# An Introduction to Optical Beams and Resonators

This is a continuation from the previous tutorial - metal-coated fibers.

## 1. Transverse Modes in Optical Resonators

Laser cavities differ in several significant ways from the closed microwave cavities that are commonly treated in electromagnetic theory tutorial.

Optical resonators first of all usually have open sides, and hence always have diffraction losses because of energy leaking out the sides of the resonator to infinity.

Optical resonators are also usually described in scalar or quasi plane-wave terms, with emphasis on the diffraction effects at apertures and mirror edges, rather than in vector terms with emphasis on matching boundary conditions.

The distinction between "longitudinal" and "transverse" modes in the resonator is also much sharper in optical than in microwave resonators.

Before beginning any detailed analysis of optical resonators, therefore, it may be useful to introduce some of the fundamental concepts we will use to describe optical resonator modes in rather broad and general terms.

### The "Recirculating Pulse" Approach

In earlier tutorials we have often emphasized how the optical radiation inside an optical cavity circulates repeatedly around the cavity, bouncing back and forth between the end mirrors (or circulating around the ring in a ring-laser cavity). In these earlier discussions we used only a plane-wave approximation, ignoring the transverse spatial variation of the waves.

To bring transverse variations into the discussion, let us next consider only that portion of the optical energy traveling in the $$+z$$ direction and contained within some short axial segment of length $$\Delta{z}$$ within the cavity.

We can think of the radiation in this segment as forming a short pulse or a thin "slab" of radiation (see Figure 14.1), whose axial thickness $$\Delta{z}$$ is small compared to the length $$L$$ of a typical laser cavity but still very large compared to an optical wavelength $$\lambda$$.

The time and space variation of the $$\mathcal{E}$$ fields within such a circulating pulse or slab as it travels through the resonator, including transverse variations, can then be written in the form

\tag{1}\begin{align}\mathcal{E}(x,y,z)&=\text{Re}\tilde{E}(x,y,z)e^{j(\omega{t}-kz)}\\&=\text{Re}|\tilde{E}(x,y,z)|e^{j(\omega{t}-kz)+j\phi(x,y,z)}\end{align}

By writing the fields in this fashion, we separate out the plane-wave aspects of the wave propagation as given by the $$e^{j\omega{t}-jkz}$$ factor, where $$\omega$$ is the optical carrier frequency and $$k=\omega/c=2\pi/\lambda$$ the associated plane-wave propagation constant, from the complex phasor amplitude $$\tilde{E}(x,y,z)$$ which describes the transverse amplitude and phase variation of the beam.

The transverse intensity profile of the beam within this particular pulse or slab is then given by $$I(x, y, z)=|\tilde{E}(x, y, z)|^2$$, whereas the transverse phase profile, or the shape of the optical wavefront is given by the transverse phase variation $$\phi(x,y, z)$$.

Although we write the phasor amplitude function $$\tilde{E}(x,y,z)$$ as a function of $$x$$, $$y$$ and also $$z$$, we will see later that the variation of this transverse beam profile with the axial or $$z$$ coordinate is generally very slow compared to the $$e^{-jkz}$$ variation that we have separated out.

The latter function goes through a complete $$e^{\pm{j}2\pi}$$ variation in just one optical wavelength. By contrast, the complex amplitude profile $$\tilde{E}(x, y, z)$$ will not change much if at all through the thickness of one "slab"; and it will also change only very slowly with distance as a particular slab propagates down the resonator, or through free space outside a resonator.

### Pulse Propagation in Stable and Unstable Resonators

If, however, we follow the transverse profile $$\tilde{E}(x,y,z)$$ of any one such slab as it travels (at the velocity of light) through one complete round trip around a laser cavity, we will definitely see the transverse field pattern in the slab change with distance as the slab propagates, diffracts, bounces off mirrors, and passes through rods, lenses and finite apertures.

These changes in the transverse pattern $$\tilde{E}(x, y, z)$$ of the slab caused by propagation and diffraction are the primary effects that determine the transverse mode properties of optical beams and resonators.

We will see later that optical resonators can usually be divided into either "geometrically stable" or "geometrically unstable" categories (where these terms refer to ray stability within the resonators, and have nothing to do with whether or not the laser is or is not stable against laser oscillation).

In such resonators, the recirculating slabs themselves may also acquire a certain macroscopic curvature caused by the focusing effects of the laser mirrors, as shown for either a typical "stable" resonator or an "unstable" resonator in Figure 14.2.

Each such pulse or slab of radiation as it travels around may thus be rather inelegantly described as a "recirculating pancake" of radiation within the resonator. An important point is that the propagation of each such slab is essentially unaffected by the radiation in the slabs immediately in front of or behind it—the optical radiation in each axial segment or "pancake" is more or less independent of the other pancakes ahead of or behind it in the resonator.

### Optical Resonators and Equivalent Periodic Lensguides

Rather than thinking of repeated round trips within a resonator, it can be helpful to think of the pulse or "pancake" as propagating instead through repeated sections of an iterated periodic optical system as in Figure 14.3.

