oscillation Conditions
Net gain greater than net loss for a circulating wave thus leads to signal build-up at the transition frequency within the laser cavity. This exponential growth will continue until the signal amplitude becomes sufficiently large that it begins to "burn up" some of the population inversion, and partially saturate the laser gain.
Steady-state oscillation within a laser cavity, just as in any other steady-state oscillator, then requires that net gain just exactly equal net losses, or that the total round-trip gain exactly equal unity, so that the recirculating signal neither grows nor decays on each round trip, but stays constant in amplitude.
In mathematical terms, using the more detailed model shown in Figure 1.32, the steady-state oscillation condition for a linear laser cavity with spacing \(L\) between the mirrors is that the total voltage gain and phase shift for a signal wave at frequency \(\omega\) in one complete round trip must satisfy the condition that
\[\tag{24}\mathcal{E}_2/\mathcal{E}_1\equiv{r_1}r_2\exp(2\alpha_mL_m-j2\omega{L/c})=1\qquad\text{at steady state}\]
where the coefficients \(r_1\) and \(r_2\) (\(|r|\le1\)) are the wave-amplitude or "voltage" reflection coefficients of the end mirrors; \(\exp(2\alpha_mL_m)\) is the round-trip voltage amplification through the laser gain medium of length \(L_m\); and \(\exp(-2j\omega{L/c})\) is the round-trip phase shift around the laser cavity of length \(L\). (For simplicity we have left out here any internal losses inside the laser cavity, and also any small additional phase-shift effects caused by the laser atoms or the cavity mirrors.)
If the laser employs instead a ring cavity of the type shown in Figure 1.33—as is becoming more common in laser systems—then this condition becomes instead
\[\tag{25}\mathcal{E}_2/\mathcal{E}_1=r_1r_2r_3\exp(\alpha_mp_m-j\omega{p/c})=1\qquad\text{at steady state}\]
where now \(p\) is the perimeter or full distance around the ring, and \(p_m\) is again the single-pass distance through the laser medium.
Round-Trip Amplitude Condition
Either of these conditions on steady-state round-trip gain then leads to two separate conditions, one on the amplitude and the other on the phase shift of the round-trip signal transmission. For example, the magnitude part of the steady-state oscillation condition expressed by Equation 1.24 requires simply that
\[\tag{26}r_1r_2\exp(2\alpha_mL_m)=1\qquad\text{or}\qquad\alpha_m=\frac{1}{4L_m}\ln\left(\frac{1}{R_1R_2}\right)\]
where \(R_1=|r_1|^2\) and \(R_2=|r_2|^2\) are the power reflectivities of the two end mirrors.
This condition determines the net gain coefficient or the minimum population inversion in the laser medium that is required to achieve oscillation in a given laser system. Using Equation 1.18 for the laser gain coefficient, we can convert this to the often-quoted threshold inversion density
\[\tag{27}\Delta{N}\equiv{N_2}-N_1\ge\Delta{N_\text{th}}\equiv\frac{\pi\Delta\omega_a}{\lambda^2\gamma_\text{rad}L_m}\ln\left(\frac{1}{R_1R_2}\right)\]
This expression on the one hand gives the minimum or threshold population inversion \(\Delta{N}_\text{th}\) that must be created by the pumping process if oscillation build-up toward sustained coherent oscillation is to be achieved. On the other hand, Equations 1.26 and 1.27 also give the saturated gain coefficient \(\alpha_m\) or the saturated inversion density \(\Delta{N}\) (atoms per unit volume) that must just be maintained to have unity net gain at steady state.
A laser oscillator will always start out with inversion somewhat greater than threshold. It will then build up to an oscillation level that just saturates the net laser gain down to equal net loss. This saturation occurs (as we will show in more detail later) when the laser oscillation begins to use up atoms from the upper level at a rate which begins to match the net pumping rate into that level; and it is just this gain saturation process which stabilizes the amplitude of a laser oscillator at its steady-state oscillation level.
Equation 1.27 makes clear that reaching laser threshold will be easiest if the laser has a narrow transition linewidth \(\Delta\omega_a\), and low cavity losses, including \(R_1, R_2\rightarrow1\). Note also that laser action generally gets more difficult to achieve as the wavelength \(\lambda\) gets shorter—infrared lasers are often easy, ultraviolet lasers are hard.
Round-Trip Phase or Frequency Condition
Equations 1.24 or 1.25 also express a round-trip phase shift condition which says that the complex gain in these equations must actually be equal to unity modulo some large factor of \(e^{-j2\pi}\); so for a linear cavity,
\[\tag{28}\exp(-j2\omega{L/c})=\exp(-jq2\pi)\qquad\text{or}\qquad\frac{2\omega{L}}{c}=q2\pi,\quad{q=\text{integer}}\]
In other words, the round-trip phase shift \(2\omega{L/c}\) inside the cavity must be some (large) integer multiple of \(2\pi\), or the round-trip path length must be an integer number of wavelengths at the oscillation frequency.
