Projection display apparatus having predetermined conditions to correct distortions

A projection display apparatus has a reflective type display for displaying a two-dimensional image on a display surface, a projection optical system for projecting an enlargement image of the two-dimensional image displayed on the display surface onto a projection image surface, the projection optical system having a diaphragm and a decentering optical element disposed between the diaphragm and the reflective type display and decentered with respect to the diaphragm, and wherein the predetermined conditions are satisfied when an image center ray is defined by a line connecting a center position of the display surface and the center position of the projection image surface.

CROSS-REFERENCE TO RELATED APPLICATIONS 
This application is based on Application No. 10-208169 filed in Japan, the 
content of which is hereby incorporated by reference. 
FIELD OF THE INVENTION 
The present invention relates to a projection display apparatus, and 
specifically relates to a compact, high resolution projection display 
apparatus for projecting on a screen a two-dimensional image of a 
reflective type display element {e.g., a reflective liquid crystal display 
(LCD) or a digital micro mirror (DMD)}. 
DESCRIPTION OF THE RELATED ART 
Recently, attention has been focused on reflective liquid crystal displays 
(LCD) which have a higher light usage efficiency than transmission type 
LCD. The reflective type LCD reflects the entering illumination light from 
its surface, and emits regular reflection projection light having the same 
reflection angle but opposite sign as the incidence angle. A projection 
display apparatus using a reflective LCD has been proposed, wherein a 
reflective LCD is illuminated from a near perpendicular direction, and the 
projection light emitted in a perpendicular direction forms an image on a 
screen via a projection optical system. 
FIG. 13 shows a conventional example of a single-panel projection display 
apparatus using a reflective LCD. The illumination light emitted from a 
light source 1 is reflected by a reflector 2, and subsequently enters a 
collimator 3. The collimator 3 improves the efficiency of the reflective 
LCD 5, and is an illumination optical system providing even illumination. 
The illumination light transmitted through the collimator 3 enters a 
polarization beam splitter 4. The polarization surface of the polarization 
beam splitter 4 reflects only the S light flux component, which enters the 
reflective LCD 5 at a right angle. The illumination light which enters the 
reflective LCD 5 at a right angle is selectively converted by the 
polarization surface to a P light flux component for each pixel by the 
reflective LCD 5. This component is reflected in regular reflection as 
projection light. The projection light reflected as regular reflection 
from the reflective LCD 5 in a perpendicular direction passes through the 
polarization beam splitter 4 without reflection, and forms an image on a 
screen 7 via a projection optical system 6. 
FIG. 14 shows a conventional example of a three-panel projection display 
apparatus using a reflective LCD. Illumination light emitted from a light 
source 11 and reflected by a reflector 12 and a mirror 13 is transmitted 
through a first lens array 14, and subsequently passes through a 
polarization separation prism 19 comprising a rectangular prism 15 and a 
parallel plane table 16. Reference number 17 refers to the total 
reflective surface, and reference number 18 refers to the polarization 
separation surface. The illumination light passing through the 
polarization separation prism 19 passes through the polarization surface 
of 1/2 wavelength panel 20, then passes through a second lens array 21 and 
an overlaid lens 22, then passes through an illumination relay optical 
system 23. The co-axial system of the projection optical system 38 
comprises the posterior part 26 of the projection optical system, a 
diaphragm SP, and the anterior part 25 of the projection optical system. 
The illumination light 36 passing through the illumination relay optical 
system 23 is reflected by a reflecting mirror 24 disposed in proximity to 
the diaphragm SP, and passes through the anterior part 25 of the 
projection optical system. 
The illumination light 36 passing through the anterior part 25 of the 
projection optical system is color separated by cross dichroic prism 27, 
and after passing through the polarization plates 33.about.35, enters the 
red, green, blue reflective LCDs 28.about.30. Reference number 31 refers 
to a red reflective surface, and reference number 32 refers to a blue 
reflective surface. The light reflected by the reflective LCDs 28.about.30 
becomes the projection light via color combination by the cross dichroic 
prism 27, which passes through the diaphragm SP position lacking the 
reflective mirror 24, and forms an image on a screen (not shown in the 
drawing). 
