Projection optical system and image projection device for projecting an image light flux onto a projection surface

Abstract

A projection optical system is an optical system for projecting an image light flux formed in an image display element onto a projection surface, and includes a transmission optical system and a reflection optical system. The transmission optical system is located on an emission surface side of the image display element and includes a stop and a plurality of lenses. The reflection optical system includes a positive-power first mirror and a second mirror. Conditional Expression (1) is satisfied
0<TL/ft<10(1)
where ft is a focal length of the transmission optical system, and TL is a distance parallel to an optical axis of the transmission optical system from a position where the first mirror reflects a principal ray of the image light flux, to the image display element.

Claims

1. A projection optical system for projecting an image light flux formed in an image display element onto a projection surface, the projection optical system comprising: a transmission optical system that is located on an emission surface side of the image display element and includes a stop and a plurality of lenses; and a reflection optical system that includes a positive-power first mirror and a second mirror, the first mirror reflecting light output from the transmission optical system, the second mirror reflecting reflected light of the first mirror toward the projection surface, wherein Conditional Expression (1) is satisfied
0<TL/ft<5.0(1) where ft is a focal length of the transmission optical system, and TL is a distance parallel to an optical axis of the transmission optical system from a position where the first mirror reflects a principal ray of the image light flux, which passes through a center in a long-side direction of the image display element and is projected onto the projection surface closest to a projection optical system side, to the image display element wherein Conditional Expression (4) is satisfied
0<T2/T1<5(4) where T1 is a total of a distance from a position of a principal ray of the image light flux reflected by the first mirror farthest from the optical axis in a short-side direction of the image display element to the optical axis and a distance from a position of a principal ray of the image light flux reflected by the second mirror farthest from the optical axis in the short-side direction of the image display element to the optical axis, and T2 is an optical path length of the principal ray from a lens located closest to the projection surface among the plurality of the lenses, to the first mirror.

2. The projection optical system according to claim 1, wherein Conditional Expression (5) is satisfied
0<T2/ft<5(5).

3. The projection optical system according to claim 1, wherein Conditional Expression (6) is satisfied
0<T1/ft<3(6).

4. The projection optical system according to claim 1, wherein Conditional Expression (7) is satisfied
0.005<Tr(T1/ft)<1(7) where Tr is a throw ratio of the projection optical system.

5. The projection optical system according to claim 1, wherein Conditional Expression (8) is satisfied
0.1<fmmax/ft<10(8) where fmmax is a maximum focal length on a surface of the first mirror.

6. The projection optical system according to claim 1, wherein the projection surface has a curvature.

7. An image projection device comprising: the projection optical system according to claim 1; and the image display element.

8. A projection optical system for projecting an image light flux formed in an image display element onto a projection surface, the projection optical system comprising: a transmission optical system that is located on an emission surface side of the image display element and includes a stop and a plurality of lenses; and a reflection optical system that includes a positive-power first mirror and a second mirror, the first mirror reflecting light output from the transmission optical system, the second mirror reflecting reflected light of the first mirror toward the projection surface, wherein Conditional Expression (1) is satisfied
0<TL/ft<5.0(1) where ft is a focal length of the transmission optical system, and TL is a distance parallel to an optical axis of the transmission optical system from a position where the first mirror reflects a principal ray of the image light flux, which passes through a center in a long-side direction of the image display element and is projected onto the projection surface closest to a projection optical system side, to the image display element, wherein Conditional Expression (9) is satisfied
0.001<fmmin/ft0.1(9) where fmmin is a minimum focal length on a surface of the first mirror.

9. The projection optical system according to claim 8, wherein Conditional Expression (8) is satisfied
0.1<fmmax/ft<10(8) where fmmax is a maximum focal length on a surface of the first mirror.

10. The projection optical system according to claim 8, wherein the projection surface has a curvature.

11. An image projection device comprising: the projection optical system according to claim 8; and the image display element.

Description

BRIEF DESCRIPTION OF DRAWINGS

(1) FIG. 1 is a diagram illustrating an image projection device according to the present disclosure.

(2) FIG. 2 is an enlarged diagram illustrating the image projection device of the present disclosure.

