BEAM PERTURBATION AND DIFFRACTION

This is a continuation from the previous tutorial - Physical properties of gaussian beams

 

1. GRATING DIFFRACTION AND SCATTERING EFFECTS

Optical elements in real life are often of good but not perfect quality. Optical beams may then be perturbed by various kinds of weak amplitude or phase perturbations, such as scratches, dust particles or blemishes on lenses or mirror surfaces, or bubbles and defects inside optical components. Optical imperfections of this kind may be modeled in many situations either as a collection of point scatterers, or as weak and more or less random amplitude or phase-perturbing screens or gratings. We will look briefly at both of these descriptions in this section, taking up the grating diffraction approach first.

Consider, for example, a perturbing screen with a random transverse amplitude or phase profile like Figure 1. The amplitude and phase variations this element will impose on a uniform beam passing through it can be described by a perturbation transmission function; and the phase and amplitude parts of this transmission function can in turn be expressed as a superposition of sinusoidal components or spatial-frequency components of the form \(\sin 2\pi x/\Lambda\) or \(\cos 2\pi x/\Lambda\) in the transverse direction.

We can analyze the scattering produced by each such spatial component separately, and then for weak gratings add their effects to describe the total scattering produced by the complete perturbation.

 

Amplitude Gratings

Let us consider first a weak sinusoidal amplitude grating oriented perpendicular to the \(z\) axis and having a sinusoidal periodicity of period \(\Lambda\) in the \(x\) direction, as in Figure 1(b). The amplitude (i.e., field) transmission through

 

 

this grating can then be written in the form 

\[\tag{1}\tilde{t}(x)=\text{exp}[-\Delta(1-\cos k_x(x-x_0))],\]

and the intensity transmission \(T(x)\equiv|\tilde{t}(x)|^2\) in the form

\[\tag{2}T(x)=\exp[-2\Delta[1-\cos k_x(x-x_0)]]=\text{exp}\left[-4\Delta\sin^2\frac{\pi(x-x_0)}{\Lambda}\right],\]

where \(k_x\equiv 2\pi/\Lambda\) is the \(k\)-vector of the periodic grating. This intensity transmission is illustrated for different values of \(\Delta\) in Figure 2.

The analytical form of Equation 1. is a convenient way of representing a sinusoidal modulation with peak-to-peak modulation depth of \(2\Delta\) in amplitude (or \(4\Delta\) in intensity), either in a spatial dimension as we are doing here, or for a sinusoidal amplitude modulation in time as we will do in a later chapter.

This form has the convenient property that the intensity transmission is always less than or equal to unity, but never goes negative even for very large \(\Delta\) (although we are most interested in the small-\(\Delta\) regime in this chapter). This particular

 

 

 

 

form also corresponds rather closely to the actual amplitude transmission versus time or space in many practical acoustooptic and electrooptic light modulators. 

A uniform plane wave passing through this grating will retain a planar wave-front, but will acquire a transverse intensity profile which we can write as 

\[\tag{3}\begin{align}&\tilde{u}=\text{exp}[-\Delta[1-\cos k_x(x-x_0)]]\\&\approx 1-\Delta+\frac{\Delta}{2}\text{exp}[-jk_x{(x-x_0)}]+\frac{\Delta}{2}\text{exp}[+jk_x(x-x_0)]\\&=\tilde{\delta_0+\tilde{\delta}_{1}e^{-jk_xx}}+\tilde{\delta}_{-1}e^{+jk_xx},\qquad\text{for}\;\Delta\ll1.\end{align}\] 

The second and third lines of this expression show that the primary effect of a weak sinusoidal amplitude grating is to scatter a small part of the original plane wave amplitude into two diffracted components or diffraction orders with transverse \(k\) vector components given by \(\pm k_x\), where 

\[\tag{4}k_x\equiv k\sin\theta_x\equiv 2\pi/\Lambda_x\approx2\pi\theta_x/\lambda.\]

The amplitudes of these sidebands are given by \(\tilde{\delta_0}=1-\Delta\) for the original wave vector component (the "carrier"), and by 

\[\tag{5}\tilde{\delta}_1=\frac{\Delta}{2}\times e^{jk_xx0}\qquad\text{and}\qquad\tilde{\delta}_{-1}=\frac{\Delta}{2}\times e^{-jk_xx_0}\]

for the amplitudes scattered into the \(+1\) and \(-1\) diffraction orders, respectively. Note that the transverse position of the amplitude grating with respect to the \(z\) axis, as contained in the xo parameter, shows up as a relative phase shift \(\text{exp}(\pm jk_xx_o)\) in the phases of these two sidebands.

The various possible relative phase angles for the amplitude grating sidebands caused by different transverse shifts of the grating are sometimes represented graphically by the kind of three-dimensional phasor diagram shown in Figure 3.

 

Phase Grating

Suppose we have instead a weak sinusoidal phase grating, with complex amplitude transmission given by

\[\tag{6}\tilde{t}(x)=\text{exp}[j\Delta\cos k_x(x-x_0)].\] 

(Note that there is no need to add a constant term in front of the cosine in this situation, since the phase shift can equally well go positive or negative.) An initially uniform plane wave passing through this grating will again be diffracted into the components 

\[\tag{7}\begin{align}\tilde{u}(x)&=e^{j\Delta\cos k_x(x-x_0)}\\&\approx 1+j\frac{\Delta}{2}\times[e^{-jk_x(x-x_0)}+e^{+jk_x(x-x_0)}]\\&=1+\tilde{\delta}_{1}e^{-jk_xx}+\tilde{\delta}_{-1}e^{+jk_xx},\qquad\text{for}\;\Delta\ll1,\end{align}\]

where \(\Delta\) is again the depth of modulation, but now in phase rather than in magnitude. 

The transmitted intensity is now constant across the aperture. The same amount of energy has, however, again been scattered (for the same modulation

 

 

depth \(\Delta\)) into two diffracted orders with the same transverse \(k\) vector components, but now with complex amplitudes given by 

\[\tag{8}\tilde{\delta}_1=j\frac{\Delta}{2}\times e^{jk_xx_0}\qquad\text{and}\qquad\tilde{\delta}_{-1}=j\frac{\Delta}{2}\times e^{-jk_xx_0}.\] 

Again the relative phase angles of these two sidebands for different transverse shifts of the phase grating can be represented by different vector diagrams as shown in Figure 4. Note, however, the distinguishing phase differences between the amplitude and phase grating results shown in Figures 3 and 4.

If the grating is a pure amplitude grating the two wave amplitudes will be related by the condition 

\[\tag{9}\tilde{\delta}_1\equiv\tilde{\delta}^*_{-1}\qquad(\text{amplitude}\;\text{grating}),\] 

whereas we have instead the condition that

\[\tag{10}\tilde{\delta}_1\equiv\tilde{\delta}^*_{-1}\qquad(\text{phase}\;\text{grating}),\]

for a pure phase grating.

 

General Two-Diffracted-Wave Situation 

Suppose a uniform plane wave passes through some general combination of a weak amplitude grating plus a weak phase grating, with both gratings having the same transverse period \(\Lambda\) (though not necessarily the same modulation depth \(\Lambda\) or offset \(x_0)\). The transmitted wave just beyond the grating can then still be  written in the general form

\[\tag{11}\tilde{u}(x)\approx\tilde{\delta}_0+\tilde{\delta}_1e^{-jk_xx}+\tilde{\delta}_{-1}e^{+jk_xx},\]

where \(\tilde{\delta}_1\)  and  \(\tilde{\delta}_{-1}\) represent the complex amplitudes of the waves scattered or diffracted into the \(+1\) and \(-1\) orders. Any arbitrary relative amplitudes and phases for the two sidebands can then always be generated by the proper combination of a weak amplitude grating plus a weak phase grating, with the resulting total wave amplitudes given by

\[\tag{12}\tilde{\delta}_1=\tilde{\delta}_{1,\text{am}}+\tilde{\delta}_{1,\text{fm}}\qquad\text{and}\qquad\tilde{\delta}_{-1}=\tilde{\delta}^*_{1,\text{am}}-\tilde{\delta}^*_{-1,\text{fm}}.\]

(Because of the obvious connection between these grating sidebands and the modulation sidebands associated with the time modulation of signals as in AM and FM radio, we use the initials AM and FM as shorthand to represent amplitude and phase gratings, respectively.)

 

Single-Sideband Gratings

An optical grating may in fact even act like a single- sideband modulator, i.e., it may produce only a single diffracted sideband which may be written in the form

\[\tag{13}\tilde{u}(x)\approx1+\tilde{\delta}_{1}e^{-jk_xx}\].

It is evident from the preceding results that such a single sideband represents an equal combination of amplitude and phase gratings, with transverse displacements \(x_0\) between them adjusted to just cancel the total sideband on one side, but reinforce the sidebands on the opposite side. The intensity variation across this beam is then

\[\tag{14}I(x)=|\tilde{u(x)|^2}\approx1+|2\tilde{\delta}_1|\cos(k_xx+\theta_1).\]

The intensity variation in this situation always has a periodic ripple with peak magnitude \(2\tilde{\delta}_1\) and period corresponding to the grating spacing, independent of the phase angle \(\theta_1\) of the single sideband.