In the linear cavity case this phase condition is met at a set of discrete and equally spaced axial-mode frequencies given by
\[\tag{29}\omega=\omega_q\equiv{q\times2\pi\times}\left(\frac{c}{2L}\right)\]
The phase shift condition thus leads to a resonance frequency condition for the laser cavity, or equivalently to an oscillation frequency condition for the laser oscillator.
The set of frequencies \(\omega_q\) are called axial modes because they represent the resonant frequencies at which there are exactly \(q\) half-wavelengths along the resonator axis between the laser mirror in the linear or standing-wave case.
This same round-trip phase shift condition becomes \(\omega{p}/c=q2\pi\) in the ring cavity case, and the resonant frequencies \(\omega_q=q\times2\pi\times(c/p)\) are then the frequencies at which the ring perimeter \(p\) is an integer number of full wavelengths. The axial-mode integer \(q\) is typically a very large number in any real laser; e.g., for the standing-wave case
\[\tag{30}q=\frac{\omega_qL}{\pi{c}}=\frac{2L}{\lambda_q}=\frac{p}{\lambda_q}\approx10^5\sim10^6\]
since \(L\) (or \(p\)) is always \(\gg\lambda\) for any except very unusual laser cavities.
The axial resonant modes of the laser cavity are thus equally spaced in frequency, with axial-mode separation \(\Delta\omega_\text{ax}\) given by
\[\tag{31}\begin{align}\Delta\omega_\text{ax}&\equiv\omega_{q+1}-\omega_q=2\pi\times\frac{c}{2L}=2\pi\times\frac{c}{p}\\&\approx2\pi\times300\text{ MHz}\qquad\text{for}\qquad{L=50\text{ cm}}\end{align}\]
For many (though not all) practical lasers, this mode spacing is smaller than the atomic linewidth \(\Delta\omega_a\); and hence there will be several axial-mode cavity resonances within the atomic gain curve, as shown in Figure 1.35. The laser may then oscillate, depending on more complex details, on just the centermost one of these axial modes, or on several (or even many) axial modes simultaneously.
Transverse Spatial Properties: The Plane Mirror Approximation
We need to consider also the transverse variation of the optical fields in the laser cavity—that is, the variation over the cross-sectional planes perpendicular to the laser axis—since it is this variation that determines the spatial coherence or the transverse-mode properties of the laser oscillator.
In the simplest description, a laser will oscillate in the form of a more or less uniform, quasi-plane-wave optical beam bouncing back and forth between carefully aligned mirrors at the two ends of the laser resonator, as in Figure 1.36. The earliest successful lasers, and even some practical lasers today, in fact used flat or planar mirrors carefully aligned exactly parallel to each other and perpendicular to the axis of the laser.
If the optical wave in Figure 1.36 travels at even a slight angle to the resonator axis running perpendicular to the two mirrors, the radiation will walk out the open sides of the cavity, past the mirror edges, after some small number of bounces, as in Figure 1.37.
This will represent a large "walk-off" loss from the laser cavity, so that only waves that are very accurately aligned with the resonator axis will remain within the cavity and be able to oscillate. Hence the beam direction for the oscillating waves will lie very accurately along the cavity axis. (This of course also requires strictly parallel alignment of the two mirrors.)
To the extent that the oscillating beam then approximates a finite diameter beam with a nearly planar (or possibly slightly spherical) wavefront, the phase of the emerging wavefront will be essentially uniform across the output mirror, a condition sometimes referred to as a "uniphase" wavefront.
There will also then be a very high degree of coherence between the instantaneous phase of the wavefront emerging from widely separated points across the output mirror (but within the overall envelope of the laser beam); and so we can also say that there is a very high degree of "spatial coherence" to the laser output.
The laser output beam coming through a partially transmitting end mirror, at least in this simplified description, will thus be a highly directional beam with a uniform phase across the mirror surface and hence essentially perfect spatial coherence in the output beam.
Transverse Modes in Real Laser Cavities
In a real laser cavity, any such quasi-plane wave, as it bounces back and forth, will of course spread transversely because of diffraction, so that some of its energy will spill over the edges of the finite laser mirrors. This spillover will represent a diffraction loss mechanism, which becomes part of the overall roundtrip losses of the laser cavity.
It is even more important to recognize, however, that such a wave, as it bounces back and forth between two mirrors, will also undergo distortion of its transverse amplitude and phase profile in each trip around the laser cavity because of these same diffraction effects.
