OPTICAL RAY TRACING PROGRAM. A simple program has been written which will enable the non-expert to

Transcription

1 OPTICAL RAY TRACING PROGRAM SLAC-TN E. A. Taylor July 1971 Introduction: A simple program has been written which will enable the non-expert to trace rays through a variety of geometric optical elements. The program is an extract of a very small part of the NASA program FOLDP by Firnett and Wilson. FOLDP was obtained from COSMIC, the software center which distributes NASA programs. The Document Number is TR This work in turn, was based on an earlier machine language program written by C. A. Lehman of Los Alamos. FOLDP (FORTRAN &ens Design Program) is in FORTRAN IV Language, for the computer. Extensive documentation, including a tape, is available for anyone wishing to use the complete design routine. The present program is about 150 statements in length, and is well suited for usage from the Wylbur terminal. Using WATFIV, 4 rays were traced through a system consisting of 8 surfaces in a total CPU time of 0.7 seconds. Scope: The Ray Tracing Program OPTIC, in its present form, has the following characteristics. 1. Type of rays traced: Meridional or skew, specified by x, y, and the corresponding slopes.. Types of surfaces a. Conic surfaces of revolution, centered on the z axis, such as spherical or paraboloidal, and including the transverse plane as a limiting case. b. Inclined planes (prisms) at any combination of angles to x and y axes.

2 -- C. Centered cylindrical surfaces with axes parallel to either the x axis or y axis. The cross section of the cylinder may be any of the conic sections in A (above). d. reflecting surfaces of the above types. 3. Colors: Each ray may be traced in up to 3 colors or wavelengths, corresponding to the given indices of refraction. 4. Input: a. One card to give the number of surfaces and number of colors. b. One card per surface giving type of surface, tilt of surface, separation of adjacent surfaces, indices of refraction for 3 colors, radius of curvature of the conic at the vertex, and eccentricity of the conic shape. C. One card for each ray to be traced, giving x, y, and two slopes at the starting point. 50 Output: a. Warnings: if a ray goes beyond the tangent to the surface, or is internally reflected, a warning is given and the trace terminates for that ray. b. Normal output consists of a tabulation, at each surface, of x, y, two slopes, total length to each vertex and the z from the vertex to the actual point of contact of ray and surface. C. The position of the final transverse focal plane may be adjusted to bring the first ray of the set to a focus, i.e. y = 0.0. Geometry: The optical system is referenced to a right hand Cartesian coordinate system, positive x measured into the paper, y vertical, and z positive toward

3 -3- the right. Each ray is additionally described by two slopes. The slope of the ray on input and output, is defined as tan x = d.x/dz and tan y = dy/dz. The rays are entered into the system by the four parameters: x, tan x, y, and tan y. All of the data is at a transverse plane z = 0.0. A surface is described by nine numbers which specify: TYPE = first number 1. Conic of revolution. This covers almost everything - spheres, paraboloids etc., with the transverse plane as a limiting case when the radius (8th number) = A centered cylinder with axis parallel to x axis. The cross section of the cylinder may be any of the conic sections. 3* A centered cylinder with axis parallel to y axis, as before with conic cross section permitted. 4. A tilted plane, as a prism face. Since this type requires non-zero entry in position or 3 below, the presence of an entry in either of these columns will cause a Type 1 to be changed to Type 4 in execution. 5. A transverse plane to terminate the trace. The system should end with a Type 5 or Type 6 surface. 60 An adjusted transverse focal plane. If the first ray to be traced has its initial y = 0.0, the final position of this plane will be adjusted to give y = 0.0, thus making this a focal plane. All successive rays will be measured at this focal distance. Tilt of Plane = nd and 3rd numbers The two slopes of the perpendicular to the tilted plane, tan x and tan y are defined the same way as the slopes of a ray.

4 -4- Separation of Vertices = 4th number The vertex of the conic section, or tilted plane, is defined as the x = 0.0, y = O.O,point of the surface. The separation is the distance from the previous vertex or initial plane to the present one. These distances are positive when measured to the right. After a reflection, as noted again later, the separation changes sign. Index of Refraction = 5th, 6th and 7th numbers This is the index of refraction of the medium preceding the present surface. The program will do 1,, or 3 colors as specified in the first card. If less than 3 colors are specified, dummy numbers must be used in 6th and/or 7th position. Radius of Curvature = 8th number The radius of curvature is taken at the vertex, x = 0.0, y = O.O,point. If R = 0.0, the surface is considered to be a plane. Positive radius indicates the center to the right, or convex surface. Negative radius indicates the center to the left, or concave surface. Eccentricity = 9th number This is zero except for paraboloidal, ellipsoidal or hyperboloidal surfaces. This subject is covered at the end of this note in reproductions from the original report. Reflection This is accomplished by changing the sign of the index of refraction after the reflective surface. The changed sign must be maintained for all indices thereafter unless another reflection is desired. The successive vertex separation distances must also be changed in sign to signify a ray movement reversed in direction.

