203/403, Astronomy Fall 1999
6. Lecture, 21 September 1999
6.1 Computer ray tracing
RayTrace version 5.0 (by J.R. Houck and T.L. Herter, professors of astronomy at Cornell) is a computer program that traces rays through optical systems in the manner discussed above. It is a powerful program, capable of tracing systems with very complicated surfaces, off-axis mirrors and lenses, and even dispersive elements such as gratings and prisms. Despite its power, its still easy to use for simple lenses and mirrors. RayTrace allows one to specify an optical system prescription (lenses and mirrors) and a bundle of rays originating in a point either lying a finite or an infinite distance from the first surface. Tracing the bundle through the system, it ends by showing the points at which the rays intersect a final surface, the focal plane (Figure 6.1). This plot of intersection points is called a spot diagram. Since the rays always originate in a point, a perfect optical system would have a point image, so there would be only one spot in the spot diagram. The actual spread of spots in the focal plane is most often what an optical Figure 6.1: schematic diagram of ray bundle, optical designer is concerned with minimizing, by system and focal-plane spot diagram. modification of the optical prescription. Many optical-design programs can even perform this optimization automatically, subject to whatever constraints or free parameters are chosen by the designer. RayTrace 5.0, its manual, a tutorial, and several example optical prescriptions, are provided, but the easiest way to learn to use it is by example; lets therefore demonstrate the program on some simple lenses and mirrors. Example 6.1 Enter the RayTrace prescription, and obtain the focal-plane spot diagram and position of best focus, for an equiconvex lens made of flint glass (n = 1.5) and used in vacuum, with an infinitely distant object. Take the focal length to be 10 cm, the diameter 1 cm, and the thickness between apices 0.2 cm. First we will need the radii of curvature for the surfaces of the lens. Since theyre equally curved in this case, r1 = r2 , and from the lensmakers equation we get r1 = 2(n 1) f = +10 cm .
Then we can turn on RayTrace by typing Ray <CR> at the command prompt in a DOS window. (Here, and henceforth, <CR> means to type a carriage return.) This should produce the screen shown in Figure 6.2; selection of the P option then produces the optical prescription entry screen shown in Figure 6.3. The front surface of our lens is spherical, has radius +10 cm and diameter 1 cm, and is followed by a medium of thickness 0.2 cm and index 1.5; these values are entered as follows:
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RayTrace by
J.R. Houck and T. Herter
Version 5.0; Last Update: 29-Aug-90 Options: P S F C D I L : : : : : : : Current Prescription mods (None) Save Prescription Fetch Prescription Color Selections Directory Initialize List Prescription on printer
T : Trace B : File to write trace info (None) Q : Quit
Select : Figure 6.2: RayTrace startup screen
SN: Surface No. .....
1
NA: Name .. 0.00000 0.00000 AP: Aperture ............ FT: Following Thickness . 1.00000 0.00000
RA: Radius (0 for flat) . EC: Eccentricity ........ Indices ---------------FI: Follow'g 0.00000 SI: Short ... 1.00000 LI: Long .... 1.00000 Decenters ---------------XD: X-Decntr 0.00000 YD: Y-Decntr 0.00000 ZS: Z-Shift 0.00000
Aconics ---------------FA: Fourth .. 0.0E+0000 SA: Sixth ... 0.0E+0000 EA: Eighth .. 0.0E+0000 Spec. Surface ---------------Special Conic Surf Coef'nt1 0.00000 Coef'nt2 0.00000 Coef'nt3 0.0E+0000
Tips ---------------AT: Alpha ... 0.00000 BT: Beta .... 0.00000 GT: Gamma ... 0.00000 Miscellaneous ---------------NS: Next Surface PS: Prev. Surface +1: Insert Surface -1: Delete Surface PN: Perscript Name EX: Exit Routine
SS: S1: S2: S3:
Select : Figure 6.3: RayTrace optical prescription entry screen.