In setting up such an iterated periodic lensguide, curved mirrors in the original resonator are replaced by thin lenses of equal focusing power, and all other elements encountered in the lensguide are made the same as those encountered in the original resonator (except that in a standing-wave cavity each element must be included twice to model a complete round trip in both directions).

The diffraction and aperturing effects that the pulse sees in a series of repeated round trips around the original laser cavity will then be the same as in propagating through an equivalent number of segments in the periodic lensguide; and the lensguide itself may be either "stable" or "unstable" in the sense discussed in the preceding.

This lensguide approach obviously adds no new physics to the problem, but it does convert the resonator problem into an equivalent waveguide problem, and it can sometimes be helpful in visualizing the behavior in an optical resonator, as we shall see.

### Optical Resonator Eigenmodes and Eigenvalues

Let us now look at how this recirculating or traveling pulse approach leads to the concept of transverse cavity modes or eigenmodes in an optical resonator.

Suppose such a pulse or slab of radiation makes one complete round trip around an optical cavity, or travels through one complete period of the equivalent lensguide.

After one complete round trip, the transverse field pattern $$\tilde{E}^{(1)}(x,y)$$ within a given slab as it arrives back at its starting plane will in general be different from its starting pattern $$\tilde{E}^{(0)}(x,y)$$ before the round trip, because of diffraction, reflection and aperturing effects; and after a second round trip the pattern $$\tilde{E}^{(2)}(x,y)$$ may again be still different.

(Note that we have dropped the $$z$$ dependence in writing these patterns, because we are only considering the transverse variation as observed at one arbitrarily chosen reference plane somewhere within the resonator, or at a set of such planes spaced one period apart in the equivalent lensguide.)

We can then ask if, to put the question in physical terms, there exist any transverse patterns, call them $$\tilde{E}_{nm}(x,y)$$, such that if a pulse or pancake starts off with one of these transverse patterns, it will return one round trip later with exactly the same pattern?

More precisely, we require that the pulse of radiation must return with exactly the same transverse form, but possibly with a reduced amplitude because of diffraction and other losses during the round trip. The wave may in general also return with an arbitrary absolute phase shift, because of the propagation distance $$p$$ around the resonator at the optical frequency $$\omega$$ of the pancake.

If we can find any such self-reproducing transverse patterns, it certainly seems reasonable to call them transverse modes of the resonator, or of the equivalent lensguide.

That is, a pulse which is launched with an initial transverse profile matching one of these transverse modes can then propagate repeatedly around the resonator, or propagate indefinitely down the lensguide, always getting weaker in amplitude, but always maintaining the same transverse profile at the same reference plane in the resonator or the lensguide.

In fact, if we add enough laser gain within the resonator to just cancel the diffraction losses, it would seem that the resonator can oscillate indefinitely in any one of these transverse modes. (We will see shortly that this is indeed true, though with some slight complications.)

### Examples of Optical Resonator Eigenmodes

Do such lossy but self-reproducing transverse eigenmodes then really exist for open-sided and finite-diameter optical cavities—especially the very long slender cavities often used in practical lasers?

The answer is that they do indeed exist, and that moreover the lowest-order transverse modes in properly designed (and aligned) laser cavities can have remarkably low diffraction losses, as well as remarkably good propagation properties.

The simplest transverse mode patterns, and the ones that are easiest to analyze, occur in the so-called geometrically stable optical.resonators or lensguides using properly curved mirrors, such as those we have illustrated earlier.

The lowest-order and higher-order modes in stable optical resonators, if expanded in rectangular transverse coordinates, are given almost (but not quite) exactly by Hermite-gaussian functions, such as those exhibited in the top part of Figure 14.4. We will discuss these gaussian modes in much greater detail in subsequent sections.

These modes, like the modes in most other optical resonators, are essentially plane waves, or slightly curved spherical waves, multiplied by the transverse amplitude and phase profiles given by the transverse mode functions $$\tilde{E}_{nm}(x,y)$$.

Although the exact vector expressions for the associated optical beams must then necessarily have some small axial $$E$$ and $$H$$ field components, the primary field components in these beams are polarized transverse to the direction of propagation, just as in ideal uniform plane or spherical waves.

These waves are very often referred to, therefore, as TEM$$_{nm}$$ optical waves, and we have used this notation in Figure 14.4. (Note, however, that a truly pure TEM electromagnetic wave can only exist in a transmission line having at least two conductors; and that these TEM$$_{nm}$$ optical waves must therefore always have some small axial $$E$$ and $$H$$ field components.)

### Planar and Unstable Resonator Modes

If we set up an optical resonator with two perfectly aligned flat mirrors— for example, two flat circular mirrors—the transverse mode patterns become more difficult to express analytically, but they nonetheless still exist.

The first four azimuthally symmetric, or $$l=0$$, modes of a typical circular plane-mirror resonator will have radial variations something like those shown in Figure 14.4(b). There also exist a large number of azimuthally varying or $$l\gt0$$ transverse modes which we have not shown here.

Again these are essentially TEM modes, with radial amplitude patterns that in this situation look approximately like lowest and higher-order Bessel functions, with a small amount of irregular diffraction ripple added.

