PHYSICAL PROPERTIES OF GAUSSIAN BEAMS

This is a continuation from the previous tutorial - Wave optics and gaussian beams

 

 

1. GAUSSIAN BEAM PROPAGATION

We first look in this section at what the analytic expressions for a lowest-order gaussian beam imply physically in terms of aperture transmission, collimated beam distances, far-field angular beam spread, and other practical aspects of gaussian beam propagation.

 

Analytical Expressions 

Let us assume a lowest-order gaussian beam characterized by a spot size Wo and a planar wavefront \(R_0=\infty\) in the transverse dimension, at a reference plane which for simplicity we take to be \(z=0\). This plane will henceforth be known for obvious reasons as the beam waist, as in Figure 1. 

 

 

 

The normalized field pattern of this gaussian beam at any other plane \(z\) will then be given by 

\[\tag{1}\begin{align}\tilde{u}(x,y,z)&=\left(\frac{2}{\pi}\right)^{1/2}\frac{\tilde{q_0}}{w_0\tilde{q}(z)}\text{exp}\left[-jkz-jk\frac{x^2+y^2}{2\tilde{q}z}\right]\\&=\left(\frac{2}{\pi}\right)^{1/2}\frac{\text{exp}[-jkz+j\psi(z)]}{w(z)}\text{exp}\left[-\frac{x^2+y^2}{w^2(z)}-jk\frac{x^2+y^2}{2R(z)}\right],\end{align}\]

where the complex radius of curvature \(\tilde{q}(z)\) is related to the spot size \(w(z)\) and the radius of curvature \(R(z)\) at any plane \(z\) by the definition

\[\tag{2}\frac{1}{\tilde{q}(z)}\equiv\frac{1}{R(z)}-j\frac{\lambda}{\pi w^2(z)}.\]

In free space this parameter obeys the propagation law 

\[\tag{3}\tilde{q}(z)=\tilde{q}_0+z=z+jzR,\]

with the initial value 

\[\tag{4}\tilde{q}_0=j\frac{\pi w^2_0}{\lambda}=jzR.\]

Note that the value of \(\lambda\) in these formulas is always the wavelength of the radiation in the medium in which the beam is propagating. 

 

 

 

All the important parameters of this gaussian beam can then be related to the waist spot size \(w_0\) and the ratio \(z/z_R\) by the formulas 

\[\tag{5}\begin{align}&w(z)=w_0\sqrt{1+\left(\frac{z}{z_R}\right)^2,}\\&R(z)=z+\frac{z^2_R}{z},\\&\psi(z)=\tan^{-1}\left(\frac{z}{z_R}\right).\end{align}\] 

In other words, the field pattern along the entire gaussian beam is characterized entirely by the single parameter \(w_)\) or \(\tilde{q}_0, or z_R)\) at the beam waist, plus the wavelength \(\lambda\) in the medium. 

 

Aperture Transmission 

Before exploring the free-space propagation properties of an ideal gaussian beam, we might consider briefly the vignetting effects of the finite apertures that will be present in any real optical system. The intensity of a gaussian beam falls off very rapidly with radius beyond the spot size \(w\).

How large must a practical aperture be before its truncation effects on a gaussian beam become negligible? 

Suppose we define the total power in an optical beam as \(P=\int\int|\tilde{u}|^2dA\) where \(dA\) integrates over the cross-sectional area. The radial intensity variation of a gaussian beam with spot size w is then given by

\[\tag{6}I(r)=\frac{2P}{\pi w^2}e^{-2r^2/w^2}.\]

The effective diameter and area of a uniform cylindrical beam (a "top hat beam") with the same peak intensity and total power as a cylindrical gaussian beam will 

 

 

then be

\[\tag{7}d_{TH}=\sqrt2w\quad\text{and}\quad A_{TH}=\frac{\pi w^2}{2}\] 

as shown in Figure 2. 

An aperture significantly larger than this will be needed, however, to pass a real gaussian beam of spot size \(w\) without serious clipping of the beam skirts. The fractional power transfer, for example, for a gaussian beam of spot size \(w\) passing through a centered circular aperture of diameter \(2a\), as in Figure 3, will be given by 

\[\tag{8}\text{power}\;\text{transmission}=\frac{2}{\pi w^2}\int^a_02\pi re^{-2r^2/w^2}dr=1-e^{-2a^2/w^2}.\]

This figure plots this transmission versus aperture radius a normalized to spot size \(w\). An aperture with radius \(a=w\) transmits \(\approx\) 86% of the total power in the gaussian beam. We will refer to this as the \(1/e\) or 86% criterion for aperture size.

A more useful rule of thumb to remember, however, is that an aperture with radius \(a=(\pi/2)\)w, or diameter \(d=\pi w\), will pass just over 99% of the gaussian beam power. We will often use this as a practical design criterion for laser beam apertures, and will refer to it as the "\(d=\pi w\)" or 99% criterion. (A criterion of \(d=3w\) which gives \(\approx\) 98.9% transmission would obviously serve equally well.) Figure 4. illustrates just where some of these significant diameters for a gaussian beam will fall on the gaussian beam profile.

 

 

 

Aperture Diffraction Effects 

Optical designers should take note, however, that sharp-edged apertures, especially circular apertures, even though they may cut off only a very small fraction of the total power in an optical beam, will also produce aperture diffraction effects like those shown in Figure 5., which will significantly distort the intensity pattern of the transmitted beam in both the near-field (Fresnel) and far-field (Fraunhofer) regions. 

We will show in the following chapter, for example, that the diffraction effects on an ideal gaussian beam of a sharp-edged circular aperture even as large as the \(d=\pi w\) criterion will cause near-field diffraction ripples with an intensity variation \(\Delta I/I\approx\pm\)17% in the near field, along with a peak intensity reduction of \(\approx\)17% on axis in the far field. We have to enlarge the aperture to \(d\approx4.6w\) to get down to \(\pm\)1% diffraction ripple effects from a sharp-edged circular aperture. 

 

Beam Collimation: The Rayleigh Range and the Confocal Parameter

Another important question is how rapidly an ideal gaussian beam will expand due to diffraction spreading as it propagates away from the waist region or,* in practical terms, over how long a distance can we propagate a collimated gaussian beam before it begins to spread significantly? 

The variation of the beam spot size w(z) with distance as given by Equation 5. is plotted in Figure 6. for two different waist spot sizes \(w_{01}\) and \(w_{02}>w_{01}\), with the transverse scale greatly enlarged.

The primary point is that as the input spot size \(w_o\) at the waist is made smaller, the beam expands more rapidly due to diffraction; remains collimated over a shorter distance in the near field; and diverges at a larger beam angle in the far field.

