Projection optical system, magnification projection optical system, magnification projection apparatus, and image projection apparatus

Abstract

In a projection optical system for use in an image projection apparatus illuminating an image display panel forming an image in accordance with a modulating signal with illumination light from a light source, the projection optical system includes first and second optical systems arranged along an optical path defining an upstream-downstream direction in the order described from upstream to downstream on the downstream side of the image display panel. The first optical system includes at least one dioptric system and has positive power. The second optical system includes at least one reflecting surface having power and has positive power. The image formed by the image display panel is formed as an intermediate image in the optical path, and the intermediate image is magnified and projected.

Description

having an optical axis 101
where X2, Y2, X2Y, Y3, X2Y2, etc. are coefficients, letting the vertical directions be the Y directions, the lateral directions be the X directions, and the depth of the curved surface be the Z directions. The vertical (upward and downward) directions and the lateral (rightward and leftward) directions are considered based on the projected image. The coefficients of the polynomial free-form surface are shown in Table 2.

TABLE 2

Coefficient Coefficient Value

X2          6.99519E−03
Y2          5.16104E−03
X2Y         6.00372E−05
Y3          4.11672E−05
X4          −1.12766E−07
X2Y2        4.40932E−07
Y4          3.71145E−07
X4Y         −3.32774E−09
X3Y2        1.09835E−09
Y5          2.32182E−09
X6          1.30492E−11
X4Y2        −4.80572E−11
X2Y4        −1.76822E−11
Y6          1.14641E−11

In Table 2, “1.14641E-11,” for instance, means “1.14641×10⁻¹¹.” The same applies to the following embodiments.

As described above, the first optical system 71 is composed of seven lenses, and the second optical system 72 is formed of two reflecting surfaces. The reflecting surface 721 is spherical, and the reflecting surface 722 is a polynomial free-form surface.

The image surface (screen) of the normal image is a plane surface parallel to the rightward and leftward directions of FIG. 7. There is a great difference in angle of incidence to the screen between a lower position (closer to the object) and a higher position (remoter from the object) of the image height. Therefore, the projected image tends to be narrowed downward and distorted. In this embodiment, distortion on the final image surface is corrected by inversely setting the distortion of the intermediate image.

FIG. 11 shows the state of image distortion on the final image surface. FIG. 11 shows the state of image distortion when an image displayed on a liquid crystal panel of approximately 0.9 inch diagonal in size is magnified to approximately 60 inches and projected. As shown in FIG. 11, grid images can be formed at approximately equal intervals, and trapezoidal distortion may be suitably corrected. Projection size is 1200×900 mm, magnification is 65× or higher, and distortion is 0.5% or lower, which values are desirable.

Embodiment 2

Embodiment 2 is a specific embodiment of an image projection apparatus and a projection optical system shown in FIG. 12. In FIG. 12, the projection optical system of the image projection apparatus is shown enlarged.

The projection optical system includes a first optical system 81 and a second optical system 82. The first optical system 81 is composed of six lenses 811 through 816. The second optical system 82 is composed of two reflecting surfaces 821 and 822. A diaphragm (not graphically represented) is provided between the lenses 813 and 814.

Like in Embodiment 1, an intermediate image is formed as an inverted image by the first optical system 81 between the reflecting surfaces 821 and 822. The reflecting surface 821, which has positive power and reflects a light beam made incident on the second optical system 82 first, has a rotationally symmetric aspheric figure. The reflecting surface 822 is a polynomial free-form surface. In Embodiment 2, the employment of a rotationally symmetric aspheric figure makes it possible to design with higher latitude.

The data of Embodiment 2 is shown in Table 3.

TABLE 3
Surface	Radius of	Surface	Refractive
No.	Curvature	Separation	Index	Dispersion	Shift	Tilt	Figure
0	∞	10.00
1	∞	30.00	1.516798	64.1983
2	∞	10.00	1
3	−108.87	8.07	1.696802	55.4597	−22.35	3.3	Spherical
4	−42.82	17.13	1				Spherical
5	46.85	11.00	1.696802	55.4597	−12.67	9.3	Spherical
6	83.16	29.98	1				Spherical
7	−157.78	0.85	1.5168	64.1673	−2.20	−38.9	Spherical
8	44.03	1.44	1				Spherical
9	∞	13.72	1
Diaphragm
10	103.77	8.23	1.846663	37.3451	−10.17		Spherical
11	−109.21	80.56	1				Spherical
12	53.50	6.62	1.7433	37.3451	−5.03		Spherical
13	186.49	16.42	1				Spherical
14	−138.57	11.00	1.487489	70.4412	1.51		Spherical
15	51.18	82.19	1				Spherical
16	1000.00	−150.00	1		−64.67	−45.0	Aspheric
17					10.0	25.0	Polynomial
							Free-Form
							Surface

