Файл: Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo .pdf

Suppose that the wavelength is changed from l to l0. Equation (19.4-4) remains valid at another image point (x00; z00). Taking the ratio of the two equations at l and l0 yields

Equating the coefficients of the terms with xi, the new focal point (x00; z00) is obtained as

where the approximations are based on 1-R 1 and zc z0.

From the above derivation, it is observed that the focal point location z00 is very close to the original z0. Along the x-direction, the dispersion relationship is given as

The image points of higher harmonics due to nonlinear encoding with zero-crossings occur when the imaging equation satisfies

Taking the ratio of Eqs. (19.4-2) and (19.4-10) within the paraxial approximation yields

Solving for x00 and z00 in the same way, the higher order harmonic image point

From the above equations, we observe that a significant move of imaging position in the z-direction occurs as z00 shrinks with increasing harmonic order. This means that the higher harmonics are forced to move towards locations very near the phased array. However, at such close distances to the phased array, the paraxial approximation is not valid. Hence, there is no longer any valid imaging equation. Consequently, the higher harmonics turn into noise. It can be argued that there may still be some imaging equation even if the paraxial approximation is not valid. However, the simulation results discussed in Section 19.4 indicate that there is no such valid imaging equation, and the conclusion that the higher harmonic images turn into noise is believed to be valid. Even if they are imaged very close to the phased array, they would appear as background noise at the relatively distant locations where the image points are. Simulations of Section 19.4 indicate that the signal-to-noise ratio in the presence of such noise is satisfactory, and remains satisfactory as the number of channels are increased.

19.4.1Dispersion Analysis

The analysis in this subsection is based on the simulation results from Eqs. (19.4-6), (19.4-7), (19.4-12), and (19.4-13) in the previous subsection.

Case 1: Spherical wave case (0.1 < zc/z0 < 10)

For the first order harmonics (m ¼ 1), the positions of the desired focal point for l0, that is, x00 and z00 have linear relationship with the wavelength l0. The slope of this relationship decreases as the ratio zc=z0 decreases. For the higher order harmonics ðm 2Þ, x00 is much greater than x0 ¼ 0 and z00 is much less than z0. Therefore, we conclude that the higher order harmonics turn into background noise as discussed in the previous subsection.

Case 2: Plane wave case (zc/z0 1)

In this case, Eqs. (19.4-6), (19.4-7), (19.4-12), and (19.4-13) can be simplified as

19.4.1.1 3-D Dispersion. The mathematical derivation for the 3-D case is very much similar to that for the 2-D case discussed before [Hu, Ersoy]. However, instead of viewing the y variables as constants, thus neglecting them in the derivation, we investigate the y variables along with the x variables, and then obtain independent equations that lead to dispersion relations in both the x-direction and the y-direction. It is concluded that if the x-coordinates and y-coordinates of the points are chosen independently, the dispersion relations are given by

19.4.2Finite-Sized Apertures

So far in the theoretical discussions, the apertures of the phased array are assumed to be point sources. In general, this assumption works well provided that the phase does not vary much within each aperture. In addition, since the zero-crossings are chosen to be the centers of the apertures, there is maximal tolerance to phase variations, for example, in the range ½ p=2; p=2&. In this section, PHASAR types of devices are considered such that phase modulation is controlled by waveguides truncated at the surface of the phased array.

We use a cylindrical coordinate system (r; f; z) to denote points on an aperture, and a spherical coordinate system (R; ; ) for points outside the aperture. In terms of these variables, the Fresnel–Kirchhoff diffraction formula for radiation fields in the Fraunhofer region is given by [Lu, Ersoy, 1993]

The transverse electric field of the LP01 mode may be accurately approximated as a Gaussian function:

where w is the waist radius of the gaussian beam. The field in the Fraunhofer region radiated by such a Gaussian field is obtained by substituting Eq. (19.4-21) into

Figure 19.7. Experimental proof of the linear dispersion relation (M ¼ 100; L ¼ 16; d ¼ 30).