Abstract:
A method for assessing the optical quality of an imaging device. The method is suitable for use in an optical system comprising a source of radiation that can output a radiation beam; and, an imaging device to be tested, which imaging device can image the radiation beam to produce an imaged beam. The steps of the method comprise: sampling the imaged beam by an opaque mask defining at least two apertures, for generating local wavefront errors at a pupil position defined by a location of the mask; and, computing an optical transfer function (OTF) for the localized wavefront errors transmitted through the sampling apertures. The method has an advantage of expanding the applicability of the optical transfer function, heretofore a measure only of global imaging device performance, so that it can provide information for correcting an imaging device defect in terms of local area modifications.

Description:
BACKGROUND OF THE INVENTION 
     This invention relates to a method for assessing the optical quality of an imaging device. 
     INTRODUCTION TO THE INVENTION 
     Attention is first directed to FIGS. 1A, 1B, 1C and 1D which show an optical system 10 and waveforms that may be used to explain the concept of assessing the optical quality of an imaging device. 
     The FIG. 1A optical system 10 includes a source of radiation comprising a point source 12 that can output a radiation beam 14; and, an imaging device comprising a conventional lens 16. The lens 16 can image the radiation beam to generate an exit pupil imaged beam 18. 
     In a first case, it may be assumed that the FIG. 1A imaging device comprising the lens 16 is ideal. For this first case, accordingly, the point source 12 can output a radiation beam comprising divergent spherical waves, which divergent spherical waves, in turn, may be imaged by the ideal lens 16 to generate convergent spherical waves. 
     Since the FIG. 1A lens 16 is assumed to be ideal, it does not introduce wavefront errors, which would otherwise deform the sphericity of the convergent waves. Therefore, a FIG. 1B graph of optical path difference (OPD) error, from assumed perfect sphericity, as a function of exit pupil position (p), is a straight line congruent with the abscissa. 
     In a second case, it may now be assumed that the imaging device is not ideal. For example, as shown in FIG. 1C, a lens 20 may have inherent defects, such as a trefoil or coma, so that in the presence of the point-source, the lens 20 generates at its exit pupil a wavefront 22 that departs from ideal sphericity. Accordingly, for this second case, a graph of optical path difference (OPD) error, from idealized perfect sphericity, as a function of exit pupil position (p), is of the type illustratively shown in FIG. 1D. 
     (Note that the FIGS. 1B, 1D graphs of OPD error may be sensed, i.e., generated, by conventional techniques, including Michelson, e.g., Twyman-Green interferometry, or Foucault knife-edge testing.) 
     SUMMARY OF THE INVENTION 
     The FIG. 1B, 1D graphs of OPD error are one measure of the optical quality of the imaging device. The graphs may be generated for the following purpose. Namely, it has been historically understood that: 
     (1) one may first mathematically interrogate the global OPD error function for its root mean square (RMS), or variance, or standard deviation, or Zernike polynomial curve fit, or peak to valley absolute difference, etc; to an end secondly, 
     (2) of ascertaining the type and the magnitude of an imaging device defect, for the purpose thirdly, 
     (3) of correcting the imaging device defect, for example, by way of a polishing or a surface smoothing technique applied to a surface element of the imaging device. 
     We have recognized a pervasive and profound problem with this three-step procedure. The problem stems from an inherent assumption, common to all of the cited theoretical mathematical interrogation techniques, which assess the optical quality of the imaging device on the basis of a global parameter, namely, the global OPD error function. Thus, the cited mathematical techniques, for example, the RMS approach, derive a wavefront error statistic from a function that has been initially developed over an entire (hence, global) imaging device exit pupil aperture. 
     Accordingly, for example, it can be the case that an infinite number of different OPD error functions can have an identical RMS value, so that there may be no direct correlation between minimizing a particular computed RMS value, on the one hand, and providing, on the other hand, a corresponding maximization of global imaging performance through a reduction of local OPD error. Indeed, it can be the case that the imperfect correlation can reduce the imaging performance when the local OPD errors are corrected on the basis of RMS minimization. 
     Restated, there may be no direct correlation between the global parameter (e.g., RMS) and a &#34;marching order&#34; to the polisher or optician as to where, locally, on the imaging device, one should directly address or remove a specific, localized defect, based on an informed or prioritized mathematical calculation. 
     For this reason, lack of direct correlation between, on the one hand, global imaging performance and, on the other hand, informed and prioritized information on local imaging device defects, as derived from the FIG. 1D OPD error function, we have been prompted to investigate a second, independent mathematical approach to assessing the optical quality or imaging performance of an imaging device. This second mathematical approach is based on the classical optical transfer function (OTF) and a related optical quality function (OQF). (See Appendix, Reference 1.) 
     The optical transfer function is a global spatial frequency-dependent complex quantity, whose modulus is a modulation transfer function (MTF), and whose phase is a phase transfer function (PTF). The MTF is a measure of a reduction in contrast from object to image, for each spatial frequency passed by an imaging device. The PTF represents commensurate relative phase shift, for each spatial frequency passed by an imaging device. The optical transfer function may be expressed by an equation (1): 
     