The figure of the rotationally symmetric aspheric surface employed as the 16^th surface is specified by giving k, A, B, and C in the following well known aspheric equation:
Z=c·r²/[1+√{1−(1+k)c²r²}]+Ar⁴+Br⁶+Cr⁸
where Z is an axial depth, c is a paraxial radius of curvature, r is the distance from an optical axis in a direction perpendicular thereto, k is a conic constant, and A, B, and C are higher-order aspheric coefficients. The same applies to the following embodiments.

The aspheric coefficients of the 16^th surface are given in Table 4.

TABLE 4

Conic Constant: k        90.301
4th-order Coefficient: A 4.12759E−08
6th-order Coefficient: B −5.10327E−12
8th-order Coefficient: C 4.43120E−16

The coefficient values of the 17^th surface, which is a polynomial free-form surface, are given in Table 5.

TABLE 5

Coefficient Coefficient Value

X2          6.11879E−03
Y2          4.61411E−03
X2Y         4.16197E−05
Y3          2.53381E−05
X4          −3.53627E−08
X2Y2        2.62702E−07
Y4          1.81518E−07
X4Y         −9.91605E−10
X3Y2        −3.16955E−11
Y5          1.39821E−09
X6          3.21795E−12
X4Y2        −5.09377E−12
X2Y4        −5.59615E−12
Y6          7.45481E−12

Embodiment 3

Embodiment 3 is a specific embodiment of an image projection apparatus and a projection optical system shown in FIG. 13.

The projection optical system includes a first optical system 91 and a second optical system 92. The first optical system 91 is composed of five lenses 911 through 915. The second optical system 92 is formed of two reflecting surfaces 921 and 922. The lens 913 is a doublet. A diaphragm (not graphically represented) is provided between the lenses 913 and 914.

Like in Embodiments 1 and 2, an intermediate image is formed between the reflecting surfaces 921 and 922. The intermediate image is formed as an inverted image by the first optical system 91. The reflecting surface 921, which has positive power and reflects a light beam made incident on the second optical system 92 first, has a spherical figure. The reflecting surface 922 is a polynomial free-form surface.

The data of Embodiment 3 is shown in Table 6.

TABLE 6
Surface	Radius of	Surface	Refractive
No.	Curvature	Separation	Index	Dispersion	Shift	Tilt	Figure
0	∞	10.00				6.5
1	∞	34.30	1.516798	64.1983
2	∞	10.00	1	0
3	−41.55	3.85	1.696802	55.4597	−9.32	2.2	Spherical
4	−30.93	1.50	1	0			Spherical
5	52.03	6.65	1.696802	55.4597	−10.85		Spherical
6	−4072.62	15.54	1	0			Spherical
7	19.64	9.07	1.487489	70.4412	−2.17		Spherical
8	174.75	3.86	1.846663	23.7848			Spherical
9	17.83	7.50	1	0			Spherical
10	∞	2.74	1	0
Diaphragm
11	36.07	3.07	1.834001	37.3451	−1.32		Spherical
12	−177.66	8.26	1	0			Spherical
13	−33.64	0.65	1.487489	70.4412	0.72		Spherical
14	−111.32	50.00	1	0			Spherical
15	3000.00	−140.00	1	0	−46.57	−45.0	Spherical
16			1	0	−64.67	−45.0	Polynomial
							Free-Form
							Surface

As is apparent from Table 6, the first surface of the lens 911 (the 3^rd surface in Table 6) is tilted 2.2 degrees. Meanwhile, the lenses 912 through 915 are not tilted but shifted with respect to the optical axis of the lens 911. The lens 913 of the dioptric system is a doublet, which acts as a group.

The coefficient values of the 16^th surface, which is a polynomial free-form surface, are shown in Table 7.