In the case of the conventional example shown in FIG. 13, a device is 
required to separate the illumination optical path and the projection 
optical path as performed by the polarization beam splitter 4 because the 
illumination light and the projection light travel on virtually the same 
optical path. A separation device such as the polarization beam splitter 4 
becomes a factor in cost increases because to manufacture such a 
separation device requires the processing of a large glass block or 
multi-layer thin film process. Since the reflective LCD 5 transmits and 
blocks light rays from the polarization surface, disturbances arise in the 
polarization surface due to irregularities of the medium within the glass 
block, such that unnecessary light components are transmitted therethrough 
and reduce the contrast of the projected image. 
In the case of the conventional example shown in FIG. 14, the F number 
required in the projection optical system 38 must be doubled for using the 
illumination light 36 and the projection light 37. Accordingly, the number 
of lens elements must be increased and the lens diameter must be increased 
to maintain the projection efficiency, and these are factors in cost 
increase and scale enlargement of the device. 
SUMMARY OF THE INVENTION 
A main object of the present invention is to provide an improved projection 
display apparatus. 
Another object of the present invention is to provide a high resolution 
projection display apparatus at low cost. 
Yet another object of the present invention is to provide a compact 
projection display apparatus at low cost. 
These objects are attained by a projection display apparatus comprising a 
reflective type display for displaying a two-dimensional image on a 
display surface, a projection optical system for projecting an enlargement 
image of the two-dimensional image displayed on the display surface onto a 
projection image surface, the projection optical system having a diaphragm 
and a decentering optical element disposed between the diaphragm and the 
reflective type display and decentered with respect to the diaphragm, and 
wherein the following conditions are satisfied when an image center ray is 
defined by a line connecting a center position of the display surface and 
the center position of the projection image surface; 
EQU .theta..sub.i &lt;10.degree. 
EQU 5.degree.&lt;.theta..sub.o &lt;15.degree. 
where .theta..sub.i represents an angle formed by a normal line of the 
projection image surface and the image center ray, and .theta..sub.o 
represents an angle formed by the normal line of the display surface and 
the image center ray.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
The projection display apparatus of the present invention is described 
hereinafter with reference to the accompanying drawings. FIGS. 1, 5, and 9 
show the constructions of the projection optical systems of the first 
through third embodiments, and FIGS. 2, 6, and 10 show the constructions 
of the projection optical systems and the illumination optical systems of 
the first through third embodiments. Each optical system structural 
drawing shows the YZ cross sectional construction in the orthogonal 
coordinate system (X, Y, Z) described below. 
The first through third embodiments comprise a reflective type display 80 
(e.g., a reflective LCD) for displaying a two-dimensional image on a 
rectangular display surface (I), and a projection optical system (FIGS. 1, 
5, 9) for projecting an enlarged two-dimensional image. The projection 
optical system of each embodiment is constructed of a plurality of blocks 
{Bi(i=1,2,3 . . . )}, one or more of which is decentered from a co-axial 
system. The projection optical system has a diaphragm SP, and a 
decentering optical element DL which is eccentrically larger between the 
diaphragm SP and the display surface I. The disposition of the decentering 
optical element DL which is eccentrically larger between the diaphragm SP 
and the display surface I allows the light flux from the display surface I 
to the projection optical system to be inclined relative to the normal 
line of the display surface I. In this way, the illumination optical 
system and the projection optical system can be separated. 
It is desirable that the optical construction of the projection display 
apparatus satisfies the following conditions. 
It is desirable that conditional equations (1) and (2) below are satisfied 
when the light rays connecting the center position of the display surface 
I and the center position of the projection image plane (equivalent to the 
screen surface) are the image center rays. 
