SYNOPSYS™ Examples: Apochromatic objective
In this example we will design an apochromatic telescope objective. Here, the purpose is to get better color correction than can be obtained with an ordinary achromatic doublet.
Background
Since the index of refraction of a lens varies with wavelength, it follows that light from a star, which is a superposition of many different wavelengths, does not all focus at the same place. A singlet lens, therefore, will not show a sharp image of the star. Indeed, the image is so bad that it is quite unusable. Here is what the image looks like with a 4-inch singlet of glass SK16, at F/10, analyzed at the C, d, and F lines:
The blue light spreads over about 0.023 inches from the center at the focal plane, while the Airy diffraction disk has a radius of 0.00028. Pretty awful!
Early astronomers found that making the focal length very long would help, so they made telescopes 60 feet long or more. They had no alternative back then.
A much better solution today is to combine two elements of differing dispersions. The dispersion is the amount the index changes with wavelength, and with a suitable combination of two glasses, one can arrange for the red and blue light (the long and short wavelengths visible to the eye) to focus at the same place. Here is the image formed by a 4-inch doublet of glass types BK7 and F2, at F/10.0, with the intensity turned up so you can see the color flair more easily:
In this image, the blur fades out beyond about 0.002 inches – 20 times better than the singlet, but it’s still not perfect, with a pronounced violet flair. If you want to examine this lens yourself, the SYNOPSYS™ input is shown below.
RLE
ID
F/10 DOUBLET
24723
FNAME 'EXAMPLE_DOUBLET.RLE '
WA1 .6500000 .6277778 .6055555 .5833333
.5611111
WA2 .5388889 .5166666 .4944444 .4722222
.4500000
CORDER
5 1 10 2 3
4 6 7 8 9
WT1 1.00000 1.00000 1.00000 1.00000 1.00000
WT2 1.00000 1.00000 1.00000 1.00000 1.00000
APS 1
UNITS INCH
OBB
0.00 1.00000 1.99999 0.00000 0.00000
0.00000 1.99999
0 AIR
1 CV
0.0414090441986 TH 0.51999730
1 N1 1.51451489 N2 1.51525748 N3 1.51607266
N4 1.51697283 N5 1.51797314
1 N6 1.51909211 N7 1.52035274 N8 1.52178374
N9 1.52342150 N10 1.52531266
1 GTB S
'BK7 '
2 RAD
-15.2390304359937 TH 0.03937000 AIR
2 AIR
3 RAD
-15.2539282493812 TH 0.33333162
3 N1 1.61541559 N2 1.61689672 N3 1.61854876
N4 1.62040241 N5 1.62249580
3 N6 1.62487671 N7 1.62760590 N8 1.63076170
N9 1.63444694 N10 1.63879937
3 GTB S
'F2 '
4 RAD
-60.1347517495283 TH 39.51318271 AIR
4 AIR
4 TH
39.51318271
4 YMT
0.00000000
5 CV
0.0000000000000 TH 0.00000000 AIR
5 AIR
END
Copy and paste this lens file into a MACro editor (type EE to open one), run it and examine the lens listing (SPEC). Then type the artificial-intelligence sentence
PLOT BACK FOR WAVL = .4 TO .8
in the Command Window. You get a plot that shows how the back focus position varies with wavelength.
Here you see that red and blue light focus in much the same place, but green light, near the center, focuses closer to the lens than they do. Correcting this defect is the goal of the apochromatic objective.
The Apochromat
A concise description of how one can proceed is given in Rutten & van Venrooij’s book Telescope Optics. The gist of it is, one must use three different kinds of glass that satisfy certain properties. They may easily be selected by inspecting the glass-table display in SYNOPSYS™. To illustrate, we will start with a design using glass types N-SK4, N-KZFS4, and N-BALF10 from the Schott catalog. (These glasses are sometimes recommended for the purpose.) Here is the starting lens file:
RLE
ID
F10 APO
24723
FNAME 'C:\SYNOPSYS\USER\APO_START.RLE '
WAVL .6500000 .5500000 .4500000
APS 3
UNITS INCH
OBB
0.00 1.00000 2.00000
-0.01194 0.00000
0.00000 2.00000
0 AIR
1 RAD
-300.4494760791975 TH 0.58187611
1 N1 1.60978880 N2 1.61494395 N3 1.62386887
1 GTB S
'N-SK4 '
2 RAD
-7.4819193194388 TH 0.31629961 AIR
2 AIR
3 RAD
-6.8555018049530 TH 0.26355283
3 N1 1.60953772 N2 1.61628830 N3 1.62823445
3 GTB S
'N-KZFS4 '
4 RAD
5.5272935517214 TH 0.04305983 AIR
4 AIR
5 RAD
5.6098999521052 TH 0.53300999
5 N1 1.66610392 N2 1.67304720 N3 1.68543133
5 GTB S
'N-BAF10 '
6 RAD
-27.9819596092866 TH 39.24611007 AIR
6 AIR
6 CV
-0.03573731
6 UMC
-0.05000000
6 TH
39.24611007
6 YMT
0.00000000
7 RAD
-11.2104527948015 TH 0.00000000 AIR
7 AIR
END
Run this file, open the SketchPAD (PAD) and then click the
GlassTable button
. Select the Schott table from the box that opens, select Spots Only and
then Preferred, to reduce the clutter. If, like me, you prefer a black
background, click the black button. Here is the display:
This shows the Schott glass map, but it is not what we want for this exercise. Click the Graph button ...
