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FINAL REPORT .' 

CORNEA OPTICAL TOPOGRAPHICAL 

SCAN SYSTEM (COTSS) 

CONTRACI' -- NAS8-34659 

THIS REPORT WAS PREPARED 

FOR 

"HE TECHNOLOGY UTILIZATION OFFICE 

AT0 1 

SUBMITTED TO: 

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 

MARSHALL SPACE FLIGHT c]GNTER 

MARSHALL SPACE FLIGHT CENTER, AL 358 12 

SUBMITTH) BY: 

11 AUGUST 1986 
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FINAL REPORT 

I CORNEA OPTICAL TOPOGRAPHICAL 

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SCAN SYSTEM (COTSS) 

mN'IRACI' -- NASS-34659 

THIS REPORT WAS PREPARED 

FOR 

THE TECHNOLOGY UTILIZATION OFFICE 

AT0 1 

SUBMITTED TO: 

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 

MARSHALL SPACE FLIGHT CENTER 

MARSHALL SPACE FLIGHT CENTEjR, AL 358 12 

SUBMITTED BY 

11 AUGUST 1986 
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The Cornea Optical Topographical Scan System is an instrument 

designed for use by opthalmologist to aid in performu surgical 

procedures such as radial keratotomy and to provide qui& accurate data 

to aid in prescribing contact lens and eyalasses. The Cornea Optical 

Topographical Scan System (COTS9 program was awarded to 13lectro-Optics 

Consultants (EOC) in October of 1983. A breadboard of the system was 

built and demonstrated in June of 1984. Additional refinements to the 

breadboard are needed to meet system requirements prior to proceeding 

with prototype development. This report gives the present status al the 

COTSS instrument 

required. 

and defines the areas in which system refinements are 

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TABLE OF C0"TS 

TITLE PAGE 

ABSTRACT.. ......................................... i 

LIST OF FIGURE .................................... iv 

BACKGROUND ........................................ 1 

INTRODUmION. ............................... i 

0BJ"IVES AND REQUIREMENTS.. 1 

PROGRAM HISTORY. ............................ 3 

............... 

COTS FUNCTIONAL DWXIPTION ...................... 4 

COTS OPERATIONAL PRINCIPAL ................ 4 

OPTICS ........................................ 7 

ELXIXONICS AND COMPUTER. ................... 7 

DEMONSTRATION BREADBOARD DESCRIPTION ........... 1 0 

COLLECTING OPTICS ............................ 10 

SCANNING ..................................... 10 

SIGNAL-TO-NOISE .............................. 10 

INITIAL BIDBOAFtD IMPROVEMENTS ........... 11 

RESULTS AND REa)MMENDATIONS. .................... 12 

ALIGNMENT SENSITIVITY ....................... 12 

ANALOG SIGNAL PROCESSING .................... 13 

ELECI'RONICALLY CONTROLLED MIRROR ........... 13 

DATA ACQUISITION ............................. 14 

DATA COLLECTED.. .............................. 14 

SYSTEM STUDIIis ..................................... 16 

COLLECTING OPTICS DESIGN AND 

SENSIVITY ANALYSIS ........................... 16 

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BEAM ALIGNMEN". ............................. .21 

BEAM STEERING.. .............................. 24 

TWO DIMENSIONAL SCANNER. ................... 26 

CONCLUSIONS. ....................................... 29 

APPENDIXA ......................................... 30 

RAY TRACE ANALYSIS.. ......................... 30 

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LISTS OF FIGURES 

FIGURE NO- TITLE PAGE 

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COTSS OBWIVE & REQUIREMENTS 

OPTICS PHOTO 

COMPUTER PHOTO 

COTSS SYSTEM DESCRIPTION 

COTSS BREADBOARD OPTICS 

COTSS BREADBOARD ELEI'RONICS 

ANDCOMPUTER 

SPATIAL DATA FROM SCANNING 

TEST LENS SURFACE 

COTSS COLLECTOR LENS 

SENSITIVITY ANALYSIS RESULTS 

TABLE 1 & TABLE 2 

QUAD DETJXTOR FOR BEAM STEERING 

BEAM STEERING SYSTEM 

DUAL GALVO SCANNER 

COTSS BREADBOARD 

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BACKGROUND 

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New surgical techniques for the treatment of human eye cornea 

