Pupil function

This is the same as (3.1) except that the numerical constant is smaller. In practice, in an acoustic microscope the ideal of a numerical aperture is never even approximately realized. It is one of the experimental frustrations of acoustic microscopy that no satisfactory method exists for measuring the pupil function of the lens (i.e. the amplitude of the waves as a function of radial position on the lens surface) but, whatever else it is, it is certainly not uniform everywhere on the lens surface and zero elsewhere. 

Since L (9, 0) and L2(9, 0) depend only on the geometry and material of the lens, it is convenient to combine them in a single pupil function, which may be defined as... 

This is the basic expression for the response of the microscope to a uniform isotropic specimen at a defocus z. It should be permanently imprinted on the mind A V(z) curve for glass calculated using this expression is shown in Fig. 7.5. The primary difficulty in achieving exact agreement with measured curves lies in knowing the exact pupil function of the lens in the experiment. 

In practice the allowed range of t may be further restricted by the pupil function of the lens. The new value of Q(t) is used for the next iteration through eqns (8.10)—(8.14), and so on until satisfactory convergence is obtained. 

Thus the data can be preprocessed by Fourier transforming V( ) 2, applying a window corresponding to the extent of the pupil function of the lens, and then inverse transforming to obtain a filtered V(m), which can then be used as the data for the Gerchberg-Saxton algorithm. 

A pupil function P 6) was measured for the microscope using the same processing method. PTFE was used as the reference material. V(z) was... 

For Vq, the situation is more complicated the deviation from linearity is described by the exp iX term in (7.39). The phase shift for each ray will depend on its angle of incidence on the solid. Because of phase cancellation, the contribution as a function of angle will have a form similar to a sine function (depending in detail on the pupil function), with the width of the central maximum decreasing with increasing z. The phase of Vq will be an average of the contributions over all angles. As the defocus is increased so the phase becomes dominated by contributions closely parallel to the z-axis and, in this limit,... 

Finally, there may be a small systematic error associated with the lens used. The origin of this is not fully understood, but it may depend on how components around the Rayleigh angle are weighted by the pupil function. Each lens may be calibrated on a well defined reference material to correct for such systematic errors. 

There are considerable difficulties in comparing theory and experiment even in such model experiments. The theoretical calculations are subject to the approximations inherent in the method, and also to uncertainties in the pupil function used to characterize the lens and in the two parameters used to characterize the crack. The experiments are subject to the difficulties of making a crack that is straight and flat to a fraction of the acoustic wavelength used, over the length measured by the line-focus-beam lens, and to the sensitivity of the results in some cases to small changes in x or z. Nevertheless, when all these considerations are taken into account it does seem... 

Fig. 12.5. A crack from an indent in glass (a) z = 0 (b) z = — 3.8 pm (c) z = —5.2 pm ELSAM, 1.5 GHz. The experimental line-scans superimposed on the images can be compared with the plots calculated using two-dimensional theory (eqns (12.2), (12.13), and (12.14)) with elastic constants from Table 6.3 and values of defocus (a) z = 0 (b) z = —4.2 pm (c) z = —6.8 pm. The values of z in the calculations were chosen for best fit the reason for the discrepancy is not known, though no doubt there are the usual uncertainties associated with thermal drift, the measurement of z, and the frequency and pupil function used (Briggs etal. 1990).

Abnormal movements Tremors, convulsions Pupil function ... 

Direct Ophthalmoscopy. The direct ophthalmoscope is perhaps an imdemtilized instmment in the assessment of glaucoma. It can provide information regarding pupil function, an estimation of the anterior chamber angle depth, spherical refractive error of the patient, presence of media opacity, and a magnified view of the optic nerve... 

The product of the pupil function and the diffraction pattern describes the light entering the objective lens. A combination of the pupil function with the inverse Fourier transform of the diffraction pattern gives an expression for the electric field at the wafer plane as... 

The effects of aherrations on lithographic performance are determined more generally hy incorporating aherrations into the aerial image models. This is done hy modifying the pupil function of the lens to include phase error due to aherrations. ... 

Mathematically, aherrations are described as wavefront deviations (or errors), i.e., the difference in phase (or path difference) of the actual wavefront emerging from the lens compared to the ideal spherical wavefront. In other words, each primary aberration will produce unique deviations within the lens pupil. This phase difference is a function of position within the lens pupil, which can be described with an aberrated pupil function. An aberrated pupil function is described in terms of wavefront deformation as ... 

In mathematical terms, the low-pass filtering function of the projection lens can he described in the following manner for the special case of a rectangular grating, where p sin 6 mX describes the positions of the discrete coherent diffraction orders. If the lens is described in terms of a two-dimensional pupil function H(u, v). 

FIGURE 12 Optical transfer functions related to incoherent illumination of square (left) and round (right) pupils. T is proportional to the convolution of the pupil function [see top diagram and Eq. (45).]... 

This illustrates the difficulties and limitations of the PSF extraction process for the observation of biological specimens. Neither theory nor experimental determinations can precisely predict the PSF that should be used for the deconvolution of a complex biological sample. Empirical measurements are clearly the most appropriate method, but comparison of the values obtained with those predicted by the theoretical model can help to solve problems in the optical setup. One good solution would be to fit an analytical model to the experimental acquisition, to make the best use of both approaches. One recent study adopted su a method [18] recovering iteratively the theoretical shape of the pupil function, including phase information, from bead acquisitions. This made it possible to model aberrations causing axial asymmetry. However, this method cannot be used to overcome problems with shift invariance, an essential requirement of deconvolution algorithms that often does not apply completely in practice. 

The pupil function of the lens is again defined as follows ... 

In this case, the pupil function may be divided into the following two parts for analysis of its acoustic fields ... 

When the acoustic waves are reflected back from the specimen, the pupil function must be considered within the region expressed as -Xa < X < Xa in accordance with the directions of wave propagation shown in Fig. 13b. Using the angular spectrum approach," we can express the intensity of the wave at the transducer of the acoustic lens as follows ... 

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