Note, however, that these transverse mode patterns, which are viewed in this figure at the end mirror surface, do not quite go to zero at the mirror edges. The amount of energy that is lost past the mirror edges represents the diffraction loss or diffraction spillover from the ends of the resonator.

Finally, there are the even more complicated geometrically unstable resonators, which we will discuss in more detail in a later tutorial. These resonators have modes in which a large amount of energy is lost on each round trip past the edges of the smaller output mirror, as illustrated for a typical situation in the bottom part of Figure 14.4. This energy in fact forms the useful output beam in unstable-resonator lasers, which must typically have large laser gain in order to operate with such large output coupling.

Unstable resonator lasers can on the other hand have important advantages for higher-power lasers, including large mode volume, good discrimination against higher-order transverse modes, all-reflective optics (which can be, for example, water-cooled in very high-power lasers), and good output beam quality.

Unstable resonators do have higher-order transverse modes, as well as the lowestorder type of mode pattern illustrated in Figure 14.4(c), but all of these modes are very difficult to express analytically and show large variations in shape with changes in the resonator length and diameter. We have therefore shown only one representative lowest-order example in Figure 14.4.

## 2. The Mathematics of Optical Resonator Modes

Let us now restate the basic problem outlined in the previous section in mathematical terms, and ask how we can calculate the propagation effects for an optical pulse through one round trip in a resonator, or one period of the periodic lensguide, and how we can find these transverse mode patterns that are self-reproducing after each such round trip or periodic step.

### The Round-Trip Propagation Integral

For essentially all the optical cavities of interest to us, the total propagation through one round trip in an optical resonator, or through one period in the equivalent lensguide, can be described mathematically by a propagation integral which will have the general form

$\tag{2}\tilde{E}^{(1)}(x,y)=e^{-jkp}\iint_{\text{Input plane}}\tilde{K}(x,y,x_0,y_0)\tilde{E}^{(0)}(x_0,y_0)dx_0dy_0$

where $$k$$ is the propagation constant at the carrier frequency of the optical signal; $$p$$ is the length of one period or round trip; and the integral is over the transverse coordinates at the reference or input plane.

The function $$\tilde{K}$$ appearing in this integral is commonly called the propagation kernel, since the field $$\tilde{E}^{(1)}(x, y)$$ after one propagation step can be obtained from the initial field $$\tilde{E}^{(0)}(x_0,y_0)$$ through the operation of the linear kernel or "propagator" $$\tilde{K}(x,y,x_0,y_0)$$.

Any arbitrary reference plane within the resonator, or within one period of the equivalent lensguide, may be chosen as the starting plane or reference plane for writing the preceding integral.

The exact form of the kernel $$\tilde{K}$$ will depend on the reference plane that is chosen. If for example, the reference plane is chosen at an aperture, and the only intervening element before the next aperture is simply free space, this propagator will be simply Huygens' integral for free space, with the integral being evaluated over the aperture at the input end of each period.

More generally the propagation kernel will contain additional factors caused by intervening lenses, apertures, and other optical elements. Evaluating the form of the kernel in Equation 14.2 will be one of our major interests in the following tutorials.

In doing resonator analyses, we will usually separate out from the propagation kernel the on-axis phase shift term $$e^{-jkp}$$, as has been done in Equation 14.2, since all the necessary information for evaluating transverse field patterns is contained in the remaining kernel $$\tilde{K}(x,y,x_0,y_0)$$, with the exponential term only furnishing a constant phase shift in front.

We will look at propagation kernels of various types in much more detail for specific situations later on. For the present all we need understand is that there is (almost always) a linear relationship like Equation 14.2 between the input field $$\tilde{E}^{(0)}(x_0,y_0)$$ and the output field $$\tilde{E}^{(1)}(x,y)$$ after one step.

### The Eigenequation for Optical Resonator Modes

In mathematical terms the propagation integral in Equation 14.2 is a linear operator equation: that is, the linear propagation operator $$\tilde{K}$$ acts on the optical field $$\tilde{E}^{(0)}(x,y)$$ at a reference plane on one round trip to produce a new optical field $$\tilde{E}^{(1)}(x,y)$$ one round trip or one period later. Given an operator equation such as this, we may then ask whether this equation has a set of eigensolutions.

That is, for a given resonator or kernel, does there exist a set of mathematical eigenmodes $$\tilde{E}_{nm}(x, y)$$ and a corresponding set of eigenvalues $$\tilde{\gamma}_{nm}$$ such that each one of these eigenmodes after one round trip satisfies the round-trip propagation expression

$\tag{3}\tilde{E}^{(1)}_{nm}(x,y)\equiv\iint\tilde{K}(x,y,x_0,y_0)\tilde{E}^{(0)}_{nm}(x_0,y_0)dx_0dy_0=\tilde{\gamma}_{nm}\tilde{E}^{(0)}_{nm}(x,y)$

or simply

$\tag{4}\tilde{\gamma}_{nm}\tilde{E}_{nm}(x,y)\equiv\iint\tilde{K}(x,y,x_0,y_0)\tilde{E}_{nm}(x_0,y_0)dx_0dy_0$

where we can drop the superscript indices.