disorders depends on accurate measurements of the shape of the outer 

surface of the cornea Radial keratotomy in which 8 to 16 radial cuts are 

made in the outer layer of the cornea to cause the surface to flatten and 

reduce myopia is an example. A very accurate mapping of the cornea is 

needed to establish the pattern 0s the cuts and to determine if the cornea is 

healing properly. In cornea transplants, the surgeon needs to have a quick 

measurement system to determine if the sutures are uniformly stretching 

the new cornea Damage from trauma such as cuts, burns, and punctures 

also require quick diagnosis and an accurate measurement of the extent of 

damage to the eye. The fitting of contact lenses can also benefit from 

faster measurement techniques with greater accuracy. Instruments now in 

use by Opthalmic surgeons will not provide the required accuracy or the 

high speed. Thus a need exists for an instrument that can provide the 

required accuracy and rapid display of the information thru real time data 

collection and data processing. 

A breadboard of such an instrument has been developed and 

demonstrated. All technical requirements have not been met at this point. 

The following is a review of the development of the breadboard instrument 

and of its status. 

The objective of the COTSS prosram is to demonstrate that an 

instrument with the capabilities described above can be built to meet the 

requirements of the medical community and still be practical for low cost, 

mass production. The more succinct technical requirements of the 

instrument are given in Figure 1. Also as indicated in Figure 1, it is 

plaMed to market the instrument for under $30,000. 

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Electro-Optics Consultants was awarded a contract in October of 1983 

to build and demonstrate a breadboard for the Cornea Optical 

Topographical Scan system (COTS). A final design of the COTS was 

agreed upon by NASA and EOC in November 1983. By March of 1984 the 

breadboard components had been procured and assembled. By June 1984 

the breadboard had been "debugged and system control software written 

for the computer. The breadboard was demonstrated at this time and the 

initial contract funding period ended. At this time the breadboard was not 

able to demonstrate that the system could meet all of the original technical 

requirements. During the latter part of 1984 and several months of 1985 

M>c performed paper studies of the system, resulm in recommendations 

for improvements and modifications that will be required for the 

breadboard to meet the original technical requirements. Funds have not 

been available for continuation of the project since fall of 1985. 

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COTS FUNCTIONAL DESCRIPTION 

SS OPERATION PRINCIPAL 

The COTS instrument consists d three primary components: (1) an 

optics package ammining a HeNe laser, modulator, scanner, lenses, mirrors, 

detectors, etc. (See Figure 21, (2) an electronics package that interfaces the 

optics package to the computer system, and (3) the computer system (See 

Figure 3) which controls the optics, acquires the data from the electronics, 

analyzes the data and displays it in a user oriented format. The 

interrelation of these components is shown schematically in Figure 4. 

The instrument in its final configuration will contain the same 

components. The optics will be packaged into the portion of the instrument 

that is placed in front oP the eye to be measured. The operator will give 

the instrument the command to collect and display data. The data 

collection time will be very short -- on the order of 1 second or less. 

Within a few seconds the topographical information’will be displayed on 

the CRT. At this point it is envisioned and the operator can choose several 

options, e.g. 

Rotate the surface to view the topography from any angle desired 

Compute cornea contours and aid in prescribing eyeglasses or 

contact lens and give an estimate of the degree of correction 

these perscriptiom will provide. 

0 Potentially provide data that can be used to determine aprikri 

the results of a proposed surgical procedure as it relates to 

topography change and in turn to vision changes. 

The system will have a special purpose computer with simplified 

controls to allow calibration, operation, etc. by a technician with minimum 

training. 

