US3302543A

US3302543A - Wide-angle dioptric system

Info

Publication number: US3302543A
Application number: US317724A
Authority: US
Inventors: Karl F Ross
Original assignee: Individual
Current assignee: Individual
Priority date: 1963-10-21
Filing date: 1963-10-21
Publication date: 1967-02-07
Anticipated expiration: 1984-02-07

Description

Feb. 7, 1967 K. F. ROSS WIDE-ANGLE DIOPTRIG SYSTEM Filed Oct. 21, 1963 8 Sheets-Sheet 1 INVENTOR: KARL E ROSS Feb. 7, 1967 oss 3,302,543

WIDE-ANGLE DIOPTRIC SYSTEM Filed Oct. 21, 1963 a Sheets-Sheet 2 :0 mln:

m '0 O 9 O' o o o 0 B1 H U /A/ o w. m r O m 0 mm? 0 Q3 1\ v: v N O o 0 O O o o o o O INVENTOR;

KARL. F ROSS Feb. 7, 1967 K. F. ROSS 3,302,543

WIDE-ANGLE DIOPTRIC SYSTEM Filed Oct. 21, 1963 8 Sheets-Sheet 5 INVENTOR. KARL F ROSS Feb. 7, 1967 K. F. ROSS WIDE-ANGLE momma SYSTEM 8 Sheets-Sheet 4 Filed Oct. 21, 1963 INVENTOR. KARL F ROSS 8 Sheets-Sheet 5 K. F. ROSS WIDE-ANGLE DIOPTRIC SYSTEM Feb. 7, 1967 Filed Oct. 21, 1963 INVENTOR: KARL F ROSS 1967 K. F. ROSS WIDE-ANGLE DIOPTRIC SYSTEM 8 Sheets-Sheet 6 Filed Oct. 21, 1963 I\VENT()R. KA R1. E POSS Feb. 7; 1967 K. F. ROSS 3,302,543

WIDE-ANGLE DIOPTRIC SYSTEM Filed Oct. 21, 1963 8 Sheets-Sheet 7 INVENTOR: KARL l-T ROSS United States Patent 3,302,543 WIDE-ANGLE DIOPTRIC SYSTEM Karl F. Ross, 5121 Post Road, Bronx, N.Y. 10471 Filed Oct. 21, 1963, Ser. No. 317,724 3 Claims. (CI. 95-16) My present invention relates to a dioptric system of the type using a lens or lens combination whose optically effective surfaces are all centered on a common point or axis. Lenses of this character have been disclosed in US. Patents Nos. 2,923,220 and 3,044,379, issued Feb. 2, 1960 and July 17, 1962 to A. Bouwers.

lenses with concentric or coaxial surfaces (for spherical or cylindrical refractivity, respectively) are particularly suitable for wide-angle or panoramic picture taking or projection, owing to the fact that rays incident at a variety of angles can be sharply focused; they also offer the designer the advantage that their principal points coincide with the center of curvature so that the focal length of any such lens, whether simple or compound, is always measured from that center. (In the present context, compound lenses are those whose internal refractive surfaces are concentric or coaxial with the outer lens surfaces.) As is the case with other leneses of circular curvature, however, these lenses provide satisfactory image definition only over a limited field angle for each direction of incidence since only the so-called paraxil rays, i.e., those passing through the lens center or in the immediate vicinity thereof, will focus sharply. It has, therefore, been necessary in the past to provide various types of diaphragms for limiting the effective field angle, with a resultant reduction in the relative aperture of the lens.

An object of my invention is to provide an improved lens of this type in which the field of converging rays is widened so that larger relative apertures may be used.

Another object of my invention is to provide means for increasing the luminous intensity of a concentric or coaxial objective of given design.

A further object of my instant invention is to provide means for effectively flattening the image plane of a spherical or cylindrical objective of the concentric or coaxial type.

It is also an object of the instant invention to provide a high-speed, wide-angle, substantially achromatic dioptric component for photographs and related optical systems.

In the following description I shall generally refer to concentric surfaces, it being understood thatunless otherwise noted-the term is intended to encompass both spherical and cylindrical lens surfaces.

It will be convenient to designated by R, the radius of curvature of the front surface of a lens to be discussed (i.e., the surface on the side of the incident or longer light rays), by R and the radius of the opposite or rear surface, and by R the radius of any concentric cemented surface therebetween. Furthermore, let

' R EK and K it being assumed that R is positive and R R are negative according to conventional notation. Also, the lens portion limited by the surfaces R and R shall have a refractive index n whereas the lens portion limited by the surfaces R R shall have an index n n It can be shown that the focal length f of a lens so de- The back-focal length f is then given by E EK, (3)

and should have a positive value if an object at infinity is to form a real image at a location beyond the lens. Since the internal radius R of the doublet must be smaller than its external radius R it is necessary that K K so that, for E 0, K n 1.

