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Fundamentals of optical design in rotating mirror cameras
Fundamentals of optical design in rotating mirror cameras M.I. FINSTON The elementary optics of rotating-mirror high-speed cameras are analysed geometrically. The images in an off-axis scanning mirror are shown to lie on Pascal’s limacon. The Miller principle by which shuttering is accomplished is described. A simple estimate of motion blurring is compared with the Rayleigh criterion. KEYWORDS:
optical instruments,
high speed photography,
Introduction
The system can produce film strips whose separate frames are exposed at intervals shorter than a single microsecond with each exposure lasting a fraction of that interval.
Ordinary photography is useful in recording scenes that do not change appreciably over time intervals of 500 ys’or longer-500 ps is the duration of the briefest exposure available on a typical professional-model 35 mm camera. Various strategies are used for the photographic recording of motions that go beyond this limitation. ‘Strobe’ lighting, for example, enables the photographer to record many stages of a rapid but complex, motion on a single frame of film. The resulting multiple exposure then represents a sequence of images that may have been recorded at intervals of 1 p.s or less.
The basic design principles of the ultra-high speed framing camera can be understood in terms of simple geometrical optics. The exact details of the optical design vary somewhat from one to another of some dozen models that are in current use in the USA This present exposition is loosely based on the Cordin Model 121 camera’, which was first publicly exhibited in 1978. This model exposes 26 frames on a strip of 70 mm film. Each frame measures 38.1 x 73.5 mm. By means of a beryllium mirror that can rotate at angular speeds up to 10 000 Hz, the camera achieves framing rates up to 2.5 million frames per second. For a more general treatment of the subject the reader is directed to Dubovik’s comprehensive book*, which is widely regarded as the standard textbook on high-speed photography.
Many situations arise in the study of aerodynamics, ballistics, and high-explosive technology when it is desirable to produce a ‘motion picture’ of an object or disturbance that travels at several times the speed of sound. In these situations, the ideal photographic record would be able to reproduce fine detail (that is, have high resolution), and be large enough for accurate measurements to be taken from it conveniently. The recording method should produce a sequence of separate images representing moments separated by a constant time interval (the interframe time). The exposure time for each individual frame should be brief enough so that the motion of the subject does not cause significant blurring of the image. Finally, the possibility of using colour film adds extra depth to the range of information that can be preserved.
Camera design The main elements of any ultra-high speed framing camera are: a stationary strip of film; a moving mirror that deflects the optical beam along the film; an arrangement of masks that shape and ‘shutter’ the light beam passing through the camera; and an unusual arrangement of lenses that functions as three separate imaging systems.
The ultra-high speed framing camera was designed to satisfy all of these requirements. This device employs ordinary photographic film, and the conditions under which it can form an image are described in the same terms as any amateur camera: focal length, illumination, film speed aperture, exposure time. At the same time, the ultra-high speed camera uses an ingenious combined optical-mechanical system to solve the problems of shuttering and film advance that limit even the most sophisticated motion-picture camera. The author is at the Lawrence Livermore National Laboratory, PO Box 808, Lwermore, CA 94550, USA Recewed 25 October 1984. Revised January 1985.
0030-3992/8~/020083-06/$03.00 OPTICS AND
LASER TECHNOLOGY.
APRIL
The overall layout of a typical framing camera is shown in simplified schematic form in Fig la. (Fig. 1b shows the actual optical layout in the Cordin Model 121 camera.) Light enters the camera through objective lens I.+. The aperture stop Pi, located at the objective lens (in this simple design), determines the size of the cone of light the camera will accept from an object point on the optical axis. Thus, Pi determines the illumination at the final image. All the rays emerging from the aperture stop that are to be used must pass through the field lens L, and reflect from the front surface of mirror M. As will be shown later, the field lens forms an image of the aperture stop in the plane
4
0 1985
systems design
1985
Butterworth
Et Co (Publishers)
Ltd 83
film. This is achieved by projecting a image onto the mirror surface. As the it casts the light from this image onto lenses, lighting up only one relay lens
Aperture stop >
stationary mirror rotates, a row of relay at a time.
The mirror rotates with a frequencyf = w/2n that may be as high as 20 000 Hz. To achieve such high angular velocities, the mirrors are turned by gas-driven turbines which are powered by compressed nitrogen or helium. The flowing gas also partially supports the moving parts, reducing friction and wear on the bearings. Furthermore, the mirrors turn in a vacuum or in an atmosphere of helium to minimize drag At such high rotational velocites, the spinning mirror must be dynamically balanced; that is, the rotational axis must coincide with a principal axis of the mirror. Often, the mirror is shaped like a triangular prism whose cross-section is an isosceles or equilateral triangle. Such a mirror may have one, two, or three reflecting surfaces.
b
The main factor that limits the time resolution of the ultra-high speed camera is the maximum rotational speed of the mirror. The centrifugal force acting on the mirror is very great. The limiting speed is reached at the point where the centrifugal force is not quite great enough to cause the mirror to deform or to burst. The greater the density of the metal of which the mirror is made, the greater is the centrifugal force at any given speed Thus in choosing a material for the mirror, the main criterion is the ratio of tensile strength to density. This problem was discussed by Schardin in a 1956 paper”. In that paper, he proved that the product of the attainable spatial and temporal resolutions of a rotating mirror camera is proportional to the square root of the strength-to-density ratio.
