Image-forming optical apparatus having a decentered optical surface

A compact image-forming optical apparatus which is free from moire fringes and capable of providing an aberration-free, clear image of minimal distortion even at a wide field angle. The image-forming optical apparatus has an objective optical system (104) and an electronic image pickup device (108). The objective optical system (104) has at least one reflecting surface (106, 107) decentered such that the whole surface is tilted with respect to the axial principal ray. The reflecting surface has a rotationally asymmetric surface configuration that corrects rotationally asymmetric decentration aberrations caused by decentration. A low-pass member (103) is disposed in the vicinity of a pupil plane (101) closer to the object than the reflecting surface. The low-pass member cuts off a high-frequency component concerning the object image in a bundle of rays led to the electronic image pickup device (108).

BACKGROUND OF THE INVENTION 
The present invention relates to an image-forming optical apparatus and, 
more particularly, to an image-forming optical apparatus which is most 
suitable for an image pickup apparatus designed to form an image 
relatively small in size and uses an objective optical system in which at 
least one reflecting surface having an image-forming power required for 
image formation is decentered. 
There has heretofore been known a compact reflecting decentered optical 
system as disclosed in Japanese Patent Application Unexamined Publication 
Number hereinafter referred to as "JP(A)"! 59-84201. This is an invention 
of a one-dimensional light-receiving lens comprising a cylindrical 
reflecting surface; therefore, two-dimensional imaging cannot be effected 
with this conventional optical system. JP(A) 62-144127 discloses an 
optical system wherein the identical cylindrical surface is used twice to 
effect reflection in order to reduce spherical aberration in the 
above-mentioned invention. 
JP(A) 62-205547 discloses the use of an aspherical reflecting surface as a 
reflecting surface, but makes no mention of the configuration of the 
reflecting surface. 
U.S. Pat. Nos. 3,810,221 and 3,836,931 both disclose an example in which a 
rotationally symmetric aspherical mirror and a lens system having a 
surface which has only one plane of symmetry are used to constitute a 
finder optical system of a reflex camera. In this example, however, the 
surface having only one plane of symmetry is utilized for the purpose of 
correcting the tilt of a virtual image for observation. 
JP(A) 1-257834 (U.S. Pat. No. 5,274,406) discloses an example in which a 
surface having only one plane of symmetry is used for a reflecting mirror 
to correct image distortion in a rear projection type television. In this 
example, however, a projection lens system is used for projection onto a 
screen, and the surface having only one plane of symmetry is used for 
correction of image distortion. An example of a back-coated mirror type 
decentered optical system using an anamorphic surface and a toric surface 
as an observation optical system is also disclosed. However, the 
decentered optical system is not sufficiently corrected for aberrations, 
including image distortion. 
None of the above-described prior arts use a surface having only one plane 
of symmetry as a back-coated mirror to form a folded optical path. 
JP(A) 8-292368, 8-292371 and 8-292372 each disclose an image pickup optical 
system (i.e. a fixed focal length optical system or a zoom optical system) 
using a surface having only one plane of symmetry as a reflecting surface. 
However, the disclosed image pickup optical system has an unfavorably long 
optical path length from an entrance surface of an optical system 
constituent element including a rotationally asymmetric surface to an exit 
surface thereof or from a rotationally asymmetric surface of the optical 
system that is closest to the object to a rotationally asymmetric surface 
thereof that is closest to the image (in an example, image formation takes 
place once in the course of travel of light along the optical path). This 
causes the optical system to increase in size. Therefore, there is no 
merit in using rotationally asymmetric surfaces, which are difficult to 
produce. 
Incidentally, to remove moire fringes appearing in an image pickup 
apparatus due to superposition of the repeating period of the pixels of an 
image pickup device and a spatial frequency component in an object image 
which is close to the repeating period, JP(A) 7-325269 proposes a low-pass 
filter which enables such moire patterns to be effectively removed by 
using a double image formed by pupil division and which is less costly and 
effective even under defocus conditions. 
In the conventional rotationally symmetric optical systems, a transmitting 
rotationally symmetric lens having a refracting power is assigned to exert 
the required refracting power. Therefore, many constituent elements are 
needed for aberration correction. In the conventional decentered optical 
systems, an imaged figure or the like is undesirably distorted and the 
correct shape cannot be recorded unless the formed image is favorably 
corrected for aberrations, particularly rotationally asymmetric 
distortion. 
In a rotationally symmetric optical system comprising a refracting lens 
which is formed from a surface rotationally symmetric about an optical 
axis, a straight-line optical path is formed. Therefore, the entire 
optical system undesirably lengthens in the direction of the optical axis, 
resulting in an unfavorably large-sized image pickup apparatus. 
SUMMARY OF THE INVENTION 
In view of the problems associated with the prior arts, an object of the 
present invention is to provide a compact image-forming optical apparatus 
which is capable of providing a clear image of minimal distortion even at 
a wide field angle and which has a low-pass filter to remove moire fringes 
appearing when an image is taken by using an electronic image pickup 
device, e.g. a CCD. 
To attain the above-described object, the present invention provides an 
image-forming optical apparatus having an objective optical system and an 
electronic image pickup device which is disposed in a plane where an image 
of an object is formed by the objective optical system. The objective 
optical system has at least one reflecting surface decentered such that 
the whole surface is tilted with respect to an axial principal ray defined 
by a light ray emanating from the center of the object and passing through 
the center of the pupil to reach the center of the object image. The 
reflecting surface has a rotationally asymmetric surface configuration 
that corrects rotationally asymmetric decentration aberrations caused by 
decentration. A low-pass member is disposed in the vicinity of a pupil 
plane closer to the object than the reflecting surface. The low-pass 
member cuts off a high-frequency component concerning the object image in 
a bundle of light rays led to the electronic image pickup device. 
In this case, it is desirable for the objective optical system to have a 
prism member formed from a medium having a refractive index (n) larger 
than 1 (n&gt;1). The prism member desirably has at least three optical 
surfaces which include a first surface disposed closest to the object and 
having an action through which a bundle of light rays enters the prism 
member, a third surface which is a reflecting surface having the 
above-described rotationally asymmetric surface configuration to reflect 
the ray bundle entering the prism member, and a second surface having an 
action through which the ray bundle exits from the prism member. 
In this case, the prism member is desirably formed such that a bundle of 
light rays from the object enters the prism member through the first 
surface, and the incident light rays are first reflected by the second 
surface and then reflected by the third surface so as to exit from the 
prism member through the second surface. 
According to the present invention, the objective optical system has at 
least one reflecting surface decentered such that the whole surface is 
tilted with respect to the axial principal ray. The reflecting surface has 
a rotationally asymmetric surface configuration that corrects rotationally 
asymmetric decentration aberrations caused by decentration. Further, a 
low-pass member adapted to cut off a high-frequency component concerning 
the object image in a bundle of light rays led to the electronic image 
pickup device is disposed in the vicinity of a pupil plane closer to the 
object than the reflecting surface. Therefore, it is possible to obtain a 
compact image-forming optical apparatus which is free from moire fringes 
and capable of providing an aberration-free, clear image of minimal 
distortion even at a wide field angle. 
Still other objects and advantages of the invention will in part be obvious 
and will in part be apparent from the specification. 
The invention accordingly comprises the features of construction, 
combinations of elements, and arrangement of parts which will be 
exemplified in the construction hereinafter set forth, and the scope of 
the invention will be indicated in the claims.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
First, the objective optical system according to the present invention will 
be described below. 
Let us explain a coordinate system used in the following description. As 
shown in FIG. 1, it is assumed that a light ray passing through the center 
of an object point and passing through the center of a stop 101 to reach 
the center of an image plane 108 is defined as an axial principal ray 102. 
It is also assumed that an optical axis defined by a straight line along 
which the axial principal ray 102 travels until it intersects a first 
surface 105 of a decentered prism optical system 104 constituting the 
objective optical system according to the present invention is defined as 
a Z-axis, and that an axis perpendicularly intersecting the Z-axis in the 
decentration plane of each surface constituting the decentered prism 
optical system 104 is defined as a Y-axis, and further that an axis 
perpendicularly intersecting the Z-axis and also perpendicularly 
intersecting the Y-axis is defined as an X-axis. 
In general, a spherical lens system comprising only a spherical lens is 
arranged such that aberrations produced by spherical surfaces, such as 
spherical aberration, coma and curvature of field, are corrected with some 
surfaces by canceling the aberrations with each other, thereby reducing 
aberrations as a whole. On the other hand, aspherical surfaces and the 
like are used to favorably effect aberration correction with a minimal 
number of surfaces. The reason for this is to reduce various aberrations 
which would be produced by spherical surfaces. However, in a decentered 
optical system, rotationally asymmetric aberrations due to decentration 
cannot be corrected by a rotationally symmetric optical system. 
When a rotationally symmetric optical system is decentered, rotationally 
asymmetric aberrations occur, and it is impossible to correct these 
aberrations only by a rotationally symmetric optical system. Rotationally 
asymmetric aberrations due to decentration include image distortion, 
curvature of field, and astigmatic and comatic aberrations, which occur 
even on the axis. FIG. 50 shows curvature of field produced by a 
decentered concave mirror M. FIG. 51 shows astigmatism produced by a 
decentered concave mirror M. FIG. 52 shows axial comatic aberration 
produced by a decentered concave mirror M. In the objective optical system 
accordingly to the present invention, a rotationally asymmetric surface is 
disposed in the optical system to correct such rotationally asymmetric 
aberrations caused by decentration. 
Rotationally asymmetric aberrations produced by a decentered concave mirror 
include rotationally asymmetric curvature of field. For example, when 
light rays from an infinitely distant object point are incident on a 
decentered concave mirror, the light rays are reflected by the concave 
mirror to form an image. In this case, the back focal length from that 
portion of the concave mirror on which the light rays strike to the image 
surface is a half the curvature of the portion on which the light rays 
strike. Consequently, an image surface tilted with respect to the axial 
principal ray is formed as shown in FIG. 50. It has heretofore been 
impossible to correct such rotationally asymmetric curvature of field by a 
rotationally symmetric optical system. The tilted curvature of field can 
be corrected by forming the concave mirror M from a rotationally 
asymmetric surface, and, in this example, arranging it such that the 
curvature is made strong (refracting power is increased) in the positive 
Y-axis direction (the upward direction in the figure), whereas the 
curvature is made weak (refracting power is reduced) in the negative 
Y-axis direction (the downward direction in the figure). It is also 
possible to obtain a flat image surface with a minimal number of 
constituent surfaces by disposing a rotationally asymmetric surface having 
the same effect as that of the above-described arrangement in the optical 
system separately from the concave mirror M. 
Next, rotationally asymmetric astigmatism will be explained. A decentered 
concave mirror M produces astigmatism even for axial rays, as shown in 
FIG. 51, as in the case of the above. The astigmatism can be corrected by 
appropriately changing the curvatures in the X- and Y-axis directions of 
the rotationally asymmetric surface as in the case of the above. 
Rotationally asymmetric coma will be explained below. A decentered concave 
mirror M produces coma even for axial rays, as shown in FIG. 52, as in the 
case of the above. The coma can be corrected by changing the tilt of the 
rotationally asymmetric surface according as the distance from the origin 
of the X-axis increases, and further appropriately changing the tilt of 
the surface according to the sign (positive or negative) of the Y-axis. 
If the objective optical system according to the present invention is 
arranged to have a folded optical path, it is possible to impart a power 
to a reflecting surface and hence possible to omit a transmission lens. 
Moreover, because the optical path is folded, the optical system can be 
formed in a compact structure. 
If the reflecting surface is a totally reflecting surface tilted with 
respect to light rays so that the light rays are incident thereon at an 
angle exceeding the critical angle, a high reflectivity can be obtained, 
and it is possible to enable the surface to have both reflecting and 
transmitting actions. The reflecting surface is preferably a reflecting 
surface having a thin film of a metal, e.g. aluminum or silver, formed 
thereon, or a reflecting surface formed from a dielectric multilayer film, 
or a semitransparent reflecting surface. In the case of a metal thin film 
having reflecting action, a high reflectivity can be readily obtained. The 
use of a dielectric reflecting film is advantageous in a case where a 
reflecting film having wavelength selectivity or a reflecting film having 
minimal absorption is to be formed. 
It is preferable to use a rotationally asymmetric surface as a reflecting 
surface. By doing so, no chromatic aberration is produced in contrast to a 
case where it is used as a transmitting surface. Moreover, even if the 
tilt of the surface is small, the surface can bend light rays. 
Accordingly, the amount of other aberrations produced by the surface is 
also small. In other words, when the same refracting power is to be 
obtained, the amount of aberration produced by a reflecting surface is 
smaller than by a refracting surface. 
In the objective optical system according to the present invention, it is 
desirable that at least one reflecting surface having a decentered 
rotationally asymmetric surface configuration should use a plane-symmetry 
free-form surface having only one plane of symmetry. A free-form surface 
(FFS) used in the present invention may be defined by the following 
equation: 
##EQU1## 
where C.sub.m (m is an integer of 2 or higher) are coefficients. 
