Huygens principle
Mathematical expression of Huygens principle, as usually presented in classical textbooks, is not of a complete form: its exploitation is limited
to the case when the intensity of the total magnetic field H is directly related to the intensity of the total electric field E in
every point of the space (i.e., we can write H = E / Z). The described situation appears at the open end of a waveguide where
intensity of electric field is related (with sufficient accuracy) to the intensity of magnetic field according to the following expressions.

${\mathbf{E}}^{tot}={\mathbf{E}}^{inc}\left(1+{\rho}_{0}\right)$
${\mathbf{H}}^{tot}={\mathbf{H}}^{inc}\left(1+{\rho}_{0}\right)$
${\mathbf{H}}^{inc}=\frac{{\mathbf{E}}^{inc}}{{Z}_{g}}$.

( 1 )

Here, the upper index inc denotes intensity of incident wave arriving to the open end of a waveguide and the upper index tot
denotes intensity of the total field (summation of the incident wave and the reflected one). Further, ρ_{0} is reflection coefficient and

${Z}_{g}=\frac{Z}{\sqrt{1{\left(\frac{{\lambda}_{0}}{{\lambda}_{m}\sqrt{{\epsilon}_{r}}}\right)}^{2}}}$

( 2 )

is impedance of the mode TE_{10} in a rectangular waveguide; Z is characteristic impedance of the environment inside a waveguide,
λ_{m} = 2a is critical wavelength for a waveguide without dielectrics and a is the dimension of a longer side of
a waveguide. Next, λ_{0} denotes wavelength in the vacuum and ε_{r} is the relative permittivity in the
waveguide. Considering the mode TE_{10}, only the component E_{y} of the intensity of electric field, and only the component
H_{x} of the intensity of magnetic field are nonzero..
Substituting E^{tot} and E^{tot}/Z_{g} the complete relation of Huygens principle,
we obtain a wellknown, simplified expression of Huygens principle (relation for recomputing magnitude of the intensity of electric field on an
elementary surface E^{(S)} to an arbitrary point, which is located in the farfield region)

${E}^{\left(P\right)}=\frac{j}{\lambda}{\displaystyle \underset{a/2}{\overset{+a/2}{\int}}{\displaystyle \underset{b/2}{\overset{+b/2}{\int}}{E}^{\left(S\right)}\mathrm{cos}\left(n,{r}_{2}\right)\frac{\mathrm{exp}\left(jkr\right)}{{r}_{2}}\text{d}y\text{\hspace{0.05em}}}\text{d}x}$.

( 3 )

Here, a and b denote dimensions of an elementary irradiated surface, λ is wavelength in a given environment,
E^{(S)} is magnitude of the intensity of electric field on an elementary surface, k denotes wavenumber, n
is a vector perpendicular to an elementary surface and r_{2} is the position vector between an elementary surface S and an
observation point P.
The complete relation can be derived from Maxwell equations for an elementary surface, where electric current density and magnetic current density
are known. For the farfield region, we get

${E}_{\vartheta}^{s}=j\frac{k}{4\pi}\left(\eta \mathbf{J}\cdot {\mathbf{T}}_{xyz\vartheta}+\mathbf{K}\cdot {\mathbf{T}}_{xyz\phi}\right)\frac{\mathrm{exp}\left(jkr\right)}{r}$,

( 4a )


${E}_{\phi}^{s}=j\frac{k}{4\pi}\left(\eta \mathbf{J}\cdot {\mathbf{T}}_{xyz\phi}+\mathbf{K}\cdot {\mathbf{T}}_{xyz\vartheta}\right)\frac{\mathrm{exp}\left(jkr\right)}{r}$.

( 4b )


Fig. A.1  Coordinate systems used in deriving complete Huygens principle 

Eqn. (4a,b) is explained in fig. 1. Here, an elementary surface of dimensions dx and dy,
where electric current J and magnetic current K are flowing, is depicted. Electric intensity of scattered field E^{s},
i.e. intensity created by the irradiated elementary surface in the point r, is expressed in spherical coordinates. Next, current densities
J and K are supposed being known in a Cartesian coordinate system. Relations for E_{ϑ}^{s} and
E_{φ}^{s} contain vectors T, which perform transformation from Cartesian coordinate system to spherical one:

${\mathbf{T}}_{xyz\phi}=\left[\begin{array}{ccc}\mathrm{sin}\left(\phi \right)& \mathrm{cos}\left(\phi \right)& 0\end{array}\right]$,

( 5a )


${\mathbf{T}}_{xyz\vartheta}=\left[\begin{array}{ccc}\mathrm{cos}\left(\vartheta \right)\mathrm{cos}\left(\phi \right)& \mathrm{cos}\left(\vartheta \right)\mathrm{sin}\left(\phi \right)& \mathrm{sin}\left(\vartheta \right)\end{array}\right]$.