A uniform plane wave coming from a finite aperture, for example, will acquire significant Fresnel diffraction ripples in even one pass down the laser cavity. When this rippled beam bounces off a finite-aperture end mirror and the truncated wavefront travels back the other way along the laser cavity, it will acquire still further distortion because of additional diffraction and propagation effects.
The simple bouncing-plane-wave description of Figure 1.36 therefore cannot be fully correct, first because the uniform plane waves will spread and distort because of diffraction, and second because real laser cavities most often employ spherically curved mirrors, as in Figure 1.38, rather than flat or planar mirrors, for reasons we will soon consider.
These mirrors have finite transverse widths or diameters, which effectively act as apertures for the circulating laser beam; and in addition there are often additional apertures elsewhere along the laser axis, either deliberately added or caused by the finite diameter of the laser tube or other intracavity elements.
To understand the transverse beam properties in real laser cavities, therefore, we must examine more carefully what happens to a propagating optical wave with a given transverse amplitude and phase pattern when it propagates through one complete round trip around a laser cavity, including all the focusing, aperturing, and diffraction effects in the round trip.
Self-Reproducing Transverse Mode Patterns
The round-trip wave propagation in a real laser cavity can be studied by carrying out analytical or computer calculations of the manner in which the transverse field pattern of the optical beam changes on repeated round trips within a given resonator. Optical resonator mode calculations of this type were first pioneered in the early 1960s by A. G. Fox and T. Li at the Bell Telephone Laboratories, and are often referred to as "Fox and Li" calculations.
Such calculations are usually carried out with the laser gain omitted for simplicity. It then turns out that for any given laser cavity, employing either finite-diameter planar or (more usually) finite-diameter curved end mirrors, one will always find a certain discrete set of transverse eigenmodes, or distinct amplitude and phase patterns for the circulating beam in the cavity, which will reproduce themselves in form, though slightly reduced in overall amplitude, after one round trip.
A typical example of such a self-reproducing transverse beam pattern is shown in Figure 1.38. These self-reproducing transverse field patterns represent the characteristic set of lowest-order and higher-order transverse eigenmodes or transverse spatial modes characteristic of that particular laser resonator.
These self-reproducing transverse eigenmodes, with amplitude and phase patterns that depend on the specific curvature and shape of the laser mirrors, are analogous to the transverse modes in a closed waveguide, or even more closely analogous to the lowest-order and higher-order propagation modes in a leaky optical lensguide.
Indeed, we can view the repeated round trips in either a standing-wave or a ring laser resonator as essentially equivalent to passage through repeated sections of an iterated periodic lensguide, with reflection from the finite-aperture cavity mirrors being replaced by transmission through equivalent finite-aperture lenses having the same focal power.
These transverse eigenmodes can then provide self-consistent oscillation beam patterns for an oscillating laser. The amplitude reduction on each pass—which is generally different for each such transverse mode—simply represents the diffraction or spillover losses for that particular mode, caused by whatever finite apertures are present in the cavity.
If the laser then begins oscillating in one of these patterns, and if the laser medium can maintain sufficient round-trip gain to overcome the diffraction losses of that particular transverse mode, along with all the other losses in the cavity, this will be one possible steady-state beam pattern or beam profile for the laser oscillation.
Planar Resonator Modes
In any reasonably well-designed laser cavity with finite-width or finite-diameter end mirrors, we will normally find that there is one such lowest-order transverse mode pattern, which is usually reasonably smooth in its transverse amplitude and phase profile, and which has the lowest diffraction loss of all the self-reproducing transverse mode patterns in that particular resonator.
In a properly aligned planar resonator, for example, the lowest-order transverse mode will generally have an amplitude profile which looks something like the upper part of figure 1.39.
That is, this mode will typically look something like the central lobe of a \(J_0(r)\) Bessel function across the mirror for circular end mirrors, or like a single lobe of a cosine wave, that is \(\mathcal{E}(x,y)\approx\cos(\pi{x}/a)\cos(\pi{y}/b)\) for rectangular mirrors of width \(2a\) by \(2b\).
The exact amplitude pattern of this lowest-order mode will, however, also have diffraction ripples, as in the upper part of Figure 1.39, whose amplitude and spacing depend on the finite mirror size; and the quasi-Bessel function or cosine variation will not drop quite to zero at the mirror edges, in agreement with the inevitable diffraction losses in such an open-sided resonator.
The phase variation of the lowest-order mode in a typical planar resonator will also exhibit some small Fresnel diffraction ripples, along with some small curvature of the wavefront along the outer edges of the resonator, as in the lower part of Figure 1.39; but over the major portion of the mode the wavefront will in fact be a very good approximation to a planar wavefront.
A plane-mirror cavity oscillating in this lowest-order transverse mode will thus in fact have output beam properties very close to those of the simple plane wave described earlier.