If eigensolutions that satisfy Equation 14.4 do exist, then these eigensolutions will provide exactly the self-reproducing transverse eigenmodes we seek, for either the optical resonator or the corresponding periodic lensguide.

That is, if we launch a "recirculating pancake" in the form of any single one of these eigenmodes $$\tilde{E}_{nm}(x,y)$$ in the proper direction at the selected reference plane, then after one round trip the field at that same plane will be

$\tag{5}\tilde{E}^{(1)}_{nm}(x,y)=\tilde{\gamma}_{nm}e^{-jkp}\tilde{E}^{(0)}_{nm}(x,y)$

The field after one period will have exactly the same transverse form, both in its phase variation $$\phi_{nm}(x, y)$$ and in its amplitude variation $$|\tilde{E}_{nm}(x, y)|$$, although if we do not include any laser gain, the transverse mode pattern will be reduced in amplitude and shifted in absolute phase by the complex eigenvalue $$\tilde{\gamma}_{nm}$$. This self-reproducing behavior is the mathematical definition of a "transverse mode" of the optical resonator or the periodic lensguide.

Note that these transverse mode patterns $$\tilde{E}_{nm}(x,y)$$ (if any exist) will have in general a different field pattern $$\tilde{E}_{nm}(x, y, z)$$ at each transverse $$z$$ plane within the resonator, i.e., the shape of each transverse mode will change (slowly) with distance as it propagates along the resonator (or returns going in the opposite direction at the same plane in a standing-wave cavity).

To put this in another way, the exact form of the kernel $$\tilde{K}(x,y,x_0,y_0)$$ and hence of the eigenmodes $$\tilde{E}_{nm}(x,y)$$ will De different if the kernel and the eigenmodes are evaluated at different reference planes, although the round-trip eigenvalues $$\tilde{\gamma}_{nm}$$ will be the same.

### Resonator Eigenvalues and Diffraction Losses

A transverse wave pattern that is bounded within a finite width will always spread out due to diffraction as it propagates. In an open-sided resonator with finite-diameter mirrors, therefore, some of the radiation will spread out past the mirror edges after each round trip, and the magnitudes of the transverse eigenvalues (again neglecting gain) will therefore always be less than unity, i.e., $$|\tilde{\gamma}_{nm}|\lt1$$.

Hence even with perfectly lossless mirrors the $$nm$$-th eigenmode of an optical resonator will always have a power loss per round trip given by

$\tag{6}\text{fractional power loss per round trip}=1-|\tilde{\gamma}_{nm}|^2$

These losses result from diffraction losses at the mirror edges or at apertures within the cavity, and will continue to occur on all subsequent round trips.

If no laser gain is present the amplitude of a given transverse mode will decay exponentially with successive round trips in the form

$\tag{7}\frac{\tilde{E}^{(k)}_{nm}(x,y)}{\tilde{E}^{(0)}_{nm}(x,y)}=\tilde{\gamma}^k_{nm}$

If we add a laser medium with transversely uniform round-trip voltage gain $$e^{\alpha_mp_m}$$ inside the optical cavity, the total round-trip amplitude gain and phase shift become

$\tag{8}\tilde{E}^{(1)}_{nm}(x,y)=\tilde{\gamma}_{nm}e^{\alpha_mp_m-jkp}\tilde{E}^{(0)}_{nm}(x,y)$

(If the gain itself has a transverse $$x$$, $$y$$ variation, this must become part of the propagation kernel determining the eigenmodes.) The amplitude condition for laser threshold or for steady-state laser oscillation then becomes

$\tag{9}\left|\frac{\tilde{E}^{(1)}_{nm}(x,y)}{\tilde{E}^{(0)}_{nm}(x,y)}\right|=|\tilde{\gamma}_{nm}e^{\alpha_mp_m-jkp}|=1$

The lowest-loss eigenmode, i.e., the one with the largest value of $$|\tilde{\gamma}_{nm}|$$ and smallest value of $$\delta_{nm}$$, will have the lowest threshold for oscillation and hence will (normally) be the dominant mode in the cavity.

### Existence of Resonator Eigenmodes

Many of you may be familiar with the electromagnetic theory of microwave cavities or microwave waveguides, where resonant eigenmodes always do exist.

That is, for closed cavities with lossless walls, such as are usually treated in electromagnetic theory texts, the wave equation describing the cavity fields is a hermitian mathematical operator; and the existence of a complete set of normal modes can therefore be rigorously proven.

The completeness property then means that any arbitrary field pattern inside the cavity can always be expanded using this set of eigenmodes as the basis set.

There is a serious mathematical difficulty for open-sided optical resonators, however, in that the round-trip propagation kernel $$\tilde{K}(x,y,x_0,y_0)$$ for such resonators is generally found not to be a hermitian operator.

This in turn means that the existence of a complete and orthogonal set of eigensolutions to Equation 14.4 is not automatically guaranteed, whereas it would be for a hermitian kernel. Such eigenmodes may exist, but we cannot guarantee in advance either their existence or, if they do exist, their completeness.

In the early days of lasers, the physical reality as well as the mathematical existence of transverse modes in open resonators was a matter of considerable debate. Even now, in fact, except for a few special situations, rigorous mathematical existence and completeness proofs for optical resonator modes do not exist.