(3) FIG. 3 is a diagram illustrating a transmission optical system of a numerical example 1.

(4) FIG. 4 is a diagram illustrating a transmission optical system of a numerical example 2.

(5) FIG. 5 is a diagram illustrating a transmission optical system of a numerical example 3.

DESCRIPTION OF EMBODIMENT

(6) Hereinafter, an exemplary embodiment will be described in detail with reference to the drawings as appropriate. However, in some cases, detailed description more than necessary may be omitted. For example, in some cases, detailed description of well-known matters or repeated description of substantially the same configuration may be omitted. This is to avoid the following description from being unnecessarily redundant, and to facilitate understanding of those skilled in the art.

(7) Note that the attached drawings and the following description are provided for those skilled in the art to fully understand the present disclosure, and are not intended to limit the subject matter as described in the appended claims.

Exemplary Embodiment

(8) Image projection device 10 according to the present disclosure will be described below with reference to FIGS. 1 to 5.

(9) FIG. 1 is a sectional diagram illustrating image projection device 10 of the present disclosure. Image projection device 10 includes projection optical system 100, image display element 130, and transmission element 140. Image projection device 10 projects an image formed by image display element 130 onto screen SC located in a non-confronting direction (oblique direction). As used herein, the non-confronting direction means the case that a direction of a normal line at a point of screen SC that a center of the projection image reaches is not matched with a direction of an optical path of the center of the image in a beam output from a final surface of projection optical system 100. In image projection device 10 of the present disclosure, a line connecting a center of stop A of transmission optical system 110 and a center of a lens located closest to a projection side is set to optical axis AZ. However, optical axis AZ may be set to an axis shared by most lens centers, or may be set while biased in a plane including an optical path (an optical path of a principal ray from the center of the image display panel to the center of a magnified image on screen SC in an optical path from image display element 130 to screen SC) of the output light with respect to the position of the image display panel.

(10) Image projection device 10 projects the image onto a region including a point at which an extended line of optical axis AZ intersects screen SC. In the case that image projection device 10 includes a reflection surface such as a prism and a mirror in transmission optical system 110, image projection device 10 projects the image onto the region including the point at which the extended line of the optical axis of the optical system intersects screen SC after the beam is reflected by the reflection surface.

(11) Image projection device 10 of the present disclosure projects the image onto screen SC having a curvature.

(12) FIG. 2 is an enlarged diagram illustrating a section of image projection device 10 of the present disclosure. Projection optical system 100 includes transmission optical system 110 having positive power as a whole and reflection optical system 120 having positive power as a whole.

(13) Transmission optical system 110 includes first lens group G1 having positive power, second lens group G2 having positive power, third lens group G3 having positive power, and fourth lens group G4 having positive power in the order from image display element 130 to screen SC, and includes prism PB between image display element 130 and first lens group G1.

(14) First lens group G1 includes one biconvex lens.

(15) Second lens group G2 includes four lens elements.

(16) Third lens group G3 includes aperture stop A. Third lens group G3 includes five lens elements on the projection side of aperture stop A, and has positive power as a whole.

(17) In transmission optical system 110, fourth lens group G4 located closest to the side of screen SC includes a positive-meniscus lens convex to the projection side, a biconcave lens, and a cemented lens of a negative lens and a positive lens in the order from screen SC to image display element 130. In fourth lens group G4, the positive-meniscus lens convex to the side of image display element 130 is disposed. In fourth lens group G4 closest to the projection side, the positive-meniscus-shape lens closest to the projection side and the positive-meniscus-shape cemented lens are disposed such that concave surfaces face each other.

(18) Fourth lens group G4 includes the lens closest to the projection side, which has a shape having a high thickness deviation ratio in transmission optical system 110, on the side closest to screen SC. This enables an increase of a difference in refractive power between the center and the periphery of the transmitted light flux, so that it is effective in correcting a field curvature.

(19) A lens adjacent to the side of image display element 130 of a lens closest to the projection side has a biconcave shape. Preferably at least one of the biconcave-shape lenses has an aspherical shape. Specifically, the lens has a shape in which a curvature decreases toward a radial direction from the center. That is, the power in a peripheral portion of the lens is smaller than that in a central portion of the lens.