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ORiGliaAL PAGE iS 

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I FIGURE 2: OPTICS PHOTO 

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OPTICS 

The optical system for COTSS is a real time phase interferometer 

(RTPI) concept (See Figure 5). A helium neon laser beam is split into an 

adjustable path reference beam and a measurement beam. The 40 MHZ 

bragg cell modulation in the measurement beam provides a reference to 

determine slope change on the cofnea. The beam is scanned across the 

cornea us@ a fast lens that matche3 the radius of curvature of the cornea 

surface. The two beams are recombined on the detector to provide data on 

the optical phase variation af the eye surface. Most RTPI systems are used 

to measure fractions of a .wavelength. This system uses the doppler 

frequency shift to obtain a larger range and still provide resolution on the 

order of 1 micron 

A Motorola 68000 based microprocessor system is used to control the 

hardware and recard the mea surface data (See Fbure 6). A Bulova scan 

mirror system provides a reference to the computer to obtain the spot 

location on the eye. The signal from the photodetector is amplified and 

limited to reduce amplitude variations. A phase lock loop provides the 

doppler frequency to the computer which intergrates the signal to obtain 

the surface heights. The spatial data is displayed as contour plots. 

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DEMONSTRATION BREADBOARD DESCRIPTION 

Upon initial assembly and checkout of the breadboard, a series of 

problem areas were discovered. The following is a description of the 

problem areas followed by the steps that were taken to correct the 

problems prior to demonstraw the CO'I'SS breadboard. 

LIjCfING OPTIQ 

Early in the program it was recognized that collection of light reflected 

from the cornea would present some design problems because of the wide 

angle over which the light is reflected. In the time available a collection 

optic system was designed using graphic and simple analytic ray trace 

techniques. It was determined after experimentation that the collection 

optics design was not adequate to maintain reflected beam alignment 

Scam@ of the cornea was to be accomplished by the combination of 

a resonant mirrot and image rotator. The image rotator mechanism could 

not maintain alignment of the beam reflected from the simulated cornea to 

the accuracy required. Thus it was not possible to do two axis scanning 

with the mechanism as initially configured because of signal drop out. 

SIGNAL-TO-NOIS 

Experimentation revealed that the signal to noise ratio of the 

hetrodyne beat signal was marginal and this problem combined with 

misalignments of the return beam caused appreciable drop in the signal 

amplitude and thus limited the accuracy of-the instrument. 

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INIT IAL BREADBOARD IM PROVEMENTS 

At the time the above problems were identified funds were not 

available for new parts. Thus breadboard modifications were implemented 

using NASA in house parts. The breadboard modifications implemented 

were: use of a lower noise preamp for the hetrodyne detector, use of a 

bandpass amplifier for further signal to noise ratio improvement and 

incorporation of a phase lock loop detector for improved detection 

threshold. The phase lock loop output voltage was digitized by an AID 

Converter board in the process control computer. Beam position data 

(position on optic surface) were derived from digitizing the drive voltage to 

the scanning mirror. A standard camera lens was used for the collection 

optics for use with the simple lens being measured. The optical paths were 

reconfigured to improve the alignment of the return beam with the 

reference beam. 

With the improved breadboard a me dimensional scan of a lens with 

less curvaturse than the cornea was performed to demonstrate 

fundamental data collection and processing. 

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Successfully collecting data with the COTSS system is dependent on 

proper optical alignment. The current optics layout is shown in figure 5. 

The lens used to simulate the -ea (test lens) is a simple biconvex of 2" 

focal length. The front surface of this lens is measured by COTSS along a 

line of approximately 1/4" in length across the front surface. This 

approximates measurement of the cornea surface. The camera lens serving 

as the collecting optic is positioned sg that the center of curvaturse of the 

front surface and a point on the scan mirror ala its axis af rotation are 

conjugates. Thew conditions ensure that the reflection from the surface 

will return along the path of the incident beam and produce a stationary 

spot on the detector. Deviations from this condition cause the spot to move 

across the face of the detector. This condition is also required for 

maintaining a nearly colinear path for the return beam and the reference 

beam. This is necessary for obtaining a strong beat signal from the 

hetrodyne process. (The signal to noise ratio is approximately 20 to 1 at 

best a@nmenL) If the test lens is moved from the position of best 

alignment, the signal to noise ratio drops over portions of the the scan. 

This is caused by misalignment of the return beam with the reference as 

well as by the spot moving off the active area of the diode. The test lens 

can be moved about 1/32" to the side of center before the signal drops 

below threshold. It can be moved about l/8" along the optical axis without 

decreasing the signal to noise before the signal drops below threshold. 

