Method of measuring spherical aberration and apparatus therefor

An optical testing method and apparatus employing a non-interferometric technique, making use of axial intensity information, in which the intensity of the light pattern along the optical axis is calibrated to achieve improved measurement of spherical aberration.

BACKGROUND OF THE INVENTION 
The present invention relates to measurement of spherical aberration, and 
more particularly, to a method of employing a non-interferometric 
technique, making use of axial intensity information, in which the 
intensity of the light pattern along the optical axis is calibrated to 
achieve improved measurement of spherical aberration, and an apparatus for 
practice of same. 
Spherical aberration is measurable via a number of experimental procedures. 
Qualitative and semi-qualitative estimates can be obtained by the visual 
inspection of the star image, or from a Foucault knife edge test. 
Quantitative methods include the Hartmann test, and a host of 
interferometric approaches, such as point diffraction, and lateral shear 
interferometry. However, such methods are known to be inaccurate, 
difficult to apply, and time-consuming in various applications. 
It is therefore an object of the present invention to provide a novel 
method for accurate direct measurement of spherical aberrations in test 
pieces. 
It is another object of the present invention to provide a novel apparatus 
for practice of the invention. 
SUMMARY OF THE INVENTION 
The present invention comprises a method and apparatus for measurement of 
spherical aberration as related to axial intensity profiles near focus for 
a lens under test. The method comprises determining the paraxial focus, 
determining actual focus, and measuring the difference therebetween. 
For the purpose of the present invention, the following terms are defined 
as presented below: optical center, optical axis, paraxial rays, paraxial 
focus, paraxial plane, and actual focus. 
The optical center of a lens is a point through which all rays of light 
pass, when going through the lens. The optical axis of a lens is an 
imaginary line which is perpendicular to the lens, and which passes 
through the optical center of the lens. 
Paraxial rays are rays of light which are near the optical axis of the lens 
such that the sines of the angles between the paraxial rays and the 
optical axis may be replaced by the value of the angles themselves in 
calculations. The paths of paraxial rays are nearly parallel with the 
optical axis. The paraxial focus refers to the point where paraxial rays 
either come to a focus or come closest to converging along the optical 
axis. The paraxial plane is the focal plane located at the paraxial focus. 
The actual focus of a lens refers to the point where rays of light leaving 
the lens actually come to a focus. Where there exists no spherical 
aberration, the actual focus and paraxial focus are identical. 
In an apparatus in practice of the invention, a source of collimated light 
is projected through a lens under test, and the far field pattern is then 
directed from a microscope to an aperture card. The lens is adjustable 
longitudinally by means of a micrometer mount, and it is adjusted to place 
the far field pattern, via the microscope, concentric to an aperture of 
the aperture card. The passed beam, via a diffuser, impinges upon a 
detector (such as a photo-multiplier tube), the output of which is 
evaluated, such as by a radiometer, the resultant irradiance signal being 
directable to a first channel of a chart recorder. This recorder is 
couplable at a second channel thereof to the micrometer, by means of which 
the x-y irradiance characteristics of the lens under test may be 
point-plotted. This point plot indicates the actual characteristics of the 
lens, which are comparable with the paraxial focus to determine spherical 
aberration thereof.

DETAILED DESCRIPTION OF THE DRAWINGS 
It is known that if the Fresnel-Kirchhoff diffraction integral is evaluated 
for a perfect lens, the intensity profile is 
EQU I.sub..delta. (0)=I.sub..theta. sin c.sup.2 (.delta.) (1) 
where .delta. is the displacement from paraxial focus. Note that 
I.sub..delta. (0) is symmetric about .delta.=0. See Appendix 1 for a 
discussion of symmetry of axial intensity. As spherical aberration is 
introduced, the axial intensity pattern shifts away from paraxial focus, 
and becomes less sin c.sup.2 looking. However, it nonetheless remains 
symmetric about an axial point defined by 
EQU W.sub..theta.2.theta. =-W.sub..theta.4.theta. (2) 
where W.sub..theta.2.theta. =defocus, and W.sub..theta.4.theta. =spherical 
aberration (as defined in the exit pupil by departure in the wavefront 
from a reference sphere). Note that equation (2) is a condition which 
minimizes the rms wavefront. It is generally valid for 
W.sub..theta.4.theta. &lt;1.lambda.. The maintenance of symmetry in this 
region can be seen from an examination of the expression for the Strehl 
ratio 
EQU I.sub..delta. (0)=1-k.sup. {&lt;w.sup.2 &gt;-&lt;w&gt;.sup.2 } (3) 
where 
##EQU1## 
Substitution of Equations 4-6 into equation (3) results in 
##EQU2## 
Completing the square, equation 8 can be rewritten as . . . 
