FINAL REPORT AUTOMATIC FOCUSING TECHNIQUES

• Document Type: CREST
• Collection: NGA Records (Formerly NIMA)
• Document Number (FOIA) /ESDN (CREST): CIA-RDP78B04770A002000030020-6
• Release Decision: RIPPUB
• Original Classification: K
• Document Page Count: 98
• Document Creation Date: December 28, 2016
• Document Release Date: December 9, 2002
• Sequence Number: 20
• Publication Date: August 1, 1966
• Content Type: REPORT

STAT
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Final Report
AUTOMATIC FOCUSING TECHNIQUES
Declass Review by
NIMA/DOD
 [ , `( ,-e 1- c t:,.-
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t
August 1966
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Final Report
AUTOMATIC FOCUSING TECHNIQUES
Copy No..........
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Various techniques for automatic focus detection and correction are
reviewed for application to a rear-projection viewer operating in a
scanning mode. The technique selected for detailed theoretical and ex-
perimental study automatically detects the sharpness of focus of an
externally generated pattern that is projected onto the film and reimaged
externally by the same optical projection system whose focus is being
controlled.
Experimental results are described in the first section, which
demonstrate the soundness of the technique over a practical range of
parameter values. A theoretical analysis in Sec. II gives a clear basis
for understanding the form of the experimental curves in Sec. I.
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ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . i i i
LIST OF ILLUSTRATIONS . . . . . . . . . . . . . . . . . . . . vii
SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . ix
I AUTOMATIC FOCUSING SYSTEM . . . . . . . . . . . . . . . . 1
A. Requirements . . . . . . . . . . . . . . . . . . . . 1
B. Focus-Detection Methods . . . . . . . . . . . . . . 5
1. Film-Plane Position or Lens-Position
Detection and Control . . . . . . . . . . . . . 5
2. Image Sharpness Detection. . . . . . . . . . . 8
C. Externally Generated Image for Focus Detection. . . 9
1. Methods of Detecting Sharp Focus of
Reflected Image of Mask. . . . . . . . . . . . 11
2. Method for Sensing Direction of Defocus. . . . 14
D. Experimental Results . . . . . . . . . . . . . . 15
E. Analysis of System Accuracy . . . . . . . . . . . . 30
F. Experimental Focus-Detection System . . . . . . . . 38
G. Conclusions . . . . . . . . . . . . . . . . . . 39
II THEORETICAL ANALYSIS . . . . . . . . . . . . . . . . . . 41
A. Single Defocus . . . . . . . . . . . . . . . . . . . 41
1. Basic Problems of Analysis . . . . . . . . . . 41
2. Geometric Form for Analysis--Disk Object . . . 45
3. Preview of Analytical Steps. . . . . . . . . . 46
4. Specular Light Loss--Point Source. . . . . . . 47
5. Disk Object--Total Light Return. . . . . . . . 55
6. Defining the Critical Values ip+ and Aq-c. . . 56
7. Defining the Critical Values ipc and AgMc. 61
8. Intercepted Light Over Range
0 AP !5: Apc . . . . . . . . . . . . . . . . . 63
9. Intercepted Light Over Range
0 AP ~pc . . . . . . . . . . . . . . . . . 65
B. Double Defocus . . . . . . . . . . . . . . . . . . . 67
APPENDIX A--DERIVATION OF FOCUS ERROR EQUATION. . . . . . . . A-1
APPENDIX B--CONVOLUTION OF CIRCLES OF DIFFERENT DIAMETERS . . B-1
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APPENDIX C--PROOF OF NO TRUNCATION LOSS IN RANGE
L\p - .p' . . . . . . . . . . . . . . . . . . C-1
APPENDIX D--PROOF OF NO TRUNCATION LOSS IN THE RANGE
D-1
~) - AP gyp
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Fig. I-1 Film-Plane Detection. . . . . . . . . . . 6
Fig. 1-2 Basic Layout for Focus Detection with
Externally Generated Image. . . . . . . . . . 10
Fig. 1-3 Method for Detecting Defocused Light
from Front of Mask . . . . . . . . . . . . . . 12
Fig. 1-4 Use of Beamsplitter and Negative Mask to
Detect Defocus . . . . . . . . . . . . . . . . 13
Fig. 1-5 Laboratory Setup Used for Experimental
Focus-Detection Studies . . . . . . . . . . . 15
Fig. 1-6 Basic Response Curve for m = 3. . . . . . . . 17
Fig. 1-7 Returned-Light Path with Mask Off Axis. . . 20
Fig. 1-8 Response with Mask Off Axis . . . . . . . . . 21
Fig. 1-9 Returned-Light Paths for Angular Film
Displacement . . . . . . . . . . . . . . . 22
Fig. 1-10 Effect of Angular Displacement of Film
Surface . . . . . . . . . . . . . . . . . 23
Fig. I-11 Restoration by Off-Axis Mask Placement
of Light Lost Because of Angular Film
Displacement . . . . . . . . . . . . . . . . . 24
Fig. 1-12 Compensation for Angular Film Displacement
by Off-Axis Mask Placement. . . . . . . . . . 25
Fig. 1-13 Return from Film Compared to Specular Return
from Plane Mirror . . . . . . . . . . . . 26
Fig. 1-14 Basic Response Curve for m = 1.0. . . . . . . 27
Fig. 1-15 Basic Response Curve for m = 10 . . . . . . . 28
Fig. 1-16 Multiple Mask Pattern . . . . . . . . . . . . 29
Fig. 1-17 Effect of Multiple Mask . . . . . . . . . . 29
Fig. 1-18 Dimensions Used in Analysis of System
Accuracy . . . . . . . . . . . . . . . . . 32
Fig. 1-19 Effect of Change in Mask Position with
Respect to Screen Position. . . . . . . . . . 33
Fig. 1-20 Effect of Difference in Focal Length for
Visual and Focus-Detection Wavelengths. . . . 34
Fig. 1-21 Effect of Compensating for Focal-Length
Difference with a Fixed Mask-Position
Correction . . . . . . . . . . . . . . . . . . 35
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Fig.
1-22 Experimental System for Simulating a
Rear-Projection Viewer .. . . . . . . . . . .
38
Fig.
II-1 Incident Cone I and Reflected Cone R from
Mirror . . . . . . . . . . . . . . . . . . . .
42
Fig.
11-2 Change in Magnification with Mirror
Movement Coq 4'1? , . . . . . . . . . . . . . . . .
42
Fig.
11-3 Detection of Intercepted Light in Object
Plane . . . . . . . . . . . . . . . . . . . .
45
Fig.
11-4 Form of Total Returned Light LT and
Intercepted Light LI as Functions of Mirror
Position aq t . . . . . . . . . . . . . . . . .
46
Fig.
II-:i Returned-Light Disk in Lens Plane . . . . . .
48
Fig.
11-6 Relationship of Light Disk and Lens Disk
with Mirror Outside Image Plane . . . . . . .
49
Fig.
11-7 Corresponding Values of AqM and AqM Leading
to Identical Shapes of A(r) Curves in
Fig. 11-5(b) . . . . . . . . . . . . . . . . .
51
Fig.
11-8 Range of r Over Which Common Area A has
Maximum Value . . . . . . . . . . . . . . . .
52
Fig.
11-9 Fraction of Returned Light for a Point
Source on Axis as a Function of pq
53
Fig.
II-10 Fraction of Returned Light Over Range
q/2 n JQll ( ` q . . . . . . . . . . . . . . .
54
Fig.
11-11 Point Source r Reimaging at r`' Causing a
Light Disk of Diameter W1 in Object Plane
with Center at Distance S' from Axis. . . . .
57
Fig.
1I-12 Relationship of Object Disk and Returned-
Light Disk Truncated by Lens. . . . . . . . .
58
Fig.
11-13 Geometrical Configuration at Critical Values
Ap and Ap. . . . . . . . . . . . . . . . . .
60
Fig.
c c
11-14 Use of Diffuser Instead of Mirror in Image
Plane . . . . . . . . . . . . . . . . . . . .
67
Fig.
B-1 Arrangement for Computing Common Area Between
Two Unequal Overlapping Disks . . . . . . . .
B-3
Fig.
B-2 Computation of Overlap Area for Small Values
of Separation . . . . . . . . . . . . . .
B-5
Fig.
B-3 Computer-Generated Curves of Common Area
as a Function of Separation S Over the Range
R - r AS t 1l -f- r for Different Radius Ratios. .
B-7
Fig.
D-1 Geometric Configuration for pp = dpc . . . . .
D-3
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Several techniques for automatic focus detection were examined in
the context of a general set of requirements for an automatic focusing
device. At an early stage in the project it was determined that the
most probable application for automatic focusing would be to the
scanning mode of operation of a rear-projection viewer.
On the basis of this application, the possible focus-detection
techniques were reviewed, and the technique of using an externally
generated image was selected for detailed study.
Section I of the report reviews this selection of a focus-detection
technique and presents the results of the analytical and experimental
work on automatic focus detection. Section II of the report contains
the detailed theoretical analysis of the basic focus-detection tech-
nique. The four appendices contain derivations of equations used in
the analysis.
(1) A focus-detection technique has been developed that
detects the sharpness of an image formed by the
optical system whose focus is being controlled. This
technique is particularly suited to application for
the automatic focusing of a rear-projection viewer
during the scanning mode of operation.
(2) The theoretical and experimental results justify the
engineering design and installation of a prototype
focus-detection system in an existing rear-projection
viewer. This will permit actual quantitative per-
formance measurements to be made and operational ex-
perience gained to confirm the advantages of automatic
focus detection,
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A p.irallel research program should be continued on extensions of
the basic focusing techniques and on modifications for other applications.
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A detailed and specific set of requirements for an automatic
focusing system will depend upon the specific type of equipment in
which an automatic focusing technique would be employed and upon the
capabilities of the given technique chosen for the selected applica-
tion. The following objectives for an automatic focusing system were
provided by the sponsor prior to the beginning of this project as an
informal guide to what is desired in an automatic focusing system for
a variety of possible applications.
(1) Resolution: Resolution sensitivity at the output
presentation (the point where the original is
examined by the eye) of greater than 10 line pairs
per millimeter.
(2) System Response: The system must sense, respond
to, and correct--by changing the projection-lens
position--variations in the object plane position
of up to ?1/4 inch. This must be accomplished at
a rate faster than the response time of the eye.
(3) Energy Required: The illumination required from
the projected image should be a minimum. The
maximum illumination reduction should be 10 percent.
This light-subtraction technique (beam-splitter in
projection path) is preferable to sensors in the
image plane.
(4) The system should be completely automatic, not re-
quiring periodic calibration. The system should
average any curvature in the object plane; how-
ever, it should consider only that portion of the
image which is actually being projected.
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(5) Magnification Ranges: The equipment that this device
(6) Equipment: This system is to be used on rear-
projection viewers, photographic enlargers, direct-
vviewing equipment, etc.
The following paragraphs interpret the preceding list of objectives
and relate them in a qualitative manner to the scope and results of our
research.
At ?.he beginning of the project work a review of the scope of the
research with the project monitor determined that the most likely appli-
cation of an automatic focusing system would be on a rear-projection
film viewer of the type used to scan large quantities of aerial film.
Therefore, the study of those automatic focusing techniques that would
have application for rear-projection film viewers would be emphasized
in the basic research work. The following comments are numbered to
correspond to the objectives listed above.
(1) and (4): The resolution of the output presentation will be
limited by the capability of the given optical system and film combina-
tion in a particular type of equipment. How close to this limiting
performance the equipment will actually operate is dependent upon many
peripheral items of which correct adjustment of focus is a primary
operational limitation. In the static situation with the film held
flat and perpendicular to the optical axis, the correct adjustment of
focus may be accomplished satisfactorily by careful manual procedures.
However, under the conditions specified for the scanning mode in a rear-
projection film viewer, both the position of the film plane and its
"flatness" are continually varying at a rate that precludes manual
focusing.