Synchronization system
Fig. 1 a Overall layout of typical framing Cordin Model 12 1 camera
camera; b -
actual layout in the
of the relay lenses LIl, to Lb. (The field lens gets its name because the diameter of this lens determines the angular field of view of the camera: that is, it determines how far an object point can be from the optical axis and still be represented in the final image). Figure la shows that the view of each lens is limited by a mask, or shutter stop. The image of Pi projected by the field lens corresponds exactly, in size and shape, to the openings in this mask Thus, as the mirror rotates about axis A, a ‘pencil’ of light is swept along the row of shutter stops, entering only one relay lens at a time. Each relay lens, in turn, receives its maximum illumination when the image of the entrance pupil is brought by the mirror’s motion into exact coincidence with it. The field lens has a second function: the combination of this lens with the objective lens I+ forms an image of the object 0 onto the mirror M. This image is intermediate: each relay lens, in turn, projects the image at the mirror into a final image on the film. Thus, the subsystem consisting of shutter stops. relay lenses, and film may be thought of as a collection of n separate cameras whose shutters are operated in sequence by the rotation of the mirror. The basic principle by which the ultra-high speed framing camera operates was invented by C.D. Miller’ in 1939. It can be summarized as follows: A rotating mirror can be used to project a sequence of time-resolved images onto a stationary piece of
84
To make the ratio of tensile strength to density as great as possible, most cameras use steel or beryllium mirrors. Because it is so light, beryllium offers a significant advantage over steel. However, the tendency of beryllium mirrors to suffer brittle fracture (possibly with explosive force) imposes safety constraints on their use.
Theoretical basis The fastest spinning mirrors are shaped as polygonal prisms. But the requirement of dynamic balance makes it impossible for the rotational axis to lie in the reflecting surface of such a mirror. Fig. la shows a mirror whose axis A lies a distance d behind the reflecting surface. As the reflecting surface moves around axis A, it varies its distance from lenses b and LF and from the relay lenses LE. How does this affect the location of the intermediate image viewed by the relay lenses? How does one calculate the correct shape for the surface containing the relay lenses, and for the strip of film behind them? If the rotational axis lay in the reflecting surface, the relay lenses would lie along a circular arc, as would the film. In practice circular arcs are sometimes used, although they are only approximations. The exact solution may be described as the locus of images formed by an off-axis rotating mirror surface. Before determining that locus, it would help to point out two consequences of the Law of Reflection: 1)
The line segment connecting a point to its image in a plane mirror is perpendicular to the plane of the mirror and is bisected by it,
OPTICS AND
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1985
circle of radius R whose centre lies at C. Let 8 denote the counterclockwise rotational angle of the mirror, starting from the vertical. Then angle CAA’ equals 8.
,a”’
Imagine a long rigid rod pivoted at point A and free to rotate about that pivot point Then gt each position 6 of the mirror, the image point A’ is located at the point where the rod crosses the circle when the rod makes angle 6 with line segment AC. a
In a real-life camera, as mentioned above, the reflecting surface of the mirror is some distance d in front of the rotational axis. In this new situation, the surface mm in the figure is the reflecting surface.
Fig. 2 In the absence of the mirror, lens L forms image a’ of object point a. The system comprising the lens together with the mirror forms a new image at a”. This image is the same as would form if the object point were moved to a”‘, the lens were moved to L”’ and the mrrror removed
Since mm is parallel-to 11, the image point again lies along the rigid rod AA’, rotating about its pivot with the same angular displacement as the mirror. However, the ray of light that proceeds from point A to the mirror along direction AA’ now has less distance to travel before striking the mirror. Thus, a new image point must be constructed as shown in Fig. 4. By Rule 1, again, the total distance 2(AB). However, the total distance Z(AB-d). In other words. point A” segment AA’. at a point within the from point A’.
AA’ is equal to AA” is equal to always lies on line circle, a distance 2d
As the mirror rotates. the curve traced out by point A” is part of the locus of those points of a line that lie a fixed distance from the intersection of the line and a stationary circle, as the line rotates about a point of the circle. Here the fixed distance is 2d, the circle has radius R, and the line rotates about point A. The definition above was first stated by Blaise Pascal, and the curve he described in those terms is known as Pascal’s limacon. Fig. 5a shows part of a ruler-andcompass construction of Pascal’s limacon. In the complete curve (Fig 5b), an internal and an external loop are visible. It is the internal loop that has relevance to the particular problem in mirror scanning illustrated here. In some cameras, the mirror is tilted relative to the rotational axis. In that case, the image trajectory is a curve in three dimensions. The projection of this curve Line segment II represents a plane reflecting surface rotating Fig. 3 about an axis normal to the plane of the paper at pornt C
2)
If a lens together with a plane mirror form a real image, then that image is identical to the one formed by moving the object and the lens to the positions of their respective mirror-images and removing the mirror (Fig. 2).