The cornea is an aspherical surface, therefore there will be some 

misalignment of the return beam. The misalignment due to the aspherical 

surface have an effect similar to translatimz the test lens as mentioned 

above. It should be possible to compensate for most of this misalignment 

by using a new collection lens design. There will be small pertubations 

that remain as a result of individual cornea curvature differences but these 

should remain small enoush to allow proper operation of the instrument. 

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With the present knowledge of the alignment sensitivities discussed 

above, it appears that a new collection lens design is a key improvement to 

the COTSS instrument. 

Additional techniques exist which can be implemented to allow the 

system to operate properly with larger misalignments. These are: 

improved andog signal process@, and the use of an electrically controlled 

mirror in the reference beam or return beam to maintain alignment during 

scan 

ANALOG SIGNAL -SING 

Analog signal processing can be used to improve the signal to noise 

ratio. This is accomplished by demeasing the bandwidth of the amplifier 

foIlowhg the detector. The present amplifier has a much larger bandwidth 

than necessary. This is to accommodate the full range of frequencies 

produced by the doppler shift as the beam scans. The signal to noise can 

be improved by using a tunable narrow band filter which tracks the 

doppler frequency. 

The return beam can be held on the hetrodene detector by using a set 

of beam steering galvonometers inconjunction with a quad detector error 

sensor as shown in Fuure 1 1 and 12. This is discussed in the section of 

Beam Steering. 

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DATA ACOUISITION 

Another area that needed improvement is the data acquisition. The 

original technique used a phase cornparitor and counted cycles. The low 

signal to noise and signal dropout due to misalignment would not allow the 

counter technique to work. A phase lock loop technique was incorporated 

which provide a better resistant to signal dropout. The phase lock loop 

requires a high speed A/D to pass the information to the computer. The 

present AID needs to be increased in rate to a 500 KHZ rate to meet the 

requirements of the phase lock loop technique. If the signal to noise is 

improved with the new optics and electronics the phase counting 

technique will be reinstated. If not a faster A/D will be incorporated. 

DATA COLLECTED 

Figure 7 is typical of the results obtained with the 1-D scan. The 

horizontal axis represents the scan across the test surface and the vertical 

axis represents the relative height of the test surface during the scan. The 

different curves were obtained by shifting the centerline of the test surface 

off the optical axis. 

There is some lack of repeatability in the data. This results from 

dropouts of the phase lock loop at points of low signal to noise ratio. This 

has been reduced to some extent by careful alignment of the optical 

system. The sample rate of the AID Converter is presently limiting the 

data to 12 samples per scan. This will be increased to 500 samples 

minimum per scan to allow for data smoothing to achieve 100 processed 

samples. A properly designed collection optics will improve the signal to 

noise ratio by improving beam alignment. These improvements coupled 

with improvement in the analog signal processing electronics will reduce 

the dropouts to a level required to achieve specified measurement 

accuracy. 

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ORIGINAL PAGE IS 

OF POOR QUALITY 

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FIGURE 7: COTSS SPATIAL DATA FROM 

SCANNING TEST LENS SURFACE 

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SYsfEMsfUDIEs 

A number of areas have been discussed for improvement of the COTSS 

breadboard to get it to the point that it can meet the technical 

specifications spelled out previously. However, the following areas are 

considered to be the most crucial and mer most of the areas discussed 

previously. 

Collecting Optics Design 

e BeamAwment 

e Beamsteering 

e Two Dimensional Scanning 

COLLWING OPTICS DES1 GN AND SENSITIVITY ANALYSIS 

A small special purpose ray trace program was written. This program 

was written in a general manner in a three dimensional, right handed, 

rectangular amdinate system. (SEE APPENDIX FOR ANALYSIS) Rays are 

specified by a point and a direction vector. Optical surfaces are specified as 

a sphere of given radius and center. The program calculates the 

intersection point of the ray and spherical surface and then uses the law of 

refraction to determine the direction vector for the refracted ray. 

Management of the analysis is accomplished through use of Fortran 

subroutines called by a mainline program. The basic analysis is 

incorporated into 'SUBROUTINE OPTIC. A second subroutine is written to 

accept the apecificationa for a lens element and use the analyeio routine. 

