Laser Mirrors and Regenerative Feedback

This is a continuation from the previous tutorial - chiral fibers

 

1. Laser Mirrors and Beam Splitters

Laser mirrors and beam splitters have certain fundamental properties that are important to understand. Before discussing the use of mirrors in laser cavities, let us therefore review the more important of these properties.

 

Single Dielectric Interface

 

 

The simplest example of a partial mirror or beam splitter is the interface between two dielectric media, as shown in Figure 11.1. Suppose we write the normalized fields for the incident and reflected waves on the two sides of this interface (labeled by subscripts \(i=1,2\)) in the form

\[\tag{1}\mathcal{E}_i(z,t)=\text{Re}\left\{\tilde{a}_i\exp[j(\omega{t}\mp\beta_iz)]+\tilde{b}_i\exp[j(\omega{t}\pm\beta_iz)]\right\},\qquad{i}=1,2\]

where \(\beta_1\) is the propagation constant in the dielectric medium on the left side of the interface, \(\tilde{a}_1\) is the normalized wave amplitude of the incident wave and \(b_1\) is the normalized wave amplitude of the reflected wave on that side of the interface, and the same expressions with subscript \(i=2\) apply on the right-hand side of the interface. The field amplitudes \(\mathcal{E}_i\) are normalized so that \(|\mathcal{E}_i|^2\) gives the intensity or power flow in the medium on either side of the interface.

Note that the upper signs in front of the \(\beta_i\) terms in the exponents apply on the left-hand side of the interface, so that \(\tilde{a}_1\) represents the complex amplitude of the incident wave traveling to the right (toward \(+z\)), and \(\tilde{b}_1\) represents the reflected or left-traveling wave. The lower signs apply on the right-hand side of the interface, so that \(\tilde{a}_2\) is again the incident but now left-traveling wave and \(\tilde{b}_2\) the right-traveling wave.

The amplitude reflection and transmission properties for a simple dielectric interface at normal incidence can then be written as

\[\tag{2}\begin{align}\tilde{b}_1&=r\tilde{a}_1+t\tilde{a}_2\\\tilde{b}_2&=t\tilde{a}_1-r\tilde{a}_2\end{align}\]

or in matrix notation

\[\tag{3}\begin{bmatrix}\tilde{b}_1\\\tilde{b}_2\end{bmatrix}=\begin{bmatrix}r&t\\t&-r\end{bmatrix}\times\begin{bmatrix}\tilde{a}_1\\\tilde{a}_2\end{bmatrix}\]

where the reflection and transmission coefficients \(r\) and \(t\) are given for this particular interface by

\[\tag{4}r=\frac{n_1-n_2}{n_1+n_2}\quad\text{and}\quad{t}=\frac{2\sqrt{n_1n_2}}{n_1+n_2}\]

and the lossless nature of the interface is expressed by \(r^2+t^2=1\).

Note that in writing these relations we are implicitly locating the \(z=0\) plane, or the reference plane for the fields given in Equation 11.1, exactly at the dielectric interface. The \(\tilde{a}_i\) and \(\tilde{b}_i\) coefficients thus express the complex wave amplitudes exactly at the interface between the two dielectrics.

For this particular interface, and this particular choice of reference plane, the coefficients \(r\) and \(t\) are purely real numbers. The reflection coefficients have opposite signs, however, depending on the direction from which the wave approaches the interface.

This change of sign can be understood in physical terms because a medium with a very high index of refraction acts essentially like a metallic surface or like a very large shunt capacitance across a transmission line.

Going from a low-index to a high-index medium (\(n_2\gt{n_1}\)) is thus like the reflection from the end of a short-circuited transmission line, with a 180° phase shift on reflection or \(r\lt0\), whereas reflection from a low index medium acts like an open-circuited transmission line, with a reflection coefficient \(r\gt0\).

 

Thin Dielectric Slab

As a more realistic model for a real laser mirror, we might consider next the reflection and transmission properties of a thin lossless dielectric slab of thickness \(L\) and index \(n\) as shown in Figure 11.2, assuming, for simplicity, air or vacuum on both sides of this slab.

 

 

We will again write the fields on both sides of the slab as in Equation 11.1, using \(\tilde{a}_i\) and \(\tilde{b}_i\) coefficients for the incident and reflected waves, except that we will now measure the incident waves and the reflected waves using separate \(z=0\) planes that are located at the outer surfaces of the slab on each side, so that the reference planes on opposite sides of the slab are offset by the thickness of the slab.

With a little calculation we can find that the general relationship between incident and reflected waves in this case can be written in the general complex matrix form

\[\tag{5}\begin{bmatrix}\tilde{b}_1\\\tilde{b}_2\end{bmatrix}=\begin{bmatrix}\tilde{r}_{11}&\tilde{t}_{12}\\\tilde{t}_{21}&\tilde{r}_{22}\end{bmatrix}\times\begin{bmatrix}\tilde{a}_1\\\tilde{a}_2\end{bmatrix}\]

where the complex reflection coefficients are now given by

\[\tag{6}\tilde{r}_{11}=\tilde{r}_{22}=r_0\frac{1-e^{-j\theta}}{1-(r_0e^{-j\theta})^2}\]

and the complex transmission coefficients by

\[\tag{7}\tilde{t}_{12}=\tilde{t}_{21}=e^{-j\theta}\frac{1-r_0^2}{1-(r_0e^{-j\theta})^2}\]

In these expressions \(\theta=n\omega{L}/c_0\) is the optical thickness of the slab, and \(r_0=(1-n)/(1+n)\) is the single-surface reflection coefficient at either surface of the slab. The scattering matrix is now complex but symmetric, and the coefficients now obey the complex condition \(|\tilde{r}|^2+|\tilde{t}|^2=1\) representing zero losses in the slab.

 

Purely Real Reflectivity

As one particularly simple example of this type of mirror, we might adjust the optical thickness nL of the slab to be an odd number of quarter wavelengths, so that \(\theta=n\omega{L}/c_0\) is an odd integer multiple of \(\pi/2\). The reflection and transmission coefficients for the mirror then take on the particularly simple form

\[\tag{8}\begin{bmatrix}\tilde{b}_1\\\tilde{b}_2\end{bmatrix}=\begin{bmatrix}r&jt\\jt&r\end{bmatrix}\times\begin{bmatrix}\tilde{a}_1\\\tilde{a}_2\end{bmatrix}\]

where \(r\) and \(t\) are again purely real and subject to \(r^2+t^2=1\).

The reflection coefficients from the two sides of this slab are now symmetric and purely real. In writing down the reflection equations we no longer have to worry about whether we are approaching this mirror or beamsplitter from its "high index side" or its "low index side."

The transmission factors in this example, however, now have an additional factor of \(j\), or a phase shift of 90°, associated with them. This phase shift arises essentially from the fact that we measure the waves at two different reference planes, on opposite sides of the slab and separated by the (small) thickness of the slab.

Note that we can, at least in principle, always adjust the index n and the thickness \(L\) of such a slab separately to obtain any desired values of \(\theta\) and \(r_0\), and hence any desired values for the magnitudes of \(r\) or \(t\). This thin-slab model might therefore be a particularly simple and symmetric model for any real lossless dielectric mirror.

 

Scattering Matrix Formalism

The formalism we have been using here is obviously a scattering matrix formalism of the kind used in circuit theory or in describing waveguide or transmission-line junctions. From this viewpoint a partially transmitting mirror is simply a two-port network connected between two waveguides or transmission lines; and the matrices we have written represent simple examples of the 2 x 2 scattering matrix \(\pmb{S}\) that might describe such a two-port network in transmission-line theory.

 

 

This scattering matrix approach is of course not limited to two-port cases. Figure 11.3 shows, for example, a partially transmitting mirror or optical beam splitter used at off-normal incidence, so that it now has four incident and outgoing waves. This is equivalent to a general four-port network, and would require a 4 x 4 scattering matrix, which we would have to write in the form

\[\tag{9}\begin{bmatrix}\tilde{b}_1\\\tilde{b}_2\\\tilde{b}_3\\\tilde{b}_4\end{bmatrix}=\begin{bmatrix}\tilde{r}_{11}&\tilde{t}_{12}&\tilde{t}_{13}&\tilde{t}_{14}\\\tilde{t}_{21}&\tilde{r}_{22}&\tilde{t}_{23}&\tilde{t}_{24}\\\tilde{t}_{31}&\tilde{t}_{32}&\tilde{r}_{33}&\tilde{t}_{34}\\\tilde{t}_{41}&\tilde{t}_{42}&\tilde{t}_{43}&\tilde{r}_{44}\end{bmatrix}\times\begin{bmatrix}\tilde{a}_1\\\tilde{a}_2\\\tilde{a}_3\\\tilde{a}_4\end{bmatrix}\]

In more compact notation we can write Equations 11.2, 11.5 or 11.9 as

\[\tag{10}\pmb{b}=\pmb{S}\times\pmb{a}\]

The column vectors \(\pmb{a}\) and \(\pmb{b}\) then contain the incident and reflected wave amplitudes. The diagonal elements of the matrix \(\pmb{S}\) give the (generally) complex reflection coefficients \(\tilde{r}_{ii}\) looking into each port of the system, and the off-diagonal elements \(\tilde{t}_{ij}\) give the amplitude transmission coefficients from, say, the wave going into port \(j\) to the wave coming out of port \(i\).

 

Multilayer Dielectric Mirrors

We now need to discuss some subtleties concerning the effective reference planes of real laser mirrors, and how we should choose these reference planes in writing scattering matrices and carrying out laser analyses.

 

 

So far we have discussed two particularly simple examples of reflecting systems and their resulting scattering matrices. Figure 11.4, however, illustrates a more typical multilayer dielectric mirror of the type often used in lasers. Such a mirror may have as many as twenty or more quarter-wavelength-thick dielectric layers of alternating high and low index of refraction, evaporated onto a transparent substrate.

(The opposite surface of this substrate usually has a high-quality antireflection coating; and the two surfaces of the substrate are often wedged by a few degrees to avoid etalon effects from any residual back-surface reflection.)

In the simple examples we have discussed so far, it may have seemed physically obvious that the reflecting surfaces are located at the physical surfaces of the dielectrics. But where should the effective mirror surface, or the effective reflecting plane \(z=0\), be located in a thick multilayer mirror like Figure 11.4 that is several optical wavelengths thick?

 

Mirror Reference Planes

In fact, there really is no unique plane which we can (or need to) identify as the exact reflecting surface, or the unique reference plane, for such a multilayer mirror. The total reflection of the mirror, as seen from outside the coating layers on either side, builds up gradually through the series of layers, which can be several wavelengths thick overall.

The choice of where to locate the reference plane in this (or any other) mirror is entirely arbitrary. We can pick any reasonable reference plane within (or even outside) the multilayer mirror as the reference plane for defining the scattering matrix coefficients. In physical terms, picking such a reference plane simply means choosing the \(z=0\) origin for measuring the electric fields \(\mathcal{E}(z,t)\) well outside the mirror, assuming the fields are expanded as in the opening equation of this section.

Even with the single dielectric interface, it is not essential that the mirror reference surface be chosen right at the physical surface between the dielectrics— especially since we very seldom if ever can position any mirror or optical element with an absolute position accuracy of better than a few optical wavelengths.

So long as we are concerned only with the amplitudes and phases of the waves \(\mathcal{E}(z,t)\) at larger distances — say, more than a few wavelengths—away from the reflecting surface, we can choose the reference surface anywhere near the physically reflecting structure.

Shifting the choice of reference plane from one axial position to another location a distance \(\Delta{z}\) away then merely rotates the phase angles of the complex wave amplitudes \(\tilde{a}_i\) and \(\tilde{b}_i\) by phase shifts \(\exp(\pm{j}\beta_i\Delta{z})\), without changing their amplitudes. This rotation of the phase angles of \(\tilde{a}_i\) and \(\tilde{b}_i\) in turn merely attaches different phase angles to each of the scattering coefficients \(\tilde{S}_{ij}\).

 

 

We can in fact even choose different reference planes at which to measure the complex incoming and outgoing waves in each of the arms, as illustrated in Figure 11.5 (although why this would be a useful choice may be open to question). Regardless of the choice of reference plane(s) and the associated phase shifts, however, the scattering matrix elements still have certain fundamental properties, which we must next consider.

 

Hermitian Matrix Notation

To understand a bit more about the basic properties of mirrors and beam splitters and their scattering matrices, it will be useful to introduce some hermitian matrix notation.

We are using the notation \(\pmb{b}\), for example, to denote a column vector with complex elements \(\tilde{b}_1,\tilde{b}_2,\tilde{b}_3,\ldots\) running from the top down. We can then define the hermitian adjoint or hermitian conjugate to this vector, denoted by \(\pmb{b}^\dagger\), as the row vector with complex conjugate elements \(\tilde{b}_1^*,\tilde{b}_2^*,\tilde{b}_3^*,\ldots\) running horizontally. Hermitian conjugation converts a column vector into a row vector, or vice versa, and takes the complex conjugate of each individual element.

More generally, suppose we have an \(m\times{n}\) matrix \(\pmb{S}\) with complex elements \(\tilde{S}_{ij}\) as given above. Then, the hermitian adjoint or hermitian conjugate \(\pmb{S}^\dagger\) of this matrix will be an \(n\times{m}\) matrix with complex elements given by \((\pmb{S}^\dagger)_{ij}=(\pmb{S})_{ji}^*\). The hermitian adjoint or hermitian conjugate is a kind of matrix generalization of the complex conjugate of an ordinary quantity. To calculate the hermitian adjoint of a complex matrix or vector, interchange subscripts and take the complex conjugate. Just as with ordinary complex conjugation, applying this operation twice restores the original matrix or vector, i.e., \(\pmb{S}^{\dagger\dagger}\equiv\pmb{S}\).

 

Power Flow Into and Out of a Scattering Matrix System

Matrix notation can then be used to write the total power flowing into or out of a multiport scattering system in a particularly compressed form. We have assumed that the wave amplitudes in our examples are normalized, so that the time-averaged power flowing into or out of any single port can be written as \(|\tilde{a}_i|^2\) or as \(|\tilde{b}_i|^2\), respectively.

