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Line-Integral Representations of the Diffraction of Scalar Fields

Traditionally, the diffraction of a scalar wave satisfying Helmholtz equation through an aperture on an otherwise black screen can be solved approximately by Kirchhoff's integral over the aperture. Rubinowicz, on the other hand, was able to split the solution into two parts: one is the geometrical part that appears only in the geometrical illuminated region, and the other representing the reflected wave is a line-integral along the edge of the aperture. However, this decomposition is not entirely satisfactory in the sense that the two separated fields are discontinuous at the boundary of the illuminated region. Also, the functional form of the line-integral is not what one would expect an ordinary reflection wave should be due to some confusing factors in the integrand. Finally, the boundary conditions on the screen imposed by Kirchhoff's approximation are mathematically inconsistent, and therefore, rigorously, this decomposition formulation must be slightly modified by taking into account the correct B.C.s. In this thesis, we use the consistent boundary conditions to derive a slightly different decomposition formula which shows that the behavior of the diffracted wave at the edge is exactly just like an ordinary reflection-realizing the conjecture of Thomas Young in the 18th century. We also derived another decomposition formula which avoids mathematical discontinuity encountered by Rubinowicz. In the last section we demonstrate that our solution is consistent with that obtained by Sommerfeld in the rigorous 2-D plane-wave diffraction problem, so our formulation in this sense may describe more accurately the behavior of diffracted wave near the edge of the aperture than Kirchhoff's formula.


Diffraction Integral Formulae
The diffraction of a scalar wave ψ satisfying the Helmholtz equation has the solution (at the field point r f ): where the integral is performed on a closed surface S which does not enclose any source of ψ, andn is the outward normal of S. Here, G is the Green's function satisfying For the problem of diffraction through an aperture on an infinite screen, one usually defines S to be the union of the aperture, the screen, and the infinity. If the aperture is finite and the screen is opaque, one expects that ψ decreases fast to zero at infinity, and thus one only has to evaluate the integral on the aperture and the screen.
where ψ s is the unperturbed source field. Also, he Kirchhoff the Green's function With these assumptions, Equation (1) can be reduced to However, the boundary conditions imposed by Kirchhoff is mathematically inconsistent, thought it gives good approximations near the axis at far field zone. Sommerfeld, on the other hand, suggested another consistent boundary conditions ψ = ψ s , on the aperture, ψ = 0, on the screen, (4) and adopted Green's function of Dirichlet type which vanishes on the aperture and the screen. For example, for a planar screen, where r * f is the mirror image of r f with respect to the screen. With these modifications, Equation (1) is reduced to