In particular, the distance which the beam travels from the waist before the beam diameter increases by \(\sqrt 2\), or before the beam area doubles, is given simply

 

 

 

 

 

by the parameter 

\[\tag{9}z=z_R\equiv\frac{\pi w^2_0}{\lambda}=\text{"Rayleight}\;\text{range."}\]

The term Rayleigh range is sometimes used in antenna theory to describe the distance \(z\approx d^2/\lambda\) that a collimated beam travels from an antenna of aperture diameter d (assuming \(d\gg\lambda)\) before the beam begins to diverge significantly. We have 

 

 

 

therefore adopted the same term here as a name for the quantity \(z_R\equiv\pi w^2_0/\lambda\). The Rayleigh range marks the approximate dividing line between the "near-field" or Fresnel and the "far-field" or Fraunhofer regions for a beam propagating out from a gaussian waist. 

To express this same point in another way, if a gaussian beam is focused from an aperture down to a waist and then expands again, the full distance between the \(\sqrt 2w_0\) spot size points is the quantity \(b\) given by 

\[\tag{10}b=2z_R=\frac{2\pi w^2_0}{\lambda}=\text{confocal}\;\text{parameter}.\]

This confocal parameter was widely used in earlier writings to characterize gaussian beams. Using the Rayleigh range \(z_R\equiv b/2\), as shown in Figure 7, seems, however, to give simpler results in most gaussian beam formulas. 

 

Collimated Gaussian Beam Propagation

Over what distance can the collimated waist region of an optical beam then extend, in practical terms? To gain some insight into this question, we might suppose that a gaussian optical beam is to be transmitted from a source aperture of diameter \(D\) with a slight initial inward convergence, as shown in Figure 8, so that the beam focuses slightly to a waist with spot size \(w_0\) at one Rayleigh range out, and then reexpands to the same diameter \(D\) two Rayleigh ranges (or one confocal parameter) out. We will choose the aperture diameter according to the \(\pi w\) or 99% criterion, i.e., we will use \(D=\pi\times\sqrt 2w_0\)at each end. 

The relation between the collimated beam distance and the transmitting aperture size using this criterion is then 

\[\tag{11}\text{collimated}\;\text{range}=2z_R=\frac{2\pi w^2_0}{\lambda}\approx\frac{D^2}{\pi\lambda}.\]

Some representative numbers for this collimated beam range at two different laser wavelengths are illustrated in Figure 8. and in Table 1. A visible laser with a 1 cm diameter aperture can project a beam having an effective diameter of a few mm with no significant diffraction spreading over a length of 50 meters or more. Such a beam can be used, for example, as a "weightless string" for alignment on a construction project. With the aid of a simple photocell array,

 

 

 

TABLE 1.  

Collimated Laser Beam Ranges 

 

the center of such a beam can easily be found to an accuracy of better than \(w/20\), or a small fraction of a mm, over the entire distance. 

 

Far-Field Beam Angle: The "Top Hat" Criterion 

Suppose we next move out into the far field, where the beam size expands linearly with distance, as in Figure 9. At what angle does a gaussian beam spread in the far field, that is, for \(z\gg z_R\)? 

From the gaussian beam equations (1-5), the \(1/e\) spot size \(w(z)\) for the field amplitude in the far field for a gaussian beam coming from a waist with spot size \(w_0\) is given by

\[\tag{12}w(z)\approx\frac{w_0z}{z_R}=\frac{\lambda z}{\pi w_0}\quad(z\gg z_R),\] 

 

 

 

which gives the simple relation 

\[\tag{13}w_0\times w(z)\approx\frac{\lambda z}{\pi}\]

connecting the spot sizes at the waist and in the far field. The far-field angular beam spread for a gaussian beam can then related to the near-field beam size or aperture area in several different ways, depending on how conservative we want to be.

The on-axis beam intensity in the far field, for example, is given by

\[\tag{14}I_\text{axis}(z)=\frac{2P}{\pi w^2(z)}\approx\frac{P}{\lambda^2z^2/2\pi w^2_0}.\] 

Hence, the on-axis intensity is the same as if the total power \(P\) were uniformly distributed over an area \(\pi w^2(z)/2=\lambda^2z^2/2\pi w^2_0\). The solid angle for an equivalent "top hat" angular distribution in the far field, call it \(\Omega_\text{TH}(z)\), is thus given by

\[\tag{15}\Omega_\text{TH}=\frac{\pi w^2(z)}{2z^2}=\frac{\lambda^2}{2\pi w^2_0}.\]

At the same time, the "equivalent top hat" definition of the source area at the waist is given from Equation 7. by \(A_\text{TH}=\pi w^2_0/2\). The product of these two quantities is thus given by 

\[\tag{16}A_\text{TH}\times\Omega_\text{TH}=\left(\frac{\lambda}{2}\right)^2.\] 

The source aperture size (at the waist) and the far-field solid angular spread thus have a product on the order of the wavelength \(\lambda\) squared, although the exact numerical factor will depend on the definitions we choose for the area and the solid angle, as we will see in more detail later. 

 

Far-Field Beam Angle: The 1/e Criterion 

Another and perhaps more reasonable definition for the far-field beam angle is to use the \(1/e\) or 86% criterion for the beam diameter, so that the far field half-angular spread is defined by the width corresponding to the \(1/e\) point for the \(E\) field amplitude at large \(z\). 

With this definition, the half-angle \(\theta_{1/e}\) out to the 1/e amplitude points in the far-field beam is given, as shown in Figure 9., by

\[\tag{17}\theta_{1/e}=\lim_{z\to\infty}\frac{w(z)}{z}=\frac{\lambda}{\pi w_0}.\]

Twice this angle then gives a full angular spread of

\[\tag{18}2\theta_{1/e}=\frac{2\lambda}{\pi w_0},\]

which can be interpreted as a more precise formulation, valid for gaussian beams, of the approximate relation \(\Delta\theta\approx\lambda/d\) that we gave in An Introduction to Lasers this tutorials.

We can then define the gaussian beam solid angle \(\Omega_{1/e}\) on this same basis as the circular cone defined by this angular spread, or

\[\tag{19}\Omega_{1/e}=\pi\theta^2_{1/e}=\frac{\lambda^2}{\pi w^2_0}.\]

This cone will, as noted in the preceding, contain 86% of the total beam power in the far field.