This routine assumes the lens is located on and normal to the 2 axis of the 

coordinate system. This subroutine is called "SUBROUTINE LENS". A third 

subroutine is written to find the minimum thickness of a lens from its 

height and two radius of curvature. This subroutine is called "SUBROUTINE 

THIW'. The mainline program is written to prompt the user to enter the 

element number and parameters or analyze. The analysis traces 10 rays 

through the system and determines the point of intersection of each ray 

with the 2 axis of the system. 

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The program accepts a distance in front of the lens from which all rays 

originate and generates 10 rays to cover the aperature of the first element. 

After tracing each ray through the lens system it prints the angle of each 

incident ray off the axis and the distance from the center of the last 

element to the intersection of the exit ray with the 2 axis. The lens is 

evaluated in terms of longitudinal spherical aberration (L.S.A.) and angle 

error. The L. S. A. is expressed as a percentage variation of each ray focus 

from the fist (low angle of paraxial) ray focus. The angle error is the 

angle that each ray makes with the normal to a sphere centered on the 

paraxial focus. 

The lens design criteria for this application are: 1) the angle error 

must be as low as possible, 2) it must collect light Over a very wide angle. 

The design approach was to cascade aplanats to produce a virtual image as 

far in front of the first element as possible and use this in conjunction with 

a focusing lens of very low L. S. A. The aplanats can be 

calculated theoretically and the virtual image can be moved to any distance 

required by adding more elements. The results obtained from the ray 

trace program indicate that the system becomes less tolerant to parameter 

variations as additional elements are added. 

It Was found that the most practical design consisted of 5 aplanats and 

a triplet focusing lens. Also it was possible to reduce the system aberration 

to below that of the triple1 by changing the front elements slightly from 

true aplanats. The resulting 8 element lens system has a L. S. A. of less 

than .009% and an angle error of less than .004 degrees. The lens design is 

shown on the following page in Figure 8. 

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FRONT FOCUS L. 5. A. 

FIGURE 8: COTS COLLECTOR LENS 

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To perform sensivity analysis the program was modified by adding an 

option to trace the rays through the lens system and then back after 

reflection from a spherical surface. Two subroutines "OPTIC B" and "LENS 

B" were written to accomplish the reflection and retracing.' When the 

retrace option is in effect the angle error is computed as the difference 

angle of the return ray from the incident ray. The analysis showed that 

the angle error would be less than .005 degrees for a range of radius of 

curvature from 5mm to 16mm as long as the lens system is focused on the 

center of curvature. This amount of error results in a fringe cycle length of 

7.2mm so that with the 2mm detector the signal loss resulting from 

misalignment will be negligable. 

For a nonspherical surface such as the cornea, different portions of the 

surface can be approximated by spherical surfaces of different radius. The 

sensitivity analysis is performed by selecting the mean radius as the 

locator for the focus of the optical system and then evaluating the 

performance of surfaces with a range of curvatures where the surfaces 

coincide along the axis. A range of curvatures from 11.6mm to 13.3mm 

were analyzed with the lens focus centered in a sphere of 12.Smm radius. 

The resulting analysis is presented in Figure 9. 

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MAXIMUM ANGLE ERROR (DEGREES) 

FIGURE 9. SENSITIVITY ANALYSIS RESULTS 

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BEAM ALIGNMENT 

Ray trace analysis indicates that the return beam from the cornea will 

have to be steared with an aurilary mechanism in order to keep it aligned 

with the reference beam. This alignment is necessary in order to maintain 

a strong hetrodyne signal so that the signal-to-noise ratio can be high 

enough for accurate measurement at the desired scanning speed. A further 

analysis indicates that a simple bi-amvex lens can be used to advantage as 

part of the beam stearing system. This analysis is as follows. The ideal 

condition is for the return beam to retrace along the incident beam and 

return to the point of origin on the first of the two scanning gafvo mirror3 

where it will exit alw the same path as the beam entering the scanner 

regardless of the position in the scan. The aspheric curvature of the cornea 

causes a deviation from the incident beam as indicated in Figure 10. 