EQU .theta..sub.i &lt;10.degree. (1) 
EQU 5.degree.&lt;.theta..sub.0 &lt;15.degree. (2) 
where .theta..sub.i represents an angle formed by a normal line 90 of the 
projection image surface S0 and the image center ray 100, and 
.theta..sub.o represents an angle formed by the normal line 110 of the 
display surface I and the image center ray (FIG. 15). 
When the value of condition (1) is outside the condition range, in order to 
correct trapezoidal distortion generated by the tilted projection, the 
number of lens elements that must be eccentric increases, thereby 
increasing the amount of decentering. The additional lens elements 
increase the manufacturing costs. When the lower limit of condition (2) is 
not met, the illumination optical system and the projection optical system 
cannot be separated. When the upper limit of condition (2) is exceeded, in 
order to correct trapezoidal distortion and coma aberration generated by 
the tilted projection, the number of lens elements that must be eccentric 
increases, thereby increasing the amount of decentering. The additional 
lens elements increase the manufacturing costs. 
In the surface of the decentering optical element DL (surface S13 in the 
present embodiments) nearest the diaphragm SP within the decentering 
direction cross section (YZ plane), it is desirable that condition (3) is 
satisfied when the positive side is the illumination light source LS side 
from the axis of symmetry of the decentering optical element DL. 
EQU -0.15&lt;-Hn/Hf&lt;0.4 (3) 
where Hn represents the distance from the axis of symmetry 120 of the 
decentering optical element DL to the position 130 nearest the 
illumination light source LS at the projection light flux transmission 
position, and Hf represents the distance from the axis of symmetry 120 of 
the decentering optical element DL to the position 140 farthest from the 
illumination light source LS at the projection light flux transmission 
position (FIG. 16). 
When the lower limit of condition (3) is not met, the decentering optical 
element DL becomes excessively decentering, and in order to correct 
distortion generated by the decentering, the number of lens elements that 
must have a decentering effect increases, thereby increasing the amount of 
decentering. When the upper limit of condition (3) is exceeded, there is 
insufficient decentering, thereby making is difficult to separate the 
illumination light flux and the projection light flux. 
It is desirable to satisfy condition (4) by providing a space filled with a 
continuous medium between the decentering optical element DL and the 
display surface I. The space is filled with a continuous medium formed by 
a block B5 corresponding to the cross dichroic prism in the first 
embodiment, and is an empty space in the second and third embodiments. 
EQU 1.0&lt;k/Ls&lt;10 (4) 
where k represents the distance along the image center ray 100 in the space 
filled with a continuous medium, and Ls represents the length of the short 
side of the display surface I (FIG. 17). Generally, the display surface I 
is a non-square, rectangular shape, because image data determines its 
shape (for example VGA of 600.times.900 pixel or XGA of 1200.times.800 
pixel). 
When the lower limit of condition (4) is not met, it becomes difficult to 
position the prism for color separation and color combination on the 
display side of the decentering optical element DL. Accordingly, the cost 
increases due to the need for several decentering optical elements DL for 
color separation and color combination since the prism for color 
separation and color combination is disposed on the diaphragm SP side of 
the decentering optical element DL. When the upper limit of condition (4) 
is exceeded, the cost increases and compactness is lost because the 
overall length of the lens increases and the lens diameter increases. It 
is desirable that a lens element having negative optical power is disposed 
on the outermost side of the projection image surface. In this way the 
lens back can be lengthened, and the distance k can be easily increased. 
It is desirable to arrange an optical system comprising both a decentering 
optical element DL and an illumination optical system between the 
decentering optical element DL and the display, such that only the optical 
element B5 (or the empty space between decentering optical element DL and 
display surface I) and the decentering optical element DL transmit both 
the illumination light illuminating the display surface I and the 
projection light reflected by the display surface I. According to this 
construction, the decentering optical element DL and the optical element 
disposed between the decentering optical element DL and the projection 
image display surface I can be common members to the illumination optical 
system and the projection optical system. Therefore, the manufacturing 
costs can be reduced. Since only a simple projection light is transmitted 
through the projection optical system on the projection image surface side 
of the decentering optical element DL, the number of lens elements and the 
lens diameter can be reduced, thereby reducing costs. 