Suppose we use the same 1/e criterion to define the effective radius of the input beam at the beam waist (ignoring the fact that an aperture of radius \(a=w_0\) at the waist would actually produce some very substantial diffraction effects on the far-field beam pattern). Then the product of the effective source aperture area \(A_{1/e}\equiv\pi w^2_0/2\) and the effective far-field solid angle \(\pi\theta^2_{1/e}\), using these 1/e definitions becomes

\[\tag{20}A_{1/e}\Omega_{1/e}=\pi w^2_0\times\pi\theta^2_{1/e}=\lambda^2.\]

This is a precise formulation for gaussian beams of a very general antenna theorem which states that

\[\tag{21}\int\int A(\Omega)d\Omega=\lambda^2\]

This theorem says in physical terms that if we measure the effective capture area \(A(\Omega)\)of an antenna for plane-wave radiation arriving from a direction specified by the vector angle \(\Omega=(\theta,\phi)\), and then integrate these measured areas over all possible arrival angles as specified by \(d\Omega\), the result (for a lossless antenna of any form) is always just the measurement wavelength \(\lambda\). This result is valid for any kind of antenna, at radio, microwave or optical wavelengths. 

 

Far-Field Beam Angle: Conservative Criterion

Finally, as a still more conservative way of expressing the same points, we might use the \(d=\pi w\) or 99% criterion instead of the 1/e criterion to define both the effective source aperture size and the effective far field solid angle.

We might then say that a source aperture of diameter \(d=\pi w_0\) transmitting a beam of initial spot size \(w_0\) will produce a far-field beam with 99% of its energy within a cone of full angular spread \(2\theta_\pi=\pi w(z)/z\). On this basis the source aperture area, call it \(A_\pi\) is \(\pi d^2/4\) and the beam far-field solid angle is \(\Omega_\pi=\pi\theta^2_\pi\); and these are related by the more conservative criterion 

\[\tag{22}A_\pi\Omega_\pi=\left(\frac{\pi}{2}\right)^4\lambda^2\approx6\lambda^2.\]

 

 

 

None of the criteria we have introduced here for defining effective aperture size and effective solid angle are divinely ordained, and which of them we use should depend largely on what objective we have in mind. 

 

Wavefront Radius of Curvature 

We can next look at how the wavefront curvature of a gaussian beam varies with distance. The radius of curvature \(R(z)\) of a gaussian beam has a variation with distance given analytically by 

\[\tag{23}R(z)=z+\frac{z^2R}{z}\approx\left\{\begin{align}\infty\quad\text{for}\quad z\ll z_R\\2z_R\quad\text{for}\quad z=z_R\\z\quad\text{for}\quad z\gg z_R\end{align}.\right.\]

This is plotted against normalized distance in Figure 10(a). 

The wavefront is flat or planar right at the waist, corresponding to an infinite radius of curvature or \(R(0)=\infty\). As the beam propagates outward, however, the wavefront gradually becomes curved, and the radius of curvature \(R(z)\) drops rather rapidly down to finite values (see Figure 10).

For distances well beyond the Rayleigh range \(z_R\) the radius then increases again as \(R(z)\approx z\), i.e., the gaussian beam becomes essentially like a spherical wave centered at the beam waist.

What this means in physical terms is that the center of curvature of the wavefront starts out at \(-\infty\) for a wavefront right at the beam waist, and then moves monotonically inward toward the waist, as the wavefront itself moves outward toward \(z\rightarrow +\infty\). 

 

Confocal Curvatures

The minimum radius of curvature occurs for the wavefront at a distance from the waist given by \(z=z_R\), with the radius value \(R=b=2z_R\). This means that at this point the center of curvature for the wavefront at \(z=+z_R\) is located at \(z=-z_R\), and vice versa, as illustrated in Figure 10. 

This particular spacing has a special significance in stable resonator theory. Suppose the curved wavefronts \(R(z)\) at \(\pm z_R\) are matched exactly by two curved mirrors of radius \(R\) and separation \(L=R=b=2z_R\). Since the focal point of a curved mirror of radius \(R\) is located at \(f=R/2\), the focal points of these two mirrors then coincide exactly at the center of the resonator.

The two mirrors are said to form a symmetric confocal resonator, thus giving rise to the confocal parameter \(b\equiv 2z_R\equiv 2\pi w^2_0/\lambda\). Such a resonator has certain particularly interesting mode properties which we will explore later.

 

 

2. GAUSSIAN BEAM FOCUSING

Besides propagating collimated gaussian beams over long distances, we are often interested in focusing such beams to very small spots, whether for recording data on optical videodisks or tapes, drilling holes in razor blades, or counting cell nuclei in a laser microscope. (Since the standard demonstration of ruby laser intensity in early days was to zap a hole in one or more razor blades with a single laser shot, pulsed laser energies were occasionally quoted in "Gillettes.") What sort of focused spot sizes and intensities can be achieved with a gaussian beam—or for that matter with any reasonably well-formed optical beam? 

 

Focused Spot Sizes

The usual situation where a collimated gaussian beam is strongly focused by a lens of focal length \(f\), as shown in Figure 11, can be viewed as simply the far-field beam problem of Figure 9 in reverse. The waist region now becomes the focal spot of spot size \(w_0\), whereas the focusing lens can be viewed as being in the far field at \(z\approx\pm f\). If \(w(f)\) is the gaussian spot size at the lens, we then have the same relationship as Equation 13. but with a reverse interpretation, namely, 

\[\tag{24}w_0\times w(f)\approx\frac{f\lambda}{\pi}.\]

What does this expression imply in practical terms?

It seems obvious that in a practical focusing problem, the incident gaussian beam should fill the aperture of the focusing lens to the largest extent possible without a severe loss of power due to the finite aperture of the lens (and also without serious edge diffraction effects). As one reasonable criterion for practical 

 

 

designs, we might adopt the \(D=\pi w(f)\) or 99% criterion for the diameter d of the focusing lens, so that we lose \(<\) 1% of the incident energy in this lens. At the same time we might adopt the 1/e or \(d_0=2w_0\) criterion for defining the effective diameter do of the focused spot, since this is a diameter which contains 86% of the focused energy, and at the edges of which the focused intensity is already down to 1/\(e^2\) \(\approx\) 14% of its peak value. Combining these criteria then gives 

\[\tag{25}d_0\approx\frac{2f\lambda}{D}\]

for the effective diameter of the focused gaussian spot.

The \(f\)-number of a focusing lens (also called the relative aperture or the speed of the lens) is defined by

\[\tag{26}f^\#\equiv\frac{f}{D}.\]

The focal spot diameter, using the rather arbitrary criteria we have just selected, will then be given by

\[\tag{27}d_0\approx2f^\#\lambda.\]

As an alternative way of reaching essentially this same conclusion, we can calculate that if a gaussian beam carries total power \(P\) and we focus it using a lens of focal length \(f\) with the same \(D=\pi w\) criterion for the lens diameter, then the peak intensity at the center of the focused spot will be given by

\[\tag{28}I_0=\frac{2P}{\pi w^2_0}\approx\frac{P}{2(f^\#\lambda)^2}.\]

The peak intensity is thus the same as if all the energy were focused into a circle with an area of \(2(f^\#\lambda)^2\), or a diameter of \((8/\pi)^{1/2}f^\#\lambda\approx1.6f^\#\lambda\).