TABLE 7

Coefficient Coefficient Value

X2          7.45075E+11
Y2          5.93127E+11
X2Y         6.02215E+07
Y3          4.60347E+07
X4          8.71082E+04
X2Y2        4.41306E+05
Y4          4.54450E+05
X4Y         3.01303E+03
X3Y2        1.30460E+03
Y5          3.27666E+03
X6          1.09946E+01
X4Y2        4.38811E+01
X2Y4        1.35219E+01
Y6          1.53991E+00

Embodiment 4

Embodiment 4 has the same optical configuration as Embodiment 3 (FIG. 13), but has different data.

The data of Embodiment 4 is shown in Table 8.

TABLE 8
Surface	Radius of	Surface	Refractive
No.	Curvature	Separation	Index	Dispersion	Shift	Tilt	Figure
0	∞	10.00				6.5
1	∞	34.30	1.516798	64.1983
2	∞	10.00	1	0
3	−43.18	2.85	1.696802	55.4597	−13.74	0.6	Spherical
4	−29.34	1.50	1	0			Spherical
5	61.46	7.25	1.696802	55.4597	−10.00		Spherical
6	−368.49	15.54	1	0			Spherical
7	20.42	8.27	1.487489	70.4412	−1.71		Spherical
8	158.23	3.75	1.846663	23.7848			Spherical
9	18.54	4.76	1	0			Spherical
10	∞	10.32	1	0	5.00
Diaphragm
11	45.42	2.98	1.834001	37.3451	−1.26		Spherical
12	−120.79	7.52	1	0			Spherical
13	−91.19	0.55	1.487489	70.4412	−4.68		Spherical
14	118.18	50.00	1	0			Spherical
15	3000.00	−140.00	1	0	−38.18	−45.0	Spherical
16			1	0	−64.67	−45.0	Polynomial
							Free-Form
							Surface

The coefficient values of the 16^th surface, which is a polynomial free-from surface, are shown in Table 9.

TABLE 9

Coefficient Coefficient Value

X2          7.77494E−03
Y2          6.11413E−03
X2Y         6.97088E−05
Y3          5.28322E−05
X4          −1.09108E−07
X2Y2        5.14945E−07
Y4          5.13271E−07
X4Y         −4.03993E−09
X3Y2        2.45390E−09
Y5          3.04301E−09
X6          9.74181E−12
X4Y2        −7.12811E−11
X2Y4        −2.11334E−11
Y6          1.03287E−11

Embodiment 5

Embodiment 5 has the same optical configuration as Embodiment 3 (FIG. 13), but has different data.

The data of Embodiment 5 is shown in Table 10.

TABLE 10
Surface	Radius of	Surface	Refractive
No.	Curvature	Separation	Index	Dispersion	Shift	Tilt	Figure
0	∞	10.00
1	∞	34.30	1.516798	64.1983
2	∞	10.00	1	0
3	−33.48	0.80	1.696802	55.4597	−9.32	−4.2	Spherical
4	−28.97	1.50	1	0			Spherical
5	116.57	8.85	1.696802	55.4597	−10.85		Spherical
6	−61.93	15.54	1	0			Spherical
7	81.87	9.38	1.487489	70.4412	−2.17		Spherical
8	−36.19	2.11	1.846663	23.7848			Spherical
9	−63.01	14.39	1	0			Spherical
10	∞	10.20	1	0
Diaphragm
11	19.82	7.50	1.834001	37.3451	−1.32		Spherical
12	16.21	21.23	1	0			Spherical
13	−11.77	7.37	1.487489	70.4412	0.72		Spherical
14	−15.25	50.00	1	0			Spherical
15	5000.00	−140.00	1	0	−44.10	−45.0	Spherical
16		700.00	1	0	10.00	27.0	Polynomial
							Free-Form
							Surface

The coefficient values of the 16^th surface, which is a polynomial free-form surface, are shown in Table 11.

TABLE 11

Coefficient Coefficient Value

X2          6.54399E−03
Y2          5.58060E−03
X2Y         5.34322E−05
Y3          4.38966E−05
X4          −6.71655E−08
X2Y2        3.76878E−07
Y4          2.54814E−07
X4Y         −1.63976E−09
X3Y2        1.70448E−09
Y5          −7.72777E−11
X6          1.01693E−11
X4Y2        −3.04830E−11
X2Y4        −1.70416E−11
Y6          −4.26270E−13

As described above, each of Embodiments 1 through 5 includes a positive-power first optical system including at least one dioptric system and a second optical system having positive power as a whole, the second optical system including at least one reflecting surface having power. The first and second optical systems are arranged in the order described from upstream to downstream on the downstream side of an object. An object image is temporarily formed as an intermediate image, and thereafter, is formed as a normal image. With respect to the optical axis of an optical element that is positioned furthest on the object side in the first optical system and has refractive power, one or more of the other optical elements are shifted or tilted. In Embodiments 3 through 5, with respect to the optical axis of the optical element (lens) 911, positioned furthest on the object side in the first optical system and having refractive power, the other optical elements 912 through 915 of the first optical system 91 are not tilted.