The ultimate in performance of any automatic focusing system will
be basically limited both by the curvature of the film and by any angular
deviation of the film from the normal film plane. Because of these
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uneven displacements of the film from the film plane, it is likely that
only a portion of the overall film image can be critically focused at
any one instant.
A focus-detection system that could examine the entire image area
and select an average plane of focus to maximize the area of the nega-
tive in or near critical focus is not practical with any of the focus-
detection ideas developed under this project.1 The best that can be
accomplished is to critically focus the center portion of the image
area--i.e., that part centered on the optical axis of the projection
system.
It is felt that this is a satisfactory compromise and that this
method may actually prove to be operationally more desirable than an
averaging approach. When the operator stops the scanning mode and de-
sires to examine a given area in more detail (and/or at a higher magni-
fication), the centering of the desired area will thus automatically
ensure critical focus for the given section being examined.
(2): For the typical lens sizes employed in a rear-projection film
viewer employing film up to 9 inches wide, the direction of the variation
of the film plane (tof 4 inch) can be sensed for magnifications up to a
value of 15 to 20. At greater magnifications the allowable range over
which the direction of defocus can be detected becomes restricted.
Section II develops the formulas that relate the range over which the
direction of defocus can be sensed to the other parameters of the system.
The speed with which the system can sense defocus is rapid--measured
in milliseconds. The correction of focus by movement of the lens. will be
slower and will be the primary limitation of the response time of the
system. The design of the servo system to control the lens was not
studied on this project. However, consideration of the problem in rela-
tion to the sizes of lenses required and the distances over which they
must move suggests that a system response measured in tenths of a second
1See discussion in Sec. I-B.
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could be realistically expected. Considering the light mass of the film
and the viscous damping of the air, the speed with which the film will
move from the film-plane position during scanning may be slow enough
that a servo response of 1/10 second will permit the focus system to
follow the film movement.. Measurement of the actual rates of displace-
ment from the film plane on a film viewer during the scanning mode
should be made and the system response specifications related to these
figures.
It. should be noted that the tension between the two rolls driving
the film will have a direct effect on both the magnitude of the film
displacement and the frequency with which it varies. As the tension
is increased, this will reduce the magnitude of the film displacement
from the film plane and will increase the frequency at which the film
will tend to vibrate or move in and out of the film plane. Reducing
the tension will tend to reduce the frequency with which the film moves
but will most likely permit it to make a wider excursion. A study of
this relationship should be made to determine the best tension and drive
characteristics for the film transport in relation to an automatic
focusing system, in terms of the range over which the focus detection
system can operate and the speed with which it can correct focus by
moving the lens. This will permit the optimum arrangement of these
parameters to achieve the best overall. system performance.
(3): The technique of imaging an external pattern onto the film
for the purpose of focus detection will require no energy from the film
image. By using a projected image of the external pattern in a wave-
length band at the edge of the visual spectrum and with the proper de-
sign of the additional optical components required to place this image
in the optical path, none or at most a very few percent of the illumina-
tion from the visual film image will be attenuated.
(5): A magnification range of 1 to 50 was specified in this pre-
liminary requirement but later modified in discussions with the project
monitor to a range of 3 to 70. As indicated above, at the present state
of development of the focus-detection system the requirements of ?1/4-
inch film deviation can be handled at low magnification ranges extending
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up to approximately the range of 15 to 20. Beyond that point the focus-
detection system continues to operate in the region near correct focus,
but the range of defocus over which the system can sense the correct
direction required to correct focus gradually diminishes. A study of
the magnifications at which the scanning mode of operation is likely to
be operated may show that it is not necessary to maintain the ?1/4-inch
requirement for the higher magnifications. Since the highest magnifica-
tions will undoubtedly be used for the static examination of a small
area of the film, the system could be manually returned to near focus
in the event that a rapid transient produced an error too large for the
system to follow. The automatic system would take over and maintain
focus for the slow variations that might occur during the static exami-
nation of a given area.
(6): The focus-detection technique developed is applicable to all
the various types of equipment mentioned, although emphasis was placed
on use during the scanning mode of the rear-projection film viewer. The
requirements for this application are in general the most difficult to
meet.
Focus-detection techniques can be divided into two basic categories:
The techniques grouped in this class do not examine the pro-
jected image for sharpness of focus. Instead, a measurement of film or
lens position is made for a given set of conditions with the system set
for correct focus. The system then monitors and controls the film or
lens position to maintain this given film-lens relationship. If all
other parameters affecting focus remain unchanged, then the correct
positioning of the film plane or lens position will result in correct
focus.
Systems employing such techniques are widely used in pro-
fessional film-enlarger equipment and in several deluxe film projectors
for 35 mm color slides. The following paragraphs illustrate how these
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In the enlarger application, the lens-to-film spacing is
varied by a carefully computed, precision-made mechanical cam system.
This cam system varies the lens-to-film spacing as a function of the
height of the enlarger head (containing lamphouse, film, and lens)
above the baseboard where the image is projected onto the printing
paper. When the enlarger is carefully focused manually at any given
height above the baseboard, then the focus is automatically maintained
over the entire range of magnifications by the mechanical cam arrange-
ment. However, such operation is assured only for the specific lens
for which the mechanical cam was designed. In addition, the placement
of the paper on the baseboard and of the film in the film holder and
the mechanical structure holding the enlarger head must remain in exact
dimensional relationship if the cam is to correctly maintain focus
throughout the operational range of the enlarger.
In the projector application a small beam of light is pro-
jected onto the center of the slide at an angle, from behind and to one
side of the main projection lens (see Fig. I-1).
'V
PROJECTION LENS
~ ~~,
Z `, LIGHT SOURCE
1
SCREEN
When the slide is in the correct position, the detector unit
receives the reflected beam centered. The lens is then manually ad-
justed Lo give a sharp image at the selected screen distance. Any
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change in the slide position due to differences in mounting or due to
bowing from heat will be sensed by the detector as the reflected beam
moves away from the center to the left or right. This signal from the
detector is used to drive a servo that repositions the slide carrier
to place the center of the slide back at the correct position and thus
return the beam to the center of the detector.
Both of these systems perform satisfactorily for the applica-
tions described, provided they are maintained and used properly within
the constraints required by their design. Neither system senses the
actual sharpness (focus) of the image and consequently neither can be
truly termed a focus-detection system.
The application of these techniques to this project was not
considered practical, since the requirements for focus control during
the scanning mode of a rear-projection film viewer preclude any control
of the film-plane position. Although it would be possible to measure
the film-plane position and in an open-loop fashion move the lens in a
precomputed relationship to the film-plane motion to achieve focus at
the projection screen, such a system would require an accurate measure-
ment of the film-plane position over the ?1/4 inch as well as a complex
functional relationship to transform this measurement into the correct
lens position to achieve focus as the film varies. While this might be
possible for a given focal-length it would become very complex with
variable focal-length lenses. Changes in lens focal lengths are re-
quired to achieve changes in magnification in rear-projection viewers
where the film-to-screen distance remains constant. This is in con-
trast to the enlarger application where changes in magnification are
achieved with a given lens by permitting the film-to-image distance to
vary.
Therefore, it is concluded that the approach offered by either
of these techniques would be both difficult and subject to inaccuracies
for application to rear-projection viewers.
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2. Image Sharpness Detection
The second category of automatic focusing techniques includes
those that detect a sharp image in the desired image plane or its optical
equivalent. The detector output can then be used in a closed-loop con-
trol system to maintain the desired critical focus by adjusting any one
of four primary parameters--film-plane position, image-screen position,
lens position, or lens focal length.
The source of the image used by the detector to detect sharp
focus can be from one of three basic sources.
.3. The Image Recorded on the Film
The projection of optical images onto a nonlinear photo-
cell to detect their sharpness is a basic technique developed in this
laboratory and previously described. The main limitation of this tech-
nique as applied to the present problem is that the film to be viewed
or scanned may contain large sections of photographs of water or uniform
cloud areas with little or no sharp detail in the film image. The focus-
detection system would see no sharp image boundaries when such sections
of the film passed the viewer and thus would not maintain sharp focus.
Therefore, small but significant pinpoint areas in the field of view
could go unnoticed because of the overall defocused position of the
image. Because of this serious limitation from an operational point of
view, this technique was considered marginal for application to the
scanning mode of a rear-projection film viewer.
The tight, sharp grain pattern that exists in modern
films makes an ideal pattern for detection by the nonlinear-photocell
J. C. Bliss and H. D. Crane, Relative Motion and Nonlinear Photocells
in Optical Image Processing, in Optical and Electro-Optical Information
Processing (M.I.T. Press, Cambridge, Massachusetts, 1965).
J. C. Bliss and H. D. Crane, "Optical Detector for Objects Within an
Adjustable Range," J. Opt. Soc. Am., Vol. 54, No. 10, pp. 1261-1266
(1964).
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technique. However, it requires magnification by 100 to 200 diameters
before projection onto the nonlinear photocell. If a rear-screen pro-
jection system were to operate at a constant magnification ratio, the
additional magnification could be optically inserted to enlarge the
grain pattern on a nonlinear photocell and make this a possible approach.
However, with the wide range of magnifications that are desirable, the
use of the film grain as a basic focusing image becomes difficult. The
secondary optical system required would have to have a magnification
that would vary inversely with the main optical system and would have to
maintain the same relative focus distances. Such a subsystem would be
in series with the focus-detection element, and any variations or shifts
in its focus with magnification change would create errors in the overall
focus-detection system. While these could be minimized by careful de-
sign and engineering, this system would be more difficult to implement
than the following method for obtaining an image source for automatic
focus detection.
c. Externally Generated Image
This technique appears to be the most versatile and offers
the best combination of advantages for application to automatic focusing
systems and in particular to the rear-projection systems during scanning
modes. The majority of the project effort was spent in examining this
technique both experimentally and analytically. The remainder of the
report will therefore deal with a description and analysis of this method
of automatic focus detection.
C. Externally Generated Image for Focus Detection
Figure 1-2 indicates the basic technique of how an exterior image
is projected onto the film and used for focus detection.
A dichroic mirror that reflects infrared but is transparent to
visual light is inserted at a 450 angle just beyond the last element of
the projection lens system. An illuminated mask is placed on the new
optical axis defined by the 450 mirror and at the same distance from
the center of the mirror as the viewing screen. This results in the
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REAR PROJECTION
VIEWING SCREEN
MASK TO GENERATE IMAGE PATTERN
FOR FOCUS DETECTION
I--
rl LIGHT SOURCE FOR
IMAGE MASK
FIG. 1-2 BASIC LAYOUT FOR FOCUS DETECTION WITH EXTERNALLY
GENERATED IMAGE
lens system imaging the illuminated pattern of the mask onto the film
in exact focus when the optical system is adjusted to project the film
image into exact focus on the screen. The surface of the film spec-
trally reflects several percent of the infrared light of the projected
pattern back through the lens and the dichroic mirror, where it is re-
focused in the plane of the illuminated mask. The use of an infrared-
reflecting filter between the film and the projection light source (not
shown in Fig. 1-2) prevents infrared light from the projection light
source from interfering with the focus-detection system. Because of the
geometry of the system (see Fig. 1-2), where x is made equal to x', the
reflected image of the mask at the location of the mask is only in focus
when the film image is in focus on the screen. r"or any other film
position, both the reflected image of the mask and the film image at the
screen are out of focus.
A discussion of the effects resulting from the infrared and visual focal
lengths of the projection optics having slight differences is given in
Sec. I-E.
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1. Methods of Detecting Sharp Focus of Reflected Image of Mask
When the reflected image of the mask is in exact focus, all
the light originating from the mask aperture returns through the aperture,
and no illumination falls on the surface of the mask facing the lens.
When the film plane is not in exact focus, the return image is also out
of focus and light now falls on this surface, surrounding the aperture.
Three methods have been considered to detect the defocused light on the
front of the mask.
of the mask is coated with photoconductive material and electrodes are
placed to measure the change in conductivity resulting from the light of
the defocused image, the change in conductivity between the electrodes
will give an indication of the amount of light impinging on the front
surface of the mask. Sharp focus will be indicated when the minimum
amount of light falls on the mask, and this will be indicated by the
minimum conductivity of the photoconductive surface. This method is
simple to implement optically but is complicated by the necessity of
fabricating a specialized mask photocell. Difficulties in obtaining
specialized photocell surfaces on various experimental masks made this
an unattractive approach for the experimental phase of the project.