Assuming we pick the right units for \(\tilde{a}_i\) and \(\tilde{b}_i\), the total power flowing out of an optical element, or out of all the ports in an \(N\)-port network, can then be written in the simplified form

\[\tag{11}P_\text{out}=\sum_{j=1}^N\tilde{b}_j^*\tilde{b}_j=\begin{bmatrix}\tilde{b}_1^*,\tilde{b}_2^*,\tilde{b}_3^*,\ldots\end{bmatrix}\times\begin{bmatrix}\tilde{b}_1\\\tilde{b}_2\\\tilde{b}_3\\\ldots\end{bmatrix}=\pmb{b}^\dagger\times\pmb{b}\]

where the final term implies matrix multiplication of \(\pmb{b}^\dagger\) times \(\pmb{b}\), carried out, as in all hermitian adjoint formulas, using the usual rules for matrix multiplication.

By using this notation, plus the usual rules for matrix multiplication, we can then relate the total power flowing out of any scattering system—for example, any optical mirror or beam splitter—to the input waves and the scattering matrix in the form

\[\tag{12}\begin{align}P_\text{out}&=\pmb{b}^\dagger\pmb{b}=(\pmb{Sa})^\dagger(\pmb{Sa})\\&=(\pmb{a}^\dagger\pmb{S}^\dagger)(\pmb{Sa})=\pmb{a}^\dagger(\pmb{S}^\dagger\pmb{S})\pmb{a}\end{align}\]

In going from the third to fourth and fourth to fifth terms, we have made use of the basic rules that (i) matrix multiplication is associative, i.e., \(\pmb{A}(\pmb{BC})=(\pmb{AB})\pmb{C}\), and (ii) the hermitian adjoint of any product is the product of the individual adjoints taken in reverse order, i.e., \((\pmb{ABC})^\dagger\equiv\pmb{C}^\dagger\pmb{B}^\dagger\pmb{A}^\dagger\).

 

Scattering Matrices for Lossless Systems

But, if a mirror or other scattering element is to be lossless, then the output power in Equation 11.12 must equal the input power, which is just \(P_\text{in}=\pmb{a}^\dagger\pmb{a}\). The only way this can be true in general, for arbitrary input signals \(\pmb{a}\), is for the product \(\pmb{S}^\dagger\pmb{S}\) in Equation 11.12 to equal the identity matrix; i.e.,

\[\tag{13}\pmb{S}^\dagger\pmb{S}=\pmb{I}\qquad\text{or}\qquad\pmb{S}^\dagger\equiv\pmb{S}^{-1}\]

In this equation \(\pmb{I}\) is the identity matrix (unity elements on the diagonal and all other elements zero), and \(\pmb{S}^{-1}\) denotes the matrix inverse of the \(\pmb{S}\) matrix. In matrix terms, this says that the scattering matrix \(\pmb{S}\) for a lossless network must be a unitary matrix, since Equation 11.13 is the definition of unitarity.

 

Matrix Forms for Lossless and Reciprocal Twoport Networks

It is also a general theorem that the scattering coefficients of a reciprocal system must obey \(|\tilde{S}_{ij}|\equiv|\tilde{S}_{ji}|\). This result comes from the symmetrical behavior of Maxwell's equations if we reverse either the \(E\) or the \(H\) fields and also reverse the sign of the time \(t\). Most common optical elements are in fact reciprocal; only elements such as optical isolators containing Faraday rotators or similar elements containing a dc magnetic field can be nonreciprocal.

If all these constraints are applied to a lossless reciprocal two-port network, the result is the set of conditions

\[\tag{14}\begin{align}|\tilde{t}_{12}|=|\tilde{t}_{21}|,\qquad|\tilde{r}_{11}|=|\tilde{r}_{22}|\\|\tilde{r}_{11}|^2+|\tilde{t}_{21}|^2=|\tilde{r}_{22}|^2+|\tilde{t}_{12}|^2=1\\\tilde{r}_{11}\tilde{t}_{12}^*+\tilde{t}_{12}\tilde{r}_{22}^*=0\end{align}\]

These are general conditions that must be obeyed by the complex reflection and transmission coefficients of any lossless two-port mirror or beam splitter.

The purely real and the complex symmetric examples we derived earlier in this section, namely,

\[\tag{15}\pmb{S}=\begin{bmatrix}r&t\\t&-r\end{bmatrix}\qquad\text{and}\qquad\pmb{S}=\begin{bmatrix}r&jt\\jt&r\end{bmatrix}\]

with \(r\) and \(t\) real, are two of the possible ways in which the four conditions of Equation 11.14 can be satisfied for a two-port system. One the other hand, the real and symmetric matrix

\[\tag{16}\pmb{S}=\begin{bmatrix}r&t\\t&r\end{bmatrix}\]

is not an allowable scattering matrix for a lossless optical mirror or beam splitter.

The conditions of unitarity plus reciprocity will always lead to a set of relationships like Equations 11.14 between the coefficients \(\tilde{S}_{ij}\) of any lossless \(N\) x \(N\) scattering matrix. We can always rotate the complex phase angles of the different matrix elements for a given physical system by choosing different reference planes in the various input and output arms.

The magnitudes of the scattering coefficients of course will not change in this process, since the power transfer from any one arm to any other arm is not changed by a different choice of reference planes. No matter how the reference planes are chosen, however, certain phase relationships between the different coefficients must be maintained, at least for lossless systems.

The exact form of the scattering matrix \(\pmb{S}\) for a real mirror or beamsplitter thus depends on where we pick the reference planes; and there is in general no unique or preferred place to pick the reference plane in a real mirror.

For all future analyses of laser cavities and interferometers in this tutorial, however, we will arbitrarily choose the complex symmetric form \(\pmb{S}=[r,\;jt,\;jt,\;r]\), with \(r\) and \(t\) purely real, as the scattering matrix form to describe all mirrors and beam splitters.

This arbitrary choice will make no difference in any of the physical conclusions we reach about laser devices. It seems easier, however, to remember that transmission coefficients always have a factor of \(j\) associated with them than to remember which side of each mirror in a laser system is the \(+r\) and which is the \(-r\) side.

 

Polarization Effects and Transverse Mode Effects

In all the examples discussed so far, we have implicitly assumed a single sense of polarization in each of the input and output directions. Optical waves can, however, have two orthogonal senses of polarization for the wave in each direction.

These may be, for example, two orthogonal linear polarizations, or positive and negative circular polarization, or whatever. If two orthogonal polarizations are present, each is in essence a separately measurable wave, with a separate wave amplitude.

If both polarizations are considered separately, therefore, the total number of ports in the scattering matrix must be doubled; i.e., a system with \(N\) input and output directions will require a \(2N\) x \(2N\) scattering matrix.

In addition, if we go to more realistic optical beams (or fibers), in which there may be both a lowest-order transverse mode and various higher-order transverse modes, then in essence each such transverse mode is a separate port or beam; and the dimensionality of the scattering matrix must be expanded to include a separate port for each different transverse mode (with possible coupling between transverse modes inside the scattering system).

 

Further Discussion

All the analysis in this section may seem an overly complicated approach to the scattering properties of a simple mirror or beam splitter. If we failed to include the unitary properties of beam splitters when we analyze more complex configurations, such as Michelson interferometers or ring-laser cavities, however, it would be easy to invent cavities or optical devices that do not conserve energy, or have other useful properties. (It is by no means unknown for such inventions to be suggested, and even to appear in research proposals.)

The dielectric mirrors and beam splitters used in most laser applications are in fact almost perfectly lossless (though the partially transmitting metal films which were often used as output mirrors for early solid-state lasers were by contrast quite lossy). Higher-power lasers require nearly lossless mirrors if the mirrors are not to be destroyed by the power they absorb; and low-gain lasers need high reflectivity and low mirror losses for good efficiency. Even a lossy mirror can usually be described as a lossless mirror which obeys the preceding restrictions, sandwiched between two thin absorbing layers.

Finally, we might also emphasize that once we choose a specific reference plane in a multilayer mirror, the phase shifts associated with the scattering matrix for that mirror are fixed at any one frequency, but may have different values at different frequencies. The different phase shifts in reflecting from a mirror at two different optical wavelengths can be quite significant when we intercompare two optical frequency standards using interferometric methods, since the exact optical length of an interferometer cavity (between the reference planes of the two end mirrors) need not be the same for two different wavelengths.

Mirror phase shifts can also be significant in nonlinear optics experiments, such as double-pass harmonic-generation experiments. Suppose a fundamental wave passes through a phase-matched nonlinear crystal, generating secondharmonic radiation; and both the fundamental and the harmonic then reflect off a mirror and back through the crystal again. This is not necessarily equivalent to a nonlinear crystal twice as long, if the relative phases of the fundamental and the harmonic are shifted in bouncing off the mirror.

 

2. Interferometers and Resonant Optical Cavities

In this section, we will introduce some of the key ideas concerning the resonance behavior of passive optical cavities, without laser gain. We will use a plane-wave or transmission-line model for the resonators, and discuss both standing-wave and ring optical resonators on an equal footing. In later sections we will then introduce laser gain to produce regenerative amplification and, eventually, laser oscillation in such structures.

 

Fabry-Perot Interferometers and Etalons

A common optical element, widely used since long before the advent of lasers is the Fabry-Perot interferometer or Fabry-Perot etalon sketched in Figure 11.6. In its original form, a Fabry-Perot interferometer consisted of two closely spaced and highly reflecting mirrors, with mirror surfaces adjusted to be as flat and parallel to each other as possible.

 

 

An alternative but conceptually equivalent element is a solid etalon made from some very low-loss material such as fused quartz or sapphire, with its two faces polished flat and parallel and perhaps coated with a metal or dielectric mirror coating.

As we will see, such Fabry-Perot interferometers or etalons can have sharp resonances or transmission passbands at discrete optical frequencies. Fabry-Perot interferometers or etalons have thus long been used as narrowband optical filters for measuring the frequency spectrum of particularly narrow optical lines, especially lines whose width was below the resolving power of prism or grating spectrometers.

In their original form, such interferometers used only flat or planar reflecting surfaces, and the spacings between the mirrors were usually smaller than, or at most on the same scale as, the transverse diameters of the mirrors.

Moreover, it was usual to illuminate such an interferometer with a converging or diverging beam having a spread of angular directions, and then look at the "Fabry-Perot rings" transmitted through the interferometer in certain discrete angular directions.

Interferometers used in this manner were generally analyzed using an infinite plane-wave model, with the plane waves assumed to be arriving either at normal incidence or at some specified angle to the normal, as in Figure 11.6.

The standard formulas in optics texts, as a result, consider the resonant frequencies and the transmission properties of Fabry-Perot etalons as a function of the mirror spacing, the optical wavelength or frequency, and the angle of incidence. The transverse width or shape of the two mirrors is generally not taken into account, and transverse field variations are neglected.

 

Optical Resonators

As the fundamental ideas of laser devices began to emerge, however, researchers began to consider the properties of interferometers formed by setting up rather small mirrors, spaced by distances large compared to the mirror sizes, as in Figure 11.7.

 

 

The waves in such long, narrow optical cavities must travel at very small angles to the optical axis of the cavity, or else the waves will very rapidly "walk off" past the edges of the mirrors; hence the off-axis angular properties of such structures are of little interest.

It was soon realized, in fact, that such structures are better thought of as optical resonators or optical cavities, with properties related as much to microwave waveguide resonators as to optical interferometers. The ideas of transverse as well as longitudinal modes in such structures, and of using curved as well as planar mirror surfaces, then began to be developed.

 

Modes in Planar and Curved-Mirror Cavities

Figure 11.8(a) shows, for example, a typical optical cavity formed by two partially transmitting mirrors set up facing each other, such as might be used in a regenerative laser amplifier or oscillator, or in a modern optical interferometer, along with two lenses being used to focus a collimated external optical beam into and out of this cavity. The one slightly unrealistic aspect of this drawing is that real laser cavities are often even longer and relatively more slender than shown here.

 

 

Most modern optical resonators and laser cavities are also designed using mirrors which are slightly curved rather than planar, as illustrated in Figures 11.7 and 11.8.

The use of such curved mirrors generally leads (as we will study in much more detail in later chapters) to the existence of very well-defined and well-behaved, low-loss transverse-mode patterns in such cavities. (By loss we mean here the leakage or diffraction losses caused by the loss of energy out the open sides of the cavity or past the edges of the finite-diameter mirrors.)

The transverse-mode patterns in many, though not all, curved-mirror optical resonators take the form of quite smooth and regular transverse patterns that resemble Hermite-gaussian or Laguerre-gaussian cross sections, and that depend to first order only on the curvature and spacing, and not on the transverse size, of the end mirrors.

Optical resonators formed by finite flat or planar mirrors have definite transverse modes also. These modes are generally more irregular than in curved-mirror cavities, and not gaussian in profile; they also typically have somewhat larger diffraction losses or power leakage.

The details of the transverse mode profiles in planar-mirror cavities also depend rather more critically on the exact size and transverse shape of the mirrors (e.g., circular, square, or whatever).

There also exist so-called unstable optical resonators, whose mirrors have a negative or divergent curvature so that the optical waves tend to be spread outward as they bounce back and forth between the end mirrors.

These unstable resonators still have definite transverse mode patterns, and in fact are very useful for high-gain lasers, though their mode properties are more complicated and their diffraction losses much higher than in planar or convergent-mirror resonators.

The basic idea that it is possible set up two aligned mirrors to create a resonant optical cavity with clearcut resonant modes may seem straightforward and obvious now, but was in fact one of the key ideas in the development of the laser.

The antecedents to this idea were the passive resonant etalons and interferometers used in classical optics; and there are many useful devices in optics today which involve passive optical cavities, or resonant mirror structures without optical gain.

 

Ring Cavities and Standing-Wave Cavities

The majority of early laser cavities, as well as passive resonant interferometers and etalons, employed just two mirrors set up facing each other to form a resonant structure, as in Figure 11.8(a).

Such a cavity is often referred to as a standing-wave cavity, since the two waves traveling in the forward and reverse directions in such a cavity form an optical standing wave. In such a standing-wave system, the signal \(E\) and \(H\) fields will have periodic spatial variations along the axis, with a period equal to one-half the optical wavelength.

(This is strictly true, of course, only if the signal inside the cavity is at a single frequency. If multiple frequencies are present, the standing waves associated with different frequency components will have different periods and spatial locations, and the summation of these will tend to wash out some of the standing-wave character.)

In more recent years, however, many laser cavities, as well as passive optical interferometers, have been designed as ring resonators, such as that in Figure 11.8(b).