 

Beam Transformations With Distance 

From our earlier discussion of plane-wave expansions of optical beams, we know that for a plane wave with transverse \(k\)-vector components given by

\[\tag{15}k_x=k\sin\theta_x\approx k\theta_x\qquad\text{and}\qquad k_y=k\sin\theta_y\approx k\theta_y\],

the propagation vector component in the z direction will be given by

\[\tag{16}k_z=\sqrt{k^2-k^2_x-k^2_y}\approx k-\frac{k}{2}[\theta^2_x+\theta^2_y].\]

If a grating produces two diffracted components traveling at angles \(\pm\theta\) in the \(x,z\) plane, these components will both propagate or rotate in relative phase as they travel outward from the grating in the form 

\[\tag{17}\tilde{\delta}_{\pm1}(z)=\tilde{\delta}_{\pm1}(0)\times\text{exp}[j(k\theta^2/2)z].\]

 

 

Note that both sidebands rotate in the same direction relative to the carrier, as illustrated in Figure 5. 

But this rotation will gradually shift the relative phases so as to convert amplitude modulation sidebands into phase modulation sidebands, and vice versa. In fact, the primary result that emerges from this is that after a distance \(d\) given by \(\text{exp}[jk\theta^2d/2]=\text{exp}[j\pi/2]\), or

\[\tag{18}d=\frac{\pi}{k\theta^2}=\frac{\lambda}{2\theta^2}=\frac{\Lambda^2}{2\lambda}\] 

the sideband components corresponding to a pure phase grating wavefront will have rotated into a phase angle relationship corresponding to a pure amplitude grating wavefront, and vice versa. In other words, a Geld distribution that starts out as, for example, a pure phase grating or phase-modulated wavefront will be converted after a certain distance into the form of a pure amplitude-modulated wavefront, and vice versa.

In addition, after twice this distance the components of either type of grating will return to the same type of grating but exactly reversed in sign.

These conversion distances depend on the square of the angle \(\theta\) or of the period \(\Lambda\) of the grating components. If we have a more complex grating with several spatial frequency components, each of these components will convert back and forth between amplitude and phase gratings with a different axial period.

The general amplitude profile of the beam in the space beyond the multiple grating will thus be quite complicated. If the conversion distances for the different spatial components are incommensurate, then the original beam profile will never be recovered completely at any distance.

 

Point Scatterers and Spherical Waves

An alternative model to the grating approach for a random scatterer or perturbation in an optical beam—one that can be a better representation for point defects like dust particles or other very small scatterers—is a single point scatterer that reradiates a weak scattered spherical wave. Such a wave then generally interferes with the primary wave to create patterns of interference rings, such as we will analyze in this section. 

Consider, for example, a weak spherical wave scattered by a point source and interfering with a primary plane wave, as in Figure 6. The total field amplitude at radius \(r\), centered on the transverse location of the point defect,

 

 

can then be written as 

\[\tag{19}\tilde{u}(x)=1+\tilde{\delta}_1\text{exp}[-jkr^2/2R(z)]\qquad\text{with}|\tilde{\delta}|\;\ll1.\]

The intensity across the transverse plane will be given by 

\[\tag{20}I(x)\approx1+|2\tilde{\delta_1}|\cos[kr^2/2R(z)+\theta_1],\]

where \(\theta_21\) is the initial phase angle of the scattered wave referred back to the scattering point. The amplitude of the scattered wave (assumed to be \(\ll1\) ) will obviously decrease as \(1/R\), but we can include this implicitly in the value of \(\tilde{\delta_1}\). 

The intensity profile in this situation will obviously consist of a "bulls-eye pattern" or series of concentric rings, with the radii \(r_n\) of the successive bright rings given by 

\[\tag{21}(kr^2_n/2R)+\theta_1=n2\pi\qquad\text{or}\qquad r_n\sqrt{\left(n-\frac{\theta_1}{2\pi}\times 2R\lambda.\right)}\]

Whether the center of the pattern will be a bright ring, a dark ring, or somewhere in between depends on the absolute value of the initial phase angle \(\theta_1\), but not on the distance \(R=z-z_0\) along the axis.

As we move farther away from the point-source origin of the rings, however, the diameter of the rings will steadily increase in proportion to \(z^{1/2}\), until all the rings walk out of a finite-sized beam, leaving only the central bright, or dim, spot filling the entire beam.

 

 

2. ABERRATED LASER BEAMS 

Let us now look briefly at what these properties of diffracted or scattered radiation imply for the overall properties of weakly perturbed or weakly aberrated laser beams. 

 

Scattered Wave Amplitude and Ripple Amplitude 

We can first note that in all of these situations, a very small amount of intensity scattered into a diffracted wave will produce a much larger relative variation or ripple amplitude in the total beam intensity. That is, if we have, for example, a scattered wave with a relative field amplitude of \(\delta_1=0.1\), the various relative quantities then become 

• amplitude of scattered wave\(=\delta_1=0.1\),
• intensity of scattered wave\(=\delta^2_1=0.01=1\)%,
• intensity ripple in total wave\(\approx\pm2\delta_1=\pm20\)%.

Only 1% power scattered into a diffracted sideband thus leads to \(\pm\)20% ripple in the total intensity across the beam profile.

Note that for \(\delta_1<1\) the intensity in the minima of the interference fringes or in the "dark" rings is not actually zero. The response of the human eye is such, however, that even a fringe pattern or ring pattern with relatively limited visibility, or relatively limited difference between "bright" and "dark" rings, will still appear visually to have a quite high contrast between these two levels.

Unless we take great pains, the near-field beam profiles associated with a coherent laser beam in almost any optical system are apt to contain a rich assortment of perturbation or dust-induced fringe patterns and ring patterns.

On the other hand, because of this enhanced visibility of interference fringes, what appears to be a terribly rippled and distorted beam profile may really represent only very small power losses into scattered waves diffracted out of the primary beam profile.

 

 

 

Low-Spatial-Frequency Beam Perturbations 

The aberrations, especially phase aberrations, that occur in real laser cavities or laser optical systems can often be separated into two broad classifications. One category consists of those phase aberrations that have only a very slow variation across the beam profile. If these aberrations are sufficiently slow, their effect is relatively trivial and easily corrected. 

A linearly varying phase profile across a beam, for example, simply represents a wedge, which bends the beam direction without distorting the beam profile as in the upper part of Figure 7. A quadratic phase variation represents a lens which causes focusing, or defocusing, of the beam.

This can be corrected by inserting an appropriate compensating lens (possibly an astigmatic lens, if the quadratic aberration is different along two transverse directions).

From another viewpoint a thin lens imposes a quadratic phase "chirp" on the spatial profile of an optical beam. This spatial chirp has many analogies to the temporal phase chirp that we discussed in this tutorial LINEAR PULSE PROPAGATION.

In fact the spatial compression and then expansion of a focused beam following a lens (Figure 8), is an exact analog in the spatial domain to the pulse compression and expansion in the time domain caused by propagation through a dispersive waveguiding system.

Slow aberrations of order higher than quadratic are then true aberrations— that is, they are not easily analyzed by paraxial techniques, nor corrected by simple lenses or spherical surfaces. These higher-order aberrations are extensively discussed in standard optics texts, and we will not attempt to cover these topics here, except to say that such aberrations with magnitudes much greater than a fraction of a wavelength across a beam profile will have serious distortion effects on a good-quality laser beam profile.

 

 

 

High-Spatial-Frequency Beam Perturbations 

The opposite limiting situation then consists of those phase or amplitude aberrations that have at least several cycles of variation across the laser beam. Suppose a laser beam which has reasonably good beam quality, or which is reasonably close to diffraction-limited in its transverse profile, is perturbed by phase or amplitude variations whose spatial frequencies are large compared to the inverse of the beam diameter \(d\), so that the spatial periods of these aberrations are given by \(\Lambda\leq d/N\), where \(N\) is some number (not necessarily an integer) at least several times unity. 

If the original laser beam is of reasonably good beam quality, this means that essentially all its energy is contained within a narrow distribution of plane waves, with a spread in \(k\) space or in angle \(\theta\) limited to roughly \(\Delta\theta\approx\lambda/d\), where \(d\) is the width or diameter of the laser beam.

The effect of higher-spatial-frequency aberrations, according to the analysis of Section 1, will then be to scatter energy from these plane-wave components into diffracted angles given by

\[\tag{22}\theta\approx\frac{\lambda}{\Lambda}\approx N\times\Delta\theta.\]

These angles are large compared to the original angular spread of the laser beam. 

The primary effect of high-spatial-frequency perturbations is thus a scattering of some amount of the original beam energy into a range of angles considerably larger than the diffraction limit for the original beam, leaving the main or central portion of the beam's angular spectrum weakened in amplitude, but otherwise unchanged in shape. Such aberrations are thus likely to weaken the central portion of the far-field profile of a laser beam, but to leave it more or less unchanged in shape.

The missing energy will then be found in a broader and more or less random pedestal of scattered energy spread over a much wider angle around the central portion of the beam, as shown schematically in Figure 9. The general 

 

 

shape and width of this background pedestal will depend on the exact shape and nature of the aberrations. 

Note the difference in viewpoint between this discussion and the discussion of strongly distorted or highly multimode beams in tutorial MULTIMODE OPTICAL BEAMS. That discussion was concerned with strongly aberrated or highly multimode beams. The present viewpoint can be used as a model for the analytical treatment of weakly aberrated or weakly non-diffraction-limited laser beams. 

 

The Intermediate situation: Serious Beam Distortion

The most difficult situation clearly arises for those aberrations whose spatial variation is roughly comparable to the laser beam diameter. These aberrations are generally too complicated to be corrected by simple lenses or wedges; and at the same time the diffraction angles produced by these aberrations are small enough that the scattered waves remain within the main beam angle, seriously distorting the main beam profile, rather than merely being scattered in a large-angle background surrounding the main laser spot. 

The designers of large-aperture laser amplifiers in laser fusion systems, where the beam profile must be very good to permit focusing onto a very small target pellet, have actually made careful point-by-point measurements of the phase aberrations across the entire laser profile, and then added special elements to correct these distortions.

