Файл: Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo .pdf
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METHOD OF IRREGULARLY SAMPLED ZERO-CROSSINGS (MISZC) |
343 |
Figure 19.4. The output of a 64-channel PHASAR at a center wavelength of 1.550 m, and channel spacing of 0.8 m [Lu, Ersoy].
[Lu, Ersoy], [Ersoy, 2005]. The beams at their focal points will be referred to as images. We will in particular discuss how to achieve the design in the presence of phase modulation corresponding to a combination of a linear and a spherical reference wave. Figure 19.5 shows a visualization of the reference waves and geometry involved with two wavelengths.
Figure 19.5. A visualization of the reference waves and geometry involved with two wavelengths in MISZC.
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DENSE WAVELENGTH DIVISION MULTIPLEXING |
Once the required phase is computed, its implementation in the case of a AWG can be done by choosing the length of each waveguide to yield the required phase. This is the way it is already done with the regular PHASAR devices with only the linear grating phase modulation in confocal or Rowland geometries. The method is first discussed below for illustration purposes with respect to a planar geometry, meaning that the phased array apertures are placed on a plane (line for the 2-D case). Generalizations of the results to confocal, Rowland and 3-D geometries are given in the subsequent sections.
The method is based on first randomly choosing the locations of the centers of radiating apertures and then either by creating the negative phase of the phasefront (possibly plus a constant) at the chosen locations so that the overall phase is zero (or a constant), or slightly adjusting locations of the centers of radiating apertures such that the total phase shift from such a center to the desired image point equals a constant value, say, zero modulo 2p. In both approaches, such points will be referred to as zero-crossings. In the second approach, they will be referred to as automatic zero-crossings. In practice, the sampling points are chosen semi-irregularly as discussed below.
The total number of zero-crossings can be a very large number, especially in the presence of linear and spherical phase modulation. Practical implementations allow only a small number of apertures, for example, 300 being a typicalnumber in the case of PHASARS. In order to avoid the problem of too many apertures, and to avoid harmonics generated due to regular sampling [Ishimaru, 1962], [Lao, 1964], we choose irregularly sampled sparse number of apertures. One way to determine zero-crossing locations is given below as a procedure.
Step 1: The aperture points are initialized by choosing one point at a time randomly along the phased array surface. In order to achieve this systematically on the total surface of the phased array, the following approach can be used:
Initial point locations ¼ uniformly spaced point locations þ small random shifts
Step 2(a): If the method of creating the negative phase of the phasefront at the chosen locations is used, the said phase is created physically, for example, by correctly choosing the lengths of the waveguides in the case of PHASAR devices.
Step 2(b): If the method of automatic zero-crossings is used, correction values are calculated for each of the initial points generated in step 1 to find the nearest zerocrossing points as
Final locations of zero-crossings ¼ Initial point locations from step 1þ correction terms
The two approaches work similarly. Below one algorithm to calculate the correction terms to generate the automatic zero-crossings is discussed.
METHOD OF IRREGULARLY SAMPLED ZERO-CROSSINGS (MISZC) |
345 |
19.3.1Computational Method for Calculating the Correction Terms
The phased array equation including the linear grating term for image formation is given by
dxi þ jiðxiÞ þ kroi ¼ 2pn þ f0 |
ð19:3-1Þ |
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where |
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jiðxiÞ ¼ krci |
ð19:3-2Þ |
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roi ¼ ðx0 xiÞ2þz02 |
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19:3-4 |
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rci ¼ ðxc xiÞ2þzc2 |
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19:3-5 |
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Equation (19.3-1) can be written as |
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dxi þ rci þ roi ¼ nl þ f0 l=2p |
ð19:3-6Þ |
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For an arbitrary position xi on the phased array, Eq. (19.3-6) becomes |
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dxi þ rci þ roi ¼ nl þ f0 l=2p þ B |
ð19:3-7Þ |
where B represents error. Let us assume that the position of the aperture is to be moved a distance in the positive x-direction such that the phase array Eq. (19.3-6) is satisfied. Then, the following is true:
dxi0 þ rci0 þ roi0 ¼ dxi þ rci þ roi B |
ð19:3-8Þ |
where roi and rci are given by Eqs. (19.3-4) and (19.3-5). Since xi0 ¼ xi þ , the following can be written:
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19:3-9 |
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19:3-10 |
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Using these equations leads to a fourth order polynomial equation for as
4F4 þ 3F3 þ 2F2 þ F1 þ F0 ¼ 0 |
ð19:3-11Þ |
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DENSE WAVELENGTH DIVISION MULTIPLEXING |
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where |
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F4 ¼ G12 1 |
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ð19:3-12Þ |
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F3 ¼ 2G1G2 þ 2Xc þ 2Xo |
ð19:3-13Þ |
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F2 ¼ G22 þ 2G1G3 4XcXo roi2 rci2 |
ð19:3-14Þ |
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F1 ¼ 2G2G3 þ 2Xcroi2 þ 2Xorci2 |
ð19:3-15Þ |
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F0 ¼ G32 roi2 rci2 |
ð19:3-16Þ |
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G1 ¼ |
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ð19:3-17Þ |
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G2 ¼ ðrci þ roi BÞd þ Xc þ Xo |
ð19:3-18Þ |
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G3 ¼ rciroi þ |
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Bðrci þ roiÞ |
ð19:3-19Þ |
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is obtained as the root of the fourth order polynomial in Eq. (19.3-11). It is interesting to observe that this equation reduces to a second order polynomial equation when there is no linear phase modulation due to a grating, as discussed in Section 15.9. The locations of the chosen zero-crossing sampling points correspond to the positions of the waveguide apertures on the phased array surface in the case of PHASAR devices.