However, were an application to be employed with a large number of iden-
tical units, serious consideration of this technique would be warranted.
placement of small-angled mirror surfaces or the use of a smooth white
diffusely reflecting surface permits a portion of the light falling on
the front of the mask to be collected by a photodetector placed off to
the side of the optical path. Figure 1-3 indicates how this approach
would be implemented with the use of small mirrored surfaces. The use
of a diffusing surface is simpler and is the technique that was em-
ployed during the laboratory experimentation phase of this project.
The diffusing surface is an inefficient reflector of light, and the
photodetector receives only a small portion of the light falling on
the front side of the mask. For this reason, a highly sensitive
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DICHROIC MIRROR
REAR PROJECTION
VIEWING SCREEN
-- r --- -------
PROJECTION
OPTICS
MIRRORS OR WHITE DIFFUSE SURFACE
PHOTODETECTOR
r
lsJ
FIG. 1-3 METHOD FOR DETECTING DEFOCUSED LIGHT FROM FRONT OF MASK
photomultiplier tube was employed during the experimental work. The use
of the diffuse surface permitted the aperture pattern in the mask to be
of any desired shape. The use of mirrored surfaces would be more diffi-
cult with the circular patterns that were employed. For vertical slits,
the mirrored surfaces would be more feasible and more efficient than the
diffuse surface. This approach separates the mask from the photodetector
and provides the flexibility desired for experimental work.
Use of Beam Sputter and Negative Mask--As shown in Fig. I-4,
a 50-percent reflection mirror is mounted at a 45`" angle in front of
mask I to reflect a portion of the reflected mask image to the location
of mask 2. Mask 2 has a pattern identical to that of mask 1 but is a
negative (transparent areas are opaque and opaque areas are transparent
ix; relation to mask 1). Thus a sharp image returning toward mask 1 is
reflected to mask 2 which intercepts all of the in-focus image. When
dul ocus occurs, some light then passes through to a photodetector lo-
cated behind mask 2. Any photodetector can be used, and it is relatively
easy to gather all the light that passes mask 2. The introduction of
the second half-transmission mirror results in a 4-to-1 light loss when
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PROJECTION
OPTICS
50% TRANSMISSION
MIRROR
REAR PROJECTION
VIEWING SCREEN
FIG. 1-4 USE OF BEAMSPLITTER AND NEGATIVE MASK TO DETECT DEFOCUS
compared to the light that would fall on the face of mask 1. Any mask
pattern, however, can be used and even with the 4-to-l light loss due to
the mirror, the efficiency in collecting the return light is probably
better than most of the arrangements for measuring light reflected from
the mask. A preliminary test setup using a positive and negative mask
consisting of parallel slits and bars was used in one of the breadboard
configurations (see Sec. I-F). The method was not used on the detailed
experimental studies of mask characteristics, however, because of the
necessity of making the inverse shape for mask 2 and the necessity of
carefully aligning the two masks and the mirror for each change in the
experimental configuration. This method has the advantage of offering
the freedom to use any standard photodetector. In addition, no
specialized mirrored surfaces are required. The light-gathering capacity
is reasonably efficient and is independent of the particular mask design,
though some limitations are imposed on mask design by the requirement of
the negative mask. This method would warrant serious consideration for
a prototype design.
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The point of exact focus is at the point where the minimum
illumination is detected by the photodetector. However, the static
photocell output does not indicate whether a minimum exists or (if the
output is not a minimum) in which direction the focus must be corrected
to approach the minimum.
This sensing was accomplished in our mobility-aid work using
the nonlinear photocell by vibrating the photocell axially at a frequency
of several hundred cycles per second and over a distance of 0.010 to
0.015 inch. The output signal from the photocell for out-of-focus images
is then a sine wave of the same frequency as the photocell vibration.
The phase of this sine wave reverses by 180' depending on which side of
the image plane the photocell is located. Because of the symmetry of
small perturbations about a sharp image plane when the photocell vibra-
tion is exactly centered on the image plane, the output signal is the
second harmonic of the basic photocell vibration frequency.
A variation of this technique is required in the present
application. The same second harmonic and phase-reversing fundamental
sine waves can be produced by small vibrations of either mask 1 or
mask 2. However, this requires planar mechanical vibration of a mask.
An alternate method is the introduction into the optical path of a
small optical component of variable focal length. A simple method has
been devised to accomplish this by using a thin-film pellicle mirror,
vibrated by air coupling from a small loudspeaker. Laboratory experi-
ments have shown this to be an ideal way to achieve small variations in
the optical path length. The vibrating mirror adds some optical com-
plexity, but eliminates the need for vibratory systems moving signi-
ficant masses--such as the masks.
An alternate method employing a three-dimensional mask and
two separate detectors was conceived. A preliminary investigation of
this approach revealed that a far more detailed study would be required
before a meaningful description or evaluation of this approach could be
made. The use of the vibrating pellicle mirror is the preferred method
at present.
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In order to experimentally measure the focus-detection system
response characteristics, the laboratory test system shown in Fig. 1-5
was assembled. This equipment permits the measurement of the light
intercepted by the mask as a function of the film position behind the
MIRROR
OR FILM
IMAGE
APERTURE
VLENS
r_V
LIGHT
SOURCE
- MASK
GEAR DRIVEN
POTENTIOMETER
X INPUT= AqM
(THE MIRROR MOTION)
I
PHOTOMULTIPLIER
Y INPUT = LIGHT OUTPUT REFLECTED
FROM MASK
FIG. 1-5 LABORATORY SETUP USED FOR EXPERIMENTAL FOCUS-DETECTION
STUDIES
lens. The film for most experiments was replaced with a high-reflectance
plane mirror. The mirror could also be replaced with a white diffusing
surface so that the effects of diffuse reflection could be evaluated.
The results of experiments confirm the theoretical analysis developed
in Sec. II and the appendices. Of particular interest are the range
over which "focus null response" is obtained and the effects of system
misalignment.
According to the theory two system parameters are of critical im-
portance in determining the response characteristics. These two
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parameters are the magnification of the optical system, which is the
ratio of mask-to-lens distance to film-to-lens distance at focus:
m = -
where p = mask-to-lens distance and q = film-to-lens distance. The
second critical parameter is the ratio of lens size to mask aperture
size. This ratio is clearly defined only if the lens and the aperture
have similar shapes. In order to permit comparison of experimental re-
sults with theoretical predictions, the aperture used in the following
experiments is of circular shape. Thus, in the following data, the ratio
of lens size to image size is
Bl
Ms W
where D = lens diameter and W = diameter of mask aperture. Once established
by a. given system, the parameters m and M S determine the range over which
focus detection can be obtained and the degree of off-axis misalignment
of the mask that can be tolerated. In conjunction with the lens f-
number, these parameters also determine the effect of angular misalign-
ment of the film. The response curve for an aligned system with a
magnification m = 3 and lens-to-mask ratio M s = 9.57 is given in Fig.
1-6. The amount of light intercepted by the mask as a function of film
distance from the lens is indicated for both specular and diffuse re-
flection. Since the light reflected from the film has been determined
to be mostly specular, it is this curve that is considered significant
and a diffuse response is included for reference only. The principal
difference between the two curves, aside from the great reduction in
returned light in the diffuse case, is that the light returned for the
diffuse case continues to increase as the film approaches the lens,
while for the specular case, the light returned reaches a constant value.
As shown in the derivation of Sec. II, the sharpness of the null is
dictated by two critical values of pq called Agi- and Aq The magni-
Me Mc'
tudes of these critical values of 4~qM depend only on m and Ms and are
given in Eqs. (1-3) and (1-4). These points are marked on the curve.
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DIFFUSE
I
Og _ Mc A4Mc
p
Q m=3
I I I I I I I I I I I
I.2f 1.4f 1.6f I.8f 2.Of
FILM DISTANCE FROM LENS, qM
FIG. 1-6 BASIC RESPONSE CURVE FOR m = 3
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{- qM
~qMe 2(mM - 1)
s
qM
-Me 2mM
S
where q m , is the in-focus distance of the film to the lens. It is clear
from Fig. 1-6 that the range of focus detection in the direction of film
motion away from the lens is limited to approximately two increments of
Me The detectable range for film motion toward the lens is approxi-
+
mately 2Iq , although the limit is not clearly defined if the total
MCI
light intercepted by the mask is measured. This may not be feasible
in a practical system since the light return as the film approaches the
lens is spread over a very large area and becomes difficult to collect.
If the light-collection area at the mask is limited in extent, the left-
hand portion of the curve of Fig. 1-6 would not reach a nearly constant
value as indicated, but would begin to decrease when the size of the
returned area of light exceeded the collection area. Thus the detec-
tion range in the negative direction of film motion can be limited by
practical difficulties as well as by the characteristics of the focus-
detection scheme.
1,gpy = 2 (q+ + LqM)
'IL
which, upon substitution of Eqs. (1-3) and (1-4), becomes
2mM - 1
_ q M s
- 1)
`,q R LmMss
or, in Terms of the lens focal length and magnification,
(m + 1) (2mM - 1)
Gqa, = f
m2M (mm S - 1)
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For M s = 9.57 (the magnitude employed in the tested system), the de-
tectable range is given for three magnifications in the table below.
DETECTION RANGE FOR M = 9.57
s
Magnification, m
,LqR
1.0 0.44f
3.0 0.095f
10.0 0.024f
The precision of focus obtainable depends primarily on the sharp-
ness of the null in the vicinity of focus, which in turn is a function
of the quality of mask construction, stray reflections from internal
lens surfaces, and aberrations of the lens at the wave length used for
detection. It is estimated that a precision of pqR/100 should be ob-
tainable with reasonable care in system design. It may be possible to
attain precisions of pqR/1000 with high-quality lenses and masks.
Also plotted in Fig. 1-6 is the total light returned through the
lens prior to masking. The "focus null" characteristic is superimposed
on this curve. As explained in Sec. II of this report, the light return
as a function of film distance from the lens (qM) is constant from
qM = 0 to qM = qMB where qMB is the point where vignetting and ray di-
vergence cause a rapid reduction in light getting through the lens.
The value of qMB is given by
qM
MB 'M mm + 1
s
When this is compared with Eq. (1-4), it is apparent that q
MB
always lies to the left of the focal point qM, and is somewhere between
AgMc and 2AgMc The interaction of the light returned and the "focus
null" curves will then always result in some degree of asymmetry in the
response. This asymmetry is most severe at low magnifications, but the
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light level at Aq- is always greater than at AgMc, even though Aq+
is slightly larger than Agrrtc.
It is next desired to investigate the effect of placing the mask
at a distance r from the optical axis. As can be seen in Fig. 1-7, if
the mask is offset sufficiently from the optical axis, a point will be
reached where no light is returned to the lens; this establishes a
maximum value for r P as given by Eq. (1-9). For the experimental system,
the maximum r equals 1.6 lens diameters.
1)
r (maix) = D/2 (m +
P
FIG. 1-7 RETURNED-LIGHT PATH WITH MASK OFF AXIS
The response curve for a mask movement approximately 50 percent of
this maximum is shown in Fig. 1-8. Note that the sharpness of the null
is reduced, but that the detection range is actually extended slightly.
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-=m=3
q
rp MAX. = 1.6D
MASK
DISPLACEMENT
rP=0.75D
= 48 % OF MAX.
III
AqMC AqMC
L I I I I I I I I I I I
I.Of I.2f 1.4f I.6f 1.8f 2.Of 2.2f
FILM DISTANCE FROM LENS, qM
FIG. 1-8 RESPONSE WITH MASK OFF AXIS
Another important possible misalignment is angular rotation of the
film. This effect is shown in Fig. 1-9. Again an extreme angle exists
such that no light is returned to the lens. This angle is given by
Eq. (I-10), where F is the lens f-number. The critical angle for the
test system is equal to 8.9?.