Maggi-Rubinowicz' Decomposition
With the Kirchhoff integral formula (Equation (3)), Rubinowicz was able to decompose the field ψ ( r f ) into two parts [1]: one that appears only in the ordinary geometrical illuminated region is the source field evaluated at the field point r f ; the other one is a line integral along the edge of the aperture: (6) This formula applies to two special cases. The first one is the plane-waveincidence case in which the source field is a plane wave: ψ s ( r) = e ikρ , where ρ is the distance measured from r to a constant phase plane of the incident field. If r happens to lie on the edge of the aperture, then we denote ρ by ρ s . Also, we denoteρ s to be the unit vector in the direction of propagation of the incident field, as shown in Figure (1). The illuminated region is an oblique cylinder, as predicted by geometrical optics. Inside the line integral, ρ f ≡ r − r f is the distance from r f to the edge r, andρ f is the unit vector of r − r f . The second one is called the point-source-incidence case, in which the source field is a spherical wave: ψ s = e ikρ /ρ, where ρ ≡ r − r s is the distance measured from r to the position of the point source r s . As before, if r happens to lie on the edge of the aperture, then we denote ρ by ρ s , and usê ρ s to denote the unit vector of r − r s . As expected, the illuminated region in this case is an oblique cone with vertex at r s , as shown in Figure ( This decomposition realized Young's interpretation for the diffraction phenomena: Young had once proposed that part of the incident field, which is called the reflected field, undergone a kind of reflection at the edge of the aperture, and the rest of the incident field, called geometrical field, just went through the aperture without any perturbation, and, the final diffraction wave was the interference of the two fields [2]. But as mentioned earlier, Equation (3) was derived based on inconsistent boundary conditions. Now, with the B.C.s proposed by Sommerfeld (Equation (4)), we are able to modify Equation (6) and obtain another similar expression which is not only mathematically-consistent, but also preserves Young's "field-splitting" interpretation.
Furthermore, Equation (6) is not entirely satisfactory in the sense that the two separated fields are not continuous at the boundary of the illuminated region. This problem had been discussed by John. S Asvestas, and he also gave an elegant "solid-angle" representation of ψ which successfully avoided such discontinuity. However, Asvestas' work was also based on Kirchhoff's formula, and therefore the mathematical inconsistency still exists. Besides that, the solid-angle representation derived by Asvestas does not reduce to electrostatic case in the long wavelength limit k → 0. In this paper, we will give a slightly different decomposition formula with consistent boundary conditions and generalize the electrostatic result to the diffraction problem.
Another unsatisfactory feature of Rubinowicz' decomposition formula Equation (6) is that the functional form of the line integral is not what one would expect an ordinary reflection wave should be, due to some confusing factors in the integrand. And therefore in this paper, we will also derive a neater representation of the line integral that mimics the behavior of ordinary reflection in geometrical optics.
In the last section, we'll explain why the boundary conditions (Equation (4)) is more suitable by compare the result of our work with Sommerfeld's 2-D straight-edge diffraction problem, which is the one of the few examples where the Helmholtz equation has an exact solution. We'll see that Kirchhoff's solution has a relative deviation from the exact solution.

Modified Expression for Rubinowicz' Decomposition Formula
With the boundary conditions Equation (4), we begin from Sommerfeld's integral formula Equation (5). Since G D (∂ψ s /∂n) = 0 on the aperture, we add it back into the integrand and rearrange a little: and in the last line we identify the first integral is nothing but Equation (3) and thus equal to Equation (6). The second integral has exactly the same functional form as the first one, except for the replacement r f → r * f . So the final result is (Note that r * f is on the opposite side of the screen, so it always lies outside of the illuminated region, and thus there is no corresponding geometrical field.) Hereρ * f is the unit vector of r − r * f . The image term G * K contributes another line integral to the final expression, which now seems more ugly. Actually, Rubinowicz had also derived this result in his paper in 1917. However, for some reason he seemed to abandon this result and used Equation (6) in his successive papers. In the following sections, we'll show that, with some deformation, Equation (8) (or equivalently, Equation (5)) can take another form which has some merits mentioned in the introduction.

Motivations from Geometrical Optics
In ordinary geometrical optics, the reflection phenomena can be comprehended this way: given a source distribution, one draw the "image source" behind the "mirror," as shown in Figure ( So, if what Young really meant (in the early days when he saw the diffraction phenomena) by "reflection at the edge" was the reflection in geometrical optics, then we expect the line integral should take the form Namely, as shown in Figure (4), we imagine that the screen has a finite thickness, and as the incident field ψ s comes in, it is reflected by the "cut" around the aperture, and thus when the reflected field reaches the field point r f , the incident field ψ s has propagated for a total optical length ρ s + ρ f , and therefore ψ reflection should take the form as Equation (9) or (10).
These expectations actually can be accomplished by some deformations of Equation (8), but let's do it another way: to derive the ψ reflection from the beginning Equation (5), and this will make derivation more neater.