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RA AP FT FI
<CR> <CR> <CR> <CR>
10 <CR> 1 <CR> 0.2 <CR> 1.5 <CR>
All of the other parameters can be left with their default values, including the eccentricity EC, for which the default value is that of a sphere. RayTrace uses the same units throughout; since we have started entering lengths in centimeters we must now continue to do so. The next surface of the lens has radius -10 cm (RayTraces sign convention is the same as ours), diameter 1 cm (still), and is followed by vacuum. In the spirit of the paraxial approximation we may expect the focus to be 10 cm from the lens, or 9.9 cm from this surface. Thus NS RA AP FT FI <CR> <CR> <CR> <CR> <CR> (to move RayTraces attention to the next surface) -10 <CR> 1 <CR> 9.9 <CR> 1 <CR>
The next surface is the focal plane. By default the following index FI of RayTraces surfaces is zero; RayTrace will trace rays until it finds a surface with FI=0, so the default surface setup will suit us just fine for the focal plane. Now type EX <CR> S <CR> T to get out of prescription-entry mode, to save the prescription (always a good idea), and to enter ray-specification mode.
This last step calls up a new menu, shown in Figure 6.4. Some of the variables listed here can be entered as numbers; some allow one to toggle through a list of options; others still are looked up from the previous data-entry steps, such as the value of BF, which is set equal to the last value of FT in the optical prescription. Now type until the Field of Object is Far Field (actually the default value); this indicates that an infinite object distance (i.e. parallel rays) are required. RP <CR> until the Ray Pattern is bullseye (a circularlysymmetric pattern of rays covering the first surface in the prescription, with the number of rays given by NR). CR <CR> until it reads List; this will produce a list of the intersections and direction cosines of the central ray in the pattern (the chief ray)before a spot diagram is drawn. PS <CR> 0.01 <CR> (actually the default value) to draw the spot diagram in a box 0.02 cm on a side, centered on the point where the chief ray intersects the focal plane. FI <CR> Were now ready to generate a spot diagram; type SD <CR>. First, the program will type a list of positions and direction cosines (, , ) for the chief ray. Note that the coordinate system in RayTrace is the same that we have been using, with the z axis pointing along the optical symmetry axis. This list will be useful in the future for debugging more complicated optical systems. Type any key to continue to the spot diagram; the result is shown in Figure 6.5.
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FI : Field of Object .............. DI DX DY SI WC : : : : :
Far Field 100.00000 0.00000 0.00000 1.00000 1
None None
Object Distance .............. X Offset (Degrees or Units) .. Y Offset (Degrees of Units) .. Starting Index ............... Wavelength Code (1, 2 or 3) .. 1 100
NP : Number of Panels ... NR : Number of Rays .....
RP : Ray Pattern ...... Circmscrbd Sqr 9.90000 0.01000
BF : Back Focal Distance .......... PS : Plot Scale (Units/100 pix.) .. CR : Central ray RX : X coord ... Ray) SD : Spot Diagram PA : Plot Again SP : System Plot SS : Start Surf ( 1) Select : No List 0.00000
RY : Y coord ...
0.00000 (Sngl
MT : Meridional Trace BE : Bend surfaces AX : X-Axis (FALSE) VH : Ver/Hor (1.00)
TS : Trace Save (FALSE) MP : Modify Presciption EX : Exit
Figure 6.4: RayTrace ray-specification menu.