In a conventional spherical or cylindrical lens lacking a cemented surface i.e. a singlet, IR |=IR I=R and n =n =n, R being of indeterminate value owing to the uniformity of the glass. In this case the formula for the back-focal length reduces to f 2(nl) (30) With such a lens, in which K =K=1, the back-focal distance s" for rays with increasing angle of incidence :1 decreases rather sharply from the value s "=f which is valid for a=0. I have found that the relationship s-f may be maintained over a relatively wide range of a if the value of K is close to unity and, preferably, ranges between approximately 0.9 and 1.1. The value of K may be roughly equal to 2K being advantageously held approximately to the range 2 K &1.8 in order to establish a relatively wide entrance pupil.

A feature of my invention resides in the provision of means for rotating a lens of this description about an axis passing through its center of curvature, at least the front surface of the lens being limited in the plane of rotation to a circular arc subtending an angle not greater than that from which sharp definition of the image is obtainable. As the lens rotates, different proportions thereof focus the same incident rays upon a given point of a concentric receiving surface so that this point is exposed to more light than would be the case if the lens were stationary. Owing to the perfect circularity of each concentric lens surface, the point of convergence of the rays from a given distant object does not change as the lens rotates. In this manner the need for a special diaphragm to restrict the effective field angle is avoided, it being however possible to reduce this field angle selectively by the provision of an adjustable light shield if the luminous intensity of the objective is to be decreased in view of existing light conditions. It will be understood that the rotation of the segmental lens need not be continuous and unidirectional but may consist of alternate sweeps in opposite directions if desired.

The above-described rotatable systems affords a wide angle of view in one plane and, for proper projection, requires that the sensitive film or other receiving surface (e.g. a ground-glass plate used for visual observation) be curved about the axis of rotation of that plane. In some instances, e.g. with panoramic cameras, the dimension of the projected image transverse to the plane of rotation may be so small that the field curvature is negligible; if it is not, a conventional flattening lens (e.g. as known from the aforementioned Bouwers Patent No. 3,044,379) may be fixedly interposed between the film and the concentric lens or may be rotatable with the latter. In accordance with a more specific feature of my invention, however, I may provide a circular flattening lens adapted to be placed in front of the (rotating or stationary) concentric lens, the parameters of the flattening lens being so chosen that the value of f changes proportionally to the secant of the angle of incidence so that the resulting image surface will be a plane.

The invention will be described in greater detail with reference to the accompanying drawing in which:

FIG. 1 is a cross-sectional view of a concentric lens according to the invention;

FIG. 2 is a set of graphs illustrating the changes in relative image distance s/R with increasing angles of incidence a as plotted for difference parameters of the lens shown in FIG. 1;

FIG. 3 is a view similar to FIG. 1, illustrating an inversion;

FIG. 4 is a cross-sectional view taken on the line IV-- IV of FIG. 3;

FIG. 5 is a set of graphs similar to FIG. 2, pertaining to the lens shown in FIGS. 3 and 4;

FIG. 6 shows the combination of a lens similar to that of FIG. 1 with a special flattening front lens therefor;

FIG. 7 is a view similar to FIG. 6, showing a modified form of flattening front lens;

FIGS. 8a-8d and 9 are graphs used in explaining the construction of the flattening lenses of FIGS. 6 and 7; and

FIG. 10 is a cross-sectional view of the lens of FIG. 1 in modified form, taken on the line XX of that figure.

Reference will first be made to FIG. 1 for a discussion of the properties of a lens adapted to be used in a wideangle exposure system according to this invention, it being understood that the lens could also be used in a projector by simple inversion of the direction of the light rays as is well known.

The lens 10 shown in FIG. 1 is a doublet composed of a positively refracting first member 11 and a negatively refracting second member 12. Member 11 has a refractive index n and is bounded by a front surface of radius R and an internal surface of radius R along which it is cemented onto member 12; the latter has as its front boundary the cemented surface of radius R and as its rear boundary an outer surface of radius R All the radii are measured from a point serving as the common center of the surfaces referred to.