         OTF (freq.)=MTF (freq.)·e.sup.j PTF(freq.).       (1) 
    
     The related global optical quality function (OQF) may be expressed by an equation (2): ##EQU1## 
     As a tool for assessing the optical quality of an imaging device, the global OTF and the OQF have the following advantage, as compared to the above elaborated optical path difference (OPD) error technique. Namely, the global OTF and OQF are directly correlated to global imaging performance of an imaging device with arbitrary aberration distributions. Accordingly, the OTF and OQF have an advantage in that they can provide the immediate parameters of concern to customers of imaging devices and systems, since the customers typically state a desired imaging device global imaging performance specification, expressed directly as a function of the OTF and OQF. 
     The advantage realized by the mathematical approach of the OTF and OQF, may be offset, however, when we point out that these global performance functions do not supply information for a corrective process in terms of required local area modifications. Restated, the OTF and OQF technique, much like the optical path difference (OPD) error technique, may be limited in usefulness, because there is no way to map these inherently global parameters to performance loss due to discrete parts of an exit pupil. 
     We have now discovered a novel method for assessing the imaging quality or performance of an optical imaging device. The novel method has an important advantage of retaining all the virtues of the mathematical approach subsumed by the global optical transfer and quality functions, while entirely overcoming their heretofore offsetting limitations, in terms of exclusivity vis-a-vis local mapping capabilities. The novel method has additional advantages, to be specified below, in terms of prioritization capabilities, and performance tailored modulation transfer function (MTF) specifications. 
     The novel method is suitable for use in an optical system comprising a source of radiation that can output a radiation beam; and, an imaging device to be tested, which imaging device can image the radiation beam to produce an imaged beam. The steps of the novel method comprise: 
     a) sampling the imaged beam by an opaque mask defining at least two apertures, for generating local wavefront errors at a pupil position defined by a location of the mask; and 
     b) computing a localized optical transfer function (OTF) for the localized wavefront errors transmitted through the sampling apertures. 
    