Reflection at the Boundary
Simplify Equation (5) a little, and we get where r ≡ r − r f , and letê z be the inward unit normal to the screen (ê z = −n). To simplify the result, we consider the following two cases: We assume the wave propagates in the direction perpendicular to the screen-and it is reasonable to make this assumption since experimentally it is the most common configuration. Under this postulation, ψ s = e ikρs where ρ s is now a constant quantity representing the distance from a constant phase plane to the screen. As shown in Figure ( where l and l f represent the magnitude of l and l f , respectively. The area element on the aperture is d a = lds × sd l, and Equation (11) can be evaluated as where φ is the angle subtended by the arc of the boundary of the aperture as measured from O. So where θ f is the angle indicated in Figure (5).
To separate the geometrical and reflected fields apart, we perform the line integral to the first term of the integrand: Since O lies inside the aperture if and only if r f ∈illuminated region, so Similar to the previous case, we attempt to assume that r f − r s is perpendicular to the screen, that is, we confine r f to lie on the central axis as shown in Figure (6). But this time the assumption is totally unreasonable-after all we cannot restrict the position of r f . We believe that this formulation can be generalized without this assumption, but for the present we'll consider this special case, and try to elucidate the idea of "reflection at the boundary." Identify ψ s = e ikρ /ρ, where ρ ≡ r − r s , and define O, l f , l as before. Here, we define a new vector l s , to be the vector from O to r s . Therefore ρ = s l − l s = s 2 l 2 + l 2 s , and Equation.
(11) can be evaluated as where θ f has the same definition as before.
Again the geometrical field can be separated out by the same method, and we get the final result: 4 Solid-Angle Representation

Motivation from Electrostatics
Although Equation (8)  Assume there is no other source charge in the half space z > 0, and therefore ∇ 2 V = 0 there. This boundary value problem has the solution where Ω f is the solid angle subtended by the region σ as observed at the field point r f . Inspired by the electrostatic result, we attempt a solution for diffraction problem of the form namely, we expect the geometrical field to take the similar form of Equation (14), while the reflected field remains a line-integral around the boundary of the aperture. The advantage of this formulation is that both the geometrical and reflected fields now vary continuously, without any jump discontinuity across the boundary of the illuminated region. Different from Asvestas' work, in this formula, we see that as the field point r f approaches to the aperture, then Ω f → 2π, and the geometrical field → ψ s while we expect the reflected field to vanish totally. That is, if we reside on the aperture, we are exposing ourself to the source ψ s without the influence of the edge.

Inverse Cone
We begin from Equation (7), and define To evaluate J, we do a trick slightly different from what Rubinowicz did. As shown in Figure (7), we make an auxiliary surface with the vertex at the field point, and make a small ball centered at the field point. Define σ 1 to be the surface of the cone outside the small ball while σ 2 to be the surface of the small ball inside the cone. Apply the divergence theorem to the region enclosed by σ 1 , σ 2 and the aperture: where F ≡ ψ s ∇G K − G K ∇ψ s , and d a is the vectorial area element pointing outwardly from the volume enclosed. If we let the radius of the small ball approach to zero, then where, as desired, Ω f is the solid angle subtended by the aperture as observed at the field point r F . So The surface integral can be evaluated by the same trick presented by Rubinowicz, as discuss in the following two cases: Case 1. Plane Wave Diffraction On the auxiliary surface, ∇G K = 0, and therefore From Figure (7), we have the relation Therefore To evaluate J * , we use the result from Section 2: Finally, we combine J and J * : Although this result fits our demand-ψ ( r f ) is now expressed in terms of the solid angle Ω f -it is still unsatisfactory since the denominator of the geometrical part is 4π instead of 2π. Accordingly, if r f approaches to the aperture, the geometrical part only gives us one half of the total source wave ψ s , and thus the reflected part must contribute the rest half part. To fix the problem, we take the long wavelength limit k → 0, and thus ψ s → e i0ρ = 1, and Equation (15) must be identical to Equation (14): So we have a line-integral representation of solid angle: This equation has also been derived by Asvestas [3], and by Yih-Yuh Chen [4] from a more elegant perspective. Note that since the wave number k vanishes, the vectorρ s now can point in an arbitrary direction, so the representation above is not unique. Finally, we construct the desired ψ ( r f ) by adding Equation (16) into