Real lasers have never had any difficulty in finding such modes in which to oscillate, however; and from a combination of empirical and experimental evidence, it is now entirely accepted that transverse eigenmodes as we have defined them in the preceding paragraphs do exist, and do provide a physically realistic and meaningful basis for describing laser oscillation in real laser resonators.

### Transverse Mode Orthogonality

A related mathematical peculiarity of optical resonator eigenmodes is that they are generally not "normal modes" in the usual sense of this term. That is, because of the nonhermitian kernel the eigenmodes $$\tilde{E}_{nm}(x,y)$$ of an optical resonator calculated at any plane $$z$$ are generally not power orthogonal in the usual fashion, i.e., for any two modes we may in general not write

$\tag{10}\iint\tilde{E}_{nm}(x,y)\tilde{E}_{pq}^*(x,y)dxdy=\delta_{np}\delta_{mq}\quad(\text{wrong})$

where $$\delta_{np}$$ is the Kronecker delta function. Rather the set of modes $$\tilde{E}_{nm}(x,y)$$ are generally biorthogonal (without complex conjugation) to an adjoint set of modes, let's call them $$\tilde{E}^{\dagger}_{pq}(x,y)$$, in the form

$\tag{11}\iint\tilde{E}_{nm}(x,y)\tilde{E}_{pq}^{\dagger}(x,y)dxdy=\delta_{np}\delta_{mq}\quad(\text{right})$

These adjoint functions $$\tilde{E}_{nm}^{\dagger}(x,y)$$ usually represent the transverse modes traveling in the opposite direction in the same cavity. The biorthogonality properties of general optical resonator modes are summarized at the end of a later tutorial.

It is also not in general possible to prove that the transverse eigenmodes of an optical resonator form a complete set. That is, it cannot be rigorously proven in advance that any field pattern within a given resonator can be written in the form

$\tag{12}\tilde{E}(x,y)\stackrel{?}{=}\sum_{nm}c_{nm}\tilde{E}_{nm}(x,y)\quad(\text{not guaranteed})$

However, the Hermite-gaussian or Laguerre-gaussian functions that approximate the eigenmodes in ideal stable resonators certainly do form a complete basis set; and in most practical situations people simply proceed as if the resonator eigenmodes always do form a complete set.

### Axial Versus Transverse Resonator Modes

It is important to understand that, once the axial phase shift term $$e^{-ikp}$$ has been factored out, the propagation kernel $$\tilde{K}(x,y,x_0,y_0)$$ in a typical optical resonator or lens waveguide depends only very slightly on the exact frequency $$\omega$$ or the exact wavelength $$\lambda$$ of the radiation in the recirculating pancake.

In physical terms, the diffraction effects experienced by a transverse mode function $$\tilde{E}_{nm}(x,y)$$ in a round trip will be essentially the same for any carrier frequency (or any axial mode frequency) within the linewidth of a single atomic transition or the oscillation bandwidth of a single laser oscillator.

Hence, the transverse mode properties and the axial frequency properties of a given optical resonator can be treated almost completely separately from each other.

The transverse eigenmodes for any given laser can then be calculated based only on the mean laser wavelength; and all of the axial modes within a given laser line will then have the same set of transverse eigenmodes and eigenvalues.

The transverse eigensolutions, in fact, might rather be viewed as the transverse propagation modes of the equivalent lensguide, for which axial resonance frequencies have no meaning.

If we shift to a different laser line which is, say, 20% different in frequency, then the diffraction effects in one round trip may change somewhat, and we can expect the form of the transverse eigenmodes to change by a noticeable amount.

By launching a continuous stream of "pancakes" one after another, nose to tail so to speak, we can fill an entire laser cavity with radiation all in one given transverse eigenmode, and all at one carrier frequency.

To satisfy the roundtrip phase-shift condition, or to make the axial variation of the fields continuous completely around the resonator, the carrier frequency of these pancakes would have to be one of the axial mode frequencies of the resonator; and having done this we would have filled the cavity with radiation in a single axial and single transverse mode.

## 3. Build-Up and Oscillation of Optical Resonator Modes

Without going into details of the exact modes for any specific resonator, we can now say some additional things about how resonator transverse modes can be calculated numerically; about their exact resonance frequencies; and about how these modes build up, compete, and decay in real lasers.

### Calculating The Lowest-Loss Eigenmode

Suppose one of our pulses or "pancakes" with an arbitrary initial field distribution $$\tilde{E}^{(0)}(x,y)$$ is launched inside a resonator with no laser gain. We will assume, without worrying about rigorous justification, that this initial distribution can be written as a sum of the transverse eigenmodes for that particular resonator, i.e.,

$\tag{13}\tilde{E}^{(0)}(x,y)=\sum_{nm}c_{nm}\tilde{E}_{nm}(x,y)$

(and we will not worry about the axial variation of the pulse, since we do not need it to calculate the round-trip propagation.)

