Scattering from Disks and Apertures
IntroductionConsider two diffraction problems: for problem A a plane wave is incident from z < 0 on a flat screen that blocks the entire plane at z=0, except for a circular hole. For problem B the same incident wave is diffracted by a circular disc, the same size and location as the aperture in problem A. Babinet's principle states that the solutions to problems A and B are closely related, and in some cases the solution of one directly yields the solution of the other. The solutions are also related to the wave generated by a moving piston flush-mounted in an infinite planar screen. Babinet's principle applies to any incident field and aperture shape, but for simplicity we only consider an incident plane wave and circularly symmetric geometry.
Born and Wolf present two derivations of Babinet's principle for the electromagnetic case, an approximate version, and a rigorous version. The sound and electromagnetics approximate derivations are similar. The rigorous derivations differ in some respects, as discussed below. The primary focus in this section is on sound waves; the term "field" is used here as a generic term for a solution of a wave equation.
Here we assume the "hard" boundary condition for a screen or a disk: the normal component of molecular velocity u is zero everywhere on the screen or disk surface. The approximation is that on the remainder of the plane the velocity is the same as an undisturbed incoming plane wave. In general, this approximation is accurate when the size of the aperture or disk is large compared to a wavelength. Given this approximation, the solution for the scattered field is given in the section on the general boundary integral solution of the wave equations, equation G18.
For problem A the the normal velocity is zero everywhere on the surface except for the aperture, where it is the same as a plane wave. For problem B the normal velocity zero on the aperture and the same as a plane wave everywhere else. Define problem C as the sum of the solutions of problem A and problem B. Thus the boundary field for problem C is simply the undisturbed plane wave, and the solution is the undisturbed plane wave for all z ≥ 0. Thus given the solution to A, the solution to B is the plane wave minus the solution to A. That, in a nutshell, is the approximate version of Babinet's principle.
A somewhat subtle point regarding scattering problems involves the distinction between the "scattered field" and "total field." The total field is equal to the scattered field plus the incident field. Consider a disk lying in the plane z=0, with the z-axis passing through its center, and a plane wave incident from z=-∞. The disk scatters a pressure pattern that is anti-symmetric with respect to z, i.e. exactly the same scattered pressure in front, and behind, the disk, except for a sign change. When the incident plane wave is added in to yield the total field, near the z-axis it tends to cancel the scattered pressure for z ≥ 0, creating a "shadow" region immediately in front of the disk. It tends to reinforce the pressure for z ≤ 0. Babinet's principle states that the scattered field of the disk equals the total field of the screen for z ≥ 0. The field for z ≤ 0 is deduced from the symmetry properties of the scattered fields.
The clearest example of this is the limiting case of a screen with no aperture. The screen scatters a plane wave in both the positive and negative z-directions. When the incident plane wave is added in, it exactly cancels the scattered plane wave for z ≥ 0. For z ≤ 0 the total field consists of the incident and reflected plane waves. At z=0 the pressure is doubled on the z ≤ 0 side of the screen, and is zero on the z ≥ 0 side of the screen. Babinet's principle says that the scattered field of the complementary disk, which in this case has zero radius and zero scattered field, equals the zero total field of the screen in z ≥ 0. It is not equal to the total field of the screen in z ≤ 0.
The solution to the piston problem does not involve an incident field, and the resulting pressure is essentially the same as the scattered field of the aperture for z ≥ 0. The normal component of the velocity u equals the piston velocity at the face of the piston, and as noted, the solution assumes that the piston excursion is negligible. This differs from the aperture case where the velocity at z=0 really is the velocity of the incident wave, as long as the size is large compared to a wavelength.
The "bright spot"
In the limiting case of a disk of zero radius the scattered field is zero, and the total field for z ≥ 0 is the incident plane wave. For a finite disk the total field still includes the incident plane wave. This causes the "bright spot" in the diffraction pattern of a disk. There is a shadow region in front of the disk, but exactly in the center the shadow is punctuated by a bright spot, in the case of light diffraction, or high pressure, in the case of sound.
Rigorous Version I
In electromagnetics the "scattered field" is generated by an electrical and/or magnetic current sheet flowing in the screen or disk. One normally does not think of sound waves arising from a current sheet, but the scattered field behaves as if this were true.