Figure 6.5: spot diagram for equiconvex lens, 9.9 cm from second surface. 1999 University of Rochester 4 All rights reserved
Astronomy 203/403, Fall 1999 The spot diagram is a plot of the intersection between approximately NR (=100 at present) rays and the plane perpendicular to the z axis, a distance BF (=9.9 cm) from the apex of the last surface. Note that it appears as a series of nested circles, about 0.004 cm in diameter at the widest. Type a key to proceed. The next screen is a collection of statistics on the spot diagram just generated; it is shown in Figure 6.6. It includes the coordinates of the spot center, that of the chief ray (with its direction cosines), and the rootmean-square (RMS) width of the collection of spots, which turns out to be 0.001394 cm. This RMS width varies along the optical path, of course, and it turns out that our present position is not the sharpest focus. RayTrace, however, has already worked the distance and size of the smallest RMS diameter, and lists this under the terms Sm_RMS (0.000398 cm) and Delta_Z (0.035592 cm; further from the lens). The program can automatically move the focal plane to this new position; and will do so if you simply type 1. This brings back the ray specification menu, with the value of BF changed from the original; it is now 9.93559 cm. Now type PA <CR> (plot again). If all that was done since the last SD was a re-focus to change BF, then RayTrace does not need to recalculate anything to plot the new set of intersections between rays and focal plane. The new spot diagram is shown in Figure 6.7. Note that indeed the spot is considerably smaller than before; this new position is a better focus. It is still not a point image, though. As we will see, this remaining blur is due to spherical aberration. Hit <CR> twice to return to the ray-specification menu. Example 6.2 Now put into the lens from Example 6.1 a parallel bundle of rays inclined 0.01 radian with respect to the optical axis, and find the position and shape of the spot diagram in the same focal plane as before. In the ray-specification screen, the parameters DX and DY function as angular offsets (in degrees) from the axis for the ray bundles; let us take the tilt to be about the y axis, and enter DX <CR> 0.572958 <CR> (thats 0.01 radians, in degrees). Type SD <CR> to generate a new spot diagram, and another key to File : None Name: None X = -0.000010 Y = 0.000002
Average Spot Position: Central Ray Values: X = 0.000000 Cx = 0.000000
Y = Cy =
0.000000 0.000000
Z Cz = =
0.000000 1.000000
Spot Statistics: ( 126 rays traced in 126 RMS = 0.001394 Sm_RMS = 0.000398 RMS_X = 0.000982 Sm_RMS_X = 0.000280 RMS_Y = 0.000989 Sm_RMS_Y = 0.000283 Bull's Eye pattern traced.
tries) : Delta_Z = Delta_Zx = Delta_Zy =
0.035592 0.035718 0.035469
Do you want to refocus? 1 - To the minimum RMS 2 - To the minimum RMS_X 3 - To the minimum RMS_Y - Any character for not refocus Select : Figure 6.6: spot statistics screen.
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Figure 6.7: spot diagram for the equiconvex lens at the position of best focus, 9.93559 cm from the apex of the second surface. Compare with Figure 6.5. get the spot statistics (Figure 6.8). Three features of this solution are worth noting. First, the spot diagram is no longer circularly symmetrical, and is slightly larger than our last result. Next, the displacement of the chief ray from the optical center, winds up very close to that predicted by the paraxial estimate of the plate scale: 1.0003 mm, compared to f = 1 mm. Finally, the position of the best focus for the present incidence angle is slightly closer to the lens (Delta_Z = -0.001382 cm) than that for the on-axis rays in Example 6.1. Example 6.3 With the same lens, place an object 20 cm in front of the lens; find the focus of the on-axis point, and in the same focal plane plot spot diagrams for images of points on the object 1 cm above and below the axis. Type FI <CR> DI <CR> 20 <CR> SD <CR> until the Field of Object is Near Field; this indicates finite object distance. to set the distance to the front of the lens to 20 cm. for a spot diagram. Only one spot will appear, because the old value of BF is very far from the new focal plane, but we need only refocus: Refocus. Note that the position of the new focal plane is 19.91557 cm, close to but not quite the paraxial value.