    
     BRIEF DESCRIPTION OF THE DRAWING 
     The invention is illustrated in the accompanying drawing, in which: 
     FIGS. 1A, 1B, 1C and 1D show an optical system and waveforms that may be used to explain the concept of assessing the optical quality of an imaging device; 
     FIGS. 2A, 2B show schematics of a sampling step of the method of the present invention comprising object to image information transfer; 
     FIGS. 3A, 3B and 3C show a basic LWPA sampling aperture transmittance function in 1-D form, generated in accordance with the steps of the present invention; 
     FIG. 4 demonstrates a full aperture performance synthesis process for a unique spatial frequency; 
     FIGS. 5A, 5B show a wavefront error contour map and corresponding profile, respectively, generated in accordance with experimental data; 
     FIG. 6 shows a schematic of an illustrative LWPA sampling geometry; and 
     FIG. 7 shows a schematic to explain a nexus between local to global performance estimation. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     As summarized above, the novel method of the present invention can provide an assessment of the optical quality or performance of an imaging device. This assessment can be done in real time or, alternatively, through software designed to or from the output of wavefront sensors. 
     The imaging device to be tested can comprise a conventional lens, or a mirror, or a diffraction grating, or a combination of these elements. Or, the imaging device can comprise an unconventional element, for example, a synthetic aperture comprising two or more monolithic apertures. The radiation source that inputs to the imaging device can comprise an extended source or a point source, and can be coherent, incoherent, or partially coherent. 
     The novel method comprises analysis of local wavefront performance, as developed over an exit pupil aperture of an imaging device, as evaluated by the local optical transfer function (OTF). For this reason, the novel method may also be referenced as a Localized Wavefront Performance Analysis (LWPA) system. 
     In the LWPA system, only the local wavefront errors that are included in LWPA spatial frequency selective sampling apertures, are used in assessing performance of an imaging device. A performance map of a full system aperture may be generated when the apertures are used to sample the wavefront at an adequate number of local areas (i.e., in satisfaction of the classical and well known Nyquist sampling rate function). The wavefront error at each sample position can therefore now be prioritized, and corrected on a basis of its impact on overall imaging performance. Further, any remaining error residuals are now optimized, for minimal impact on imaging device performance. 
     Restating this last paragraph, but with more particularity, a LWPA system prioritization sequence evaluates an imaging device quality on a basis of a local optical quality function (OQF) that is specific to discrete regions of an exit pupil aperture defined by an imaging device. This information, in turn, may be used to generate a corresponding OQF map over the entire exit pupil aperture. An optician or polisher, or adaptive optical control system, can therefore now use this OQF map, to maximize imaging performance, by correcting those localized wavefront error regions with the lowest local OQF values. The net result of this action of the novel method is to generate a substantially monotonic-type convergence to an acceptable performance, at a given spatial frequency, thus securing a performance tailored modulation transfer function (MTF). 
     The method of the present invention is based on a first principle that an arbitrary object can be constructed from a knowledge of its Fourier spatial components, and on a second principle that a spatial filter, utilizing two dilute apertures, can be used to extract selected spatial frequency information from the object. 
     The two principles provide an understanding for the first step of the present invention, namely, sampling an imaged beam (generated by an imaging device under test), by way of an opaque screen defining at least two apertures. 
     The at least two apertures may each comprise a geometric shape defining a rectangle or a star, but preferably comprise a circle or a square. The apertures can comprise areas of appreciable extent, for example, an appreciable percentage of the global exit pupil, or, in a limiting case, may comprise a pinhole dimensionality. (See Appendix, Reference 3.) 
     Attention is now directed to FIGS. 2-7, which are provided as heuristics to demonstrate preferred aspects of the steps of the method of the present invention. 
     A Lens Sampling Process, Object-To-Image 
     FIG. 2A symbolically shows an imaging process of the present invention, in terms of the Fourier transform theory, and with a 1-D object for an input signal. The two headed arrow indicates a Fourier transformation from the space domain (X) to the frequency domain (f). The spectrum of this FIG. 2A object is schematically shown as the well-known sinc function. A specification frequency (f*) may be selected for performance evaluation in FIG. 2B, where a Young&#39;s two-slit aperture is shown sampling the image quality at a pupil position X1. The specification frequency is directly proportional to a distance D between the sample apertures, where D=λzf*, λ is the light wavelength, and z the distance between the fringe observation and pupil planes. 
     This FIG. 2 example indicates how the LWPA apertures extract information on lens quality at critical (specification) spatial frequencies. The transform shown by the double headed arrow is, in general, complex, which indicates that input signal fringe contrast (fringe modulation depth) and phase contribute to the image quality. Local area Xl&#39;s contribution is shown as the small black rectangle at the OQF function at frequency f*. Multiple fringe patterns are obtained from the several pupil regions that are sampled. When they are superimposed, the composite fringe obtained from their superposition, is indicative of global performance at frequency f*. 
     Global performance values over the total lens bandwidth are indicated by the solid line in the OQF plot of the FIG. 2B, and may be obtained by exploring the pupil with LWPA sample aperture sets having separation distances that cover the bandpass of interest. However, in developing the performance results for a specific spatial frequency, such as f*, a fixed LWPA sample aperture separation distance is required. Each sampled position supplies fully redundant (constant separation distance) data in order to obtain a robust estimate of the global performance loss for information transfer at a unique spatial frequency. (cf. this procedure to Appendix, Reference 4, where the object is the elimination of aberrations.) 