Traveling-wave or ring cavities are not really different in principle from linear or standing-wave cavities, since the round-trip optical path in going once down a standing-wave cavity and back is essentially equivalent to going once around a ring cavity of the same overall path length.

Ring resonators do have the special property, however, of having separate and independent resonances in the two opposite directions around the ring. Despite their slightly greater complexity, ring cavities offer several practical advantages in different applications, and are being increasingly used in practical devices.

One of these major advantages is that when a ring resonator is driven by an external signal, the cavity is excited with signals going in only one direction around the ring, and there is no reflection directly back into the external signal source.

This can be important because many laser devices do not function well when looking directly into a back reflection. Ring laser oscillators can also, with proper design, be made to oscillate in one direction only, so that there is no standing-wave character to the fields inside the cavity; and this can give advantages in power output and in mode stability.

 

Mode-Matching Optics

To excite any such optical cavity in just one of its transverse modes, it is necessary to shape and focus the input beam using lenses and other so-called mode-matching optics in order to couple properly into the desired transverse mode of the cavity.

The desired mode is usually the lowest-order transverse mode of the cavity, since this is (by definition) the transverse mode with the most highly confined transverse field pattern and the lowest leakage or diffraction losses.

If the input beam is not properly aligned and mode-matched to the transverse pattern of the lowest-order mode, the input wave will excite some mixture of lowest-order and higher-order transverse modes in the cavity.

Since these higher-order transverse modes usually have slightly different resonance frequencies, tuning the input signal may excite a number of separate and frequency-shifted resonances in different transverse modes as the frequency is varied; but since the higher-order modes often have larger diffraction losses and thus lower Q values, the cavity response in the higher-order modes is often weaker than in the lowest-order transverse mode.

 

Uniform Plane-Wave (Transmission-Line) Approximation

The transverse field patterns inside most practical laser resonators and interferometer cavities, even when excited in a single transverse mode, are still very close to ideal plane waves.

The fields propagate along the axial direction of the resonator essentially like uniform plane waves, with only minor or second-order effects due to the finite transverse width and transverse mode profile of the fields.

We will therefore disregard all these transverse-mode complications and analyze the resonant properties of the signals inside and outside such cavities or interferometers using only a simple on-axis plane-wave approach.

That is, we will consider the variations of the fields only in the axial or \(z\) direction, and ignore any variations in the transverse or \(x\) and \(y\) directions. This is equivalent to using essentially a transmission-line model to describe all the cavity resonance effects.

 

3. Resonance Properties of Passive Optical Cavities

Let us develop therefore an elementary analysis for the resonance properties of either a linear (standing-wave) or a ring (traveling-wave) optical cavity, using the plane-wave or transmission-line analytical models shown in Figure 11.9.

 

 

Basic Cavity Analysis: The Circulating Intensity

To do this, we will suppose that a steady-state sinusoidal optical signal is incident on one of the cavity mirrors, call it mirror \(M_1\), using the notations \(\tilde{E}_\text{inc}\) and \(\tilde{E}_\text{refl}\) to denote the incident and reflected complex signal amplitudes, respectively, as measured just outside this mirror. We will also use \(\tilde{E}_\text{circ}\) to denote the circulating signal amplitude inside the cavity, as measured just inside the same mirror.

The circulating signal just inside the input mirror then consists of the vector sum of that portion of the incident signal which is transmitted through the input mirror, and thus has the value \(jt_1\tilde{E}_\text{inc}\); plus a contribution representing the circulating signal \(\tilde{E}_\text{circ}\) which left this same point one round-trip time earlier, traveled once around the cavity, and has returned to the same point after passing through all the elements (twice, in the standing-wave model) and bouncing off mirror \(M_1\) as well as all the other mirrors in the cavity. The total circulating signal just inside mirror \(M_1\) can thus be written in the form

\[\tag{17}\tilde{E}_\text{circ}=jt_1\tilde{E}_\text{inc}+\tilde{g}_\text{rt}(\omega)\tilde{E}_\text{circ}\]

where \(\tilde{g}_\text{rt}(\omega)\) ls the net complex round-trip gain for a wave making one complete transit around the interior of the resonant cavity, whether it be a standing-wave or a ring-type cavity. Equation 11.17 is the key equation for calculating the resonance properties of any resonant optical cavity, optical interferometer, or oscillating laser system.

 

Passive Lossy Optical Cavities

In analyzing optical cavities we will consistently use \(L\) for the one-way length of a standing-wave cavity, and \(p\) (\(\equiv2L\)) for the perimeter or the roundtrip path length in either the ring or the standing-wave cavities. By using this notation, and by always considering round-trip gains, losses, and phase shifts, we can develop a unified analysis that treats standing-wave or ring cavities on an equal footing.

Suppose then that the round-trip optical path in either type of cavity contains material with voltage absorption coefficient \(\alpha_0\) (or possibly other kinds of internal losses), so that the attenuation of the signal amplitude or signal voltage in one round trip is \(\exp(-\alpha_0p)=\exp(-2\alpha_0L)\), and the round-trip power reduction is \(\exp(-2\alpha_0p)=\exp(-4\alpha_0L)\).

We are, of course, considering here sinusoidal optical signals with frequency \(\omega\) and propagation constant \(\beta=\beta(\omega)=\omega/c\), where \(c\) is the velocity of light in the material inside the cavity. Hence there is also a phase shift or propagation factor \(\exp(-j\omega{p}/c)\) associated with the round trip. (Let's leave out any atomic phase shifts \(\Delta\beta_m\) for the moment.)

The circulating signal after one complete round trip in either type of cavity will then return to the reference plane just inside mirror \(M_1\) with a net roundtrip transmission factor, or complex round-trip gain, which is given for a passive lossy cavity by

\[\tag{18}\tilde{g}_\text{rt}(\omega)\equiv{r_1}r_2(r_3\ldots)\times\exp[-\alpha_0p-j\omega{p}/c]\]

In writing this expression we put the \((r_3\ldots)\) factor inside brackets because there may or may not be a third or fourth mirror in the cavity, depending on whether we are considering a simple two-mirror standing-wave resonator or some kind of multimirror ring (or folded linear) cavity. We refer to \(\tilde{g}_\text{rt}\) as the "complex round-trip gain" inside the cavity, even though of course in any passive optical cavity (or even in any laser cavity below oscillation threshold) the magnitude of this round-trip gain will be less than unity, i.e., \(|\tilde{g}_\text{rt}|\lt1\).

We can then write Equation 11.17 as 

\[\tag{19}\tilde{E}_\text{circ}=jt_1\tilde{E}_\text{inc}+r_1r_2(r_3\ldots)\exp[-\alpha_0p-j\omega{p}/c]\tilde{E}_\text{circ}\]

This expression then applies equally well to either a ring or a standing-wave cavity, if we simply replace \(p\) by \(2L\) for the standing wave.

 

Cavity Resonances

This derivation says we can relate the circulating signal inside the cavity to the incident signal outside the cavity by

\[\tag{20}\frac{\tilde{E}_\text{circ}}{\tilde{E}_\text{inc}}=\frac{jt_1}{1-\tilde{g}_\text{rt}(\omega)}=\frac{jt_1}{1-r_1r_2(r_3\ldots)\exp[-\alpha_0p-j\omega{p/c}]}\]

What does this equation tell us? To help answer this question, Figure 11.10(a) shows several examples of how the circulating intensity \(I_\text{circ}\equiv|\tilde{E}_\text{circ}|^2\) inside such an optical resonator varies with frequency \(\omega\) or round-trip phase shift \(\omega{p/c}\), assuming unit incident intensity, a round-trip internal power loss of \(2\alpha_0p=2\%\), and symmetric mirror reflectivities \(R_1=R_2=R\) which vary from \(R\) = 70% to \(R\) = 98%.

It is obvious from these plots, as well as from Equation 11.20, that the signal inside the optical resonator exhibits a strong resonance behavior each time the round-trip phase shift \(\omega{p/c}\) equals an integer multiple of \(2\pi\), i.e., each time \(\omega=\omega_q\equiv{q}\times2\pi\times(c/p)\), with \(q\) being an integer.

In fact, the circulating intensity inside the cavity at these resonances becomes many times larger than the intensity incident on the cavity from outside. As we will discuss in more detail in the following sections, these resonant frequencies are known as cavity axial modes, and the frequency interval between resonances is known as the axial mode spacing or the free spectral range of the cavity.

 

 

Rotating Vector Interpretation

The resonance behavior that is evident in these plots can perhaps be most easily understood from the following graphical analysis. The denominator in the ratio of circulating to incident signal amplitudes in Equation 11.20 is given by the complex factor \(1-\tilde{g}_\text{rt}(\omega)\equiv1-r_1r_2(r_3\ldots)\exp[-\alpha_0p-j\omega{p/c}]\).

The quantity \(\tilde{g}_\text{rt}(\omega)\) is a complex vector with a magnitude that is less—but perhaps not much less— than unity. This complex gain has a phase angle \(\omega{p/c}\) such that \(\tilde{g}_\text{rt}(\omega)\) rotates through one complete revolution in the complex plane every time \(\omega{p/c}\) increases by \(2\pi\). Since the cavity perimeter \(p\) is many optical wavelengths in length, the rotation of \(\omega{p/c}\) through many complete cycles with increasing frequency is quite rapid.

Suppose we plot this denominator in a complex plane. The complex vector representing \(\tilde{g}_\text{rt}(\omega)\) then rotates about the point \(1+j0\) as shown in Figure 11.11.

Every time the tip of the rotating \(1-\tilde{g}_\text{rt}(\omega)\) vector sweeps close to the origin in this sketch, the denominator \(1-\tilde{g}_\text{rt}(\omega)\) of the \(\tilde{E}_\text{circ}/\tilde{E}_\text{inc}\) ratio becomes very small, and the value of the circulating field becomes correspondingly large. This occurs, of course, every time \(\omega{p/c}\) passes through another integer multiple of \(2\pi\).

 

Circulating Intensity Magnification

Let us examine how large the circulating signal inside the cavity can become at one of these peaks. Consider as a simple example a symmetric linear cavity with equal end-mirror reflectivities \(r_1=r_2=r\), and assume negligible internal losses, or \(\alpha_0p\approx0\). The peak value of the circulating field at resonance is then given by

\[\tag{21}\left.\frac{\tilde{E}_\text{circ}}{\tilde{E}_\text{inc}}\right|_{\omega=\omega_q}=\frac{jt}{1-r_1r_2e^{-\alpha_0p}}\approx\frac{jt}{1-r^2}=\frac{j}{t}\]

where we have used \(t_1=t_2=\sqrt{1-r^2}\) for lossless mirrors. (Note that the 90° phase shift between the incident and circulating fields is inherent in our choice of matrix representation for the partially transmitting mirror.)

The ratio of circulating intensity to incident intensity for a symmetric cavity with negligible internal losses is thus given by

\[\tag{22}\left.\frac{I_\text{circ}}{I_\text{inc}}\right|_{\omega=\omega_q}\approx\left|\frac{1}{t}\right|^2=\frac{1}{T}\]

where \(T\equiv{t^2}\) is the power transmission of either end mirror. If we assume, for example, end mirrors which are 99% reflecting and 1% transmitting, so that \(T\) = 1% = 0.01 (note that mirrors are usually characterized by their power reflection and transmission values), then this gives

\[\tag{23}I_\text{circ}\approx100\times{I_\text{inc}}\quad\text{for}\quad\begin{cases}R_1=R_2=0.99\\\alpha_0p\ll0.01\end{cases}\]

In other words, 1 watt of laser power incident on this cavity from outside will build up a circulating power of \(\approx\) 100 watts traveling in each direction inside the laser cavity, as illustrated in Figure 11.12.

 

 

The circulating power inside a passive cavity resonator can thus be much larger than the power incident on the cavity end mirror from outside. There is, of course, no way that this magnified circulating power can be usefully extracted (at least not continuously), since energy conservation still must be obeyed!

This form of power magnification can be used, however, in testing the damage thresholds of low-loss optical elements placed inside such a cavity. This circulating stored energy can also be switched out of the cavity on a transient basis, using a fast switch, to give a short pulse of energy at the magnified intensity level. The latter technique is sometimes referred to as "cavity dumping".

The intensity magnification in this symmetric and lossless example has a maximum value of \(1/t^2=1/(1-R)\), where \(R\) is the mirror reflectivity at each end.

Finite internal losses or increased mirror transmission will in general reduce this resonance enhancement, the circulating intensity at resonance being given more generally by

\[\tag{24}\left.\frac{I_\text{circ}}{I_\text{inc}}\right|_{\omega=\omega_q}=\frac{t_1t_2}{[1-r_1r_2(r_3\ldots)e^{-\alpha_0p}]^2}\]

If, for example, \(r_1r_2=0.99\) (and \(r_3\equiv1\)), so that the round-trip power loss through the end mirrors is 1%, then giving the internal losses the same value of 1% by making \(\alpha_0p\) = 0.01 will cut the circulating field amplitude in half, and decrease the circulating intensity by four times, or a reduction of approximately 6 dB.

 

Transmitted Cavity Fields

When the 100 watts of power circulating inside the cavity in Figure 11.12 impinge on the 1% transmitting output mirror, this means that a net transmitted power of 1 watt—equal to the incident signal—must be transmitted out through the output mirror at the opposite end of the cavity, as shown in Figure 11.12.

In other words, this particular cavity, at resonance, has a resonant transmission from input to output of essentially unity. As the frequency \(\omega\) is tuned off resonance, however, both the circulating and the transmitted signal intensities will drop rapidly, as illustrated in Figure 11.10.

To develop a more general formula for the transmitted field \(\tilde{E}_\text{trans}\) coming out the other end or through the other mirror \(M_2\), in either the ring or the linear example, we can suppose that a portion \(p_1\) of the cavity path is in the leg between mirrors \(M_1\) and \(M_2\) (\(p_1\equiv{L}\) in the linear example).