Case 2. Point Source Diffraction
Again, on the auxiliary surface, ∇G K = 0, and Therefore Again, we use the result from Section 2: Combine J and J * , we have To construct the correct factor 1/2π, we use Equation (16) to add another (Ω f /4π) ψ s to the geometric wave. But note thatρ s in Equation (16) is an arbitrary constant vector, and in Equation (17)ρ s represents a varying vector that changes its direction as we integrate along the edge. Thus we must specify one direction forρ s in Equation (16) so that we can insert it into Equation (17). The result is most symmetric if we adoptρ s ≡ρ 0 in Equation (16) ψ We have seen that in both cases the field ψ ( r f ) has the form and it is obvious that both the geometrical and the line integral parts of ψ are now continuous across the surface of illuminated region. However, the integrand in [· · · ] now depends on the type of the source. There is another point to be mentioned: since Ω f ∝ 1/ρ f 2 , in the far zone the geometrical field is overwhelmed by the reflected field, which is proportional to 1/ρ f of the source wave. The situation is reversed in the near zone, of course.

Comparison of Boundary Conditions
As discussed in previous sections, the boundary conditions based on Kirchhoff's theory is mathematically inconsistent, and by using the proper Green's function, the diffraction theory can be transformed into a boundary value problem of Dirichlet type which is mathematically admissible. However, this is not the whole story. As the source wave ψ s propagates toward the aperture, the wave must be modified by the presence of the opaque screen, and thus ψ is not exactly equal to ψ s , the unperturbed source, on the aperture. So the boundary values Equation (4) imposed earlier is still, unsatisfactory in the physical sense.
However, Sommerfeld has solved a 2−D straight edge diffraction problem rigorously without using the unperturbed source wave as boundary values [5], and we'll see in a moment that, by some deformation of Equation (8), the functional form of our solution is very closed to that of Sommerfeld's, and, therefore, we may regard Equation (4) as an acceptable approximation to the real, rigorous solution.

Approximated Solution for a Point Source
Consider an infinite half plane lying on z = 0 and x > 0, with a point source lying in the region z < 0 as before. The solution ψ in the space z > 0 can be solved by Equation (8) ψ where the line integral is performed along the infinite straight edge 1 . The amplitude A of the source field is now expressed explicitly for later convenience.
In the far field region kρ f ≫ 1, we apply stationary-phase approximation to evaluate the reflected field: The the stationary point occurs when ∇ (ρ s + ρ f ) · d l = 0, and we expand the phase at the stationary point: where the subscript 0 denotes the stationary point, which in this special case So finally, The factor e i π 4 can explain the reason why the diffraction pattern in the water has a phase delay compared to the incident wave.

Approximated Solution for Plane Waves
In the case of plane-wave-incidence, we can obtain the solution from Equation (20) by taking the limit ρ s → ∞, A → ∞, while keeping A ρ s → finite number, taken to be 1 The result is Here e ikρs has been dropped since Sommerfeld assumed that the plane wave has phase 0 right at ρ f = 0.
So the total field is In comparison with Sommerfeld's solution [5] ψ ( r f ) ≃ we see that, apart from the factor cos φ 2 sin α 2 , the two representations are similar.
The discrepancy results from the different boundary conditions as discussed before.
If, however, we use the Kirchhoff's integral formula (with mathematically inconsistent B.C.s), we would obtain which has a different functional form from Sommerfeld's solution.

Conclusion
By using mathematically consistent boundary conditions, we have seen that Rubinowicz' decomposition formulation can be more useful: the functional form of the line integral becomes much neater and admits a simple interpretation of reflection at edges. The formulation also provides us a different approach that makes the diffraction phenomena similar to the electrostatic problem by using solid angle representation for the geometrical field. Finally, the diffracted field predicted by this formulation is much closer to the physical solution, as discussed in the last section.