In a projection display apparatus provided with an illumination light 
source LS for illuminating a display surface I as in each of the present 
embodiments, it is desirable that the decentering optical element DL has a 
positive optical power, and parallel and decentering relative to the 
diaphragm SP on the illumination light source LS side. According to this 
construction, since the principal ray from the decentering optical element 
DL to the display surface I approaches being telecentric, the trapezoidal 
distortion is readily corrected, and the lens back is easily lengthened. 
Furthermore, the part DL1 of decentering optical element DL through which 
illumination light in the illumination opitcal system on the opposite side 
from the part DL2 used as the projection optical system in the decentering 
optical element DL can be jointly used as the collimator of the 
illumination optical system. 
It is desirable that the positive optical power of the decentering optical 
element DL satisfies condition (5). 
EQU 0.03&lt;-Ph.times.Si.times..beta.&lt;0.7 (5) 
where Ph represents the positive optical power of the decentering optical 
element DL, Si represents the distance from the projection image surface 
to the surface (S2) nearest the projection image surface side of the 
projection optical system (see FIG. 18), and .beta. represents the 
reduction magnification in the decentering direction of the decentering 
optical element DL. 
When the lower limit of condition (5) is not met, the amount of parallel 
decentering becomes excessive due to the prism effect in inclined 
projection on the display surface I. Accordingly, the cost is increased 
due to necessity of increasing the diameter and the core thickness of the 
decentering optical element DL. When the upper limit of condition (5) is 
exceeded, it becomes difficult to ensure optical efficiency due to the 
excessive distortion generated by the decentering optical element DL, and 
the amount of decentering is insufficient due to the prism effect, such 
that it becomes difficult to join the illumination optical system and the 
decentering optical element DL. 
It is desirable that condition 6 is satisfied as shown below. 
EQU 2.degree.&lt;.theta..sub.p &lt;15.degree. (6) 
where .theta. represents the angle formed by the tangent plane 150 at the 
position at which the image center ray 100 passes through a surface (S13) 
of the decentering optical element DL nearest the projection image surface 
side (S0), and the tangent plane 160 at the position at which the image 
center ray passes through a surface (S14) of the decentering optical 
element DL nearest the display surface I side. 
The prism effect of the decentering optical element DL makes it possible to 
obtain tilted illumination and tilted projection relative to the display 
surface I. When the lower limit of condition (6) is not met, it is 
difficult to bend the light rays at an angle required for tilted 
illumination, thus making it difficult to separate the illumination light 
and the projection light. When the upper limit of condition (6) is 
exceeded, the separation of the illumination light and the projection 
light becomes excessively large due to bending of the light rays more than 
is necessary. As a result, the prism and mirror used for color separation 
and color combination become larger, and the tilted projection becomes 
larger. Accordingly, the cost increases due to the additional lens 
elements needed to maintain optical performance. 
It is desirable that the principal ray emitted from the four corners of the 
display surface I toward the projection optical system satisfy condition 
(7). 
EQU .vertline..theta..sub.k -.theta..sub.o .vertline.max&lt;5.degree.(7) 
where .theta..sub.k represents the angle formed by the principal rays 
emitted from the four corners of the element surface I and the normal line 
110 of the display surface I. and .vertline..theta..sub.k -.theta..sub.o 
.vertline.max represents the maximum value of .vertline..theta..sub.k 
-.theta..sub.o .vertline. of the principal rays as there are four angles, 
.theta.k.sub.1, .theta.k.sub.2, .theta.k.sub.3 and .theta.k.sub.4 for a 
rectangular display surface I. It should be noted that all of the rays 
emitted from the four corners are not at the same angle .theta..sub.k 
because the projection optical system is tilted relative to the display 
surface. 