 

Influence of the Lens f Number and the Lens Fresnel Number

Whatever the choice of definitions, it is evident that an ideal gaussian beam can be focused down to a spot that is roughly one to two optical wavelengths in diameter, multiplied by the \(f\)-number of the focusing lens. Note that a long focal length lens, say, an \(f/10\) lens, will generally be simple, inexpensive and easy to obtain, with quite small aberration coefficients. Lenses with /-numbers less than 2, and especially with \(f^\#\leq 1\), on the other hand, generally require complex and expensive multielement designs, and can become very expensive.

Some optics workers also like to characterize a simple lens of diameter \(D=2a\) and focal length \(f\) by its lens Fresnel number \(N_f\), given by

\[\tag{29}N_f\equiv\frac{a^2}{f\lambda}.\]

In terms of this quantity plus our arbitrary criteria for beam and spot diameters, the focal spot diameter and the lens diameter are then related by

\[\tag{30}\frac{d_0}{D}\approx\frac{1}{2N_f}.\]

Whereas the \(f\)-number of a given lens is independent of wavelength, the Fresnel number depends on wavelength. The limitation expressed by Equation 30 can become significant particularly for longer wavelengths,, for example, in focusing infrared beams using \(\text{IR}\) lenses. Strong focusing, down to a spot size much less than the lens diameter, requires a lens with an adequately large Fresnel number \(N_f\).

A crucial condition for accomplishing strong focusing, regardless of definitions, is that the incident gaussian beam properly fill the focusing lens aperture, since it is the gaussian beam diameter and not the lens diameter that is the critical dimension in determining the focal spot size of the gaussian beam.

 

Depth of Focus

The depth of focus of a gaussian beam is obviously given by the Rayleigh range \(z_R\) of the gaussian waist, or perhaps by \(2z_R\), depending upon just how we want to define the depth of focus. If we use the latter definition, along with the lens diameter criterion \(D=\pi w(f)\), then the depth of focus can be written as 

\[\tag{31}\text{depth}\;\text{of}\;\text{focus}=2z_R\approx2\pi f^{\#^2}\lambda\approx\frac{\pi}{2}\left(\frac{d_0}{\lambda}\right)^2\lambda.\]

If the beam is focused down to a spot \(N\) wavelengths in diameter, the depth of focus will be \(\approx N^2\) wavelengths in length. 

All these expressions for focused spot size and depth of focus do of course assume \((a)\) that the gaussian beam entering the lens is more or less collimated, with a planar wavefront, so that the beam focuses approximately at the focal point \(f\); and \((b)\) that the beam is in fact "strongly focused," in the sense that \(w_0\ll w(f)\), or \(z_R\ll f\), or \(N_f\gg 1\).

This latter point is equivalent to saying that the lens is in the far field, as seen looking backward from the waist or the focal point. If either of these assumptions is not entirely valid, corrections must be applied in calculating the exact location and size of the focused spot, as illustrated in several of the Problems following this section.

 

Focal Spot Deviation

When a collimated beam is focused by an ideal lens, the actual focal spot, meaning the position of minimum spot size and maximum energy density, does 

 

 

not in fact occur exactly at the geometrical focus of the lens; but rather is located just slightly inside the lens focal length. The amount of this focal spot deviation—which is typically very small—can be easily calculated for a focused gaussian beam from Figure 12.

Using the notation of Figure 12, we can let the distance from the lens to the beam waist, or the actual focal spot, be \(z\), whereas the focal length of the lens is \(f\). A collimated beam passing through a thin lens of focal length \(f\) acquires (by definition) a wavefront radius of curvature equal to \(f\).

The wavefront curvature just beyond the thin lens must therefore be given, from the combination of gaussian beam theory and lens theory, by 

\[\tag{32}R(z)=z+z^2_R/z=f.\]

The difference between the focal length / and the actual distance z to the waist can then be written as

\[\tag{33}\Delta f\equiv f-z=z^2_R/z\approx z^2_R/f.\]

Since the Rayleigh range \(z_R\) of the focused beam is normally much less than the focal length \(f\), the focal deviation is generally much less than the depth of focus (which means that in fact it is really quite negligible). One way of expressing this criterion is

\[\tag{34}\frac{\Delta f}{f}\approx\frac{1}{2N^2_f}.\] 

As a practical matter, when adjusting an optical setup one very seldom knows the exact value of the lens focal length, or the exact location of the lens focal point, or the exact degree of collimation of the input beam to sufficiently high accuracy that this focal spot deviation is of any practical significance.

We usually find the best focal adjustment in any optical system by small experimental adjustments over a small adjustment range, after the system is assembled. 

 

3. LENS LAWS AND GAUSSIAN MODE MATCHING 

A common requirement in laser optical systems is to propagate a gaussian beam through a cascaded sequence of lenses, free-space regions, and other optical elements, as shown in Figure 14, perhaps in order to match a gaussian beam coming from a waist with specified spot size \(w_1\) at location \(z_1\) into another laser cavity or interferometer requiring waist spot size \(w_2\) at location \(z_2\). The design steps necessary to accomplish this are usually referred to as gaussian beam mode matching. 

Such problems if they become at all complicated are probably best handled by the general \(\text{ABCD}\) methods we will introduce later. A quick introduction to elementary gaussian mode matching techniques at this point may, however, be useful.

 

Lens Laws and Collins Charts 

The lens law for purely spherical waves passing through an ideal thin lens of focal length \(f\) (Figure 13) is 

\[\tag{35}\frac{1}{R_2}=\frac{1}{R_1}-\frac{1}{f}.\]

We follow the standard convention in this book of using positive \(R\) for diverging waves going in the \(+z\) direction, and positive \(f\) for converging or positive lenses. A gaussian spherical beam passing through such a thin lens then has its radius of curvature \(R\) changed in exactly the same way, whereas its spot size \(w\) is unchanged. The lens law for gaussian beams is therefore the direct analog, that is,

\[\tag{36}\frac{1}{\tilde{q}_2}=\frac{1}{\tilde{q}_1}-\frac{1}{f},\]

where \(\tilde{q}\) is the complex curvature parameter defined in Equation 2. 

By applying this lens law, plus the propagation rule \(\tilde{q}_2=\tilde{q}_1+z_2-z_1\) for a free-space section, we can then propagate a gaussian beam forward or backward through any sequence of thin lenses and spaces. It can be helpful to plot this propagation as a trajectory in the complex \(1/\tilde{q}\) plane or, more conveniently, in the complex \(j/\tilde{q}\) plane with rectangular coordinates \(x\) and \(y\) corresponding to \(x\equiv\lambda/\pi w^2(z)\) and \(y\equiv 1/R(z)\), respectively.