Now suppose line segment 11(Fig 3) represents a plane reflecting surface that rotates about an axis normal to the plane of the paper, at point C. What is the locus of images of A formed in the mirror 11 as it rotates? The mirror forms image A’ of point A. By Rule 1. I I is the perpendicular bisector of line segment AA’. Thus point A’ can be constructed easily. The midpoint of AA’ is labelled B in the figure. It is easy to prove that A’ lies on the circle: triangles ABC and A’BC are right-angled triangles with a common leg; thus they are congruent. Hence. length CA’ equals length CA the radius of the circle.
Thus as the mirror rotates. the distance between C and A’ remains constant and the image point A’ follows a
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1985
Fig. 4
Reflecting surface is now mm
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Fig. 6
Construction of trajectory of point P”
In Fig 7, the mirror has been deleted from the optical schematic of the camera to simplify a discussion of the refractive optics alone. In this figure. 0’ is the intermediate image at the mirror and 0” is the final image on the film. For simplicity, the objective lens has been moved within the aperture stop.
b
Fig. 5a Ruler and compass construction of Pascal’s limacon; b complete curve
in the plane limacon.
the
of Fig 4 is, again, part of Pascal’s
What is the trajectory of an image formed by the combination of a lens and a rotating plane mirror? The intermediate image, to a good approximation, is always located at the mirror surface. The image of the aperture stop Pi in lens LR though, is swept through a curve as the mirror rotates; the trajectory of this image determines the disposition of the relay lenses. Let s1 denote the fixed distance between Pi (the ‘object’ for lens Ld and the field lens. Let s2 denote the distance from lens LF at which the image is formed. By Rule 2, then, an equivalent optical diagram like that of Fig. 6 can be drawn. Finally, there is a simple method to construct the trajectory of point P”, the image of Pi in LF and the mirror. It is expressed in terms of an imaginary mechanical linkage: points P and P” are connected by a rigid rod of fixed length s1 + s? This rod passes through a loose swivelling collar at point R As the mirror turns, point R slides along line PA and point P moves along its trajectory (the internal loop of Pascal’s limacon), taking its end of the rod along with it. The free end of the rod then marks point P”. The trajectory of point P’ also falls along a Pascal’s limacon. In the simple but unrealistic case where the mirror surface is coincident with the rotational axis, point R is fixed and coincides with point A. In that case, Points P’ and P” sweep out simple circular arcs.
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Two rays (the ‘peripheral,’ or ‘extreme’ rays) are shown emanating from object point 0; the combined lenses L,, and LF bring them to a focus at intermediate image point 0’. Since the intermediate image at 0’ serves as the object for relay lens LR. the rays emanating from 0 that are captured by L, are brought to a focus on the film at 0”. Meanwhile. lens L, performs a second function: to collect the rays passing through L,, and project them into an image of the aperture stop at LR. Thus, the two extreme rays shown entering the top and bottom of the aperture stop are brought, respectively, to the bottom and top of the aperture for LR. Any ray entering lens I+ between the two extreme rays will exit through LR.
Operation Figure 8 illustrates what happens as the mirror rotates. If light from the intermediate image 0’ fell on all the relay lenses L,,, LR2, etc, at the same time, all the images of the film would form simultaneously. This is prevented from happening by the field lens LF. By projecting an image of the camera’s entrance aperture. this lens makes sure that every ray that enters the camera ends up in the small area where that image lies. To illustrate this, Fig. 8 shows rays emerging parallel to
I-
I Fig. 7
Simplified schematic of formatlon of image on film at 0”
OPTICS AND LASER TECHNOLOGY.
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The time required for the image of Pi to pass from exact coincidence with one relay lens to the next is called the interframe time. The intensity of light illuminating the film is not constant during the exposure of a single frame; this illumination reaches maximum value when exact coincidence occurs.
its
The effective exposure time is the amount of time during the exposure of a single frame that the illumination of the film is near its maximum value. It is desirable to keep this time interval very short-provided that there is sufficient illumination of the subject, the effective exposure time limits the greatest speed that can be ‘frozen’ by the camera Many cameras use a simple device that shortens this time to some fraction of the interframe time: the entrance and exit stops are shaped as diamonds (Fig. 9). Thus as the aperture stop’s image sweeps across the shutter stops there is very little overlap until exact coincidence is almost reached. The graph of film illumination against time rises in a parabola as coincidence is approached, and falls in a symmetrical manner after coincidence is passed (Fig 10). This treatment has ignored the wave nature of light. But as apertures become smaller, this nature asserts itself in the phenomenon of diffraction. In fact, this diffraction limit forced the designers of the Model 121 camera to use rectangular apertures instead of the preferred diamond shape, sacrificing some time resolution for the sake of improved spatial resolution.