Then, on each round trip inside the resonator each transverse mode component will be multiplied by its eigenvalue $$\tilde{\gamma}_{nm}$$; and hence the field at the same reference plane $$k$$ round trips later will be given by

$\tag{14}\tilde{E}^{(k)}(x,y)=\sum_{nm}c_{nm}\tilde{\gamma}^{k}_{nm}\tilde{E}_{nm}(x,y)$

The relative amplitude of each transverse mode will thus have attenuated after $$k$$ successive round trips as $$|\tilde{\gamma}_{nm}|^k$$.

Suppose we index the transverse eigenmodes so that $$nm=00$$ labels the transverse mode with the largest eigenvalue or the smallest loss per round trip. All other $$nm$$ combinations will then have smaller eigenvalues and hence larger mode losses. Suppose we run the field distribution $$\tilde{E}(x,y)$$ through many repeated round trips, letting $$k$$ in Equation 14.14 become large.

Then the amplitudes of the various eigenmodes will attenuate or die out with different rates on repeated round trips (see Figure 14.5).

It is apparent that, whatever may be the initial mode distribution, after a sufficient number of round trips the lowest-loss or $$00$$ mode will become dominant compared to all the other transverse modes. There is a chance that two modes will have exactly the same magnitude, and hence both will persist, but we can handle this as an unusual special situation.

We can also dismiss as extremely unlikely the chance of any real initial distribution containing no initial component of the $$00$$ mode whatsoever. Starting with any arbitrary initial transverse field pattern and following it through enough round trips in the resonator is thus a prescription for finding the lowest-order transverse mode of an optical resonator or lensguide.

### The Fox and Li Approach

This conceptual approach to finding the lowest-order resonator transverse modes is often called the "Fox and Li" approach, since it describes not only the real physical situation in an optical cavity, but also the numerical modecalculation procedure pioneered by A. G. Fox and T. Li at Bell Telephone Laboratories around 1960, in the earliest days of the laser.

Fox and Li simulated the iterative round trips of a wavefront $$\tilde{E}(x,y)$$ in a resonator by using numerical computation on a digital computer. In these computations they repeatedly integrated the propagation equation (14.2) using the Huygens' integral kernels for plane-mirror resonators and other simple situations. Figure 14.6 shows some typical results from this kind of calculation.

Fox and Li's first calculations were made assuming, for simplicity, a "strip resonator," that is, a resonator with end mirrors in the form of two parallel flat strips having transverse variations in the $$x$$ direction only (strip width = $$2a$$), spaced by a distance $$L$$ in the $$z$$ direction, and with no variations in the $$y$$ direction along the strips. The starting field on one end mirror was simply a uniform field pattern $$\tilde{E}^{(0)}(x,y)=1$$ across the mirror, as in Figure 14.6(a).

The two curves in Figure 14.6(b) then show the resulting field pattern or diffraction pattern $$\tilde{E}^{(1)}(x,y)$$ after the first propagation step from one end of the laser cavity to the other.

(Fox and Li's initial calculations involved axially symmetrical laser cavities, and were phrased in terms of propagation steps from one end to the other, rather than complete round trips; but the essential ideas remain unchanged.)

A beam propagating away from an aperture with sharp edges can be expected to exhibit Fresnel diffraction ripples in its near-field pattern, and the conventional Fresnel diffraction ripples in the field pattern after this first step are very evident.

### Convergence to the Lowest-Order Mode

Initially we do not know the eigenmodes $$\tilde{E}_{nm}(x,y)$$ of the resonator and we thus have no way of separating an arbitrary starting function $$\tilde{E}^{(0)}(x,y)$$ into eigenmodes. After a sufficient number of bounces, however, the wavefunction $$\tilde{E}^{(k)}(x,y)$$ in the computer should converge in form to the lowest-order eigenmode $$\tilde{E}_{00}(x,y)$$, for the reasons given in the preceding; and the eigenvalue for this mode should be given from the computer iterations by

$\tag{15}\tilde{\gamma}_{00}=\lim_{k\rightarrow\infty}\frac{\tilde{E}^{(k+1)}(x,y)}{\tilde{E}^{(k)}(x,y)}$

The field distribution $$\tilde{E}^{(k)}(x,y)$$ in the computer of course decreases steadily in overall amplitude with each successive bounce because of diffraction losses, but this is handled in the calculations simply by rescaling the overall signal level back upward by a constant amount after each iteration, or each few iterations.

Figure 14.6(c) then shows the steady-state, unchanging amplitude and phase pattern that the resonator mode in this particular example settles into after $$k\approx$$ 250 to 300 round trips. (This is a comparatively low-loss resonator, and the higher-order modes only die out quite slowly.)

The finite value of the steady-state mode pattern just at the mirror edge indicates that the mode does still have some diffraction losses past the edges of the end mirror.

The smoothed shape and tapered profile of the steady-state pattern also indicate, however, that higher spatial frequency components are rapidly lost past the edges of the resonator, and that this lowest-loss transverse mode pattern has a very typical ability to "pull in its edges" and minimize its diffraction losses due to diffraction spreading.

The exact shape of this mode pattern changes, and the mode losses decrease or increase, as the width of the planar end mirrors is changed, or the cavity length L is changed.