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L 1 + mMs
2F (m + 1) Ms
The effect of angular misalignment for various angles is given in Fig.
1-10. It is apparent that satisfactory response is maintained for mis-
alignments of as much as one-half of the critical angle. It is conceivable
FIG. 1-9 RETURNED-LIGHT PATHS FOR ANGULAR FILM DISPLACEMENT
that off-axis points can be intentionally included to help compensate
for angular movement of the film. This is shown in Fig. I-11, in which
for a given film rotation the off-axis mask causes the light to be re-
turned to the lens. A test was made in which an angular deflection of
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I I I I I I I
1.4f I .6f I.6f 2.Of 2.2f
FILM DISTANCE FROM LENS, qM
FIG. 1-10 EFFECT OF ANGULAR DISPLACEMENT OF FILM SURFACE
12 percent greater than critical was produced and then compensated by
moving the mask 1.4 lens diameters off-axis. The response is shown in
Fig. 1-12. Notice that Fig. 1-12 shows the offset of the focal point
as would be expected from inspection of Fig. I-11. As can be seen in
Fig. I-12, this compensating effect has very nearly restored the
response to that of the aligned system.
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FOCAL PLANE SHIFT
DUE TO FILM ROTATION
WITH OFF-AXIS MASK
FIG. I-11 RESTORATION BY OFF-AXIS MASK PLACEMENT OF LIGHT LOST
BECAUSE OF ANGULAR FILM DISPLACEMENT
In the previous curves the "film" is a highly reflective plane
mirror. In order to estimate the total amount of light available from
the film and to compare its response with pure spectral characteristics,
the mirror was replaced with an actual piece of film. The results of
this test are indicated in Fig. 1-13. The lower curve indicates the
film response with the source intensity unchanged from that used for
the mirror curve. The source intensity was then increased so as to
bring the total light return to nearly the level produced by the mirror.
It is clear that there is some reduction in the depth of the null, but
it is felt that most of this is due to increased background level as a
result of increasing the light intensity. In a well-designed physical
system, the stray light can be kept to a much lower level. Figures 1-14
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1.126, (10?) FILM MISALIGNMENT CORRECTED BY
MASK DISPLACEMENT OF 1.42D (93% rp MAX)
AgMc ~qMc+
1 I I I I I I I I I I
I.Of 1.2f I.4f I.6t I .8f 2.Of 2.2f
FILM DISTANCE FROM LENS, qM
FIG. 1-12 COMPENSATION FOR ANGULAR FILM DISPLACEMENT
BY OFF-AXIS MASK PLACEMENT
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FILM (LIGHT LEVEL
INCREASED TO EQUALIZE
TOTAL RETURN)
V FILM (NO CHANGE IN
LIGHT LEVEL)
~qMe AgMc
P:m:3
q
I I I I I- I I I I I I I I
.Of 1.2f I.4f I.6f 1.8f 2.Of 2.2f
FILM DISTANCE FROM LENS, qM
FIG. 1-13 RETURN FROM FILM COMPARED TO SPECULAR RETURN
FROM PLANE MIRROR
and 1-15 are basic response curves made with the same lens system but
at different magnifications. In Fig. 1-14 m = 1.0, and in Fig. I-15
m = 10. It is clear from both of these figures that the sharp break in
the returned light characteristics occurs when the film moves a distance
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qMB
P
V =m= 1.0
,LqMc AqMc
1.6f 1.8f 2.Of 2.2f
FILM DISTANCE FROM LENS, qM
of 2AgMc, and this is independent of the magnification used. This vali-
dates the use of the figure corresponding to a displacement of 2AgMc as
the limitation-of-detection range in the positive direction of film
motion.
Finally, as a method of increasing total range while at the same
time maintaining the high null sensitivity, the possibility of using a
mask consisting of a distributed array of small images was considered.
A possible mask is shown in Fig. 1-16. For this mask AgMc is very
clearly defined for the small circles making up the bulk of the mask.
However, since the array of circles also has an overall circular shape,
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mo
Ile- TOTAL LIGHT RETURN
AgMe AgMc+
FILM DISTANCE FROM LENS, qM
FIG. I-15 BASIC RESPONSE CURVE FOR m = 10
a p'gMc can be defined for the larger circle. Thus one would expect to
see a combined effect which is shown in the actual response curve in
Fig. 1-17. Note that in this figure the null sharpness is governed by
AgMc for the small circles, while the overall range is dictated by A'q Mc
of the large circle. Where the break occurs between the two ranges is
a function of the separation between the individual apertures. If
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000
08000
~boooo
spacing is very regular and equal, then the edge transition between the
two slopes will have a number of ripples due to mutual interference of
the adjacent holes. The resulting ripples, if large enough, can result
in false nulls which could in turn cause an error in the focus-detection
I.If
FILM DISTANCE FROM LENS, qM
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system. However, if the holes are irregularly distributed, these ripples
will average to the relatively smooth curve shown in Fig. 1-17.
E. Analysis of System Accuracy
As discussed in Sec. I-D, the experimental curves indicate that with
careful design and engineering, the focus-detection system and associated
lens servo should be capable of providing high precision for the focusing
of the external image. This means that the mask image will be focused
by the lens precisely at the front reflective surface of the film and
the refocused reflected image exactly at the mask plane. With the
assumption of exact focus by the focus-detection system, the following
analysis will quantitatively describe resulting errors in the visual
film image. Two factors cause the visual screen image for the film to
be slightly out of focus when the external focus-detection image is in
exact focus.
The depth of the film image behind the front surface of the film
base has been measured to be between 0.003 and 0.004 inch from the
sample aerial film supplied by the sponsor. This is a distance greater
than the allowable depth of field for a significant portion of the
operating range of a rear-projection viewer. In other applications,
such as an enlarger, the externally generated image would be reflected
from the emulsion side of the film rather than the other side of the film
base. In these applications the dimensional difference between the
surface and the emulsion will be smaller.
The second source of error results from the difference in the focal
length of the projection lens in the visual wavelengths that are used
to project the film image, and the focal length in the ultraviolet or
infrared range used for the focus-detection system. The use of a band
of wavelengths on the edge of the visual spectrum is dictated by the
application to rear-projection viewers where it is desirable that the
pattern used for focus detection not be visible to the viewer. In other
applications where the external image would cause no interference with
the system function, the focus-detection system could, if desired,
operate in the visual region.
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In the following analysis, the error of focus is defined as the dis-
tance from the plane of the film image to the plane where the image of the
screen is focused by the visual wavelengths, when the mask and film surface
are in exact focus as determined by the focus-detection system. Knowing
this dimensional error between the film plane and the plane of sharp focus of
the projection lens, we can compare this error with the depth of field of
the projection lens at any given f-number as defined by the circle of con-
fusion allowed for the resolution the system must have at the viewing screen.
The equation for the error as derived in Appendix A [Eq. (A-7)] has
been normalized with respect to the distance L which is defined as the
distance from the film surface to the projection screen. This is a
constant in a.iy given system (with the exception of the minor variations
due to the motion of the film plane about the nominal value; these small
changes in L ,would not significantly affect the resulting error values
and thus are not included in this derivation). The resulting equation
(qn + tn) (1 - qn) - fn(1 + tn)
En = (1 - q1) - f n (I-11)
Figure 1-18 (not to scale) shows the relationship of the basic
geometric parameters used in the above equation. The remaining terms
are defined as follows:
4 k fn
qn = Z 1- Vi - Y
k = f, f _ t
f L to = L
The unprimed terms always refer to the visual lens parameters, and
the primed terms refer to those lens parameters related to the lens
characteristics in the wavelength band used for the focus-detection
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PLANE THAT IS IN
FOCUS VISUALLY
ON THE SCREEN
(VARIABLE)
FILM IMAGE-
PLANE
FOCUS ERROR
_q' ---,~i
D
FRONT SURFACE OF FILM--HELD IN FOCUS
ON MASK BY FOCUS DETECTION SYSTEM
MASK POSITION
(ADJUSTABLE) I
SCREEN ',*I
FIG. I-18 DIMENSIONS USED IN ANALYSIS OF SYSTEM ACCURACY
system. The equation and its derivation assume that the focus-detection
servo adjusts precisely the lens positions so that p' is always at the
film surface and q' is always at the mask.
The mask location relative to the screen is considered a constant,
but subject to selection as a design parameter to be used as a means to
reduce the error of the visual focus. The difference in lens focal
length between the visual range and the band used for the detection
system may or may not be subject to design control during the engineering
of the focus-detection system. Some flexibility is provided by the choice
of working the focus-detection system at either the ultraviolet or the
infrared edge of the visual band.
For a selection of system parameters the error equation was computed
on a digital computer and the focus error is plotted as a function of
magnification for these parameters in Figs. 1-19, 1-20, and 1-21. The
magnitude of the error is plotted in terms of film-thickness units t.
For the values of L, K', and y assumed, the value of a t unit is 0.004
inch.
The error curve plotted in Fig. 1-19 is for the case when the lens
has the same focal length in both the visual and focus-detection wave-
length bands. For this case the mask position can be located to provide
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/ f/8.0
/ I I I I I I I I I I I I 1 I I 1
3 4 5 6 7 8 9 10 20 30 40 50 60 80 100
APPROXIMATE MAGNIFICATION, m*
FIG. 1-19 EFFECT OF CHANGE IN MASK POSITION WITH RESPECT TO SCREEN POSITION
no error for only a single magnification. For greater magnifications,
the error will increase and reach a maximum value of t as m approaches
infinity. At lower magnifications than the selected zero error value,
the error will increase without limits.
The error curves plotted in Fig. 1-20 are for the case where the
mask position is held constant at the screen location (N = 1) and the
ratio of the lens focal lengths in the visual and focus-detection bands
is allowed to vary (k / 1). When the lens has a different focal length
in the focus-detection band than in the visual and the mask position
is moved from the screen, two cases are examined:
(1) focus-detection focal length is longer than the
visual and the mask is moved further from the lens
than the screen (y > 1)
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FIG. 1-20 EFFECT OF DIFFERENCE IN FOCAL LENGTH FOR VISUAL AND FOCUS-
DETECTION WAVELENGTHS
4 5 6 7 8 9 10 20 30
APPROXIMATE MAGNIFICATION, m*
(2) focus-detection focal length is shorter than the
visual and the mask is moved toward the lens from
the screen (y < 1).
The curves of Fig. 1-21 show that zero error can be achieved for
two values of magnification.
A large number of combinations of parameters y and k were computed.
The ones plotted in Fig. 1-21 indicate the most promising results that
were achieved over the general magnification range of 3 to 70. In a
given system one or more lenses would be used to achieve this range;
the freedom to select different values of k for each lens would, be-
cause of the smaller range of magnification covered by each lens, per-
mit better error correction. The selection of the magnification at which
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k = 0.9990
Y = 0.9980
k = 0.9985 k = 0.9960
y = 0.9970 y = 0.9840
4 5 6 7 8 9 10 20 30 40 50 60 80 100
APPROXIMATE MAGNIFICATION, m*
FIG. 1-21 EFFECT OF COMPENSATING FOR FOCAL-LENGTH DIFFERENCE
WITH A FIXED MASK-POSITION CORRECTION
zero errors are desired is a function of the system operating require-
ments, as is the magnitude of error permitted at other magnifications.
How well these requirements can be met will depend on the freedom to
select differences in focal lengths (k values) and the values required
for the other system parameters (i.e., range of magnification, allowable
error, lens aperture, etc.).
To relate these focusing errors to overall performance, a first-
order estimate of the allowable magnitude of error was derived and is
shown in Figs. 1-19, 1-20, and 1-21 by the dotted lines for 2 values of
aperture f-numbers. The requirement of a screen resolution of ten line
pairs per millimeter is assumed to require a circle of confusion of
0.1 mm as the maximum allowable when referred to the screen side of the
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optical system. The allowable circle of confusion referred to the film
side is the circle of confusion on the screen side divided by the magni-
fication. Therefore, at the film side the allowable circle of confusion
in this system is 0.1/m millimeters.