(20) In projection optical system 100, focusing is achieved using two lens groups, namely, second lens group G2 and fourth lens group G4. Fourth lens group G4 includes at least one surface having an aspherical shape, and suppresses image distortion or degradation of resolution, which is generated during the focusing. Therefore, good optical performance is satisfied even if a projection length varies.

(21) An intermediate image is formed between transmission optical system 110 and screen SC. Therefore, since a conjugate position of the light beam output from transmission optical system 110 and first mirror 121 located on the side closest to screen SC is lengthened, an angle of the light beam incident on first mirror 121 decreases, and it is advantageous for the downsizing of the reflection optical system.

(22) Reflection optical system 120 reflects a light flux output from transmission optical system 110, and projects the light flux onto screen SC. Reflection optical system 120 includes two mirrors, that is, first mirror 121 and second mirror 122. The reflection surface of first mirror 121 has a concave free-form surface shape, and has positive power as a whole. Reflection optical system 120 is not limited to the two mirrors, but may include at least one mirror.

(23) Image display element 130 forms the image, which is projected onto screen SC, based on an image signal. Spatial modulation elements such as a DMD (Digital Micromirror Device) and a transmission or reflection type liquid crystal panel can be used as the image display element. Image display element 130 of the present disclosure has a rectangular shape in which an X-axis direction (a direction perpendicular to a paper plane) in FIG. 2 is a long side while a Y-axis direction is a short side.

(24) Transmission element 140 is disposed between reflection optical system 120 and screen SC. The light flux reflected by reflection optical system 120 is transmitted through transmission element 140, and projected onto screen SC. Transmission element 140 is formed into a toroidal shape having surfaces of different curvatures in directions corresponding to the long-side direction and the short-side direction of image display element 130, and has a shape convex to the side of screen SC. That is, the curvature in the X-axis direction (the direction perpendicular to the paper plane in FIG. 2) corresponding to the long-side direction of image display element 130 in the incident surface of transmission element 140 is larger than the curvature in the Y-axis direction corresponding to the short-side direction.

(25) Desirably, in reflection optical system 120, a free-form surface shape is provided in first mirror 121 on the side of image display element 130. When the free-form surface having the positive power is disposed in first mirror 121, a height of the light beam incident on second mirror 122 can be suppressed while the image distortion is corrected, so that it is advantageous for the downsizing.

(26) A distance between fourteenth lens L14 disposed on the side closest to screen SC and first mirror 121 having a reflection surface of a free-form surface is longer than a distance between first mirror 121 and second mirror 122. Therefore, the distance between first mirror 121 and second mirror 122 can be shortened, and a low profile in the Y-axis direction of projection optical system 100 can be achieved.

(27) An angle formed between optical axis AZ and a line connecting a reflection position of second mirror 122 farthest from optical axis AZ of transmission optical system 110 and a closest outermost shell of fourteenth lens L14 is smaller than an angle formed between optical axis AZ and a line connecting an outermost reflection position of first mirror 121 and a closest outermost shell of first lens L1.

(28) A preferable condition satisfied by the projection optical system of the exemplary embodiment will be described below. A plurality of conditions are defined with respect to the projection optical systems of the exemplary embodiment, and the projection optical system satisfying the plurality of conditions is most preferable. However, the projection optical system can also obtain the corresponding effect by satisfying the individual condition.

(29) FIG. 3 is a diagram illustrating a transmission optical system of a numerical example 1. FIG. 4 is a diagram illustrating a transmission optical system of a numerical example 2. FIG. 5 is a diagram illustrating a transmission optical system of a numerical example 3.

(30) Preferably the projection optical system of the present disclosure satisfies Conditional Expression (1).
0<TL/ft<10(1)
where

(31) ft is a focal length of the transmission optical system, and

(32) TL is a distance parallel to optical axis AZ from a position where the first mirror reflects a principal ray of the light flux, which passes through the center in the long-side direction of the image display element and is projected onto the projection surface closest to the projection device side, to the image display element.

