Translation formulas for both scalar and vector spherical wave solutions of the Helmboltz equation are developed in a straightforward manner using differential operator representations for the modal functions and well-known expressions for the scalar and dyadic free-space Green's functions. The expansion coefficients are given in compact integral or differential operator forms useful for analytic investigation.
As antennas increase in size or operating frequency, it becomes increasingly difficult to obtain adequate real estate to measure antenna properties in the far field. Typically, a distance of nD 2 /λ is specified as the beginning of the far-field region, where n is often 2 for routine work, D is the diameter of the smallest sphere that encloses the antenna's radiating parts, and λ is the wavelength. At this distance and for a spherical wavefront, the phase differs by π/8 between the center and edge of the antenna. For precision measurements, n often must be much greater than 2 [1, Chap. 14]. Table 19.1 shows the far-field distances for some sample antenna diameters and operating frequencies. We see that even for antennas of modest size, the far-field distance can become prohibitively large when the frequency is high enough. Thus there are situations where measuring an antenna in the near field would be advantageous because large amounts of real estate are not required. There are other advantages as well. Since the near-field method is usually employed inside a chamber, it is not subject to the effects of the weather and also provides a more secure environment.Determining the far-field pattern of an antenna from near-field measurements requires a mathematical transformation and correction for the characteristics of the measuring antenna (hereafter referred to as the probe).
The general receiving antenna is represented as a linear differential operator converting the incident field and its spatial derivatives at a single point in space to an output voltage. T h e differential operator is specified explicitly in terms of the multipole coefficients of the antenna's complex receiving pattern. When the linear operator representation is applied to the special probes used in spherical near-field measurements, a probe-corrected spherical transmission formula is revealed that retains the form, applicability, and simplicity of the nonprobe-corrected equations. The new spherical transmission formula is shown to be consistent with the previous transmission formula derived from the rotational and translational addition theorems for spherical waves.
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