In general, the above-described free-form surface does not have planes of 
symmetry in both the XZ- and YZ-planes. In the present invention, a 
free-form surface having only one plane of symmetry parallel to the 
YZ-plane is obtained by making all terms with odd-numbered powers of x 
zero. For example, in the above defining equation (a), the coefficients of 
the terms C.sub.4, C.sub.6, C.sub.9, C.sub.11, C.sub.13, C.sub.15, 
C.sub.18, C.sub.20, C.sub.22, C.sub.24, C.sub.26, C.sub.28, C.sub.31, 
C.sub.33, C.sub.35, C.sub.37, . . . are set equal to zero. By doing so, it 
is possible to obtain a free-form surface having only one plane of 
symmetry parallel to the YZ-plane. 
A free-form surface having only one plane of symmetry parallel to the 
XZ-plane is obtained by making all terms with odd-numbered powers of y 
zero. For example, in the above defining equation (a), the coefficients of 
the terms C.sub.3, C.sub.6, C.sub.8, C.sub.10, C.sub.13, C.sub.15, 
C.sub.17, C.sub.19, C.sub.21, C.sub.24, C.sub.26, C.sub.28, C.sub.30, 
C.sub.32, C.sub.34, C.sub.36, . . . are set equal to zero. By doing so, it 
is possible to obtain a free-form surface having only one plane of 
symmetry parallel to the XZ-plane. The use of a free-form surface having 
such a plane of symmetry makes it possible to improve the productivity. 
Rotationally asymmetric aberrations due to decentration can be effectively 
corrected by using a free-form surface having either a plane of symmetry 
parallel to the YZ-plane or a plane of symmetry parallel to the XZ-plane. 
It should be noted that the above defining equation is shown as merely an 
example, and that the feature of the present invention resides in that 
rotationally asymmetric aberrations due to decentration are corrected by a 
rotationally asymmetric surface having only one plane of symmetry. 
Therefore, the same advantageous effect can be obtained for any other 
defining equation. 
Regarding an objective optical system designed to form an image relatively 
small in size as in the objective optical system according to the present 
invention, the objective optical system can be made compact in size on 
drawings by the principle of coefficient multiplication. However, in view 
of the actual production, it is not preferable to make the objective 
optical system excessively small in size because the thickness of the edge 
and center of the lens would be excessively reduced and the lens diameter 
would become excessively small, causing the production cost to increase 
unfavorably. In the case of a conventional optical system comprising a 
refracting lens system, if the optical system is constructed in conformity 
to a producible size, an unfavorably long distance must be ensured between 
refracting surfaces having power because the optical axis is straight, 
resulting in a waste of space. If the optical axis is spatially folded by 
using reflecting surfaces, an optical path necessary for image formation 
can be ensured by effectively utilizing a relatively small space. In this 
case, if the optical path length of the objective optical system is 
unnecessarily long, the optical system increases in size contrary to the 
purpose of effectively using the space by employing an arrangement in 
which the optical axis is folded by decentration. In addition, if the 
optical path length is excessively long in comparison to the image formed 
by the optical system, it is difficult to ensure the back focus required 
for disposing an electronic image pickup device for capturing an optical 
image. 
The following is a description of a low-pass filter disclosed in JP(A) 
7-325269, which is used in combination with the decentered prism optical 
system 104, which constitutes the above-described objective optical system 
according to the present invention. 
The low-pass filter disclosed in JP(A) 7-325269 is capable of effectively 
removing causes of reduction in the image quality, such as moire fringes 
appearing when an endoscopic image is displayed on a monitor, and possible 
to implement at low cost. As shown in FIG. 16, the low-pass filter has a 
polyhedral structure consisting essentially of surface portions a, b, c 
and d formed by dividing the surfaces on both sides A and B into a 
plurality of surface portions and slanting them properly. Two lines normal 
to the portions a and b are in a skew relation to each other with an equal 
angle .theta.a to an optical axis O. Lines normal to the other portions c 
and d are also in a skew relation to each other. The low-pass filter is 
given a filter function ito remove a spatial frequency component 
corresponding to double the distance between two images formed through the 
two surfaces A and B, thereby removing moire or the like due to an array 
of fibers in a fiber bundle or an array of light-receiving elements, for 
example. 
FIG. 16 shows a double-sided polyhedral lens 1 as an optical element having 
an optical low-pass filter function in a first embodiment of JP(A) 
7-325269 (hereinafter referred to as "prior application"). The 
double-sided polyhedral lens 1 has a polyhedral lens formed on each side 
thereof. The polyhedral lens has two split surfaces which are skewed 
relative to each other like the blades of a propeller, as in the case of 
one side of a single-sided polyhedral lens 2 in a modification of the 
first embodiment of the prior application, which is shown in FIG. 17. 
First, the single-sided polyhedral lens 2 in the modification, which has a 
simpler structure, will be described. 
As shown in FIG. 17, one side (assumed to be a side B, for example) of the 
single-sided polyhedral lens 2 is provided with two semicircular portions 
a and b which have an optical axis O in common with each other and have 
slant surfaces which are slanted in opposite directions to each other. 
Lines normal to the two slant surfaces are in a skew relation to each 
other and slant at an angle .theta. to each other. 
Because the portions a and b are formed in a skew relation to each other as 
described above, the filter function to remove moire or the like can be 
satisfactorily exhibited even under defocus conditions as described later 
(see FIG. 22). 
Assuming that the tilt angles of the lines normal to the two slant surfaces 
with respect to the optical axis O are .theta.1 and .theta.2, 
respectively, the following relationship holds: 
EQU .theta.=.theta.1-.theta.2 (1) 
(it should be noted that .theta.1 and .theta.2 express angles, inclusive of 
signs; in FIG. 17, .theta.1=-.vertline..theta.2.vertline.) 
Assuming that a Z-axis is taken in a direction parallel to the optical axis 
O, and X- and Y-axes are taken in a plane perpendicular to the Z-axis, for 
example, and further that the X-axis is taken in the direction of a 
boundary line l between the portions a and 2, data concerning the 
configuration of the side B of the single-sided polyhedral lens 2 is as 
follows: 
For the portion a, Y is zero or positive (i.e. Y.gtoreq.0). For the surface 
of the portion a, Z=P.multidot.X. For the portion b, Y is negative (i.e. 
Y&lt;0). For the surface of the portion b, Z=-P.multidot.X. The parameter P 
expressing a slant surface is, for example, P=tan 1'.apprxeq.0.00029. 
Here, 1'.apprxeq.0.00029 rad. In this case, the angle .theta. is assumed 
to be .theta.=2'. As shown in part (b) of FIG. 17, the size of the 
single-sided polyhedral lens 2 is as follows: The diameter .phi. is 8 
millimeters; the thickness T.sub.0 is 1 millimeter, for example; and the 
refractive index n is 1.51633, for example. 
It should be noted that the right-hand half of part (b) of FIG. 17 is a 
plan view, and the left-hand half is a side view. As shown in part (b) of 
FIG. 17, the side B is provided with a mark M for identifying the 
orientation of the optical element. The other side of the single-sided 
polyhedral lens 2, that is, side A, is a flat surface. 
FIG. 18 shows an endoscope apparatus 3 as an optical apparatus according to 
a modification of the first embodiment of the prior application which uses 
the single-sided polyhedral lens 2. The endoscope apparatus 3 includes a 
hard endoscope 5 having an illumination optical system and an observation 
optical system; a television camera 6 attached to the hard endoscope 5 and 
containing an image pickup device; a light source unit 7 which supplies 
illuminating light to the hard endoscope 5; a CCU 9 which executes signal 
processing for a solid-state image pickup device 8, e.g. a CCD, contained 
in the television camera 6; and a color monitor 10 connected to the CCU 9 
to display image signals. 
The endoscope apparatus 3 is arranged to remove moire fringes due to the 
two-dimensional array of light-receiving elements (including mosaic 
filters) in the solid-state image pickup device 8 as an optical member, 
such as moire fringes caused by interference with the period of the array, 
and moire fringes caused by interference with color modulation. 
The hard endoscope 5 has an insert part 11 formed from a hard sheathed 
tube. A grip portion 12 with an enlarged diameter is formed at the rear 
end of the insert part 11 so as to be gripped. An eyepiece portion 13 is 
formed at the rear end of the grip portion 12. The grip portion 12 is 
provided with a light guide socket 14. The hard endoscope 5 is detachably 
connected to the light source unit 7 through a light guide cable 15. 
White illuminating light from a lamp 16 in the light source unit 7 is 
transmitted through a light guide serving as an illuminating light 
transmitting device in the light guide cable 15. The illuminating light is 
supplied through the light guide socket 14 to a light guide 17 in the hard 
endoscope 5. The transmitted illuminating light is emitted forward from an 
end surface attached to an illuminating window in the distal end portion 
of the insert part 11. Thus, an illumination optical system is formed. 
An objective lens system 18 is mounted in a viewing window formed adjacent 
to the illuminating window. An image of a subject illuminated is formed by 
the objective lens system 18. The subject image is relayed by a relay lens 
system 19 as an image transmitting optical system which is disposed in the 
insert part 11 along an optical axis of the objective lens system 18 in 
coaxial relation to it so that a final image is formed in the vicinity of 
the eyepiece portion 13. 
The image can be observed with the naked eye through an ocular lens (ocular 
optical system) 20 which forms an observation optical system. In a case 
where the television camera 6 is attached to the eyepiece portion 13, the 
image transmitted through the relay lens system 19 can be taken through 
the ocular lens 20 by the television camera 6. More specifically, the 
television camera 6 contains a single-sided polyhedral lens 2 having the 
function of an optical low-pass filter, an image-forming lens (imaging 
lens) 21, and a solid-state image pickup device 8 having color separating 
filters, e.g. mosaic filters. Thus, light passing through the ocular lens 
20 passes through the single-sided polyhedral lens 2 and the imaging lens 
21 to form an image on the solid-state image pickup device 8. 
The image is photoelectrically converted by the solid-state image pickup 
device 8 into electric signals, which are then converted into a standard 
video signal by the CCU 9 to display an image on the color monitor 10. As 
shown for example in FIG. 19, the solid-state image pickup device 8 has 
pixels regularly arranged in a two-dimensional matrix on a photoelectric 
conversion surface. The pixels serve as light-receiving elements having a 
photoelectric conversion function. 
FIG. 19 shows the pixel array on the solid-state image pickup device 8. In 
the figure, R, G and B show that mosaic filters of R (red), G (green) and 
B (blue) are placed in front of the pixels. Light that is separated into 
R, G and B colors is photoelectrically converted to thereby perform a 
color image pickup operation. Reference symbols in the figure represent 
dimensions and periods as follows: 
Px . . . the dimension in the horizontal direction of one pixel of the 
solid-state image pickup device 8; 
Py . . . the dimension in the vertical direction of one pixel of the 
solid-state image pickup device 8; 
Wy . . . the dimension in the vertical direction of the effective image 
pickup area of the solid-state image pickup device 8; 
Wx . . . the dimension in the horizontal direction of the effective image 
pickup area of the solid-state image pickup device 8; 
M . . . the period in the horizontal direction of the mosaic filters of the 
solid-state image pickup device 8, expressed in units of pixels; in the 
example shown in FIG. 19, M=2; 
N . . . the period in the vertical direction of the mosaic filters of the 
solid-state image pickup device 8, expressed in units of pixels; in the 
example shown in FIG. 19, N=2. 
It should be noted that M and N are regarded as M=N=1 in the case of a 
solid-state image pickup device 8 with no mosaic filters. 
FIG. 20 shows the image pickup optical system of the television camera 6. 
Part (a) of FIG. 20 shows the orientation of the single-sided polyhedral 
lens 2. As shown in the figure, the single-sided polyhedral lens 2 is 
disposed such that the boundary line (dividing line) l between the 
portions a and b lies in the horizontal direction, that is, in parallel to 
the horizontal (breadthwise) direction of the solid-state image pickup 
device 8. In this case, the image separates into two images in the 
horizontal scanning direction of the solid-state image pickup device 8. 
More specifically, part (b) of FIG. 20 shows a double image formed through 
the single-sided polyhedral lens 2. The two images are formed apart from 
each other in the horizontal direction by a distance d. It is assumed in 
the present invention that the horizontal (breadthwise) direction of the 
solid-state image pickup device 8 is set to the X-axis, and the vertical 
(lengthwise) direction to the Y-axis. 
FIG. 21 is a diagram for explaining the function of removing moire fringes 
by forming a double image using the single-sided polyhedral lens 2 in the 
image pickup optical system. The basic idea of removing moire fringes is 
as follows: In sampling of an object image, if the sampling frequency is 
close to a frequency component contained in the object image, moire 
fringes appear. Therefore, the optical low-pass filter is set to frequency 
characteristics with which the relevant frequency component is removed. 