FIGURE 10 

The ray trace program was used to determine 0, and A as a function 

of 0, for worst case mnea.curvature. The distance R was then calculated 

using the formula: 

The galvo scanner is located at the cordinate system origin and the 

beam will exit along the X axis if it returns to the origin. R represents the 

distance to an apparent point from which the divergent path appears to 

orginate. The data resulting from this analysis is tabulated in Table 1. 

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The distance R was used as an on axis source point for the ray trace 

program with a simple bi-convex lens. A nominal focal plane was 

established and the height off axis of the lens output determined for each 

ray defined by and R from Table 1. The result is Table 2 for lens of 1.52 

index and both radius of 15Omm. Table 2 indicated that there is a 

maximum of approximately .2mm separation of beams at the focal plane. 

This separation allows the use of small mirror galvos for the beam steering 

and assures that the parallel beams exiting the beam steerer will stay 

within the 2mm square hetrodyne detector. 

The output of the beam steerer goes to a parabolic collimator which 

has a quadrature detector at its focal point. This detector provides the 

error signal for a servo loop that drives the beam stearer to hold the beam 

at the focal point of the arabola. This condition assures that the beams exit 

the beam stearer parallel. 

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I TABLE 1 I ................................................................................. 

................................................................................... A R 

0.5214 I 0.03441 f 97.267 f 1029.34 

1.0429 f 0.11076 I 98.059 i 1021.3 

98.988 f 1003.57 

983.45 

1.5643 i 0.17115 i 

2.0857 ! 0.23788 ! 100.71 f 

2.6072 i 0.31379 i 102.86 i 957.08 

3.1286 i 0.40246 i 105.71 i 926.88 

109.47 893.92 

858.54 

3.65 f 0.50885 i 

4.1715 i 0.64019 i 114.36 i 

4.6929 f 0.80899 i 121.32 I 823.85 

5.2143 ! 1.0378 ! 131.16 ! 788.63 

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TABLE2 f ............................................................... 

............................................................... R H 

I 

............................................................... 1.0398 ! 788.63 i 0.1 

0.80899 i 823.85 i 0.027 

0.50885 ! 893.92 i -0.0745 

0.40246 926.88 i -0.097 

0.31379 i 957.08 i -0.104 

0.23788 f 983.45 i -0.098 

0.17115 f 1003.57 i -0.08 

............................................................... 

............................................................... 

............................................................... 0.64019 i 858.54 f -0.032 

............................................................... 

............................................................... 

............................................................... 

............................................................... 

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BEAM STEERIN G 

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The previous sensitivity analysis shows a worst case angle error of 1.3 

degrees for the returning beam. It will be necessary to compensate for 

this error by use of galvo driven or pezoelectric driven mirrors in the 

return beam so that it will be on the detector and aligned with the 

reference beam. The mirrors can be controlled by an analog servo loop 

using a 4 quadrant detector as an error sensor. This is illustrated below in 

Figure 11. The loop will ensure that the beam centroid remains centered 

on the detector. 

Two servos are necessary because both angle and displacement must 

be corrected for . The first servo returns the beam to a fixer poiint and the 

second maintains a fixed direction. The beam stearing system is illustrated 

below in Figure 12. 

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XY 

MIRROR / I RETURN BEAM ' 

S I -u 

QUAD 

DETECTOR 

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QUAD 

DETECTOR 

FIGURE 12: BEAM STEERING SYSTEM 

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TWO DIMENSIONAL SCANNER 

The technique choosen is to use two galvanometer scanners in the 

arrangement of Figure 13. This seems to be the best available with off the 

- shelf hardware. This design which has been choosen for the proof of 

principle breadboard will be limited to a 1 second frame scan time. It is 

limited by the use of the large galvo mirror for the fast axis scan The 

small galvos which are used to correct the angle of the return beam have a 

.6ms step response and as such are adequate for a Ims line scan time. If a 

suitable scanner were to replace the fast axis galvo, the frame time could 

be reduced to .I second. A possibility for future develoment is to use 

piezoelectric driven mirrors to replace the small galvos which correct the 

angle of the return beam. These are capable of 1 Oms. Also with -1 ms line 

time the polygon wheel could be used to obtain a .1 sec frame time. The 

use of .ms line time would require a redesign of the analog electronics to 

accommodate the increased signal bandwidth and an increase in laser 

power to maintain signal to noise ratio. 