In each of the aforesaid embodiments, the light is inclined as it enters 
the display surface I. Accordingly, when the principal ray is outside the 
condition range of condition (7), a large amount of trapezoidal distortion 
is generated. To correct this distortion requires many decentering lens 
elements, which increases the amount of decentering. These additional lens 
elements increase the manufacturing costs. 
It is desirable that a lens (the lens element having surfaces S9 and S10) 
having negative optical power is disposed adjacent to the diaphragm on the 
display surface side. According to this construction, it is possible the 
rays between the decentering optical element DL and the display surface I 
are telecentric. For this reason, it is possible to minimize the color 
irregularity generated by color separation and color combination by the 
cross dichroic prism, so as to attain excellent color reproduction in the 
projection image. 
It is desirable that a lens (the lens element having surfaces S15 and S16) 
having positive optical power is disposed in proximity to the display. 
According to this construction, it is unnecessary for the rays to be 
telecentric between the decentering optical element DL and the positive 
optical power lens near the display element. For this reason, the overall 
lens length can be reduced even when the spacing is increased, thereby 
obtaining a compact projection optical system. 
It is desirable that the optical power of the positive lens disposed in 
proximity to the display element satisfies condition (8). 
EQU 0.05&lt;-Pc.times.Si.times..beta.&lt;0.4 (8) 
where Pc represents the optical power of the positive lens disposed near 
the display. 
When the lower limit of condition (8) is not met, the telecentric effect is 
weakened between the decentering optical element DL and the positive 
optical power lens in proximity to the display surface I, and increasing 
the distance therebetween increases the overall length of the lens. When 
the upper limit of condition (8) is exceeded, the illumination light pupil 
position is too close to the display surface I, such that the angle must 
be increased to separate the illumination light and the projection light. 
For this reason, in order to correct aberration caused by the inclined 
projection, the number of lens elements and the amount of decentering must 
be increased, thereby increasing costs. 
The construction of the projection display apparatus of the present 
invention is described below by way of specific examples via construction 
data, spot diagrams and the like of the projection optical system. 
In the construction data of the projection optical system of the various 
embodiments, Si (i=1,2,3 . . . ) represents the No. i surface counting 
from the projection image surface side (S0; screen surface), and ri 
(i=0,1,2,3 . . . ) represents the radius of curvature of the surface Si in 
a system including the decentering standard surface (SI; without optical 
effect) and the display surface I. Furthermore, di (i=0,1,2,3 . . . ) 
represents the axial distance of the No. i surface counting from the 
projection image surface (S0) side. Ni (i=1,2,3 . . . ) and vi (i=1,2,3 . 
. . ) respectively represent the refraction index (Nd) and Abbe number (d) 
relative to the d-line of the No. i optical element counting from the 
projection image surface (S0) side. Since the illumination optical system 
jointly uses part of the projection optical system, these construction 
data are omitted. 
In the orthogonal coordinate system (X,Y,Z), the center position of the 
decentering standard surface SI is the surface peak coordinate 
(XDE,YDE,ZDE)=(parallel decentering position in the X direction, parallel 
decentering position in the Y direction, parallel decentering position in 
the Z direction) designated the origin (0,0,0), and expresses both the 
position of the lead parallel decentering surface and rotational 
decentering (the counterclockwise direction facing this sheet is 
designated positive) at a rotation angle ADE (.degree.) around the X-axis 
as the center of the surface peak of that surface. The X-axis direction is 
a direction perpendicular to the sheet surface (the back surface direction 
of the sheet is designated positive), the Y-axis direction is a linear 
direction intersecting the decentering standard surface S1 and the sheet 
surface (the upward direction in the drawing is designated positive), and 
the Z-axis direction is the normal line direction of the decentering 
standard surface S1 (the display surface side is designated positive). 