Fig. 8 As the mirror rotates, lens L, ensures the image is projected through only one relay lens at a time
the optical axis from the top and bottom of the entrance aperture and being refracted by LF onto the bottom and top, respectively, of the second relay lens. Tke image of the entrance aperture Pi sweeps sequentially across the row of relay lenses. Each relay lens receives light from the intermediate image only when the image of Pi lies over it In other words, the field lens shapes the light from 0’ into a narrow beam which lights up only one relay lens at a time.
1
As the mirror rotates. the intermediate image viewed by the relay lenses actually rotates as well. How does this affect the final image at the film? In Fig. 11, the image has formed just at the surface of the vertical mirror, and then the mirror has turned through angle 8. As shown, the intermediate image has turned through angle 28. This is true not only for the intermediate image; the beam of light passing from the mirror to the relay lens is also swept through twice the rotational angle of the mirror (Fig. 12).
a
b
Fig. 9a The diamond on the left is sweeping with velocity v across the (stationary) diamond on the right. The area of each diamond is N//2. The doubly shaded area in Fig. 9b has area M/2 = vr.(W/B)H/2. The amount of film illumination, expressed as a fraction of its maximum value, is M/B/f, or S$H/B.(l/BH) = [(v)*/($j]?. This result is true while the illumination is rising; it falls symmetrically
OPTICS AND
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b Fig. 10
’
B/v Plot of film illumination against time
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kr e
’
estimate of how much the yardstick’s image appears to change will give an idea of how much blurring (from this cause) to expect. Plane of mirror, 8 = 0
Each exit stop subtends an angle, typically, of 5” at the mirror. If 20% of the exposure time of one frame is ‘effective’, then the intermediate image rotates through l”, or 0.5” on either side of coincidence during the effective exposure time. If the final image fills a side-toside distance of 35 mm on the film, then a 0.5” rotation will shorten this distance by (see Fig 13): 35 mm X (1 - cos 0.5”) = 35 mm = 1.33 x lop3 mm.
Fig. 1 1
Intermediate
image formation
as the mirror rotates
X
l/2
X
(0.00873)’
The limit that diffraction places on a camera’s resolution can be expressed as the product: Relative
Aperture
x
0.68 x lo-’
mm
It can be seen that for the values of the relative aperture that are in common use, the rotation of the mirror is no more important than diffraction in limiting the camera’s resolution. In practice, the factors other than image motion and diffraction that may limit the spatial resolution of the camera include lens aberrations, bending of the mirror surface by centrifugal force, and, especially when helium is used to drive the turbine, refraction by turbulent driving gas interposed in the optical path.
Acknowledgements The author wishes to thank Dr Larry L. Shaw for critically reading the manuscript, and for his continued encouragement and advice. Fig. 12
Fig. 13
The intermediate
image rotates at twice the rate of the mirror
The effect of image translation
during exposure is small
Therefore, when each relay lens is fully illuminated the relay lens is viewing the intermediate image face-on (that is, it is unrotated from the relay lens’s point of view). How much blurring of the image occurs due to the rotation of the mirror during the exposure of a single frame? This question will be answered with a simple numerical estimate, which will show that virtually no blurring occurs purely as a result of correct mirror rotation. Suppose that the object is a ‘yardstick’ (of arbitrary length), so orientated that its image extends fully across the frame of film, along the scan direction. During the exposure of each frame, the mirror’s rotation causes a translation of the spot on the film at which the rays from a specific object point come to a focus. This translation cannot be bigger than the apparent change in size of the yardstick’s image during the exposure of the frame. Since the translation of the image spot gives rise to the blurring in question, an
88
Work performed under the auspices of the US Department of Energy by the Lawrence Livermore National Laboratory under Contract W-7405-Eng-48. Disclaimer. This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor the University of California nor any of their employees, makes any warranty. express or implied, or assumes any legal liability or responsibility for the accuracy. completeness, or usefulness of any information apparatus. product, or process disclosed. or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial products. process, or service by trade name. trademark. manufacturer, or otherwise, does not necessarily constitute or imply its endorsement recommendation. or favouring by the United States Government or the University of California. The views and opinions of author expressed herein do not necessarily state or reflect those of the United States Government or the University of California, and shall not be used for advertising or product endorsement purposes.
References Shaw, LL, Sonderman, J.&, Seppala, LG. ‘Large-format 70-mm high speed framing camera’. Proc 13th Intern Cong High Speed Photography and Photonics (Tokyo. 1978). Shin-ichi Hyodo, editor, The Japan Society of Precision Engineering Tokyo (1979) 218-221 Dubovik, A. The photographic recording of high-speed processes, John Wiley, New York (1981) Miller, C.P. US Patent Report 2 400 887, May 28, 1946: J Sot Mar Picf Eng 53 (1949) 479. A description of the Miller camera and an informative historical note appear in: Nebeker, S. ‘Rotating mirror cameras’, High-Speed Photography and Photoniu (1983) 31-37
Nnvslette,:
in Phoromerhod.,:
26 (5)
Schardin, H. ‘The relationship between maximum frame frequency and resolution in rotating-mirror framing cameras’, Proc Third Intern Cong High-Speed Photography, edited by R.B. Collins, Academic Press, New York. and Butterworths Scientific Publications, London (I 957) 3 16-3
OPTICS AND LASER TECHNOLOGY.