Experimenters working with other kinds of high-power laser oscillators have used real-time measurement and correction of beam aberrations by adaptive optical techniques within the laser resonator, for example by servo-controlled deformable laser mirrors.

 

Amplitude Versus Phase Perturbations

The discussions at the beginning of this tutorial demonstrate that weak or small-amplitude perturbations with high spatial frequencies do about the same amount of damage to an optical beam—that is, about the same amount of energy is scattered out into larger angles—for either amplitude or phase gratings having the same peak-to-peak modulation index \(\Delta\).

As beam profile variations or beam perturbing effects become stronger, however, particularly for beam profile variations with slow to intermediate spatial frequencies, a general principle is that phase perturbations are generally more serious, and cause more reduction in far-field beam intensity, than do amplitude variations. 

If an optical beam with fixed total power is transmitted through an aperture having an arbitrary transverse shape, the highest on-axis intensity in the far field will be obtained if the transmitted power is distributed as a collimated plane wave with uniform intensity over the entire transmitting aperture, regardless of its shape.

Suppose, however, that the phase and amplitude distribution in the transmitting aperture is to be modified (keeping the total transmitted power fixed), perhaps in order to obtain other advantages, such as reduced intensity in certain side lobes, or perhaps simply because a uniform intensity distribution is not available from the beam source.

In the jargon used by some workers, modification of the amplitude or intensity distribution within the aperture is referred to as apodization, whereas modification of the phase profile or wavefront shape is referred to as (phase) aberration. 

The general rule then is that large but slow amplitude variations or apodization will not greatly reduce the far-field brightness; whereas large but slow phase variations or aberration will quite strongly reduce the on-axis brightness in the far field. To demonstrate this, we can calculate the on-axis far-field intensity for a slit or circular aperture using various transverse or radial amplitude variations, such as \(1-(x/a)^n\) or \(\cos^n(\pi x/2a)\) (at fixed total power), and see that the brightess is generally reduced by factors of the order of 50% or less.

Large phase variations across the beam, however, represent strong lenses or prisms, which either strongly defocus the beam in the far field, or bend the direction of different parts of the beam to different positions in the far field. It is also possible to show using fundamental mathematical inequalities (see the References) that applying phase aberration to an already apodized beam always makes the far-field brightness become still lower, whereas in the reverse situation, applying apodization to an already phase-aberrated beam can sometimes make the farfield brightness get (somewhat) better.

Phase aberrations always make things worse; apodization sometimes can make them even a little better.

 

Nonlinear Wavefront Perturbations: Small-Scale Self-Focusing

The grating description introduced in this section is particularly relevant to the nonlinear type of self-induced perturbation known as "small-scale self-focusing" which occurs in higher-power laser beams passing through almost any optical material (and which we have also mentioned in an earlier section).

In almost any transparent optical material, the local index of refraction of the material will increase slightly with increasing optical intensity at high enough optical field strengths, in the form \(n=n_0+n_2|\tilde{E}|^2\).

This is referred to as the optical Kerr effect, since it represents a kind of Kerr effect induced in the medium by the optical fields themselves. The coefficient \(n_2\) is present, though typically very weak, and has a positive sign in nearly all optical materials. (The optical intensities needed to produce a noticeable index change are typically in the range of 1 to 10 Gw/\(\text{cm}^2\).)

Suppose a high-power laser beam has an amplitude profile with some initial weak amplitude ripples, produced by any sort of initial perturbation. These ripples in the optical intensity will then cause a periodic variation in the local index of refraction, thus producing a weak phase grating having the same period, through the \(n_2|\tilde{E}|^2\) effect.

But this phase grating will then diffract additional energy from the main beam into phase-grating sidebands; and these sidebands will, after a certain distance as discussed in the preceding, convert into an additional amplitude grating, which can produce additional index changes and additional scattering.

Analysis of this effect shows that in fact the feedback in this process is positive: the strength of the amplitude ripples on the beam and of the index perturbation in the material will grow exponentially with distance as the beam propagates forward through the medium.

The growth rate itself depends on the intensity of the laser beam. For beams with intensity levels in the range \(\geq 1\) Gw/\(\text{cm}^2\)—which are not uncommon in high-power Nd:glass lasers, for example—this exponential growth can cause initially very small ripples to grow to a level sufficient to destroy the material within a very short distance. Elimination of these small-scale selffocusing effects, both by carefully filtering out the initial ripples and by selecting low Kerr coefficient materials, is one of the primary objectives in designing a high-power and high-intensity laser system.

 

 

 

3. APERTURE DIFFRACTION: RECTANGULAR APERTURES 

All real optical systems will have finite apertures, with some sort of boundaries or edges. In this section, therefore, we will consider the near-field and far-field diffraction patterns that are produced when paraxial optical beams pass through simple apertures having sharp edges, with emphasis on slits and square or rectangular apertures. (Circular apertures will be taken up in the following section.)

The primary objective in presenting these examples is again to gain insight into the kinds of physical diffraction effects that can occur in simple situations, as well as some familiarity with the common analytical techniques for describing these diffraction effects. 

 

Fresnel Diffraction Effects

Much of the material in this section will be a replay of the optics section from the freshman physics curriculum. Let us remind ourselves therefore (at least for those who need reminding) of the elementary physics of Fresnel diffraction and Fresnel zones. 

The Huygens-Fresnel integral says, as we have already noted, that the wave amplitude at an observation point \(r\equiv(s,z)\) as in Figure 10 is produced by the vector sum of wavelets coming from source points \(r_0\equiv(s_0,z_0)\) with a net phase delay, over and above the on-axis phase shift term, given by

\[\tag{23}\text{exp}[-jk\rho(\boldsymbol{r},\boldsymbol{r_0})]=\text{exp}\left[-j\frac{\pi|s-s_0|^2}{(z-z_0)\lambda}\right].\]

Wavelets coming from different source points \(\boldsymbol{r}_0\) thus add with different relative phases depending on the magnitude of this additional phase shift. 

One way to visualize this is to drop a perpendicular from the observation point r back to the source plane, as shown in Figure 10, and then to imagine the intercept of this line as surrounded by circles of constant \(s-s_0\), or constant additional phase shift, as shown in Figure 10. All those source points lying within the central circle for which \(k\rho\leq\pi\), or \(|s-s_0|^2\leq(z-z_0)\lambda\), will then

 

 

produce phasor contributions that add up at the observation point with phasor angles that are more or less in phase, or at least in the same half-plane, as shown in the phasor diagram in Figure 11.

However, all those points lying in the next annular region for which \(\pi\leq k\rho\leq 2\pi\), or \((z-z_0)\lambda\leq|s-s_0|^2\leq 2(z-z_0)\lambda\), will add with phasor angles that tend to cancel the contributions coming from wavelets in the inner circle. 

Suppose the wave passing through the source plane is a uniform plane wave with constant phase, and that we gradually open up a circular aperture of increasing diameter \(2a\) in the source plane surrounding the projected observation point \((s,z_0)\).

Examination of the phasor diagram in Figure 11., or a more formal integration of Huygens' integral, will then show that the intensity at the observation point 8,\(z\) increases steadily with increasing source aperture diameter, up to a maximum value when the aperture just contains the entire inner disk for which \(k\rho\leq\pi\).

The intensity at the observation point will then decrease as the source aperture is opened further, and in fact will drop entirely to zero when the diameter \(2a\) is such that \(k\rho=2\pi\). (A dramatic demonstration of this effect using a microwave source and detector can be found in the optics text by Andrews.)

 

Fresnel Zones and the Fresnel Number 

The intensity at the observation point in fact will oscillate periodically from zero to maximum and back again as successive additional annular rings or Fresnel zones are contained within the aperture.

The radii \(s_n\) that define the boundaries of these successive rings or so-called Fresnel zones about the projected source point \(s, z_0\) are thus given by 

\[\tag{24}s^2_n=nL\lambda,\qquad L\equiv z-z_0.\]

An important point here is that each successive Fresnel zone has equal area, so that each contributes an equal positive or negative contribution (assuming the entire circular aperture is uniformly illuminated). 

To look at the consequences of this in another way, suppose we consider an observation point situated on the \(z\) axis a distance \(L\) out from the center of an aperture of width or diameter \(2a\). The number of Fresnel zones contained within the aperture, as seen from the observation point, is then given by the Fresnel number defined by

\[\tag{25}N\equiv\frac{a^2}{L\lambda}.\] 

The importance of this Fresnel number parameter both for beam propagation problems and for optical resonators will become apparent in this section and in later chapters.

 

Far-Field Intensity and the Rayleigh Range

A second useful parameter in characterizing the diffraction properties of an aperture is the Rayleigh range of a collimated beam emerging from that aperture. Let us consider a definition of this parameter for an aperture of arbitrary crosssectional shape.

Suppose an aperture of any arbitrary cross-sectional shape is illuminated by a collimated and uniform-intensity plane wave. We can then show quite generally from Huygens' integral that the peak intensity in the far field occurs exactly on the beam axis, and that this on-axis intensity \(I(z)\) at distance \(z\) is related to the uniform intensity \(I_0\) in the transmitting aperture itself by

\[\tag{26}\frac{I(z)}{I_0}=\left(\frac{A}{z\lambda}\right)^2,\]

where \(A\) is the total area of the input aperture. 

For a gaussian beam, on the other hand, the peak or on-axis intensities at the beam waist and in the far field \((z\gg z_R)\) are related by

\[\tag{27}\frac{I(z)}{I_0}=\left(\frac{z_R}{z}\right)^2,\]

where \(z_R\) is the Rayleigh range for a gaussian beam, given by \(z_R\equiv\pi w^2_0/\lambda\) where \(w_0\) is the gaussian spot size at the beam waist.