For the present, only tne cross-section of the lens visible in FIG. 1 is to be considered. In its transverse plane the lens 10 may have the cross-section illistrated in FIG. 10 where its front and internal surfaces are shown to have the same radii R and R (being thus segments of concentric spheres) whereas the rear surface has an aspherical configuration more fully described hereinafter. In some instances, however, all the lens surfaces may be spherical, e.g. in the event that the lens 10 is used together with one of the flattening lenses described hereinbelow.

As shown in FIG. 1, R is positive, R is negative and smaller than R so that The axis 100 passes through the center 0 and lies at the intersection of two mutually perpendicular planes of symmetry of the lens. At 101 a light ray parallel to axis 100 impinges on the front surface 110 of lens member 11, including an angle a with the radius R thereof. Upon refraction, the ray continues inside lens member 11 at an angle ,8 with reference to radius R and strikes the internal surface 111 between

lens members

11 and 12 at an angle 7 with reference to its radius R This internal surface, being negatively refracting, further deflects the ray so that its angle with radius R within member 12 will have the value 5. As the ray strikes the rear surface 112 of member 12, it includes with the radius R of that surface an angle 6 which changes to (p as the ray emerges after further refraction. If and n are the angles which the radii R R respectively include with axis 100, the emerging ray approaches this axis at an angle (p--i1 and intersects it at a point 102 which, for small values of a, is in the vicinity of point F representing the focal point of the lens on axis 100. The distance s" between point 102 and the rear vertex 103 of the lens along axis is thus substantially equal to the back-focal length f of the lens measured between this vertex and focal point F It will be apparent that, owing to the circularity of the lens surfaces, other focal points will exist which have the same distance f from the rear lens surface 112 and which lie on other axes passing in different directions through the center 0, such as the

axes

100 and 100". The locus of all these focal points is the curve F which is a circular arc of radius j centered on O, 1 being thefocal length of the system.

In accordance with the laws of geometry and optical refraction it can be shown that the following relationships obtain:

As will be readily apparent from FIG. 1, point 102 is separated from center 0 by a distance Sin 1 R +s -R cos 1 +R (11) For small values of a we can write:

lp- Sin gu=K sin a fiK a (8a) and s n; .22 {g s 1 m a a where 1+K K -K, E- -1 E 1 2 Thus, with sin 1 :1 and tan 1 p'q, we can simplify Equation 11 for paraxial rays to obtain whence, for 11:0,

The lens 10 is shown bounded along its sides by rear- Wardly converging

opaque strips

13, 14 of nonreflecting character. The separation of these strips, i.e. the thick ness of the lens in the plane seen in FIG. 1, is so selected that the arcs intercepted by them on the entrance and exit surfaces 110, 112 of the lens are limited to regions within which s"-f When the lens is rotated about its center 0, its axis 100 together with the field defined by the limiting

rays

100, 100" (ca 1a will sweep acros the surface F through an angle which may be much larger than 2a and in practice may approach In specific cases, e.g. with slanted incident rays as shown in the two Bouwers patents referred to, the angle of exposure and projection may be extended to a full 360.

Two opaque

segmental bodies

15 and 16 are secured to the lens 10 0utside its

surfaces

13 and 14 to block the passage of stray light rays when the lens is rotated about a transverse axis through point 0 as described hereinafter with reference to FIG. 4. Segment 16 is formed with a slot 17 adapted to receive an arcuate light shield 23 which isadjustable with reference to the lens.

Since, from equations 4 and 5,

K sin a SIIl the value of a is also determined from the relationship SIIl a -K1 inasmuch as sin 7 cannot be larger than unity. This is further borne out by inspection of FIG. 1 which shows that a considerable reduction of R below the magnitude illustrated would require a sharper convergence of the lens sides 13 and 14 with the result that some of the rays incident upon the front surface of the lens would strike these sides before reaching the cemented surface. With n =1.5, for example, a could not be substantially greater than 45 if K =2, regardless of any improvement in the deviation of s" from f that may be effected by suitable selection of the other parameters of the system.

In FIG. 2 I have shown a series of graphs II to VII in which s"/R has been plotted against a for different values of K and K Curve I, included for purposes of comparison, relates to a homogeneous spherical lens wherein R =R =R (there being no internal refractive surface of radius R and n =n =1.5, the ratio f /R being computed as 0.5. It will be noted that the value of s"/R drops at a steadily increasing rate with rising values of a. Curves II to VII, on the other hand, apply to lenses of the type illustrated in FIG. 1 having the following It will be seen from FIG. 2 that, for values of K and K in the vicinity of 2 and 1, respectively, the value of s"/R remains substantially constant for angles of incidence a up to approximately 0.6 and above. It is interesting to note that, for higher values of at, the curve tends to droop when K is small and/0r K /K is large (curves II and V) but tends to rise when the situation is reversed (curves III, VI, VII). Particularly good results are observed with K -L9 and K ranging within about 140% of unity, as is true of the systems represented by curves III, IV and V; the last-mentioned curve has exceptional flatness up to and beyond a=0.8.