In Embodiments 3 and 4, the first optical system 91 is composed of two or more groups. Of the two or more groups, the lens 913 forming a group as a doublet is shifted.

In each of Embodiments 1 through 5, at least one of the reflecting surfaces included in the second optical system is a free-form surface. Of the reflecting surfaces included in the second optical system, the reflecting surface positioned furthest on the side of the position at which the normal image is formed is a free-form surface. Further, in Embodiments 1 through 5, the reflecting surface having positive power and reflecting a light beam made incident on the second optical system first is rotationally symmetric. In Embodiments 1 and 3 through 5, the rotationally symmetric reflecting surface is a spherical reflecting surface.

In each of Embodiments 1 through 5, the first optical system is formed of only a dioptric system, and the dioptric system of the first optical system excludes an aspheric surface figure.

Accordingly, an image projection apparatus is realized by combining an object with the projection optical system of any of Embodiments 1 through 5.

Embodiment 6

Embodiment 6 is a specific embodiment of the projection optical system and the image projection apparatus described with reference to FIGS. 9 and 10.

The data of Embodiment 6 is shown in Table 12.

TABLE 12
Surface	Radius of	Surface	Refractive
No.	Curvature	Separation	Index	Dispersion	Figure	Shift	Tilt
0	∞	22	1		Plane
1	∞	34.3	1.62	59.51	Plane
2	∞	7.1	1		Plane
3	−94	5.06	1.74	44.9	Spherical	8.22	0.264
4	−42.9	0.1	1		Spherical
5	140.1	5.02	1.53	66.01	Spherical
6	−106.3	1.95	1		Spherical
7	54.2	15.69	1.5	69.24	Aspheric
8	−39.6	17.47	1.76	27.59	Spherical
9	41.2	21.71	1		Spherical
10	−79.3	5.74	1.75	34.1	Spherical
11	−33	1.35	1		Spherical
12	∞	93.3	1		Plane
Diaphragm
13	−50.6	25	1.63	57.93	Spherical
14	−159.1	42.86	1		Spherical
15	186.7	25	1.68	31.56	Aspheric
16	96.1	47.89	1.56	63.6	Spherical
17	−553.4	50	1		Spherical
18	212.4	25	1.72	35.45	Spherical
19	157.6	185	1		Polynomial
					Free-Form
					Surface
20	10000	−266	1		Axially	13.68	46
					Symmetric
					Reflecting
					Surface
21	0	780.85	1		Polynomial	77.69	−32.9
					Free-Form
					Surface

The aspheric coefficients of the 7^th and 15^th surfaces are given in Table 13.

	TABLE 13
	Coefficient	7th Surface	15th Surface
	K	0	0.286791
	A	−1.35E−06	−6.00E−09
	B	−1.56E−09	−1.22E−11
	C	2.72E−13	8.92E−16
	D	−4.35E−15	1.40E−20
	E	0.00E+00	4.55E−23
	F	0.00E+00	−1.29E−26
	G	0.00E+00	7.10E−31
	H	0.00E+00	1.73E−34
	I	0.00E+00	−1.83E−38

The coefficient values of the 19^th and 21^st surfaces, which are polynomial free-form surfaces, are given in Table 14.

	TABLE 14
	Coefficient	19th Surface	21st Surface
	X2	−0.00095	0.002876
	Y2	−0.00096	0.00173
	X2Y	−1.94E−07	−1.03E−05
	Y3	−3.80E−07	−5.02E−06
	X4	−1.06E−07	3.07E−09
	X2Y2	−2.11E−07	5.56E−08
	Y4	−1.01E−07	2.42E−08
	X4Y	5.54E−11	1.17E−11
	Y5	−4.67E−11	−1.45E−10
	X6	−7.62E−14	−1.72E−15
	X4Y2	−8.23E−13	−7.17E−13
	X2Y4	−1.19E−16	1.19E−13
	Y6	1.20E−13	6.52E−13

MTF performance and distortion on a screen by the projection optical system of Embodiment 6 are 60% or higher and 2% or lower, respectively, at a frequency of 0.5 c/mm.