From Equation 36 the effect of a thin lens is to cause a vertical jump of magnitude \(-1/f\) in the \(j/\tilde{q}\) plane. 

To those familiar with bilateral transformations as used in electrical circuit theory and elsewhere, it will be obvious that the transformation law through a free-space section, as given by \(\tilde{q}(z)=\tilde{q}_0+z=z+jz_R\) corresponds to a 

 

 

 

 

 

transformation around a circular arc in the complex \(j/\tilde{q}\) plane, as shown in Figure 14. In these so-called gaussian beam charts or Collins charts—which are very similar in form to the Smith charts of transmission line theory—different gaussian beam waists correspond to different points \(x=1/z_R\), or \(y=0\) on the \(x\) axis.

Free space propagation then corresponds to circular arcs passing through these points and the origin (which corresponds to the far field at \(z\rightarrow\infty\)); whereas lines of constant \(z/z_R\) for different \(z_R\) are also circles passing through the origin. Thin lenses are then vertical transitions on the same chart as shown. 

Charts of this type may be of some use for visualizing gaussian beam propagation problems or for diagramming solutions. With widespread access to computers, however, their practical uses as calculational tools are negligible.

 

 

4. AXIAL PHASE SHIFTS: THE GUOY EFFECT 

The propagation of a gaussian beam also involves a subtle but sometimes important added phase shift through the waist region, which we will briefly describe in this section. 

 

Axial Phase Shift 

The propagation equation (3 or 5) for a lowest-order gaussian beam includes both a spot size variation and a cumulative phase shift variation with axial distance \(z\) which are given on the optical axis \((x=y=0)\) by the factors 

\[\tag{37}\tilde{u}\varpropto\frac{\tilde{q}_0e^{-jkz}}{\tilde{q}(z)}=\frac{e^{-jkz}}{1-jz/z_R}=\frac{\text{exp}[-jkz+j\psi(z)]}{w(z)}.\]

In addition to the free-space or plane-wave phase shift given by the \(e^{-jkz}\) term, there is also an added axially-varying phase shift \(\psi(z)\) given by

\[\tag{38}\psi(z)=\tan^{-1}\left(\frac{z}{z_R}\right)\]

assuming we measure this added phase shift with respect to the beam waist location.

 

 

 

The net effect of this added phase shift \(\psi(z)\)for the lowest-order gaussian mode, as plotted in Figure 15, is to give an additional cumulative phase shift of \(\pm 90^\circ\) on either side of the waist, or a total added phase shift of \(180^\circ\) in passing through the waist, with most of this additional phase shift occurring within one or two Rayleigh ranges on either side of the waist. 

This added phase shift means in physical terms that the effective axial propagation constant in the waist region is slightly smaller, i.e., \(k_{eq}(z)=k-\Delta k\), or that the phase velocity and the spacing between phase fronts are slightly larger, i.e., \(v_\phi(z)=c+\Delta  v\), than for an ideal plane wave. The phase fronts for a gaussian beam passing through a waist will thus shift forward by a total amount of half a wavelength compared to an ideal plane wave, as illustrated in Figure 16.

A mathematical understanding of this additional phase shift can be obtained by rewriting the paraxial wave equation (7) in the form

\[\tag{39}\frac{\partial\tilde{u}(x,y,z)}{\partial z}=-\frac{j}{2k}\nabla^2_{xy}\tilde{u}(x,y,z),\]

where \(\nabla^2_{xy}\) is the Laplacian in \(x,y\) coordinates. The transverse second derivatives of the wave amplitude u thus lead, through the wave equation, to a small but significant additional phase shift per unit length in the axial direction.

The resulting increased phase velocity in the axial direction is exactly like the increased phase velocity in a closed waveguide. The transverse derivatives are the largest, and hence the added phase shift term is most significant, within one or two focal depths on either side of a focus, more or less independent of the exact transverse amplitude profile of the focused beam. 

 

The Guoy Effect

This result is in fact simply the gaussian beam version of the Guoy effect, which is valid for any kind of optical (or microwave) beam passing through a focal region. This effect, which was first discovered experimentally by Guoy in 

 

 

 

 

1890, says that a beam with any reasonably simple cross section will acquire an extra half-cycle of phase shift in passing through a focal region. 

Figure 17. shows the simple apparatus employed by Guoy to demonstrate this effect. In the original experiment the light diverging from a small pinhole was reflected into two overlapping beams reflected from both a planar and a curved mirror. Interference effects between the two beams then produced a set of circular interference fringes between the two beams which could be observed at transverse planes near the first image of the pinhole.

Guoy noticed that the centermost fringe in this "bulls-eye pattern" changed sign from dark to light (or vice versa) if he observed the fringes at observation planes just before or just after the focal point. This change of sign implied that the focused beam had somehow picked up an extra n phase shift in passing through the focus. 

We will see shortly that higher-order transverse modes, because they have more complicated transverse second derivatives in Equation 39, have larger Guoy phase shifts in passing through the waist region.

In fact, if the lowest-order gaussian mode has Guoy phase shift \(\psi(zz)\) at any plane \(z\), measured relative to the focal point, then an \(nm\)-th order Hermite-gaussian mode with the same \(\tilde{q}\) parameter will have a Guoy phase shift of \((n+m+1)\times\psi(z)\). 

These differing phase shifts are directly responsible for the slightly different resonance frequencies and mode beats of different \(nm\)-th order transverse modes in stable laser cavities.

The Guoy phase shift also explains the possibly somewhat puzzling \(90^\circ\) phase shift associated with the factor of j in the \(j/L\lambda\) constant that occurs as part of the kernel in Huygens' integral (Equations 17 or 19).

The physical interpretation of Huygens' integral considers the Huygens' wavelets as being ideal spherical wavelets diverging from each source point on the wavefront in the input plane, except that there is apparently a \(90^\circ\) phase shift between the incident wavefront and the diverging wavelet.

The Guoy effect says that this occurs because each wavelet will acquire exactly \(90^\circ\) of extra phase shift in diverging from its point source or focus to the far field, thus accounting exactly for the \(j\) factor in the \(j/L\lambda\) term.

 

 

5. HIGHER-ORDER GAUSSIAN MODES

Let us now look in somewhat more detail at the higher-order Hermite-gaussian modes we derived in the previous chapter. In doing this we will consider only the "standard" set of higher-order Hermite-gaussians discussed in HIGHER-ORDER GAUSSIAN MODES tutorial, since they usually match up most closely with the actual higher-order modes in simple optical resonators (at least in optical resonators which do not have "soft" apertures or radially varying gains or losses). 