The indicated values for n and n are not critical but have been chosen near the lower and upper limits of the range of available refractive indices; if the difference between n and n is materially reduced, the lens begins to approximate a homogeneous one with the characteristic illustrated by graph I in FIG. 2.

In accordance with Equation 12 the field of view is limited by the value of K to an angle u as given in the foregoing table. Thus, K cannot be substantially greater than 2 if a field close to 90 is desired.

With the listed values for n and n E as derived from Equation 2 assumes the value in order that both E and f /R be positive, K, (which of course must be smaller than K has to be greater than 1/4 for K =2 and greater than 1/3 for K =1.5. Actually, the lower limit of K is also determined by the requirement that g and 1; as given in Equations 9 and 10, be greater than zero for all angles of incidence up to m under the assumed conditions of refractivity it then becomes necessary that, e.g. for a field of view encom passed by a ,,,,=:1, K 0.47 if K =l.5. In practice it will usually be desirable to make f equal to a major fraction of R preferably upwards of about R 2, which imposes further limitations upon the range within which K can be varied; thus, with K =1.8 and a field of view as stated above, K must be chosen greater than 0.72 if the back-focal length is to exceed half the radius of the rear surface 112 of the lens.

The Equation 3 for the back-focal length may also be written in the form is proportional to d o for the optical glasses considered, with n n EA n so that If, now, the refractive power of the lens 10 is to be the same for light rays of the D line and of the C line of the spectrum, then Equation 3b must be valid not only for n an and n zn but also for n A =n and n A =n Hence,

1 1/11 can be calculated. Following are several representative values:

TABLE B 111 m K1 K1 Az/Ai m/v:

The first, second and fourth lines of the foregoing Table B correspond to curves V, III and VI of Table A, respectively. It will be seen that the dispersion ratio of the two glasses decreases generally inversely wit-h the ratio K /K the requirement for achromatism being thus more easily satisfied with parameters which do not afford optimum focusing over the largest field of view. Curve III of FIG. 2 may be regarded as a suitable compromise if, for example, v =25 and 67.

The lenses heretofore described are reversible, i.e. their optical behavior is similar When the direction of the light rays passing through them is inverted. Thus, I have shown in FIGS. 3 and 4 a lens 10' which is substantially an inversion of the lens 10 of FIG. 1, its front and rear portions 11', 12' being separated by a forwardly convex internal surface .111 while the refractive index of front member 11' is greater than the index n of rear member 12'. An incident ray 101, parallel toaxis 100, includes with the radius R of front surface 110 an angle a which changes to 13' within the member 11': at the internal surface 111' the ray includes with the radius R of that surface an angle 7' within member 11' and an angle 6 within member 12'. The angle of incidence at the rear surface 112', with reference to its radius R is 12' ahead of that surface and t beyond it; by analogy with FIG. 1, the angles included by the radii R R with the axis 100 have been designated 5 and 1;, respec- The following table lists, by way of example, the parameters of three lenses constituting inversions of the systems represented by the curves II, IV and V of FIG. 2 (see also Table A); the deviation curves of these three inverted lenses have been shown at H, IV and V in FIG. 5.

TABLE Curve n1 m K1 K MR fad-Rial Rr/ II 1. 75 l. 1. 8 1 O. 696 0. 696 IV 1. 75 l. 5 1.9 1 0.725 0.725 V 1. 75 I. 5 l. 73 0. 91 0. 6 0. 76

The final column of Table C lists the value which must be identical with the value f /R given for the corresponding system in Table A and which is, of course, equal to f /R whenever, as with the first two systems, K =l. The curve V has been included in FIG. 5 to show, for purposes of better comparison with the curve of FIG. 2, the parameter plotted against a. It will be seen that the graphs II, IV and V of FIG. 5 are generally similar to the graphs II, IV and V, respectively, of FIG. 2.

The lens as also the lens 10 of FIG. 1 (cf. FIG. 10), is armored along its sides with opaque and nonre- 8 fleeting shielding 61, 62 (FIG. 4) in addition to the light shields 13', 14 visible in FIG. 3.

The means for rotating the sectoral lens 10 about its axis have been shown diagrammatically in FIG. 4 as comprising a drive motor 22 coupled with a two-

part lens shaft

21a, 21b via

gears

32, 44.