The transmitted signal intensity coming out through mirror \(M_2\) will then be given by

\[\tag{25}\tilde{E}_\text{trans}=jt_2\exp[-\alpha_0p_1-j\omega{p_1/c}]\times\tilde{E}_\text{circ}\]

Hence the net transmission through the cavity or interferometer, from input to output, is given by

\[\tag{26}\frac{\tilde{E}_\text{trans}}{\tilde{E}_\text{inc}}=\frac{-t_1t_2\exp[-\alpha_0p_1-j\omega{p_1/c}]}{1-r_1r_2(r_3\ldots)\exp[-\alpha_0p-j\omega{p/c}]}=\frac{-t_1t_2\exp[-\alpha_0p_1-j\omega{p_1/c}]}{1-\tilde{g}_\text{rt}(\omega)}\]

for the ring example, or by the essentially equivalent formula

\[\tag{27}\frac{\tilde{E}_\text{trans}}{\tilde{E}_\text{inc}}=\frac{-t_1t_2\exp[-\alpha_0L-j\omega{L/c}]}{1-r_1r_2\exp[-2\alpha_0L-2j\omega{L/c}]}=-\frac{t_1t_2}{\sqrt{r_1r_2}}\frac{\sqrt{\tilde{g}_\text{rt}(\omega)}}{1-\tilde{g}_\text{rt}(\omega)}\]

for the standing-wave cavity. The minus sign in front of either expression is a basically irrelevant additional phase shift of \(\pi\) that arises because we insist on making a certain choice of reference planes at each mirror, as discussed in the preceding section.

Both Figure 11.10(b) and Figure 11.13(a) plot the transmitted intensity versus frequency for various choices of mirror reflectivities and losses. It is evident that the resonant cavity acts as a narrowband transmission filter, with a periodically spaced set of transmission passbands whose bandwidth and peak transmission depend on the cavity losses and the balance between input and output coupling to the cavity. Figure 11.13(b) also shows the transmission phase angle versus frequency for a typical case.

 

 

Dielectric Etalons

The resonant transmission properties of optical interferometers or Fabry- Perot etalons have long been used as passive optical filters for incoherent light sources.

In the laser field, thin dielectric etalons, with or without additional reflective coatings, are also often used as filters inside laser cavities, in order to tune the laser, to obtain wavelength or frequency selection, or to reduce the gain bandwidth and thus limit the number of oscillating axial modes inside a laser cavity. The intracavity application of such an etalon is illustrated in Figure 11.14.

In this application the etalon is usually tilted to an angle large enough that the external reflected beams from the etalon are deflected away from the cavity axis, so that they do not set up any unwanted resonances with the other mirrors in the resonator.

At the same time the angle is made small enough that the waves bouncing back and forth inside the etalon are shifted transversely by only a very small amount compared to the beam diameter on each bounce, thus keeping the "walk-off losses" of the etalon interferometer small.

The resonant transmission peaks of the etalon can then be tuned by small changes in the etalon angle. By using several etalons of different thickness in cascade, it is possible to combine the narrow linewidth but small free spectral range obtained from a longer etalon, with the wider linewidth but also wider free spectral range of a thinner etalon, as shown in Figure 11.14(b).

 

Reflected Cavity Fields

Let us examine finally the reflected wave that comes back from a resonant cavity or etalon at the cavity input mirror, as illustrated in Figure 11.15.

 

 

Note that in in a standing-wave cavity (Figure 11.15(a)) the reflected wave goes straight back along the same direction as the incident wave, and hence presumably straight back into whatever source generated the incident wave; whereas in the ring cavity (Figure 11.15(b)) this reflected wave goes off in a different direction, like a specular reflection from mirror \(M_1\).

This can be a major advantage of a ring as compared to a standing-wave cavity, since many laser devices do not function well when looking into even a relatively weak back-reflection.

Suppose we look first at the reflected wave from the symmetric cavity example with the 100 watts of circulating power in Figure 11.12. Since the input mirror \(M_1\) in this example also has a 1% power transmission, it might appear that at resonance another 1 watt of power must be transmitted back out of the cavity in the reverse direction, because of transmission from the 100 watts of circulating power back through the 1% mirror at the input end.

This seems to say that with 1 watt of incident power, 1 watt of power can appear in the reflected wave as well as in the transmitted wave coming out of the cavity. Obtaining 2 watts of total output power in the transmitted plus reflected waves from the cavity, with only 1 watt of input power, does pose some conceptual difficulties, however; and it would seem that \(\approx\) 0 watts in the reflected wave would be a more reasonable result for this example.

 

Reflected Signal Formulas

The significant point here is, of course, that for both the traveling-wave and standing-wave cases the total "reflected" wave \(\tilde{E}_\text{refl}\) coming from mirror \(M_1\) must consist of a component \(r_1\tilde{E}_\text{inc}\) that is due to straightforward reflection from the outer surface of mirror \(M_1\), plus a second component that represents the circulating signal \(\tilde{E}_\text{circ}\) inside the cavity that is transmitted out through the mirror \(M_1\) into the same direction, as illustrated graphically in Figures 11.15(a) and (b).

The latter component comes from the circulating signal \(\tilde{E}_\text{circ}\) that left the reference plane one round-trip time earlier; traveled once around the cavity except for bouncing off mirror \(M_1\); and then is transmitted out through the input mirror. The value of this component is thus given by \(jt_1(\tilde{g}_\text{rt}/r_1)\times\tilde{E}_\text{circ}\). (The round-trip gain must be divided by \(r_1\) because the wave does not bounce off mirror \(M_1\), it goes through it.)

The total reflected wave thus consists of

\[\tag{28}\tilde{E}_\text{refl}=r_1\tilde{E}_\text{inc}+jt_1(\tilde{g}_\text{rt}/r_1)\tilde{E}_\text{circ}\]

Using our earlier expressions for \(\tilde{E}_\text{circ}\), we can then write the total reflection coefficient from mirror \(M_1\) as

\[\tag{29}\frac{\tilde{E}_\text{refl}}{\tilde{E}_\text{inc}}=r_1-\left[\frac{t_1^2r_2e^{-\alpha_0p-j\omega{p/c}}}{1-r_1r_2(r_3\ldots)e^{-\alpha_0p-j\omega{p/c}}}\right]=r_1-\frac{t_1^2}{r_1}\frac{\tilde{g}_\text{rt}(\omega)}{1-\tilde{g}_\text{rt}(\omega)}\]

These expressions, with their two separate terms, are useful in emphasizing that the reflected signal does consist of the directly reflected component, plus a transmitted component coming from the circulating signal inside the cavity, as shown in Figure 11.15.

By using the lossless mirror expression that \(r_1^2+t_1^2=1\), however, we can also convert these expressions into the slightly simpler forms

\[\tag{30}\frac{\tilde{E}_\text{refl}}{\tilde{E}_\text{inc}}=\frac{r_1-r_2e^{-\alpha_0p-j\omega{p/c}}}{1-r_1r_2e^{-\alpha_0p-j\omega{p/c}}}=\frac{1}{r_1}\times\frac{r_1^2-\tilde{g}_\text{rt}(\omega)}{1-\tilde{g}_\text{rt}(\omega)}\]

The second form of this expression makes it evident that the reflectivity from mirror \(M_1\) of the cavity depends only on the amplitude reflectivity \(r_1\) of that mirror and the round-trip gain \(\tilde{g}_\text{rt}(\omega)\), and that the reflectivity can go to zero if these become exactly equal at resonance.

Figures 11.16 and 11.17 show several curves of the power reflectivity \(I_\text{refl}/I_\text{inc}\) from a resonant cavity or interferometer versus frequency, assuming a fixed reflectivity \(R_1\) for the front mirror and varying values of the additional losses due to \(R_2e^{-2\alpha_0p}\).

Note that only the product \(R_2e^{-2\alpha_0p}\) counts; it makes no difference to the total reflectivity at mirror \(M_1\) how the additional losses in the rest of the cavity are divided between the second mirror reflectivity \(R_2\) and the internal losses \(e^{-2\alpha_0p}\).

In plotting Figure 11.17, we have plotted the intensity-reflection curves for \(R_1\ge{R}_2e^{-2\alpha_0p}\) above the axis and the curves for \(R_1\le{R}_2e^{-2\alpha_0p}\) below the axis, to indicate that there is indeed a 180° phase shift in the total reflectivity at resonance as the interferometer goes from one situation to the other. We have also shown some examples of the rather complex phase-angle variations with frequency exhibited by the reflected wave.

 

 

 

Matched Input Conditions

The special situation when the input mirror reflectivity \(R_1\) exactly equals the additional loss terms given by \(R_2e^{-2\alpha_0p}\) causes the two terms in the reflection expression to exactly cancel, and the net reflection coefficient to become exactly zero at resonance.

This is often called the impedance-matched situation, since it corresponds to looking into an impedance-matched resonant circuit or cavity (i.e., load impedance = characteristic impedance) on an ordinary transmission line. The numerical example that we considered in Figure 11.12, with the 100 watts of circulating power, was an input-matched situation, as is any symmetric interferometer (i.e., \(r_1=r_2\)) with very small internal losses.

 

EtaIon Mirrors

Even when the reflectivities from each individual mirror surface are comparatively low, i.e., \(R_1\) and \(R_2\ll1\), the combined reflectivity in the backward direction from an interferometer cavity over the off-resonance part of the reflection curve can be considerably larger than the individual reflections from either surface alone, as illustrated in Figure 11.18.

 

 

As one application of this, polished etalons made from dielectric materials such as quartz (\(n\approx\) 1.46) or clear sapphire (\(n\approx\) 1.76) with highly parallel faces are often used as the output mirrors for pulsed high-power solid-state lasers— that is, the lasers are operated with a 100% mirror on one end and a polished etalon a few mm or a cm thick, generally with no additional reflective coatings, as the output mirror on the other end.

Since these lasers typically have large round-trip gains, they operate best with low-reflectivity output mirrors, and the uncoated dielectric etalon provides a simple way of achieving the necessary output mirror reflectivity.

These uncoated etalons are also simple to fabricate, can have very high optical-damage thresholds, and the reflectivity peaks can serve a useful purpose in narrowing the oscillation bandwidth of a wide-line laser medium, such as a Nd:glass or dye laser.

Note that the reflectivity peaks in these mirrors occur not at resonance, but rather half-way between the axial-mode resonances of the etalon itself.

 

Transient Cavity Reflections

The fact that the total reflected signal \(\tilde{E}_\text{refl}\) from a resonant cavity is formed from the vector combination of the directly reflected input signal \(r_1\tilde{E}_\text{inc}\), plus a transmitted portion of the circulating signal, or \(jt_1(\tilde{g}_\text{rt}/r_1)\tilde{E}_\text{circ}\) means that the transient response of the cavity reflection, if we suddenly change either one of these signals, can be rather complex.

If an input signal is very suddenly turned on, for example, the directly reflected component appears immediately; but the circulating component only appears more gradually, after the circulating field inside the cavity has time to build up.

This makes it possible to devise various clever schemes for modulating the reflected signal on a transient or pulsed basis. Suppose, for example, that a matched steady-state on-resonance situation with these two terms essentially canceling each other has been established, and that the incident signal is then suddenly turned off by means of some kind of fast electro-optic modulator.

The net reflected signal will then suddenly jump from near zero to a value equal to \(-r_1\) times the originally incident signal, with a step-function leading edge; and will then gradually die away, with the cavity decay rate \(\omega/Q_c\) as the stored energy drains out of the resonant cavity.

If we can instead suddenly shift the phase of the incident signal by 180°, the total reflected signal will suddenly jump up to a value \(\tilde{E}_\text{refl}\approx-2r_1\tilde{E}_\text{inc}\), or a reflected power equal to four times the incident power, at least in the leading edge of the resulting transient response.

 

4. "Delta Notation" for Cavity Gains and Losses

Before we continue our general discussion of cavity resonance properties, let us introduce in this section a unified notation for describing cavity gain and loss factors that will be useful throughout the rest of this book. We can also then use this notation to simplify some of the formulas from the preceding section.

 

Mirror Reflectivities: The Delta Notation

The usual practice in optics is to describe mirrors by their power reflection and transmission values; "95% reflectivity," for example, means a mirror with \(R_1=0.95\) and hence, for a lossless mirror, \(T_1=1-R_1=0.05\).

In the early days of lasers, the available gains in most laser systems were very small, and oscillation could be obtained only with very high-reflectivity mirrors. It then became conventional to describe the small difference between the mirror power reflectivity and unity by the symbol \(\delta\), so that a mirror with 95% reflectivity would be described by \(R_1=1-\delta_1=0.95\), or \(\delta_1=T_1=0.05\).

As a more convenient and general definition, however, one useful for both high- and low-reflectivity mirrors, and hence for either small or large output coupling, we will in the remainder of this text relate the reflectivity \(R_1\) of any mirror to a "mirror coupling coefficient" \(\delta_1\) by means of the definition

\[\tag{31}\begin{align}R_1&\equiv{e}^{-\delta_1}\quad\qquad(\text{exact definition, arbitrary }\delta_1),\\&\approx1-\delta_1\qquad(\text{approximate definition, }\delta_1\ll1).\end{align}\]

Thus, if we have a laser cavity with two end mirrors having reflectivities \(R_1\) and \(R_2\), we will write these as \(R_1\equiv{r_1}^2\equiv{e}^{-\delta_1}\) and \(R_2\equiv{r_2}^2\equiv{e}^{-\delta_2}\).

The general definition of \(\delta_i\) is thus

\[\tag{32}\delta_i\equiv\ln\left(\frac{1}{R_i}\right)=2\ln\left(\frac{1}{r_i}\right)\]

for mirrors with arbitrarily low reflectivity and thus arbitrarily large output coupling.

In the high-reflectivity, low-coupling limit we can still write the mirror transmission as \(T=1-R\), with the approximation that \(T\approx\delta\) and hence \(t\approx\sqrt{\delta}\) for \(\delta\ll1\).

For a mirror having, say, \(R\) = 80% reflectivity and \(T\) = 20% power transmission, the exact value of \(\delta\) is given by \(\delta=\ln(1.25)=0.223\), not far distant from the approximate value of \(1-R=0.20\). Hence the approximation that \(\delta\approx1-R=T\) remains reasonably accurate even for mirror reflectivities as low as \(R\approx\) 80% and \(T\approx\) 20%.