APRIL
18
1985
optical instruments,
high speed photography,
Introduction
The system can produce film strips whose separate frames are exposed at intervals shorter than a single microsecond with each exposure lasting a fraction of that interval.
Ordinary photography is useful in recording scenes that do not change appreciably over time intervals of 500 ys’or longer-500 ps is the duration of the briefest exposure available on a typical professional-model 35 mm camera. Various strategies are used for the photographic recording of motions that go beyond this limitation. ‘Strobe’ lighting, for example, enables the photographer to record many stages of a rapid but complex, motion on a single frame of film. The resulting multiple exposure then represents a sequence of images that may have been recorded at intervals of 1 p.s or less.
The basic design principles of the ultra-high speed framing camera can be understood in terms of simple geometrical optics. The exact details of the optical design vary somewhat from one to another of some dozen models that are in current use in the USA This present exposition is loosely based on the Cordin Model 121 camera’, which was first publicly exhibited in 1978. This model exposes 26 frames on a strip of 70 mm film. Each frame measures 38.1 x 73.5 mm. By means of a beryllium mirror that can rotate at angular speeds up to 10 000 Hz, the camera achieves framing rates up to 2.5 million frames per second. For a more general treatment of the subject the reader is directed to Dubovik’s comprehensive book*, which is widely regarded as the standard textbook on high-speed photography.
Many situations arise in the study of aerodynamics, ballistics, and high-explosive technology when it is desirable to produce a ‘motion picture’ of an object or disturbance that travels at several times the speed of sound. In these situations, the ideal photographic record would be able to reproduce fine detail (that is, have high resolution), and be large enough for accurate measurements to be taken from it conveniently. The recording method should produce a sequence of separate images representing moments separated by a constant time interval (the interframe time). The exposure time for each individual frame should be brief enough so that the motion of the subject does not cause significant blurring of the image. Finally, the possibility of using colour film adds extra depth to the range of information that can be preserved.
Camera design The main elements of any ultra-high speed framing camera are: a stationary strip of film; a moving mirror that deflects the optical beam along the film; an arrangement of masks that shape and ‘shutter’ the light beam passing through the camera; and an unusual arrangement of lenses that functions as three separate imaging systems.
The ultra-high speed framing camera was designed to satisfy all of these requirements. This device employs ordinary photographic film, and the conditions under which it can form an image are described in the same terms as any amateur camera: focal length, illumination, film speed aperture, exposure time. At the same time, the ultra-high speed camera uses an ingenious combined optical-mechanical system to solve the problems of shuttering and film advance that limit even the most sophisticated motion-picture camera. The author is at the Lawrence Livermore National Laboratory, PO Box 808, Lwermore, CA 94550, USA Recewed 25 October 1984. Revised January 1985.
0030-3992/8~/020083-06/$03.00 OPTICS AND
LASER TECHNOLOGY.
APRIL
The overall layout of a typical framing camera is shown in simplified schematic form in Fig la. (Fig. 1b shows the actual optical layout in the Cordin Model 121 camera.) Light enters the camera through objective lens I.+. The aperture stop Pi, located at the objective lens (in this simple design), determines the size of the cone of light the camera will accept from an object point on the optical axis. Thus, Pi determines the illumination at the final image. All the rays emerging from the aperture stop that are to be used must pass through the field lens L, and reflect from the front surface of mirror M. As will be shown later, the field lens forms an image of the aperture stop in the plane
4
0 1985
systems design
1985
Butterworth
Et Co (Publishers)
Ltd 83
film. This is achieved by projecting a image onto the mirror surface. As the it casts the light from this image onto lenses, lighting up only one relay lens
Aperture stop >
stationary mirror rotates, a row of relay at a time.
The mirror rotates with a frequencyf = w/2n that may be as high as 20 000 Hz. To achieve such high angular velocities, the mirrors are turned by gas-driven turbines which are powered by compressed nitrogen or helium. The flowing gas also partially supports the moving parts, reducing friction and wear on the bearings. Furthermore, the mirrors turn in a vacuum or in an atmosphere of helium to minimize drag At such high rotational velocites, the spinning mirror must be dynamically balanced; that is, the rotational axis must coincide with a principal axis of the mirror. Often, the mirror is shaped like a triangular prism whose cross-section is an isosceles or equilateral triangle. Such a mirror may have one, two, or three reflecting surfaces.
b
The main factor that limits the time resolution of the ultra-high speed camera is the maximum rotational speed of the mirror. The centrifugal force acting on the mirror is very great. The limiting speed is reached at the point where the centrifugal force is not quite great enough to cause the mirror to deform or to burst. The greater the density of the metal of which the mirror is made, the greater is the centrifugal force at any given speed Thus in choosing a material for the mirror, the main criterion is the ratio of tensile strength to density. This problem was discussed by Schardin in a 1956 paper”. In that paper, he proved that the product of the attainable spatial and temporal resolutions of a rotating mirror camera is proportional to the square root of the strength-to-density ratio.