Now, there does not seem to be any universally accepted way of defining a similar Rayleigh range \(z_R\) for other beam profiles or aperture shapes, although it is generally accepted that the Rayleigh range should mark in some sense the boundary between the "near-field" and "far field" diffraction regions for the beam emerging from the aperture.

As a convenient and simple definition, we will adopt the convention that the Rayleigh range for a transmitting aperture of any shape is given by equating the two preceding equations for \(I(z)/I_0\). That is, for a transmitting aperture of area \(A\), the Rayleigh range is given by

\[\tag{28}z_R\equiv\frac{A}{\lambda}\qquad(\text{arbitrary}\;\text{aperture}\;\text{shape}).\]

This definition will lead to slightly different expressions for the relationship between the aperture diameter or width and the Rayleigh range, or between the aperture Fresnel number and Rayleigh range, depending on the exact shape of the aperture, as we will see in later sections. It does, however, seem to be a convenient and meaningful way of defining a Rayleigh range for any shape of aperture.

 

 

 

 

Apertures and Scalar Diffraction Theory 

Let us now turn to the diffraction properties of sharp-edged apertures. The primary mathematical foundation for this discussion is an analytical solution published by Sommerfeld around 1896 for the diffraction of an infinite plane wave incident at an arbitrary angle on a perfectly conducting half plane. Notable aspects of this Sommerfeld solution (which we will not examine in detail here) are that it is a vector solution to the full electromagnetic problem; it assumes as boundary condition an infinitely thin and perfectly conducting sheet covering half of the transverse plane; part of the diffracted signal appears as a nonuniform cylindrical wave radiating from the discontinuous edge of this half plane; and the results are expressed in terms of Fresnel integral functions of the coordinates.

This Sommerfeld solution (along with earlier work by Huygens, Rayleigh, Fresnel, and Kirchoff) provides the mathematical underpinnings for the scalar Huygens- Fresnel integral theory we will use to describe diffraction effects in a variety of situations. 

We might note that the Fresnel-Kirchoff and the Rayleigh-Sommerfeld formulations of diffraction theory predict slightly different forms for the obliquity factor as a function of angle in Huygens' integral. These two forms, however, both go to the same limit of unity in the paraxial limit which is of interest to us.

We might also note again that Sommerfeld's theory is based on a diffracting surface which is perfectly conducting and infinitely thin. The diffracting apertures that we encounter in most real diffraction problems, on the other hand, are made from materials that have a finite thickness and are certainly not perfectly conducting.

The Huygens-Fresnel integral, however, is calculated using only the incident field values in the open part of the aperture. We normally do not even consider what are the electromagnetic properties of the surrounding material, therefore, and in practice the nature of the aperture does not seem to make any significant difference either in the theory or in the experimental results that are obtained under conditions typical of paraxial optical beams. 

 

 

 

Edge Waves or Boundary-Wave Diffraction Theory 

One important concept that emerges both in the Sommerfeld theory and in the discussions further on in this section is that diffraction from a sharpedged aperture creates a series of nonuniform spherical waves (or more precisely, cylindrical waves) that appear to be scattered from line sources exactly along the edges of the aperture, as sketched in Figure 12.

This concept in fact traces back to suggestions by Young and observations by Newton more than a century earlier, noting that if we observe the diffracted light from an aperture, we can indeed see that the edge of the aperture appears to be a brightly illuminated line source. 

The edge-wave interpretation of the Sommerfeld diffraction results was also extended around 1957 by Keller into a more general formulation of diffraction theory entirely in terms of edges waves scattered from aperture boundaries (see References). This interpretation is sometimes referred to as the "Keller edge-wave theory," or the "boundary diffraction wave theory" of optics.

We must use this edge-wave theory with some caution, however, as the simple boundary-wave picture applies rigorously only to edges or apertures that are illuminated with uniform-intensity plane or spherical waves, and not to other more general types of illumination. 

 

Single-Slit Diffraction Formula 

One classic problem in diffraction theory is the diffraction of a uniform collimated plane wave by a single slit of width \(2a\), as illustrated in Figure 13. The Huygens-Fresnel integral in one dimension that applies to this problem is 

\[\tag{29}\tilde{u}(x,z)=\sqrt{\frac{j}{z\lambda}}\int^a_{-a}\tilde{u}_0(x_0,z_0)\text{exp}\left[-j\frac{\pi}{(z-z_0)\lambda}(x-x_0)^2\right]dx_0.\] 

Suppose we assume a uniform incident plane wave, and define the normalized variables 

\[\tag{30}y\equiv\frac{x}{a}\qquad\text{and}\qquad N\equiv\frac{a^2}{(z-z_0)\lambda}=\text{Fresnel}\;\text{number},\]

where N is the number of circular Fresnel zones that will be visible in the slit, as seen from a distance \(z-z_0\)beyond the slit. The Huygens-Fresnel integral then takes on the normalized form 

\[\tag{31}\tilde{u}(y)=\sqrt{jN}\int^1_{-1}e^{-j\pi N(y-y_0)^2}dy_0.\]

This expression makes it apparent that the normalized diffraction pattern depends on distance \(z\) only through the dimensionless Fresnel number \(N\) and no other parameters. 

 

The Complex Fresnel Integral 

It is convenient to interpret this single-slit result using the complex Fresnel integral function \(\tilde{F}(x)\) defined by 

\[\tag{32}\tilde{F}(x)\equiv C(x)+jS(x)\equiv\int^x_0e^{j\pi t^2/2}dt.\]

The real and imaginary parts of this are the well-known Fresnel cosine and sine integrals, which are conventionally defined as 

\[\tag{34}\tilde{F}(x)\approx\left\{\begin{array}{c}x\;e^{j\pi x^2/2}\qquad\qquad\quad\text{for}|x|\ll1,\\\frac{1+j}{2}-\frac{j}{\pi x}e^{j\pi x^2/2}\qquad\text{for}|x|\gg1.\end{array}\right.\]

The large argument form is particularly useful, and is in fact a quite reasonably good approximation even for arguments only slightly greater than unity.

 

Delta Function Property of the Fresnel Kernel 

The integrand in the Fresnel integral function or in the sine and cosine integrals is obviously a normalized version of the Huygens' integral kernel, and it can be useful to examine briefly the form of this integrand.

The functions \(\cos(\pi/2)x^2\) and \(\sin(\pi/2)x^2\), if plotted versus \(x\), appear as in Figure 15.

We can note that the total area under either of these functions is finite and also equal, as evidenced by the result that

 

 

 

\[\tag{35}\int^\infty_{-\infty}e^{j(\pi/2)x^2}dx=\int^\infty_{-\infty}e^{-ax}dx\left|_{a\equiv-j\pi/2}=\sqrt{\frac{\pi}{a}}=\sqrt{2j}\right..\]

(We obviously have to add a very small positive real part to the constant a, or perhaps to the value of \(\pi\), to make the integral converge properly.) 

This complex Fresnel integrand, moreover, despite the fact that it neither goes to infinity at \(x=0\) nor goes to small values as \(x\rightarrow\infty\), still operates something like a rather crude Dirac delta function.

If we multiply the curves shown in Figure 15. by any slowly varying function \(f(x)\) and integrate from \(-\infty\) to \(\infty\), the integral will pick out primarily the value of \(f(x)\) within the central lobes of the kernel around \(x=0\).

At larger values of \(x\) the \(\cos(\pi x^2/2)\) and \(\sin(\pi x^2/2)\) functions oscillate so furiously between positive and negative values that they average out to 0 over any slow variations of the function \(f(x)\).

We must emphasize, however, that the Fresnel kernel really is a quite poor delta function, with a considerable response to values of \(f(x)\)—and especially to discontinuous changes in \(f(x)\)—at values of \(x\) well away from zero. 

 

Rational Approximation to the Fresnel Integral

The Fresnel integral function is essentially identical to an error function of complex argument. Analytical expressions and numerical results related to the error function can thus be very useful in working with the Fresnel integral. 

Abramowitz and Stegun also give a rational approximation—that is, a purely emperical polynomial approximation—to the Fresnel integral function which can be very useful for numerical calculations. The complex Fresnel integral can first

 

 

 

be written as 

\[\tag{36}\tilde{F}(x)=\frac{1+j}{2}-[g(x)+jf(x)]e^{j\pi x^2/2},\]

and the two functions \(f(x)\) and \(g(x)\) can then be approximated by 

\[\tag{37}\begin{align}&f(x)\approx\frac{1+0.962x}{2+1.792x+3.104x^2},\\&g(x)\approx\frac{1}{2+4.142x+3.492x^2+6.670x^3}.\end{align}\]

The resulting error in the calculation of \(\tilde{F}x\) is \(\leq2\times10^{-3}\) for all values of \(0\leq x<\infty\). (Similar polynomial approximations exist for Bessel functions and many of the other higher transcendental functions.)

 

Single-Slit Diffraction Results

The diffraction pattern predicted by Huygens' integral for a uniformly illuminated slit then becomes, in terms of the Fresnel integral function, 

\[\tag{38}\tilde{u}(y)=\sqrt{\frac{j}{2}}[\tilde{F}{[\sqrt{2N}(1-y)]-\tilde{F}^*}[-\sqrt{2N}(1+y)]].\] 

From the analytical results for the Fresnel integral function, we can deduce the following characteristics of the single-slit diffraction patterns in the near and far-field regions: 

 

 

(1) Far-field diffraction pattern. In the far-field region, where \(z\gg a^2/\lambda\) or \(N\ll 1\), the analytical form for the beam pattern becomes 

\[\tag{39}\tilde{u}(x)\approx(4jN)^{1/2}\frac{\sin\text{c}(2\pi Nx/a)}{2\pi Nx/a}=(4jN)^{1/2}\sin\text{c}\left(\frac{2\pi ax}{z\lambda}\right)\]

for \(z\gg a^2\lambda\) or \(N\ll 1\). [We use the definition that \(sinc(x)\equiv(\sin x)/x\).] The diffraction pattern thus stabilizes into a single central peak, plus substantially weaker sidelobes, as shown for a rectangular aperture (equivalent to two crossed slits) in Figure 16.