The primary conclusion from this numerical simulation or "computer experiment" is that even a simple optical resonator consisting only of two flat end mirrors, with completely open sides, still has a lowest-order transverse mode which will reproduce itself on repeated round trips.

This mode is in fact almost like a half-cosine in appearance, rather similar to more familiar waveguide cavity modes. The effects of the finite mirror edges and diffraction losses do show up, however, in the small diffraction ripples on the mode wavefront and on the mode amplitude pattern, and in the finite diffraction losses characteristic of the mode.

Note that this same analysis also means that a lens waveguide consisting simply of a series of slit apertures, without any lenses (see Figure 14.7), will also propagate exactly the same transverse mode pattern as a traveling mode pattern in the lensguide.

The diffraction effects of the aperture edges in this lensguide are exactly equivalent to the cutting off of the transverse mode pattern on each bounce by the finite mirror width in the Fox and Li resonator calculation.

Fox and Li, and many others since them, have done many more such calculations for resonators with curved mirrors, mirrors of more complex shape, mirrors with central holes, planar but tilted mirrors, and so forth.

In every situation a lowest-order mode with some sort of self-reproducing mode pattern and associated eigenvalue has resulted from this sort of calculation.

### Finding the Higher-Order Transverse Modes

More sophisticated numerical procedures then allow us to obtain higherorder eigenmodes from the Fox and Li iterative procedure as well.

For example, even in the simple Fox and Li procedure if we reach a stage in the iterative calculation where only two dominant modes are left, then there will be only two terms left in Equation 14.14.

The field amplitude at any fixed point on the mirror surface will then display a periodic beating between the two modes (see Figure 14.8).

This periodic interference occurs because the fields of the two modes combine with different phases on successive round trips, since the different eigenmodes have eigenvalues $$\tilde{\gamma}_{nm}$$ with different phase angles $$\psi_{nm}$$.

The eigenvalue of the next-highest eigenmode can then be deduced from the rate and period with which this "mode beating" between the two modes dies out.

A more sophisticated procedure known as the Prony method is one among several numerical techniques that allow us to start with an initial distribution containing a mixture of many eigenmodes, and after $$N$$ iterations to deduce the $$N$$ lowest-loss eigenvalues $$\tilde{\gamma}_{nm}$$ and eigenmodes $$\tilde{E}_{nm}(x,y)$$.

### Resonator Eigenfrequencies

Having found the transverse eigenmodes $$\tilde{E}_{nm}(x, y)$$ and eigenvalues $$\tilde{\gamma}_{nm}$$ of a given cavity or lensguide, we can also find the exact resonant frequencies, or axial-plus-transverse mode resonances, of the cavity in the following manner.

The exact resonance frequency of a given axial-plus-transverse mode in a laser cavity is determined by the resonance condition that the round-trip phase shift in the cavity must be an integer multiple of $$2\pi$$.

Suppose a real regenerative laser cavity has round-trip phase shift due to the laser medium given by $$\exp[-j\Delta\beta_mp_m]$$, and suppose we consider a particular transverse mode $$\tilde{E}_{nm}(x, y)$$ with a complex eigenvalue $$\tilde{\gamma}_{nm}\equiv|\tilde{\gamma}_{nm}|\exp[j\psi_{nm}]$$.

Regenerative feedback or laser oscillation for this particular transverse mode can occur only at frequencies for which the total round-trip phase shift is given by

$\tag{16}\exp[-jkp-j\Delta\beta_mp_m+j\psi_{nm}]=\exp[-jq2\pi]$

where we use $$\psi_{nm}$$ for the phase angle of $$\tilde{\gamma}_{nm}$$. The axial phase shift term $$e^{-jkp}$$ has been brought back into this expression, with $$k=\omega/c$$, and $$q$$ being an axial-mode integer.

Equating the phase angles on opposite sides of Equation 14.16 then gives

$\tag{17}\frac{\omega{p}}{c}+\Delta\beta_mp_m-\psi_{nm}=q\times2\pi$

The resonance frequencies $$\omega_{qnm}$$ of the axial-plus-transverse modes in this cavity are thus given by

$\tag{18}\omega=\omega_{qnm}\equiv\frac{2\pi{c}}{p}\left[q+\frac{\psi_{nm}}{2\pi}-\frac{\Delta\beta_mp_m}{2\pi}\right]$

Since $$q$$ is normally a very large integer ($$\approx{p/\lambda}$$), the transverse mode factor $$\psi_{nm}/2\pi$$ represents only a small correction to the plane-wave resonance frequency $$\omega_q\equiv{q}\times2\pi(c/p)$$.

This correction will be in general slightly different for each specific $$nm$$-th transverse mode. As we already know, the $$\Delta\beta_mp_m/2\pi$$ factor is an additional (and usually still smaller) atomic frequency pulling effect caused by the reactive or $$\chi'$$ part of the laser susceptibility.

### Transverse Mode "Beats"

Different transverse modes $$\tilde{E}_{nm}$$ thus lead to slightly different resonance frequencies $$\omega_{qnm}$$, with small relative frequency shifts which are determined by the phase angles of the transverse mode eigenvalues $$\tilde{\gamma}_{nm}$$ in real laser cavities.