For the screen, problem A, we assume the usual "hard" boundary condition for sound waves: the normal component of molecular velocity is zero on the surface of the screen. For the disk, problem B, we assume the more-or-less theoretical "soft" boundary condition: the pressure is zero on the surface of the disk. These correspond to the electromagnetic cases of a perfect electrical, and magnetic, conductor respectively.
For a hard screen in the plane z=0, the scattered pressure is an odd function of z, and the scattered field z-component of molecular velocity is an even function of z. For a soft disk the symmetry is reversed.
Let ρ denote the cylindrical coordinate radius, and let a denote the radius of the hole and disk. A superscript I, S or T denotes incident, scattered, and total fields respectively; a subscript A or B denotes problem A or problem B. The variable u denotes the z-component of molecular velocity, and all references to "velocity" refer to the normal component. The variable p denotes pressure.
A rigorous boundary value solution must satisfy two conditions: (1) the scattered field must be a solution of the wave equations (the incident field is of course assumed to be a solution); and (2) the total field must satisfy the boundary conditions.
On the surface of the hard screen and soft disk the boundary conditions for the total field at z=0 are:
Away from the scattering surfaces the boundary conditions are that both velocity and pressure are continuous. The incident wave is continuous everywhere. The scattered fields that are an even function of z will be continuous at z=0, but the fields with odd symmetry must be zero at z=0, in the regions on the plane away from a scattering surface:
Suppose we have a solution for problem A. Comparing equations (1) and (2), if the incident field is the same for both problems, and if we have, at z=0
Then the fields of problem B will also satisfy the boundary conditions at z=0. If the incident field sources are in z < 0, we can set the scattered field of the disk equal to the total field of the screen for all z ≥ 0, and we have a solution to problem B for z ≥ 0. As noted above, the scattered fields for z ≤ 0 for problem B are then deduced from the symmetry relations for scattered fields.
This version of Babinet's principle is the sound analogue of the electromagnetic version. Clearly the boundary conditions for the screen and disk can also be interchanged. In the derivation given by Born and Wolf the disk and screen are both perfect electrical conductors, and the incident field polarization is rotated 900 between problems A and B. Babinet's principle is presented in the form of the total electric fields in problem A plus the total magnetic fields in problem B. Using the symmetry property of electromagnetic fields this is essentially equivalent to the derivation given here. The problem in the case of sound is that there is no direct analog for the case of a "hard" boundary condition for both the screen and the disk, which is really the most interesting case. That is the subject of the next section.
Rigorous Version II
This derivation is unfortunately a bit messier than the previous one. We can represent any sound wave in the region z ≥ 0 as a summation of cylindrical waves, in terms of the potential function
where k is the free space propagation constant, f0=1, and β0=0. If the values of βn are equally spaced, as the spacing approaches zero and N→ ∞ this summation approaches a continuous wave spectrum that is a general representation of any sound wave traveling in the +z direction. (See the section on Wave Spectra). Pressure is proportional to φ , and the z-velocity component is proportional to the partial derivative of φ with respect to z. For the case of a0=1, and an=0 for n > 0, equation (4) represents a plane wave of unity amplitude propagating in the +z direction, which we denote φ 0; this is the incident field. The boundary conditions for a "hard" screen and disk at z=0 are
We again assume a solution for problem A, meaning a set of coefficients an such that
A set of coefficients bn is then a solution of problem B if
Each term in the summation physically represents a conical wave propagating at an angle θn with respect to the z-axis, where
If the aperture is large compared to a wavelength, the values of the coefficients an are negligible for large angles. Therefore for waves propagating at small angles βn/k << 1 and equation (7) is satisfied for
To the same level of approximation the left hand side of equation (9) is the scattered field for problem B, and the right hand side is the total field for problem A, so this is the same as equation (3). This rigorously confirms the approximate result, and provides a quantitative indication of when the approximation will be accurate. Equation (7) could be considered to be an exact formulation of Babinet's principle for this case, in the sense that it provides an exact relation between the two solutions.
Equations (6) and (7) can be formulated as matrix equations, which can be solved for the coefficients an and bn. An example for an aperture and disk of radius 2 wavelengths is shown in this figure. If the approximate Babinet's principle was exact for a hard screen and a hard disk, all of the dots would lie inside the circles. They don't, but they are fairly close even at this relatively small size. For larger radii the agreement rapidly improves.
Revised (and hopefully more understandable) version posted July 2006.
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