Optical window interferograms: a simple method for their evaluation Joseph M. Geary A simple procedure for evaluating interferograms based on the calculation of wavefront deviation is demon- strated in the context of image analysis, specifically as applied to optical windows. The purpose of this short paper is to demonstrate a simple hand technique for the evaluation of interfer- ograms which the author found useful in the context of image analysis. The particular application dealt with the selection of optical windowsfor photographic aerial reconnaissance. Suppose we have an interferogram whose fringe pattern represents phase variations across some aper- ture or pupil of an optical component or system. Al- though the OTF of this interferogram can be deter- mined, the autocorrelation procedure is tedius and time consuming if done by hand. Even with a computer programmed to perform the autocorrelation, many data points must be located and measured on the interfero- gram, usually via some semiautomated technique re- quiring human interaction and judgment. This in itself can become a source of operator tension. There are circumstances, of course, when the task requires an OTF calculation. On the other hand, there are times when life is permitted to be simpler. On these latter occasions it would be convenient to assign some number to the interferogram whose value bore a direct correlation with the image quality obtained via the component. Once this is demonstrated, one can rely solely on the inter- ferometric data and eliminate the need for an actual imaging test. Wavefont variance (or rms deviation) is such a com- monly used single number quality indicator for inter- ferograms, but here again we have a quantity not easily calculated. An alternative method is now described. Keeping in mind that fringe departure from paral- lelism and equispacing implies image degradation, we When this work was done, the author was with the University of Arizona, Optical Sciences Center; he is now with the U.S. Bureau of Radiological Health, Division of Electronic Products, Medical Physics Branch, Rockville, Maryland 20857. Received 10 January 1977. wish to find a simple measure of this degradation. Consider the interferogram in Fig. 1. Through its centralmost fringe, the reference fringe, we have drawn a straight line. The line is drawn such that the cross- hatched area is roughly minimal (eyeball approxima- tion). We then determine the area of the cross hatch by use of a planimeter (a device for determining en- closed areas by tracing around the perimeter). The arrows indicate the direction taken by the planimeter, i.e., clockwise around the enclosing curve. We divide this area by the length of the straight line whose ends are defined by its intersection with the edge of the pupil. The resulting number s is the average deviation from the straight line. This takes care of one fringe. To obtain an average value, several fringes must be ana- lyzed. The number depends on how the fringes and their spacing vary over the pupil. For Fig. 2(J) three were thought sufficient. For Fig. 2(F), five fringes were so treated. The positioning of additional lines cannot be arbi- trary but must reflect the wedging present in the in- terferogram. To find the average spacing S between lines, we count the total number of spaces between fringes and divide this into the pupil diameter. The additional lines then must be parallel to the first line with a separation NS, where N is an integer (which in Fig. 1 is 4). The fringe to be associated with this aux- iliary line is the fourth fringe over from the reference fringe, which is counted as zero. Finding the s value for the auxiliary lines of Fig. 1 and then averaging over all the lines analyzed yield = Nisi/. We now want to relate this average deviation s to optical path difference. Each fringe in the inter- ferogram is separated from its neighbor by an optical path difference of /2n (for a double pass interferometer and where n is the refractive index of the window glass). Therefore S corresponds to /2n. We can then set up the ratio S/(X/2n) = /(X/x). Solving for x, we have x = 2nS/s. For Fig. 1, we found the average magnitude of the wavefront deviation to be approximately /9 for n = 1.5. 2134 APPLIED OPTICS/ Vol. 16, No. 8 / August 1977
Transcript
Optical window interferograms: a simple method fortheir evaluation
Joseph M. Geary
A simple procedure for evaluating interferograms based on the calculation of wavefront deviation is demon-strated in the context of image analysis, specifically as applied to optical windows.
The purpose of this short paper is to demonstratea simple hand technique for the evaluation of interfer-ograms which the author found useful in the context ofimage analysis. The particular application dealt withthe selection of optical windows for photographic aerialreconnaissance.
Suppose we have an interferogram whose fringepattern represents phase variations across some aper-ture or pupil of an optical component or system. Al-though the OTF of this interferogram can be deter-mined, the autocorrelation procedure is tedius and timeconsuming if done by hand. Even with a computerprogrammed to perform the autocorrelation, many datapoints must be located and measured on the interfero-gram, usually via some semiautomated technique re-quiring human interaction and judgment. This in itselfcan become a source of operator tension. There arecircumstances, of course, when the task requires an OTFcalculation. On the other hand, there are times whenlife is permitted to be simpler. On these latter occasionsit would be convenient to assign some number to theinterferogram whose value bore a direct correlation withthe image quality obtained via the component. Oncethis is demonstrated, one can rely solely on the inter-ferometric data and eliminate the need for an actualimaging test.
Wavefont variance (or rms deviation) is such a com-monly used single number quality indicator for inter-ferograms, but here again we have a quantity not easilycalculated. An alternative method is now described.
Keeping in mind that fringe departure from paral-lelism and equispacing implies image degradation, we
When this work was done, the author was with the University ofArizona, Optical Sciences Center; he is now with the U.S. Bureau ofRadiological Health, Division of Electronic Products, Medical PhysicsBranch, Rockville, Maryland 20857.
Received 10 January 1977.
wish to find a simple measure of this degradation.Consider the interferogram in Fig. 1. Through itscentralmost fringe, the reference fringe, we have drawna straight line. The line is drawn such that the cross-hatched area is roughly minimal (eyeball approxima-tion). We then determine the area of the cross hatchby use of a planimeter (a device for determining en-closed areas by tracing around the perimeter). Thearrows indicate the direction taken by the planimeter,i.e., clockwise around the enclosing curve. We dividethis area by the length of the straight line whose endsare defined by its intersection with the edge of the pupil.The resulting number s is the average deviation fromthe straight line. This takes care of one fringe. Toobtain an average value, several fringes must be ana-lyzed. The number depends on how the fringes andtheir spacing vary over the pupil. For Fig. 2(J) threewere thought sufficient. For Fig. 2(F), five fringes wereso treated.