One-half the allowable depth of field is given by the equation
1 d = FD
2 c
where F = lens aperture in f-stop units, and Dr= diameter of the circle
Substituting 0-=-1 for D in the above equation gives the allowable
rn c
error between the film image and the point of exact visual focus of the
lens in millimeters if the system resolution requirements are to be
approximately met. The resulting equation is
I d = 0.1i
2 m
Multiplying by 10 to convert to film thickness units results in
1 d _ F
2 m
A study of the error in visual focus as a function magnification
in Figs. 1-19, 1-20, and 1-21 shows that the error in visual focus
cannot be kept within the required depth of field over the full magnifi-
cation range of 3 to 70 when restricted to fixed values of y. If y is
allowed to vary as a function of the magnification (physically this
means moving the mask with respect to the screen) a zero error could be
achieved over a wide range of magnifications. A solution of Eq. (i-11)
for y over the desired range of m with the requirement that the error
be kept at zero and for a selected value of k would provide data giving
the required motion of the mask in relation to the screen as a function
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of magnification. The magnification is not explicit in Eq. (I-11) but
is defined in terms of f n in Appendix A.
Moving the mask as magnification is changed is worth serious con-
sideration as a means of reducing the focusing error. The motion re-
quired need not be precise, since small errors in mask position are
only second-order effects in the accuracy of focus. For example, at a
magnification of three, the depth of field in the film plane for an
f/4.0 lens is ?0.005 inch. Referred to the mask position the allowable
error corresponding to this depth of field is m2 times 0.005 inch or
0.045 inch. For greater magnifications, the required accuracy of mask
location decreases. At m = 70, for example, the depth of field at the
film for an f/4.0 lens is ?0.00023 inch. The corresponding allowable
error in mask placement is (0.00023) times (70)2 or 1.1 inches.
Therefore, a mechanism to move the mask as a function of magnifica-
tion could be easily designed to provide for precise correction to the
visual focus. Errors in the motion of the mask are only second-order
effects in the focus correction. This is in contrast to the system
described in Sec. I-B-1 where the open-loop lens position is controlled
by a cam. In these systems the cam error must be less than the depth
of the field permitted at the film throughout the operating range.
This analysis of error in the automatic focusing system using an
external image reflected from the front surface of the film indicates
that for low magnifications (for example m = 3 to 10), system accuracy
can meet the requirements with a selected but stationary mask position.
From an operational point of view, automatic focusing during the scanning
mode is very desirable, since in this mode it is not possible to manually
focus with sufficient speed or accuracy. If operational use of future
rear-projection viewers plan significant, use of static high-magnification
viewing, two courses of action are suggested.
(1) During the high magnifications, when the film is
stationary, the operator can use manual focusing
since the images are stationary during close
scrutiny. This is recommended if only occasional
high magnification viewing will be performed.
37
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with the change
automatic focus
magnifications.
Experimental Focus-Detection System
Figure 1-22 shows schematically a breadboard system that was built
and tested to confirm that the focus-detection system would work in a
in the flat position, the other optical path lengths were adjusted so
IR REFLECTING
FILTER
PROJECTION
LIGHT SOURCE
00
X >I
-k a+ C+ d+e = X --
PROJECTION
FRONT SURFACE
MIRRORS
50% TRANSMISSION
MIRROR
MASK I
REAR PROJECTION
VIEWING SCREEN
that mask and screen were at equal distances from the film. At the low
magnifications used, the depth of field of the lens was sufficient to
place the visual projection image in focus when sharp focus was indicated
by the signal from the photodetector. The system employing two masks,
one positive and one negative, was utilized to confirm that this approach
could be easily implemented. A dielectric film reflecting filter was
placed between the projection light source and the film to prevent infra-
red from this source reaching the photomultiplier tube. The combination
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of this infrared reflecting filter at the projection light source and an
infrared pass filter placed in front of the photodetector provided
sufficient separation of wavelength bands for independent operation of
the photodetection system and the visual projection system.
A small audio speaker was used to vibrate the pellicle mirror to
produce output sine waves from the photodetector whose frequency doubles
at exact focus. When the image is out of focus, the phase indicates
the direction required to correct focus.
The use of a microscope to examine the projected image at the screen
position revealed sharp, critical focus as determined by viewing the
grain pattern of the projected image, when the system was adjusted by
visual reference to an oscilloscope showing a second-harmonic signal
from the photodetector.
The laboratory tests of this breadboard confirmed that the external-
image focus-detection technique could be applied to a rear-projection
viewer system of the size and layout required for viewing 9-inch wide
aerial film at magnifications from 3 to 10. Returned-light levels for
the photodetector were more than adequate to provide an excellent signal-
to-noise ratio from the photomultiplier. The use of two multilayer
dielectric filters, one to exclude infrared from the projected image and
the other to exclude the visible spectrum from the photodetector were
more than adequate to isolate the two systems when operating at realistic
illumination levels as would be required in a rear-projection viewing
system.
(1) A focus detection system has been developed offering
the following advantages:
(a) The system detects the actual sharpness
of an image formed by the optical system
in which focus is being measured.
(b) The signal from the detection system can
be used to control any one of the three
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basic parameters, lens position, film position,
or image-plane position.
(c) The actual detection of image focus, combined
with closed-loop control of any one of the
basic focus parameters, permits the system to
be applied to a variety of applications and
equipment.
(co The system is particularly suited to applica-
tion to the dynamic focusing of the rear-
projection viewer in the scanning mode at low
magnifications.
(,a) A direct-application project in which a working
prototype focus-detection unit would be engineered
Lo an existing rear-projection viewer. The com-
pleted system could be used to measure the quanti-
tative technical performance achieved in a real
system, and the viewer could be returned to the
sponsor i'or operational use to determine its
potential functional benefits.
(b) Continuation of the basic theoretical and
laboratory work to extend and refine this tech-
nique and to develop extensions for application
to a variety of optical systems requiring auto-
matic focus detection and correction. Specific
areas of further study should include the ex-
tension of each design to achieve increased
accuracy and range of focus detection. An
investigation of three-dimensional masks and
dual detectors for detecting the direction of
focus error would, if successful, provide an
alternative to the vibrating pellicle mirror
now used for this purpose.
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The purpose of this section is to develop an analytic basis for
the experimental curves of Sec. I.
Suppose that we form a real image of an object, and reflect the
image in a perfect mirror. The reflected image will act as a new source
which is in turn imaged by the lens. If the mirror were located in the
in-focus image plane, then ideally the new image (by optical reciprocity)
would fall exactly on top of the original object. When the mirror is
shifted along the optical axis, the position of the first real image
also shifts axially, and there is a corresponding axial shift of the
position of the second image with regard to the original object plane.
Viewed from the object plane, the second image appears defocused, and
we define this situation single defocus.
Substituting a light diffuser, such as a piece of white paper, in
place of the mirror then the first image is itself defocused whenever
the diffuser is outside of the image plane. The second image formed
by the lens is a focused version of the first defocused image. Viewed
from the original object plane we see a defocused version of the second
image, and this we define double defocus. Single-defocus analysis is
the most relevant here, since the total incident light returned from
ordinary film is primarily specular (the film acting as a mirror) and
only slightly diffuse.
A. Single Defocus
1. Basic Problems of Analysis
A mirror located in the image plane of a lens system will re-
flect the real image, which will in turn act as a new source to be
imaged by the lens. Assuming a distortionless lens, optical reciprocity
shows that the second image will fall exactly on the original object.
Thus, in Fig. II-1, point P is imaged as point Q. All of the light
reaching Q in incident bundle I is reflected about the normal N drawn
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FIG. II-1 INCIDENT CONE I AND REFLECTED CONE R
FROM MIRROR
to the mirror at Q. A first important effect to note is that of all
the light in the reflected bundle R, only a portion R1 is actually re-
imaged at point P. The remainder (R-R') misses the lens entirely and
is lost. The percentage of light lost increases with the angular dis-
placement e of the object.
If the mirror moves axially by an amount. ~qM (Fig. 11-2) then
the image of the source forms not at Q, but at Q', a distance Aq = 2AgM
in front of the original image plane. Point Q1 is displaced from the
FIG. 11-2 CHANGE IN MAGNIFICATION WITH MIRROR
MOVEMENT AqM
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optical axis by exactly the same amount as '
point Q, but since Q1 is
closer to the lens, we see that the angular displacement e' of the
image is larger than the angular displacement e of the original image.
Q' is reimaged as P', displaced an amount ip from the object plane.
We can show that the change in size of the second image is
such that P' falls on a line drawn through the original point P and the
intersection of the focal plane and the optical axis. Let r be the
displacement of P from the optical axis. From the simple lateral magni-
fication relation for a lens, we see that the first image Q is displaced
r' from the optical axis, where
r' _ (q/p)r
As the mirror is moved, the displacement of the new image point Q' from
the axis is the same as for point Q. The moving mirror thus creates a
moving point image of constant displacement r' which is reimaged in some
plane p' = p + Lip. The displacement r" of the returned image is given
r'~ = _p ' ' (r~) = PI (a rl
q
where q' = q + Lq. From the basic lens formula
1 1 + 1
f p q
we readily derive the relation between p' and q', namely
p'f
Thus,
p/ - f
r" - p + Lip - 1 q r
f p
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+ p
r 1 r p
p f
/
, r 1 + 1 q (II-6b)
all of the forms show that r" is a linear function of Op that inter-
cepts zero (the optical axis) at p' = f. In other words, the final image
point P' moves along a straight-line locus drawn between the original
point P and a point on the optical axis at the focal length of the lens,
as shown in Fig. 11-2.
To express r,r its terms of Aq, we simply subtract Eq. (11-3)
from the relation for the second imaging, namely
1 1
p+~p + q+Qq
1~p - p/q
p q P
Lq + f
Lq - q/p
q p q
pp + f
Substituting Eq. (II-8a) into Eq. (II-6b) we find
or in terms of mirror movement AqM = 1/2 L1q
,
r = r
2L1gM p`)
1 + q
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We see then that an accurate analysis of the single-defocus
scheme requires attention to loss of reimaged light with increasing
angular displacement e in Fig. II-1, and the change in size of the second
image, according to Fig. 11-2.
Assume a large opaque sheet in the object plane with a circular
cutout of diameter W that is uniformly illuminated from the rear, Fig.
11-3. Assume further that the opaque sheet has a white diffusing sur-
face facing the lens and that a photodetector monitors the total light
falling onto the white surface.
01 0 02
ILLUMINATION
H
PHOTO MI M M2
- DETECTOR I . I
FIG. 11-3 DETECTION OF INTERCEPTED LIGHT IN OBJECT PLANE
With mirror M in the image plane, the real image of the illu-
minated aperture is reflected by the mirror and exactly reimaged on the
aperture, thus no light is reflected into the detector. If M shifts
towards the lens by an amount AqM, to position Ml, the second image shifts
an amount Op, to position 01, and is enlarged according to Eq. (11-6).
Similarly, if M moves away from the lens to M2, the now smaller second
image shifts towards the lens to 02. With any shift in mirror position
from focus, the second image no longer matches the object, and some light
reaches the detector.
What we are interested in is the magnitude of detected light
as a function of mirror shift AqM. For purposes of analysis, we will
assume that the photodetector collects all of the light intercepted by
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the opaque sheet. We also are interested in how this function changes
with system parameters; in particular, the magnification ratio q/p, the
width W of the object, the lateral displacement of the object, and
angular changes of the mirror with respect to the optical axis.
3. Preview of Analytical Steps
We see from Sec. I that the experimental curves have the
general form sketched in Fig. II-4(a). We hope to understand from the
analysis the reasons for the shapes of different parts of the curve,
and in particular to be able to predict how the curve will vary with
changes in magnification ratio (p/q) and changes in object-to-lens
ratio (W/D).
AqM
(a)
(b)
FIG. 11-4 FORM OF TOTAL RETURNED LIGHT LT AND INTERCEPTED LIGHT Li
AS FUNCTIONS OF MIRROR POSITION Aq.