##EQU3## 
Let . . . 
EQU W.sub.020 =-W.sub.040 .+-..xi., (10) 
and Equation 9 becomes . . . 
##EQU4## 
Equation 11 shows that I.sub..delta. (0) is symmetric about W.sub.020 
=-W.sub.040. This turns out to be true in general, even when Equation 2 is 
no longer valid. The proof is shown in Appendix 2. 
The Fresnel-Kirchhoff diffraction integral was modeled to obtain axial 
intensity profiles in the presence of varying amounts of spherical 
aberration. The latter was generated by considering a plano-convex lens 
(plano side facing collimated input beam) whose pupil size controlled the 
amount of spherical aberration. This is found from . . . 
##EQU5## 
where .phi.=0.020383 
.SIGMA..sub.1 =40.54 
y=pupil radius 
A brief description of the program can be found in Appendix 3. The results 
are shown in FIG. 1a-0. These are axial intensity plots. FIG. 1a-1 shows 
profiles from .lambda./4 to .lambda.3 in .lambda./4 increments. FIGS. 1m, 
n, O are for W.sub.040 =4, 5, and 10 .lambda. respectively. (Note 
.lambda.=0.6328 .mu.m). The paraxial plane is located at W.sub.020 =0.0 
microns in allm plots. It is important to note that despite the bizarre 
appearance of some of these profiles, they are all symmetric about the 
point defined by Equation 2. This point need not be a peak as can be seen 
from FIGS. 1k, l, and m. In these simulations, the x-axis is plotted in 
terms of W.sub.020, which is related to .delta. via Equation 7. (In a 
laboratory experiment, it is .delta. which would be measured.) 
The intensities in the FIG. 1 plots are normalized to the diffraction 
limited axial intensity. However, pupil size does not affect the shape or 
relative position of the axial intensity plots for a given amount of 
spherical aberration. If we have a perfect lens of fixed diameter and 
introduced spherical aberration introduced via the input beam, the plots 
for different amounts of spherical aberration would be identical to those 
in FIG. 1. 
The analyses in the preceding sections indicate a novel means of 
determining spherical aberration in practice. An embodiment in practice of 
the invention for measuring axial intensity is illustrated in FIG. 2. In 
this arrangement, light rays from a source of collimated light are 
projected through a beam expander, such as a Tropel beam expander, through 
lens L (the lens under test), through a microscope having an objective O, 
such as a 40.times. objective, and an eyepiece, such as a 7.times. 
eyepiece, through an aperture card AC, and through aperture A defined in 
card AC via diffuser D, to photo-multiplier tube (PMT). Lens L is 
displaceable along the optical axis of the apparatus by means of a 
micrometer stage M. The output of the PMT is coupled to a radiometer and 
the detected irradiance signal therefrom is supplied to a chart recorder 
C, which also receives positional data from micrometer stage M, so as to 
render an x-y point plot of the tested lens L. 
In operation, data is taken in discrete steps. First, the lens is shifted 
by adjustment of micrometer M longitudinally through some increment. Then 
the far field pattern as seen at the aperture card in front of the 
detector (photo-multiplier tube PMT) is adjusted laterally so that it is 
concentric with the sampling aperture. Finally, the intensity through the 
sampling aperture, via diffuser D and the PMT, is measured. This lens 
position-intensity information is recorded as a point on the chart 
recorder. The process is repeated until the entire axial intensity curve 
is point-plotted. 
The amount of spherical aberration introduced in the tests can be 
determined by various sized circular apertures being placed in front of 
the lens. For example, sizes were determined from Equation 12 for 
W.sub.040 =0.5, 1, 2, 3, 4, 5, and 10 waves respectively. FIG. 3a-g shows 
the results, where the numeric scale represents an arbitrary linear scale 
of the micrometer M. Differences between these plots vs the theoretical 
patterns of FIG. 1 are due to the effective finite sampling aperture size 
(.about.4 .mu.m diameter) employed in the experiment. The axial scans in 
FIG. 1 are point intensity calculations. In addition, the data in FIG. 3 
is not locally normalized. There is real oscillation in the maximum 
intensity peak as a function of pupil radius as shown in FIG. 4. Aside 
from these caveats, the plots in FIG. 1 are fairly well represented by the 
corresponding scans in FIG. 3. (Note that the latter are plotted against 
.delta., not W.sub.020.) 