( 5b )

Currents J and K do not need to physically flow on an elementary surface (we speak about so called equivalent currents, which are
induced by electric field and magnetic one on the elementary surface). These currents can be evaluated using

$\mathbf{J}=\mathbf{n}\times \mathbf{H}$,

( 6a )


$\mathbf{K}=\mathbf{n}\times \mathbf{E}$.

( 6b )

Here, n is perpendicular to the elementary surface (identical with the unitary vector u_{z}).
Using eqns. (6a,b), current densities flowing on electric and magnetic walls can be computed in a real case.
Unfortunately, an examined irradiated surface does not need being neither electric nor magnetic one.
Direct application of the described principle is conditioned by the requirement that no other objects are present in a halfspace defined by the
elementary surface and the positive direction of vector z. If this condition is not met, Huygens approach and principle of equivalent currents
can be applied, but the situation is complicated by reflections from those objects which further influences irradiation of the elementary surface.
This case is not analyzed here.
Thanks to the fact that vector products n × E, n × H (n is perpendicular to the surface) are computed
when evaluating equivalent currents, only those components of currents J and K have to be known, which are tangential to the elementary surface.
The last symbol, which has not been explained yet, is freespace characteristic impedance η.
That is the end of a brief note on the complete form of Huygens principle. In the following paragraphs, we show the simplification of the complete
form in the case when electric intensity is known on an elementary surface and magnetic intensity is related to it via impedance (this approach can be
used for an open end of a waveguide but it cannot be applied to a surface of a microstrip antenna because its surface impedance is not constant).
Before performing the described simplification, let us turn our attention to the signs in eqns. (4a,b). Note that the righthand
sides of those relations start by the sign minus. The sign minus in (4a) is changed to plus if H is expressed as
–E/Z_{g} (see direction of E and H in fig. 1). This is further explained by the
following example.
Assume an elementary electric dipole with current density J. The current is expected to flow in the direction x. Considering such component of
E, which is oriented in parallel with the current J, then the relation E ~ +jJ has to be valid in the farfield region.
Fig. 2 illustrates the situation.

Fig. A.2  Explaining sign convections in complete Huygens principle 

Now, let us investigate the situation of magnetic intensity and the behavior of E, H in the nearfield zone of an elementary dipole.
In fig. 2, both the vector E and the vector H are depicted. Complex amplitude of components E_{ϑ}
and H_{φ} depend on the distance r by the following way:

${E}_{\vartheta}\approx \frac{j}{kr}+\frac{1}{{\left(kr\right)}^{2}}\frac{j}{{\left(kr\right)}^{3}},$,
${H}_{\phi}\approx \frac{j}{kr}+\frac{1}{{\left(kr\right)}^{2}}$.


Magnetic field on the surface of the dipole (thanks to the small distance r, the term 1/k^{2}r^{2}
is dominant) is of the same phase than the current. On the contrary, phase of electric field (the term j/k^{3}r^{3})
is dominant) is shifted for 90°. In the farfield zone, (thanks to the large distance r, the terms j/kr are dominant) phase
of both components is identical (90°). Phase course of E_{ϑ} and H_{φ} is depicted in fig. 2.
Dealing with the nearfield zone of an elementary dipole  the zone is given by the distance λ_{0}/(2π),
where the amplitude of the longitudinal component E_{r} and the amplitude of the transversal one Ej are of the same magnitude.
Now, let us concentrate on the derivation of the simplified Huygens principle (relations for E_{ϑ} and E_{φ}).
Assume an open end of a waveguide (fig. 3). In the waveguide, mode TE_{10} propagates. The electric intensity
of the total field at the end of the waveguide is denoted by E, the magnetic intensity by H. In fig. 3,
even relations among intensities E and H and current densities J and K are given.

