## HUYGENS’S PRINCIPLE

The principle proposed by Christian Huygens (1629-1695) is of fundamental importance in the development of the wave theory. Huygens’s principle states that, ‘‘Each point on a primary wavefront serves as the source of spherical secondary wavelets that advance with a speed and frequency equal to those of the primary wave. The primary wavefront at some later time is the envelope of these wavelets’’ (9,10). This is illustrated in Fig. 1 for spherical and plane waves modeled

Incoming

wave

Figure 2. Diffraction of waves through a slit based on the Huygens principle.

as a construction of Huygens secondary waves. Actually, the intensities of the secondary spherical wavelets are not uniform in all directions, but vary continuously from a maximum in the direction of wave propagation to a minimum of zero in the backward direction. As a result, there is no backward propagating wavefront. The Huygens source approximation is based on the assumption that the magnetic and electrical fields are related as a plane wave in the aperture.

Let us consider the situation shown in Fig. 2, in which an infinite electromagnetic plane wave is incident on an infinite flat sheet that is opaque to the waves. This sheet has an opening that is very small in terms of wavelengths. Accordingly, the outgoing wave corresponds to a spherical wavefront propagating from a point source. That is, when an incoming wave comes against a barrier with a small opening, all but one of the effective Huygens point sources are blocked, and the energy coming through the opening behaves as a single point source. In addition, the outgoing wave emerges in all directions, instead of just passing straight through the slit.

On the other hand, consider an infinite plane electromagnetic wave incident on an infinite opaque sheet shown in Fig. 3 that has an opening a. The field everywhere to the right of the sheet is the result of the section of the wave that passes through the slot. If a is large in terms of wavelengths, the field distribution across the slot is assumed, to a first approximation, to be uniform. The total electromagnetic field at a point to the right of the opening is obtained by integrating the contributions from an array of Huygens sources distributed over the length a. We calculate the electrical field at point P on a reference plane located at a distance R0 behind

(a) Spherical |

Figure 1. Spherical and plane-wave fronts constructed with Huygens secondary waves. |

P |

(b) Plane |

Figure 3. Plane wave incident on an opaque sheet with a slot of width a. |

Incoming wave |

the plane by Huygens’s principle (11): |

S‘ |

e |

E = E. |

-dy |

(1) |

T |

For points near to the array, the integral does not simplify but can be reduced to the form of Fresnel integrals. The actual evaluation of this integral is best achieved on a PC computer, which reduces the integral to the summation of N Huygens sources: |

Distance along the у axis (cm) (a) R0 = 2.5 cm |

-jkri |

e |

E = £ E0 |

(2) |

T |

where Ті is the distance from the ith source to point P. The field variation near the slot that is obtained in this way is commonly called a Fresnel diffraction pattern (4). For example, let us consider the case in which the slot length a is 5 cm and the wavelength is 1.5 cm (20 GHz). We can use Eq. (2) to compute the field along a straight line parallel to the slot and distance R0 from it. The field variation for R0 = 2.5 cm shown in Fig. 4(a) is well within the near field (Fresnel region). As we continue to increase R0 the shape of the field variation along this line continues to vary with R0 until we reach the far field or Fraunhofer region. [See the trends in Figs. 4(b), 4(c), and 4(d)]. Once we have entered the Fraunhofer region, the pattern is invariant to range. For the point to be in the far field, the following relationship must exist: |

Distance along the у axis (cm) (b) R0 = 5 cm |

Distance along the у axis (cm) |

2 a2 Y |

Rn> — |

(3) |

where a is the width of the slot and A is the wavelength. Thus, the larger the aperture or the shorter the wavelength, the greater the distance at which the pattern must be measured if we wish to avoid the effects of Fresnel diffraction. Huygens’s principle is not without limitations as it neglects the vector nature of the electromagnetic field space. It also neglects the effect of the currents that flow at the slot edges. However, if the aperture is sufficiently large and we confine our attention to directions roughly normal to the aperture, the scalar theory of Huygens’s principle gives very satisfactory results. Geometric optic techniques are commonly applied in reflector antennas to establish the fields in the reflector aperture plane. This procedure is referred to as the aperture field method and it is employed as an alternative to the so-called induced current method, which is based upon an approximation for the electric current distribution on the reflector surface. The fields in the aperture plane can be thought of as an ensemble of Huygens sources. The radiation pattern can be computed via a numerical summation of the sources. |

(c) Ro |

15 cm |

Figure 4. Electromagnetic field versus distance along the Y axis. |

Distance along the у axis (cm) (d) R0 = 20 cm |

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