Synchronization system
Fig. 1 a Overall layout of typical framing Cordin Model 12 1 camera
camera; b -
actual layout in the
of the relay lenses LIl, to Lb. (The field lens gets its name because the diameter of this lens determines the angular field of view of the camera: that is, it determines how far an object point can be from the optical axis and still be represented in the final image). Figure la shows that the view of each lens is limited by a mask, or shutter stop. The image of Pi projected by the field lens corresponds exactly, in size and shape, to the openings in this mask Thus, as the mirror rotates about axis A, a ‘pencil’ of light is swept along the row of shutter stops, entering only one relay lens at a time. Each relay lens, in turn, receives its maximum illumination when the image of the entrance pupil is brought by the mirror’s motion into exact coincidence with it. The field lens has a second function: the combination of this lens with the objective lens I+ forms an image of the object 0 onto the mirror M. This image is intermediate: each relay lens, in turn, projects the image at the mirror into a final image on the film. Thus, the subsystem consisting of shutter stops. relay lenses, and film may be thought of as a collection of n separate cameras whose shutters are operated in sequence by the rotation of the mirror. The basic principle by which the ultra-high speed framing camera operates was invented by C.D. Miller’ in 1939. It can be summarized as follows: A rotating mirror can be used to project a sequence of time-resolved images onto a stationary piece of
84
To make the ratio of tensile strength to density as great as possible, most cameras use steel or beryllium mirrors. Because it is so light, beryllium offers a significant advantage over steel. However, the tendency of beryllium mirrors to suffer brittle fracture (possibly with explosive force) imposes safety constraints on their use.
Theoretical basis The fastest spinning mirrors are shaped as polygonal prisms. But the requirement of dynamic balance makes it impossible for the rotational axis to lie in the reflecting surface of such a mirror. Fig. la shows a mirror whose axis A lies a distance d behind the reflecting surface. As the reflecting surface moves around axis A, it varies its distance from lenses b and LF and from the relay lenses LE. How does this affect the location of the intermediate image viewed by the relay lenses? How does one calculate the correct shape for the surface containing the relay lenses, and for the strip of film behind them? If the rotational axis lay in the reflecting surface, the relay lenses would lie along a circular arc, as would the film. In practice circular arcs are sometimes used, although they are only approximations. The exact solution may be described as the locus of images formed by an off-axis rotating mirror surface. Before determining that locus, it would help to point out two consequences of the Law of Reflection: 1)
The line segment connecting a point to its image in a plane mirror is perpendicular to the plane of the mirror and is bisected by it,
OPTICS AND
LASER TECHNOLOGY.
APRIL
1985
circle of radius R whose centre lies at C. Let 8 denote the counterclockwise rotational angle of the mirror, starting from the vertical. Then angle CAA’ equals 8.
,a”’
Imagine a long rigid rod pivoted at point A and free to rotate about that pivot point Then gt each position 6 of the mirror, the image point A’ is located at the point where the rod crosses the circle when the rod makes angle 6 with line segment AC. a
In a real-life camera, as mentioned above, the reflecting surface of the mirror is some distance d in front of the rotational axis. In this new situation, the surface mm in the figure is the reflecting surface.
Fig. 2 In the absence of the mirror, lens L forms image a’ of object point a. The system comprising the lens together with the mirror forms a new image at a”. This image is the same as would form if the object point were moved to a”‘, the lens were moved to L”’ and the mrrror removed
Since mm is parallel-to 11, the image point again lies along the rigid rod AA’, rotating about its pivot with the same angular displacement as the mirror. However, the ray of light that proceeds from point A to the mirror along direction AA’ now has less distance to travel before striking the mirror. Thus, a new image point must be constructed as shown in Fig. 4. By Rule 1, again, the total distance 2(AB). However, the total distance Z(AB-d). In other words. point A” segment AA’. at a point within the from point A’.
AA’ is equal to AA” is equal to always lies on line circle, a distance 2d
As the mirror rotates. the curve traced out by point A” is part of the locus of those points of a line that lie a fixed distance from the intersection of the line and a stationary circle, as the line rotates about a point of the circle. Here the fixed distance is 2d, the circle has radius R, and the line rotates about point A. The definition above was first stated by Blaise Pascal, and the curve he described in those terms is known as Pascal’s limacon. Fig. 5a shows part of a ruler-andcompass construction of Pascal’s limacon. In the complete curve (Fig 5b), an internal and an external loop are visible. It is the internal loop that has relevance to the particular problem in mirror scanning illustrated here. In some cameras, the mirror is tilted relative to the rotational axis. In that case, the image trajectory is a curve in three dimensions. The projection of this curve Line segment II represents a plane reflecting surface rotating Fig. 3 about an axis normal to the plane of the paper at pornt C
2)
If a lens together with a plane mirror form a real image, then that image is identical to the one formed by moving the object and the lens to the positions of their respective mirror-images and removing the mirror (Fig. 2).