 

Internal Cavity Gain and Loss Factors

If we use this notation for the end-mirror coupling factors \(\delta_1\) and \(\delta_2\), the round-trip power gain inside any cavity or interferometer with a perimeter \(p\), internal loss coefficient \(\alpha_0\), and internal gain coefficient \(\alpha_m\), can be conveniently condensed into the form

\[\tag{33}|\tilde{g}_\text{rt}|^2=R_1R_2e^{2\alpha_mp_m-2\alpha_0p}=e^{\delta_m-\delta_0-\delta_1-\delta_2}\]

That is, we can make a natural extension of the "\(\delta\) notation" by also writing the total round-trip power gain and loss in the cavity due to the internal loss coefficient \(\alpha_0\) and the laser gain medium in the forms

\[\tag{34}\delta_0\equiv2\alpha_0p\qquad\text{and}\qquad\delta_m\equiv2\alpha_mp_m\]

As a further extension we can include within the internal loss coefficient \(\delta_0\) not only the round-trip power reduction arising from any distributed attenuation \(\alpha_0p\), but also any additional discrete losses that may occur inside the cavity because of lossy interfaces, imperfect Brewster windows, internal scattering elements, or whatever. The significant quantity is thus not \(\alpha_0\) or \(p\) separately, but the total internal power loss in one complete round trip, as expressed by \(e^{-\delta_0}\).

From here on we will thus generally express any kind of round-trip power gain or power loss in an optical cavity by the notation

\[\tag{35}\delta_x\equiv\ln[\text{power gain, or power loss, ratio per round trip}]\]

In the small gain or loss limit, \(\delta_x\) is essentially the fractional power gain or loss per round trip due to mechanism \(x\), and we will often speak loosely of \(\delta_x\) in those terms, e.g., \(\delta_x=0.20\) means \(\approx\) 20% power gain or loss per round trip.

 

Total Cavity Gains and Losses

As one final bit of notation, it will often be convenient to combine all the round-trip losses contained in the factor \(\delta_0\)—which we will call internal cavity losses—with all the loss factors \(\delta_1\) and \(\delta_2\) due to the cavity mirrors—which we will call external coupling losses—to give a total cavity-loss factor \(\delta_c\) defined by

\[\tag{36}\delta_c\equiv\delta_0+\delta_1+\delta_2=2\alpha_0p+\ln\left(\frac{1}{R_1R_2}\right)\]

(plus additional mirror reflectivities \(R_3\) if needed). With this notation the roundtrip power gain inside any cavity also containing a laser gain medium can then be written in the simple form

\[\tag{37}|\tilde{g}_\text{rt}|^2=e^{\delta_m-\delta_c}\approx1+\delta_m-\delta_c\quad\text{if}\quad|\delta_m-\delta_c|\ll1\]

The net growth (or decay) rate for a signal circulating around inside the laser cavity (with no injected signal) is thus simply the difference between the total (saturated) laser gain factor \(\delta_m\) and the total cavity loss factor \(\delta_c\).

With all of these gain and loss factors \(\delta_x\) defined in terms of a round trip, most of our formulas will apply equally well to either standing-wave or ring-type laser cavities. Note that similar notation is used in much of the laser literature, but published papers are not always consistent about whether \(\delta_x\) means power gain or loss per one-way pass or per round trip. In consulting the literature, watch out for possible factors of two, depending on which definition is employed.

 

Cavity Q Values

It can also be useful in some situations to relate cavity gain or loss factors to cavity \(Q\) factors, defined in the following manner.

Suppose some initially injected energy is circulating around inside a laser cavity, with no further injected signal being applied. If we consider only a "cold" laser cavity (no gain present), this circulating energy will decrease after a number \(N\) of round trips in the exponential fashion

\[\tag{38}I_\text{circ}(t)=I_\text{circ}(t_0)\times\exp[-N\delta_c]=I_\text{circ}(t_0)\times\exp\left[-\frac{\delta_c}{T_\text{rt}}(t-t_0)\right]\]

where \(T_\text{rt}\equiv{p/c}\) is the round-trip transit time in the laser cavity, and \(N=(t-t_0)/T_\text{rt}\) is the number of round trips in time \(t-t_0\).

Another way of expressing this exponential decay that is commonly used in many engineering fields is to write it in the form

\[\tag{39}I_\text{circ}(t)=I_\text{circ}(t_0)\times\exp\left[-\frac{\omega_a}{Q_c}(t-t_0)\right]\]

where \(Q_c\) is sometimes called the "cold cavity \(Q\)" of the laser cavity due to internal losses plus external coupling. This \(Q\) value plays the same role in an optical cavity as does, for example, the familiar \(Q=\omega{L}/R\) value characteristic of a series RLC electrical circuit.

The \(Q\) factor (sometimes called the "quality factor") of an optical cavity due to its internal losses plus external coupling through the mirrors can thus be calculated from

\[\tag{40}Q_c=\frac{\omega_aT_\text{rt}}{\delta_c}=\frac{2\pi{p}}{\lambda}\frac{1}{\delta_c}\]

(We could also define a negative \(Q_m\) value representing the laser gain, by replacing the loss factor \(\delta_c\) by \(\delta_m\) in this expression.) Real laser cavities typically have very large \(Q_c\) values, even in very lossy optical cavities.

Suppose for example that an optical cavity is very lossy, with 90% power loss per round trip, corresponding to \(\delta_c=\ln(1/0.1)\approx2.3\). The \(Q_c\) value will then still be very large, even though 90% of the circulating energy is lost out of the cavity on every round trip, because the cavity perimeter \(p\) will (except in very special cases) be larger than the optical wavelength \(\lambda\) by a factor typically somewhere between \(10^4\) and \(10^6\). In physical terms, the power loss per round trip is large, but the fractional power loss per cycle (which determines the \(Q_c\) value) is very small.

 

Field Values in Low-Loss Cavities

By using the delta notation, plus the low-loss approximations, we can write some useful simplified forms of the analytical results given earlier in this tutorial. The on-resonance value of the denominator \(1-\tilde{g}_\text{rt}\) that appears in all the resonant-cavity expressions can first be simplified to the form

\[\tag{41}1-\tilde{g}_\text{rt}\equiv1-r_1r_2e^{\alpha_mp_m-\alpha_0p}\approx\frac{\delta_c-\delta_m}{2}\]

The peak value for the circulating intensity in a purely passive cavity (\(\delta_m=0\)) at resonance can then be written as

\[\tag{42}\left.\frac{I_\text{circ}}{I_\text{inc}}\right|_{\omega=\omega_q}\approx\frac{4\delta_1}{(\delta_1+\delta_2+\delta_0)^2}\approx\begin{cases}4/\delta_1\qquad\text{if }\delta_2+\delta_0\ll\delta_1\text{ and }\delta_1\ll1\\1/\delta_1\qquad\text{if }\delta_2+\delta_0=\delta_1\end{cases}\]

The power increase of the signals inside the cavity at resonance is thus of order \(4\delta_1/\delta_c^2\approx1/\delta_c\), where again \(\delta_c\equiv\delta_1+\delta_2+\delta_0\). For maximum enhancement, the only loss mechanism in the cavity should be the external mirror transmission or coupling \(\delta_1\) through which the external signal is injected.

Similarly the peak signal transmission through a passive low-loss interferometer or cavity at resonance can be written in the form

\[\tag{43}\left.\frac{I_\text{trans}}{I_\text{inc}}\right|_{\omega=\omega_q}\approx\frac{4\delta_1\delta_2}{(\delta_1+\delta_2+\delta_0)^2}=\frac{4\delta_1\delta_2}{\delta_c^2}\]

A little examination shows that this gives \(I_\text{trans}/I_\text{inc}\approx1\) if \(\delta_1\approx\delta_2\) and \(\delta_0\ll\delta_1,\delta_2\). The peak transmission through a Fabry-Perot etalon can thus approach 100%, provided that (a) the end-mirror reflectivities are closely enough matched, and (b) the internal loss \(\delta_0\) is small compared to the end-mirror couplings. The actual mirror-transmission values \(\delta_1\) and \(\delta_2\) are not important for high peak transmission (though of course they have a critical effect on the bandwidth of the transmission peak).

 

Reflected Waves For Low-Loss Cavities

The somewhat more complex behavior of the reflected waves for a passive cavity or interferometer can be emphasized for the low-loss situation, where all the \(\delta\)'s are \(\ll1\), by writing the on-resonance voltage reflectivity in the form

\[\tag{44}\left.\frac{\tilde{E}_\text{refl}}{\tilde{E}_\text{inc}}\right|_{\omega=\omega_q}\approx\frac{\delta_2+\delta_0-\delta_1}{\delta_2+\delta_0+\delta_1}\qquad\text{if all }\delta\text{'s }\ll1\]

This gives the limiting values

\[\tag{45}\left.\frac{\tilde{E}_\text{refl}}{\tilde{E}_\text{inc}}\right|_{\omega=\omega_q}\approx\begin{cases}+1\qquad\text{if }\delta_2+\delta_0\gg\delta_1\\0\quad\qquad\text{if }\delta_2+\delta_0=\delta_1\\-1\qquad\text{if }\delta_2+\delta_0\ll\delta_1\end{cases}\]

These three limiting cases may be described as follows.

1. If the internal cavity losses plus output mirror coupling are significantly larger than the input mirror coupling, i.e., \(\delta_2+\delta_0\gg\delta_1\), then this represents an undercoupled cavity. The circulating intensity inside the cavity does not build up to a large value, and the net reflectivity for the external signal is essentially just the normal reflectivity \(r_1\approx+1\) due to the input mirror alone.

2. If the input coupling is exactly equal to all the other cavity losses, i.e., if \(\delta_1=\delta_2+\delta_0\), then this is the impedance-matched situation, in which the normal reflection component of \(+r_1\) from the mirror itself is just matched by a net component of \(-r_1\) from the circulating energy inside the cavity. The input reflectivity is zero, and all the power delivered by the external source onto mirror \(M_1\) goes into either the internal cavity losses or the transmitted output through mirror \(M_2\).

3. Finally, in the overcoupled situation, all the other losses and coupling are small compared to the coupling at mirror \(M_1\), i.e., \(\delta_2+\delta_0\ll\delta_1\). The on-resonance circulating intensity builds up to its largest possible value \((\tilde{E}_\text{circ}/\tilde{E}_\text{inc}\approx4j/\delta_1)\); and the out-coupled portion of this circulating intensity completely reverses the \(r_1\approx+1\) term to give a total reflectivity of \(\approx-1\) instead.

Obviously, all these reflected and transmitted cavity field expressions can be generalized to active laser cavities if we use \(\delta_c-\delta_m\) rather than simply \(\delta_c\) in the denominator.

 

5. Optical-Cavity Mode Frequencies

Since the resonance frequencies of optical cavities and interferometers are of particular interest, let us also examine in somewhat more detail the frequency properties of passive optical resonators, including the axial-mode spacing, the resonance bandwidths, and the frequency tuning or scanning possibilities in optical resonators.

 

Axial-Mode Spacing

The axial modes in a passive optical resonator (without laser gain) occur, as we have already seen, at those frequencies \(\omega\) which satisfy the round-trip phase condition \(\phi(\omega)\equiv\omega{p/c}=q\times2\pi\) in a ring cavity of perimeter \(p\), or \(\phi(\omega)\equiv2\omega{L/c}=q\times2\pi\) in a standing-wave cavity of length \(L\), with \(q\) being a (large) integer. (We have left out any atomic pulling effects at this point, but will include them in the next section.)

The resonant frequencies of the optical cavity are thus given by

\[\tag{46}\omega=\omega_q=q\times2\pi\times\frac{c}{p}=q\times2\pi\times\frac{c}{2L},\qquad{q=\text{integer}}\]

These axial modes form an equally spaced comb of resonant frequencies labeled by index \(q\), as in Figure 11.19, with each mode separated by the axiai-mode spacing or axial-mode interval

\[\tag{47}\Delta\omega_{ax}\equiv\omega_{q+1}-\omega_q=2\pi\times\frac{c}{p}=2\pi\times\frac{c}{2L}\]

 

 

The frequency spacing between the axial modes of a standing-wave cavity, expressed in Hz or cycles/second, is thus \(\Delta{f}_\text{ax}\equiv\Delta\omega_\text{ax}/2\pi=c/p\) or \(c/2L\). (Since most cavities in the earlier days of lasers were standing-wave cavities, many laser workers routinely speak of the "\(c/2L\) mode spacing" in a laser cavity.) These quantities must be written as \(c_0/np\) or \(c_0/2nL\) if it is necessary to take explicitly into account the index of refraction of any dielectric material inside the cavity.

For typical cavities in laser oscillators, this axial mode spacing will have values in the range

\[\tag{48}\Delta{f}_\text{ax}=\frac{c}{2L}\approx\begin{cases}150\text{ MHz}\quad\qquad\text{if }L=1\text{ m}\\500\text{ MHz}\quad\qquad\text{if }L=30\text{ cm}\\2,000\text{ MHz}\qquad\text{if }L=5\text{ cm and }n=1.5\end{cases}\]

The axial-mode intervals for laser cavities are thus typically a hundred Mhz or less for a one- or two-meter-long argon-ion or CO₂ laser cavity; rising to 500 MHz for a shorter, 30-cm-long He-Ne or Nd:YAG laser; and increasing to several GHz for a very short laser a few cm long. An example of the latter category might be a very simple solid-state laser with the mirror coatings evaporated directly on the ends of the rod.

Note also that semiconductor injection lasers (and also certain very short dye-laser cavities) can have cavity lengths \(L\) of only a few tens to a few hundreds of microns. These axial modes become so widely spaced that it may make more sense to specify their axial-mode spacing as a wavelength spacing in Å than as a frequency spacing in MHz.

A typical GaAs semiconductor diode laser with length \(L\) = 100 μm and index of refraction \(n\) = 3.6 has an axial-mode spacing of \(\Delta{f}_\text{ax}\approx4.2\times10^{11}\) Hz, corresponding to a wavelength interval \(\Delta\lambda\approx10\) Å at a center wavelength of \(\lambda\approx8600\) Å.

Figure 11.20 shows, as a similar example, the amplified spontaneous emission spectrum from a thin film of optically pumped organic dye medium filling the space between two very closely spaced high-reflectivity mirrors. 

 

 

Interferometers and Free Spectral Range

In the jargon of optical interferometry, the \(c/p\) or \(c/2L\) frequency spacing is commonly called the free spectral range, since it represents the frequency interval between transmission peaks of a resonant interferometer.

