MTF generates a table of system response at designated frequencies at one or more focal positions. Separate tables are generated for each field and zoom positon. The default is to compute the diffraction sine wave response.
MTF> MFR 100. MTF> PLO FRE Y
| MFR | maximum lines/mm for computation, measured on the image surface |
|---|---|
| PLO FRE | Plot response vs. frequency graphs. |
Table 1. Tabular output
WARNING - At least 3 focal positions are recommended for finding best focus
APERTURE STOP
SEMI-DIAMETER = 4.489186
(Based on the maximum reference ray height at the stop.)
X and Y focal lengths for each field angle
X 0.499969E+02 0.517701E+02 0.538569E+02
Y 0.499969E+02 0.556946E+02 0.633075E+02
X and Y F-numbers for each field angle
X 4.499692 4.659279 4.847088
Y 4.499692 5.153604 7.145553
Reference sphere radius for each field angle
0.523489E+02 0.541108E+02 0.561527E+02
Relative execution time = 1.06 UNITS
(skipped output)
POSITION 1 DIFFRACTION MTF ORA 1-Jun-99
Cooke Triplet f/4.5
WAVELENGTH WEIGHT NO. OF RAYS
FIELD (X,Y)=( 0.00, 1.00)MAX, ( 0.00, 20.00)DEG 656.3 NM 1 864
RELATIVE ILLUMINATION = 61.4 PER CENT 546.1 NM 2 1240
ILLUMINATION (UNIT BRIGHTNESS) = 0.023762 486.1 NM 1 1578
DISTORTION = 1.25 PER CENT
DIFFRACTION LIMIT FOCUS POSITION
Formula Actual 0.00000
L/MM f/4.500 RAD TAN RAD TAN
---- ---------------- ---------
0 .999 .999 .999 .999 .999
8 .974 .974 .960 .815 .710
16 .949 .947 .921 .457 .344
24 .923 .921 .881 .138 .264
32 .898 .894 .841 .013 .200
40 .872 .868 .802 .002 .169
48 .847 .842 .762 .073 .149
56 .821 .816 .723 .117 .121
64 .796 .790 .684 .104 .095
72 .771 .764 .644 .060 .086
80 .746 .737 .605 .022 .085
88 .721 .711 .567 .006 .079
96 .696 .685 .529 .010 .071
MTF lists the aperture shapes on the aperture stop surface, as these will limit the ray bundles. If no aperture is specified for the aperture stop surface, then MTF uses the CODE V default aperture for that surface.
The X and Y focal lengths for each field are computed as the ratio of differential pupil position to differential numerical aperture for each pupil position. This computes the instantaneous magnification of the lens at that field position, and takes into account any vignetting and image surface tilt or curvature. For finite conjugate systems, the magnification is listed in place of the focal lengths. The X and Y Vnumbers are computed as the inverse of the difference in direction cosines of the plus and minus reference rays for that field position. Note that if you have not executed a SET VIG command, the reference rays may not correspond to the real edges of the pupil, and the f/number calculations may be in error. The MTF calculations will, however, be correct in any case.
For each field position specified, information is printed for the number of rays traced in the MTF grid for each wavelength, and the relative illumination and distortion at that field position. The relative illumination calculation assumes unit brightness at the exit pupil in units of W/cm2-steradian. This calculation is done by counting rays traced and taking into account the image obliquity. The distortion calculation is the same as is output in ANA.
Next comes the MTF table. The first column is the spatial frequency in line pairs/mm at the image surface. The next column is the diffraction limit of the optical system based on the Vnumber in the specification data (or Vnumber computed from the specification data and first-order ray tracing). The third column, which may be a pair of columns, is the diffraction limit of the actual optical system in the radial and tangential directions, based on the actual vignetting, obscurations, and pupil aberration factors in the lens for that field position. After that comes the computed radial and tangential (or X and Y) MTFs for each focal position. Radial (or X) MTF is horizontal MTF (vertical bars) in local image surface coordinates; tangential (or Y) MTF is vertical MTF (horizontal bars) in local image surface coordinates.
The followoing figure shows the MTF vs. spatial frequency. Each field is plotted, with radial and tangential MTFs plotted separately.