The width of the central peak is inversely proportional to the slit width, and eventually diverges linearly with increasing distance from the source aperture, yet remains essentially constant in shape. 

(2) Fresnel number and Rayleigh range. The on-axis field amplitude in the far field can thus be written in the form

\[\tag{40}\frac{\tilde{u}(z)}{\tilde{u}_0}=\sqrt{4jN}=\sqrt{jz_R/z},\]

 

 

where the first equality comes from Huygens' integral and the second from our convention for defining \(z_R\) for an aperture. We thus have the relation 

\[\tag{41}\frac{z}{z_R}=\frac{1}{4N}\qquad\text{or}\qquad z_R\equiv\frac{4a^2}{\lambda}\qquad\text{(single-slit}\;\text{aperture)}\]

between the Fresnel number and the Rayleigh range for a slit or a square aperture. (Other aperture shapes will give a slightly different numerical constant in this relation, as we will see later.)

(3) Near-field diffraction ripples. The near-field diffraction patterns closer in to a uniformly illuminated slit are much more complex and variable than the far-field pattern, which has a constant shape, and merely expands linearly in size with increasing distance. 

Figure 17, for example, along with Figure 18, shows plots of the normalized beam intensity profile \(|\tilde{u}(y)|^2\) across the normalized slit width \(y=x/a\) at various Fresnel numbers \(N\) or normalized distances \(z/z_R\) from the input slit. (These results were plotted using the rational approximations given in Equation 37; note that the distance \(z\) goes inversely with the Fresnel number \(N\).) 

The diffraction behavior clearly separates into a far-field region where \(N\ll1\) and \(z\gg z_R\), and a near-field region for which \(N\geq1\) and \(z\leq z_R\). For \(N=0.5\), for example, which is roughly the boundary region between near and far fields, the beam has essentially only one smooth central lobe, similar to the far field, with weak ripples or side lobes in the outer tails.

From Figure 17 is may seem that in the boundary region around \(N\approx 0.5\), the central lobe of the beam appears to be considerably narrower than the slit from which it came, even though no focusing is present, and only diffraction spreading is taking place.

A substantially fraction of the beam energy, however, has actually diverged out into the strong tails of the beam, extending well beyond the projected slit width, so that the rms width of the beam continually increases. Figure 18 shows that beyond this

 

 

 

distance the beam does diverge into a linearly spreading beam with an essentially constant profile. 

In the near-field region, on the other hand, as we move closer to the aperture and the Fresnel number N increases, the diffraction patterns acquire an increasingly complex structure, with increasing number of diffraction ripples, or "Fresnel ripples." The pattern becomes increasing rectangular in shape as we move closer to the aperture, especially for \(N\geq3\) or 4, but the Fresnel ripples remain strong and increase rapidly in frequency as we move in toward the aperture. 

(4) Ripple interpretation. The strong Fresnel ripples observed on the beam in the near field can be interpreted using the asymptotic form of the Fresnel integral function, as follows. Consider the two terms in the Huygens' integral solution, using the large-argument approximation for the complex Fresnel integral given in Equations 34 or 36. The two exp\((j\pi x^2/2)\) terms, when evaluated for \(x=\sqrt{2N}(1\pm y)\), then have the forms

\[\tag{42}\tilde{u}(y)\approx\text{exp}[-j\pi N(y\pm1)^2]=\text{exp}\left[-j\frac{\pi(x\pm a)^2}{z\lambda}\right].\]

But we can recognize these as exactly the analytical forms for two spherical (or, more precisely, cylindrical) waves emanating from the edges of the slit, as in Figure 13. That is, Equation 43 corresponds to two cylindrical waves coming from source points \(x_0=\pm a\), and observed at an observation point \((x, z)\) beyond the slit.

The Fresnel ripples seen in Figure 17 and 18 result from the presence of two Sommerfeld edge waves. These waves interfere with the primary geometric part of the transmitted wave (which is represented by the constant term in the asymptotic expansion), producing the large-scale or large-amplitude ripples; and also interfere with each other, thus leading to a complex pattern of finer ripples on top of the coarser ripples.

(5) Number of ripples.  Inspection will show in fact that for \(N\geq1\), the diffraction patterns in Figure 17 have essentially \(N\) large-scale ripples across the aperture width. These ripples come, more or less, from each edge wave interfering with the geometric plane wave coming through the aperture on the same side.

These larger ripples are then modified by smaller-amplitude but higher-frequency ripples, especially near the outer edges of the aperture. These higherfrequency ripples are clearly produced primarily by interference with the spherical wave that comes from the opposite side of the aperture, so that we are on the \(4N\)-th Fresnel zone from that source point or, rather, that source line. (Why is this \(4N\) and not \(2N\)?)

(6) Geometric shadow terms.  We can also note that if we are inside the projected slit aperture, so that \(|x|\leq a\) and hence \(|y|\leq 1\), and also in the near-field region where \(N\geq1\) or \(z\leq 4z_R\), then both of the arguments in the complex Fresnel integral of Equation 38 will be positive.

Hence, the two constant terms in the large-argument expansion will add in phase. These two terms combine in fact to give a value just equal to unity, i.e., they give the constant geometric projection of the incident beam through the aperture.

As soon as we move outside the geometric shadow, however, on either side of the beam, one or the other of these arguments changes sign and becomes negative. The two constant terms then cancel, and give the expected value of zero for the geometric value of the amplitude outside of \(x=\pm a\) or \(y=\pm1\).

 

Conclusions: Single-Slit Diffraction Behavior

All in all, the complex diffraction ripples in the near field can thus be viewed as resulting from a complicated interference between the simple geometric transmission of the aperture, and the spherical edge waves scattered from the two edges.

As we move in ever closer to the aperture (yet still remain within the underlying paraxial approximation), the beam profile in Figure 17 becomes more and more "square" and the Fresnel ripples become higher and higher in frequency, although they never disappear.

The Fresnel ripple behavior near the aperture, in fact, is very analogous to the Gibbs phenomena that occur in the Fourier analysis of a square pulse as we take an increasing number of Fourier components. We can show that for large N the outermost ripple, just inside the beam boundary, is always the largest, and that this ripple has a peak overshoot of \(\approx 18\)% in amplitude or twice this in intensity. 

 

Numerical Beam Calculations: Number of Sample Points 

In practical beam or resonator problems, we often want to calculate the exact diffraction pattern produced by some source wave \(\tilde{u}_0(x_0)\) which is more complicated than a simple plane wave, after that wave has passed through a hard-edged aperture such as a slit of width \(2a\). We must then evaluate Huygens' integral (Equation 29) with the appropriate source function \(\tilde{u_0}(x_0)\) included; and the resulting calculations must usually be done numerically.

The Sommerfeld edge waves from the aperture edges will still play a role in the diffraction patterns with apertures other than simple slits, and with source functions other than a simple plane wave.

The exact diffraction patterns will be more complex, however, than just a geometric projection of the incident source function, plus edge waves proportional to the amplitude of the source function at the aperture edges. (The Sommerfeld edge waves in some sense have a physical reality as cylindrical waves diverging, or appearing to diverge, from the slit edges; but in a more accurate picture they represent a mathematical effect of sharply truncating a uniform plane wavefunction at the aperture edges.)

From the uniform plane wave solutions discussed in this section, we can at least draw the insight that the diffraction patterns for reasonably smooth source functions passing through apertures of width \(2a\) are likely to exhibit on the order of \(N\) large-scale ripples across the beam aperture, plus smaller-amplitude ripples with spatial periods ranging down to approximately \(2z/4N\). (Again, why \(4N\)?)

From the sampling theorems of transform theory, we then know that to describe the diffracted wave pattern accurately we will need at least 2 sample points per ripple period.

Depending upon the accuracy that is required, we will therefore need somewhere between \(2N\) and \(8N\) total sample points in each transverse direction across the aperture—or a comparable number of terms in any kind of series expansion— in order to do numerical calculations of the diffraction patterns with adequate accuracy. (We can derive this same criterion by asking how many points we will need to evaluate the kernal in Huygens' integral accurately for all pairs of points across the aperture.)

As we move inward toward a source aperture, the Fresnel number \(N\) goes up: it takes more mathematical work to propagate a beam a short distance beyond an aperture than a long distance! (assuming that we want to describe the detailed near-field function accurately). 

 

Beam Spillover and Guard Bands

In doing numerical calculations of beam propagation beyond a slit or other hard-edged aperture, we must also realize that the diffracted wave function even in the near-field region spreads out for a sizeable distance into the shadow region outside the aperture width, i.e., into the region \(x>a\) or \(y>1\). (Look again at Figure 17 to confirm this.)

Although the field amplitude obviously dies out as we move transversely outward, a significant amount of energy can reside outside the geometric beam boundary, and this region must be taken into account, particularly if we are making a series of forward propagation steps beyond the aperture. 

It is thus necessary to include in the numerical calculations the field values in a "guard band" that extends into the shadow region, out to a distance of perhaps \(\approx1.2\) to \(\approx1.5\) times the half-width of the aperture itself. Additional guard-band space may also be required by whatever numerical technique is being employed, in order to avoid spurious results due to aliasing effects in the numerical procedure. This additional requirement will increase still further the total number of sample points required in a numerical calculation procedure.