19.3.2Extension of MISZC to 3-D Geometry
Extension to 3-D geometry is useful because other technologies can also be used. For example, the arrayed waveguides can be arranged in a 2-D plane or a 2-D curvature, instead of along a 1-D line discussed above. Other technologies such as scanning electron beam microscopy [Ersoy, 1979] and reactive ion etching which are used for manufacturing diffractive optical elements could also be potentially used. The basic method in 3-D is conceptually the same as before. In other words, the locations of the centers of radiating apertures are first (semi)randomly chosen; then either the negative phase of the phasefront (possibly plus a constant) at the chosen locations is physically generated so that the overall phase is zero (or a constant), or the locations of the centers of radiating apertures are slightly adjusted so that the total phase shift from such a center to the desired image point equals a constant value, say, zero modulo 2p.
In the case of choosing automatic zero-crossings, Eq. (19.3-11) is still valid if adjustment is done only along the x-direction, and the following replacements are made:
z20 ! z20 þ ðy0 yiÞ2
ð19:3-20Þ
z2c ! z2c þ ðyc yiÞ2
ANALYSIS OF MISZC |
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19.4ANALYSIS OF MISZC
In the method discussed above, the problem of higher order harmonic images is minimized. In this section, an analysis in 3-D is provided to explain why this is the case. In planar devices such as optical PHASARS, two dimensions are used. The 2-D analysis needed in planar devices such as optical PHASARS is achieved simply by skipping one dimension, say, the y variable from the equations.
The MISZC is a nonlinear encoding method. In general, with such an encoding technique, the harmonic images are generated due to two mechanisms: (1) regular sampling and (2) nonlinear encoding. In MISZC, harmonic images due to regular sampling are converted into tolerable background noise by irregular sampling [Doles, 1988]. The analysis of why the harmonic images due to nonlinear encoding with zerocrossings are also eliminated in the presence of phase modulation is given below.
Equation (19.3-1) can be written more generally as
jðxi; yiÞ þ yðxi; yiÞ þ kroi ¼ 2np þ j0 |
ð19:4-1Þ |
where jðxi; yiÞ is the phase shift caused by the wave propagation from the origin of the spherical reference wave ðxc; yc; zcÞ to the ith coupling aperture ðxi; yiÞ on the surface of the phased array; yðxi; yiÞ is another phase shift, for example, the linear phase shift in Eq. (19.3-1); kroi is the phase shift caused by the wave propagation from the aperture ðxi; yiÞ on the surface of the phased array to the image point (object point) located at ðxo; yo; zoÞ. In a PHASAR device, yðxi; yiÞ can be expressed as ncakxi, where nc is the effective index of refraction inside the waveguide.
For the center wavelength l, Eq. (19.4-1) is written as
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krci þ nkxia þ kroi |
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ð19:4-2Þ |
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Based on paraxial approximation, we write |
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x0xi þ y0yi |
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ci ¼ ð c |
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ð19:4-3Þ |
Substituting Eq. (19.4-3) into Eq. (19.4-2) and neglecting constant phase terms results in
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xid xi |
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ð19:4-4Þ |
zc |
zo |
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where d ¼ nca.