Now suppose line segment 11(Fig 3) represents a plane reflecting surface that rotates about an axis normal to the plane of the paper, at point C. What is the locus of images of A formed in the mirror 11 as it rotates? The mirror forms image A’ of point A. By Rule 1. I I is the perpendicular bisector of line segment AA’. Thus point A’ can be constructed easily. The midpoint of AA’ is labelled B in the figure. It is easy to prove that A’ lies on the circle: triangles ABC and A’BC are right-angled triangles with a common leg; thus they are congruent. Hence. length CA’ equals length CA the radius of the circle.
Thus as the mirror rotates. the distance between C and A’ remains constant and the image point A’ follows a
OPTICS
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Fig. 4
Reflecting surface is now mm
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Fig. 6
Construction of trajectory of point P”
In Fig 7, the mirror has been deleted from the optical schematic of the camera to simplify a discussion of the refractive optics alone. In this figure. 0’ is the intermediate image at the mirror and 0” is the final image on the film. For simplicity, the objective lens has been moved within the aperture stop.
b
Fig. 5a Ruler and compass construction of Pascal’s limacon; b complete curve
in the plane limacon.
the
of Fig 4 is, again, part of Pascal’s
What is the trajectory of an image formed by the combination of a lens and a rotating plane mirror? The intermediate image, to a good approximation, is always located at the mirror surface. The image of the aperture stop Pi in lens LR though, is swept through a curve as the mirror rotates; the trajectory of this image determines the disposition of the relay lenses. Let s1 denote the fixed distance between Pi (the ‘object’ for lens Ld and the field lens. Let s2 denote the distance from lens LF at which the image is formed. By Rule 2, then, an equivalent optical diagram like that of Fig. 6 can be drawn. Finally, there is a simple method to construct the trajectory of point P”, the image of Pi in LF and the mirror. It is expressed in terms of an imaginary mechanical linkage: points P and P” are connected by a rigid rod of fixed length s1 + s? This rod passes through a loose swivelling collar at point R As the mirror turns, point R slides along line PA and point P moves along its trajectory (the internal loop of Pascal’s limacon), taking its end of the rod along with it. The free end of the rod then marks point P”. The trajectory of point P’ also falls along a Pascal’s limacon. In the simple but unrealistic case where the mirror surface is coincident with the rotational axis, point R is fixed and coincides with point A. In that case, Points P’ and P” sweep out simple circular arcs.
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Two rays (the ‘peripheral,’ or ‘extreme’ rays) are shown emanating from object point 0; the combined lenses L,, and LF bring them to a focus at intermediate image point 0’. Since the intermediate image at 0’ serves as the object for relay lens LR. the rays emanating from 0 that are captured by L, are brought to a focus on the film at 0”. Meanwhile. lens L, performs a second function: to collect the rays passing through L,, and project them into an image of the aperture stop at LR. Thus, the two extreme rays shown entering the top and bottom of the aperture stop are brought, respectively, to the bottom and top of the aperture for LR. Any ray entering lens I+ between the two extreme rays will exit through LR.
Operation Figure 8 illustrates what happens as the mirror rotates. If light from the intermediate image 0’ fell on all the relay lenses L,,, LR2, etc, at the same time, all the images of the film would form simultaneously. This is prevented from happening by the field lens LF. By projecting an image of the camera’s entrance aperture. this lens makes sure that every ray that enters the camera ends up in the small area where that image lies. To illustrate this, Fig. 8 shows rays emerging parallel to
I-
I Fig. 7
Simplified schematic of formatlon of image on film at 0”
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1985
The time required for the image of Pi to pass from exact coincidence with one relay lens to the next is called the interframe time. The intensity of light illuminating the film is not constant during the exposure of a single frame; this illumination reaches maximum value when exact coincidence occurs.
its
The effective exposure time is the amount of time during the exposure of a single frame that the illumination of the film is near its maximum value. It is desirable to keep this time interval very short-provided that there is sufficient illumination of the subject, the effective exposure time limits the greatest speed that can be ‘frozen’ by the camera Many cameras use a simple device that shortens this time to some fraction of the interframe time: the entrance and exit stops are shaped as diamonds (Fig. 9). Thus as the aperture stop’s image sweeps across the shutter stops there is very little overlap until exact coincidence is almost reached. The graph of film illumination against time rises in a parabola as coincidence is approached, and falls in a symmetrical manner after coincidence is passed (Fig 10). This treatment has ignored the wave nature of light. But as apertures become smaller, this nature asserts itself in the phenomenon of diffraction. In fact, this diffraction limit forced the designers of the Model 121 camera to use rectangular apertures instead of the preferred diamond shape, sacrificing some time resolution for the sake of improved spatial resolution.
Fig. 8 As the mirror rotates, lens L, ensures the image is projected through only one relay lens at a time
the optical axis from the top and bottom of the entrance aperture and being refracted by LF onto the bottom and top, respectively, of the second relay lens. Tke image of the entrance aperture Pi sweeps sequentially across the row of relay lenses. Each relay lens receives light from the intermediate image only when the image of Pi lies over it In other words, the field lens shapes the light from 0’ into a narrow beam which lights up only one relay lens at a time.