The important feature to recognize in each of the experimental scans is the 
point of "symmetry". This position is easily identified. By measuring the 
separation, .delta..sub.s, between this point and the paraxial plane, we 
obtain a measure of the amount of spherical aberration present in the 
system. (Recall that .delta..sub.s is related to W.sub.040 through 
Equation 7.) 
In FIG. 1, the paraxial plane is easily identified. It is found 
experimentally by doing an axial scan for the system stopped way down, 
i.e., small aperture. FIG. 5 shows the axial scan for the lens stopped 
down to f/45.5. Using Equation 12, the amount of spherical aberration is 
0.014. This is very insignificant. The center of symmetry in FIG. 5 is 
easily identified, and marks the experimentally determined paraxial plane. 
The f number (f/45.5) is the number obtained after mathematically dividing 
the focal length o the lens by the length of the lens aperture. The axial 
scan results yields the focal plane location along the optical axis as 
follows. The length from the the lens will have its highest intensity at 
its actual focus. The highest intensity of light observed along the 
optical axis occurs at the paraxial focus, the location of which is 
yielded by the axial scan. 
As mentioned above, the actual focus of the lens is identical to the 
paraxial focus only where there exists no spherical aberration in the 
lens. Where slight spherical aberration exists, the actual focus of the 
lens is near the paraxial focus. Although a variety of techniques are used 
in the art to locate the actual focus of a lens, a particular technique is 
as follows. First, the paraxial focus is determined by performing an axial 
scan and observing the location where light has its highest intensity 
along the optical axis of the lens. Once the paraxial focus has been 
located, the actual focus is located empirically by taking scans in 
proximity to the paraxial focus and observing if there is a point where 
the light from the lens increases in intensity above that observed in the 
paraxial focus. 
The circular apertures actually used did not quite have the calculated 
values. Table I lists their measured values along with revised F-numbers 
and recalculated W.sub.040 's via Equation 12. Table I also includes 
measured .delta..sub.s, and the determination of the amount of spherical 
from Equation 7. Note that the measured spherical is always less than the 
corresponding calculated value. The Tropel collimating lens of the 6" beam 
expander used in one embodiment was found to have -0.3 of spherical. 
However, even for the maximum aperture used in these tests, the Tropel 
would have contributed only -0.005 to out results . . . not a significant 
amount. Perhaps of greater import is the irradiance profile in the exit 
pupil. The beam exiting the Tropel has a truncated gaussian intensity 
rather than a flat or uniform profile. [Note the computer results were for 
the latter scenario.] 
TABLE 1 
______________________________________ 
Comparison of calculated vs measured spherical aberration 
Calculated 
Measured 
r (cm) f/# .delta..sub.s (mm) 
W.sub.040 (.lambda.) 
W.sub.040 (.lambda.) 
.DELTA.(W(.lambda.) 
______________________________________ 
1.38 17.78 0.75 0.62 0.47 .15 
1.62 15.14 1.02 1.18 0.88 .30 
1.85 13.26 1.46 1.99 1.64 .35 
2.05 11.97 1.90 3.00 2.62 .38 
2.20 11.15 2.14 3.97 3.40 .57 
2.37 10.35 2.63 5.35 4.85 .50 
2.77 8.86 3.70 9.98 9.31 .67 
______________________________________ 
In addition to the experimental scans presented in FIG. 3, a brief test was 
run for a circular aperture with a central obscuration. The data is 
presented in FIG. 6 for W.sub.040 =.lambda./2 respectively. (The dashed 
data represents the unobscured 3 .lambda. case.) The general shape of the 
curves is better matched at .lambda./2 than at 3.lambda.. Nevertheless, in 
both instances .delta..sub.s is larger for the obscured case. 
Thus it is shown that axial intensity scans are symmetric about the point 
defined by W.sub.020 =-W.sub.040 for unobscured circular pupils. This 
feature forms the basis of our new method of measuring spherical 
aberration. This technique provides a direct measurement of spherical 
aberration. Interferometric methods need considerably more labor to 
extract the quantity of interest, and often require the use of 
sophisticated fringe reduction techniques, and analysis codes. Thus, the 
axial scan method offers definite advantages in simplicity. 