The proposed design for the 2-D scanning breadboard which will be used to 

demonstrate data collection from a surface typical of the cornea is shown in 

Figure 14. 

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CONCLUSIONS 

The COTSS has been demonstrated for a simple, one-dimensional 

scan. These results are documented in this report. Additional studies 

of the system have revealed techniques and modifications that must be 

implemented before the COTSS system can achieve all of the technical 

objectives that have been set forth. Based on these studies, the COTSS 

system still seems to be a viable system to meet the needs of the 

ophthamolic surgeon. 

It is recommended that the required hardware be procured and the 

CO'rSS breadboard upgraded to demonstrate the two-dimensional scan 

capability. A list of hardware has been compiled that will be needed 

for these modifications. Also, shown in Figure 14 is a schematic layout 

of the breadboard with the modifications discussed in this report. 

29 
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APPENDIX A 

RAY TRACE ANALYSIS 

The following is a description of a computer algorithm which is used to 

analyze optical systems consisting d spherical surfaces. All vectors are 

referenced to the right hand Cartesian coordinate system show. Y 

A ray is described by the vector equation 

a a a..L R=P+SA 

where: d is position vector of any point along the ray.‘ B 

d 

P is a constant specifying a given position on the ray such as the 

intersection of the ray with an optical surface. 

ob 

A is a unit vector in the direction of progress of the ray. 

d 

S is the scalar independent variable 

An optical surface-performs the function of transforming an incident 

ray into a departing ray in accordance with the law of reflection or 

refraction. This means that new vectors and x are established at each 

surface. The vector F is evaluated by solving the equation for the 

intersection of the incident ray with a sphere. The vector A is computed 

by using the law of 

reflection or refraction as the case may be. Thus, the vector constants 

of the ray equation are evaluated at teach encounter of a ray with an 

optical surface. This constitutes the algorithm. P (I) and A (I) are arrays in 

the Fortran program which are recomputed by SUBROUTINE OPTIC for each 

surface encountered. 

The solution for the intersection of a ray with a spherical surface is 

obtained as follows. 

d 

Q specifies the center of the sphere 

R is the radius of the sphere 

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First the equation for the ray is transformed to a parallel coordinate 

system having its origin at the center of the sphere. 

The equation for the sphere is: 

Xz+ Y2+ 2'= R2 

Expansion of the ray equation is: 

x = x,+ sxq 

Y = Yp+ SI& 

2= z,+Sz, 

so: 

a tr (Xt+ Sq) + (Yp+ S&f + (2p+ Sv = R 

or: 

This expression is solved for S using the quadratic equation. The variable S 

is then substituted into the ray equation to find X, Y, and 2. 

Since the vector 5 = J + + & is referenced to a coordinate system 

whose origin is the center on the sphere, it is in the direction of the 

outward normal to .the surface. Thus, the surface normal is: 

Finally, the position vector is transformed back to the original 

coordinate system by the equation: 

& at& 

P=P +Q 

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The direction of the departing ray is found using the law of reflection or 

refraction. These laws are illustrated below. 

- / 

.... *.% ......... 

.............. .~ ......... 

... 

. ..................... __ - 

.............. ........ - __ .................... .. - 

-. ... ... _____ '34- - .... . - ....... - . ........... --- 

.... . .. ...... - -. ... .- .~ 

- ......... .~ .. ~ . ~ .. - .. - . .- -_ -. . 

. 

....... -. ... 

p W=RA c7-lo N - - -- - -- 

4 

In the refraction diagram above 0 is the surface normal unit vector. A,is 

the incident ray direction vector and x1 is the departing ray direction 

vector. Both-$ukd &are unit vectors. 

z 

M is a unit vector perpendicular to the incident ray. The angle "S" is the 

difference between the angle of departure and the angle of incidence. 

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From the diagram: 

-L 

u =Xi sin - A, cos # 

so: = cb +A, cos#)/sin Q 

Then the departing ray direction vector is given by: 

OT: 

The angle functions appearing in this equation are evaluated as foltows. 

From snells law: 

SCX 

and: 

The equations which have been presented have been incorporated into the 

Fortran subroutine OPTIC. 

33