 

Higher-Order Hermite-Gaussian Mode Functions

The free-space Hermite-gaussian \(\text{TEM}_{nm}\) solutions derived in the preceding chapter can be written, in either the \(x\) or \(y\) transverse dimensions, and with the plane-wave \(e^{-jkz}\) phase shift factor included for completeness, in the normalized form 

\[\tag{40}\begin{align}\tilde{u}_n(x,z)=\left(\frac{2}{\pi}\right)^{1/4}\left(\frac{1}{2_nn!w_0}\right)^{1/2}\left(\frac{\tilde{q}_0}{\tilde{q}(z)}\right)^{1/2}\left[\frac{\tilde{q}_0}{\tilde{q}^*_0}\frac{\tilde{q}^*(z)}{\tilde{q}(z)}\right]^{n/2}\\\times H_n\left(\frac{\sqrt2x}{w(z)}\right)\text{exp}\left[-jkz-j\frac{kx^2}{2\tilde{q}(z)}\right],\end{align}\] 

where the \(H_n\)'s are the Hermite polynomials of order \(n\), and the parameters \(\tilde{q}(z)\), \(w(z)\) and \(\psi(z)\) are exactly the same as for the lowest-order gaussian mode as given in Equation 5. These same functions can be written in alternative form, emphasing the spot size \(w(z)\) and Guoy phase shift \(\psi(z)\), in the form

\[\tag{41}\begin{align}\tilde{u}_n(x,z)=\left(\frac{2}{\pi}\right)^{1/4}&\left(\frac{\text{exp}[j(2n+1)\psi(z)]}{2_nn!w(z)}\right)^{1/2}\\&\times H_n\left(\frac{\sqrt2x}{w(z)}\right)\text{exp}\left[-jkz-j\frac{kx^2}{2R(z)}-\frac{x^2}{w^2(z)}\right],\end{align}\]

where \(\psi(z)\) is still given by \(\psi(z)=\tan^{-1}(z/z_R)\).

Note the important point that the higher-order modes, because of their more rapid transverse variation, have a net Guoy phase shift of \((n+1/2)\psi(z)\) in traveling from the waist to any other plane z, as compared to only \(\psi(z)\) for the lowest-order mode. This differential phase shift between Hermite-gaussian modes of different orders is of fundamental importance in explaining, for example, why higher-order transverse modes in a stable laser cavity will have different oscillation frequencies; or how the Hermite-gaussian components that add up to make a uniform rectangular or strip beam in one transverse dimension at an input plane located in the near field (at a beam waist) can add up to give a \((\sin x)/x\) transverse variation for the same beam in the far field.

 

Hermite-Gaussian Mode Patterns

Figure 18. illustrates the transverse amplitude variations for the first six even and odd Hermite-gaussian modes. Note that the first few (unnormalized) Hermite polynomials are given by

\[\tag{42}\begin{align}&H_0=1\qquad\qquad\qquad H_1(x)=2x\\&H_2(x)=4x^2-2\qquad H_3(x)=8x^3-12x\end{align}\]

These polynomials obey the recursion relation

\[\tag{43}H_{n+1}(x)=2xH_n(x)-2nH_{n-1}(x)\]

which can provide a useful way of calculating the higher-order polynomials in numerical computations.

The Hermite-gaussian beam functions alternate between even and odd symmetry with alternating index \(n\). The \(n\)-\(th\) order function has \(n\) nulls and \(n+1\)

 

 

 

 

 

peaks. These same Hermite-gaussian functions are also the quantum mechanical eigenfunctions for the linear quantum harmonic oscillator. Figure 19.  illustrates the intensity variation, or the wave amplitude squared, for the \(n=10\) eigenmode, showing how the wave distribution approaches the classical proba-

 

 

 

bility density for a linear harmonic oscillator. It can also be seen that for larger values of n the outermost peaks become noticeably more intense than the inner peaks. 

The complete set of Hermite-gaussian transverse modes for a beam in two transverse dimensions can then be written as \(\tilde{u}_{nm}(x,y,z)=\tilde{u}_n(x,z)\times\tilde{u}_m(y,z)\), where in the most general situation a different \(\tilde{q}(z)\) parameter, and even a different waist location, may apply to the \(x\) and the \(y\) variations.

Figure 20. shows how the intensity patterns of various higher-order modes appear if the output beam from a laser oscillating in one of these higher-order modes is projected onto a screen. Note that the Hermite-gaussian functions are everywhere scaled to the spot size w through the arguments \(x/w\) and \(y/w\). Hence, the intensity pattern of any given \(\text{TEM}_\text{nm}\) mode changes size but not shape as it propagates forward in \(z-a\) given \(\text{TEM}_\text{nm}\) mode looks exactly the same, except for scaling, at every point along the \(z\) axis. 

The higher-order Laguerre-gaussian mode patterns also described in this tutorial HIGHER-ORDER GAUSSIAN MODES (Equation 16.64) are characterized by azimuthal and radial symmetry, rather than by the rectangular symmetry of the Hermite-gaussian modes, as illustrated in Figure 21. As explained earlier, most real lasers prefer to oscillate in modes of rectangular rather than cylindrical symmetry, although with very 

 

 

 

 

 

careful adjustment, certain internal-mirror lasers can be made to oscillate in the cylindrical Hermite-gaussian modes. 

 

The "Donut Mode" 

In many laser experiments with stable laser resonators, the experimental procedure is to stop down an adjustable circular aperture inside the laser cavity until higher-order mode oscillation is completely suppressed and the laser oscillates only in the desired \(\text{TEM}_{00}\) mode.

For aperture diameters slightly larger than this value, lasers are often observed to produce an output beam in the form of a circularly symmetric ring with a dark spot on axis, as illustrated in Figure 22. 

This mode, often referred to as the "donut mode," cannot be an \(m=0\) mode, since an \(m=0\) Laguerre-gaussian mode can never have a null on axis. It might be interpreted as a higher-order \(\tilde{u}_{pm}(r,\theta)\) Laguerre-gaussian mode with \(p=1\) and an azimuthal variation like \(e^{jm\theta}\) with \(m\geq p\). In most practical lasers, however, 

 

 

 

this "mode" is more likely to represent a linear combination of the \(\text{TEM}_{10}\) and \(\text{TEM}_{01}\) Hermite-gaussian modes oscillating separately and independently, with slightly different oscillation frequencies because of the astigmatism introduced by the Brewster windows in the laser. The time-averaged total power output is then still circularly symmetric about the axis.

 

Higher-Order Mode Sizes 

It is obvious from inspection as well as from analytical approximations that higher-order Hermite-gaussian or Laguerre-gaussian modes spread out further in diameter as the mode index \(n\) (or \(p)\) increases. The mode pattern of the \(n=10\) mode function shown in Figure 23, for example, spreads out considerably farther than the lowest-order or \(n=0\) gaussian mode. This increase in mode diameter with increasing index \(n\) can be put on a quantitative footing as follows. 