Surfaces Si marked with an asterisk (*) represent aspherical surfaces, and 
are defined by equation AS below which represents the surface shape (each 
surface peak standard) of the aspherical surface. Aspherical surface data 
are included with other data. 
EQU Z=(c.h.sup.2)/[1+.sqroot.{1-(1+K).multidot.c.sup.2 .multidot.h.sup.2 
}]+(A.multidot.h.sup.4 +B.multidot.h.sup.6 +C.multidot.h.sup.8 
+D.multidot.h.sup.10 +E.multidot.h.sup.12) (AS) 
where Z represents an aspherical displacement amount from a reference 
surface in aspherical surface along the optical axis direction, h 
represents a height perpendicular to the optical axis (defined by h.sup.2 
=X.sup.2 +Y.sup.2), c represents a curvature and K, A, B, C, D and E 
represent aspherical coefficients. 
______________________________________ 
Embodiment 1 projection optical system 
______________________________________ 
[sur- [radius [refractive 
face] of curvature] 
[axial distance] 
index] [Abbe number] 
______________________________________ 
S0 r0 = .infin. 
d0 = 845.000 
S1 r1 = .infin. 
S2 (XDE,YDE,ZDE,ADE) = (0.0,4.551,0.190,-1.916) 
r2 = 53.383 
d2 = 1.100 N1 = 1.7545 
.nu.1 = 51.570 
S3 r3 = 29.413 
d3 = 13.480 
S4 r4 = 209.287 
d4 = 1.557 N2 = 1.4875 
.nu.2 = 70.440 
S5* r5 = 28.406 
d5 = 53.053 
S6 r6 = 87.096 
d6 = 4.986 N3 = 1.8195 
.nu.3 = 25.842 
S7 r7 = -450.877 
S8 (SP)(XDE,YDE,ZDE,ADE) = (0.000,0.000,120.000,0.000) 
r8 = .infin. (effective radius: 11.359) 
S9 (XDE,YDE,ZDE,ADE) = (0.000,2.440,123.000,0.977) 
r9 = -391.745 
d9 = 10.000 
N4 = 1.8473 
.nu.4 = 25.736 
S10 r10 = 47.089 
d10 = 0.100 
S11* r11 = 41.324 
d11 = 7.106 
N5 = 1.5139 
.nu.5 = 66.880 
S12 r12 = -50.378 
S13* (XDE,YDE,ZDE,ADE) = (0.000,11.125,204.808,2.827) 
r13 = 269.577 
d11 = 16.679 
N6 = 1.5168 
.nu.6 = 65.261 
S14* r14 = -51.560 
S15 (XDE,YDE,ZDE,ADE) = (0.000,11.997,222.465,-4.759) 
r15 = .infin. 
d15 = 90.000 
N7 = 1.5168 
.nu.7 = 65.261 
S16 r16 = .infin. 
S17(I) (XDE,YDE,ZDE,ADE) = (0.000,4.447,313.152,-2.455) 
r17 = .infin. 
Aspherical Data of the fifth surface (S5) 
K = -0.983596 
A = -0.941346 .times. 10.sup.-6 
B = -0.382769 .times. 10.sup.-8 
C = 0.361847 .times. 10.sup.-11 
D = -0.479956 .times. 10.sup.-14 
Aspherical Data of the eleventh surface (S11) 
K = 0.000000 
A = -0.180239 .times. 10.sup.-5 
B = -0.869663 .times. 10.sup.-8 
C = 0.650313 .times. 10.sup.-10 
D = -0.225062 .times. 10.sup.-12 
E = 0.294746 .times. 10.sup.-15 
Aspherical Data of the thirteenth surface (S13) 
K = 0.000000 
A = 0.910200 .times. 10.sup.-7 
B = -0.374901 .times. 10.sup.-9 
C = 0.319043 .times. 10.sup.-12 
D = -0.452650 .times. 10.sup.-16 
E = -0.175222 .times. 10.sup.-19 
Aspherical Data of the fourteenth surface (S14) 
K = 0.000000 
A = 0.144980 .times. 10.sup.-5 
B = -0.146171 .times. 10.sup.-9 
C = 0.124804 .times. 10.sup.-12 
D = 0.116210 .times. 10.sup.-15 
E = -0.412338 .times. 10.sup.-19 
______________________________________ 
______________________________________ 
Embodiment 2 projection optical system 
______________________________________ 
[sur- [radius [refractive 
face] of curvature] 
[axial distance] 
index] [Abbe number] 
______________________________________ 
S0 r0 = .infin. 