Furthermore, in contradistinction to lens 10, the lens 10 is formed with a narrow air gap 63 along its internal surface 111, this air gap separating the two portions 11', 12 from each other. The effect of the air gap 63 is to cut off the light rays incident at an angle greater than the limiting angle a' as illustrated for the ray 101a, as these rays undergo internal reflection on reaching the surface 111. The value of a is given by the relationship 1 7 max= 1 max whence Thus, the gap 63 acts as a virtual diaphragm limiting the field of incident rays for each beam direction. If this gap is sufficiently narrow, the rays that are not internally reflected will have substantially the .path they would have if the

lens members

11, 12 were cemented together.

It will be apparent that the gap 63 need not be provided at the junction of the two differently refracting lens members, though this is most convenient, but could also be located within either of these members forwardly or rearwardly of surface 111'. In the first case, i.e. if member 11' is split concentrically, the value of a' will be greater than that given by Equation 12b; in the second case, i.e. if the gap is within member 12', it will be less. Naturally, such a gap could also be provided within the member 11 of lens 10 (FIG. 1), in order to reduce the maximum field angle to less than the value a given in Table A.

The use of a beam-limiting concentric gap in a spherical or cylindrical system is, of course, not limited to rotating sectoral lenses; in a stationary lens the gap could extend, "for example, over an arc of 180 or even 360.

Finally, I have shown in FIGS. 3 and 4 a stationary dispersive flattening lens 158, to the read of lens 10', whose front and

back surfaces

159, 160 are concentric with those of the rotating lens 10' and whose optical effect derives from its varying refractive index 12,. The radius of surface 159 is here shown but slightly greater than R (this being, of course, possible only if R is at least equal to R in order that lens 158' should not interfere with the rotation of lens 10'), the radius of surface 160 having been designated R Within the central plane of rotation visible in FIG. 3, which includes the axis 100, the value of n is relatively small and may substantially equal In; in a direction transverse to that plane, as seen in FIG. 4, this index increases progressively to a relatively large value which may be substantially equal to In. In order to afford the desired flattening of the image as projected upon the film 20, the back-focal distance s of the overall system should vary according to the relationship cost) (17) 0 being the angle of incidence of a ray a with reference to axis 100 in the plane of FIG. 4; the relationship between s" and n may be expressed as From these equations the law of n as a function of 0 may be derived, yet this law need not be strictly adhered to because, in practice, the small spread of refractive in dices limits the field angle to relatively small values 20 if the lens 158' is to be of moderate width. Thus, ug la, 5; if %,=2, o z8 FIG. 6 illustrates the combination of lens 10 with a special flattening lens 210 fixedly positioned in front thereof. Lens 210 has a planar forward face 211 in a concave rear face 212 of noncircular cross-section, described more fully hereinafter with reference to FIG. 9, which is so shaped that the rays emerging from the rear surface 112 of lens 10 are focused on a fiat surface 220 transverse to axis 100. To this end it is necessary that the incident rays 201, taken to originate at an infinitely distant object and therefore to be parallel, are so refracted by the surface 212 as to diverge to a different extent for different angles of incidence so that the secondary focus P" will always fall on the plane 220. It is assumed here that both

lenses

210 and 10 have identical curvatures in all planes which include their common axis 100, yet the extent of lens 210 in the plane of rotation of lens 10 about axis 21 may be considerably greater than in a plane transverse thereto. With this arrangement it is, of course, no longer necessary to provide a flattening lens behind the lens 10, e.g. as shown at 158 in FIG. 4.

At 230 I have shown, in dot-dash lines, an auxiliary lens of conventional sperical curvature (not necessarily concentric with lens 10) which may be used in addition to lens 210, or as integral part thereof, to focus the system upon objects at finite distances, or to allow it to project images upon a screen or the like from, say, a film passing along plane 220. Lens 230 may, of course, be designed in keeping with the usual technique as a singlet, a doublet or any suitable combination of members with the desired focal length; its refractive index and/or Abb number 11 may or may not be equal to those of lens 210. The sole requirement, insofar as the present invention is concerned, is its ability to let the light rays of a given beam pass parallel between the

surfaces

211 and 212 of lens 210.

It will be noted that the optically effective surface 212 of flattening lens 210 extends, in the plane of rotation about axis 21, over an angle which is substantially greater than the sectional field angle of the circularly curved focal surface F (FIG; 1) of lens 10, i.e. the angle defined by

lines

100 and 100 in FIG. 1.