The analysis proceeds as follows: In the following two sec-
tions, we show that the total light LT returned through the lens has
the form sketched in Fig. II-4(b). For large defocus (i.e., large
values of Al), JgMthe intercepted light LI will have the same general
form, as a function of 6gbi, as the total light LT. However, we are
primarily interested in intercepted light LI for small values of 6q M.
Of special importance is the existence of a range on either side of
6qM = 0 for which the computation of intercepted light is relatively
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simple. In Sees. II-A-6 and II-A-7, we define the value of these ranges,
and in Sees. II-A-8 and II-A-9 we discuss the form of the LI curves over
In general we are able to predict analytically all of the
prominent features of the curve of Fig. II-4(a).
We noted in connection with Fig. II-1 that some portion of
the light reflected from the mirror will cross the lens plane outside
the limits of the lens and therefore will be lost. What we wish to
calculate here is how the proportion of light lost in this fashion
varies with the magnitude of displacement of the object point from the
optical axis, and also on the axial position of the mirror.
Consider a point located a distance p from the lens, and a
distance r from the optical axis, Fig. II-5(a). The image of this
point, which is at a distance q from the lens, will have a displacement
from the optical axis of r(q/p) from Eq. II-1. The circular cone of
light coming to focus at this image point is reflected by the mirror,
and intersects the lens plane with a disk having exactly the same
diameter as the lens but displaced from the optical axis by a distance
S. where
(=)
The light that passes back through the lens is represented by the common
cross-hatched area. The magnitude of the common area A, as a function
of the separation S is derived in Appendix B and plotted as the AqM = 0
curve in Fig. II-5(b). (The derivation in Appendix B is for two disks
of arbitrary radii, (r,R) though we are concerned at the moment only
with the case of equal radii, r = R). Note that A reaches zero when
S = D, where D is the diameter of the lens. From Eq. (II-11) we see
that the value of r, (which we call rc) for which S = D is
_ D p
rc 2 q
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a (Aq +=QqM . -_2 M CO)
I
pqM=-5q// (ogM=2q)
AgM=-a (4 qM q/2)
AqMO(AgM=0)
FIG. 11-5 RETURNED-LIGHT DISK IN LENS PLANE
(a) Separation S of light disk and lens disk
(b) Common area A as a function of normalized object distance
from lens axis, for different mirror positions
At this limiting condition the lowest ray of the focusing cone is exactly
parallel to the optical axis. This ray is reflected back on itself with
the remainder of the cone completely below it.
Let us consider now the effect of moving the mirror along the
optical axis. With mirror movement away from the lens, the diameter of
the reflected light cone in the lens plane is increased to
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q + 2AgM
D D
q
2AgM
= D 1 +
q
For positive values of AqM, D' > D and we have the situation
of Fig. II-6(a). The common area remains constant for a lateral shift
up to S = (R' - R) at which point the two circles become internally
(b)
T4-5884-6
FIG. 11-6 RELATIONSHIP OF LIGHT DISK AND LENS DISK
WITH MIRROR OUTSIDE IMAGE PLANE
(a) Mirror beyond image plane
(b) Mirror between lens and image plane
tangent. The common area decreases monotonically for larger separations,
becoming zero at S = R' + R at which point the two circles are externally
tangent. For negative values of AqM, DI < D, and we have the situation
of Fig. II-6(b).
We can readily generalize the expression (II-11) to include
the effects of different mirror positions, namely
(q + AqM) Aq
S = 2r = 2r q } M
p p q
D =2R
D'= 2R'
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Using Eq. (I1-14), we can find the value of r for which A = 0 (implying
S - R -+ Ii') by setting
f GqM AqM\
2r - 11+ q = R + R 1+2 (II-15)
p ` q
,
1) p
2 q
i.e., the same critical value r that we met before. This result is
c
intuitively clear, namely that this zero overlap condition is reached
when the lowest converging ray is parallel to the optical axis. The
result (11-16) is true for positive and negative values of Lq Thus,
all A(r) curves of Fig. II-5 become zero for r = r .
We see from Fig. 11-5 that each curve is labeled with two
values of LqM. The reason is that the form of the convolution depends
only on the ratio of the circle diameters. Thus, the convolution curve
for a reflected circle of diameter D' = kD, where k > 1 (which occurs
for positive values of GqI), has exactly the same shape, though not the
same amplitude (we will discuss this further), as the curve for a re-
flected circle of diameter D' = D/k, which occurs for negative values
of pqM. We can use Eq. (11-13) to find the corresponding values of
OqM and AqM, by setting
D
Aq M
1 + 2 -
1 + 2 = 0
oqM
C,qM q
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The corresponding values of and AqM are shown plotted in Fig. 11-7.
Note that AqM has an asymptote at (-q/2), at which point the reflected
circle has a diameter D' = 0.
1 I I I I I I I 1 1
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
FIG. 11-7 CORRESPONDING VALUES OF Aq- AND AqM LEADING TO IDENTICAL
SHAPES OF A(r) CURVES IN FIG. 11-5(b)
Let us next find the maximum value of r, namely r , over
which A has its maximum value. Using Eq. (11-14) and setting S = R - R`,
we find
'q- 'q
2r* P 1+ qM =R - [R 1+2qM
- AqM/q
r = r -
c - c
1 + (LqM/q) q + 1
For positive values of AqM, Eq. (11-19) is instead set up as S = R1 -
or
~q 2pq
2r* q 1+ M R + M - R
P q q
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From Eq. (11-18) we find that Eqs. (11-20) and (11-22) yield identical
values of r for corresponding values of AqM and AqM. Equation (11-22)
is plotted in Fig. 11-8. Note that r"K/rc approaches unity for large
AqM, which from Eq. (11-18) corresponds to AqM q/2. As already
noted, the condition iq%, = q/2 corresponds to D' = 0 (a circle of zero
diameter), for which case we should expect the convolution curve to be
constant for 0 !~ r 5 r , and then drop abruptly to zero, as seen in
t:
* AqM/q 1
(AqM/q) c q F + 1
Aq
I I I I I I I I
i.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
M +
q
FIG. 11.8 RANGE OF r OVER WHICH COMMON AREA A HAS MAXIMUM VALUE
One final point. about the curves of Fig. 11-5; though the
A(r) curves have identical shape for corresponding values of AqM and
AqM, their amplitudes differ considerably. For r = 0, all of the re-
flected light is collected for negative AqM, since D' < D. For positive
AqM, however, D' > D, and the proportion of collected light C is only
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2
C =
(2D--')
or from Eq. (11-13),
which is plotted in Fig. 11-9. Thus the curves for positive values of
AgMj1 which are noted in parentheses in Fig. 11-5, have rapidly decreasing
amplitudes in accordance with Fig. 11-9.
0.6
C
0.4
r
1.0 2.0 3.0 4.0 5.0
?M+
g
FIG. 11-9 FRACTION OF RETURNED
LIGHT FOR A POINT SOURCE
ON AXIS AS A FUNCTION
OF OqM
We have discussed the results only for - q/2 Aq M S 00. We
must also consider the range - q :r AqM - q/2; this implies mirror
movements right up to the lens. In Fig. II-10(a) we illustrate the
case of AqM = - q/2., where the reflected circle has a diameter D' = 0.
For even larger magnitudes of AqM we see from Fig. II-10(b) that D'
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2
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AqM: -q/2
FIG. 11-10 FRACTION OF RETURNED LIGHT OVER RANGE q/2 Aq- q
increases in amplitude again, and that for r = r C the reflected circle
again lies tangent to the lens circle, but inside the lens circle. Thus,
the common area remains constant for 0 S r r for all values of
c
- q pqM q/2. As r increases beyond rc the value of A decreases
until A = 0 at a value rwhich has identically the same form as r*
in Eq. (1I-20). From Fig. II-10(b) we can write
M
q + AqM - q ;
D/2 _ ** q
r -
p
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** 1
r = r
c q + 1
AqM
The resulting curves for - q ' AqM S - q/2 are shown in Fig. II-10(c).
5. Disk Object--Total Light Return
We have calculated the fraction of light returned from a
single point source at a distance r from the optical axis, in terms of
the magnification ratio p/q, lens diameter D. and mirror position AqM.
For an object extending over an area we can integrate the fraction of
light returned from each point of the object, and obtain a measure of
the fraction of light from the entire object that passes back through
the lens. Because of the rotational symmetry this integral is parti-
cularly simple for a circular disk object and a circular lens aperture.
The total fraction of light F passing back through the lens
fora circular disk object of diameter W, Fig. II-3, is simply
W/2
F = 1 2'rrr A(r)dr (11-27)
(rrW2 ) (TrD2 ) [17(R') 2 ] 0
where A(r) is the fraction of light returned from a point source located
a distance r from the optical axis. The function F(AgM) has the general
form shown in Fig. II-4(b); a total fraction of returned light that is
less than unity at perfect focus (AqM = 0), increasing to unity at some
value of AqM, and decreasing monotonically with increasing AqM. The
function F reaches unity at that value of AqM for which all of the light
is returned for all values of r up to r = W/2, which we find from
Eq. 11-20 to be
1 _
D p 1 1 + CY
W q
where we substitute for the common expression W/D q/p. Thus all
curves start at unity for sufficiently large AqM and monotonically
decrease for increasing AqM But for sufficiently large AqM, all
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curves again become identical in shape. This occurs at that value of
AqM for which the lens is completely surrounded by the reflected circle
for all values of r up to r = W/2. To find this value we recall the
relations
~q
D3 = R 1 + 2 M
q
Aq
S 2rq (l+ M
The critical value occurs where S = R1 - R, for r = W/2,
Qq-i-
22 Oq-f-
p (1 qM = 2R qM (11-29)
-i-
q
ql
q Dp - 1 (Y
1 -
Wq
Thus, for all (pqM/q) z CY/(l - cY) the fraction of light returned is
identical for all points of the object, and therefore for the object as
a whole, and the resulting curve should have the same shape as that of
Fig. 11-9 for a single point source.
We are primarily interested, however, in the magnitude of
intercepted light LI as a function of pqM. We know that this function
is zero (at least ideally) at AqM = 0, rising on either side of AqM = 0.
But because the total amount of returned light decreases monotonically
with OqM, we can expect that L1(i) would have the form sketched in
Fig. II-4(a), as actually found in the curves of Sec. I.
6. Defining the Critical Values Lp? and AgMc
In the last section, we calculated the total amount of light
returned through the lens. We concentrate now on the more complex
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problem of the total light intercepted, outside of the disk object, which
is the output measure that we are really interested in.
Consider a point r on the disk object, which has a diameter W.
This point images in the q-plane, a distance r (q/p) from the optical
axis, Fig. II-11. If the mirror moves towards the lens by an amount
FIG. II-11 POINT SOURCE r REIMAGING AT r": CAUSING A LIGHT DISK
OF DIAMETER W' IN OBJECT PLANE WITH CENTER
AT DISTANCE S' FROM AXIS
oqM, then the point image moves (2AgM) towards the lens to point r',
and the cone radiating through r' forms a disk of diameter D' in the
lens plane, whose center is displaced a distance S from the axis [see
Eqs. (11-13) and (II-14)]. If S > R - R' then part of the R' disk is
lost outside of the lens [Fig. II-12(a)]. This truncated cone in turn
images at point r", which as we saw earlier lies along a locus drawn
through the object point r and the focal point f. As this truncated
cone passes the object plane (the p-plane) some of the light is inter-
cepted beyond the edge of the original disk, namely the cross-hatched
region in Fig. II-12(b). If the imaging cone were a pure circular disk,
the fraction of intercepted light would be easily calculated. However,
if the focusing cone is sufficiently truncated, then the intercepted
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light is reduced by the darkened areas A and A' in Fig. II-12(b), and
precise calculation is difficult.
FIG. 11-12 RELATIONSHIP OF OBJECT DISK AND RETURNED-LIGHT DISK TRUNCATED
BY LENS
movement, AgMc, and corresponding value Ap`, such that if AgMc s AqM `= 0,
the truncation can be neglected without error in calculating total inter-
cepted light. We show in Appendix C that at the limit AqM AgMe the
truncation points just reach the disk rim as in Fig. II-12(c), so that
the areas A and Al of Fig. II-12(b) are zero.