... and then select Plot P(F,e) vs. Ve
The display changes, and now the abscissa is the V-number at the e line (0.54607 um) and the ordinate is the quantity (NF – Ne)/(NF – NC). Shift-click in the center of this display (which zooms in), and you will see the picture below.
The theory of the apochromat says that you must select three glasses that do not lie on a straight line. They must form a triangle, and the larger the area the better. The green circles show the present glasses in the triplet. They work very well – but we can do somewhat better.
Click on the green circle with the number “1” beside it. That is the glass currently on surface 1, N-SK4. Now click the Properties button, to see the properties of this glass:
Hmmm ... this glass is not all that stable: humidity rating of 3 and acid sensitivity of 5. Let’s see if we can find a better glass for the first element. (This is exposed to the environment, so it’s important.) Click the Graph button again, and then click the radio button for Acid Sensitivity and then OK. Shift-click as needed near the green circle, so things get much bigger, and then click the Full Name button.
Now you see a red vertical line through the glass locations, showing the acid sensitivity. The line for glass N-SK4 is rather long, since that glass is not very resistant. To the left you see N-BAK2, with no line at all (it’s in the best category). Click that spot, and when the glass name appears in the window on the right ...
... click the Properties button again. Aha! This glass has an acid rating of 1, better humidity tolerance, and a lower price as well. There’s no reason we can’t use it instead of the previous N-SK4, since it makes just as good a triangle with the other glasses. Type the surface number (1) into the Surface box and click “\Apply/”. Glass N-BAK2 is now assigned to surface 1.
Now clean up the display by deleting the names (click Spots Only), and then Graph, and select No Graph and OK. Our triangle is just as nice as before.
Of course the lens is not optimized for this glass, so we have to run the optimization program. Here is a MACro that will do the job.
PANT
VLIST RAD 1 2 3 4 5 7
VLIST TH 2 4
END
AANT
AEC
ACC
GSO 0 1 4 1 0 0
GSO 0 1 4 2 0 0
GSO 0 1 4 3 0 0
GNO 0 .2 3 1 .75 0
GNO 0 .1 3 1 1.0 0
GNO 0 .2 3 2 .75 0
GNO 0 .1 3 2 1.0 0
GNO 0 .2 3 3 .75 0
GNO 0 .1 3 3 1.0 0
END
SNAP
SYNO 30
Run this MACro, and now the correction is better than 1/10 wave on axis. We have a better design that is cheaper to make, more resistant to the elements, and corrected over the range 0.45 to 0.65 um. Here is the RLE file for that design:
RLE
ID
F10 APO
24723
FNAME 'C:\SYNOPSYS\USER\APO_START.RLE '
WAVL .6500000 .5500000 .4500000
APS 3
UNITS INCH
OBB
0.00 1.00000 2.00000
-0.01249 0.00000 0.00000
2.00000
0 AIR
1 RAD
-282.3206120622309 TH 0.58187611
1 N1 1.53742490 N2 1.54188880 N3 1.54960358
1 GTB S
'N-BAK2 '
2 RAD
-7.1715443994768 TH 0.33067176 AIR
2 AIR
3 RAD
-6.6987409012966 TH 0.26355283
3 N1 1.60953772 N2 1.61628830 N3 1.62823445
3 GTB S
'N-KZFS4 '
4 RAD
5.4860309728572 TH 0.03937000 AIR
4 AIR
5
RAD 5.5795739858116 TH
0.53300999
5 N1 1.66610392 N2 1.67304720 N3 1.68543133
5 GTB S
'N-BAF10 '
6 RAD
-20.4792159953376 TH 39.45248745 AIR
6 AIR
6 CV
-0.04882999
6 UMC
-0.05000000
6 TH
39.45248745
6 YMT
0.00000000
7 RAD
-11.3257437577435 TH 0.00000000 AIR
7 AIR
END
Let’s see how the focal length varies with color in the new design.
CHG
NOP
END
PLOT DELF FOR WAVL = .45 TO .65
This analysis shows a defocus of about 0.0061 inches over the design range, and a perfect Airy disk. (The latter was calculated by Image Tools (MIT) with 10 wavelengths assigned to the lens to make a good white in the center.) The defocus is not zero because the program has balanced a small shift against the change of spherical aberration with wavelength. Both are small.
The unique tools in SYNOPSYS™ make this kind of job rather easy.