(33) Conditional Expression (1) defines suitable ranges of the focal length and a total length of transmission optical system 110. The compact projection optical system in which the image distortion is reduced can be obtained when Conditional Expression (1) is satisfied. When Conditional Expression (1) exceeds an upper limit, the total length relative to the transmission optical system is lengthened, and the transmission optical system 110 is hardly downsized. On the other hand, when Conditional Expression (1) is below a lower limit, the total length relative to the transmission optical system is excessively shortened, and generation of various errors is hardly suppressed.

(34) The above effect can further be obtained by satisfying Conditional Expression (1).
0.05<TL/ft<7.5(1)

(35) The above effect can further be obtained by satisfying Conditional Expression (1).
0.07<TL/ft<5.0(1)

(36) Preferably the projection optical system of the present disclosure satisfies Conditional Expression (2).
0.1<ft/fmx<60(2)
where

(37) fmx is a focal length in the X-direction (a long-side direction of the image) at the position of the first mirror that reflects the principal ray of the light flux, which passes through the center in the long-side direction of the image display element and is projected onto the projection surface closest to the projection device side.

(38) Conditional Expression (2) defines a suitable range of the shape of the first mirror. When Conditional Expression (2) is below the lower limit, the distortion of the projection image is increased in the long-side direction of the image display element. On the other hand, when Conditional Expression (2) exceeds the upper limit, it is disadvantageous for the downsizing because the distance between the first mirror and the second mirror is excessively increased, and a coma aberration and the field curvature are generated in the transmission optical system because the power of the transmission optical system is excessively increased.

(39) Assuming that dz/dx is a change (inclination) in sag amount in the X-axis direction of the reflection surface of first mirror 121, and that d.sup.2z/dx.sup.2 is a change in inclination in the X-axis direction of first mirror 121, focal length fmx in the X-axis direction at the position of first mirror 121 that reflects the principal ray of the light flux, which passes through the center in the long-side direction of image display element 130 and is projected onto the projection surface closest to the image projection device side, can be given by fmx=1/(2(d.sup.2z/dx.sup.2)).

(40) The above effect can further be obtained by satisfying Conditional Expression (2).
3<ft/fmx<50(2)

(41) The above effect can further be obtained by satisfying Conditional Expression (2).
5<ft/fmx<40(2)

(42) Preferably the projection optical system of the present disclosure satisfies Conditional Expression (3).
0.1<ft/fmy<60(3)
where

(43) fmy is a focal length in the Y-direction (a short direction of the image) at the position of the first mirror that reflects the principal ray of the light flux, which passes through the center of the long side of the image display element and is projected onto the projection surface closest to the image projection device side.

(44) Conditional Expression (3) defines a suitable range of the shape of first mirror 121. When Conditional Expression (3) is below the lower limit, the image distortion is increased in the short direction of the image display element. On the other hand, when Conditional Expression (3) exceeds the upper limit, it is disadvantageous for the downsizing because the distance between first mirror 121 and second mirror 122 is excessively increased, and the coma aberration and the field curvature are generated in the transmission optical system because the power of the transmission optical system is excessively increased.

(45) Assuming that dz/dy is a change (inclination) in sag amount in the Y-axis direction of the surface of the first mirror, and that d.sup.2z/dy.sup.2 is a change in inclination in the Y-axis direction of the first mirror, focal length fmy in the Y-axis direction at the position of first mirror 121 that reflects the principal ray of the light flux, which passes through the center section in the long-side direction of image display element 130 and is projected onto the projection surface closest to the image projection device side, can be given by fmy=1/(2(d.sup.2z/dy.sup.2)).

(46) The above effect can further be obtained by satisfying Conditional Expression (3).
3<ft/fmy<50(3)

(47) The above effect can further be obtained by satisfying Conditional Expression (3).
5<ft/fmy<45(3)

(48) Preferably the projection optical system of the present disclosure satisfies Conditional Expression (4).
0<T2/T1<5(4)
where

(49) T1 is a total of a distance from a position farthest from the optical axis in the short direction of the image in the light flux reflected by the first mirror to the optical axis and a distance from a position farthest from the optical axis in the short direction of the image in the light flux reflected by the second mirror to the optical axis, and

(50) T2 is an optical path length from a lens located closest to the projection surface to the first mirror in the light beam, which passes through the center in the long-side direction of the image display element and is projected onto the projection surface closest to the image projection device side.