In Embodiment 6, the screen onto which a normal image is projected is 60 inches in size. The maximum width of the projection optical system in a direction perpendicular to the screen is 472 mm.

MTF performance at an evaluation frequency of 0.5 c/mm was examined, setting grid lines of ±1.0Y, ±0.5Y, 0.0Y, ±1.0X, ±0.5X, and 0.0X along the X axis (rightward and leftward directions) and the Y axis (upward and downward directions) on the screen as shown in FIG. 14. Table 15 shows the results (MTF values) of the examination.

TABLE 15
		0.0 X	0.5 X	1.0 X
	1.0 Y	76.7%	74.3%	73.4%
	0.5 Y	71.5%	74.7%	75.4%
	0.0 Y	83.1%	82.1%	79.7%
	−0.5 Y	91.2%	85.0%	83.9%
	−0.1 Y	92.6%	71.6%	83.1%

FIG. 15 shows MTF characteristics in the saggital direction (s) and the meridional direction (m) at ±1.0Y and 0.0Y in the range of frequencies of 0 to 0.5 c/mm at X=0.0X. FIG. 16 shows MTF characteristics in the saggital direction (s) and the meridional direction (m) at ±1.0Y and 0.0Y in the range of frequencies of 0 to 0.5 c/mm at X=0.5X. FIG. 17 shows MTF characteristics in the saggital direction (s) and the meridional direction (m) at ±1.0Y and 0.0Y in the range of frequencies of 0 to 0.5 c/mm at X=1.0X. FIGS. 15 through 17 show that Embodiment 6 has good MTF characteristics.

According to the projection optical system of Embodiment 6, a reflection dioptric system includes first and second reflecting mirrors arranged in the order described from upstream to downstream on the downstream side of the transmission dioptric system, and the intermediate image surface of a projected object surface is positioned between the first and second reflecting mirrors. The first reflecting mirror has a negative-power, axially symmetric reflecting surface (the 22^nd surface). The second reflecting mirror has an anamorphic polynomial free-form surface having different vertical and lateral powers (the 23^rd surface). An anamorphic polynomial free-form surface having different vertical and lateral powers (the 19^th surface) is provided in the transmission dioptric system as a part correcting the aspect ratio of the intermediate image surface of a projected object surface.

It is possible to correct the aspect ratio of the intermediate image through the figure of a reflecting mirror in the reflection dioptric system. However, it is desirable that the figure of the reflecting mirror of the reflection dioptric system be determined mainly in terms of distortion correction. Accordingly, it is desirable that the aspect ratio be adjustable beforehand in the transmission dioptric system. Accordingly, it is effective to employ the above-described anamorphic polynomial free-form surface as a part correcting the aspect ratio in the transmission dioptric system.

The number of polynomial free-form surfaces employed in the transmission dioptric system is not limited to one. However, according to Embodiment 6, the employment of only the single polynomial free-form surface (the 19^th surface) in the transmission dioptric system produced sufficient correction effect. The polynomial free-form surface employed in the transmission dioptric system may be positioned close to the projected object surface. However, it is desirable that the polynomial free-form surface be positioned close to the projection surface side in the transmission dioptric system in order to increase the correction effect.

In the transmission dioptric system, an NA (=0.143) on the projected object surface side is greater than an NA (=0.01) on the intermediate image surface side. The magnification of the intermediate image M1 (=1.5) falls within the range of 1 to 5. The magnification of projection (=75×) is 40× or higher. The angle of projection to a projection surface θ (=11 degrees) is 5 degrees or greater.

In the case of forming the transmission dioptric system, its NA on the projected object surface side (hereinafter, an NA1) is determined by the orientation distribution characteristics of an illumination system, while its NA on the intermediate image surface side (hereinafter, an NA2) is changeable by the arrangement and the configuration of the transmission dioptric system. In order to increase magnification of projection, it is effective to increase the power of the reflection dioptric system. This, however, reduces the focal length of the reflection dioptric system on its image or downstream side so that the focal point of light beams is shifted to the reflecting mirror side of the reflection dioptric system. As a result, only a small-size normal image can be formed. That is, magnification is reduced. In order to eliminate this disadvantage, the NA2 of light beams incident on the reflection dioptric system was focused on. As a result, it was determined that making the NA2 smaller than the NA1 had a remarkable effect in increasing the projection optical system magnification.