Let us use the peak of the outermost ripple in the Hermite-gaussian pattern, call its location \(x_n\), as a convenient and fairly realistic measure of the spread or half-width of the Hermite-gaussian function. Numerically calculating and plotting the location of this outmost peak versus the mode index \(n\), as in Figure 23—or alternatively, exploring more advanced descriptions of the mathematical properties of the Hermite-gaussian functions—then shows that this width increases with \(n\) in approximately the form

\[\tag{44}\text{mode}\;\text{half}-\text{width},x_n\approx\sqrt n\times w.\] 

In addition, since these higher-order modes have \(n/2\) full ripples or periods of approximately equal width across the full width \(2\sqrt n w\) of an \(n\)-\(th\) order Hermitegaussian function, the spatial period \(\Lambda_n\) of the quasi-sinusoidal ripples associated with, or describable by, a Hermite-gaussian function of order \(n\) and spot size \(w\) is given by 

\[\tag{45}\text{spatial}\;\text{period},\Lambda_n\approx\frac{4w}{\sqrt n}.\]

Both of these criteria are very reasonable approximations to the mathematical properties of the Hermite-gaussian functions, especially for larger \(n\). 

 

Higher-Order Transverse Mode Aperturing

To illustrate the use of these quantities, suppose that we have an aperture of width or diameter \(2a\), corresponding perhaps to a mode control aperture or an end mirror inside a laser cavity; and that we are considering expanding the amplitude distribution across that aperture using a set of Hermite-gaussian modes of spot size \(w\) at the plane of the aperture. It is then obvious that only those Hermite-gaussian modes of orders low enough so that \(x_n\leq a\), or with indices \(n\) less than the value given by 

\[\tag{46}n\leq N_\text{max}\approx\left(\frac{a}{w}\right)^2\]

will pass through this aperture, or oscillate inside this cavity with relatively negligible mode losses. Modes with higher mode indices will spill over past the edges of the aperture; and we can expect a rapid increase in energy losses caused by the aperture for all modes with indices larger than this value. (Obviously this criteria is the most accurate for apertures at least several times larger than \(w\), since the sharpness of the outer edge transition becomes increasingly apparent at higher mode numbers.)

Larger-diameter lasers often choose to oscillate in multiple transverse modes extending up to and including the highest-order transverse modes that will "fit" inside the laser tube or the laser mirrors according to this criterion, since all of these transverse modes will have comparatively low diffraction losses at the laser tube walls or mirror edges.

Transverse mode-control apertures are often placed inside stable laser cavities in order to attenuate or block higher-order modes from oscillating while producing minimal loss for the lowest-order \(\text{TEM}_{00}\) modes.

A common rule of thumb for the necessary aperture size in low-gain lasers, such as for example He-Ne lasers, is that the mode control aperture should have an aperture size of diameter \(2a\approx3.5\) to \(4.0\times w\), or slightly larger than the \(2a=\pi w\) or 99% criterion we introduced at the beginning of this section.

 

Numerical Hermjte-Gaussian Mode Expansions

Suppose we wish to carry out a numerical expansion of some given (or perhaps unknown) function \(f(x)\) across an aperture or strip of width \(2a\) using a

 

 

Hermite-gaussian basis set in the form 

\[\tag{47}f(x)=\sum^N_{n=0}c_n\tilde{u}_n(x;w),\quad-a\leq x\leq a,\]

where \(\tilde{u}_{n}(x;w)\) refers to an \(n\)-\(th\) order Hermite-gaussian function characterized by spot size \(w\), and \(N\) is the maximum index value to be kept in a finite expansion.

Let us explore some of the numerical considerations involved in this expansion, such as the optimum choice of the gaussian spot size \(w\) (assuming this to be a free parameter), and the number of terms \(N\) that we will need to keep in the summation. 

The expansion coefficients for a given function \(f(x)\) will be given by the overlap integrals

\[\tag{48}c_n=\int^a_{-a}f(x)\tilde{u}^*_n(x)dx.\]

Figure 24. shows, for example, how the expansion coefficient magnitudes \(|cn|\) will decrease in amplitude with increasing mode index \(n\) if we expand .a simple rectangular function of width \(2a\) using Hermite-gaussian basis sets of different fundamental spot size \(w\).

The dashed lines represent the values \(N_\text{max}=(a/w)^2\) in each situation. It is obvious from these plots that the amplitude of the expansion coefficients drops off rapidly in each situation as soon as n increases slightly beyond this value. This fall-off obviously occurs because the Hermite-gaussian modes of order higher than this extend past the edges of the aperture, or the square input function, and hence less and less of the Hermite-gaussian function falls within the overlap integral given in the preceding.

 

 

As a slightly different illustration of the same point, Figure 25 shows similar results for the expansion of a half-square (or displaced square) function covering the range \(0\leq x\leq a\) using Hermite-gaussian basis functions.

The quantity plotted in this situation is the residual mean-square error in the series approximation to the function \(f(x)\) caused by truncating the series expansion at a maximum value \(N\), for different choices of the ratio \(a/w\).

Again we see that the maximum error drops rapidly with increasing number of terms in the series expansion, but only up to a rather surprisingly sharp corner at \(N\approx N_\text{max}=(a/w)^2\).

Beyond this value, keeping additional terms only causes a very slow further improvement in the accuracy with which the function is approximated by the finite series. 

 

Spatial Frequency Considerations 

Suppose as a more general example that we wish to describe an arbitrary function \(f(x)\) across an aperture of width \(2a\) with a finite sum of \(N+1\) Hermite-gaussian functions \(\tilde{u}_n(x;w)\), of arbitrary spot size \(w\), for \(0\leq n\leq N\). How then should we select the spot size \(w\) to use in the expansion, and the maximum index \(N\) at which the series expansion is to be truncated? 

To do this sensibly, we must first calculate (from experimental or other evidence) what is the maximum spatial frequency or spatial period \(\Lambda\) of the fluctuations in the function to be expanded across the interval \(-a\leq x\leq a\) That is, we must pick a value of \(\Lambda\) such that the significant variations in \(f(x)\) will be no more rapid than \(\approx\cos 2\pi x/\Lambda\) at most.

