Aperture Theory and the Equivalence Principle

I would like to present a slightly different perspective on the problem below. The concept is simple, as illustrated below in Figure 1. Here we have physical aperture antenna, a horn antenna in this case, which is replaced by an equivalent surface current J_c on the conducting part of the horn antenna, and aperture sources J_s1  and  M_s1  radiating in free space. These currents produce the horn antenna radiated fields in Region 1, and a null field internal to the equivalent surface S (Region 2, Love's equivalence). We'll next consider a general example of the equivalence principle, as for instance discussed in Constantine Balanis' Antenna Theory - Analysis and Design, Second Edition, Chapter 12 (pp. 575-581).

In Figure 2 we have arbitrary sources J₁ and M₁ radiating in free space, in the presence of an electric conductor C. We can apply the surface equivalence principle, also referred to as the equivalence theorem, and replace the conductor with an equivalent surface current J_c, which now radiates in free space, as do sources J₁ and M₁. The free space region outside the conductor is designated as Region 1, and the region inside the conductor as Region 2. Balanis and others now argue that when the fields internal to the equivalent surface S are zero, the fields 

"cannot be disturbed if the properties of the medium within it are changed",

for example by replacing it with a perfect electrical conductor. It is then stated that

"The introduction of the perfect conductor will have an effect on the equivalent source J_s (J_c in Figure 2) and will prohibit the use of [the free space radiation integrals] since the current densities no longer radiate into an unbounded medium.  Imagine that the geometrical configuration of the electric conductor is identical to the profile of the imaginary surface S, over which J_s and M_s exist. As the electric conductor takes its place, ... the electric current density J_s, which is tangent to the surface S, is short-circuited by the electric conductor. Thus the equivalent problem [reduces to a problem in which there] exists only a magnetic current density M_s over S, and it radiates in the presence of the electric conductor producing outside S the original fiels E₁ and H₁. Within S the fields are zero but, as before, this is not a region of interest. The difficulty in trying to use the equivalent problem [where the conductor has been placed inside S] is that [the free space radiation integrals] cannot be used, because the current densities do not radiate into an unbounded medium. The problem of a magnetic current density radiating in the presence of an electric conducting surface must be solved. So it seems that the equivalent problem is just as difficult as the original problem itself."

During my investigation into the problem, I discovered that contrary to the above statement, it is indeed possible to use the free space radiation integrals for an arbitrary equivalent surface S, with the aperture short circuited and only M_s retained, but for completely different reasons.



                    Figure 1. An aperture antenna replaced by equivalent surface currents radiating in free space



                    Figure 2. Electric and magnetic current sources illuminating an electric conductor in free space


We now take a closer look at the general surface equivalence problem depicted in Figure 2. From the surface equivalence principle we know that  J_c, J₁ and M₁ together will produce the scattered field in Region 1, but a null field in Region 2 (Love's equivalence). The surface current J_c can be expressed as

J_c = J_c (J₁) + J_c (M₁)                                                                 (1)

and the radiated field as 

E (or H) =   ∫_J J₁    +    ∫_M M₁   +   ∫_J J_c                                    (2)

where  ∫_J  represents the electric current radiation integral and ∫_M  the magnetic current radiation integral for E or H, respectively.

Subsituting (1) into (2) we find

E (or H) =   ∫_J J₁    +    ∫_M M₁   +   ∫_J J_c (J₁)   +   ∫_J J_c (M₁)     (3)

If we now increase the radius of the conductor C to infinity, C will become a flat plane behind sources J₁ and M₁. If the conductor C is brought close to sources J₁ and M₁, J₁ will effectively be short-circuited, i.e. the negative mirror image of J₁ will be induced on C and we can state 

J_c (J₁) = − J₁                                                                              (4)

If the conductor placed behind the sources  J₁ and M₁ is an infinete flat plane, the positive mirror image of  M₁ will be induced on C, implying that 

∫_J J_c (M₁)   =   ∫_M M₁                                                               (5)  

Substitution of (4) and (5) into (3) yields

E (or H) =   ∫_J J₁    +   ∫_M M₁   –   ∫_J J₁    +   ∫_M M₁                

             =  2  ∫_M M₁                                                                                       (6)

In other words, when  aribtrary sources J₁ and M₁ radiated in free space and an infinite flat electric ground plane is placed behind these sources, the electric current will be short-circuited and the magnetic current will be doubled.  This concept is invariably used in aperture theory, as for instance discussed by Balanis on pp. 580-581. 

It will now be shown that this approach is fundamentally flawed when applied to aperture theory and the equivalence principle.

Figure 3 shows just the aperture field region of the aperture antenna problem of Figure 1.  The problem is the same as that depicted in Figure 2, with the difference that while the true non-zero radiatied fields are present in Region 1, the fields internal to S (Region 2) are identically zero (E₂ ≡ H₂ ≡ 0). The aperture sources are given by J_s1  and  M_s1 and correspond to sources J₁ and M₁ in Figure 2. If we now place a conductor C inside Region 2, this simply becomes another equivalence problem - we can replace the conductor with an equivalent surface current J_c (with the region inside this conductor designated as Region 3) and solve for J_c by means of the Method of Moments, for instance. However, due to the fact that the fields illuminating this conductor are always zero from Love's equivalence, J_c ≡ 0, always. This remains true irrespective of how close to S the conductor may be brought, and irrespective of its shape. In this case J_c is no longer composed of two separately induced parts as in (1), and hence no cancellation or 'short circuiting' of any of the aperture sources can ever occur. It was for the purpose of demonstrating how the the electric current can be short-circuited by placing a conductor behind it that I initially devised the problem discussed in my paper "An Illustrative Equivalence Theorem Example". Since the electric and magnetic surface currents are known, I hoped to demonstrate that by placing an electric conductor in the null field region and solving for the equivalent electric current J_c by means of the Method of Moments, I would be able to show definitively that J_c = − J₁ and also what current would be induced by M₁ . However, much to my surprise I found that J_c = 0, and it was only after pondering this result intensively that I realized what was going on. The concept is flawed.

In my aperture theory paper I show that if we separate J_s1  and  M_s1 with a very small gap, n × E₁ = 0 in this gap, where n is the unit vector normal to S in the aperture. In other words, there appears to be an electric conductor behind M_s1 and we can effectively short circuit the aperture. This approach is demonstrated to yield the correct results in the article, where only free space radiation integrals are used for a variety of problems.




    Figure 3. Electric and magnetic equivalent surface currents illuminating an electric conductor in the null region