Fabry-Perot etalons, which are often used in laser experiments as resonant mirrors, filters, or bandwidth narrowing elements, typically have lengths ranging from \(L\) = 1 cm down to 100 μm, and are often made of materials like fused quartz, with an index of refraction \(n\approx\) 1.46, or sapphire, with \(n\approx\) 1.76. Their axial-mode spacings or free spectral ranges therefore typically have values more like

\[\tag{49}\Delta{f}_\text{ax}=\frac{c_0}{2nL}\approx\begin{cases}0.33\text{ cm}^{-1}\approx10\text{ GHz}\qquad\text{if }L=1\text{ cm}\\3.3\text{ cm}^{-1}\approx100\text{ GHz}\qquad\text{if }L=1\text{ mm}\\33\text{ cm}^{-1}\approx10^{12}\text{ Hz}\qquad\text{if }L=100\text{ μm}\end{cases}\]

assuming \(n\) = 1.5 in each case. When mode spacings become this large, it is more convenient to express them in wavenumber units, or

\[\tag{50}\Delta\nu_\text{ax}\equiv\Delta\left(\frac{1}{\lambda_0}\right)_\text{ax}=\frac{c_0^2}{np}\quad\text{or}\quad\frac{c_0^2}{2nL}\]

A convenient rule of thumb to remember is that 1 wavenumber or 1 cm\(^{-1}\) equals 30 GHz.

 

Axial-Mode Number

The axial mode index \(q\) in an optical cavity or interferometer is given by

\[\tag{51}q=\frac{\omega_q}{\Delta\omega_\text{ax}}=\frac{p}{\lambda_q}=\frac{L}{\lambda_q/2}\]

Since the perimeter \(p\) of an optical cavity is typically much longer than the optical wavelength \(\lambda\), the value of \(q\) is typically a very large integer, on the order of 10\(^6\) to 10\(^7\) for typical laser cavity lengths.

In a ring cavity the index q represents simply the number of optical wavelengths or optical cycles around the cavity perimeter, and in a standing-wave cavity it represents the number of half-optical-wavelengths along the cavity axis as illustrated in Figure 11.19(b). The values of the mode integer \(q\) in very short cavities or thin etalons become more like 10\(^3\) to 10\(^5\).

Going from mode \(q\) to mode \(q+1\) thus corresponds to increasing the optical frequency or decreasing the optical wavelength \(\lambda\) just enough to squeeze one more half-wavelength into the standing-wave cavity length, as shown in Figure 11.19(b).

Note that the standing-wave patterns of two adjacent axial modes in a linear cavity are spatially in phase at the ends of the cavity, but exactly out of phase at the center of the cavity. The spatial offset of the fields can have important implications for axial-mode competition in various kinds of lasers.

 

Bandwidth, Resolving Power, and Finesse

Let us next look at the resonance bandwidths of optical cavities or interferometers. The dominant frequency dependence for both the circulating and the transmitted signals in a resonant cavity is obviously contained in the resonance denominator \(1-\tilde{g}_\text{rt}(\omega)\equiv1-r_1r_2(r_3\ldots)e^{-\alpha_0p-j\omega{p/c}}\). The magnitudes of \(\tilde{E}_\text{circ}/\tilde{E}_\text{inc}\) and \(\tilde{E}_\text{trans}/\tilde{E}_\text{inc}\) will be decreased by \(\sqrt{2}\), or the corresponding intensities reduced to half their maximum values, at those frequencies for which the quantity \(|1-\tilde{g}_\text{rt}(\omega)|^2\) doubles compared to its value at a resonance peak.

With a little algebra, this gives a FWHM bandwidth for the resonance peaks of

\[\tag{52}\begin{align}\Delta\omega_\text{cav}&=\frac{4c}{p}\sin^{-1}\left[\frac{1-g_\text{rt}}{2\sqrt{g_\text{rt}}}\right]\\&\approx\frac{2\pi{c}}{p}\times\left[\frac{1-g_\text{rt}}{\pi\sqrt{g_\text{rt}}}\right]=\left[\frac{1-g_\text{rt}}{\pi\sqrt{g_\text{rt}}}\right]\times\Delta\omega_\text{ax}\end{align}\]

where the assumption in the second line is that the magnitude of the round-trip gain, \(g_\text{rt}\equiv|\tilde{g}_\text{rt}(\omega)|\), is not too much less than unity. The resonance bandwidth in general is obviously only a fraction of the axial-mode spacing or free spectral range, and becomes narrower the closer the round-trip gain \(\tilde{g}_\text{rt}\) comes to unity.

In classical optics the power transmission through an etalon or interferometer is often written in the form

\[\tag{53}\left|\frac{\tilde{E}_\text{trans}(\omega)}{\tilde{E}_\text{inc}(\omega)}\right|^2=\frac{T_\text{max}}{1+(2\mathcal{F}/\pi)^2\sin^2(\pi\omega/\Delta\omega_\text{ax})}\]

where \(T_\text{max}\) is the peak transmission through the etalon; \(\Delta\omega_\text{ax}\equiv2\pi{c/p}\) is the free spectral range or axial-mode interval between resonances; and \(\mathcal{F}\) is the so-called finesse of the interferometer.

By comparing this with Equation 11.27 for \(\tilde{E}_\text{trans}/\tilde{E}_\text{inc}\) we can see that the finesse is in fact just the ratio of the free spectral range to the cavity bandwidth, as given by

\[\tag{54}\text{finesse},\mathcal{F}\equiv\frac{\pi\sqrt{g_\text{rt}}}{1-g_\text{rt}}\approx\frac{\Delta\omega_\text{ax}}{\Delta\omega_\text{cav}}\]

The finesse thus gives the resolving power of the etalon used as a transmission filter. This resolving power obviously becomes largest in the limit of mirror reflectivities approaching unity (\(r_1,r_2\rightarrow1\)) and very small internal losses (\(\alpha_0p\rightarrow0\)).

As a practical matter, a finesse of \(\mathcal{F}\approx\) 100 for a passive interferometer or optical cavity in the visible is considered extremely good; and a finesse this large clearly requires \(1-g_\text{rt}\le0.03\), or less than 3% round-trip voltage loss.

If we use the delta factors defined in the preceding section, and include gain as well as cavity losses, the finesse can be written as

\[\tag{55}\mathcal{F}\equiv\frac{\pi\sqrt{g_\text{rt}}}{1-g_\text{rt}}\approx\frac{2\pi}{\delta_c-\delta_m}\]

where \(\delta_c\equiv\delta_1+\delta_2+\delta_0\) is the total fractional power loss per one round trip in the cavity due to all the cavity-loss mechanisms—mirror reflectivities plus internal losses.

The laser gain, if any is present, then appears as a kind of "negative loss" term. The resonance bandwidth for the circulating signals then becomes

\[\tag{56}\Delta\omega_\text{cav}\approx\frac{\Delta\omega_\text{ax}}{\mathcal{F}}=\frac{\delta_c-\delta_m}{2\pi}\times\Delta\omega_\text{ax}\]

Obviously by adding laser gain \(\delta_m\) to a passive cavity with total losses \(\delta_c\) we can make the finesse \(\mathcal{F}\) approach infinity, and the resonance bandwidth approach zero.

 

Axial Modes in Dispersive Optical Cavities

The resonance frequency formulas given in Equation 11.46 above become slightly more complicated for dispersive optical cavities—cavities in which the velocity of light c or the index of refraction n are themselves functions of frequency.

The round-trip phase-shift condition for the \(q\)-th axial mode in this case (again with atomic pulling or \(\Delta\beta_n\) effects neglected) becomes

\[\tag{57}\frac{n(\omega)\omega{p}}{c_0}\equiv\frac{2n(\omega)\omega{L}}{c_0}=q\times2\pi\]

where \(c_0\) is the velocity of light in free space and \(n(\omega)\) the frequency-dependent refractive index.

Since the fractional spacing between axial modes is normally very small, and the index variation with frequency is also small, we can almost always expand the index of refraction about its value at some central mode \(\omega_q\) in the form \(n(\omega_{q+1})\approx{n}(\omega_q)+n'(\omega_q)\times\Delta\omega_\text{ax}\), where \(n'\equiv{d}n(\omega)/d\omega\).

The axial-mode spacing is then given, to first order of approximation in \(n(\omega)\), by

\[\tag{58}\Delta\omega_\text{ax}\approx\frac{2\pi{c_0}}{(n+n'\omega)p}=2\pi\times\frac{1}{1+(\omega/n)(dn/d\omega)}\times\frac{c}{p}\]

where \(n\) and \(n'\equiv{dn/d\omega}\); are midband values.

The correction term \((\omega/n)(dn/d\omega)\) for transparent dielectrics is usually positive, so that the effective axial-mode spacing is slightly reduced by this term. The resulting correction factor can become as large as a 10-percent reduction in axial-mode spacing over the \(2\pi{c/p}\) value for the special case of GaAs injection lasers, in which the GaAs crystal fills the entire cavity and has an unusually large dispersion at the lasing wavelength. For most other lasers, even solid dielectric etalons, this correction is very small and is usually neglected.

 

Optical Cavity Tuning

Very small changes in the length of an optical cavity can be used to tune the resonant frequencies of the cavity by sizable amounts. From the resonant-frequency expressions given earlier, we can see that changing the cavity perimeter by a small amount \(\delta{p}\) at fixed \(q\) tunes each of the axial-mode resonant frequencies by an amount

\[\tag{59}\frac{\delta\omega_q}{\omega_q}\approx-\frac{\delta{p}}{p}\approx-\frac{\delta{L}}{L}\]

which can be rewritten as

\[\tag{60}\delta\omega_q\approx-\frac{\delta{p}}{\lambda}\times\Delta\omega_\text{ax}\approx-\frac{\delta{L}}{\lambda/2}\times\Delta\omega_\text{ax}\]

In other words, changing the ring-cavity perimeter by one optical wavelength, or the standing-wave cavity length by one half-wavelength, shifts each of the axial modes over by an amount just equal to the spacing between axial modes. A round-trip length change of \(\lambda\) causes mode \(q\) to be tuned over to the frequency previously occupied by mode \(q\pm1\), depending on whether the cavity is shortened or lengthened.

 

Cavity Tuning Methods

Interferometer cavities and laser oscillators are commonly tuned (or stabilized) in absolute frequency by a combination of temperature tuning (to be described below), plus the use of a piezoelectric mounting on one cavity mirror to move the mirror back and forth by a few optical wavelengths, thus scanning the absolute frequency of each axial mode by a few axial-mode intervals.

With typical piezoelectric "stacks," a few hundred volts applied to the piezoelectric element is usually sufficient to tune each resonance through one axial-mode interval. (Note that moving one of the mirrors in a ring cavity by a distance \(\Delta{z}\) actually increases the ring perimeter by an amount \(\approx2\Delta{z}\), depending on how the ring is laid out; so a mirror motion of \(\Delta{z}\) accomplishes approximately the same frequency tuning in either the ring or the standing-wave cavity.)

The absolute amount of frequency tuning for an increase of one wavelength in the cavity perimeter, namely, \(\Delta\omega_\text{ax}\), is itself inversely proportional to the cavity length; so the absolute amount of frequency tuning can become very large for very short interferometer cavities.

Magnetic drivers—in the simplest case, converted loudspeaker coils—can also be used to obtain larger mirror motions, for example, for long-wavelength infrared lasers.

To first order the spacing \(\Delta\omega_\text{ax}\) between adjacent axial modes is hardly changed by adding a few half-wavelengths \(\lambda/2\) to the cavity length \(L\) or perimeter \(p\). Hence, to first order, changing the cavity length by a few wavelengths simply tunes the entire comb of axial modes back and forth underneath the atomic line, without noticeably changing the axial-mode spacing.

 

Temperature Tuning and Thermal Drifts

Note also that in a typical optical cavity or laser structure a temperature change \(\delta{T}\) of a few degrees or less will produce enough thermal expansion of the cavity to give a \(\delta{p}\) of one half-wavelength or more. Optical cavities thus generally have a large thermal tuning or thermal drift rate, unless carefully stabilized in temperature.

Highly stable laser cavities are often made with the mirror spacing controlled by a rod of Invar, a steel alloy having small or even zero expansion coefficient at room temperature.

Rods of quartz, carbon fiber, or zero-expansion ceramics can also be used for the same purpose. Unwanted tuning and frequency jitter of lasers caused by mechanical vibrations and acoustic noise is another very serious issue in any laser where high-frequency stability is required, and careful shock mounting and acoustic isolation may be required for highest stability.

 

Scanning Optical Interferometers

Tunable interferometer cavities are often used as passive tunable filters, or as so-called scanning interferometers, for measuring laser output spectra or other optical signals.

To measure a laser signal in this fashion, we can send the signal through a passive optical cavity or scanning interferometer as illustrated in Figure 11.21, and then scan the axial modes of the interferometer back and forth in frequency across the laser spectrum by changing the passive cavity length (typically at an audio frequency rate or slower).

A strong optical signal is transmitted through the passive cavity each time one of its axial-mode resonances coincides with an input laser signal. This transmitted signal is then detected and displayed on an oscilloscope as illustrated in Figure 11.21.

Note that if the scanning interferometer is scanned by more than one of its free spectral ranges, another transmission resonance will be observed each time another axial mode of the scanning interferometer cavity scans across any one of the incident laser frequencies.

Hence to prevent confusion or ambiguity in the results, the axial-mode spacing or free spectral range of the interferometer should be wider than the full oscillation range of the laser signal being measured, as illustrated in Figure 11.21.

This requires that the interferometer cavity be shorter than the cavity of the laser generating the signals—and often considerably shorter. The shorter the scanning cavity, however, the wider its resonance linewidth and the poorer its frequency resolution for a given amount of loss.

Scanning interferometer design is thus a tradeoff between free spectral range and resolving power, with a high premium given to minimizing the losses in the scanning cavity.

 

 

Confocal Fabry-Perot Interferometers

With a conventional Fabry-Perot interferometer using planar or only slightly curved mirrors, the incident laser beam must be very precisely aligned with the axis of the interferometer in order not to excite many higher-order transverse modes of the interferometer cavity, and thus obtain very confused resonance signals.

For interferometer cavities which are exactly confocal (meaning that the center of curvature of each mirror lies exactly on the other mirror) this difficulty does not occur, for reasons we will explain in a later tutorial.

Hence confocal optical cavities, because of their relative freedom from alignment restrictions, are widely used for scanning interferometers, including several commercially available instruments of this type.