 

On-Axis Intensity: Square Aperture

It is also often of interest to calculate the field amplitude or the intensity as a function of distance z exactly on the aperture axis in both the near and far fields for a wave diffracted by apertures of various shapes. For a uniformly illuminated single slit, the diffracted amplitude on axis is given in the near and far fields by

The diffraction pattern for a rectangular aperture is then simply the product of diffraction patterns for two slits at right angles with Fresnel numbers \(N_x=a^2/z\lambda\) and \(N_y=b^2/z\lambda\) in the two transverse directions, as illustrated in Figure 16. 

The intensity on axis for a square aperture is thus equal to the amplitude for a slit aperture raised to the fourth power. Figure 19 shows this on-axis intensity versus normalized distance \(z/z_R\equiv(4N)^{-1}\) for both square and circular apertures of width or diameter \(2a\). For both apertures the intensity (by definition) asymptotically approaches the value \((z_R/z)^2\) at large \(z\).

For a square aperture (upper plot) the intensity oscillates periodically with period \(2N\) as we move inward toward the source aperture, i.e., peaks occur roughly near \(N=1,3,5\cdots\) and minima occur near \(N=2,4,6\cdots\), The phase factors in the second term of the large-argument expansion for the Fresnel integral show, however, that the extrema will not occur exactly at integer values of \(N\), though the periodicity goes as \(N\pi\).

Note that in contrast to the circular aperture (which we will discuss shortly), the amplitude of the on-axis variation for the square aperture fades out as we move closer in to the source aperture. 

 

 

 

 

4. APERTURE DIFFRACTION: CIRCULAR APERTURES 

Another classic diffraction problem is the diffraction of a uniform plane wave by a circular aperture of diameter \(2a\). We will explore some of the practical implications of this problem for laser-beam behavior in this section. 

 

Circular Aperture With Cylindrical Symmetry

Suppose a circular aperture of diameter \(2a\) located at plane \(z_0\) is illuminated by an arbitrary but cylindically symmetric source function \(\tilde{u}_0(r_0)\). The general Huygens' integral giving the field amplitude \(\tilde{u}(r)\) at plane \(z\) can then be written in radial coordinates in the form 

\[\tag{44}\tilde{u}(r)=j2\pi Ne^{-j\pi N(r/a)^2}\int^1_0\frac{r_0\tilde{u}_0(r_0)e^{-j\pi N(r_0/a)^2}}{a}J_0\left(\frac{2\pi N_{rr_0}}{a^2}\right)d\left(\frac{r_0}{a}\right),\]

where the Fresnel number \(N\) for the circular case is defined by 

\[\tag{45}N\equiv\frac{a^2}{(z-z_0)\lambda}.\]

Note that this again depends only on the normalized variable \(r/a\) and the Fresnel number \(N\).

 

Circular Aperture, Noncylindrical Symmetry 

If the input function \(\tilde{u}_0(r_0,\theta_0)\) is not cylindrically symmetric, we must first expand this input function into a set of functions \(\tilde{u}_{0m}(r_0)\) of increasing azimuthal order \(m\) in the form

\[\tag{46}\tilde{u}_0(r_0,\theta_0)=\sum^\infty_{m=-\infty}\tilde{u}_{0m}(r_0)e^{jm\theta}.\]

The functions \(\tilde{u}_{om}(r_0)\) can be found by Fourier-transforming the source function \(\tilde{u}_0(r_0,\theta_0)\) in the azimuthal variable \(\theta_0\) in the form 

\[\tag{47}\tilde{u}_{om}(r_0)\equiv(2\pi)^{-1}\int^{2\pi}_{0}\tilde{u}_0(r_0,\theta_0)e^{-jm\theta_0}d\theta_0.\]

Each individual azimuthal component \(\tilde{u}_m(r)\)can then be propagated to distance \(z-z_0\) by a generalized form of Equation 44, namely

\[\tag{48}\tilde{u}_m(r)=j^{m+1}2\pi Ne^{-j\pi N(r/a)^2}\int^1_0\frac{r_0\tilde{u}_{0m}(r_0)e^{-j\pi N(r_0/a)^2}}{a}J_m\left(\frac{2\pi N_{rr_0}}{a^2}\right)d\left(\frac{r_0}{a}\right),\]

where \(J_m\) is Bessel function of order \(m\). The final output function is obtained by reassembling the azimuthal components in the same series, i.e.,

\[\tag{49}\tilde{u}(r,\theta)=\sum^\infty_{m=-\infty}\tilde{u}_m(r)e^{jm\theta}.\]

The integral transforms involved in these calculations are referred to as the Fourier-Bessel or Hankel transforms.

No analytical solutions as convenient as the Fresnel integral function exist for these circular aperture situations, even for uniform plane wave excitation. The circular analogs of the Fresnel integral function are the Lommel functions, which are not much discussed in standard references.

The general features of the near and far-field diffraction patterns for a circular aperture are, however, generally similar to the slit or square aperture situations, although there are also some very significant differences, as we will now discuss. 

 

Circular Aperture: Far-Field Diffraction Pattern 

The properties of circular-aperture mode functions become successively more complex as we go to higher azimuthal orders; and in practice the first effort we make with any laser having cylindrical symmetry is to obtain at least an azimuthally symmetric beam. In the remainder of this section, therefore, we will limit our discussion almost entirely to the lowest-order \(m=0\) or azimuthally uniform beams. 

The far-field diffraction pattern for a uniformly illuminated circular aperture, i.e., the solution to Equation 44 for \(N\ll1\), can be obtained using the Bessel function relation

\[\tag{50}\frac{d}{dz}[zJ_1(z)]=zJ_0(z).\] 

The result is the well-known Airy disk pattern given by

\[\tag{51}\tilde{u}(r,z)\approx j\pi Ne^{-j\pi N(r/a)^2}\times\frac{2J_1(2\pi Nr/a)}{2\pi Nr/a},\]

 

 

 

which we can rewrite in terms of the far-field angle \(\theta\) in the form 

\[\tag{52}\tilde{u}(r,z)\approx j4\pi Ne^{-j\pi N(r/a)^2}\times\frac{2J_1(2\pi a\theta/\lambda)}{(2\pi a\theta/\lambda)}\]

which is valid in the far field for \(z\gg z_R\) or \(N\rightarrow 0\).  By analogy to the sine function which is defined as \((\text{sin}x)/x\), this function is sometimes referred to as the "jinc" function, with the definition that \(\text{jinc}(r)\equiv2J_1(r)/r.\)

This pattern has a single dominant central lobe, which in the circular situation contains \(\approx 86\)% of the total energy, surrounded by a series of increasingly weaker circular rings, as shown in Figure 20. The first null of this pattern occurs at a half-angle \(\theta_1\) or a radius \(r_1\) in the far field given by

\[\tag{53}\theta_1=\frac{r_1}{z}\approx\frac{1.22\lambda}{d},\]

with successive nulls \(r_n\) or \(\theta_n\) denned by successive zeros of the \(J_1\) Bessel function.

Since the limiting value of the first-order Bessel function for small argument is \(J_1(r)\approx r/2\), the far-field on-axis intensity for a circular aperture is given by

\[\tag{54}\frac{I(0,z)}{I_0}=\left(\frac{A}{z\lambda}\right)^2=\left(\frac{\pi a^2}{z\lambda}\right)^2=\left(\frac{z}{z_R}\right)^2,\]

where we have again made use of our general definition of Rayleigh range. The relationship between Rayleigh range and Fresnel number for the circular aperture is thus

\[\tag{55}\frac{z}{z_R}=\frac{1}{\pi N}\qquad\text{or}\qquad z_r\equiv\frac{\pi a^2}{\lambda}\qquad\text{(circular}\;\text{aperture)}.\]

This differs by a small numerical factor (i.e., \(4/\pi)\) from the comparable expression given in Equation 41. for the square aperture situation.

 

Circular Aperture: Near-Field Diffraction Patterns

The near-field or Fresnel diffraction patterns for a uniformly illuminated circular aperture, for \(N>1\) or \(z<z_R\), consist of a series of circular rings 

 

 

 

modulating a constant-amplitude geometric background or "top hat" pedestal, in a fashion generally similar to the single-slit diffraction patterns shown in Figures 17 and 18. The analogous near-field patterns for the uniformly illuminated circular aperture are somewhat more difficult to calculate, however, and do not seem to be given in many elementary optics texts.

Figure 21 shows a few examples of plots of intensity versus radius for near-field diffraction patterns from a circular aperture at various Fresnel numbers greater than unity. (Note that in Figure 21 one side of the plot represents a theoretical calculation from the Huygens' integral, whereas the other side represents a careful experimental measurement for the same conditions using a laser beam setup.)

It is again apparent that these near-field diffraction patterns have approximately \(N\) large-amplitude Fresnel ripples across the full width of the beam, and that these larger fringes are then modulated by many smaller-amplitude but higher-frequency Fresnel ripples on top of them.

A significant difference from the single-slit situation, however, is that the centermost ripple or spike (which occurs for every odd Fresnel number \(N)\) is now significantly larger than all the other Fresnel ripples. The central minimum (which occurs for every even \(N)\) is also significantly deeper, and in fact goes exactly to zero. (Recall that in the slit or square aperture situations, the strongest near-field Fresnel ripples occurred at the edges of the aperture.)

 

 

 

 

Circular Aperture: On-Axis Intensity 

This difference between rectangular and circular apertures also shows up in the behavior of the intensity on axis, which has already been plotted for both square and circular apertures in Figure 19. As we move in toward the uniformly illuminated circular aperture, the on-axis intensity oscillates with increasing frequency between limiting values of zero and of four times the intensity in the transmitting aperture.