The unwanted diffraction losses past the mirror edges for this lowest-order transverse mode will also typically be very small, unless the mirror sizes are made very small. The lower-order self-consistent transverse modes in almost any type of resonator in fact exhibit an uncanny ability to shape their amplitude and phase patterns in ways that minimize their diffraction losses on each round trip.
Higher-Order Modes
This same laser cavity will generally also have many higher-order transverse modes. These will generally have larger diffraction losses and also more complex transverse amplitude and phase variations, like the higher-order transverse modes in waveguides. And they will generally have several transverse nulls and phase reversals, with either even or odd symmetry in simple cases.
Their transverse spread inside the cavity is generally larger, which makes their diffraction losses larger than those of the lowest-order transverse mode; and their diffraction spread or beam spread outside the cavity is also generally larger than that for the lowest-order transverse mode. For these reasons, laser oscillation in these higher-order modes is generally considered undesirable.
Stable and Unstable Laser Resonators
Practical laser cavities most often employ curved rather than planar end mirrors, in order to shape the transverse modes of the cavity and control the diffraction losses.
There is one broad class of such curved-mirror resonator designs, the so-called stable laser resonators, in which the diffraction losses are generally very small, and the lowest-order and higher-order modes have the form (very nearly) of Hermite-gaussian functions, as in Figure 1.40, with the lowest-order mode having a gaussian transverse profile of the form \(\mathcal{E}(r)=\exp(-r^2/w^2)\).
Such gaussian modes and the resulting gaussian output beams are particularly easy to handle both analytically and in experiments, and practical lasers are very often designed in this fashion.
On the other hand, these Hermite-gaussian modes in realistic laser cavities do turn out to be very slender in diameter, so that they do not readily fill all the volume of larger-diameter laser tubes or rods.
The laser must then oscillate in a mixture of lowest-order and higher-order modes (which tends to spoil the beam collimation properties) in order to fill and extract all the available power from the laser volume.
There is also a class of so-called unstable optical resonators, which make use of deliberately diverging laser wavefronts as shown in Figure 1.41. These resonators have transverse mode patterns that much more readily fill large laser volumes, but still suppress higher-order transverse modes.
These unstable optical resonators necessarily have much larger output coupling or lower effective mirror reflectivity than stable resonators, since the diffraction spread past the output mirror edges is used as the output coupling mechanism. This property limits the usefulness of unstable resonators for low-gain laser systems.
The mode properties of such unstable resonators are also rather more complex and esoteric than the simple Hermite-gaussian stable modes. (Note that the "stability" referred to in these resonator classifications is that of geometrical rays bouncing back and forth in the cavity designs in question, and has nothing directly to do with the stability or instability of the laser oscillation in the resulting transverse eigenmodes.)
Perhaps the most useful class of laser resonator modes in the future will be the geometrically unstable but still Hermite-gaussian modes that can be obtained in so-called "complex paraxial" resonators by using variable reflectivity mirrors.
General Transverse-Mode Oscillation Properties
Each different optical-resonator design, whether planar, stable, unstable, or still more complex, will thus possess some lowest-order transverse mode pattern which can circulate repeatedly around the laser cavity without changing its amplitude or phase profile.
The phase profile of this lowest-order transverse mode will usually be comparatively smooth and regular across the output mirror of the laser cavity (as well as at any other transverse plane within the cavity).
The phase front is often quasi spherical across the output plane of the laser, but this spherical curvature can be removed by a simple lens to convert the output beam into a fairly well-collimated plane wave.
A laser cavity which oscillates only in this lowest-order transverse mode will thus generally produce an output beam with good transverse characteristics and with a nearly uniphase character across the output mirror.
If the laser oscillates simultaneously in several transverse modes, however, as can readily happen in real lasers, the output wavefront will no longer be "uniphase," and the collimation and focusing properties of the beam will generally deteriorate.
Forcing laser oscillation to occur only in the lowest-order transverse mode is thus a practical design objective, which is achieved in some though not all practical lasers.
The primary obstacle to achieving single-transverse-mode oscillation in higher-power (or higher-gain) lasers is that the diffraction losses of the lowest- and higher-order modes in a large-diameter cavity are all small and nearly identical; so there is little or no loss discrimination between the different transverse modes.
A designer must then add mode-control apertures, employ unstable resonator designs, or use other tricks to suppress the unwanted higher-order transverse modes.
Note also that the transverse mode properties we have just been discussing, and the axial mode or resonant frequency properties we discussed earlier, are almost independent of each other. There are some important secondary connections between these properties.
In simplified terms, however, the round-trip propagation length determines the resonant axial-mode frequencies of the laser, whereas the focusing and diffraction effects associated with mirrors and apertures in the round-trip propagation determine the transverse mode patterns.
The next tutorial discusses about performance of representative lenses.