d0 = 845.000 
S1 r1 = .infin. 
S2 (XDE,YDE,ZDE,ADE) = (0.000,13.027,0.100,2.593) 
r2 = 103.355 
d2 = 1.500 N1 = 1.6958 
.nu.1 = 53.789 
S3 r3 = 44.929 
d3 = 16.377 
S4 r4 = 162.886 
d4 = 15.000 
N2 = 1.4875 
.nu.2 = 70.440 
S5* r5 = 46.791 
S6 (XDE,YDE,ZDE,ADE) = (0.000,0.312,179.525,-1.277) 
r6 = 69.314 
d6 = 4.067 N3 = 1.7985 
.nu.3 = 22.600 
S7 r7 = 265.314 
d7 = 21.594 
S8(SP) r8 = .infin.(effective radius = 16.018) 
d8 = 10.561 
S9 r9 = -83.217 
d9 = 15.000 
N4 = 1.8267 
.nu.4 = 24.179 
S10 r10 = 68.657 
d10 = 0.100 
S11* r11 = 59.816 
d11 = 15.000 
N5 = 1.6543 
.nu.5 = 55.745 
S12 r12 = -63.017 
S13 (XDE,YDE,ZDE,ADE) = (0.000,18.067,248.261,-7.984) 
r13 = 127.713 
d11 = 6.809 
N6 = 1.5168 
.nu.6 = 65.261 
S14 r14 = 746.072 
d14 = 110.000 
S15 r15 = 58.547 
d15 = 15.000 
N7 = 1.7545 
.nu.7 = 51.570 
S16 r16 = 101.156 
d16 = 3.000 
S17(I) 
r17 = .infin. 
Aspherical Data of the fifth surface (S5) 
K = -0.800000 
A = -0.956207 .times. 10.sup.-6 
B = -0.236775 .times. 10.sup.-9 
C = -0.968458 .times. 10.sup.-13 
D = -0.418944 .times. 10.sup.-16 
Aspherical Data of the eleventh surface (S11) 
K = 0.000000 
A = -0.102161 .times. 10.sup.-5 
B = -0.285814 .times. 10.sup.-8 
C = 0.131048 .times. 10.sup.-10 
D = -0.256918 .times. 10.sup.-13 
E = 0.190853 .times. 10.sup.-16 
______________________________________ 
______________________________________ 
Embodiment 3 projection optical system 
______________________________________ 
[sur- [radius [refractive 
face] of curvature] 
[axial distance] 
index] [Abbe number] 
______________________________________ 
S0 r0 = .infin. 
d0 = 845.000 
S1 r1 = .infin. 