As shown in FIG. 21, if an object which is bright and dark at a repeating 
period is imaged (on the image pickup surface of the solid-state image 
pickup device 8) through the single-sided polyhedral lens 2 and the 
image-forming lens 21, a first image is formed through the portion a, and 
a second image is formed through the portion b. In a case where the 
distance by which the first and second images are apart from each other is 
set to 1/2 of the period, if the intensity distributions of the two images 
are superimposed on one another, the peaks of the intensity distribution 
of one image fill the troughs of the intensity distribution of the other 
image, resulting in uniform intensity distributions. Consequently, the 
presence of moire fringes is unrecognizable. In other words, if such a 
double image is formed, a frequency component having a repeating period 
which is double the separation distance between the two images disappears. 
Accordingly, moire fringes can be removed by properly setting the 
relationship between the pixel sampling pitch (repeating period) and the 
image separation distance. 
FIG. 22 is a diagram for explaining differences in action between an 
optical element satisfying the skew relationship between slant surfaces, 
which is employed in the embodiments and modifications of the prior 
application, and an optical element having surface portions which are 
merely symmetric with respect to a point. Part (a) of FIG. 22 shows the 
optical element in FIG. 17, that is, a portion of an optical system where 
the single-sided polyhedral lens 2, in which the slant surfaces satisfy 
the skew relationship, is disposed. Part (b) of FIG. 22 shows an optical 
system in which a prism-shaped member an element shown in FIG. 1 of JP(A) 
3-248695! having a flat surface at one side thereof and an angular surface 
at the other side thereof is disposed together with a lens. 
With such an optical element disposed in an optical path, an image-forming 
optical system forms a double image of an object on an image plane. When 
the image plane is in focus, the same effect is obtained (a double image 
is formed) no matter which optical element is used. Under defocus 
conditions, the two optical elements differ in function from each other. 
To check the difference in function under defocus conditions, 
image-formation conditions at a position away from the image plane should 
be examined. 
With the arrangement shown in part (b) of FIG. 22, light beams refracted by 
the slant surfaces of the prism-shaped member form respective images on 
the opposite sides of the optical axis for example, in part (b) of FIG. 
22, light on the upper side of the optical axis forms an image on the 
lower side of the optical axis at the image plane!. In this case, because 
light beams from the upper side of the optical axis and light beams from 
the lower side of the optical axis intersect each other, light gathers in 
a relatively narrow area in the vicinity of the optical axis. Accordingly, 
at a defocus position, image separation may become impossible to effect 
(or may become insufficient), and the low-pass filter function may become 
lost (or may become insufficient). That is, in part (b) of FIG. 22, a 
double image is formed at a defocus position away from the focus position 
in the rearward direction. However, at defocus position away from the 
focus position in the forward direction, the object image does not 
separate into two images. Thus, the low-pass filter function becomes lost 
(or insufficient). In other words, the shape of point images changes 
according to the defocus position; therefore, MTF (Modulation Transfer 
Function) changes undesirably. 
In part (a) of FIG. 22, because the slant surfaces are in a skew relation 
to each other, light beams which form two images can be kept separate from 
each other even under considerably defocused conditions. Therefore, the 
low-pass filter function is not lost. That is, in part (a) of FIG. 22, a 
double image is formed at a defocus position rearwardly away from the 
focus position, and a double image is also formed at a defocus position 
forwardly away from the focus position. Moreover, there is substantially 
no change in the image separation distance (the distance between the two 
images). Therefore, the arrangement shown in part (a) of FIG. 22 exhibits 
the low-pass filter function even for a defocused image. Accordingly, the 
arrangement has the function of preventing the occurrence of moire fringes 
not only for an image formed in an in-focus state but also for an image 
formed in a defocused state. In other words, even under defocus 
conditions, the point images maintain a double-image configuration, and 
MTF is determined by multiplying the double image by the effect of 
defocus. 
The single-sided polyhedral lens 2 having slant surfaces as shown in FIG. 
17 can be produced by plastic or glass molding process using a metal form. 
Alternatively, the single-sided polyhedral lens 2 may be produced by 
coating a planar substrate non-uniformly. 
In such a case, the surface configuration of a boundary portion (a portion 
surrounded by the dashed-and-dotted lines in FIG. 25, described later) 
between the two slant surfaces of the single-sided polyhedral lens 2 is 
not conformable to the design values but disordered. The low-pass filter 
function degrades at the disordered portion, and light passing through the 
disordered portion causes flare. To prevent the occurrence of these 
problems, for example, a substance which does not transmit light is 
provided on the disordered portion to form a light-blocking portion. 
In a case where the polyhedral lens 2 is used in the television camera 6 
(or a consumer VTR camera, a general TV camera, etc.), which is attached 
to the hard endoscope 5 as shown in FIG. 18, it is desirable to satisfy at 
least one of conditions (2) to (21) described below. In this modification 
of the first embodiment of the prior application, the polyhedral lens 2 is 
set so as to satisfy these conditions, thereby enabling moire fringes to 
be effectively removed even under defocus conditions in a case where moire 
fringes appear because of the sampling period or other cause. For 
conditional expressions, the magnification .beta.r, the distance d between 
two images, etc. are defined as follows: 
.beta.r . . . the magnification of a lens lying between the polyhedral lens 
2 and the image-formation plane (the solid-state image pickup device 8 in 
this case); 
Sf . . . the distance from the polyhedral lens 2 to an image formed by a 
lens forward of the polyhedral lens 2 (in the figure, the rightward 
direction is assumed to be a positive direction); 
d . . . the distance between two images formed by the polyhedral lens 2 
see part (b) of FIG. 20!; 
n . . . the refractive index of the polyhedral lens 2. 
It should be noted that in FIG. 18 there is another lens (e.g. the ocular 
lens 20 of the endoscope 5) in front of the polyhedral lens 2; if there is 
no lens in front of the polyhedral lens 2 (i.e. an image of an object is 
directly taken), Sf is the distance to the object. 
First, to remove moire fringes due to the horizontal sampling of the mosaic 
filters, it is desirable to satisfy the following condition: 
EQU 1/.vertline.2(n-1).theta.Sf.beta.r.vertline.=1/(P.times.M) (2) 
In the condition (2), the denominator of the left-hand side member 
expresses double the separation distance between the two images on the 
image plane, and the denominator of the right-hand side member expresses 
the distance of the sampling period. Moire fringes are removed by setting 
the variables so that these distances are equal to each other. 
From the viewpoint of practical application, moire fringes may remain to 
some extent. Therefore, the condition (2) may be relaxed to give the 
following condition: 
EQU 0.75/(P.times.M).ltoreq.1/.vertline.2(n-1).theta.Sf.beta.r.vertline..ltoreq 
.1.5/(P.times.M) (3) 
The lower limit of the condition (3) corresponds to a frequency at which 
the value of MTF is about 40%. If the image separation distance is greater 
than this value, MTF on the low-frequency side becomes small, and the 
image contrast lowers to such an extent as to give rise to a problem. The 
upper limit corresponds to a frequency at which the value of MTF is about 
70%. If the image separation distance is smaller than this value, the 
function of removing moire fringes is deteriorated. 
To remove moire fringes due to luminance sampling, M=N=1 should be set in 
the conditions (2) and (3). 
In the case of an NTSC television camera, electron-scope, etc., moire 
fringes due to the modulation of color signals appear. If it is necessary 
to remove such moire fringes, because the color subcarrier frequency is 
3.58 MHz, it is desirable to satisfy the following condition: 
EQU 0.75.multidot.40.multidot.3.58/Wy.ltoreq.1/.vertline.2(n-1).theta.Sf.beta.r 
.vertline..ltoreq.1.5.multidot.40.multidot.3.58/Wy (4) 
That is, 
EQU 107.4/Wy.ltoreq.1/.vertline.2(n-1).theta.Sf.beta.r.vertline..ltoreq.214.8/W 
y(5) 
Here, the fact that 1 MHz is equivalent to 80 TV lines is used. 
If the single-sided polyhedral lens 2 is placed as shown in part (a) of 
FIG. 20, trap lines (lines where MTF=0) lie as shown by the dotted lines 
in FIG. 23 (which shows an operation of removing moire fringes due to 
color modulation by trap lines in a spatial frequency plane). However, 
moire fringes caused by color modulation appear at points marked with 
circles in FIG. 23. Therefore, the trap lines should pass through these 
points. If the dividing direction of the polyhedral lens 2 is tilted with 
respect to the horizontal direction by .omega.+90.degree., trap lines lie 
as shown by the solid lines in FIG. 23. Accordingly, assuming that .phi. 
is defined by 
EQU .phi.=arc tan 1.64/3.58 i.e. 1.64/3.58=tan .phi. (6) 
it is desirable to satisfy the following condition: 
EQU cos(90.degree.-.omega.-.phi.).multidot.0.75.multidot.40.multidot.A/Wy.ltore 
q.1/.vertline.2(n-1).theta.Sf.beta.r.vertline..ltoreq.cos(90.degree.-.omega 
.-.phi.).multidot.1.5.multidot.40.multidot.A (7) 
where A=.sqroot. (1.64.multidot.1.64+3.58.multidot.3.58) here .sqroot. ( 
) expresses the square root of the sum of the terms inside the 
parentheses!; in terms of square, 
A.multidot.A=1.64.multidot.1.64+3.58.multidot.3.58. It should be noted 
that the angle is assumed to be positive when measured clockwise from a 
coordinate axis. 
The condition (7) may be rewritten as follows: 
EQU sin(.omega.+.phi.)118.multidot.136/Wy.ltoreq.1/.vertline.2(n-1).theta.Sf.be 
ta.r.vertline..ltoreq.sin(.omega.-.phi.)236.27/Wy (8) 
Therefore, it is desirable to satisfy either the condition (7) or (8). That 
is, moire fringes due to color modulation can be removed by setting the 
dividing direction (the direction of the boundary line l) and so forth so 
that the condition (7) or (8) is satisfied. It should be noted that Ux and 
Uy in FIG. 23 denote spatial frequencies in the X and Y directions, 
respectively, on the image plane (in this case, the photoelectric 
conversion surface of the solid-state image pickup device 8). 
Let us give a supplementary explanation of the above-mentioned trap lines. 
In an optical system, the relationship between the spatial frequency the 
number of repetitions of bright and dark of an object (image) per 
millimeter! and the intensity is referred to as frequency characteristics 
as in the case of an electric circuit. A graph expressing frequency 
characteristics is called MTF (Modulation Transfer Function). 
In the case of a lens, unlike an electric signal, an object (image) 
corresponding to it is two-dimensional. Therefore, trap lines are 
considered in a frequency plane. FIG. 23 shows a frequency plane. A 
coordinate axis representing the size of frequency response is 
perpendicular to the plane of the figure (accordingly, the size of 
frequency response at each frequency is not recognizable in FIG. 23) A 
trap line is a line connecting points where MTF=0, i.e. the frequency 
response is zero. 
It should be noted that in the case of a TV camera or the like, 3.58 
MHz in the conditions (4) and (7) should be replaced with 4.43 MHz. 
In the case of an image pickup apparatus, e.g. a television camera of a 
high-definition television (abbreviated as "HD-TV"), an electron-scope, 
etc., the effective sample number per line (scanning line) is specified as 
1920 (the January 1991 issue of the Journal of Television Technology, p. 
20). Therefore, moire fringes due to luminance digital sampling can be 
removed if the following condition is satisfied: 
EQU 0.75.multidot.(1920/Wx).ltoreq.1/.vertline.2(n-1).theta.Sf.beta.r.vertline. 
.ltoreq.1.5.multidot.(1920/Wx) (9) 
To remove moire fringes due to color signal sampling, it is desirable to 
satisfy the following condition because the color digital sample number is 
specified as 960 (the above-mentioned Journal of Television Technology): 
EQU 0.75.multidot.(960/Wx).ltoreq.1/.vertline.2(n-1).theta.Sf.beta.r.vertline.. 
ltoreq.1.5.multidot.(960/Wx) (10) 
In a case where the number npx of horizontal pixels of the solid-state 
image pickup device 8 is short of 1920, the left-hand term and the 
right-hand term of each of the conditions (9) and (10) should be 
multiplied by 
EQU npx/1920 (11) 
(In this description, when a reference is made to the condition (11), it 
means simply an inequality in which the condition (9) or (10) is 
multiplied by npx/1920). 
The polyhedral lens 2 can be used to remove moire fringes appearing in a 
combination of a fiber-scope using as an image transmitting device an 
image guide serving as an optical member which transmits pixels through 
fibers of a two-dimensional array of fibers in a fiber bundle, and an 
electronic image pickup system, e.g. a TV camera. FIG. 24 shows an image 
of an array of fibers in a fiber bundle. As illustrated in the figure, 
fibers are arranged in a staggered format (the distance between each pair 
of adjacent fibers is equal to each other). Assuming that the fiber pitch 
in the fiber bundle image is Pf (in this case, the image may be considered 
to be the real image on the solid-state image pickup device 8), it is 
desirable to satisfy the following condition: 
EQU 0.75/(Pf.multidot.sin 
60.degree.).ltoreq.1/.vertline.2(n-1).theta.Sf.beta.r.vertline..ltoreq.1.5 
/(Pf.multidot.sin 60.degree.) (12) 
In a case where some fiber-scopes and an electronic image pickup system are 
combined together, the Pf of any of the fiber-scopes or an approximate 
mean of Pf values of the fiber-scopes should satisfy the condition (12). 