     (Note that in the LWPA process, it is assumed that a wavefront obtained during testing or performance assessment is supplied as input. In this case, the locally sampled signals are due totally to lens aberrations, and are not confounded by object phase variations. This is, indeed, the desirable condition for our purpose which involves lens performance assessment, and not image restoration, as in Reference 4.) 
     Local-To-Global Performance Connection 
     The two-aperture sampling method, discussed earlier, can be considered as comprising a narrow-band filter that extracts from an input wavefront those components that are of maximal importance for object information, transmitted at a specific spatial frequency. The LWPA system combines this operation of frequency selection with the operation of performance assessment. This may be accomplished by obtaining the OTF of the system defined by the two-aperture pupil and the local wavefront errors that are circumscribed therein. FIGS. 3A, 3B and 3C schematically present this critical concept. (See Appendix, Reference 3.) 
     FIG. 3A presents the basic LWPA sampling aperture transmittance function in 1-D form. The distance between the two sample apertures (D) determines the filtered spatial frequency band of interest, in this case denoted f*. The local OTF result may be obtained by the autocorrelation of this complex pupil function, and gives the MTF (FIG. 3B) and PTF (FIG. 3C) components. 
     The dashed MTF curve represents the diffraction limited case of no aberrations, the solid curve the aberration perturbed case. For a two aperture filter, the diffraction limited MTF is always 0.5 at frequency f*. Note that the ratio of (MTF(f*)-perturbed)/(MTF(f*)-diffraction limited) is roughly 0.50 in the above example. This value represents the OQF(f*) in the aperture sample region. A correction in this region will increase the elemental performance at spatial frequency f* by nearly 50%. 
     The local OTF obtained from each sample location gives that region&#39;s differential contribution to full aperture imaging performance for an object spatial frequency that is directly related to the separation distance between the two apertures. This concept is schematically shown in FIG. 4. 
     Full Aperture Performance Synthesis Process 
     As shown in FIG. 4, the two apertures, separated by a distance D, give Young&#39;s fringes at the diffraction plane, for each sample position (1 through N). Local image quality is indicated by fringe contrast degradation (MTF) due to local aberrations, or from phasing (PTF) errors as indicated by translation of a main lobe from a reference position (zero (0) in figures). Such local aberrations can tend to reduce the composite fringe generated from the superposition of the N sample patterns of unit normalized amplitude. The resulting composite fringe is indicative of global quality. 
     In summary, the FIG. 4 represents a schematic version of the situation for a perfect full aperture optic. In this case, the fringes from the N sample areas will have maximal amplitudes and be perfectly phased (exact alignment of all central lobes). This will result in a composite unattenuated fringe of maximal contrast of N. A key point in this development is that each pattern can be evaluated separately for affect on full aperture performance. In essence, the Young&#39;s fringes form a linear system which allows superposition--the total effect due to the N local fringes is given by their superposition. This yields the desired connection between the local and full aperture performance. 
     LWPA uses a somewhat modified form of Young&#39;s interferometer, in that the sample apertures may be of appreciable extent. This expands the sensitivity of the system to any higher-order (2nd degree or higher) aberrations that are contained in the sample region, because they are nonlinear in nature, and thus have a strong impact on the local fringe pattern quality and resulting local OQF. 
     A Fourier transform of the sample pupil gives the above fringe pattern in the far-field plane. A second transform gives the local OQF, the primary output. In the results presented herein, the local OQF values are computed directly from the autocorrelation of the wavefront errors circumscribed in the two sampling apertures. 
     In summary, the LWPA concept can be connected with the Young&#39;s aperture sampling of a wavefront and its transfer to the diffraction (image) plane. A direct sampling by two dilute apertures, separated by appropriate distances, gives the information necessary to obtain a local image quality in terms of critical specification frequencies. The performance at different pupil regions can be obtained with a generalized &#34;Young&#39;s interferometer&#34; sampling process, in that finite area sampling apertures are used. An additional generalization may entail the use of more than two apertures, in order to obtain results pertinent to several spatial frequencies simultaneously (multi-band spatial frequency filtering). (See Appendix, Reference 5.) 
     Experimental Data 
     In order to test the LWPA concept, a wave front affected with 1/10 wave (peak to valley) of spherical aberration was evaluated. The data consisted of a single strip through the wavefront error map which comprised approximately 10% of the full aperture. A contour (shaded) wavefront error map is presented in FIGS. 5A, 5B. The wavefront error contour map is shown in FIG. 5A, and the corresponding profile in FIG. 5B. 
     The local performance contribution for the example aberration (pure Zernike spherical) is well known and thus allows assessment of LWPA accuracy. A strip format was selected for ease of presentation, but the results are directly extendable to the complete pupil. 
     The corresponding LWPA sampling geometry is schematically shown in FIG. 6. 
     Specific sampling parameters are as follows: 
     1) Rectangular sample area (both apertures): about 10% of input aperture area. 
     2) Sample aperture separation distance D is about 14% of input aperture width - commensurate with 10c/mm information. 
     3) Filter shift distance (sample spacing every 4 data points) - about 8% of input aperture diameter. 
     The above parameters resulted in a LWPA consisting of 13 uniformally spaced samples that spanned the pupil region. Since each sample consisted of two dilute sample apertures, the total data base comprised 26 local regions. The performance map of table 1 gives test-case results in terms of OQF versus aperture position. 
     