(51) Conditional Expression (4) defines a suitable range of a size of a projection area caused by a distance between the transmission optical system and the reflection optical system and a size of the light flux reflected by the second mirror. When Conditional Expression (4) is below the lower limit, the distance between the lens closest to the projection side and first mirror 121 is decreased, the intermediate image having good aberration performance cannot be formed, and the field curvature cannot properly be corrected in the projection surface. On the other hand, when Conditional Expression (4) exceeds the upper limit, the distance between the lens closest to the projection side and first mirror 121 is increased, the light beam incident on reflection optical system 120 is excessively spread, and the whole size of projection optical system 100 is increased.

(52) The above effect can further be obtained by satisfying Conditional Expression (4).
0.2<T2/T1<4(4)

(53) The above effect can further be obtained by satisfying Conditional Expression (4).
0.4<T2/T1<2(4)

(54) Preferably the projection optical system of the present disclosure satisfies Conditional Expression (5).
0<T2/ft<5(5)

(55) Conditional Expression (5) defines suitable ranges of the focal length of the transmission optical system and the distance between transmission optical system and the reflection optical system. When Conditional Expression (5) is below the lower limit, the distance between the lens closest to the projection side and the first mirror is decreased relative to the transmission optical system to hardly form the intermediate image in which the image without distortion can be projected onto the screen. On the other hand, when Conditional Expression (5) exceeds the upper limit, the distance between the lens closest to the projection side and first mirror is increased, the spread of the light beam incident on the reflection optical system is increased, and the whole size of the optical system is increased.

(56) The above effect can further be obtained by satisfying Conditional Expression (5).
0.05<T2/ft<3(5)

(57) The above effect can further be obtained by satisfying Conditional Expression (5).
0.10<T2/ft<2(5)

(58) Preferably the projection optical system of the present disclosure satisfies Conditional Expression (6).
0<T1/ft<3(6)

(59) Conditional Expression (6) defines a suitable range of the size of the projection optical system from transmission optical system 110 and second mirror 122. When Conditional Expression (6) is below the lower limit, the size of second mirror 122 is decreased, but the projection area cannot be enlarged. On the other hand, when Conditional Expression (6) exceeds the upper limit, the size of second mirror 122 is excessively increased, and the size of projection optical system 100 is increased in the direction of the distance (height) from optical axis AZ.

(60) The above effect can further be obtained by satisfying Conditional Expression (6).
0.05<T1/ft<3(6)

(61) The above effect can further be obtained by satisfying Conditional Expression (6).
0.10<T1/ft<2(6)

(62) Preferably the projection optical system of the present disclosure satisfies Conditional Expression (7).
0.005<Tr(T1/ft)<1(7)
where

(63) Tr is a throw ratio of the projection optical system.

(64) Conditional Expression (7) defines suitable ranges of the throw ratio of projection optical system 100 and the size of reflection optical system 120. As used herein, the throw ratio means a value in which a projection distance of the projection optical system is divided by the size in a lengthwise direction of the image projected onto screen SC. The projection distance means a distance from an upper end of second mirror 122 to screen SC. When Conditional Expression (7) is below the lower limit, it is difficult to properly correct a distortion aberration in the projection surface. On the other hand, when Conditional Expression (7) exceeds the upper limit, an exit pupil position of transmission optical system 110 comes close to the side of reflection optical system 120, and the angle of the light beam incident on reflection optical system 120 from transmission optical system 110 is spread, so that it is not suitable for the low profile of projection optical system 100.

(65) The above effect can further be obtained by satisfying Conditional Expression (7).
0.010<Tr(T1/ft)<0.50(7)

(66) The above effect can further be obtained by satisfying Conditional Expression (7).
0.020<Tr(T1/ft)<0.30(7)

(67) Preferably the projection optical system of the present disclosure satisfies Conditional Expression (8).
0.1<fmmax/ft<10(8)
where

(68) fmmax is a maximum focal length on a surface of the first mirror.