In FIG. 7 I have shown a modified system similar to that of FIG. 6 wherein, however, a field of parallel rays between

surfaces

311, 312 of a lens member 310 is produced (for a primary focus at infinity) with the aid of an auxiliary lens member 330 bounded by

spherical surfaces

311, 331 concentric with the surfaces 110, 111of lens 10. This is accomplished by choosing the radii R, and R of

surfaces

331 and 311, respectively, so that the power of member 330 is zero, it being assumed that the refractive index of member 310 is less than that of member 330. If, for convenience, we again assign the values n =l.5 and n =l.75 to these two members, we find that the power of member 330 is given as 0.75 g2 5 O.75(R,R

R, R 7R,,R

whence, for zero power,

Rg gRb may be reproduced by projection onto a screen through the

lens system

10, 310, 330). Lens 330 is shown composed of two

portions

330a, 330b separated by a narrow,

forwardly convex spherical gap 363 which acts as a virtual diaphragm, limiting the field of incident rays, in the manner described in conjunction with FIGS. 3 and 4.

The curvature of the concave

rear surfaces

212 and 312 of

lens members

210 and 310 in FIGS. 6 and 7 will now be described with reference to FIGS. 8a-8d and 9.

According to the classical theory of optics, a planar beam of parallel light rays traversing a circular boundary between two media of different optical densities is caused to converge upon a real focus beyond the boundary if the latter is concave toward the denser medium but is caused to diverge, seemingly originating at a virtual focus before the boundary, if the latter is of the opposite curvature. If the refractive indices of the two media are 1 and n, respectively, the distance between the focus and the boundary is given as r/n-l if the beam comes from the side of the more permeable medium (e,g. air) and as nr/n-1 if it originates within the denser medium. The first situation is illustrated in FIGS. 8a and 8d, the second in FIGS. 8b and 80.

FIG. 8a shows a beam B composed of rays parallel to an axis A, impinging upon a positively refracting boundary between an optically thinner and an optically denser medium, the latter having the refractive index n.

The boundary may be considered in first approximation,

for rays close to the axis, a circle C illustrated in dotdash lines. More exactly, however, this boundary is constituted by an ellipse E, the focal length f, constituting the distance between the vertex V, and the distal focus F, of the ellipse. The eccentricity e of the ellipse, whose half-axes bear the usual designations a and b, is given by the relationship Curve C is, of course, the osculatory circle of the ellipse at its vertex V,,, the radius a of this circle being equal to, a /b. With n=1.5, the focal length f,,=a+e=3r.

FIG. 8b shows a similar positively refracting surface represented in first approximation, for paraxial rays, by a circle C The beam B parallel to axis A originates within the denser medium to the left of the boundary and is focused on a point F whose distance from vertex V is given (for the previously assumed value of n) as f =2r, r being again the radius of the circle. It can be shown that F is the distal focus of a hyperbola H which osculates the circle C and whose half-axes a, b and eccentricity e satisfy the relationship FIG. 8 shows the case of a negatively retracting boundary traversed by a beam parallel to axis A the bound ary being defined by a circle C osculating a hyperbola H The virtual focus F is again the distal focal point of hyperbola H its distance f from the vertex V being given by Equation 20. In this figure, as in FIG. 8b, the beam impinges upon the boundary from the side of the denser medium.

A reversal of the situation shown in FIG. has been illustrated in FIG. 8d wherein a beam B centered on an axis A enters the denser medium by way of a negatively refracting boundary defined by a circle C osculating an ellipse E The parameters of the ellipse E are the same as those of ellipse E, in FIG. 8a, point F being one of the foci of the ellipse whose distance f from the distal vertex V is defined by Equation 19.

It should be noted that the ellipses and hyperbolas described in connection with FIGS. 8a-8d apply only to parallel rays; the more general case, in which rays originating at one point are focused upon another, cannot be solved with any second-order curve. Moreover, these conic sections are of but limited utility in practical optical applications since their focusing action extends only to beams incident in a certain direction, i.e. parallel to the axis.

In my copending application Ser. No. 850,628, filed November 3, 1959, now Patent No. 3,112,355, issued November 26, 1963. I have shown in connection with a catoptric system that it is possible to construct a curve which osculates in each point a different conic (specifically a hyperbola), the locus of the foci of all these conics being a predetermined curve which is the conjugate of a straight line. In the present instance, as will be more fully described with reference to FIG. 9, I propose to utilize the same technique in plotting the curvature of lens surface 212 or 312 (FIGS. 6 and 7) to insure that the rays projected through lens 10 converge upon the

plane

220 or 320 at all angles of incidence.