S2 (XDE,YDE,ZDE,ADE) = (0.000,10.734,0.100,1.192) 
r2 = 64.918 
d2 = 1.600 N1 = 1.7545 
.nu.1 = 51.570 
S3 r3 = 37.730 
d3 = 14.893 
S4 r4 = 119.934 
d4 = 7.302 N2 = 1.4875 
.nu.2 = 70.440 
S5* r5 = 31.045 
S6 (XDE,YDE,ZDE,ADE) = (0.000,-2.308,177.602,0.426) 
r6 = 60.731 
d6 = 8.370 N3 = 1.6723 
.nu.3 = 28.247 
S7 r7 = -1247.768 
S8(SP) (XDE,YDE,ZDE,ADE) = (0.000,0.000,205.186,0.000) 
r8 = .infin.(effective radius = 17.312) 
S9 (XDE,YDE,ZDE,ADE) = (0.000,-2.027,208.186,0.572) 
r9 = -99.624 
d9 = 15.000 
N4 = 1.8068 
.nu.4 = 25.248 
S10 r10 = 49.796 
d10 = 0.100 
S11* r11 = 45.425 
d11 = 14.079 
N5 = 1.5502 
.nu.5 = 63.013 
S12 r12 = -69.877 
S13 (XDE,YDE,ZDE,ADE) = (0.000,20.537,237.241,-3.369) 
r13 = 138.420 
d11 = 8.985 
N6 = 1.5168 
.nu.6 = 65.261 
S14 r14 = -830.085 
S15 (XDE,YDE,ZDE,ADE) = (0.000,3.561,355.433,-6.760) 
r15 = 77.998 
d15 = 5.252 
N7 = 1.4875 
.nu.7 = 70.440 
S16 r16 = .infin. 
S17(I) (XDE,YDE,ZDE,ADE) = (0.000,2.825,361.641,-8.225) 
r17 = .infin. 
Aspherical Data of the fifth surface (S5) 
K = -0.800000 
A = -0.688237 .times. 10.sup.-6 
B = -0.836011 .times. 10.sup.-9 
C = -0.171182 .times. 10.sup.-13 
D = -0.514561 .times. 10.sup.-15 
Aspherical Data of the eleventh surface (S11) 
K = 0.000000 
A = -0.146304 .times. 10.sup.-5 
B = -0.179349 .times. 10.sup.-8 
C = 0.927093 .times. 10.sup.-11 
D = -0.201469 .times. 10.sup.-13 
E = 0.163826 .times. 10.sup.-16 
______________________________________ 
The optical performance of the projection optical system of each embodiment 
is shown in spot diagrams (FIGS. 3, 7, 11), and aberration diagrams (FIGS. 
4, 8, 12). The evaluation fields were (566.0 mm in the X-axis direction 
and (340.0 mm in the Y-axis direction on the screen surface S0, and each 
evaluation position is represented by coordinates (X,Y) in Table. In 
essence, the screen surface S0 is the projection image surface, and the 
display element surface I is the object surface, but in the present 
embodiments, the screen surface S0 is regarded as the object surface, and 
the display element surface I is evaluated for optical performance as a 
condensed system of optical statistics. 
The following table shows values corresponding to the conditional equations 
of each of the embodiments. 
TABLE 
__________________________________________________________________________ 
Cond. Cond. 
Cond. 
Cond. 
Cond. 
Cond. 
Cond. 
(5) Cond. 
(7) (8) 
Embodi- 
(1) (2) (3) (4) 
Ph.sub.x Si 
(6) .vertline..theta..sub.k -.theta..sub.o .vertline. 
Pc.sub.x Si 
ments 
.theta..sub.i 
.theta..sub.o 
Hn/Hf 
k/Ls 
.sub.x .beta. 
.theta..sub.p 
max .sub.x .beta. 
__________________________________________________________________________ 
1 0.200 
10.310 
0.241 
3.9 0.3395 
15.83 
0.668 
-- 
2 0.662 
10.274 
-0.013 
4.8 0.0963 
7.63 
0.356 
0.1789 
3 0.577 
10.463 
-0.066 
4.7 0.1250 
7.65 
0.268 
0.1799 
__________________________________________________________________________ 
As described above, the present invention realizes a low cost, compact, and 
high resolution projection display apparatus by providing a decentering 
optical element which becomes larger decentering between a diaphragm and a 
display under predetermined conditions. 
Although the present invention has been fully described by way of examples 
with reference to the accompanying drawings, it is to be noted that 
various changes and modifications will be apparent to those skilled in the 
art. Therefore, unless such changes and modifications depart from the 
scope of the present invention, they should be construed as being included 
therein.