It was determined that the NA2 might be 0.005 to 0.01 to realize a magnification projection optical system magnifying and projecting the image of a projected object surface of 0.9 inch diagonal size onto a 60-inch screen with a thickness of 500 mm or less. If the NA2 is overly reduced, the overall length of the transmission dioptric system increases. Accordingly, the NA2 is desirably 0.005 to 0.01, considering the downsizing of the entire apparatus.

If the NA2 is set to 0.01 or greater, the transmission dioptric system is made compact. However, an increase in the NA tends to make it difficult to perform distortion correction on the projection screen or ensure magnification performance. The upper limit of the NA2 may be 0.01 or greater for a screen size smaller than 60 inches.

The image projection apparatus of the present invention may be of a front projector type or of a rear projection type with a folding mirror folding back an imaging optical path.

The present invention is not limited to the specifically disclosed embodiments, and variations and modifications may be made without departing from the scope of the present invention.

The present application is based on Japanese priority patent applications No. 2003-029595, filed on Feb. 6, 2003, No. 2003-029602, filed on Feb. 6, 2003, and No. 2003-409304, filed on Dec. 8, 2003, the entire contents of which are hereby incorporated by reference.

Claims

1. A projection optical system for use in an image projection apparatus projecting an image of an object and forming a final image on a plane, the projection optical system comprising:

first and second optical systems arranged so that the first optical system is closer to the object than is the second optical system;

wherein the first optical system includes at least one dioptric system and has positive power,

the second optical system includes at least one reflecting surface having positive power,

the image of the object is formed as an intermediate image in an optical path of the first and second optical systems, and the intermediate image is magnified and projected, and

the second optical system is configured so that an angle of incidence to a projection surface on the plane is greater at a position farthest from the object than at a position closest to the object on the final image.

2. The projection optical system as claimed in claim 1, wherein a difference between the angle of incidence at the position of the final image farthest from the object and the angle of incidence at the position of the final image closest to the object is 53 degrees or more.

3. The projection optical system as claimed in claim 1, wherein the first optical system is configured to distort the intermediate image so as to offset a possible trapezoidal distortion of the final image.

4. The projection optical system as claimed in claim 1, wherein the first optical system is configured to form the intermediate image with a distortion inverse to a possible trapezoidal distortion of the final image.

5. The projection optical system as claimed in claim 1, wherein the first optical system is configured to form the intermediate image distorted inversely relative to the final image narrowed downward and distorted.

6. The projection optical system as claimed in claim 2, wherein the first optical system is configured to distort the intermediate image so as to offset a possible trapezoidal distortion of the final image.

7. The projection optical system as claimed in claim 2, wherein the first optical system is configured to form the intermediate image with a distortion inverse to a possible trapezoidal distortion of the final image.

8. The projection optical system as claimed in claim 2, wherein the first optical system is configured to form the intermediate image distorted inversely relative to the final image narrowed downward and distorted.

9. The projection optical system as claimed in claim 1, wherein the first optical system is configured to tilt the intermediate image with respect to a principal ray of a light beam emitted from a center of a surface of the object surface.

10. A projection optical system for use in an image projection apparatus projecting an image of an object and forming a final image on a plane, the projection optical system comprising:

first and second optical systems arranged so that the first optical system is closer to the object than the second optical system;

wherein the first optical system includes at least one dioptric system and has positive power,

the second optical system includes at least one reflecting surface having positive power,

the image of the object is formed as an intermediate image in an optical path of the first and second optical systems, and the intermediate image is magnified and projected, and

the second optical system is configured so that an optical path from the reflecting surface having the positive power to the final image is longer at a position thereof farthest from the object than at a position thereof closest to the object.

11. The projection optical system as claimed in claim 10, wherein a difference between the optical path from the reflecting surface having the positive power to the position of the final image farthest from the object and the optical path from the reflecting surface having the positive power to the position of the final image closest to the object is 663 mm or more.

12. The projection optical system as claimed in claim 10, wherein the first optical system is configured to distort the intermediate image so as to offset a possible trapezoidal distortion of the final image.

13. The projection optical system as claimed 10, wherein the first optical system is configured to form the intermediate image with a distortion inverse to a possible trapezoidal distortion of the final image.

14. The projection optical system as claimed in claim 10, wherein the first optical system is configured to form the intermediate image distorted inversely relative to the final image narrowed downward and distorted.