These peaks and nulls occur exactly at integer values of the Fresnel number \(N\), and the magnitude of these oscillations does not decrease as we get closer in to the aperture, in distinct contrast to the slit or square aperture case.

This magnification of the intensity on axis can sometimes cause significant damage problems in higher-power lasers, and we will explore it further below. 

 

General Formula: Circular Aperture With Gaussian Illumination 

Before carrying this discussion further, it will be convenient to generalize the azimuthally uniform circular-aperture situation slightly by supposing that the source field in the aperture is a centered gaussian plane-wavefunction with a radial amplitude function given by

\[\tag{56}\tilde{u}_0(r_0)=e^{-r^2_0/w^2_0},\]

as shown in Figure 22.

It is then possible, after some manipulation, to expand the general Huygens' integral given in Equation 44. into an infinite series of terms containing successively higher-order Bessel functions (see References). The two leading terms

in this series expansion are given by

\[\tag{57}\tilde{u}(rz)\approx\frac{\tilde{q}_0\;e^{-j\pi N(r/a)^2}}{\tilde{q}z}\times[1-e^{-j\pi N}e^{-a^2/w^2_0}J_0(2\pi Nr/a)],\]

where \(\tilde{q}_0=j\pi w^2_0/\lambda\),  and \(\tilde{q}(z)=\tilde{q}_0+z\) is given by the usual gaussian-beam propagation expressions. This approximation is valid only for values of r near the optical axis, i.e., for \(r/a\ll1\), but it is valid on axis, i.e., at \(r=0\), for any arbitrary distance \(z\) in the near or far fields beyond the aperture.

To make Equation 57 more transparent, we can define the amplitude \(\delta_a\equiv e^{-a^2/w^2_0}\) to represent the wave amplitude of the input gaussian beam at the point where it is cut off at the edge of the circular aperture, as in Figure 22. The magnitude of the wave on and near the \(z\) axis in the near and far fields then has the form

\[\tag{58}I(r,z)\approx\left[\frac{w_0}{z(z)}\right]^2\times|1-\delta_{a}e^{-j\pi N}J_0(2\pi Nr/a)|^2\qquad(r\ll a),\]

where \(\delta_a\leq 1\), with \(\delta_a=1\) corresponding to a uniform plane-wave input.

 

Near-Axis Fresnel Ripples: Uniform Illumination 

Equation 58 says that for either gaussian or uniform plane wave illumination, the wave amplitude on and near the axis consists in essence of the amplitude corresponding to the unperturbed gaussian beam, with a relative magnitude of unity times \(\tilde{q}_0/\tilde{q}(z)\), plus a Bessel function contribution of the form \(J_0(2\pi Nr/a)\) which has relative magnitude \(\delta_a\leq1\), and which adds to the unperturbed beam with relative phase given by \(e^{-j\pi N}\). 

Consider first the uniform plane-wave situation, with \(w_0\rightarrow\infty\) and \(\delta_a\rightarrow 1\). The centermost ripples in the uniform circular-aperture diffraction patterns given in the preceding have the form of a \(J_0(r)\) Bessel function centered on the axis whose central lobe has magnitude of unity.

This Bessel function subtracts from the background value for even \(N\) values, thus giving an exact null on axis, but adds to the background value for odd \(N\) values, thus giving twice the amplitude. The central spikes occurring at even Fresnel numbers should thus have an intensity equal to four times the peak intensity in the transmitting aperture itself.

The surrounding rings are substantially weaker than the central spike, because the higher-order maxima of the \(J_0\) function become successively smaller compared to unity. The central spikes continue to reoccur at more and more closely spaced axial distances as we move toward the source aperture, as seen in the near-field patterns in Figure 21, and also in the on-axis intensity results for the uniformly illuminated circular aperture given in Figure 19. They also become much narrower or sharper as we move in to larger Fresnel numbers, because the \(J_0(2\pi Nr/a)\) function becomes much narrower as \(N\) increases.

 

Near-Axis Fresnel Ripples: Truncated Gaussian Illumination

We can easily extend this discussion to the circularly truncated gaussian beam situation. The \(J_0(r)\) function responsible for the central spike is simply reduced in amplitude in this situation by exactly the relative field strength \(\delta_a=\text{exp}(-a^2/w^2_0)\) at the edge of the circular aperture. For \(\delta_a<1\) this smooths out 

 

 

 

the ripple variation along the z axis in the near field, filling in the nulls and trimming down the peaks. 

Figure 23.  shows a few experimental and theoretical near-field diffraction patterns for the intensity versus radius when a gaussian beam is truncated by a circular aperture at the \(1/e\) intensity points (i.e., at \(a=w_0/\sqrt{2})\). The central null at even \(N\) and central peak at odd \(N\) are still apparent, but the null no longer touches bottom, and the peak is now substantially reduced in intensity.

We clearly have here an example of the general point discussed in the previous section: Truncating the gaussian beam at a fractional amplitude given by \(\delta_a\), and hence at a much smaller fractional intensity given by \(\delta^2_0\), produces central diffraction ripples in the near-field intensity pattern whose peak-to-peak amplitude variation is given by \(1\pm\delta_a\) The ripples in the near field thus have an intensity variation given by \(\approx\pm2\delta_a\).

Truncating the gaussian beam even as far out as its 1% intensity point, for example, corresponding to \(\delta^2_0=0.01\) or \(\delta_a=0.1\), will still cause an intensity ripple of magnitude \(\approx\pm0.2\) or \(\pm\)20% in the near field. 

 

Far-field Intensity For a Truncated Gaussian 

It is worth noting that in the truncated gaussian situation, the change in on-axis intensity caused by the central diffraction fringe or intensity ripple persists out to arbitrary distance in the far field; and this ripple is a negative ripple, or an intensity reduction, as \(z\rightarrow\infty\).

Truncating a gaussian beam with a circular aperture at the \(\delta^2_a\) intensity radius will reduce the on-axis far-field intensity—and in fact the whole central far-field lobe intensity—by a fractional amount \((1-\delta_a)^2\approx 1-2\delta^a\). 

 

 

 

Suppose we wish to reduce the intensity ripples in the near field, or the brightness reducton in the far field to \(\leq1\)%.

This then requires \(\delta_a=\text{exp}(-a^2/w^2_0)\leq1/200\), which in turn requires an aperture with diameter \(d=2a\geq4.6w_0\). This is substantially larger than the \(d=\pi w_0\) criterion we have introduced earlier on the basis of \(\geq99\)% power transmission through the aperture itself. 

 

Optimum Aperture for a Gaussian Beam

As an extension of this point, suppose we have a gaussian beam of fixed total power \(P_0\) whose spot size \(w_0\) we can adjust to any desired value by passing this beam through a magnifying or demagnifying telescope.

We wish to transmit this beam through a circular aperture whose diameter \(2a\) is fixed by practical considerations, such as available lens or mirror sizes, in such a way as to maximize the on-axis intensity in the far-field beam from the aperture. Increasing the beam spot size \(w_0\) to better fill the aperture will thus decrease the far-field angular spread in this situation, but beyond a certain point will also cause increasing power loss as the gaussian beam is truncated by the finite-diameter aperture.

Figure 24.  illustrates the optimum choice of the ratio \(w_0/a\) for optimum farfield intensity in this situation. The maximum far-field intensity is \(\approx81\)% of what could be obtained if the same total power could be uniformly distributed over the circular aperture; and this condition occurs for \(w_0/a\approx0.89\), or for an aperture diameter \(d\approx2.25w_0\).

Note that the amplitude reduction at the aperture edge in this situation has a value of only \(\delta_a\approx0.28\), so that this particular situation will lead to quite substantial near-field ripples.

 

The Poisson Spot, or the Spot of Arago

Exactly the same kind of intense but narrow on-axis Fresnel structure in the near field that we have discussed in the preceding can also be seen, not only 

 

 

 

from a beam passing through a circular aperture, but also from the edge-wave diffraction effects when a larger beam is transmitted past or around a sharp-edged circular obstacle.

The resulting sharp narrow spike, with surrounding rings, that appears on the axis in the shadow region behind a circular aperture is generally known either as the Poisson spot or, as is becoming more common in recent discussions, the spot of Arago. 

Suppose a collimated and centered gaussian beam of spot size \(w_0\) is transmitted past an opaque, sharp-edged circular obstacle of diameter \(2a\). In accordance with Babinet's principle, the transmitted amplitude in this situation is then equal to the amplitude of the original gaussian beam, minus the diffraction pattern that would result from sending the same gaussian beam through a circular aperture of the same diameter.

We can then use exactly the same approximate analysis as given in Equation 58, except that we no longer have the direct background term. Instead, we see only the scattered Fresnel terms, so that the intensity on and near the axis, in the shadow region behind the obstacle, is given approximately, near the axis, by

\[\tag{59}\tilde{u}(r,z)\approx-\frac{\tilde{q}_0}{\tilde{q}(z)}e^{-j\pi N-a^2/w^2_0}\times e^{-j\pi N(r/a)^2}J_0(2\pi Nr/a).\]

This expression produces a central spike of amplitude \(\approx\delta_a\), and surrounding Bessel function rings, which are present everywhere along the axis in the exact center of the shadow region of the circular object. The central intense spike is often called the Poisson spot in classical optics texts.

This spot has also come to be called the "spot of Arago" by many laser workers. An example of this spot of Arago for a gaussian beam coming past a circular obstacle is shown in Figure 25.

Note again that for a beam coming through a circular apertures the on-axis bright spot only occurs at distances corresponding to even Fresnel numbers, although these locations become very closely spaced on axis as we move in closer to the aperture. For a beam coming past a circular obstacle, however, or for almost any kind of annular aperture, this narrow but intense spot occurs essentially everywhere along the axis in the near field.