While the present invention has been described in connection with rather 
specific embodiments thereof, it will be understood that many 
modifications and variations will be readily apparent to those of ordinary 
skill in the art and that this application is intended to cover any 
adaptation or variation thereof. Therefore, it is manifestly intended that 
this invention be only limited by the claims and the equivalents thereof. 
APPENDIX 1 
The Bessel-Fourier transform is given by . . . 
##EQU6## 
If we let .delta.=f-z, assume that .delta./f&lt;&lt;1, and express things in 
normalize pupil coordinates, when Equation A1.1 can be rewritten as . . . 
##EQU7## 
The axial amplitude can be found from . . . 
##EQU8## 
where 
EQU J.sub..theta. (0)=1 
and 
##EQU9## 
By letting X=R.sup.2, Equation A1.4 becomes . . . 
##EQU10## 
Integration yields . . . 
##EQU11## 
which can be rewritten . . . 
##EQU12## 
therefore 
##EQU13## 
The axial Fraunhoffer amplitude is given by . . . 
##EQU14## 
where B is complex. The axial intensity is found from . . . 
EQU I(0)=u.sub.F (0)u.sub.F *(0) (A1.9) 
Consequently . . . 
##EQU15## 
The axial intensity is therefore symmetric for the conditions described 
However, if the assumption .delta./f&lt;&lt;1 cannot be made, as in the case of 
very large F-numbers, .sup.1 then I(0) is no longer symmetric. 
APPENDIX 2 
The symmetry in the axial intensity about W.sub.2 =-W.sub.4 is easily 
obtained by working in Fourier space. The axial field is 
##EQU16## 
where z is the distance from the lens to the image plane and is related to 
the paraxial focal distance by f=z+.delta. where .delta.=-z(f/a).sup.2 
W.sub.2. For our work we assume that .delta.&lt;&lt;f so that z.about.f and the 
coefficient in front of the integral is independent of W.sub.2. 
Eq. (A2.1) can be cast into the form of a Fourier transform by letting 
r.sup.2 =x. Thus, 
##EQU17## 
where .omega.=W.sub.2 /.lambda., -.infin..ltoreq..omega..ltoreq.+.infin., 
c=.lambda./2W.sub.4, and the finite limits have been incorporated into the 
rect function. This equation is equivalent to the convolution 
##EQU18## 
The next step consists of manipulating the argument of the exponential. 
With a little work, Eq. (A2.3) takes the form 
##EQU19## 
Lastly, we transform to the Strehl plane, through W.sub.2 =-W.sub.4 
+.epsilon., and form the intensity, 
##EQU20## 
Eq. (A2.5) shows that the axial intensity is symmetric about the Strehl 
plane .epsilon.=0 or W.sub.2 =-W.sub.4. In passing we note that as W.sub.4 
.fwdarw.0 the exponential approaches .delta.(W.sub.2), and consequently, 
I(W.sub.2).fwdarw.[sin (.pi.W.sub.2 /.lambda.)/(.pi.W.sub.2 
/.lambda.].sup.2, as it should. Also, for small W.sub.4 the exponential is 
approximately .delta.(.xi.-(W.sub.2 +W.sub.4)) and the intensity 
approaches [sin c (W.sub.2 +W.sub.4)].sup.2. This behavior is shown in the 
figures for W.sub.4 .gtoreq.0.1.lambda.. 
APPENDIX 3 
For circularly symmetric aberrations the lens induced propageted field is 
##EQU21## 
where the second and third exponentials express the phase induced by the 
lens. By making the approximation (1/z=1/f)=.lambda./f.sup.2 for z&lt;&lt;f and 
employing the defocus shift equation .delta.=2(f/a).sup.2 W.sub.2, Eq. 
(A3.1) becomes 
##EQU22## 
where a is the aperture radius and the intensity is u u. Equation A3.2 is 
used for determining the axial intensity by setting .rho.=0. In order to 
solve Eq. A3.2, it is written in its differential form and a variable 
order Adams predictor-corrector method is applied using the Gear package. 
The intensity is normalized to the axial focal plane intensity I.sub.o, 
for =W.sub.4 =0, which is related to the aperture intensity I.sub.a by 
I.sub..phi. =(.pi.a.sup.2 /.lambda.f).sup.2 I.sub.a. While this the 
preferred mode of solving Eq. A3.2, other means known to those skilled in 
the art are equally within the scope of the present invention.