We must then select values of \(w\) and \(N\) so that the highest-order Hermite-gaussian functions to be employed will simultaneously satisfy two criteria: they must at least fill the aperture, and they must at least handle the highest spatial

variations in the signal. But these are equivalent to the two conditions 

\[\tag{49}N\geq N_\text{max}\equiv\left(\frac{a}{w}\right)^2\quad\text{and}\quad\Lambda_N\approx\frac{4w}{\sqrt N}\leq\Lambda.\]

Satisfying these conditions simultaneously then leads to the spot-size and maximum-index criteria

\[\tag{50}w\leq\sqrt{\frac{a\Lambda}{4}}\quad\text{and}\quad N\geq\frac{4a}{\Lambda}.\]

The second of these criteria is obviously a Hermite-gaussian version of the familiar sampling theorem of Fourier transform theory, which says that to describe an arbitrary function which is bandlimited to a spatial frequency \(2\pi/\Lambda\), we need at least two sample points per spatial period \(\Lambda\).

In the Hermite-gaussian analog we need \(N=4a/\Lambda\) samples in space across a width of 2a, or equivalently \(N=4a/\Lambda\) coefficients in a Hermite-gaussian expansion. 

 

 

6. MULTIMODE OPTICAL BEAMS

Lasers that oscillate in multiple higher-order transverse modes are almost always considered as "bad" lasers, since they will have a far-field beam spread cosider-ably larger than a well-behaved lowest-order single-transverse-mode laser. The quasi analytic results that we have just obtained for Hermite-gaussian mode expansions can also be applied to give a useful description of multimode or non-diffraction- limited laser beams, in the following fashion. 

 

Description of a Multimode or Non-Diffraction-Limited Beam 

Suppose that an oscillating laser emits a reasonably well-collimated but obviously multimode optical beam which occupies a width or diameter \(2a\) in the transverse direction at the output from the laser. (By "collimated" we mean simply that any overall spherical curvature of the wavefronts emitted from the laser has been corrected by a suitable collimating lens.) 

The far-field angular spread of the multimode beam coming from this laser will then be substantially larger than the value \(\Delta\theta\approx\lambda/2a\) that would be characteristic of a more or less diffraction-limited optical beam.

From another viewpoint, if the output beam from this laser consists of a mixture of a sizable number of different transverse modes, the wavefront at the laser output is likely to be quite random in character, with considerable spatial incoherence or variation in local amplitude and phase from point to point across the aperture.

How can such a strongly non-diffraction-limited laser beam then be described analytically—especially in situations where little information may be available concerning the detailed mode characteristics of the laser, and where all that is known for certain may be the near-field aperture width and the far-field angular spread of this beam?

 

Hermite-Gaussian Analysis of Multimode Optical Beams

One useful approach can be to analyze this beam as if the output fields in the beam are made up of, or can be analyzed as, a superposition of Hermitegaussian modes having a characteristic spot size \(w_0\). This assumption might apply quite well, for example, to the output beam from a laser with a stable gaussian resonator, such as we will describe in the following chapter, in which the natural resonator spot size is \(w_0\), but the laser tube diameter or mirror diameter \(2a\) is substantially larger than \(w_0\). This laser may then oscillate simultaneously in multiple transverse modes which fill the entire diameter \(2a\). 

More generally, consider an arbitrary, irregular, multimode beam coming from any kind of laser cavity, stable or not; and assume that this beam has sizable fluctuations in amplitude and especially in phase across its diameter \(2a\). Regardless of whether the underlying mode structure in this beam is gaussian, we can still use a set of Hermite-gaussian modes to expand the fields.

Following the procedure outlined in the previous section, we can first ask what spot size \(w_0\) and number of modes \(N\) we would need to choose so that the spatial frequencies and the spatial resolution of the set of Hermite-gaussian modes would be just adequate to describe the most rapid spatial variations across the aperture of that particular beam.

We can then use these values to calculate the basis set of Hermite-gaussian functions which we can employ to describe that particular beam with adequate accuracy.

For an aperture of width \(2a\), where a is at least a few times larger than \(w_0\), the maximum number of Hermite-gaussian modes that will "fit" within the aperture, or the number of modes that will be needed to describe the fields in the aperture, will then be given by

\[\tag{51}N\approx N_\text{max}\equiv(a/w_0)^2.\] 

The corresponding maximum half-angle spread of the overall beam in the far field, using a near-field spot size of \(w_0\) and modes of index running up to \(n=N\), will then be

\[\tag{52}\theta_\text{max}\approx\sqrt N\times \theta_{1/e}=\frac{N^{1/2}\lambda}{\pi w_0}=\frac{a\lambda}{\pi w^2_0}.\]

If we consider for simplicity a circular aperture of diameter \(2a\), then the far field beam will also have a circular cross section of angular diameter \(2\theta_\text{max}\). The product of the source aperture area \(A\equiv\pi a^2\)  and the far-field solid angular spread \(\Omega\equiv\pi\theta^2_\text{max}\) will then be given by 

\[A\times\Omega\equiv(\pi a^2)\times(\pi\theta^2_\text{max})\approx(N\lambda)^2.\]

This product for the multimode or non-diffraction-limited beam is then \(N^2\) times the diffraction-limited value \(A\times\Omega=\lambda^2\) we derived earlier for an ideal lowestorder gaussian beam. 

 

"Times Diffraction Limited" \(\text{(TDL)}\)

An irregular or multimode laser beam which can be described in the fashion leading up to Equation 53 is often said to be "\(N\) times diffraction limited." That is, its far-field angular spread is \(\approx N\) times as large in one dimension (or \(N^2\) times in solid angle) as the diffraction-limited angular spread that would be obtained from a uniphase beam with a reasonably regular amplitude variation filling the same aperture.

The quantity \(N\) is sometimes referred to as the "times diffraction limited" or \(\text{"TDL"}\) of the beam. Note that if this same beam is focused to a spot with a suitable lens, the diameter of the focused spot will also be \(\approx N\) times the spot size that would be obtained with an ideal diffraction-limited beam.

This argument can also be applied in the reverse direction. That is, given a beam known to be N times diffraction limited (based on experimental data on its initial aperture size and its far-field beam spread), we can treat this beam analytically as if it were made up of a mixture of \(N^2\) Hermite-gaussian modes, with spot size given by \(w_0\approx a/N^{1/2}\) and with mode amplitude coefficients assumed to be approximately equal or perhaps randomly distributed in amplitude.

The relatively simple mathematical properties of the Hermite-gaussian modes then make it possible to calculate or at least estimate other properties of this beam that might be of interest (for example, perhaps the amount of harmonic generation it would produce in a given crystal).

It may seem somewhat inconsistent here to employ the Hermite-gaussian modes, which are characteristic of rectangular coordinates, and then compute areas and solid angles assuming circular beams, which might more accurately be described using cylindrical coordinates and Laguerre-gaussian functions. The only excuse is that the Hermite-gaussian properties are perhaps simpler and more familiar than the Laguerre-gaussians, and the right answer comes out by using formulas based on a circular aperture.