Figure 14.9 illustrates how each axial mode frequency $$\omega_q$$ in the plane-wave approximation splits into a set of different axial-plus-transverse mode resonances $$\omega_{qnm}$$ in a typical resonator. (The actual magnitude of this splitting can be quite different for different types of real resonators.)

Heterodyne interference effects or "beats" at the difference frequencies between these transverse modes can often be detected by examining the output signal of a laser oscillator with any kind of standard square-law photodetector (i.e., any detector whose response is proportional to the optical intensity, or to the optical $$E$$ field squared, such as a photomultiplier tube or solid-state photodiode).

These "transverse mode beats" can provide a sensitive test for the presence of multiple transverse modes. In addition, since the inter-mode beat frequency can be easily measured, and since this frequency depends in a sensitive fashion on the phase angles of the mode eigenvalues, the agreement between measured and theoretical frequencies can provide a test for the validity of the transverse mode calculations.

### The Buildup of Laser Oscillation

The Fox and Li numerical approach simulates mathematically what actually happens physically in a real optical resonator with an initially injected field distribution and no gain.

Each transverse mode component circulates around and dies out at a rate determined by its eigenvalue. With a slight change in viewpoint, this same picture also describes what happens in a real laser oscillator at turn-on.

When a laser oscillator is turned on from a cold start, an initial mode distribution $$\tilde{E}^{(0)}(x,y)$$ (determined in most real situations by noise or spontaneous emission in the laser cavity) begins to circulate repeatedly around the cavity, and to grow in amplitude if the cavity is above threshold.

If the gain medium is spatially uniform so that all modes $$\tilde{E}_{nm}(x,y)$$ see the same gain, then the lowest-loss or $$00$$ mode grows the fastest, since it has the highest value of net gain minus loss.

In simple situations the dominant or $$00$$ mode will eventually grow to a level where it saturates the gain down until the gain for this particular mode just equals the loss.

This mode will then stay at a steady-state level, whereas all the higher-loss transverse modes die out, in the same way as sketched earlier. This initial growth and eventual stabilization process is illustrated in Figure 14.10.

### Transverse Mode Competition Effects

There are many factors that complicate this picture in real lasers. In a more realistic picture, for example, the $$n=0$$, $$m=0$$ mode may still build up most rapidly; but this mode will then saturate the gain medium strongly only in those regions of the transverse plane where the field amplitude $$|\tilde{E}_{00}(x,y)|^2$$ is large.

This may leave unsaturated gain at other transverse positions $$x$$, $$y$$, and this may allow other higher-order transverse modes to oscillate simultaneously.

In very high-gain but short-pulse lasers the entire laser pulse may be over in so few round trips that the $$00$$ mode has insufficient time to grow to where it dominates over higher-order modes. The transverse mode selection may thus be less effective in a $$Q$$-switched laser than in a cw steady-state laser.

Even in a cw laser, the differences in loss and in growth rate for different eigenmodes $$\tilde{E}_{nm}$$ may be very small, so that the competition between modes is very weak.

The gain may not be uniform across the laser, so that different eigenmodes $$\tilde{E}_{nm}(x,y)$$ actually see different gains. If the atomic linewidth is particularly narrow, and the laser is tuned so that the lowest-order $$q00$$ resonant frequency is off line center, whereas some other higher-order $$qnm$$ transverse mode is tuned closer to line center, the laser may then also see higher net gain in the higher-order mode even though it has higher diffraction losses.

Interference effects between the fields of different transverse modes may also modulate the gain differently at different transverse positions and thus cross-couple the different transverse modes.

In all these different situations, several transverse modes may then oscillate simultaneously, or the laser may jump back and forth between transverse modes

### Single Transverse Mode Operation

All in all, it is often a considerable struggle to force a large or high-gain laser oscillator to oscillate only in a single lowest-loss transverse mode.

One of the main considerations in the design of a practical laser resonator is to have simultaneously both minimum unwanted loss for the lowest-order transverse eigenmode, and also high mode discrimination—that is, a large increase in diffraction losses—for all the higher-order transverse modes.

This is often accomplished by putting an adjustable aperture inside the laser cavity and reducing its size until it attenuates and if possible kills the higher-order modes, but still has negligible effect on the desired lowest-order modes. The unstable resonator provides another and somewhat different method for accomplishing the same goal.

Despite these complexities, which we will discuss in more detail in later sections, there are many lasers which do operate in a single lowest-order transverse mode according to the description presented above.

Even in more complex situations the transverse eigenmode picture generally provides a solid and useful basis for describing the more complex multimode and coupled-mode phenomena that may occur in real lasers.

### Conclusions

Evaluating the round-trip wave propagation in an optical resonator, using the appropriate round-trip kernel or mathematical transformation, is obviously the primary step in evaluating and understanding the transverse modes, their losses, and their resonant frequencies, in any real laser resonator.

In the following two tutorials we introduce two primary tools for accomplishing this: ray matrix methods for treating ray propagation without diffraction, and paraxial wave optics for treating wave propagation including diffraction in most real laser beams and cavities.

The next tutorial discusses about timing synchronization in coherent optical transmission systems.