This also applies to Pf in the conditions (22), (23) and (24) described 
later. 
Assuming that 
EQU U.sub.0 =1/.vertline.2(n-1).theta.Sf.beta.r.vertline. (13) 
MTF is given by 
EQU MTF=cos(U/U.sub.0 .multidot..pi./2) (14) 
Therefore, if the equal sign of the condition (12) is valid, MTF at the 
frequency of 1/(Pf.multidot.sin 60.degree.) at that time is 0.5. Thus, the 
extent of moire fringes can be reduced to a half or less. 
In many cases, a polyhedral lens 2 such as that shown in FIG. 17 is 
produced by plastic or glass molding process using a metal form. 
Alternatively, the polyhedral lens 2 may be produced by coating a planar 
substrate non-uniformly. In such a case, the surface configuration of a 
boundary portion between the two slant surfaces of the polyhedral lens 2, 
which is surrounded by the dashed-and-dotted lines in part (a) of FIG. 25, 
is not conformable to the design values but disordered. Assuming that an 
area on the polyhedral lens 2 which is occupied by marginal rays is Sm 
the portion surrounded by the dotted line in part (a) of FIG. 25!, the 
area S.alpha. occupied by the disordered portion the hatched portion in 
part (a) of FIG. 25! is desirably set so as to satisfy the following 
condition: 
EQU S.alpha./Sm&lt;0.3 (15) 
If S.alpha./Sm is not smaller than 0.3, the image is disturbed by flare, 
which is unfavorable for practical use. 
In the case of an endoscope of high-grade optical performance which uses a 
relay lens and an image fiber bundle having a large number of fibers, for 
example, an image of better contrast can be obtained by satisfying the 
following condition: 
EQU S.alpha./Sm&lt;0.12 (16) 
The foregoing matter was experimentally confirmed by using the single-sided 
polyhedral lens 2 shown in FIG. 17. The experiment revealed that the 
maximum value of the deviation in surface configuration of the defective 
portion S.alpha. was not more than 10 micrometers. 
Part (a) of FIG. 26 is a detailed view of a portion having a disordered 
surface configuration. Part (b) of FIG. 26 is a detailed view of a portion 
having a disordered surface configuration in a case where there is a 
difference in height between two slant surfaces. Part (a) of FIG. 26 shows 
the condition of a section taken along the line A in part (c) of FIG. 26. 
Part (b) of FIG. 26 shows the condition of a section taken along the line 
B in part (c) of FIG. 26. That is, part (a) of FIG. 26 shows a portion 
where the two slant surfaces are at the same height, whereas part (b) of 
FIG. 26 shows a portion where the two slant surfaces are different in 
height from each other. 
Properly speaking, the section taken along the line A should be a straight 
line. However, a recess is undesirably formed in the middle between the 
two slant surfaces as shown in part (a) of FIG. 26. The section taken 
along the line B should have a squarely bent shape. However, the corners 
are deformed as shown in part (b) of FIG. 26. First, a portion having a 
disordered surface configuration in the case of part (a) of FIG. 26 is 
defined as follows: A portion is defined as one that has a disordered 
surface configuration when the amount of deviation H.alpha. from a plane 
obtained by smoothly extending a portion which is away from the boundary 
line satisfies the following condition: 
EQU H.alpha.&gt;.lambda. (17) 
where .lambda. is a mean of working wavelengths. 
Alternatively, a portion is defined as one that has a disordered surface 
configuration when the angle .alpha. formed between a tangential plane at 
a point on the surface and the above-described extension plane satisfies 
the following condition: 
EQU .alpha.&gt;1.degree. (18) 
In a case where there is a difference in height between two surfaces as 
shown in part (b) of FIG. 26, H.alpha. is defined by the amount of 
deviation from a smooth extension (the dotted line in the figure) of each 
surface. The same is the case with the angle .alpha.. It is desirable that 
the height difference G in part (b) of FIG. 26! between the two surfaces 
should satisfy the following condition: 
EQU G&lt;10 micrometers (19) 
If G is not smaller than 10 micrometers, when a bright point-like object is 
seen, intense emission lines unfavorably appear around the object. 
A low-pass filter used in the present invention, as described later, is 
designed such that as the size of the effective diameter increases, the 
height difference between two slant surfaces becomes larger. When the 
low-pass filter is used in an endoscope, because the size of the pupil is 
about 7 millimeters at the most, the size of the low-pass filter is also 
of the order of 7 millimeters. In such a case, it is preferable to satisfy 
the condition (19). 
A substance which does not transmit light may be provided to cover 
substantially the portion S.alpha. in FIG. 25 to prevent flare which would 
otherwise be caused by light passing through the portion having a 
disordered surface configuration. Examples of such a substance include 
CrO.sub.2 --Cr--CrO.sub.2 coating, black paint, etc. The light-blocking 
portion may be provided on the reverse side of the polyhedral lens 2 at a 
position which substantially covers the portion S.alpha.. Part (b) of FIG. 
25 shows an example in which a light-blocking portion 23 is provided by 
coating. 
To avoid the disordered surface configuration from causing problems, the 
boundary portion may be decentered with respect to the bundle of marginal 
rays as shown in FIG. 27. By doing so, when the bundle of marginal rays is 
large in diameter, the portion S.alpha. having a disordered surface 
configuration becomes small relative to Sm (A in FIG. 27), so that 
problems will not arise. When Sm is small (B in FIG. 27), the boundary 
line l lies outside the ray bundle. Therefore, the contrast of the image 
can be kept, although the moire removing function is lost. 
It is desirable that the amount of eccentricity (displacement) e of the 
boundary line should satisfy the following condition: 
EQU e/Da.ltoreq.0.25 (20) 
If e/Da exceeds 0.25, the moire removing function becomes deteriorated even 
when Sm is in the state A in FIG. 27. 
Alternatively, as shown in FIG. 28, the surface of the polyhedral lens 2 
may be divided into three surface portions (i.e. a portion a, a portion b, 
and a central portion e) such that the boundary between the surface 
portions does not lie in the central portion of the polyhedral lens 2. 
This may be realized as follows: In making of a mold for forming the 
polyhedral lens 2, after both surfaces have been ground, only the central 
portion is polished to form a portion e having disorder removed from the 
surface thereof. In this case, the boundaries between the three split 
surfaces do not always need to be clear. 
Alternatively, a portion of the mold which corresponds to the area S.alpha. 
in part (a) of FIG. 25 may be smoothed by regrinding or polishing, thereby 
removing a portion of the mold which is higher than the configuration 
according to the design values. The examples shown in FIGS. 25, 27 and 28 
are particularly effective when combined with an image pickup apparatus 
whose pupil diameter is variable, such as a television camera, an 
electron-scope, an adapter, a hard endoscope, or a fiber-scope. 
When placed in an optical system, the polyhedral lens 2 is preferably 
disposed in the vicinity of the pupil position, as has already been stated 
above. Let us examine this more specifically. FIG. 29 shows a positional 
relationship between the pupil and the polyhedral lens 2. In FIG. 29, hm 
and hc denote the marginal ray height and the extra-axial principal ray 
height, respectively, at the polyhedral lens surface of the polyhedral 
lens 2. Here, it is desirable to satisfy the following condition: 
EQU .vertline.hc/hm.vertline.&lt;0.8 (21) 
If hc is large to such an extent that the condition (21) is no longer 
satisfied, a large difference is produced between the proportions of the 
areas where extra-axial rays pass through the two surfaces of the 
polyhedral lens 2. Consequently, the moire removing function is 
deteriorated. Even when the condition (21) is not satisfied, if the 
polyhedral lens 2 is moved to a position close to the pupil, the condition 
(21) can be satisfied. Thus, it is desirable to set the polyhedral lens 2 
in the vicinity of the pupil position. 
It should be noted that the polyhedral lens 2 need not perpendicularly 
intersect the optical axis O, but may be tilted with respect to the 
optical axis O up to 10-odd degrees. According to the foregoing 
modification of the first embodiment of the prior application, the 
endoscope apparatus employs a single-sided polyhedral lens 2 with slant 
surfaces that are in a skew relation to each other, which is allowed to 
exhibit a great optical low-pass filter function with respect to moire 
fringes due to the pixel array of the solid-state image pickup device 8, 
the sampling period, etc. by setting the polyhedral lens 2 so that the 
conditions for moire removal are satisfied. Accordingly, moire fringes can 
be satisfactorily removed from not only an image formed on the image 
pickup surface of the solid-state image pickup device 8 in an in-focus 
state but also an image formed in a defocus state (the moire removing 
function is greater than that of the point-symmetry optical filter 
disclosed in JP(A) 3-248695). 
Accordingly, the endoscopic image displayed on the color monitor 10 is free 
from moire fringes and of good image quality. Moreover, the moire removing 
function can be realized at much lower cost than in the case of using a 
crystal filter. 
The following is a description of a double-sided polyhedral lens 1 which 
enables the filter function to be enhanced in comparison to the 
single-sided polyhedral lens 2, together with an endoscope apparatus 
according to the first embodiment of the prior application which uses the 
double-sided polyhedral lens 1. 
FIG. 16 shows a double-sided polyhedral lens 1 used in the first embodiment 
of the prior application. The double-sided polyhedral lens 1 has two split 
surfaces on each side thereof. The two surfaces are skewed relative to 
each other like the blades of a propeller, as in the case of the side B of 
the single-sided polyhedral lens 2. That is, as shown in part (a) of FIG. 
16, the double-sided polyhedral lens 1 has two semicircular slant surface 
portions a and b formed on one side A thereof as in the case of the 
single-sided polyhedral lens 2. The portions a and b have an optical axis 
O substantially in common with each other and lie on both sides of a 
boundary line la passing perpendicularly to the optical axis O. 
As shown in part (b) of FIG. 16, the double-sided polyhedral lens 1 has two 
semicircular slant surface portions c and d formed on the other side B 
thereof as in the case of the single-sided polyhedral lens 2. The portions 
c and d have the optical axis O substantially in common with each other 
and lie on both sides of a boundary line lb passing perpendicularly to the 
optical axis O. As shown in the right-hand half of part (c) of FIG. 16, 
the boundary line la on the side A and the boundary line lb on the side B 
are substantially perpendicular to each other. 
Lines normal to the two slant surfaces on the side A substantially at the 
respective centers are in a skew relation to each other and slant at an 
angle .theta.a to each other. On the side B also, lines normal to the two 
slant surfaces substantially at the respective centers are in a skew 
relation to each other and slant at an angle .theta.b to each other. 
Because the slant surface portions are formed in a skew relation to each 
other as described above, the low-pass filter function can be 
satisfactorily exhibited even in the case of an out-of-focus (defocused) 
image, as stated above. In FIG. 16, the angles .theta.a and .theta.b are 
set, for example, to .theta.a=.theta.b=2'40". 
Assuming that a Z-axis is taken in a direction parallel to the optical axis 
O, and X- and Y-axes are taken in a plane perpendicular to the Z-axis, and 
further that the X-axis is taken in the direction of the boundary between 
the portions a and b, for example, data concerning the configuration of 
the side A of the double-sided polyhedral lens 1 is as follows: 
For the portion a, Y is zero or positive (i.e. Y.gtoreq.0). For the surface 
of the portion a, Z=P.multidot.X. For the portion b, Y is negative (i.e. 
Y&lt;0). For the surface of the portion b, Z=-P.multidot.X. The parameter P 
expressing a slant surface is, for example, P=tan 1'20".apprxeq.0.0004. As 
shown in part (c) of FIG. 16, the size of the double-sided polyhedral lens 
1 is as follows: The diameter .phi. is 8 millimeters; the thickness 
T.sub.0 is 1 millimeter, for example; and the refractive index n is 
1.51633, for example. 
It should be noted that the right-hand half of part (c) of FIG. 16 is a 
plan view, and the left-hand half is a side view. As shown in part (c) of 
FIG. 16, the side A is provided with a mark M for identifying the 
orientation of the optical element. 
It should be noted that the angles .theta.a and .theta.b of the 
double-sided polyhedral lens 1 may be unequal to each other and should be 
properly selected in conformity to the range of Pf values of some 
fiber-scopes to be combined and Px, Py, N, M and so forth of the 
solid-state image pickup device 8. 