(69) Conditional Expression (8) defines a suitable range of a relationship between the maximum focal length in the surface of first mirror 121 and the focal length of transmission optical system 110. When Conditional Expression (8) is below the lower limit, it is difficult to favorably correct astigmatism near the center on the image projection device side in the projection surface. On the other hand, when Conditional Expression (8) exceeds the upper limit, the distortion is hardly corrected in the peripheral region of the projection surface.

(70) At the position of first mirror 121 that reflects the light beam output from image display element 130, fmmax means the maximum focal length in fmx=1/(2(d.sup.2z/dx.sup.2)) and fmy=1/(2(d.sup.2z/dy.sup.2)).

(71) The above effect can further be obtained by satisfying Conditional Expression (8).
0.130<fmmax/ft<5(8)

(72) The above effect can further be obtained by satisfying Conditional Expression (8).
0.160<fmmax/ft<3(8)

(73) Preferably the projection optical system of the exemplary embodiment satisfies Conditional Expression (9).
0.001<fmmin/ft0.1(9)
where

(74) fmmin is a minimum focal length at the position where each light beam is reflected by the reflection surface of the first mirror.

(75) Conditional Expression (9) defines a suitable range of the minimum focal length on the first mirror surface in focal lengths that are obtained from the positions where the light beams output from the image display element are reflected by the first mirror reflection surface. When Conditional Expression (9) is below the lower limit, it is difficult to properly correct the astigmatism near the center on the image projection device side in the projection surface. On the other hand, when Conditional Expression (9) exceeds the upper limit, it is difficult to properly correct the distortion near the center of the projection surface.

(76) At the position of the first mirror that reflects each light beam output from the image display element, fmmin means the minimum focal length in fmx=1/(2(d.sup.2z/dx.sup.2)) and fmy=1/(2(d.sup.2z/dy.sup.2)).

(77) The above effect can further be obtained by satisfying Conditional Expression (9).
0.010<fmmin/ft0.09(9)

(78) The above effect can further be obtained by satisfying Conditional Expression (9).
0.015<fmmin/ft0.08(9)

(79) Table 1 shows a corresponding value of each conditional expression, which is obtained for projection optical systems of numerical examples 1 to 3.

(80) (Corresponding Value of Conditional Expression)

(81) TABLE-US-00001 TABLE 1 NUMERICAL NUMERICAL NUMERICAL EXAMPLE 1 EXAMPLE 2 EXAMPLE 3 TL 223.936 223.778 221.598 fmx 6.706 6.910 8.080 fmy 6.147 6.320 6.751 T1 44.188 44.191 41.612 T2 49.368 50.323 51.441 fmmax 49.942 50.105 61.698 fmmin 4.850 5.159 4.396 CONDITIONAL 0.922 1.362 3.422 EXPRESSION (1) CONDITIONAL 36.233 23.775 8.016 EXPRESSION (2) CONDITIONAL 39.524 25.995 9.593 EXPRESSION (3) CONDITIONAL 1.117 1.139 1.236 EXPRESSION (4) CONDITIONAL 0.203 0.306 0.794 EXPRESSION (5) CONDITIONAL 0.182 0.269 0.643 EXPRESSION (6) CONDITIONAL 0.032 0.047 0.118 EXPRESSION (7) CONDITIONAL 0.206 0.305 0.953 EXPRESSION (8) CONDITIONAL 0.020 0.031 0.068 EXPRESSION (9)

(82) Numerical examples in which the projection optical system of the exemplary embodiment is specifically implemented will be described below. In each numerical example, a unit of a length is (mm), and a unit of an angle of view is (degrees) in Tables. In each numerical example, r is a curvature radius, d is an interplanar spacing, nd is a refractive index to the d line, and vd is an Abbe number to the d line. In each numerical example, a surface denoted by an asterisk is an aspherical surface or a free-form surface, and an aspherical shape is defined by the following equation.