${\mathbf{J}}_{ekv}=\mathbf{n}\times \mathbf{H}=\frac{\mathbf{E}}{Z}=\frac{\left({E}_{x},{E}_{y}\mathrm{,0}\right)}{Z}=\frac{\left(\mathrm{0,}{E}_{y}\mathrm{,0}\right)}{Z}$
${\mathbf{K}}_{ekv}=\mathbf{n}\times \mathbf{E}=\left({E}_{y},{E}_{x}\mathrm{,0}\right)=\left({E}_{y}\mathrm{,0,0}\right)$

Fig. A.3  Simplification of general Huygens principle for openended waveguide 

Let us substitute for J and K to the complete relation for E_{ϑ} and E_{φ}, and assume this moment that
E and H are mutually perpendicular and their components x and y are nonzero. At the end of the derivation, this assumption
will be cancelled.
The described substituting and integrating waves from all the elementary surfaces yield

${E}_{\vartheta}^{s}=j\frac{k}{4\pi}\frac{\mathrm{exp}\left(jkr\right)}{r}{\displaystyle \underset{S}{\int}\left\{\eta \left[\begin{array}{cc}{E}_{x}& {E}_{y}\end{array}\right]\text{\hspace{0.05em}}\left[\begin{array}{c}\mathrm{cos}\left(\vartheta \right)\mathrm{cos}\left(\phi \right)\\ \mathrm{cos}\left(\vartheta \right)\mathrm{sin}\left(\phi \right)\end{array}\right]+\left[\begin{array}{cc}{E}_{y}& {E}_{x}\end{array}\right]\text{\hspace{0.05em}}\left[\begin{array}{c}\mathrm{sin}\left(\phi \right)\\ \mathrm{cos}\left(\phi \right)\end{array}\right]\right\}\text{d}S}$,

( 7a )


${E}_{\vartheta}^{s}=\frac{j}{2\lambda}\frac{\mathrm{exp}\left(jkr\right)}{r}\left[1+\frac{\eta}{Z}\mathrm{cos}\left(\vartheta \right)\right]\text{\hspace{0.05em}}\left[{N}_{x}\mathrm{cos}\left(\phi \right)+{N}_{y}\mathrm{sin}\left(\phi \right)\right]$,

( 7b )

Here, N_{x}, N_{y} are double integrals, which form own radiation pattern of the open end of the waveguide

${N}_{x}={\displaystyle \underset{S}{\int}{E}_{x}\mathrm{exp}\left(jk\text{\Delta r}\right)\text{d}S}$,



${N}_{y}={\displaystyle \underset{S}{\int}{E}_{y}\mathrm{exp}\left(jk\text{\Delta r}\right)\text{d}S}$.


Analogically for the component E_{φ}

${E}_{\phi}^{s}=j\frac{k}{4\pi}\frac{\mathrm{exp}\left(jkr\right)}{r}{\displaystyle \underset{S}{\int}\left\{\eta \left[\begin{array}{cc}{E}_{x}& {E}_{y}\end{array}\right]\text{\hspace{0.05em}}\left[\begin{array}{c}\mathrm{sin}\left(\phi \right)\\ \mathrm{cos}\left(\phi \right)\end{array}\right]+\left[\begin{array}{cc}{E}_{y}& {E}_{x}\end{array}\right]\text{\hspace{0.05em}}\left[\begin{array}{c}\mathrm{cos}\left(\vartheta \right)\mathrm{cos}\left(\phi \right)\\ \mathrm{cos}\left(\vartheta \right)\mathrm{sin}\left(\phi \right)\end{array}\right]\right\}\text{d}S}$,

( 8a )


${E}_{\phi}^{s}=\frac{j}{2\lambda}\frac{\mathrm{exp}\left(jkr\right)}{r}\left[1+\frac{\eta}{Z}\mathrm{cos}\left(\vartheta \right)\right]\text{\hspace{0.05em}}\left[{N}_{x}\mathrm{sin}\left(\phi \right){N}_{y}\mathrm{cos}\left(\phi \right)\right]$.

( 8b )

When deriving eqns. (8a,b), the observation point was expected in the farfield region. We could therefore take the exponential
term out of the integral and we integrated space shifts Δr only.
In the case of the open end of the waveguide, the situation becomes even simpler because the component E_{x} is zero, and therefore
even N_{x} is zero.
And finally, two concluding notes:
 Simplified Huygens principle expects that intensity of incident field only E^{inc} (or H^{inc})
is substituted into it, and that the intensity of the total field is approximately twice higher (the total field is exactly twice higher for perfectly
conducting wall and approximately twice higher for the open end of the waveguide, e.g.).
 Simplified Huygens principle contains the term cos(ϑ) instead of the correct (1 + η cos(ϑ)/Z. Therefore, the simplified
principle introduces an approximation of the real directivity pattern. This approximation can be used when beamwidth is going to be determined,
but it is infeasible when sidelobe level is investigated.