It is desirable to set the angles .theta.a and .theta.b so that the angle 
.theta.a or .theta.b satisfies at least two of the conditions (2), (3), 
(5), (7), (8), (9), (10) and (11), in which the angle .theta.a or .theta.b 
is substituted for the angle .theta.. By doing so, it is possible to allow 
the double-sided polyhedral lens 1 to give advantageous effects similar to 
those of the modification shown in FIG. 17 and to have different filter 
functions at the two sides thereof. Therefore, the double-sided polyhedral 
lens 1 can remove more causes of reduction in the image quality in 
observation or the like than the single-sided polyhedral lens 2. Thus, the 
double-sided polyhedral lens 1 gives even more remarkable effects than the 
single-sided polyhedral lens 2. The double-sided polyhedral lens 1 can be 
produced at much lower cost than a crystal filter having comparable action 
and effect as in the case of the modification. 
It is desirable to set each of the sides A and B of the double-sided 
polyhedral lens 1 so that at least one of the conditions (15), (16), (19), 
(20) and (21) is satisfied. Advantageous effects produced by doing so are 
the same as those in the case of the single-sided polyhedral lens 2 shown 
in FIG. 17. 
MTF of the double-sided polyhedral lens 1 is given by the product of MTFs 
of the two sides thereof. Accordingly, the double-sided polyhedral lens 1 
enables the number of trap lines to be increased and makes it possible to 
further enhance the low-pass filter function. This is particularly 
effective in a combination of a fiber-scope in which intense moire fringes 
appear and a television camera. FIG. 30 shows an endoscope apparatus 31 
according to the first embodiment of the prior application as an optical 
apparatus having such an arrangement (more specifically, an image pickup 
apparatus having an image pickup function). 
The endoscope apparatus 31 includes a fiber-scope 32 serving as a soft 
endoscope, which has an illumination optical system and an observation 
optical system; a pickup lens adapter 33 removably attached to the 
fiber-scope 32 and containing a pickup lens; a television camera 34 
removably attached to the pickup lens adapter 33 and containing an image 
pickup device; a light source unit 35 which supplies illuminating light to 
an illuminating light transmitting device of the fiber-scope 32; a CCU 36 
which executes signal processing for a solid-state image pickup device 8, 
e.g. a CCD, contained in the television camera 34; and a color monitor 37 
connected to the CCU 36 to display image signals. 
The fiber-scope 32 has a soft, long and narrow insert part 41 which has 
flexibility and is inserted into the body cavity or the like. A thick 
control part 42 is formed at the rear end of the insert part 41 and 
provided with a bending control device (not shown). An eyepiece portion 43 
is formed at the rear end of the control part 42. A light guide cable 45 
extends from the control part 42 and is detachably connected to the light 
source unit 35. 
White illuminating light from a lamp 46 in the light source unit 35 is 
supplied to a light guide 47 in the light guide cable 45. The transmitted 
illuminating light is emitted forward from an end surface attached to an 
illuminating window in the distal end portion of the insert part 41. Thus, 
an illumination optical system which illuminates a subject such as an 
affected part is formed. 
An objective lens 48 is mounted in a viewing window formed in the distal 
end portion at a position adjacent to the illuminating window. An image of 
a subject illuminated is formed by the objective lens 48. A distal end 
surface of an image guide 49 is disposed at a position where the subject 
image is formed by the objective lens 48. The image guide 49 is formed 
from a bundle of fibers to have an image transmitting function. The image 
guide 49 inserted in the insert part 41 transmits the image to a rear end 
surface thereof at the rear side of the insert part 41. The image 
transmitted to the rear end surface can be observed with the naked eye 
through an ocular lens 50 which is provided in an eyepiece window of the 
eyepiece portion 43 to form an observation optical system. 
In a case where the television camera 34 is attached to the eyepiece 
portion 43 through the pickup lens adapter 33, the image transmitted 
through the image guide 49, which serves as an image transmitting optical 
system, is taken through the ocular lens 50, an iris 52, a second 
double-sided polyhedral lens 51 and a pickup lens 53, which are provided 
in the pickup lens adapter 33, and a final image is formed on the 
solid-state image pickup device 8 having color separation filters, e.g. 
mosaic filters, through a first double-sided polyhedral lens 1 disposed in 
the television camera 34. Thus, an image pickup device (or an image pickup 
apparatus) is formed. The image is photoelectrically converted by the 
solid-state image pickup device 8 into electrical signals, which are then 
converted into a standard video signal by the CCU 36 to display a color 
image on the color monitor 37, which serves as a color display device. 
The CCU 36 generates a light-control signal as an average brightness value, 
for example, by integrating the luminance signal over one frame period, 
and outputs the signal to an iris driver 54 provided in the pickup lens 
adapter 33 to vary the amount of opening of the iris 52. The iris 52 is 
controlled such that when the average level of the luminance signal is 
high, the iris 52 is narrowed down, whereas, when it is low, the iris 52 
is opened wider. That is, an auto-iris control mechanism is formed. Thus, 
the brightness of the endoscopic image displayed on the color monitor 37 
is automatically controlled to a level suitable for observation at all 
times. 
In the first double-sided polyhedral lens 1 used in the first embodiment of 
the prior application, the polyhedral lens surfaces on the sides A and B 
are each set so as to satisfy the condition (2) or the condition (3)! and 
the condition (4) or the condition (5) or (7) or (8)!, thereby removing 
moire fringes due to mosaic filters and moire fringes due to color signal 
modulation. 
In the second double-sided polyhedral lens 51, the polyhedral lens surfaces 
on the sides A and B are each set so as to satisfy the condition (12), 
thereby two-dimensionally removing moire fringes due to the fiber pitch Pf 
of the fiber bundle image as shown in FIG. 24. 
The second double-sided polyhedral lens 51 may be set as shown in FIG. 35 
(described later), and trap lines may be set as shown in FIG. 36, thereby 
removing dark due to clad portions of the fibers in a fiber-scope image 
which would otherwise appear conspicuously. The single-sided polyhedral 
lens 2 may be used in place of the second double-sided polyhedral lens 51 
and set so as to satisfy the condition (12). 
The endoscope apparatus 31 according to the first embodiment of the prior 
application has the function of satisfactorily removing moire fringes from 
not only an in-focus image but also a defocused image as in the case of 
the modification that uses the single-sided polyhedral lens 2. Therefore, 
it is possible to obtain an image of good quality without moire fringes 
even when an affected part or the like is observed. 
Moreover, because the first embodiment of the prior application uses a 
double-sided polyhedral lens, which has a polyhedral lens formed on each 
side thereof, it is possible to dissolve more causes of reduction in the 
image quality than in the case of the modification that uses the 
single-sided polyhedral lens 2, and hence possible to obtain an image of 
good quality. In addition, advantageous effects similar to those obtained 
by using a crystal filter can be produced at much lower cost. 
In the first embodiment of the prior application, shown in FIG. 30, the 
television camera 34 may be combined with the a hard endoscope 5 using a 
relay lens system in which moire does not appear to a great extent, by 
replacing the pickup lens adapter 33 with a pickup lens adapter 56 which 
is provided with a cover glass 55 in place of the double-sided polyhedral 
lens 51. 
By doing so, it is possible to implement an endoscope apparatus 60 having 
an optimal low-pass filter function simply by exchanging the pickup lens 
adapter 33 for the pickup lens adapter 56. FIG. 31 is a diagram showing 
the arrangement of an endoscope system 61 according to a second embodiment 
of the optical apparatus of the prior application which enables the 
endoscope apparatuses 31 and 60 to be implemented (in FIG. 31, 
illustration of the light source unit is omitted). 
In FIG. 31, the endoscope apparatus 60 is shown by the solid lines. In the 
endoscope apparatus 60, the hard endoscope 5 is used in place of the 
fiber-scope 32 in FIG. 30, and the pickup lens adapter 33 is replaced with 
the pickup lens adapter 56 correspondingly. The television camera 34, 
which is used in common, is removably attached to the pickup lens adapter 
56. 
In the endoscope apparatus 60, it is also possible to remove moire fringes 
due to mosaic filters and moire fringes due to color signal modulation and 
hence possible to obtain an image of good quality. It should be noted that 
the hard endoscope 5 has basically the same arrangement as that shown in 
FIG. 18, for example. 
FIG. 32 shows a fiber-scope 70 according to a third embodiment of the 
optical apparatus of the prior application. In the fiber-scope 70, a 
double-sided polyhedral lens 71 is provided in the vicinity of the pupil 
of the ocular lens 50. The angles .theta.a and .theta.b of the 
double-sided polyhedral lens 71 are set so as to satisfy the following 
conditions: 
EQU 0.75/Pf.ltoreq.1/.vertline.2(n-1).theta.aSf.beta.r.vertline..ltoreq.1.5/Pf( 
22) 
EQU 0.75/Pf.ltoreq.1/.vertline.2(n-1).theta.bSf.beta.r.vertline..ltoreq.1.5/Pf( 
23) 
Further, the double-sided polyhedral lens 71 is set so that the angle .psi. 
formed between the boundary lines la and lb satisfies the following 
condition: 
EQU 45.degree..ltoreq..psi..ltoreq.75.degree. (24) 
Thus, the double-sided polyhedral lens 71 is arranged to be capable of 
removing all fundamental spatial frequency spectra in the fiber bundle 
array. It should be noted that .beta.r is the magnification of an ocular 
lens 50 lying between the polyhedral lens 71 and the position of the 
image-formation plane, and Sf is the distance from the polyhedral lens 71 
to an image formed by the ocular lens 50 forward of the polyhedral lens 
71. 
More specifically, as shown in parts (a) and (b) of FIG. 33, the directions 
of the boundary lines la and lb of the double-sided polyhedral lens 71 are 
set parallel to the array directions of the fiber bundle (i.e. an array 
direction J in a horizontal direction and an array direction K in an 
obliquely upward direction; the boundary line la is parallel to the 
direction J, and the boundary line lb is parallel to the direction K). 
Accordingly, the angle .PSI. formed between the array directions J and K 
of the fiber bundle, which are horizontal and obliquely upward directions, 
respectively, is equal to the angle .psi. between the boundary lines la 
and lb (.psi.=.PSI.). 
FIG. 34 shows spatial frequency spectra (marked with circles) of a fiber 
bundle image obtained with the above-described arrangement, together with 
trap lines (dotted lines) obtained by the double-sided polyhedral lens 71. 
It should be noted that in the case of image fibers arrayed at random, a 
mean value should be taken as Pf. Alternatively, the double-sided 
polyhedral lens 1 shown in FIG. 16 my be employed in place of the 
double-sided polyhedral lens 71 in FIG. 32, thereby forming an arrangement 
similar to that of the first modification. 
That is, the double-sided polyhedral lens 1 shown in FIG. 16, in which the 
boundary lines la and lb are arranged to intersect each other at right 
angles as shown in part (a) of FIG. 35, may be disposed such that the 
boundary line la is parallel to the direction J with respect to the fiber 
bundle array shown in part (b) of FIG. 35. At this time, one of the angles 
.theta.a and .theta.b should preferably satisfy the condition (22) or 
(23). The other of the angles .theta.a and .theta.b should preferably 
satisfy the condition (12) as rewritten by substituting the angle .theta.a 
or .theta.b for the angle .theta.. 
FIG. 36 shows spatial frequency spectra (marked with circles) of a fiber 
bundle image obtained with the above-described arrangement, together with 
trap lines (dotted lines) obtained by the double-sided polyhedral lens 1. 
As shown in FIG. 36, the fundamental frequency of the fiber bundle can be 
reduced close to zero. Therefore, the clad portions are inconspicuous, and 
moire fringes are also inconspicuous when the fiber-scope is combined with 
a TV camera. 
Although in the embodiments shown in FIGS. 18, 33 and 35 the trap lines 
where MTF of a polyhedral lens is zero are parallel to the X-axis or the 
Y-axis, it should be noted that the trap lines do not always need to be 
parallel to the X-axis or the Y-axis for the purpose of removing moire 
fringes when a television camera is attached to the fiber-scope or for the 
purpose of removing dark due to the clad portions or of removing dark due 
to the clad portions during observation through the fiber-scope with the 
naked eye. 
The arrangement may be such that, as shown in part (a) of FIG. 37, the 
boundary lines la and lb are disposed at .+-.45.degree. to the X-axis so 
that trap lines (dotted lines) pass in the vicinities of points (marked 
with circles) of the fundamental frequency (or higher-order frequencies) 
of the fibers. 
It is also possible to adopt the structure of a fourth embodiment of the 
optical apparatus of the prior application as shown in FIG. 38, thereby 
removing dark due to the clad portions during observation through the 
fiber-scope with the naked eye, for example. That Is, the double-sided 
polyhedral lens 1, for example, which is accommodated in a dark removing 
eyepiece adapter 75 shown in FIG. 38 is arranged to be rotatable, thereby 
enabling the low-pass filter function to be variable. 