Diffraction MTF is calculated utilizing H. H. Hopkins' method [1] for the numerical evaluation of the auto correlation integral of the pupil function. The canonical pupil coordinates, suggested by Hopkins, are also employed. An adjustment in the calculation is made so that the frequencies are measured on the image surface itself.
The square wave response is calculated from a series summation of sine wave response values (see W. Smith, Modem Optical Engineering, McGraw-Hill, N.Y., 1966, p. 318).
The relative illumination of each field point is computed including all effects of vignetting, pupil expansion and cosine4 (but not variations of transmission or angular sensitivity of coatings).
Any calculation that deals with extended objects depends upon isoplanatism constancy of the point image patch as the field changes over the width of the object, e.g. over enough cycles of the MTF target to integrate properly. Isoplanatism is connected with Offense Against Sine Condition (OSC) and is destroyed by significant OSC residuals. OSC can also lead to invalid distribution of rays over the exit pupil, if not properly compensated.
In all CODE V options that deal with diffraction or accurate intensity calculations (ALI, LSF, MTF, PAR, PMA, PSF, TOR, TRA, and WAY), special programming has been incorporated to represent the local intensity and magnification changes that occur with OSC, so that any wavefront and point image is correctly apodized and calculated, even in the presence of large OSC; for example, an f/0.3 parabola would have massive OSC. This special process is valid in all forms of calculation (convolution, FFT, etc.) and has little effect on compute time.
Even though the calculation is accurate, the lack of isoplanatism will still cast doubt on the usefulness of any option results dealing with extended objects, when the image is changing so rapidly across the target. In these options (LSF, MTF, PAR, PMA, PSF), a test is made to see if OSC is sufficient to affect results. If so, a warning is issued to alert you that the system is not isoplanatic at the field in question.
In some of the CODE V options that deal with diffraction calculations (LSF, MTF, PAR, PMA, and PSF), the listed output includes focal lengths (or reduction ratios if the object distance is finite) and f-numbers based on the references rays at each field. A value for both the X and Y directions is given. These numbers vary with field and are useful for relating image dimensions to object dimensions and for simple relative illumination calculations. Also included are the reference sphere radii for each field. Diffraction is assumed to take place at the reference sphere.
The f-numbers are based on a calculation of the image intensity. Thus, if the pupils are approximately elliptical, an estimate of the relative illumination can be obtained from the product of the X and Y numerical apertures (the numerical aperture is the reciprocal of the f-number). This calculation assumes a Lambertian object and a uniformly illuminated exit pupil and ignores transmission losses in the lens. It includes the effect of a non-flat image surface. Note that, for a perfect thin lens with stop in contact and the object at infinity, the X numerical aperture is proportional to the cosine of the field angle while the Y numerical aperture is proportional to the 3rd power of the cosine, thus giving the cosine 4th law.
The reduction ratio, which is listed for a system with a finite object distance, is the ratio of an incremental distance on the image to the corresponding distance on the object, measured in a coordinate system normal to the surface. In general, a square patch on the image can represent a parallelogram shaped patch on the object, with the X and Y reduction ratios used to calculate the sides. The angle of the parallelogram is not listed although it is included in the internal calculation.
The focal length, which is listed for a system with an infinite object distance, relates the image patch size to an angle in object space. It can be thought of as the product of the f-number and pupil diameter, similar to the paraxial definition. Consistent with the other calculations, it assumes a measurement normal to the object surface which is typically flat for an object at infinity. Thus, dividing the image patch size by the focal length gives the change in the sine of the field angle. To get the change in angle, divide this number by the cosine of the field angle.
The reference sphere radius is also listed for each field. It is used to calculate the wave aberrations and is also used in the calculation of the focal lengths and f-numbers although it normally has very little influence on these numbers. It can be used to identify potential problems when the radius is so short that the reference sphere is within the caustic of the image.
Excerpted from the CODE V Reference Manual. (c) Copyright 2002 by Optical Research Associates. Excerpted by permission of Optical Research Associates. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from Optical Research Associates.
Maintained by John Loomis, last updated 1 June 1999