15. The projection optical system as claimed in claim 11, wherein the first optical system is configured to distort the intermediate image so as to offset a possible trapezoidal distortion of the final image.

16. The projection optical system as claimed in claim 11, wherein the first optical system is configured to form the intermediate image with a distortion inverse to a possible trapezoidal distortion of the final image.

17. The projection optical system as claimed in claim 11, wherein the first optical system is configured to form the intermediate image distorted inversely relative to the final image narrowed downward and distorted.

18. A projection optical system for use in an image projection apparatus projecting an image of an object and forming a final image on a plane, the projection optical system comprising:

first and second optical systems arranged so that the first optical system is closer to the object than the second optical system;

wherein the first optical system includes at least one dioptric system and has positive power,

the second optical system includes at least one reflecting surface having positive power,

the image of the object is formed as an intermediate image in an optical path of the first and second optical systems, and the intermediate image is magnified and projected, and

the first optical system is configured to form the intermediate image with a distortion inverse to a possible trapezoidal distortion of the final image.

19. The projection optical system as claimed in claim 18, wherein the first optical system is configured to distort the intermediate image so as to offset the possible trapezoidal distortion of the final image.

20. The projection optical system as claimed in claim 18, wherein the first optical system is configured to form the intermediate image distorted inversely relative to the final image narrowed downward and distorted.

21. An image projection apparatus configured to project an image of an object and form a final image on a plane, the image projection apparatus comprising:

an image forming part configured to be illuminated with illumination light and to form the image of the object;

first and second optical systems arranged so that the first optical system is closer to the image forming part than is the second optical system;

wherein the first optical system includes at least one dioptric system and has positive power,

the second optical system includes at least one reflecting surface having positive power,

the image of the object is formed as an intermediate image in an optical path of the first and second optical systems, and the intermediate image is magnified and projected, and

the second optical system is configured so that an angle of incidence to a projection surface on the plane is greater at a position farthest from the object than at a position closest to the object on the final image.

22. An image projection apparatus configured to project an image of an object and form a final image on a plane, the image projection apparatus comprising:

an image forming part configured to be illuminated with illumination light and to form the image of the object;

first and second optical systems arranged so that the first optical system is closer to the image forming part than is the second optical system,

wherein the first optical system includes at least one dioptric system and has positive power,

the second optical system includes at least one reflecting surface having positive power,

the image of the object is formed as an intermediate image in an optical path of the first and second optical systems, and the intermediate image is magnified and projected, and

the second optical system is configured so that an optical path from the reflecting surface having the positive power to the final image is longer at a position thereof farthest from the object than at a position thereof closest to the object.

23. An image projection apparatus configured to project an image of an object and form a final image on a plane, the image projection apparatus comprising:

an image forming part configured to be illuminated with illumination light and to form the image of the object;

first and second optical systems arranged so that the first optical system is closer to the image forming part than is the second optical system,

wherein the first optical system includes at least one dioptric system and has positive power,

the second optical system includes at least one reflecting surface having positive power,

the image of the object is formed as an intermediate image in an optical path of the first and second optical systems, and the intermediate image is magnified and projected, and

the first optical system is configured to form the intermediate image with a distortion inverse to a possible trapezoidal distortion of the final image.

24. An image projection apparatus, comprising:

a liquid crystal panel or a digital micromirror device (DMD);

a plurality of lenses; and

a concave mirror;

wherein a picture formed in the liquid crystal panel or the DMD exits from the plurality of lenses to be incident on and reflected from the concave mirror to be magnified and projected,

an intermediate image is formed between the liquid crystal panel or the DMD and the concave mirror,

the intermediate image is tilted and curved with respect to a principal ray of a light beam emitted from a center of the liquid crystal panel or the DMD,

the liquid crystal panel or the DMD is disposed shifted in a plane perpendicular to an optical axis of the plurality of lenses,

an optical path length between the concave mirror and a projected image of the picture increases as an angle between the light beam incident on the concave mirror and the light beam reflected from the concave mirror increases,

the light beam from concave mirror converges, and

the light beam reflected from the concave mirror crosses the optical axis of the plurality of lenses.

25. The image projection apparatus according to claim 24, wherein the plurality of lenses and the concave mirror are shifted in a same direction relative to the liquid crystal panel or the DMD.

26. The image projection apparatus according to claim 25, wherein a perpendicular of a surface of the liquid crystal panel or the DMD is parallel to the optical axis of the plurality of lenses.