The eyepiece adapter 75 has a ring-shaped frame member 76 serving as a 
mounting member for the eyepiece adapter 75. The forward end portion of 
the frame member 76 is fitted to the eyepiece portion 43 of an existing 
fiber-scope (e.g. denoted by reference numeral 32 in FIG. 30) and secured 
by using a fixing screw 77, thereby enabling the eyepiece adapter 75 to be 
removably attached to the eyepiece portion 43). A lens frame 78 having the 
double-sided polyhedral lens 1 mounted thereon is fitted to the inner 
peripheral surface of the rear end of the frame member 76, thereby being 
rotatably accommodated in the frame member 76. 
The frame member 76 is provided with a groove 76a extending 
circumferentially over 90 degrees. A pin 78a projecting from the lens 
frame 78 extends through the groove 76a to project from the frame member 
76. By rotating a projecting portion of the pin 78a which projects from 
the groove 76a, the double-sided polyhedral lens 1 can be rotated. 
If the pin 78a is rotated through 45 degrees from a reference position 
shown in FIG. 35 (e.g. a position where the pin 78a projects upwardly), 
for example, the double-sided polyhedral lens 1 can be set to the position 
shown in part (a) of FIG. 37. If the pin 78a is further rotated, the 
double-sided polyhedral lens 1 can be set to the position shown in part 
(a) of FIG. 39. Even in a state where each of the boundary lines la and lb 
intersects the X-axis at an angle other than .+-.45 degrees as shown in 
part (a) of FIG. 39, the low-pass filter function can be obtained as long 
as the trap lines pass in the vicinities of the frequency components 
(marked with circles) of the fibers. 
According to this embodiment, dark due to the clad portions, which 
interfere with observation and degrade the image quality, can be removed 
by attaching the eyepiece adapter 75 to the eyepiece portion 43 of the 
existing fiber-scope 32. Thus, an image suitable for observation can be 
obtained. The polyhedral lens adapter may be attached to a TV camera or a 
TV adapter. 
Moreover, by adjusting the amount of rotation, the double-sided polyhedral 
lens 1 can be set to an image condition most suitable for observation. 
That is, if the double-sided polyhedral lens 1 can not be rotated and is 
placed in a fixed-angle position (orientation), there may be cases where 
the double-sided polyhedral lens 1 in the position shown in part (a) of 
FIG. 35 cannot be set to a desired position where dark can be removed as 
shown in FIG. 36, or the double-sided polyhedral lens 1 in the position 
shown in part (a) of FIG. 37 cannot be set to a desired position where 
dark can be removed as shown in part (b) of FIG. 37. Even in such cases, 
the double-sided polyhedral lens 1 can be set to a position where an image 
most favorable for observation can be obtained by adjusting the amount of 
rotation. 
Incidentally, to apply the double-sided polyhedral lens 1, 51, or 71, which 
has a polyhedral lens provided on each side thereof, to all image pickup 
optical systems, it is necessary to take care that the refracting actions 
of the surfaces on both sides do not cancel each other in a case where the 
angle .psi. formed between the boundary lines on the two sides is 
.psi..ltoreq.30.degree.. 
That is, it is necessary to satisfy the following condition: 
EQU .psi..ltoreq.30.degree. at the same time .theta.a-.theta.b.noteq.0(25) 
If .psi.=0 and, at the same time, .theta.a-.theta.b=0, when the difference 
.psi. between the directions of refraction of one light beam at the sides 
A and B is .psi..ltoreq.30.degree., as shown in FIG. 40, the angles 
.theta.a and .theta.b undesirably cancel each other at the two sides. 
When .psi..ltoreq.30.degree., the angles .theta.a and .theta.b can be 
selected almost freely. It should be noted that the low-pass filter 
function may be made variable by arranging the single-sided polyhedral 
lens 2 or the double-sided polyhedral lens 1 such that the amount of 
decentration relative to the ray bundle is variable. 
Part (a) of FIG. 41 shows an optical element 62 in a fifth embodiment of 
the prior application which has both an optical low-pass filter function 
and an image-forming function. The optical element 62 has one surface 
thereof formed into a polyhedral lens 62A to have a low-pass filter 
function. The other surface of the optical element 62 is formed into an 
ordinary convex lens 62B or other lens to have an image-forming function. 
It is also possible to employ a polyhedral lens configuration as shown in 
part (b) of FIG. 41, which has advantageous effects similar to those of 
the double-sided polyhedral lens 1. That is, a polyhedral lens 65 shown in 
part (b) of FIG. 41 has a polyhedral lens surface 65A on one side thereof. 
The polyhedral lens surface 65A has four split surfaces which are skewed 
in the shape of the blades of a fan to form a polyhedral lens. The 
polyhedral lens 65 has an ordinary convex lens 65B on the other side. 
By providing an optical element with both an optical low-pass filter 
function and a lens function as described above, it becomes unnecessary to 
provide another lens and it is possible to eliminate time and labor which 
would otherwise be needed to assemble separate optical parts and to adjust 
them. Consequently, the costs can be reduced to a considerable extent. It 
is also possible to minimize variations in products and to make the 
optical element even more compact. The optical element 62 can be used, for 
example, in the ocular lens 50 or the pickup lens 53. Although the number 
of split surfaces of the polyhedral lens 65 is four, it may be 3, 5 or any 
desired number. 
FIG. 42 shows an example of an optical system according to a sixth 
embodiment of the prior application in which two polyhedral lenses 68 and 
69 are disposed such that the directions in which MTF reduces are varied 
from each other. For example, the polyhedral lenses 68 and 69 may be used 
in place of the double-sided polyhedral lens 51 and the pickup lens 53 in 
FIG. 30. The polyhedral lenses 68 and 69 have polyhedral lens surfaces 68A 
and 69A formed on one side of each of them and ordinary lens surfaces 68B 
and 69B on the other sides thereof. The optical system in which two 
single-sided polyhedral lenses are disposed as described above can exhibit 
almost the same function as that of the double-sided polyhedral lens 1 in 
the first embodiment shown in FIG. 16, and is superior in that it can be 
formed by machining more easily than in the case of forming polyhedral 
lenses on both sides of an optical element. 
Moreover, by forming the ordinary lens surfaces 68B and 69B on the other 
sides, it is possible to eliminate the need to provide separate lens 
elements. It is also possible to form an optical system by placing three 
or more single-sided polyhedral lenses in a line, or to form an optical 
system by combining together one double-sided polyhedral lens and one or 
more single-sided polyhedral lenses. It is also possible to form an 
optical system by combining together one single-sided polyhedral lens and 
one or more double-sided polyhedral lenses. 
As in a modification shown in part (a) of FIG. 43, the polyhedral lens-side 
split surfaces on at least one side of the double-sided polyhedral lens 1 
may be formed into aspherical surfaces 81a to construct a double-sided 
polyhedral lens 81, thereby controlling MTF. By doing so, the low-pass 
filter function can be changed as desired. 
Regarding the single-sided polyhedral lens 2 shown in FIG. 17, the split 
surfaces (two slant surfaces) may be formed into aspherical surfaces 82a 
as shown in part (b) of FIG. 43 to control MTF or low-pass filter 
function. Alternatively, this technique may be used to remove dark due to 
the clad portions of fibers in a fiber-scope which would otherwise appear 
conspicuously. Although machining for the formation is difficult, because 
the surface on one side may be a spherical or aspherical surface, the 
degree of freedom of designing an optical system increases and a 
high-performance optical system can be obtained. In this example also, at 
least one of the conditions (15), (16), (19), (20) and (21) can be 
applied, and the above-described advantageous effects can be obtained. If 
the polyhedral lens shown in part (b) of FIG. 43 is combined with a camera 
with an auto-iris, the low-pass filter function can be changed with the 
change in the stop diameter, which is very convenient. 
FIG. 44 shows a double-sided polyhedral lens 85 according to a seventh 
embodiment of the prior application. The double-sided polyhedral lens 85 
is an example in which the boundary lines la and lb on both sides lie at a 
plurality of positions away from the optical axis as viewed from the -Z 
direction. When thick, the bundle of light rays is split into three, and 
three split images are formed, but a thin bundle of rays is not split. 
Therefore, in an optical system whose stop diameter is variable or in a 
combination of optical systems, the double-sided polyhedral lens 85 
advantageously enables the low-pass filter function to be changed 
according to the stop diameter. It should be noted that the surface on one 
side of the double-sided polyhedral lens 85 may be divided into three or 
more surface portions. 
FIG. 45 shows an example of a polyhedral lens 86 in which the sectional 
configuration in a direction perpendicular to the surface dividing 
boundary line l is aspherical. Although in the modification shown in FIG. 
17 only a pair of parallel trap lines of MTF can be obtained, this 
embodiment enables MTF to be reduced also in a direction parallel to the 
boundary line l as shown in part (d) of FIG. 45. That is, parts (a), (b) 
and (c) of FIG. 45 show a single-sided polyhedral lens 86 in an eighth 
embodiment of the prior application. In the polyhedral lens 86 shown in 
parts (a), (b) and (c) of FIG. 45, an aspherical surface configuration is 
used to increase the number of trap lines, thus realizing spatial 
frequency characteristics as shown in part (d) of FIG. 45. In this 
embodiment, the polyhedral lens 86 has the function of splitting an image 
into a plurality of images substantially at the aspherical portion. 
Therefore, the aspherical surface may be, for example, one that has an 
inflection point, and it is also possible to employ an aspherical surface 
with a multiplicity of optical axes as shown in part (a) of FIG. 46, or 
angular aspherical surfaces as shown in parts (b) and (c) of FIG. 46. 
FIG. 47 shows a single-sided polyhedral lens 92 in a ninth embodiment of 
the prior application. The single-sided polyhedral lens 92 is an example 
that has four split surfaces. The single-sided polyhedral lens 92 differs 
from those shown in FIG. 17 and part (b) of FIG. 41 in that no step is 
produced at the boundaries. In this embodiment also, it is desirable to 
satisfy at least one of the conditions (15), (16), (20) and (21). Of the 
four split surfaces of the single-sided polyhedral lens 92, each pair of 
surfaces which are not adjacent to each other are in a skew relation to 
each other like the blades of a propeller. 
FIG. 48 shows a single-sided polyhedral lens 94 in a modification which has 
split surfaces 94a which are approximately parallel to each other. MTF can 
be controlled by selecting a normal direction for each of the surfaces 
94a. In this modification also, it is desirable to satisfy at least one of 
the conditions (15), (16), (19), (20) and (21). In a case where the 
polyhedral lens 94 as shown in FIG. 48 is produced by plastic or glass 
molding process, it is desirable for the split surfaces to have a 
configuration as shown in part (b) of FIG. 48 from the viewpoint of 
facilitating the mold making process and preventing a mold grinding wheel 
from touching a ground surface. 
In the above discussion of low-pass filters, light is handled in a 
geometrical-optical manner. However, in either of the examples shown in 
FIGS. 16 and 26, the height difference between the split surfaces is of 
the order of from 1 micrometer to several micrometers. In such a case, 
wave-optical examination is needed. That is, polyhedral lenses have a 
phase filter effect in addition to the MTF of a wedge-shaped prism. For 
example, FIG. 49 shows the optical path length Lo when the double-sided 
polyhedral lens 1 in the embodiment shown in FIG. 16 is seen in the Z 
direction. The straight lines show contour lines of the optical path 
length Lo, which is given by 
EQU Lo=Tz(n-1)/.lambda.c (26) 
where Tz is the thickness of the double-sided polyhedral lens 1 in the Z 
direction, which is a function of X and Y; n is the refractive index of 
the double-sided polyhedral lens 1; and .lambda.c is a working wavelength 
or a mean thereof. 
Specifically, Tz is as follows: 
When X.gtoreq.0 and Y.gtoreq.0, Tz=(-Y+X)P+T.sub.0 
When X.gtoreq.0 and Y&lt;0, Tz=(-Y-X)P+T.sub.0 
When X&lt;0 and Y.gtoreq.0, Tz=(Y+X)P+T.sub.0 
When X&lt;0 and Y&lt;0, Tz=(Y-X)P+T.sub.0 
In the above expressions, T.sub.0 =1 millimeter; T.sub.0 represents the 
thickness of the double-sided polyhedral lens 1 when X=Y=0. 
Wave-optical MTF R(Ux',Uy') in this case is given approximately by 
EQU H(X,Y)=A(X,Y)exp2.pi.iLo(X,Y)! (27) 
The pupil function is defined by H(X,Y). A(X,Y) is the amplitude 
transmittance of the pupil. Using the pupil function H(X,Y), 
EQU R(Ux',Uy')=(1/C).intg..intg.H(X,Y)H*(X-Xo,Y-Yo)dXdY (28) 
In the above expression, the integration is performed over the whole pupil, 
and * represents the complex conjugate of H(X,Y). Further, C is a constant 
for standardization. 
EQU Xo=.lambda.cUx'S, Yo=.lambda.cUy'S (29) 
where S is the distance from the surface of the polyhedral lens that has a 
low-pass filter function to an intermediate image formed by light passing 
through that surface i.e. an image formed on the assumption that there is 
no lens system behind (on the exit side) of the polyhedral lens!; and Ux' 
and Uy' represent spatial frequencies in the intermediate image. 