Although this spot contains little total energy, its peak intensity as we noted in the preceding can be as high as four times the average intensity in the transmitting aperture.

Such spots of Arago have in fact been responsible for many small damage spots and even holes drilled in the center of lenses or optical windows placed in the near-field output region in front of unstable-resonator lasers. Note also that analogous very bright on-axis spots or Fresnel diffraction peaks may occur at certain points along the axis inside a laser resonator, where the large circulating intensity may do similar unexpected (and unwanted!) optical damage.

 

Diffraction Patterns for Annular Apertures

By applying Babinet's principle—that is, by combining the approximate analytic formulas for one or more truncated or untruncated gaussian beams— we can then compute the on-axis and near-axis intensities for a wide variety of annular apertures which have either uniform or radially tapered illumination, and which may be truncated on both their inner and outer edges.

Because this type of illumination provides a reasonably good model for the output beam coming from a circular unstable resonator, these results can be of some interest in the design and evaluation of unstable optical resonators.

The kind of annular output beam that comes from a simple hard-edged unstable resonator in fact generally produces both a strong spot of Arago in the near field, and a far-field beam spread which is proportional more to the radial width of the annulus than to the overall beam diameter.

Proposals are often made, therefore, to find some way of combining the standard diffraction coupling past the edge of the unstable resonator with partial transmission through the central part of the output mirror.

One difficulty with this idea, however, lies in controlling the absolute phase angle which the centrally transmitted wavefront will have relative to the wavefront transmitted past the output mirror edge.

To illustrate the kind of behavior we can expect in these situations, Figure 26 shows a few examples of the axial intensity variation we can expect from apertures of this type with different intensities and relative phase angles for the wavefront coming throught the central and annular regions of the output aperture.

 

The Distinction Between Circular and Other Apertures

The intense but narrow on-axis spikes, rings and nulls that we have seen in Figures 21, 23, and 25 are very commonly seen whenever a coherent optical wavefront passes through an aperture having circular symmetry, but are much attenuated for essentially any other aperture shape.

A physical explanation for the large increase in sharpness and in peak intensity of these central spike phenomena seen with a circular aperture as compared to to other shapes might be argued as follows.

We can note, for example, that in a square aperture the effective Fresnel number or Fresnel phase shift, as seen from an on-axis point, is different in different radial directions, as illustrated in Figure 27.

For the square aperture, the effective Fresnel number is equal to \(a^2/z\lambda\) only exactly along the \(z\) or \(y\) axes; and the Fresnel phase shift changes as we consider scattered waves coming from points near the corners of the aperture.

In other words, the edge waves from various points along the perimeter of a rectangular aperture (or a slit) will arrive at the observation point with somewhat different phase shifts, so that they will not all add in phase.

 

 

 

 

 

In the circular situation, by contrast—and in fact only in this situation—the Fresnel phase shifts between a point on axis and the scattered wavelets coming from the aperture edge are the same for every point around the perimeter of the aperture, since the Fresnel number is independent of azimuth.

In a circular aperture, therefore, all of the edge-wave contributions from the entire perimeter can arrive at an on-axis point exactly in phase, and thus they can add up to produce the maximum possible Fresnel ripple fluctuation. 

The smearing or averaging out of the Fresnel number in other situations seems to be the obvious physical reason for the substantial softening or smoothing out of the on-axis peaks as compared to the circular situation, expecially at larger Fresnel numbers. (This smearing out of the Fresnel number along different 

 

 

 

azimuthal directions is also responsible for the extrema of the square-aperture on-axis intensity not occurring exactly at integer values of the Fresnel number \(N\).) 

Anything that tends to reduce the in-phase vector addition characteristic of the circular aperture will tend to damp out the peak amplitude of the Fresnel ripple effects.

Techniques for reducing Fresnel ripple effects can include tapering the aperture transmission at the aperture edge ("soft" versus "hard" apertures); or using any aperture shape other than a perfect circle (rippled or serrated apertures, ellipses, etc.). Several of these approaches have in fact been used in laser beam applications. 

 

"Supergaussians" and Other Smoothed Beam Shapes 

As an extension of this idea, we can note that the simple gaussian beam profile, though it may be attractive to the mathematical analyst, is generally not attractive to the designer of laser fusion amplifiers, or other high-power laser systems, for two related reasons. If the gaussian spot size is made relatively small compared to the amplifier aperture, in order to avoid edge diffraction effects, then the energy extraction from the laser medium is poor—the outer part of the beam does not saturate and extract energy from the laser medium, which has been pumped or excited at great trouble and expense.

If the gaussian beam is expanded to more effectively fill the aperture, however, this will produce strong edge diffraction ripples, which are not only unsightly but will produce serious self-focusing effects at high powers, thus causing even more troublesome and expensive damage effects. 

To get around this, laser designers have explored a variety of other potential transverse beam profiles, which may be more difficult to generate, but which will on the one hand more uniformly fill the aperture, and on the other hand taper smoothly to zero or near-zero at the aperture edge, on the principle that the way to avoid edge waves is to set the optical amplitude \(\delta_a\) to zero at the aperture edge. (This principle is approximately but not exactly valid.) One set of transverse functions that have been proposed are the "supergaussians," with

 

 

the analytical form 

\[\tag{60}\tilde{u}(r)=\text{exp}[-c_nr^n],\qquad n\geq2.\] 

Intensity profiles for some of these supergaussians are shown in Figure 28. Profiles of this type with indices ranging from \(n=5\) to 10 have been analyzed in various laser fusion programs.

We can show, for example, that a TEMoo gaussian beam profile only fills \(\approx22\)% of the volume of a laser rod or tube with diameter \(d=3w\), whereas a supergaussian with \(n=8\) will fill \(\approx86\)% of the rod volume at the same diameter. 

Figure 29 shows, on one side, another smoothed intensity profile, in this situation one obtained by passing a gaussian beam through a strongly saturating amplifier so that the center of the gaussian is much reduced compared to the outer edges.

The other side of Figure 29 then shows the calculated near-field diffraction pattern for this input intensity for a particular outer edge truncation, at a distance corresponding to a Fresnel number of \(N=10\). Diffraction ripples still exist, but the elimination of the central spike and general smoothing of the near-field pattern is very evident.

 

Relay Imaging of Apertured Laser Beams

As an extension of this topic, we can note that multistage laser amplifiers often use beam expanding telescopes between successively higher-power stages, combined with spatial filters consisting of small apertures at the telescope focus which remove the amplitude components scattered or diffracted into larger angles by imperfections, ripples, edges, or small-scale self-focusing in each successive amplifier stage.

A concept found very useful in the design of these systems is the use of image relaying, as illustrated in Figure 30. In this technique, the

 

 

locations and focal lengths of the relay lenses are chosen so that they simultaneously achieve the desired beam magnification, namely, \(M=f_2/f_1\), and achieve direct imaging from an object plane located at -\(d_1\) to an image plane located at \(+d_2\) as shown in Figure 30.

This means that, except for the (we hope) small diffraction effects of intervening apertures, the beam profile at the aperture plane or midplane of one amplifier can be imaged as an essentially identical but magnified beam profile at the midplane or image plane of the next amplifier. The intensity cutoff at one aperture is thus automatically converted into an (almost) zero amplitude at the edge of the next aperture.

By thus successively relaying the aperture images from amplifier to amplifier through the system, we can hope to fill the diameter of each stage, and yet simultaneously reduce or mimimize the edge diffraction effects produced by the finite aperture of each stage. 

 

Far-Field Angular Beam Spread: Arbitrary Aperture Shapes 

Finally, for use in predicting the far-field diffraction effects of beams coming from more imperfect or irregular apertures, we can present a rather remarkable expression for the far-field angular beam spread from uniformly illuminated apertures of arbitrary transverse cross section which has recently been published by Clark, Howard, and Freniere (see References).

The derivation of this formula, which is rather subtle, makes use of the concepts that the normalized intensity near the beam axis in the far field from a uniformly illuminated aperture will be directly proportional to the aperture area \(A\) (as we have already shown); but because of edge diffraction effects, the amount of intensity scattered out into larger angles away from the axis should be directly proportional to the total perimeter length \(p\) of the aperture. 

Using this approach, these authors obtain an approximate expression for the fractional encircled power \(P(\theta)/P_\text{tot}\) contained within a far-field cone of halfangle \(\theta\), in the form

\[\tag{61}\lim_{\theta\rightarrow\infty}\frac{P(\theta)}{P_\text{tot}}\approx1-\approx\frac{p\lambda}{2\pi^2A}\frac{1}{\theta}=1-\frac{1}{2}\frac{\theta_\text{hp}}{\theta}.\] 

The half-angle \(\theta_\text{hp}\) which contains (approximately) half of the total power in the far-field beam will thus be given by

\[\tag{62}\theta_\text{hp}\equiv\frac{p\lambda}{\pi^2A}.\]

 

 

 

If we define a solid angle \(\Omega_\text{hp}=\pi\theta^2_\text{hp}\) based on this half angle, the product of the transmitting aperture area and the far-field solid angle which contains half of the total power can be written as 

For simple circular or square apertures, therefore, Equation 63 gives results in agreement with the Equation 21 derived earlier. 

Equation 61 is more remarkable, however, in that it appears to be quite accurate even at small encircled angles for a variety of different aperture shapes, as illustrated in Figure 31.

The four different plots show, respectively, the approximate analytical expression, and its comparison with exact encircled energies for a circular aperture, a rectangular aperture, and an annular aperture with four radial struts. Note in particular the substantially larger far-field angular spread for the annular aperture with struts, due to the relatively larger perimeter to area ratio of this aperture.