 

6. Regenerative Laser Amplification

In this section we will finally add laser gain as well as mirrors to a laser cavity, and thus finally achieve true regenerative feedback and regenerative amplification in a laser amplifier. Doing this will bring us closer to the threshold of laser oscillation—a threshold we will finally cross in the following tutorial.

 

Regenerative Gain Formula

Suppose that we add a laser gain medium with gain coefficient \(\alpha_m(\omega)p_m\) and added phase shift \(-j\Delta\beta_m(\omega)p_m\) to the interferometer model we have already analyzed.

Then the formulas we have already developed will all remain valid, except that the round-trip gain \(\tilde{g}_\text{rt}(\omega)\) inside the laser cavity will be modified to

\[\tag{61}\tilde{g}_\text{rt}(\omega)=r_1r_2(r_3\ldots)\times\exp[\alpha_mp_m-\alpha_0p-j\omega{p/c}-j\Delta\beta_m(\omega)p_m]\]

The length \(p_m\) here is the total length of the active laser medium in a ring laser cavity, or twice the length of the laser medium (i.e., \(p_m= 2L_m\)) in a standing-wave laser cavity.

The circulating power in the laser cavity will then still be given by Equation 11.20, except that \(\tilde{g}_\text{rt}(\omega)\) will now be given by Equation 11.61. The overall regenerative gain through the cavity, or the transmission from input to output, will thus be given by

\[\tag{62}\begin{align}\frac{\tilde{E}_\text{trans}}{\tilde{E}_\text{inc}}&=-\frac{t_1t_2\exp[(\alpha_mp_m-\alpha_0p-j\omega{p/c}-j\Delta\beta_mp_m)/2]}{1-r_1r_2\exp[\alpha_mp_m-\alpha_0p-j\omega{p/c}-j\Delta\beta_mp_m]}\\&=-\frac{t_1t_2}{\sqrt{r_1r_2}}\times\frac{\sqrt{\tilde{g}_\text{rt}(\omega)}}{1-\tilde{g}_\text{rt}(\omega)}\end{align}\]

This is the formula for transmission gain through the regenerative amplifier. We could write another, slightly more complex formula for the reflection gain \(\tilde{E}_\text{refl}/\tilde{E}_\text{inc}\) coming back out the input end of the amplifier (and this reflection gain would in fact be a more useful way to employ a regenerative ring-laser amplifier).

 

Gain Properties of Regenerative Amplifiers

We are now going to demonstrate that as we turn up the magnitude of the round-trip gain inside the interferometer or laser cavity toward a limiting value of unity, the peak value of the transmission (and also the reflection) gain through the laser cavity will shoot upward toward infinity, as a result of regeneration in the laser cavity.

Figure 11.22 plots the log of the transmitted power gain \(|\tilde{E}_\text{trans}/\tilde{E}_\text{inc}|^2\) versus frequency, as given by Equation 11.62, assuming a fixed gain medium with the rather largish midband gain value \(\exp[\alpha_mp_m-\alpha_0p_0]=2\), and with increasing end-mirror reflectivities ranging from \(R_1=R_2=0\) (i.e., no mirrors) to \(R_1=R_2\approx\) 35%.

 

 

Two aspects of the amplification behavior in this system are immediately apparent. First, when mirrors with even rather small reflectivity are added, the overall power gain \(|\tilde{g}(\omega)|^2\) at certain frequencies within the atomic-gain curve can become larger, and eventually very much larger, than the single-pass gain of the laser medium itself; and second, these high-gain frequencies occur only in very narrow bands located at the regularly spaced axial-mode resonances of the cavity.

The round-trip cavity length \(p\) for these particular calculations has been chosen so that one of the axial-mode resonances lies exactly at the line-center frequency \(\omega_a\), and several adjoining axial modes are nearby within the atomic linewidth.

The regenerative amplification of these off-line-center axial modes is clearly reduced relative to the centermost mode, especially at higher mirror reflectivities, because of the decreasing atomic gain away from line center.

We have used a fairly high atomic-gain value in order to make all the regenerative gain peaks broader and thus easier to plot; and we have used log scales in order to exhibit the large increases that the overall gain can acquire.

Figure 11.23 plots a similar set of transmission gain curves versus frequency, but this time on a linear scale and assuming fixed mirror reflectivities of \(R_1=R_2=40\%\), with increasing amounts of internal laser gain.

Figure 11.23(a) shows the overall power gain or intensity transmission through the cavity with no internal laser gain and 4% internal round-trip power losses, demonstrating the typical resonance behavior of a passive interferometer cavity or etalon.

The peak overall transmission is slightly less than unity, and the transmission peaks are spaced in frequency by the usual free spectral range of the interferometer. Adding small amounts of internal gain then rapidly converts these resonance peaks into overall regenerative gain peaks, with peak gain substantially larger than unity, as shown in Figure 11.23(b).

 

 

Regenerative Feedback Model

To readers familiar with regenerative feedback systems, the reasons for the behavior shown in Figures 11.22 and 11.23 will seem obvious. The laser cavity can be modeled by a typical control-system or feedback-system block diagram, as in Figure 11.24.

 

 

Near the axial-mode resonances the system has a round-trip phase shift which passes through an integer multiple of \(2\pi\), thus producing positive or regenerative feedback.

This condition reoccurs at each axial mode; and because the round-trip path length in a laser is very long in units of wavelengths, these axial modes are very closely spaced in frequency.

As the magnitude of the round-trip gain in this feedback loop approaches unity, the overall gain from input to output of the feedback system approaches infinity; and the system in fact becomes unstable and breaks into self-oscillation when the round-trip gain just becomes unity.

This same kind of regenerative feedback can always be used to obtain large overall gain in any kind of amplifying system, using a single-pass amplifier with comparatively small gain, but applying positive feedback to obtain a very large overall gain.

This feedback mechanism is effective, however, only over a very limited bandwidth, where the feedback signal has the correct phase. For example, at frequencies halfway between the axial modes, the round-trip phase shift changes so that the feedback produces instead negative feedback or degeneration.

This in turn actually reduces the overall gain below what would otherwise be the net transmission through the two mirrors and the gain medium. (Note the demonstration of this in Figure 11.22.)

 

Physical Interpretation: The Approach to Threshold

Another physical interpretation of the large regenerative gain observed at resonance can be given as follows. Suppose as an extreme example that the input mirror to a regenerative amplifier has a large reflectivity, say, \(R\) = 98%. It may then appear that 98% of an input signal is immediately reflected back from the laser input and wasted, with only 2% entering the amplifier to be amplified.

The high-reflectivity mirrors plus the internal gain inside the cavity, however, permit any energy inside the cavity to recirculate or reverberate inside the cavity many times, extracting energy from the laser medium on each bounce, so that the recirculating wave also builds up to very large amplitudes inside the cavity, relative to the incident wave amplitude outside the cavity.

This build-up plus coherent reinforcement leads to a very large increase of the internal circulating energy relative to the incident energy striking the mirror from outside.

On each end-mirror reflection, a portion of this very large circulating energy is also transmitted back out through both the input and the output mirrors, leading to the large overall transmission and reflections gains that occur at resonance.

 

Geometric Interpretation

Looking again at the vector or geometric interpretation given in Figure 11.11 may also be helpful in explaining the high overall gain obtained at resonance. The crucial aspect of the overall gain expression is again the feedback denominator \(1-\tilde{g}_\text{rt}(\omega)\).

The round-trip gain \(\tilde{g}_\text{rt}(\omega)\) has a magnitude just less than unity (for cavities below threshold), and a phase angle which rotates rapidly with frequency in the complex plane. The length of this vector may also change slowly as it rotates because of the change in laser gain as the frequency is tuned off line center.

Figure 11.25 shows again how this vector is pivoted at the point \(1+j0\) in the complex plane, and rotates rapidly about that point. The cavity transmission gain is inversely proportional to the distance from the origin to the tip of this vector.

Hence each time the tip of \(1-\tilde{g}_\text{rt}(\omega)\) sweeps close to the origin the gain becomes very high—but over only a brief section of the rotation cycle—and another axial-mode resonance is generated.

 

 

Experimental Illustration

The existence of axial cavity modes, and especially the regenerative amplification which occurs at axial-mode peaks in a laser cavity below threshold, shows up most dramatically perhaps in semiconductor diode lasers.

The widely spaced axial modes in these lasers (\(\Delta\omega_\text{ax}\approx10Å\)) are located within an even wider atomic transition (atomic linewidth greater than 100Å), and it becomes possible to observe these modes with a relatively low-resolution optical spectrometer.

Moreover, rather than making a measurement of regenerative laser gain versus frequency (which requires a tunable signal source and other complexities), we can simply measure the amplified internal spontaneous emission coming from within the laser diode itself, as this radiation is regeneratively amplified inside the laser cavity and transmitted through the end mirrors. Measurements of this type are also aided by the particularly strong spontaneous emission in semiconductor lasers.

A typical example of this kind of measurement is shown in Figure 11.26. Note that the curve in part (b) of this figure represents a slightly higher driving current through the laser, which produces slightly more internal gain, thus bringing the laser closer to oscillation threshold.

(Note also the difference between the vertical scales of the two parts.) The regenerative increase of the centermost modes relative to the modes further out on the atomic gain curve thus becomes quite marked in part (b).

Note also that this kind of axial-mode structure will develop only in the emission traveling along the axial direction and coming out through the ends of the laser cavity.

The spontaneous emission from the atoms coming out through the sides of the laser cavity will not exhibit this structure (except insofar as internal defects or spurious reflections may scatter some of the axial radiation out through the sides of the cavity).

 

 

7. Approaching Threshold: The Highly Regenerative Limit

As the laser gain is turned up (or the cavity losses are turned down) in a regenerative laser cavity, and the laser cavity approaches oscillation threshold, we can observe that:

• The regenerative gain peaks become very high (especially the centermost one);
• These regenerative gain peaks also become very narrow;
• Each regenerative gain peak approaches (as we will now show) a fixed gain-bandwidth product.

This section analyzes this limiting situation when an optical cavity is highly regenerative and just below threshold.

 

The Approach to Threshold

Let us note once again that the quantity \(1-\tilde{g}_\text{rt}(\omega)\) appearing in the denominator of Equation 11.62 is one minus the round-trip voltage gain for a wave circulating around inside the laser cavity, including bouncing off the mirrors at each end.

If this round-trip gain has magnitude less than unity, then the laser cavity is below threshold. This means that unless new energy is continually injected into the cavity by an external injected signal, the recirculating signal energy inside the cavity will decay in amplitude on successive round trips, so that any signals in the cavity will gradually die out. The cavity can thus not oscillate so long as \(\tilde{g}_\text{rt}\lt1\). It can, however, function as a regenerative amplifier, with potentially very high transmission or reflection gain.

If the magnitude of the internal round-trip gain approaches and then exceeds unity, however, then any circulating signals inside the cavity will grow in amplitude on each successive round trip, eventually building up to unlimited amplitudes. Of course when the signal amplitude inside the cavity grows to a large enough value, the signal fields will begin to saturate the population inversion and reduce the atomic gain.

The round-trip gain will then be driven back down toward the value of exactly unity, at which point the circulating signals inside the cavity neither grow nor decay on successive round trips. The laser can then maintain a steady-state self-sustained oscillation, without any externally injected signal. The condition for the build-up of such a steady-state self-sustained oscillation in a laser cavity (starting from an injected signal, or just from spontaneous emission noise) is thus \(|\tilde{g}_\text{rt}|\gt1\).

The line where the round-trip gain magnitude \(|\tilde{g}_\text{rt}|\) becomes just equal to unity, as shown in Figure 11.27, thus marks a boundary line between the stable, below-threshold, finite regenerative-gain region, and the unstable, above-threshold region where no steady-state operation is possible. This boundary line thus represents both oscillation threshold (where oscillation can just start) and the steady-state oscillation condition for an oscillating laser.

 

 

The High-Gain Near-Threshold Limit

Some interesting calculations can then be made as the round-trip gain in the cavity approaches the threshold limit from below. To show this, suppose we write the round-trip gain inside a regenerative cavity in the phase-amplitude form

\[\tag{63}\tilde{g}_\text{rt}(\omega)\equiv{g}_\text{rt}(\omega)e^{-j\phi(\omega)}\]

Then we can generally assume that the round-trip gain magnitude \(g_\text{rt}(\omega)\) will be essentially constant across any one axial-mode peak at \(\omega=\omega_q\), although the value of \(g_\text{rt,q}\equiv{g}_\text{rt}(\omega_q)\) will change from one axial mode peak to the next, depending on where each individual peak is located within the atomic linewidth.

This is equivalent to saying that the laser gain coefficient \(\alpha_m(\omega)\) within any one axial-mode peak may be approximated by its value at the center of that peak; i.e., \(\alpha_m(\omega)\equiv\alpha_m(\omega_q)\equiv\alpha_{mq}\) for \(\omega\approx\omega_q\) for the \(q\)-th axial mode. We must, however, keep track of the slightly different gain values \(\alpha_{mq}\) at different axial modes \(q\), because very small differences in \(\exp(2\alpha_{mq}p_m)\) between different axial modes can lead to large differences in the height of the overall gain peaks, especially as the centermost mode approaches threshold.

Near any single high-gain axial-mode peak located at \(\omega=\omega_q\), we can also approximate the round-trip phase shift inside the cavity by

\[\tag{64}\phi(\omega)\approx\frac{\omega{p}}{c}=\frac{\omega_qp}{c}+\frac{(\omega-\omega_q)p}{c}=q\times2\pi+\delta\phi(\omega)\]

where \(q\times2\pi\) is the phase shift exactly at the axial-mode peak, and the additional phase shift \(\delta\phi(\omega)\) given by

\[\tag{65}\delta\phi(\omega)\equiv\frac{\omega-\omega_q}{c}p\approx2\pi\times\frac{\omega-\omega_q}{\Delta\omega_\text{ax}}\]

is the small phase deviation as we tune away from resonance. In writing these expressions, we have supposed that any small atomic pulling effects due to the \(\Delta\beta_mp_m\) term are incorporated into a slightly pulled value for the axial-mode frequency \(\omega_q\).