                       TABLE I______________________________________Performance Map (10 c/mm)POS     MTF(10)       PTF(10)  OOF(10)______________________________________1       0.4883        +15.5    0.97662       0.4959        -4.8     0.99183       0.4992        -15.4    0.99844       0.5000        -18.3    1.0005       0.4995        -15.4    0.99906       0.4988        -8.7     0.99767       0.4985        0.0      0.99708       0.4988        +8.7     0.99769       0.4995        +15.5    0.999010      0.5000        +18.3    1.00011      0.4992        +15.4    0.998412      0.4959        +4.8     0.991813      0.4883        -15.5    0.9766______________________________________ 
    
     The table values are ordered in the form of a 10 c/mm imaging performance map (OQF). The results are shown at 13 equally spaced points over this aperture. Table parameters include: 
     POS: the position of the local error in the aperture-position is along the X-direction; left-to-right with position 1 leftmost (see FIG. 6); 
     MTF(10): local MTF at 10 c/mm; 
     PTF(10): local phase error at 10 c/mm in degrees; 
     OQF(10): local optical quality factor at 10 c/mm-ratio of MTF(10) to the corresponding diffraction limited MTF. 
     This result shows that wavefront errors at positions 1 and 13 had the greatest impact on performance (lowest OQF), while 4 and 10 had the least (highest OQF). Since local positions 1 and 13 had the lowest OQF values, they would be modified first with the expectation that their correction results in the maximal global OQF improvement at 10 c/mm. Simulated correction was accomplished for these maximally high loss positions in addition to those at 2 and 12 and 4 and 10. Correction of errors at 2 and 12 effected the second greatest improvement because they generated the second lowest OQF, whereas correction at 4 and 10 did not contribute to improvement, in that their OQF was already at a peak limiting value of 1.000. After each correction, the global OQF was recomputed. Corrected results and calculated improvement ratios are shown in Table 2. 
     
                       TABLE II______________________________________Results at Three PositionSets (10 c/mm) (Input MTF 0.8405)      Positions Corrected:      1 &amp; 13    2 &amp; 12  4 &amp; 10______________________________________MTF-after corr:        0.8584      0.8429  0.8394Improve. Ratio:        1.022       1.015   1.000______________________________________ 
    
     As seen, the LWPA dictated corrections in regions 1 and 13 gave the greatest improvement, 2 and 12 the next greatest improvement, and, 4 and 10 little or no improvement. These results are in complete agreement with the prioritization indicated by the Table I LWPA performance map. The maximal 2+ % gained with the correction of errors in positions 1 and 13 may represent a significant improvement for 10 c/mm information transfer. On this basis, the example case has demonstrated a localized performance directed optimization for full aperture QQF. 
     It is important to assess the effect of the above LWPA dictated correction on other frequencies in order to determine impact on lens wideband performance. A qualitative assessment is presented in Table III. 
     
                       TABLE III______________________________________Filter Results vs. Frequency(10 c/mm)Frequency c/mm        D1MTF      D2MTF    %-IMP______________________________________10           0.0318     0.0179   56.320           0.0563     0.0088   15.630           0.0548     0.0263   47.9______________________________________ D1MTF: The difference between the diffraction limited MTF and the perturbed input MTF. D2MTF: The difference between the corrected and input MTF. %IMP: The percent improvement calculated as the ratio D2MTF/D1MTF. 
    
     The table indicates the performance improvement for a band of spatial frequencies given LWPA dictated correction at positions 1 and 13, regions contributing maximal performance loss for 10 c/mm information transfer. 
     The 56.3% improvement in the 10 c/mm global OQF is as expected. The interesting aspect of the data concerns a non-uniform enhancement associated with the data at frequencies of 20 and 30 c/mm. The fact that 20 and 30 c/mm performance is also enhanced is not unexpected, given the high spatial correlation associated with the input aberration function. However, the relative differential in the size of the improvements (47.9%-vs-15.6% ) suggests that performance improvement at collateral frequencies will not be uniform. The non-randomness of arbitrary aberrations, particularly the highly deterministic form for the example test case, supports the differential shown in the Table III data. 
     Local to Global Performance Estimation 
     This section considers the connection between local and global performance in greater detail. It would be highly desirable to show that there is more than a correlation between local and global performance. The following results will show, that with appropriately spaced samples, the average vector sum (AVS) of the local complex OQF values gives a good approximation to the global result. This concept is shown schematically in a FIG. 7. 
     Note that the global OQF is the average vector sum (AVS) of the local OQF values; arrow denotes vector in diagram. This formula ties the full aperture performance, denoted the global OQF, to the local OQF values represented by the m OQF(f*)&#39;s under the summation symbol. In this model, the OQF is a complex number (treated in vector form) which is comprised of modulus and phase parts that are indicative of performance. Thus, the global OQF at a given spatial frequency is the phasor or vector average of the m local OQF values. A mathematical derivation of this result is given in Appendix A. Table IV presents the AVS calculation data for the Table II performance map. 
     