<CR> 1 <CR>
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File : None
Name: None X = 0.100215 Y = -0.000000
Average Spot Position: Central Ray Values: X = 0.100030 Cx = 0.009933
Y = Cy =
0.000000 0.000000
Z = Cz =
0.000000 0.999951
Spot Statistics: ( 126 rays traced in 126 RMS = 0.000431 Sm_RMS = 0.000428 RMS_X = 0.000315 Sm_RMS_X = 0.000311 RMS_Y = 0.000294 Sm_RMS_Y = 0.000293 Bull's Eye pattern traced.
tries) : Delta_Z = -0.001382 Delta_Zx = -0.001827 Delta_Zy = -0.000950
Do you want to refocus? 1 - To the minimum RMS 2 - To the minimum RMS_X 3 - To the minimum RMS_Y - Any character for not refocus Select : Figure 6.8: spot diagram and spot statistics for Example 6.2.
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Astronomy 203/403, Fall 1999 Now we have the best BF for on axis points. We can plot spot diagrams and obtain statistics for all three object points requested in one ray trace, by typing DX <CR> 1 <CR> NP <CR> 5 <CR> DX and DY are lengths, measured in the current distance units, when FI is set to near field. to set the number of panels to plot. This will produce spot diagrams for points at +DX, 0 and -DX. It should be NP = 3; this represents a minor bug in the program. for a spot diagrams, and for statistics.
SD <CR> <CR>
The results are shown in Figure 6.9. The spot diagrams for the off-axis points are larger and oblong compared to central spots; this stretched-out appearance is due primarily to astigmatism. Note from the statistics that the central rays originating at x = 1 cm wind up at x = 0.996 cm the image is inverted, and the lateral magnification is very close to -1, as expected.
File : None
Name: None Panel Mode: 3 panels computed.
126 rays attempted in bull's eye pattern. DX -1.000000 0.000000 1.000000 DY NumRays Avg. X Avg. Y 0.000000 118 0.996116 -0.000000 0.000000 126 0.000005 -0.000001 0.000000 117 -0.996077 -0.000004 Z-Center 0.000000 0.000000 0.000000 RMS Delta_Z 0.002447 -0.121745 0.000544 -0.000000 0.002419 -0.120653
1 2 3
Done with Stats Press <cr> to continue Figure 6.9: finite imagery with the equiconvex lens in Example 6.3. Left to right: DX = -1, 0, 1 cm. Example 6.4 Obtain spot diagrams and the best focal position for a Cassegrain telescope with a primary mirror 20 cm in diameter 10 cm and apex curvature 60 cm, and a secondary with eccentricity 1.66667, apex curvature 18.66660 cm, and diameter 3 cm, for incident light from a distant object at angles 0, 0.1 degree, and 0.2 degree.
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Astronomy 203/403, Fall 1999 First, we need to know the focal lengths. That of the paraboloid is 30 cm, half the apex curvature radius (see Equations 3.8 and 3.9). For the hyperboloid we have Equations 3.11 and 3.12, r= b2 a ,
=
Using the first of these to eliminate a, we get r b4
2
a2 + b 2 a
.
=
whence
r b2
+ b2 b=
r
2 1
,
(6.1)
b2 r = 2 r 1 c = a a= f = ca = 1
(6.2) .
a f 2 r 1 = r 1 = 28 and 7 cm
This convex miror faces the concave paraboloid in a nesting fashion, with its nearer focal point coincident with that of the paraboloid; thus the apices are 23 cm apart. Type EX to escape back to the startup menu, I to initialize (throw away the lens prescription), and P to enter the prescription. First the primary mirror: RA <CR> -60 <CR> -- negative, because its center of curvature lies at smaller z than the surface (i.e. its concave from the point of view of the incident light); for a paraboloid; to signal reflection rather than refraction; for the next surface.
EC FI AP NS
<CR> 1 <CR> <CR> -1 <CR> <CR> 10 <CR> <CR>
RayTrace allows specification of the shadow cast by the secondary mirror on the primary; we have left the FT of this first surface zero because it is conventional to place this shadow right on the primary. It is specified as follows: SS <CR> CO <CR> S1 <CR> 3 <CR> to specify the special central obscuration surface; when a central obscuration is specified, this parameter gives its diameter; all rays inside this diameter are blocked; all rays outside this diameter are blocked; ...