Assuming that Pf' is the fiber pitch in the fiber bundle image of the 
intermediate image, if Tz(X,Y) is selected so that the following 
conditions (30) and (31) are satisfied in place of the conditions (12), 
(22) and (23), moire fringes appearing in a combination with a fiber-scope 
can be reduced to 50% or less: 
EQU 0&lt;R(Ux',Uy')&lt;0.5 (30) 
EQU .sqroot. (Ux'.multidot.Ux'+Uy'.multidot.Uy')=1/(Pf' sin 60.degree.)(31) 
It should be noted that .sqroot. on the left-hand side of the condition 
(31) expresses the square root. 
It is not necessary to satisfy the condition (30) with respect to all Ux' 
and Uy' that satisfy the condition (31), but the condition (30) is only 
necessary to satisfy with respect to Ux' and Uy' satisfying the condition 
(31) in the vicinity of the fundamental spatial frequency spectra of the 
fiber bundle image. The error in production of fibers alone is about 
several % and the magnification error of the lens is several %. Therefore, 
considering these errors, a range defined by the fundamental spatial 
frequency of the fiber bundle image .+-. about 10% is the vicinity of the 
fundamental spatial frequency spectra. 
Similarly, the following condition (32) should preferably be satisfied in 
place of the condition (3): 
EQU 0&lt;R(1/PxM,Uy')&lt;0.5 (32) 
where Px' is the image size of one pixel of the solid-state image pickup 
device 8 in the X direction at the position of the above-described 
intermediate image. 
Similarly, the following condition (33) should preferably be satisfied in 
place of the condition (5): 
EQU 0&lt;R(40.multidot.3.58/Wy',Uy')&lt;0.5 (33) 
where Wy' is the vertical dimension of the effective portion of the 
solid-state image pickup device 8 in terms of the measure at the position 
of the intermediate image. 
Similarly, the following condition (34) should preferably be satisfied in 
place of the condition (9): 
EQU 0&lt;R(1920/Wx',Uy')&lt;0.5 (34) 
where Wx' is the horizontal dimension of the effective portion of the 
solid-state image pickup device 8 in terms of the measure at the position 
of the intermediate image. 
Similarly, the following condition (35) should preferably be satisfied in 
place of the condition (10): 
EQU 0&lt;R(960/Wx',Uy')&lt;0.5 (35) 
When the number of pixels in the horizontal direction is insufficient, the 
first argument of R in the condition (34) or (35) should be multiplied by 
npx/1920. 
Let us examine the functional form of the optical path length Lo(X,Y). It 
is preferable that no astigmatism should occur on the optical axis. To 
prevent the occurrence of astigmatism, Lo(X,Y) should coincide with 
Lo(X,Y) when it is rotated through 360.degree./nr around the Z-axis, where 
nr is a natural number and 
EQU nr.gtoreq.3 (36) 
The double-sided polyhedral lens 1 is an example in which nr=4. 
Thus, even more diverse low-pass filter performance can be elicited by 
handling polyhedral lenses in a wave-optical manner as stated above. 
Examples 1 and 2 of the image-forming optical apparatus according to the 
present invention will be described below. Constituent parameters of each 
example will be described later. In the constituent parameters in each 
example, as shown in FIG. 1, one plane (the plane of a stop 101 in the 
case of FIG. 1) specified as a reference plane of an optical system is 
defined as the origin of a decentration plane, and an axial principal ray 
102 is defined by a light ray emanating from the center of an object (not 
shown) and passing through the center of the stop 101. A Z-axis is taken 
in a direction in which the light ray from the object center travels along 
the axial principal ray 102 until it reaches the first surface of the 
optical system. A plane containing both the Z-axis and the center of an 
image plane 108 is defined as a YZ-plane. A Y-axis is taken in a direction 
perpendicularly intersecting the Z-axis in the YZ-plane. A direction in 
which the Z-axis extends from the object point to the first surface of the 
optical system is defined as a positive direction of the Z-axis. The 
upward direction of the Y-axis as viewed in the figure (i.e. a direction 
in which light rays are reflected by a first reflecting surface 106) is 
defined as a positive direction of the Y-axis. An axis which constitutes a 
right-handed orthogonal coordinate system in combination with the Y- and 
Z-axes is defined as an X-axis. 
In Examples 1 and 2, each surface is decentered in the YZ-plane, and the 
only one plane of symmetry of each rotationally asymmetric free-form 
surface is the YZ-plane. 
Regarding decentered surfaces, each surface is given displacements (x, y 
and z, respectively) in the X-, Y- and Z-axis directions of the vertex 
position of the surface from the origin of the optical system and tilt 
angles of the center axis of the surface the Z-axis of the above equation 
(a) in regard to free-form surfaces! with respect to the X-, Y- and Z-axes 
(.alpha., .beta. and .gamma., respectively). In this case, positive 
.alpha. and .beta. mean counterclockwise rotation relative to the positive 
directions of the corresponding axes, and positive .gamma. means clockwise 
rotation relative to the positive direction of the Z-axis. 
Among optical surfaces constituting the optical systems according to 
Examples 1 and 2, each pair of adjacent surfaces which form a coaxial 
system is given a surface separation. In addition, the refractive index of 
each medium, together with Abbe's number, is given according to the 
conventional method. 
The surface configuration of each free-form surface is defined by the above 
equation (a). The Z-axis of the defining equation (a) is the axis of a 
free-form surface. 
It should be noted that terms concerning aspherical surfaces for which no 
data is shown are zero. The refractive index is expressed by the 
refractive index for the spectral d-line (wavelength: 587.56 nanometers). 
Lengths are given in millimeters. 
Free-form surfaces may also be defined by Zernike polynomials. That is, the 
configuration of a free-form surface may be defined by the following 
equation (b). The Z-axis of the defining equation (b) is the axis of the 
Zernike polynomial. 
X=R.times.cos(A) 
Y=R.times.sin(A) 
##EQU2## 
Examples 1 and 2 will be described below. FIGS. 1 and 2 are sectional views 
of Examples 1 and 2, taken along the YZ-plane containing the axial 
principal ray 102. In each figure, reference numeral 101 denotes a stop; 
102 denotes an axial principal ray; 103 denotes a low-pass filter; 104 
denotes a decentered prism optical system constituting an objective 
optical system; 105 denotes a first surface of the decentered prism 
optical system 104; 106 denotes a second surface of the decentered prism 
optical system 104; 107 denotes a third surface of the decentered prism 
optical system 104; 108 denotes an image plane where an image pickup 
surface of an electronic image pickup device, e.g. a CCD, is disposed; and 
109 denotes a filter unit including an infrared cutoff filter, a cover 
glass, etc. The optical system according to these examples uses three 
rotationally asymmetric free-form surfaces each having one plane of 
symmetry. The optical system has, in order from the object side thereof, a 
stop 101, a first transmitting surface 105, a first reflecting surface 
106, a second reflecting surface 107, and a second transmitting surface 
106. The first reflecting surface and the second transmitting surface are 
formed from the identical surface 106, and the first reflecting surface 
utilizes total reflection. Therefore, the optical system comprises as 
small a number of surfaces as three. 
Light rays emanating from an object (not shown) pass through the low-pass 
filter 103, which is disposed in the vicinity of the stop 101 on the 
object side thereof, and further pass through the aperture of the stop 101 
to enter the decentered prism optical system 104 through the first surface 
105 thereof. The light rays are reflected by the second surface 106 and 
then reflected by the third surface 107. The reflected light rays come out 
of the decentered prism optical system 104 through the second surface 106 
and pass through the filter unit 109 to form an object image on the image 
plane 108. 
In these examples, the stop 101 and the low-pass filter 103, which also 
serves as a protective glass, are provided on the object side of the 
decentered prism optical system 104 including decentered rotationally 
asymmetric reflecting surfaces. As the low-pass filter 103, a low-pass 
filter such as a polyhedral lens described in JP(A) 7-325269 is used. The 
space can be efficiently utilized by adding various functions such as the 
function of controlling the light quantity, for example, by disposing a 
wavelength selecting or ND filter in front of the stop 101. 
In the decentered prism optical system 104 according to the present 
invention, which is arranged as shown in FIG. 1, the low-pass filter 103 
is disposed in the vicinity of the stop 101 on the object side of the 
decentered prism optical system 104 because there are cases where it is 
difficult to ensure a sufficient back focus. As the low-pass filter 103, 
it is desirable to use a low-pass filter proposed in JP(A) 7-325269, which 
forms a double image by pupil division and which is less costly and 
effective even under defocus conditions. It is also possible to use a 
low-pass filter having a plane surface at one side thereof and an angular 
surface at the other side thereof, as disclosed in JP(A) 3-248695. It is 
also possible to use a known low-pass filter, e.g. a crystal low-pass 
filter. Regarding the position of a low-pass filter that forms a double 
image by pupil division, it is desirable from the viewpoint of minimizing 
the size of the low-pass filter and minimizing the unevenness of low-pass 
effect at the image plane to place the low-pass filter at a position apart 
from the pupil (stop 101) position by a distance not more than f/2, where 
f is the focal length of the decentered prism optical system 104. 
The specifications of Example 1 are as follows: The horizontal half field 
angle is 21.32 degrees; the vertical half field angle is 16.31 degrees; 
the entrance pupil diameter is 1.785 millimeters; and the image size is 
3.83.times.2.93 millimeters. The specifications of Example 2 are as 
follows: The horizontal half field angle is 21.32 degrees; the vertical 
half field angle is 16.31 degrees; the entrance pupil diameter is 1.785 
millimeters; and the image size is 3.90.times.2.89 millimeters. It should 
be noted that, in Examples 1 and 2, the image size is optimized on the 
assumption that the system uses an image pickup device of 1/4 inch size 
which has an image field size of about 4.times.3 millimeters, and that the 
optical system according to each example can be applied to other sizes by 
coefficient-multiplying the entire optical system. 
The constituent parameters will be shown later. Displacements of each 
surface are expressed by amounts of displacement from the surface No. 3. 
The surface No. 8 is a hypothetic plane. The surface Nos. 9 and higher 
represent various optical members (filter unit 109) including an infrared 
cutoff filter, a cover glass, etc. 
In Example 1, powers in the vicinities of points where the axial principal 
ray intersects each surface are, in order from the first transmitting 
surface, 0.183, -1.497, 2.654 and -0.275 in the decentration plane (Y) and 
-0.483, -1.057, 2.426 and -0.241 in the direction (X) perpendicularly 
intersecting the decentration plane. Thus, each transmitting surface is 
assigned a small power, whereas each reflecting surface is assigned a 
large power, thereby effectively utilizing the point at which aberrations 
produced by the reflecting surface become small, and reducing chromatic 
aberrations produced by the transmitting surface. In broad perspective, 
the optical system has a retrofocus type power distribution in which a 
negative power and a positive power are distributed in order from the 
object side. Further, in Example 1, the first transmitting surface is 
assigned a negative power or a small positive power, and the negative 
power of the first reflecting surface and the positive power of the second 
reflecting surface are made strong, thereby ensuring a long back focus in 
comparison to Example 2. 
Lateral aberrations with respect each field angle in Example 1 are 
graphically shown in FIG. 3, and the condition of distortion in Example 1 
is shown in FIG. 4. In the aberrational diagram of FIG. 3, the 
parenthesized numerals denote horizontal (X-direction) field angle, 
vertical (Y-direction) field angle!, and lateral aberrations at the field 
angles are shown. As will be clear from the sectional view of FIG. 1 and 
the aberrational diagrams of FIGS. 3 and 4, Example 1 attains favorable 
optical performance with a simple arrangement comprising a single block 
having a small size of about 8.times.6.times.6 millimeters despite the 
wide horizontal field angle of 42.6 degrees. 
Constituent parameters in the foregoing Examples 1 and 2 are shown below. 
It should be noted that each free-form surface is denoted by "FFS". 
EXAMPLE 
______________________________________ 
Surface 
Radius of 
Surface Displacement 
Refractive 
Abbe's 
No. curvature 
separation 
and tilt index No. 
______________________________________ 
Object 
.infin. .infin. 
plane 
1 .infin. 1.50 1.5163 64.2 
2 .infin. 0.75 
3 .infin.(Stop)) 
(Reference 
plane) 
4 FFS 1! (1) 1.8061 40.9 
5 FFS 2! (2) 1.8061 40.9 
6 FFS 3! (3) 1.8061 40.9 
7 FFS 2! (2) 
8 .infin. 0.00 (4) 
9 .infin. 1.00 1.5163 64.1 
10 .infin. 0.40 1.5163 64.1 
Image .infin. 
plane 
______________________________________ 
FFS 1! 
C.sub.5 
1.3831 .times. 10.sup.-2 
C.sub.7 
-3.6090 .times. 10.sup.-2 
C.sub.8 
-7.3679 .times. 10-3 
C.sub.10 
1.5720 .times. 10-3 
FFS 2! 