Because of the closed-loop nature of our automatic focus-
control system, we are not really concerned with the exact shape of the
L1() curve on both sides of AqM = 0. However, we are very concerned
with its width, since this defines the operating range of the servo
system. It turns out that the AgMc value gives a good measure of the
range, the curve being approximately linear for 0 s Aq s AgMc and
falling off for AqM > AgMc. The AgMe points are marked on all of the
experimental curves of Sec. I.
We can define a mirror position AgMc (and corresponding value
Apc) which also gives an estimate of operating range for AqM > 0. But
whereas specular light loss beyond the lens is ineffective over the
range 0 S AqM AgMc we will see that it does have an effect over the
range 0 5 AqM AgMc
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To find the critical value Lp+ let r = 0 in Fig. II-11. Now
e
as the mirror moves towards the lens, we have two effects. First, the
image point moves in a +Lp direction along the optical axis. Second,
the intercepted cone radius R' decreases so that RI < R. At some value
of AqM, the geometry is such that the intercepted disk diameter in the
p-plane exactly matches the object diameter; that is, the converging
cone has diameter W in the object plane (p-plane). This condition is
shown in Fig. 11-13(a) (the cone from C2 has a width W in the object
D~ + (Lpc) = W
p + Lpc
Substituting Eqs. (11-13) and (11-7), and noting that pq = 2qM, we find
AP +
c 1
p = D q (11-32)
W f
or, equivalently, using Eq. (II-8a),
&qc
Note that because p and q are defined as positive in the direction away
from the lens, a (+pp) corresponds to a (-pq) and vice versa.
For small Lp only the light from the edge of the disk contri-
butes intercepted light. The zone of contribution increases monotonically
with Lp. When Lp = pp+, the zone of contribution just reaches to the
c
center of the disk. This is one way to interpret the significance of
Lpc.
gyp? [Fig. II-13(a)]. Consider the center C, of the disk in the object
plane. Only when Lpc reaches the magnitude pp' is there a contribution
at C1 from every cone focusing on the disk in the Ap image plane.
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FIG. 11-13 GEOMETRICAL CONFIGURATION AT CRITICAL VALUES App AND
stituting AgMe/q = 1/2 Aqc/q - a/2 into Eqs. (11-13) and (11-14),
we find
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Consider the extreme cones E and Ef from the edge of the disk. Sub-
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S + R' - R = R(a - a2) (11-36)
which is positive since cy < 1 in all cases of interest to us. In other
words S + R' > R so that the cones to the edge points extend beyond the
lens. Therefore the extreme rays to E and E' emanate from the edge of
the lens. For these extreme rays to cross at the center C1 of the disk,
D W
P Apc
Substituting Eq. (II-6b) for W', we again find the result to be Eq.
(11-32) for ppc which demonstrates the alternate derivation.
7. Defining the Critical Values Lp. and Aq+
As the mirror moves away from the lens in the QqM direction,
the image moves towards the lens in the -pp direction. As we did in the
previous section for ApC, we can find a critical value pp c from the
following geometric ratio in Fig. II-13(b),
D
P + Apc - tpc
which in effect states that the reimaging cone from the center of the
disk object, which comes to focus in the center of the image in the
ipc plane (defined by extreme rays V and L), creates a blur circle in
the object plane whose diameter is equal to the disk diameter, namely
W. (Note that ppC will be a negative quantity.) Equation (11-38) can
be put in the form
Apc W
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which, by means of Eq. (I1-8b), we can convert to the ~q system:
_ - q/p _ cx
1 -
P q Cy
Apc
A
q +
For the sake of completeness, we show that the second type of
derivation used in the previous section for AgMc can be used for Aq+c
as well. In this case, we ask that each cone coming to focus in the
/~p plane contribute a ray at the center of the object in the 6p = 0
c
plane. Let us consider the cones coming to the edge of the image in the
pp plane [Fig. II-13(b)]. In particular, we consider the extreme ray
c
U' which represents the topmost ray in the lens plane of the returning
cone. In this case, where the mirror is moving away from the lens, the
intercepted disk in the lens plane is larger than the lens disk (i.e.,
R' > R). The uppermost ray would occur at a distance R' above the axis
if there were no displacement of the disk, but with a shift S the upper-
most ray is a distance (R' - S) above the axis.
From Fig. II-1.3(b) we see that (R' - S) must satisfy the
following geometric condition
(R' - S) = W'/2 p
Apc
where W' is the image size in the Lpc plane. Using Eq. (II-6b) for W',
Eq. (11-42) takes the form
(R' - S) 2 (....E...- + f
~p
c
or in terms of Aq+, using Eq. (II-8a),
(R' - S) = W q q = D (a) q
2 qc p 2 Lq+
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Now, using Eqs. (11-13) and (II-14), we can evaluate R' - S as
Aq+ Aq+
R ~ - S = R 1 + qc - 2Rc 1 + 1 qc
Equating Eqs. (11-42) and (11-45), we can solve for Aq+, which is
c
identical to Eq. (11-4).
"8. Intercepted Light Over Range 0 Ap S Ap+
c
We have already noted that the exact shape of the intercepted
light curve is not too important in the region about AqM = 0. However,
we can readily demonstrate the ease of computation in case it should be
necessary to compute this function.
For any given mirror position we can evaluate LI in the form
W/2
LI = f 2rrrI (r)dr
0
assuming a disk object of diameter W, where I represents the fraction
of. the final intercepted light from a point source within W, a distance
r from the axis.
The intercepted light I is generally a complicated function of
r because of the two-step process of (1) truncation by the lens of the
returning light cone; and (2) intersection of the smaller truncated disk
with the original object. But a great simplification is achieved over
the range 0 5 Ap 5 Ap? because the truncation has no effect on the
calculation, as we show in Appendix C. Over this range, then, I(r)
involves only the convolution of the intercepted cone disk with the
object disk.
The convolution of two circles involves three parameters,
namely the radii of the two disks and their separation. The object
disk has a diameter W. The intercepted disk has a diameter W1, where
from Fig. II-11
D Ap
p+Ap
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W/ D k 1 -} q q) p Lp 1)
Substituting Eq. (II-8b), Eq. (11-48) becomes
i) q
Ap I
(11-48)
As for the separation distance S', from Fig. II-11 we can
S + r AP
p+Ap
which, with Eqs. (II-6b) and (11-14) can be manipulated to the form
P + 1
r rp
~pf
In summary then we have the relations:
I(W, W`, S')
I 1
W = D
P cI
Ap i'
P -I- l
LP
r
c
P
pp + f
We can use Eq. (II-8a) to convert the expressions for W' and S' into
functions of Aq:
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W i _ - Dp Aq W Aq
q q A' q
S' = r AS + 1
Thus, for each value of Aq, W, and W' are fixed, and we can
integrate Eq. (11-46) since for each value of r we have a unique value
of S' and can readily evaluate I (W, W', S'). We must include a nor-
malization factor to account for the fact that as the blur circle in
the object plane changes size, the average intensity varies inversely
as the square of its diameter. Thus the expression for I must contain
the factor 1/[(Tr/4) (W')2].
9. Intercepted Light Over Range 0
Ap e Apc
Fcr negative Ap the return disk in the lens plane is larger
than the lens disk and, in the range of interest here, does not com-
pletely illuminate the lens disk. In Appendix D we show that if the
light returning through the lens is treated as coming from an illumi-
nated lens disk, but with a small truncated section (i.e., the part
not illuminated in the first place by the returning light cone) then
the same conclusion holds over the range 0 Ap S Apc as over the range
0 e Ap E Ap+; namely, that any light in the truncated section would
c
have passed through the object aperture anyway and therefore need not
be accounted for in computing the intercepted light.
For the negative range of Ap (or positive range of Aq) we
can set up an integral for the intercepted light LI in the same form
as Eq. (II-46), namely
W/2
LI = J 2TTr I(r)dr
0
To evaluate I(r), which is a simple convolution of two disks, we must
know the two disk diameters and their separation. The convolution is
in the object plane, so again we have the object disk of diameter W
for one of the disks. The second disk is a blur disk of diameter W',
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which is seen from Fig. D-1 (in Appendix D) to have the form
D Ap
P1' =
p+Ap
The separation S' of these two disks as we again see from Fig. D-1, has
p
_ r
p + Ap
Using Eq. (II-8a) these equations can be converted to the Aq system as
W 1
rV
+ q+
A,q
q 4-
4q pl
q + 1
Aq
Thus, for each value of Aq, W, and W' are fixed and we can
integrate Eq. (11-54) since for each value of r we have a unique value
of S' and can readily evaluate I(W, W', S'). Again, as in the ApF
range, we must include a normalizing factor in the expression for I,
namely l/[(,-r/4) (W')21, in order to give the proper weight to the common
area of W and W' in terms of light intensity.
However, another important normalizing factor is needed in
the Ap range that was not required in the Ap+ range. In the present
derivation we treat the return light cone from the lens as arising from
the lens disk itself (with a small "truncated," but really not illumi-
nated, sector). However, the light intensity over the lens disk itself
varies as (D/D')2, which from Eq. (11-13) has the form
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This factor, which depends only on the value of Aq; is independent of
r and therefore can be factored outside of the integral in Eq. (II-54).
If it were not for this factor, the LI(Aq) curves would tend to be
similar over the ranges 0 Aq < Aqc and 0 Aq < Aqc. This extra
factor tends to bend the curve over in the former range, as Aq increases,
and we therefore understand the experimentally found asymmetry in these
two ranges.
With a diffusing surface instead of a specular reflector, the
analysis for intercepted light as a function of AqD (the position of
the diffusing surface) is different in detail than with the mirror.
To illustrate, consider again a cut-out disk of diameter W. at a dis-
tance p from the lens, with an image distance q. If the diffuser moves
axially by an amount AqD, then a blurred image appears on the diffusing
surface. (Movement of the mirror simply caused an axial shift of the
in-focus image.) Light re-emitted from the diffuser is reformed in an
in-focus version (in some plane p + Ap) of the out-of-focus image; see
Fig. 11-14.
q
P
FIG. 11-14 USE OF DIFFUSER INSTEAD OF MIRROR IN IMAGE PLANE
Consider a single point on the disk object, in particular the
uppermost point r of the disk. Light from point r focuses in the image
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plane at r'. Movement of the diffuser by Aqn results in a blurred disk.
Light reradiated in turn from each point of this blur disk tends to re-
focus in the p + pp plane. Consider the upper and lower points B and A
of the blur disk. These tend to reimage as points B' and A' in the
p + Lp plane. Now consider the light intercepted outside of the object
disk in the p-plane. We see from the figure that in the p-plane the
intercepted disk to B' is externally tangent to the object disk, and the
intercepted disk to A' is internally tangent, so that the entire cone
converging on A' is intercepted, whereas none of the converging cone to
B' is intercepted. It is clear that to find the total intercepted
light, we could perform the convolution for the entire disk.
We do not pursue the analysis of double defocus further here,
however; first, because as we noted earlier it is not particularly
relevant to this application, and second because there is at least
some rudimentary analysis along these lines available.
STAT
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DERIVATION OF FOCUS ERROR EQUATION
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DERIVATION OF FOCUS ERROR EQUATION
The distance between the plane of the real image on the film and
the plane which is in focus on the screen is the focusing error. The
focus-detection system is assumed to position the lens so that at all
magnifications the mask is in exact focus on the film reflective sur-
face. The resulting position of the lens as measured from the film
surface is given by
qi _ K~ 1 - 1 - 4f'/K'
2 (A-1)
where fl is the focal length of the lens for the spectral band of light
used for the focus-detection system, (f is the visual focal length
of the lens). The position of the mask is given by K', which is the
distance from the film surface to the mask. Reference to Fig. 1-18
will aid in visualizing these distances. The distance from the lens
to the screen is now
where L is the distance from film surface to screen. The in-focus
point for the visual screen is given by
pf
p - f
where q is measured from the lens. The error in focus is then
E = q' + t - q (A-4)
This formula is obtained by solution of the lens formula under the
constraint that the distance between image and object is fixed at K1.