5 -5- EXAMPLE: Initial Focal I LJ I I System Cards: 3 surfaces, 3 colors Type Tilt x Tilt y Vertex Sep. Color 1 Color Color 3 Rad Eccent Ray Cards: (Any number of these permitted) X Tan X Y Tan Y CL 065 A Input to this program is Format-free, with one or more spaces between successive numbers. The above problem input then is: i o o o 6 o o o o o o o

6 -6- Output: (First ray only shown) X Ray, Color = Surf Type =l 0.0 Surf Type =l 0.0 Focal Plane 0.0 Tan X Y Tan Y Z Length In this example the final vertex distance was changed from to in order to bring the first ray to a focus. All succeeding rays will use this new focal distance. Messages: The program assumes the surfaces are unlimited in size, i.e. a spherical surface is a complete sphere, regardless of how thin the lens may really be, so that the designer is obliged to examine all the lenses and apertures to make sure the numbers are consistent. If a wild ray misses the entire sphere or cylinder, etc, a message is given and the trace stops for that ray. Similarly if the angles become so oblique that internal reflection occurs, a message is given, and the trace terminates. A message is given if a designed reflection occurs. It is placed immediately ahead of the reflective surface.

7 1. : Conic -- A conic of revolution about the optic (z) axis having its surface vertex at (0, 0, 0) is described by the equation: x+ y-i+ -Rj = R t bz (1.007) where R is the radius of curvature at (0, 0, 0) and b is an ccc-entricity dependent parameter, b = b(c), whose value determines the type of, conic as follows: - b=o b=l Sphere Paraboloid O<b<l Oblate Ellipsoid - 1 <b<o Prolate Ellipsoid b>l Hyperboloid. If R>O the ccntctr of curvature is to the right of the origin and the surface is convex. If R<O the center of curvature is to the left of the origin and the surface is cnncavc. The conic of revoll;fi.on dcfil?c c! by equation (1, 007) is pro- duced by rotating the plane conic, y -t (z -R) = R+ bz, (1. OOS) about the z-axis.

8 1.. ; 1 Circle (b = 0). Se tting b = 0 in equation (1.006) yields: -T,* y t (z -l-q =R,.-:- (1.009) _.- which is a circle of radius R in the y-z plane with center at (R, 0), as illustrated in Figure 5. Y, _ Convex (R > 0) Concave (R < 0),. Figure 5. Circle (b = 0) Since E = 0 for a circle it follows that: b =c... (1.010) 1... Parabola (b = 1). Setting b = 1 in equation ( ) yields: (1.011),. which is the standard equation of a parabola in the y-z plane. (Se e,figure 6.).

9 l-n-67-7go- i O- - Convex (R > 0).- Concave (R C 0) Figure 6. Parabola (b = 1) Since E = 1 for a parabola, then: b = t. (1. 01) Ellipse (O< lb 1 < 1). The stand- crd equation of an ellipse in the y-z plane whose vcr,tex is at the origin is: : ( - a) 5+ =l al a -(l. 013). :.-... where al > 0 and a # 0. C omparing (1. 013) to ( ), it follows that: I R=- al a (1. 01-i) b= 1:, al (1.015)

10 or, = 1 -b (1.016) R al = i-1, (1.017) 4 I a >.a1 Oblate Ellipse I _ Eccentricity is given by: (1.018) so that b = Ci (1.019) where O<b<l. The oblate ellipse is illustrated in Figure 7. Convex (R> 0) Concave (R < 0), Figure 7. Oblatc Elli.poc (0 < b < 1)

11 :..~. -_--, ~-...T-R P&-p I-1-9. _..- _ _. _.. b)e,j--.l[&[ < al Prolate Ellipse _ ,. --. ]Eccent?icity is given by: _ _: -: -:.-.. e a = -- l al : where O<e<l. It folio&s~~fro& Equation (1. 015) that: (1.00).I - --, : and - - b = il. 01) -b c = -6 l 1 (1. 0) The prolate ellipse, which is specified by -1 <b< 0, is illustrated in Figure * Convex (R > 0) Concave (R < 0) Figure 8. Prolate Ellipse (-1 < b < 0)

12 -- i _.-.f, _.._. I-R-07-7~0..1,..e o-7. Pa,p-f-lG-C- : ---.m-&,- -se :4 WcrboL? - --(1) > 1 ). The standard equntion of a hypcrtioia in the y-z plane whose vertex is at the origin is:...-. ( -, a$ * y. a --= 1 al <q-s-- -- (1.03) where al >O and ai rf 0. Comparing (1. 03) and (1. 008)~ R=i -a a b = l+~ al 3. (1.04) - (1. 05). az,= -R b-l (1.06) R ai=b-l.. (1.07) The eccentricity c (E > 1) is given by: (1.08) so that b = 6. (I. 09)

13 , The two types of hyj~c.rbolas arc illustrated in Figure _ \: - \. \.-. -Y\ \ -F-, / Y \.. _. : \. Convex (R > 0) Concave (R < 0) Figure 9. Hyperbola (b > 1) In addition to the conic of revolution, defined by Equation (1. 007j, the program will accept the two types of conic cylinders described belo\\!. * The horizontal conic cylinder (axis of cylinder parallel to x axis) is defined by: Y+ ( z - R). = R t bz. (1.030) y axis) is defined by:.. The vertical conic cylinder (axis of cylinder parallel to : x t (z - R) = R t bz P. -.(1031).