C.sub.5 
2.5299 .times. 10.sup.-2 
C.sub.7 
1.7622 .times. 10.sup.-2 
C.sub.8 
-8.4559 .times. 10-4 
C.sub.10 
2.4760 .times. 10-4 
FFS 3! 
C.sub.5 
-4.4848 .times. 10.sup.-2 
C.sub.7 
-4.0459 .times. 10.sup.-2 
C.sub.8 
5.2867 .times. 10-4 
C.sub.10 
4.0114 .times. 10.sup.-4 
C.sub.12 
-1.0522 .times. 10.sup.-4 
C.sub.4 
-2.5073 .times. 10-4 
C.sub.16 
-1.0204 .times. 10.sup.-4 
C.sub.17 
1.2722 .times. 10.sup.-5 
C.sub.19 
2.1529 .times. 10.sup.-5 
C.sub.21 
4.3005 .times. 10.sup.-6 
______________________________________ 
Displacement and tilt (1) 
x 0.000 y 0.000 z 0.803 
.alpha. 15.34 .beta. 0.00 .gamma. 
0.00 
Displacement and tilt (2) 
x 0.000 y 0.333 z 3.551 
.alpha. -40.74 .beta. 0.00 .gamma. 
0.00 
Displacement and tilt (3) 
x 0.000 y 3.101 z 3.473 
.alpha. 109.99 .beta. 0.00 .gamma. 
0.00 
Displacement and tilt (4) 
x 0.000 y -2.637 z 7.430 
.alpha. -56.89 .beta. 0.00 .gamma. 
0.00 
______________________________________ 
EXAMPLE 
______________________________________ 
Surface 
Radius of 
Surface Displacement 
Refractive 
Abbe's 
No. curvature 
separation 
and tilt index No. 
______________________________________ 
Object 
.infin. .infin. 
plane 
1 .infin. 1.50 1.5163 64.2 
2 .infin. 0.75 
3 .infin.(Stop)) 
(Reference 
plane) 
4 FFS1! (1) 1.5400 59.4 
5 FFS2! (2) 1.5400 59.4 
6 FFS3! (3) 1.5400 59.4 
7 FFS2! (2) 
8 .infin. 0.00 (4) 
9 .infin. 1.50 1.5163 64.2 
10 .infin. 0.75 1.5163 64.2 
Image 
plane 
______________________________________ 
FFS 1! 
C.sub.5 
5.5611 .times. 10.sup.-2 
C.sub.7 
7.3954 .times. 10.sup.-2 
C.sub.8 
-2.2062 .times. 10-3 
C.sub.10 
-9.3451 .times. 10.sup.-3 
FFS 2! 
C.sub.5 
7.1877 .times. 10.sup.-4 
C.sub.7 
1.4712 .times. 10.sup.-2 
C.sub.8 
-3.4549 .times. 10-4 
C.sub.10 
-3.3241 .times. 10.sup.-3 
FFS 3! 
C.sub.5 
2.4606 .times. 10.sup.-2 
C.sub.7 
-3.6544 .times. 10.sup.-2 
C.sub.8 
7.8613 .times. 10-4 
C.sub.10 
-2.4000 .times. 10.sup.-4 
C12 2.2952 .times. 10-4 
C14 9.1669 .times. 10-4 
C16 1.2169 .times. 10-4 
C17 1.8425 .times. 10.sup.-5 
C19 7.3383 .times. 10.sup.-5 
C21 4.7013 .times. 10.sup.-5 
______________________________________ 
Displacement and tilt (1) 
x 0.000 y 0.000 z 0.815 
.alpha. 23.06 .beta. 0.00 .gamma. 
0.00 
Displacement and tilt (2) 
x 0.000 y 0.433 z 3.776 
.alpha. -36.70 .beta. 0.00 .gamma. 
0.00 
Displacement and tilt (3) 
x 0.000 y 3.776 z 3.290 
.alpha. 118.34 .beta. 0.00 .gamma. 
0.00 
Displacement and tilt (4) 
x 0.000 y 1.487 z 5.797 
.alpha. -44.31 .beta. 0.00 .gamma. 
0.00 
______________________________________ 
In the decentered prism optical system 104 as shown in FIG. 1 or 2, the 
pupil position may be set inside the decentered prism optical system 104, 
not on the object side of the first surface 105. In such a case, a 
transmitting or reflecting surface which is present in the vicinity of the 
pupil position is formed into a polyhedral configuration on the basis of 
the description in JP(A) 7-325269. By doing so, it is possible to obtain a 
low-pass function whereby moire fringes are prevented from appearing. FIG. 
5 is a schematic perspective view showing the first transmitting surface 
105 of the decentered prism optical system 104 as formed into a polyhedral 
configuration in which lines normal to a plurality of surfaces are in a 
skew relation with respect to the optical axis 102 and which has a 
low-pass function. 
An image-forming optical apparatus according to the present invention 
arranged as described above may be used in an image pickup apparatus such 
as a compact TV camera using, for example, a CCD as an image pickup 
device. FIG. 6 is a conceptual view showing an arrangement in which an 
image-forming optical apparatus according to the present invention is 
incorporated into an image pickup apparatus using a CCD 111 as an 
electronic image pickup device. In this case, an objective optical system 
110 has a decentered prism optical system 104 used as a rear unit disposed 
on the image side of a stop 101. A front unit 120 including a refracting 
optical system is disposed on the object side of the stop 101. An object 
image is formed on the CCD 111 disposed in the image plane through the 
front unit 120, the low-pass filter 103 and the rear unit 104. The object 
image is converted into an image signal by the CCD 111. The image signal 
is processed by a processing device 112 and displayed directly on a CRT 
113 operating as an electronic finder. In addition, the image signal is 
recorded on a recording medium 114 contained in the image pickup 
apparatus. Further, the image pickup apparatus has a microphone 115 to 
record sound information at the same time as the image signal is recorded. 
The image pickup apparatus may be arranged such that the processing device 
112 corrects distortion and lateral chromatic aberrations produced in the 
optical system 110 by using a digital image processing technique on the 
basis of information concerning distortion and lateral chromatic 
aberrations of the optical system 110 previously stored in the recording 
medium 114 or a memory or the like attached to the processing device 112. 
The size and production cost of the image pickup apparatus can be reduced 
by reducing the number of constituent elements and size of the objective 
optical system 110 on the basis of the present invention. 
An image-forming optical apparatus according to the present invention 
arranged as described above may be used in a video endoscope system 
arranged as shown in part (a) of FIG. 7. The video endoscope system 
includes a video endoscope 171, a light source unit 172 for supplying 
illuminating light, a video processor 173 for executing processing of 
signals associated with the video endoscope 171, a monitor 174 for 
displaying video signals outputted from the video processor 173, a VTR 
deck 175 and a video disk 176, which are connected to the video processor 
173 to record video signals and so forth, and a video printer 177 for 
printing out video signals in the form of images. The video endoscope 171 
has an insert part 178 with a distal end portion 179. An image-forming 
optical apparatus according to the present invention as shown for example 
in part (b) of FIG. 7 is incorporated in the distal end portion 179 of the 
insert part 178 to form a direct-view video endoscope. 
In part (b) of FIG. 7, an image pickup apparatus used in the direct-view 
video endoscope comprises an objective optical system consisting 
essentially of a decentered prism optical system 104, and a CCD 111 
disposed in an image plane of the decentered prism optical system 104. A 
stop 101 is positioned on the object side of the decentered prism optical 
system 104. A low-pass filter 103 is disposed on the image side of the 
stop 101. An object image is formed on the CCD 111 through the low-pass 
filter 103 and the decentered prism optical system 104. The object image 
is converted into an image signal by the CCD 111. The image signal is 
displayed directly on the monitor 174 by the video processor 173. In 
addition, the image signal is recorded in the VTR deck 175 and on the 
video disk 176 and also printed out in the form of an image from the video 
printer 177. 
The objective optical system according to the present invention is 
applicable to various other forms shown below in addition to the foregoing 
examples. 
FIG. 8 shows an arrangement in which a two-unit zoom lens system is 
provided as an objective optical system. A first lens unit G1 comprises a 
decentered prism optical system 200 having a negative power as a whole. A 
second lens unit G2 comprises a lens system having a positive power as a 
whole. To effect zooming from a wide-angle end to a telephoto end, the 
second lens unit G2 moves toward the object side. The decentered prism 
optical system 200 is in inverse relation the decentered prism optical 
system 104 according to Example 1 (FIG. 1) in terms of the arrangement of 
the optical path. That is, the pupil side of the decentered prism optical 
system 104 is the image side of the decentered prism optical system 200, 
and the image side of the former is the pupil side of the latter. At least 
one surface of the decentered prism optical system 200 is formed from a 
rotationally asymmetric free-form surface having one plane of symmetry. It 
is desirable that all the three surfaces of the decentered prism optical 
system 200 should be formed from rotationally asymmetric free-form 
surfaces each having one plane of symmetry. 
The arrangement shown in FIG. 8 is an example in which the objective 
optical system according to the present invention is disposed in an 
electronic camera 201. An object image formed on an image plane 108 of a 
CCD 202 is converted into an image signal. The image signal is inputted 
into an image signal processing device 203. The image signal processed in 
the processing device 203 is inputted to an LCD (Liquid Crystal Display) 
204, thereby displaying the object image on the LCD 204. An enlarged image 
of the display image is projected into a photographer's eye through an 
ocular prism 205 formed from a decentered optical system. Meanwhile, the 
image signal is recorded in a recording device 206 connected to the 
processing device 203. The low-pass filter 103 and the filter unit 109, 
which includes an infrared cutoff filter, etc., are similar to those in 
Example 1. 
The objective optical system shown in FIG. 8 may be a three-unit zoom lens 
system having a third lens unit G3 (not shown) in addition to the two lens 
units, or a four-unit zoom lens system having a fourth lens unit G4 (not 
shown) provided in addition to the three lens units. 
A plurality of decentered prism optical systems may be used in an objective 
optical system. An example of such an arrangement is shown in FIG. 9. In 
this example, a first lens unit G1 and a fourth lens unit G4 are formed 
from decentered prism optical systems 210 and 211, respectively. A second 
lens unit G2 and a third lens unit G3 are disposed between the first lens 
unit G1 and the fourth lens unit G4. The decentered prism optical systems 
210 and 211 are different in configuration from the decentered prism 
optical systems shown in FIGS. 1 and 8. That is, light enters the 
decentered prism optical system 210 (211) through a first surface 212 and 
is reflected successively by a second surface 213 and a third surface 214 
so as to exit from the decentered prism optical system 210 (211) through a 
fourth surface 215. In each of the decentered prism optical systems 210 
and 211, at least one surface, desirably every surface, is formed from a 
rotationally asymmetric free-form surface having one plane of symmetry. 
The low-pass filter 103 is disposed in the vicinity of a pupil closer to 
the object than the reflecting surface 213 of the decentered prism optical 
system 211. An aperture stop 101 is provided in the pupil plane. 
The decentered prism optical system may also be formed as shown in FIGS. 10 
to 15. In these decentered prism optical systems, at least one surface, 
desirably every surface, is formed from a rotationally asymmetric 
free-form surface having one plane of symmetry. 
A decentered prism optical system 220 shown in FIG. 10 is arranged such 
that light enters it through a first surface 221 and is reflected 
successively by a second surface 222, a third surface 223 and the first 
surface 221 so as to exit from the optical system 220 through the second 
surface 222. 
A decentered prism optical system 230 shown in FIG. 11 is arranged such 
that light enters it through a first surface 231 and is reflected 
successively by a second surface 232 and a third surface 233 so as to exit 
from the optical system 230 through a fourth surface 234. 
A decentered prism optical system 240 shown in FIG. 12 is arranged such 
that light enters it through a first surface 241 and is reflected 
successively by a second surface 242, a third surface 243 and a fourth 
surface 244 so as to exit from the optical system 240 through the third 
surface 243. 
A decentered prism optical system 250 shown in FIG. 13 is arranged such 
that light enters it through a first surface 251 and is reflected 
successively by a second surface 252, a third surface 253 and the second 
surface 252 so as to exit from the optical system 250 through a fourth 
surface 254. 
A decentered prism optical system 260 shown in FIG. 14 is arranged such 
that light enters it through a first surface 261 and is reflected 
successively by a second surface 262, a third surface 263, the second 
surface 262 and a fourth surface 264 so as to exit from the optical system 
260 through the second surface 262. 
A decentered prism optical system 270 shown in FIG. 15 is arranged such 
that light enters it through a first surface 271 and is reflected 
successively by a second surface 272, the first surface 271, a third 
surface 273 and the first surface 271 so as to exit from the optical 
system 270 through the third surface 273. 
As will be clear from the foregoing description, it is possible according 
to the present invention to obtain a compact image-forming optical 
apparatus which is free from moire fringes and capable of providing an 
aberration-free, clear image of minimal distortion even at a wide field 
angle.