                       TABLE IV______________________________________LWPA VECTOR SUM ESTIMATE OF GLOBAL OQF                                OOFPOS. PTF(10)  OOF(10)  C(PTF) S(PTF) X C(PTF)______________________________________1    +15.5    0.9766   0.9636 0.2672 0.94132    -4.8     0.9918   0.9965 -0.0837                                0.98833    -15.4    0.9984   0.9641 -0.2656                                0.96264    -18.3    1.000    0.9494 -0.314 0.94945    -15.4    0.9990   0.9641 -0.2656                                0.96316    -8.7     0.9976   0.9885 -0.1513                                0.98617    0.0      0.9970   1.0000 0.0000 0.99708    +8.7     0.9976   0.9885 0.1513 0.98619    +15.5    0.9990   0.9641 0.2656 0.963110   +18.3    1.000    0.9494 0.3140 0.949411   +15.4    0.9984   0.9641 0.2656 0.962612   +4.8     0.9918   0.9965 0.0837 0.988313   -15.5    0.9766   0.9636 -0.2672                                0.9413______________________________________ Vector Sum: 12.5786 
    
     Table entries present the vector summation components for each position in the aperture with the following nomenclature: 
     POS: aperture position; 
     PTF(10): local phase error at 10 c/mm in degrees; 
     OQF(10): local optical quality factor at 10 c/mm-ratio for MTF(10) to its corresponding diffraction limited MTF; 
     C(PTF)=Cosine of PTF, local cosine vector component; 
     S(PTF)=Sine of PTF, local sine vector component; 
     OQF X C(PTF)=the product of the local OQF and the cosine of the PTF. 
     This represents the cosine component contribution of the local region to the global OQF. 
     The C(PTF) and the S(PTF) are the vector cosine and sine components (real and imaginary axis projections) associated with the local OQF&#39;s. According to the math model shown in FIG. 4 and derived in Appendix A, the AVS of the OQf&#39;s is a good estimator of the corresponding global result. In algebraic summation of the sine and cosine vector local OQF components gives corresponding global components. The global OQF is then computed at the Root-Sum-Square (RSS) of the two sums. Note that since the tabular results are for an optic with spherical aberration, a symmetrical or even function, the S(PTF) factors sum to zero and it is not necessary to compute OQF X S(PTF). However, the OQF X C(PTF) summation total is 12.5786 and the estimated global OQF is calculated directly as 12.5786/13 or 0.9676. This compares with a conventional global result of 0.9635 as computed from the autocorrelation of the full aperture complex pupil function - well within our computational accuracy. 
     The above example highlights the strength of the localized approach in maximizing the rate of convergence to acceptable performance. We note that the performance map of Table I is a satisfactory indicator of the performance contribution from local pupil regions in that results are in agreement with the local wavefront error contributions expected from the test case deterministic aberration. For example, the maximum error slope differential for the input spherical aberration is located in the regions associated with positions 1 and 13 for 10 c/mm information transfer - maximal loss would be expected in these regions. 
     A good counter-example is afforded by the minimal loss indicated for errors at positions 4 and 10. In this case, the slope differentials - again for 10 c/mm information transfer - are negligible, again lending credence to the LWPA generated performance map. This result is also mathematically supported by the fact that the second derivative of the wavefront error for 10 c/mm information, as characterized by the differential of the PTF(10) function, is approximately zero at these positions. The presence of an inflection point suggests that error modification will be relatively ineffective as a means of increasing imaging performance at the evaluation frequency. It should be noted, however, that the parts of the pupil circumscribed by the 4 and 10 samples will likely have significant impact on information transfer on other specification spatial frequencies. ##SPC1##