A spot diagram is a collection of ray data resulting from tracing a large number of rays from a single object point through several aperture coordinates. In OSLO, as shown in the previous section, the aperture coordinates are normally set up to form a square grid in the entrance pupil. Spot diagrams can be processed in a variety of ways to provide either geometrical or diffraction analyses of optical images. Although spot diagrams involve ray tracing, many aspects of spot diagram analysis involve considerations of statistics or numerical analysis that have little to do with the techniques used to trace rays.
OSLO uses enhanced spot diagrams that contain angles and path lengths in addition to the intersection coordinates of rays with the image surface. This enables the program to carry out focus-shifting operations without tracing additional rays. Spot diagrams in OSLO are stored in memory, making computations of image evaluations that use them very fast.
In order to compute a spot diagram, it is first necessary to define an object point using the set_object_point command described in the previous section. A spot diagram can then be computed using the trace_spot_diagram command. The grid used to set up the aperture coordinates of rays in the spot diagram is determined by the value of the number of aperture divisions across the diameter of the entrance pupil. For systems that have a large numerical aperture on the object side, the grid cells are equally spaced in numerical aperture, not in entrance pupil coordinates.
The figure below shows the grid pattern used in OSLO for 10 aperture divisions. The number of aperture divisions is not restricted to integer values, but is a continuous parameter that determines the size of grid cells in the spot diagram.
The general concept of spot diagram analysis involves tracing enough rays so that the data for any particular ray can be treated statistically. Each ray is considered to carry a weight proportional to the area of its cell in the aperture of the system. As can be seen from the picture, with a square grid of rays and a circular aperture, rays at the edge of the aperture may carry too much or too little weight, depending on their exact coordinates. By adjusting the exact value of the number of aperture divisions, an ad hoc compensation for this effect can sometimes be obtained. For systems that have meridional symmetry, only half the rays need to be traced, as shown in the figure. OSLO automatically tests whether the system has such symmetry and traces the appropriate number of rays.
The diameter of the beam transmitted through the system in computing a spot diagram is normally determined by the entrance beam radius ebr. However, the spot diagram ray trace in OSLO can also use aperture checking on selected surfaces to determine whether rays pass through the system or not. When a ray is found to hit a surface outside a checked aperture (or inside an obstruction), it is blocked so that it does not appear in the spot diagram, as shown in the previous section.
The number of rays needed to make a satisfactory spot diagram depends on several factors, including the quality of the system and the intended application. The number of rays is determined by an operating condition called sdad, which determines the grid spacing on the entrance pupil. If the system is vignetted or has a central obstruction, sdad can be set to large values, since only rays that are not blocked are included in the total count.
*SPOT DIAGRAM: POLYCHROMATIC
APDIV 17.030000
WAV WEIGHTS:
WW1 WW2 WW3
1.000000 1.000000 1.000000
NUMBER OF RAYS TRACED:
WV1 WV2 WV3
104 104 104
PER CENT WEIGHTED RAY TRANSMISSION: 44.827586
*SPOT SIZES
GEO RMS Y GEO RMS X GEO RMS R DIFFR LIMIT CENTY CENTX
0.021401 0.007555 0.022696 0.003426 -0.002879 --
*WAVEFRONT RS
WAVELENGTH 1
PKVAL OPD RMS OPD STREHL RATIO RSY RSX RSZ
2.217030 0.485008 0.089483 -0.004131 -- --
The top set of data summarizes the input conditions and shows how many rays passed through the system. The number of rays traced divided by the number of entering rays is used together with the wavelength and ray weights to determine the approximate transmittance of the system.
The next two sets of data show the statistical data specifying the ray distribution and the wavefront corresponding to the spot diagram.
For the ray distribution, the spot sizes, defined to be the rms values of the ray deviations of the rays from the centroid of the distribution are specified in the x, y, and r directions. The diffraction limit (the radius of the Airy disc of an equivalent perfect system) is displayed so that the performance of the actual system can be compared to a diffraction-limited system. Finally, the centroid coordinates (CENTY,CENTX) of the distribution on the image surface, with respect to the reference ray height (which can be found from the preceding set_object_point command), is given.
For the wavefront, the peak-to-valley optical path difference (PKVAL OPD) among all the rays in the spot diagram is shown, along with the RMS value of the OPD. The next term, called STREHL RATIO, is the ratio of the intensity at the center of the actual diffraction image, divided by the intensity that would be obtained with an aberration-free wavefront.
Finally, the coordinates (RSY,RSX,RSZ) of the best-fit reference sphere used for OPD data are shown. This sphere is found by fitting a spherical surface to the actual wavefront emerging from the system, the criterion being that the variance of the wavefront from the reference sphere be a minimum.
In general, diffraction-related data in spot diagram routines are measured relative to the best-fit reference sphere, while geometrical data are measured relative to the centroid of the ray intersections. The following figures show the essential quantities involved.
When a spot diagram has been computed, it remains in memory and serves as the basis for a variety of image evaluation commands, including several that simply display the ray and wavefront data in various formats. For example, Calculate >> Display Spot Diagram >> Print spot diagram lists the current spot diagram in the text window, in the following format.
*SPOT DATA
RAY FY FX DY DX DYA DXA
WV1
1 -0.410926 0.058704 0.054844 0.000402 0.050487 -0.007340
2 -0.410926 0.176111 0.053551 0.001322 0.050555 -0.022024
3 -0.293519 0.058704 0.039044 -6.5163e-05 0.035935 -0.007321
4 -0.293519 0.176111 0.038106 -0.000120 0.036004 -0.021967
5 -0.293519 0.293519 0.036152 0.000123 0.036143 -0.036624
.
. etc.
.
WV2
WV3
FOCUS: --
The data from this command are also written into the spreadsheet buffer, so they can be used by macro programs or saved to disc. Since many spot diagrams have more than 1000 rays, it may be necessary to divide the output into blocks to prevent overwriting the buffer.
To determine more detail about the quality of the image, other analyses are available. These are shown on the Calculate menu under the Setup Spd/Wavefront command. As mentioned at the beginning of this section, these analyses differ from the ordinary ray trace routines in that they attempt to provide physically measurable results. In order to obtain accurate results, you must understand the model being used and provide realistic input data. In this connection, there are two factors that are extremely important.
The first factor is the description of apertures in the system. You must ensure that checked apertures are properly placed and specified so that the rays that make up the underlying spot diagram are in fact the ones that get through the system, as discussed in the preceding section.
The second factor is specifying the correct number of aperture divisions (sdad). Since most of the image evaluation routines perform statistical analysis of the spot diagram data, it is crucial that the aperture be adequately sampled. For example, the computation of the point spread function involves adding up Huygens wavelets from each grid cell in the spot diagram. For this to be accurate, the phase difference between adjacent cells must be a small fraction of a wavelength. If the phase difference were larger than a wavelength, the results would be meaningless. In the Setup Spd/Wavefront dialog box, the Aperture Divisions argument is supported by an optional list (you can also type in any value you want). The values on the list, which are not integers, are chosen to provide ray grids that provide maximum accuracy in computing the cutoff frequency due to diffraction effects.
To ensure the accuracy of image evaluation based on a spot diagram, you should always re-compute the evaluation with a larger number of aperture divisions and make sure that the results do not change!
Next Page | Previous Page | Chapter Summary | Table of Contents
Copyright © 1997 Sinclair Optics Inc. All rights reserved.
Page last updated 19970601