The positioning of additional lines cannot be arbi-trary but must reflect the wedging present in the in-terferogram. To find the average spacing S betweenlines, we count the total number of spaces betweenfringes and divide this into the pupil diameter. Theadditional lines then must be parallel to the first linewith a separation NS, where N is an integer (which inFig. 1 is 4). The fringe to be associated with this aux-iliary line is the fourth fringe over from the referencefringe, which is counted as zero.
Finding the s value for the auxiliary lines of Fig. 1 andthen averaging over all the lines analyzed yield =Nisi/. We now want to relate this average deviations to optical path difference. Each fringe in the inter-ferogram is separated from its neighbor by an opticalpath difference of /2n (for a double pass interferometerand where n is the refractive index of the window glass).Therefore S corresponds to /2n.
We can then set up the ratio S/(X/2n) = /(X/x).Solving for x, we have x = 2nS/s. For Fig. 1, we foundthe average magnitude of the wavefront deviation to beapproximately /9 for n = 1.5.
The nine window interferograms in Fig. 2 were ana-lyzed in this way. The results are shown in Table I, fora 9-cm (3.6-in.) diam aperture with n = 1.5. Also shownare the respective air-image average limiting resolutions(for high contrast targets) obtained on a 254-cm (100-in.) f/8.4 collimator (parabolic primary mirror) utilizinga 60.9-cm (24-in.) f/6.7 reconnaissance lens.
Comparison of the imagery in Fig. 2 shows that imagequality follows the same ranking. [The last three arehard to distinguish from each other. However, theirquality is excellent when compared to that of Fig. 2(A),which represents imagery without any window in theoptical path.]
This brings us to the important question of criterion.In the present case and consequent upon the precedingimage comparisons, a window was judged acceptable ifthe average magnitude of its wavefront deviation wasX/5 or less and unacceptable if the deviation was greaterthan X/5.
Of the nine windows analyzed, only (H), (I), and (J)were considered acceptable for use.
We note that this criterion is independent of therecon camera lens focal length. The only thing thatneeds to be known about a potential camera system isthe maximum aperture that its lens will subtend on thewindow, e.g., a window may be entirely acceptable fora maximum aperture of 5 cm (2 in.) but totally unac-ceptable for an aperture of 10 cm or 12.7 cm (4 in. or 5in.).
Now that the criterion has been established, futuretests on similar components need only involve inter-ferometry. Imaging tests can be dispensed with.
As already mentioned, wavefront distortions in in-terferograms are more commonly discussed in terms ofvariance V, specifically rms = v/V. In polar coordi-nates, variance across a pupil is defined as
I 152d ( pd -
|1f2, pdpdO
where 0 is the phase distortion.The function we have calculated using the planimeter
technique is the deviation, defined mathematically as
1 - 01pdpdO (f ' , I -_f pdpda) .7r 00
Fig. 2. Imagery obtained through windows of varying quality.2 pdpdO
Table II. Results of Root Mean Square and Deviation Comparison
Abberration 0 G V/V GA/V
Defocus A p2 %AA 0.250A 0.289A 0.865Spherical B p4 %1B 0.257B 0.298B 0.862Astigmatism C p2 COS20 'AC 0.206C 0.250C 0.824Astigmatism D p2 cos2O 0 0.318D 0.408D 0.779
with defocusComa E p
3COSO 0 0.255E 0.354E 0.720
Fig. 3. Planimeter path for bull's eye and hyperbolic type fringepatterns.
To obtain an idea of how VV and G compare witheach other, calculations for several aberrations weremade. The results are presented in Table II. (Note:The normalized chief ray height in the image plane hasbeen set at its maximum value, i.e., 1.)
It should be noted that the above expression for G isstrictly valid when the fringe spacing in the interfero-gram is fairly uniform across the pupil. When this isnot true, G is less well defined. The bull's eye and hy-perbolic fringe patterns fall into the latter category.However, the planimeter technique was applied to thesepatterns successfully. The main question in this regardwas what path to follow. The paths selected by theauthor for these patterns in his evaluation are illustratedin Fig. 3. Determination of the pure wedge grid is thesame as previously described, and again the planimeterpath is clockwise. The determined areas are indicatedby cross hatching in Fig. 3.
The analysis reported in this paper was extractedfrom a master's thesis written under the guidance of J.Wyant, submitted and approved by the faculty of theOptical Science Center, University of Arizona. Ex-perimental data were collected using the facilities of thephotoreconnaissance laboratory at the Naval Air De-velopment Center.
BibliographyP. F. Foreman, "The Photographic Window," in New Developments
for Aerial Photography; a Discussion of Applied Research forAerial Photography, presented by the Perkin-Elmer Corporationto OSA, ASP, SPIE, and SPSE at a joint technical meeting, 16September 1964, Wilton, Conn. (The Perkin-Elmer Corporation,Electro-Optical Division, Norwalk, Conn., 1964), pp. 2-1-2-21.
P. S. Young, F. E. Roberts, E. A. Strouse, et al., ReconnaissanceWindow Study; Final Report, Tech. Rept. AFAL-TR-72-193 (ThePerkin-Elmer Corporation, Norwalk, Conn., 1972), 320 pp.
J. M. Geary, "Aerial Photographic Reconnaisance Windows," un-published Master's Thesis, U. Arizona (1975).
R. Berggren, Opt. Spectra 4, 22 (1970).M. P. Rimmer, C. M. King, and D. G. Fox, Appl. Opt. 11, 2790