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E = q ' + t. - L LL-7
f
L - ' q (q' + t) - q' - t - L + q'
L - q' _ 1
f
E _ ( q r + t ) (L - q ' ) - f(L + t)
L - q' - f
K'
E (qt' + tn) (1 - q'n ) - (1 + tn) fn
E =
n L
f = f K'
n L Y L
To check, let y = k = 1; then
1 - qn - f n
qn _ 2 1 1 2 1 - 4f
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1 - qn 2 + 2 1 - 4fn
qn (1 - qn) = 4 - 4 (1 - 4fn) = fn (A-11)
Substituting Eqs. (A-9), (A-10), and (A-11) into Eq. (A-7):
E _
n
t
fn + 2n (1 + 1 - 4fn) - fn - to fn
2+2 1-4ff
t
n (1 + 1 - 4f ) - t f
2 n n n
2 (1 + \1 -4f ) - fn
2 (1 + 1 - 4fn) - fn
to 2 (1 + 1 - 4fn) - fn
E = t
n n
Thus in the case where visual focal length and detection-system
focal length are the
same (a perfect achromatic lens), and the mask
and screen are exactly the same distance from the film, the focusing
error is constant and equal to the film thickness.
For purposes of discussion, it is useful to define a quantity m
such that
(m* + 1)2 = f
n
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This quantity is very close to the actual magnification at which the
system is operating.
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CONVOLUTION OF CIRCLES OF DIFFERENT DIAMETERS
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APPENDIX B
CONVOLUTION OF CIRCLES OF DIFFERENT DIAMETERS
We are interested here in deriving the common area A between two
circles having different radii r and R as their relative separation S
is varied (see Fig. B-1). The method of approach is to set up a double
FIG. B-1 ARRANGEMENT FOR
COMPUTING COMMON
AREA BETWEEN TWO
UNEQUAL OVERLAPPING
DISKS
integration in order to calculate the cross-hatched common area:
RcosO 2_x2
A = 2 ~ dx dy (B_l)
0 S- r -x
where 0 is the angle shown in Fig. B-1. The relation between 0 and S
is derived from
S = r sin cY + R sin 0 (B-2)
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R cos e = r cos a (B-3)
Substituting Eq. (B-3) into Eq. (B-2) we find
` 2
cost 8 + R
(R ) -t
which can be manipulated to the form
-?-R
5 - r
r2 2
2SR
Let us continue with the evaluation of Eq. (B-1) and then substi-
H from Eq. (B-5). The indefinite integral of the
2 x a2 - 2 + a2 sin-1 (ax / (B-6)
Applying Eq. (B-5) to Eq. (B-1), we obtain
A = R cos e
s26+ V1R/
cos2 e (B-4)
+ R2 sin 1 (cos 0)
cos2e - RS
+ r2 sin-1 (R cos (B-7)
r J
which can be converted by means of Eq. (B-5) to the form
A RS cos 0 + R 2 sin-1 (cos 0) + r 2 sin-1 (R cos e~ (B-8)
= - r
To check the validity of Eq. (B-8), we note that for r = R and S = 0,
Eq. (B-5) correctly predicts that cos 0 = 1 (or e = 0), and Eq. (B-8)
2
correctly predicts A = TTR. Also, for r = R, and S = 2R (i.e.,
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externally tangent circles), Eq. (B-5) correctly predicts cos 9 = 0
(or 9 = TT/2) and Eq. (B-8) correctly predicts A = 0.
Now we must note that Eq. (B-6) applies only for S > S where
c
S c R - r2. How this limitation comes about can be seen from
Fig. B-2(a). When S = Sc, the center of the smaller circle lies exactly
on the common chord of intersection. This case represents the largest
(a)
FIG. B-2 COMPUTATION OF OVERLAP AREA FOR SMALL VALUES OF
SEPARATION
possible chord of intersection, namely the diameter of the smaller
circle. For S < Sc, the integral in Eq. (B-1) is incomplete since it
accounts only for the area bounded by the heavy lines in Fig. B-2(b),
but not the cross-hatched areas. The area A' of the cross-hatched
regions is readily evaluated from
r
A= 4 J - x dx (B-9)
Rcos6
which by application of Eq. (B-6) reduces to
F(R A TTr2 - 2R2 cos 9 -
l
- 2r2 sin-1 (R cos Af
r
(B- 10)
cost 6
Adding A' to Eq. (B-8) and simplifying, we obtain for 0 iS S R - r
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R2 - r2 - 2S2
S
+ R2 sin I (cos 8) - r2 sin-1 rR cos 8
1 r 1
To verify these relations note that for S = 4R-T-r2
Eq. (B-5)
correctly predicts cos H = r/R, and Eqs. (B-7) and (B-11) predict the
identical value of area, namely
T? Y' (
A = - r 1 R2 - r2 + R2 sin-_1 IR) (B-12)
which predicts the correct value of A as R -? w, since the common arc
through the smaller circle then becomes a straight line, and we expect
simply that A TTr2/2.
Figure B-3 shows a set of computer results for various ratios of
R/r. Note that for R/r = 20 the curve is almost symmetric, as it should
be for R/r -. For R = r, the curve is remarkably linear. The curve
shape changes rapidly with small increases in R/r, the R/r = 2 curve
lying almost halfway between the R/r = 1 and R/r = 20 curves.
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_> 40[S-(R-r)
FIG. B-3 COMPUTER-GENERATED CURVES OF COMMON AREA AS A FUNCTION
OF SEPARATION S OVER THE RANGE R-r < S < R+r
FOR DIFFERENT RADIUS RATIOS r
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Apc
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PROOF OF NO TRUNCATION LOSS IN RANGE 0 e ~p N Apc
Our goal here is to prove that over the range 0 Lp _2 Lp+ the
c c
truncation by the lens of the reimaging cone does not have to be con-
sidered in calculating intercepted light.
Referring back to Fig. II-11, assume first that the lens does not
truncate the reimaging cone at r", and that a pure disk of diameter W'
is intercepted in the object plane, with the center of the disk dis-
placed S' from the optical axis. Let us immediately jump to considera-
tion of the Ap = ip+ condition, in which case W' = W (that is, the
c
intercepted disk is exactly the same size as the object disk). Sub-
stituting Eq. (11-32) for pp?/p into Eq. (11-50) for S' we obtain
S~ = r (1 - cx)
which for the largest value of r, namely r = W/2, becomes
S` = 2 (1 - (Y)
Before we can use these relations for W' and S' we must return to
the lens plane and determine the form of the truncated cone formed by
lens blockage, in particular the width z of the truncated zone in
Fig. II-12(a).
From Eq. (11-33), we see that in terms of mirror position
AgMc tr
q 2
Substituting Eq. (11-33) into Eqs. (11-13) and (11-14), we have
= D (1 - a) , (C-4)
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cy (2 - cr)
Now the truncation distance z is simply
r = (; + H') - R = 7 cx (1 - a) (C-6)
The fraction of the radius of disk R' that is truncated is
D
c (1 - cx)
With this result, we can now return to the object plane and complete
our proof.
What we wish to show now is that the truncating arc, which is part
of a larger (dashed) circle of diameter W/2 (1/1 - cy) falls exactly on
the intersection point of the two equal-area circles of diameter W. To
demonstrate this we consider the darkened triangle and show that the
value x (which is the distance from the center of the larger circle to
the intersection point) is just equal to the diameter of the larger
circle, so that the large circle does in fact pass through the inter-
section point.
To solve for x, we must find y and e. One point that we can note
immediately is that the truncation arc passes exactly through the center
of the lower circle, since the truncation distance z' = W/2 CY exactly
compensates for the amount by which the separation S' [Eq. (C-2)] is
short of a complete radius of separation; this amount is W/2 a . With
this simple observation, we can write
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. WI
y = 2 - S
where W'/2 is the radius of the larger circle, or
y = 2 1 j - 2 (1 - 01) = 2 11 cx/ a
2
x2 = y2 + 12 2y RD cos (2 + 9)
" 2 2
cost
2SR
where R and r are the general radii of the two intersecting circles. In
our case, the circles have equal radii, namely R = r = W/2. Furthermore,
we are really interested in sin A, since cos [(77/2) + A] _ - sin g, and
we can write
si
n2 6 = 1 - cost 9 = (TSX (C-12)
But in our case S = S' = W/2 (1 - CY), and R = W/2, so that
x2 (2W
ll - 01)2 U2 -f- 1 + 2 (1 - Q) a (1 2 ~a,(C-14)
X 1 - CY
which is precisely what we wished to show.
C-5
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What we have in effect shown is that for Op = App the truncation
c
portion of the converging cone would have passed completely back through
the object aperture anyway, and so does not affect the calculation of
intercepi.ed light. We have shown this-for the extreme value r = W/2.
If it is true for r = W/2, it is clearly true for smaller values of r,
see the conclusion is true for every point on the disk. Furthermore,
it the conclusion is true for Ap = ApE it is also clearly true for all
Smaller values of Up over the range 0 Ap Ap .
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PROOF OF NO TRUNCATION LOSS IN THE RANGE 0 5 Lp < ppc
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PROOF OF NO TRUNCATION LOSS IN THE RANGE 0 5 Qp < Qp
c
In Appendix C we showed that over the range 0 5 Qp < Qp~,
any re-
turn light that does not pass the lens will have passed through the
object aperture anyway, and therefore need not be accounted for in com-
puting the intercepted light LI.
Interestingly, a similar conclusion can be drawn for the range
0 pp Qpc if we properly define the "truncated" returning disk of
light. However, there is an important new intensity effect that must
be included in the integration for intercepted light LI.
In the derivation of Appendix Cl for the range 0 < Qp Qp+, the
c
return disk of light in the lens plane is smaller than the lens in
diameter and has a portion truncated by the lens [Fig. 12(a)]. For the
range 0 Qp < Qpc the return disk in the lens plane is larger than the
lens, and it turns out to be more convenient in this case to view the
returning light that passes the lens (cross-hatched region in Fig. D-1)
as deriving from the lens disk itself, but with a small truncated seg-
ment of height z. (Actually, it is not a truncation in the earlier
sense; rather there is no light in this region.)
FIG. D-1 GEOMETRIC CONFIGURATION FOR Ap =AP__
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Note that for an object point above the axis [Fig. D-l(a)], there
is a downward displacement S of the returning disk, and the truncated
portion occurs at the top of the lens disk. The truncated cone comes
to focus in the Ap plane and then spreads out again so that in the
c
object plane the truncated portion now appears at the bottom of the
disk. What we would like to show is that at the extreme of the range,
namely Op = Apc, exactly the same condition holds as in Fig. II-12(c).
From Eqs. (11-39) and (11-40) we see that
gyp
I)
- B- +
For this condition, we have the following expressions for D' and S,
from Eqs. (11-13) and (11-14):
Aq)
S Wq
D 2 - a
2 < 1 - fY
We readily compute the truncation distance z as
z = 1~ - (R' - S) = 2 a
or, as a fraction of the radius of the circle being truncated (namely
the lens disk) we have, in analogy to Eq. (C-7),
z
tx
R
Now by the definition of pp we know that the diameter of the blur
disk in the object plane is equal to that of the disk object itself,
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+
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namely W. We have then just one other quantity to determine, namely the
displacement S' of the truncated blur disk in the object plane.
From Fig. D-1(a) we have the geometric relation
S' W"/2
p + Lpc
where W' is the diameter of the in-focus disk in the Lp plane, which
c
from Eq. (11-6) takes the form
W,/
W p + ip - f
W p + Apc - f
S
2 p - f
p + Apc
But these are exactly the same conditions as in the proof of
Appendix C, namely two equal circles (our circles of diameter W), dis-
placed by a distance S' = W/2 (1 - (v), with a truncation fraction that
derives from a pair of circles whose diametric ratio is (1 - cx); i.e.,
our lens-plane circles D and D'.
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