In FIG. 9 the spherical lens 10, whose center is shown at O, has not been illustrated since its only parameter of interest for the subsequent analysis is its focal length f. In fact, for purposes of this discussion the lens 10 could even be a homogeneous spherical segment having a characteristic as shown in graph I of FIG. 2, provided of course that its field of view were limited to a solid angle small enough to reduce the defocusing eflfect for slanting rays to a tolerable value.

The curve 412 in FIG. 9 represents the profile of the

surfaces

212 and 312 of FIGS. 6 and 7, it being understood that this curve could also be the outline of a cylindrical surface if focusing in only one plane were desired. The line 4 20, perpendicular to axis 100, is representative of the projection surfaces 220 and 320 of the two preceding figures.

If a lens centered on point 0, having a focal length 1, is to concentrate incident parallel rays upon the line 420 at all angles of incidence, these rays must reach the lens as divergent beams originating from point F on a curve 400 having a distance s from O, the distance s" of from the conjugate point on line 420 being given by the known relationship and s= cos 0 The curve 412, representing the dispersive boundary between two media of refractive indices n and 1, respectively, is so designed as to osculate in each point P(x, y) a respective hyperbola H, H" whose axis A is parallel to the direction of a ray 410 incident at P which is refracted by the boundary 412 toward the point 0; axis A intersects the curve 400 in a point F whose distance from the vertex V proximal to point P satisfies the relationship established with reference to focus F and vertex V of hyperbola H in FIG. 80 as given by Equation 20; thus, the distance Wequals a+e=a(n+1), a and bzpa being the half-axes of the hyperbola whose asymptotes are shown at HA, HA" and which is centered on a point M. In a co-ordinate system based upon the axes a and b of the hyperbola, with M as the origin, point P has the abscissa X and the ordinate Y bearing the well-known relationship X Y" X Y W a"? m 23 From Equations 20 and 23 we obtain 12 whence 1 2 (Ta-T 5) and At 450 I have indicated the perpendicular to curve 412 in point P. This perpendicular includes an angle I with the direction of the axis A and that of the incident ray 401 parallel thereto. The refracted ray 401' lies on a straight line between points F and O, the angle included between this line and the axis A being designated 11. The radius of curvature p, which of course coincides with the perpendicular 450, includes with the principal axis an angle -r and with ray 401 an angle a. The angle of incidence between ray 401 and axis 100 has been designated w.

The following relationships obtain between the various angles identified above:

Point P can also be defined in polar co-ordinates by its distance R from point 0 and the angle 0, with R cos 0=X and R sin 0=Y. The distance A between F and P will be found to equal nX-i-a, with nX+a 31 and an+X nX+a 32 Furthermore, as will be clearly apparent from the drawing,

AznX-i-a s-R (33) From the foregoing relationships, in conjunction with the well-known formula for the radius of curvature R -dT and R F Equation 37 can be rewritten, using the value of .9 taken from Equation 2, in the following form:

14 but on a straight line 520 tangent thereto, this line corresponding for example to a generatrix of the transparent Since curve 412 is to be symmetrical about

axis

100, 10 cylinder 20 in FIG. 5. For this purpose the radius p R =O for :0, t=0. If, for convenience, k= n so that s has the finite value nf/nl', the value of R is found to be Ru R,,4nR f 1 It thus follows that, if R is selected to equal f, fi also goes to zero.

The dependent variable R of the polar co-ordinate system R, 0 may now be expressed as polynomial according to Maclaurins formula, the coefficient of the odd-numbered powers of t being zero. The higher derivatives of R at the point t=0 are found by repeated differentiation to have the following values:

11 Ro dt which is reasonably accurate for angles up to about 0=60.

It will thus be apparent that the system illustrated in FIGS. 6, 7 and 9 has an axis of symmetry 100 which passes through the middle of the dispersive

stationary lens

210 or 310 and through the center of the rotatable collective lens 10, the

rear face

212 or 312 of the dispersive lens being provided with a noncircular curvature (at least in a plane transverse to axis 21) which converts incident beams 201, 301 of parallel light rays into diverging beams 201', 301' with axes 200, 300 trained upon the lens axis 21, these axes coinciding in FIG. 9 with the ray 401' which constitutes a principal ray in a bundle of parallel light rays incident upon the boundary 412 in the vicinity of the point P; the intersection between the beam axis or principal ray and the locus 400, represented by point F in FIG. 9, is one of the foci of a hyperbola (H', H") which osculates the boundary 412 at the point P but which has different parameters for different such points, except that the relationship expressed by Equation 20 holds true for all these hyperbolas.