 

Gain and Bandwidth Near Any One Axial-Mode Peak

Suppose we consider only those frequencies within a narrow bandwidth about one such axial mode, so that \(\omega\approx\omega_q\) and \(|\omega-\omega_q|\ll\Delta\omega_\text{ax}\). We may then make the approximation that

\[\tag{66}e^{-j\phi(\omega)}=e^{-j\delta\phi(\omega)}\approx1-j\delta\phi(\omega)=1-j2\pi\frac{\omega-\omega_q}{\Delta\omega_\text{ax}}\]

The transmission gain given by Equation 11.62 near this one axial-mode peak (assuming for simplicity that the round-trip path length is evenly divided between the forward and reverse paths, as in a standing-wave cavity) can then be put into the form

\[\tag{67}\begin{align}\left.\frac{\tilde{E}_\text{trans}}{\tilde{E}_\text{inc}}\right|_{\omega\approx\omega_q}&=-\frac{t_1t_2}{\sqrt{r_1r_2}}\frac{g_\text{rt}^{1/2}(\omega)e^{-j\phi(\omega)/2}}{1-g_\text{rt}(\omega)e^{-j\phi(\omega)}}\\&\approx-\frac{t_1t_2}{\sqrt{r_1r_2}}\frac{g_\text{rt,q}^{1/2}e^{-j\phi(\omega)/2}}{1-g_{\text{rt},q}+j(2\pi{g_{\text{rt},q}}/\Delta\omega_\text{ax})\times(\omega-\omega_q)}\end{align}\]

The overall gain profile for this one axial mode can evidently be well approximated by a complex lorentzian resonance lineshape, so that we can rewrite this expression in the form

\[\tag{68}\left.\frac{\tilde{E}_\text{trans}}{\tilde{E}_\text{inc}}\right|_{\omega\approx\omega_q}=-e^{-j\phi(\omega)/2}\frac{g_{0,q}}{1+2j(\omega-\omega_q)/\Delta\omega_{3\text{dB},q}}\]

The minus sign in front of this expression comes from our convention for mirror transmissions, and the \(e^{-j\phi(\omega)/2}\) term is simply a phase shift term representing the net optical path length \(\omega{L/c}\) from mirror \(M_1\) to mirror \(M_2\). The significant part of this expression is the remaining portion, which is a complex lorentzian lineshape with a peak voltage gain from input to output of \(g_{0,q}\) for the \(q\)-th axial-mode gain peak, and a FWHM bandwidth of \(\Delta\omega_{3\text{dB},q}\) for that same gain peak.

By comparing Equations 11.67 and 11.68, we see that in the highly regenerative limit any single axial-mode peak thus has a midband voltage gain magnitude given by

\[\tag{69}g_{0,q}\equiv\frac{t_1t_2}{\sqrt{r_1r_2}}\frac{g_{\text{rt},q}^{1/2}}{1-g_{\text{rt},q}}\]

and a 3 dB amplification bandwidth given by

\[\tag{70}\Delta\omega_{3\text{dB},q}\approx\frac{1-g_{\text{rt},q}}{g_{\text{rt},q}}\times\frac{\Delta\omega_\text{ax}}{\pi}\]

As the round-trip gain magnitude \(g_{\text{rt},q}\) inside the cavity approaches unity, the corresponding regenerative transmission gain through the cavity obviously becomes very high, so that \(g_{0,q}\rightarrow\infty\), and the bandwidth of that gain peak becomes very narrow, so that \(\Delta\omega_{3\text{dB},q}\rightarrow0\).

 

Gain-Bandwidth Product

But more than this, as the peak gain becomes very large and the bandwidth very small, their product approaches a fixed gain-bandwidth product given by

\[\tag{71}[g_0\Delta\omega_{3\text{dB}}]_q\approx{g}_{\text{rt},q}^{-1/2}\times\frac{t_1t_2}{\sqrt{r_1r_2}}\times\frac{\Delta\omega_\text{ax}}{\pi}\]

But since in the high-gain limit \(g_{\text{rt},q}\rightarrow1\), we can further simplify this

\[\tag{72}g_0\Delta\omega_{3\text{dB}}\approx\frac{t_1t_2}{\sqrt{r_1r_2}}\times\frac{\Delta\omega_\text{ax}}{\pi}\]

In this final result, therefore, the dependence on the atomic gain and cavity losses, and even on the axial-mode coefficient \(q\), drops out entirely, leaving only the cavity external-coupling parameters \(r_1\), \(r_2\) and \(t_1\), \(t_2\) in the formula.

We conclude that there is a fixed gain-bandwidth product in the high-gain limit for each axial-mode peak. Moreover, this gain-bandwidth product is the same for all axial modes, and depends only on the cavity coupling parameters, i.e., on \(r_1\), \(r_2\), \(t_1\), and \(t_2\), and not on either the laser gain or the internal cavity losses.

Note that the peak transmission gain values \(g_{0,q}\) for the different axial modes across an atomic-gain curve will have significantly different values, because of the slightly different values of \(\alpha_{mq}\) or \(g_{\text{rt},q}\) in the resonance denominators. The centermost mode will rapidly outstrip the off-center modes as its value of roundtrip gain \(g_{\text{rt},q}\) comes closest to unity.

Earlier figures have illustrated how the peak gain of the most favored mode races up to infinity, and the bandwidth heads toward zero, as the mode approaches threshold. The off-center axial modes will not come quite as close to the threshold limit, but all will have the same gain-bandwidth product.

 

Numerical Example

Does this kind of regenerative gain enhancement have practical applications in laser devices?

The answer is generally no, first because the practical gain-bandwidth products are too small to be useful, and second because the necessary adjustments of the round-trip gain to achieve high overall gain are too delicate to be controlled in practical applications. Unless a laser medium has enough single-pass gain to be used without regeneration, it is probably not useful as a laser amplifier.

On the other hand, the theory of regenerative laser amplification is very useful in understanding laser physics and particularly in understanding the manner in which laser oscillators approach oscillation threshold.

As a representative numerical example for gain-bandwidth product, we might consider a typical low-loss laser cavity with the parameters

\[\tag{73}\left.\begin{array}{ll}r_1r_2=R=0.97\\t_1t_2=T=0.03\\L=30\text{ cm}\\\Delta\omega_\text{ax}=2\pi\times500\text{ MHz}\quad\end{array}\right\}\qquad{g_0\Delta{f}_\text{3dB}}\approx5\text{ MHz}\]

Suppose we want to place a 30-cm-long He-Ne laser tube, which might be able to produce somewhere between 5% and 10% power gain per one-way pass, inside this cavity, and then magnify this up by regeneration to obtain a peak-transmission gain for the centermost axial mode of \(g_0=10\) or \(g_0^2=100=20\) dB.

Since this cavity has about 6% power loss through the end mirrors per round trip, and perhaps a few percent more of internal losses, the He-Ne laser tube will easily be able to bring the cavity arbitrarily close to threshold, and produce the desired 20 dB of overall gain from input to output on the centermost axial mode.

The amplification bandwidth of this axial mode will then, however, turn out to be only \(\Delta{f}_\text{3dB}\approx500\) kHz! The usefulness of a bandwidth of a few hundred kHz, even though it may be at an optical carrier frequency, seems dubious. In fact, even to measure this bandpass will require a frequency stability of \(\Delta{f}/f_0\approx5\times10^5/5\times10^{14}\approx1\times10^{-9}\) between the signal source and the laser amplifier. Regenerative optical amplifiers are thus useful as a source of insight into the physics of laser oscillation, but seem not to have practical applications.

The concept of a fixed gain-bandwidth product which we have derived here applies, of course, not only to laser amplifiers, but also to any type of regenerative amplifier in any frequency range. Given any kind of electronic or acoustic or mechanical amplification process, no matter how weak its gain, we can always employ positive feedback to increase the overall gain by any desired amount.

If the feedback loop has a long time delay, however, as is inherent in a laser merely from the propagation time around the cavity, the total phase shift in the feedback loop will be large and will change rapidly with frequency. This in turn will inherently limit the bandwidth or, more precisely, the gain-bandwidth product of the regeneratively magnified amplification.

 

Regenerative Noise Amplification in a Laser

The fixed gain-bandwidth product for a regenerative laser amplifier operating just below threshold can be used to derive, at least in a heuristic fashion, one of the most famous formulas in laser theory, the so-called Schawlow-Townes formula for the spectral linewidth caused by quantum noise of a laser oscillator operating above threshold.

To derive this, we must first note that a regenerative laser amplifier—or indeed any other kind of coherent optical amplifier—will have a certain finite amount of noise because of spontaneous emission from the upper-laser-level atoms inside the amplifier.

For a laser amplifier of any kind, regenerative or single pass, this finite amount can be represented by an equivalent input noise power, which we view as coming into the input of the amplifier, with an input noise power spectral density given by

\[\tag{74}\frac{dP_n}{d\omega}=\frac{N_2}{N_2-N_1}\times\hbar\omega\]

In other words, the equivalent input noise power to the amplifier is equivalent to one input photon per second per cycle of bandwidth, multiplied by an excess noise factor \(N_2/(N_2-N_1)\). This excess noise factor is unity if the lower laser level is empty, so that \(N_2-N_1=N_2\). It becomes larger than unity if \(N_1\) is finite, because then more upper-level atoms and hence more spontaneous emission will be present for the same net inversion and gain.

Consider now a regenerative laser amplifier pumped right up to the oscillation threshold point, with no coherent input signal applied. Even with no input signal present, this laser will still amplify the equivalent input noise within its 3 dB amplification bandwidth \(\Delta\omega_\text{3dB}\). (Really, of course, it is regeneratively amplifying its own spontaneous emission generated within the laser cavity.)

The effective rectangular bandwidth of a lorentzian amplifier with FWHM bandwidth \(\Delta\omega_\text{3dB}\) is in fact \((\pi/2)\times\Delta\omega_\text{3dB}\).

Hence the total amplified noise output power from the amplifier very close to threshold will be given by

\[\tag{75}\begin{align}P_\text{out}&=G_0\times\frac{\pi\Delta\omega_\text{3dB}}{2}\times\frac{dP_n}{d\omega}\\&=\frac{N_2}{N_2-N_1}\times\frac{\pi{G_0}\Delta\omega_\text{3dB}\hbar\omega}{2}\end{align}\]

where \(G_0\equiv{g_0}^2\) is the midband overall transmission gain through the laser cavity.

As we bring the cavity closer and closer to threshold, the overall gain \(G_0\) will become very large; the bandwidth \(\Delta\omega_\text{3dB}\) will become very narrow; and the noise output from the laser will become an increasingly powerful but increasingly narrowband amplified noise signal.

Suppose we bring the cavity extremely close to oscillation threshold, as shown in Figure 11.28. One partially correct, but incomplete, way of describing what happens as the laser comes very close to the threshold point is the following: When the amplified noise output power comes close to the available power that can be extracted from the laser gain medium, then the laser gain medium begins to saturate.

As a result of this gain saturation, the overall power gain \(G_0\) will no longer increase beyond the point where the (very) narrowband amplified noise output equals the potential output oscillation power from the laser.

 

 

The Schawlow-Townes Formula

From this (partially correct) viewpoint, the laser oscillator is simply a very high-gain, very narrow bandwidth, amplified spontaneous emission noise source operating just the slightest bit below the exact threshold point, with an overall power output given by Equation 11.75, with \(P_\text{out}=P_\text{osc}\), where \(P_\text{osc}\) is the free-running power output that the laser oscillator can deliver.

But we also have a constant gain-bandwidth expression connecting \(g_0\equiv{G}_0^{1/2}\) and \(\Delta\omega_\text{3dB}\). For simplicity let us assume a laser cavity with very small internal losses and reasonably small external coupling, so that we can write \(r_1r_2\approx1\) and \(t_1t_2\approx\delta_c\), where \(\delta_c\) is the cavity loss factor derived earlier.

By using the cavity \(Q_c\) definition from Equation 11.40, we can then rewrite the gain-bandwidth product in the form

\[\tag{76}G_0^{1/2}\Delta\omega_\text{3dB}\approx\frac{2\omega_a}{Q_c}=2\Delta\omega_c\]

where \(\Delta\omega_c\equiv\omega/Q_c\) is the "cold cavity" bandwidth of any one axial mode in the laser cavity due to its external coupling.

Combining Equations 11.75 and 11.76 then gives the interesting result that

\[\tag{77}\Delta\omega_\text{osc}=\Delta\omega_\text{3dB}\approx(2)\times\frac{N_2}{N_2-N_1}\times\frac{\pi\hbar\omega\Delta\omega_c^2}{P_\text{osc}}\]

where \(\Delta\omega_c\) is to be interpreted as the spectral width of the highly amplified noise coming out of the laser operating at or above threshold. This formula for the "noise bandwidth" of the laser output is commonly referred to as the "Schawlow- Townes formula" for a laser oscillator. Note that this linewidth depends only on the cold-cavity bandwidth \(\Delta\omega_c\) of the laser cavity, and on the power level \(P_\text{osc}\) at which the laser oscillates above threshold.

  

More Correct Description

We have put the factor of 2 in Equation 11.77 in brackets to emphasize the way in which this formula is partially correct and partially incorrect. When a laser reaches its oscillation threshold, there is a qualitative change in the character of the laser output spectrum.

The laser changes over just at threshold from being a very narrowband but still essentially incoherent gaussian noise source, with large (but slow) fluctuations in both amplitude and phase, to being a coherent sinusoidal oscillator, with a highly stabilized phasor amplitude, but still with random noise-like but very slow fluctuations or drifts in the oscillation phase.

If we simply delete the bracketed factor of 2 in Equation 11.77, this equation still correctly predicts the spectral bandwidth of the coherent laser oscillation above threshold, as caused by random phase fluctuations in the laser output.

In other words, the Schawlow-Townes result, reduced by a factor of two, is still correct in the nonlinear region above threshold, even though it is derived by using a linear below-threshold model.

The phase fluctuations and the consequent oscillation spectral broadening caused by spontaneous emission in a laser oscillator above threshold (along with some small but still observable residual amplitude fluctuations) are commonly referred to as quantum noise fluctuations in the laser.

These quantum amplitude and frequency fluctuations in ordinary lasers are usually completely masked by much larger fluctuations due to mechanical vibrations, acoustic noise, thermal drift, and other "technical noise sources."

Quantum amplitude and frequency fluctuations have been measured, however, in excellent agreement with the Schawlow-Townes formula, by careful measurements both on highly stabilized gas lasers, and on semiconductor injection lasers, where these fluctuations can be substantially more noticeable.

 

The next tutorial discusses about analytical modeling of the impact of fiber non-linear propagation on coherent systems and networks.