1
As the mirror rotates. the intermediate image viewed by the relay lenses actually rotates as well. How does this affect the final image at the film? In Fig. 11, the image has formed just at the surface of the vertical mirror, and then the mirror has turned through angle 8. As shown, the intermediate image has turned through angle 28. This is true not only for the intermediate image; the beam of light passing from the mirror to the relay lens is also swept through twice the rotational angle of the mirror (Fig. 12).
a
b
Fig. 9a The diamond on the left is sweeping with velocity v across the (stationary) diamond on the right. The area of each diamond is N//2. The doubly shaded area in Fig. 9b has area M/2 = vr.(W/B)H/2. The amount of film illumination, expressed as a fraction of its maximum value, is M/B/f, or S$H/B.(l/BH) = [(v)*/($j]?. This result is true while the illumination is rising; it falls symmetrically
OPTICS AND
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b Fig. 10
’
B/v Plot of film illumination against time
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kr e
’
estimate of how much the yardstick’s image appears to change will give an idea of how much blurring (from this cause) to expect. Plane of mirror, 8 = 0
Each exit stop subtends an angle, typically, of 5” at the mirror. If 20% of the exposure time of one frame is ‘effective’, then the intermediate image rotates through l”, or 0.5” on either side of coincidence during the effective exposure time. If the final image fills a side-toside distance of 35 mm on the film, then a 0.5” rotation will shorten this distance by (see Fig 13): 35 mm X (1 - cos 0.5”) = 35 mm = 1.33 x lop3 mm.
Fig. 1 1
Intermediate
image formation
as the mirror rotates
X
l/2
X
(0.00873)’
The limit that diffraction places on a camera’s resolution can be expressed as the product: Relative
Aperture
x
0.68 x lo-’
mm
It can be seen that for the values of the relative aperture that are in common use, the rotation of the mirror is no more important than diffraction in limiting the camera’s resolution. In practice, the factors other than image motion and diffraction that may limit the spatial resolution of the camera include lens aberrations, bending of the mirror surface by centrifugal force, and, especially when helium is used to drive the turbine, refraction by turbulent driving gas interposed in the optical path.
Acknowledgements The author wishes to thank Dr Larry L. Shaw for critically reading the manuscript, and for his continued encouragement and advice. Fig. 12
Fig. 13
The intermediate
image rotates at twice the rate of the mirror
The effect of image translation
during exposure is small
Therefore, when each relay lens is fully illuminated the relay lens is viewing the intermediate image face-on (that is, it is unrotated from the relay lens’s point of view). How much blurring of the image occurs due to the rotation of the mirror during the exposure of a single frame? This question will be answered with a simple numerical estimate, which will show that virtually no blurring occurs purely as a result of correct mirror rotation. Suppose that the object is a ‘yardstick’ (of arbitrary length), so orientated that its image extends fully across the frame of film, along the scan direction. During the exposure of each frame, the mirror’s rotation causes a translation of the spot on the film at which the rays from a specific object point come to a focus. This translation cannot be bigger than the apparent change in size of the yardstick’s image during the exposure of the frame. Since the translation of the image spot gives rise to the blurring in question, an
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Work performed under the auspices of the US Department of Energy by the Lawrence Livermore National Laboratory under Contract W-7405-Eng-48. Disclaimer. This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor the University of California nor any of their employees, makes any warranty. express or implied, or assumes any legal liability or responsibility for the accuracy. completeness, or usefulness of any information apparatus. product, or process disclosed. or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial products. process, or service by trade name. trademark. manufacturer, or otherwise, does not necessarily constitute or imply its endorsement recommendation. or favouring by the United States Government or the University of California. The views and opinions of author expressed herein do not necessarily state or reflect those of the United States Government or the University of California, and shall not be used for advertising or product endorsement purposes.
References Shaw, LL, Sonderman, J.&, Seppala, LG. ‘Large-format 70-mm high speed framing camera’. Proc 13th Intern Cong High Speed Photography and Photonics (Tokyo. 1978). Shin-ichi Hyodo, editor, The Japan Society of Precision Engineering Tokyo (1979) 218-221 Dubovik, A. The photographic recording of high-speed processes, John Wiley, New York (1981) Miller, C.P. US Patent Report 2 400 887, May 28, 1946: J Sot Mar Picf Eng 53 (1949) 479. A description of the Miller camera and an informative historical note appear in: Nebeker, S. ‘Rotating mirror cameras’, High-Speed Photography and Photoniu (1983) 31-37
Nnvslette,:
in Phoromerhod.,:
26 (5)
Schardin, H. ‘The relationship between maximum frame frequency and resolution in rotating-mirror framing cameras’, Proc Third Intern Cong High-Speed Photography, edited by R.B. Collins, Academic Press, New York. and Butterworths Scientific Publications, London (I 957) 3 16-3
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