In FIG. 10 I have shown, in transverse section, a modification of lens 10 in which the rear surface of lens member 12 is not a spherical segment, as heretofore assumed, but has a noncircular curvature 112" in a plane including the axis of shafts 221a, 221b; in a plane transverse thereto, i.e. the one visible in FIGS. 6 and 7, this face still is bounded by a circular are centered on the point 0. The radius p of curvature 112" is so chosen that slanting beams 501 are caused to converge not along the arc F which suggests that, for a focal length varying inversely with cos 0 where 6 (by analogy with FIG. 9) is the angle between beam axis 500 and the lens axis 100,

wherein K (p) and E(p) have the same significance as K2 and E but with substitution of p for R For the specific values for n and n heretofore assumed, Equation 44 yields the relationship The foregoing formula for .1 is, however, not precise because, on the other hand, the noncircular curvature 112" causes a deflection of the beam axis as exaggeratedly indicated at 500x, thereby reducing the effective length of f, and because, on the other hand, the cross-section of the rear surface of lens member 12 in a plane including the axis 500 is, strictly speaking, not a circular arc the the arc of an ellipse with the major half-axis p and the minor half-axis R hence the curvature of the rear lens face within that cross-section has a radius smaller than p. Since these two aberrations tend to balance each other for small angles 0, the expression for given in Equation 46 or, more generally, derived from Equation 44 may be considered suitable for most cases.

It will be apparent that a flattening lens of the general type shown at 158 in FIGS. 3 and 4 could also be given an aspherical rear surface similar to the surface 112" of FIG. 10, in lieu of or in addition to its varying index of refraction, with consequent enlargement of its field angle 0; conversely, the lens 158, with or without this modification, could be made integral with lens 10' for rotation therewith.

My invention is, of course, not limited to the specific embodiments as described and illustrated but may be modified in various respects, including the combination of compatible features from different embodiments, without departing from the spirit and scope of the appended claims.

I claim:

1. In an optical objective system, in combination, supporting means forming a substantially flat receiving surface for projected light rays and dioptric means adapted to focus incident rays upon said receiving surface, said dioptric means including a sectoral lens having two outer surfaces and at least one inner surface all circularly curved around an axis parallel to said receiving surface within a limited sectional sectoral field angle, said lens being ro- 15 tatable about said axis, and further including imageflattening lens means fixedly positioned with reference to said receiving surface, said lens means having at least one optically'eifective curved surface extending over an angle substantially larger than said field angle in the plane of rotation; and means for rotating said sectoral lens about said axis.

2. The combination defined in claim 1 wherein said sectoral lens is a doublet.

3. The combination defined in claim 1 wherein said image-flattening lens means comprises a dispersive lens element bounded by surfaces that are substantially concentric with those of said sectoral lens, said lens element being disposed on the side of said lens element opposite said receiving surface.

References Cited by the Examiner UNITED STATES PATENTS 3/1959 Blaisse 95-16 2/ 1960 Bouwers 9516 12/1960 Van Heel 88-57 7/ 1962 Bouwers 95-16 10/ 1964 Bouwers 8857 2/1965 Campbell 95l6 FOREIGN PATENTS 4/1959 Great Britain.

JOHN M. HORAN, Primary Examiner.

Claims

1. IN AN OPTICAL OBJECTIVE SYSTEM, IN COMBINATION, SUPPORTING MEANS FORMING A SUBSTANTIALLY FLAT RECEIVING SURFACE FOR PROJECTED LIGHT RAYS AND DIOPTRIC MEANS ADAPTED TO FOCUS INCIDENT RAYS UPON SAID RECEIVING SURFACE, SAID DIOPTRIC MEANS INCLUDING A SECTORAL LENS HAVING TWO OUTER SURFACES AND AT LEAST ONE INNER SURFACE ALL CIRCULARY CURVED AROUND AN AXIS PARALLEL TO SAID RECEIVING SURFACE WITHIN A LIMITED SECTIONAL SECTORAL FIELD ANGLE, SAID LENS BEING ROTATABLE ABOUT SAID AXIS, AND FURTHER INCLUDING IMAGEFLATTENING LENS MEANS FIXEDLY POSITIONED WITH REFERENCE TO SAID RECEIVING SURFACE, SAID LENS MEANS HAVING AT LEAST ONE OPTICALLY EFFECTIVE CURVED SURFACE EXTENDING OVER AN ANGLE SUBSTANTIALLY LARGER THAN SAID FIELD ANGLE IN THE PLANE OF ROTATION; AND MEANS FOR ROTATING SAID SECTORAL LENS ABOUT SAID AXIS.