(83) $\begin{array}{cc}z=\frac{{\mathrm{cr}}^2}{1+\sqrt{1-\left(1+k\right)c^2{r}^2}}+{\mathrm{Ar}}^4+{\mathrm{Br}}^6+{\mathrm{Cr}}^8+{\mathrm{Dr}}^{10}+{\mathrm{Er}}^{12}+{\mathrm{Fr}}^{14}+{\mathrm{Gr}}^{16}& \left[\mathrm{Mathematical}\mathrm{Formula}1\right]\end{array}$
Where

(84) z is a sag amount of a surface parallel to a z-axis,

(85) r is a radial distance (=(x.sup.2+y.sup.2)),

(86) c is a curvature at a surface vertex, and

(87) k is a conic constant.

(88) Only an aspherical coefficient except for zero is written in addition to a conic constant k. In the lens group data, a lens configuration length is a distance from a first surface to a final surface, a front-side principal point position is a distance from the first surface, and a rear-side principal point position is a distance from the first surface.

(89) A free-form surface shape is defined by the following equation using a local rectangular coordinate system (x, y, z) in which the surface vertex of the free-form surface is set to an origin.

(90) $\begin{array}{cc}z=\frac{{\mathrm{cr}}^2}{1+\sqrt{1-\left(1+k\right)c^2{r}^2}}+\underset{j=2}{\overset{137}{.Math.}}{C}_j{x}^m{y}^n& \left[\mathrm{Mathematical}\mathrm{Formula}2\right]\\ j=\frac{{\left(m+n\right)}^2+m+3n}{2}+2& \left[\mathrm{Mathematical}\mathrm{Formula}3\right]\end{array}$
Where

(91) z is a sag amount of a surface parallel to a z-axis,

(92) r is a radial distance (=(x.sup.2+y.sup.2)),

(93) c is a curvature at a surface vertex, and

(94) k is a conic constant, and

(95) Cj is a coefficient of a monomial x.sup.my.sup.n.

(96) In the following data, an i-degree member of x and a j-degree member of y, which are a free-form surface coefficient in a polynomial expression, are written as xiyj for convenience. For example, X2Y indicates the free-form surface coefficient of the second-degree member of x and the first-degree member of y in the polynomial expression.

Numerical Example 1

(97) Tables 2 to 7 show specific data of the transmission optical system of the numerical example 1. In the numerical example 1, the throw ratio is 0.175. The projection magnification ranges from 111.79 to 217.09. For a size of the image display element used, a long-side direction is 9.856 mm, and a short-side direction is 6.162 mm.

(98) Table 2 shows surface data of each optical element of example 1.

(99) TABLE-US-00002 TABLE 2 SURFACE NUMBER DMD r (Y-AXIS r (X-AXIS EFFECTIVE ECCENTRICITY SURFACE DIRECTION) DIRECTION) d nd vd DIAMETER Y Tilt REMARK 1 INFINITY INFINITY 1.000 1.230 2 INFINITY INFINITY 15.000 1.51680 64.20 3 INFINITY INFINITY 12.389 4* 18.143 18.143 8.000 1.59349 67.00 5* 1071.887 1071.887 VARIABLE 6 16.711 16.711 0.900 1.95375 32.32 7 9.455 9.455 7.600 1.49700 81.61 8 16.609 16.609 1.500 1.90366 31.31 9 54.556 54.556 0.419 10 30.215 30.215 3.500 1.84666 23.78 11 134.074 134.074 VARIABLE 12 (STOP) INFINITY INFINITY 1.000 9.964 13 141.786 141.786 3.000 1.72825 28.32 14 44.735 44.735 25.696 15 14.911 14.911 0.800 1.71300 53.94 16 26.716 26.716 0.300 17 38.576 38.576 4.000 1.70154 41.15 18 61.536 61.536 1.788 19 53.406 53.406 6.600 1.75211 25.05 20 112.460 112.460 3.966 21 47.828 47.828 1.500 1.59349 67.00 22 174.726 174.726 VARIABLE 23 26.615 26.615 13.000 1.72916 54.67 24 84.826 84.826 1.500 1.92286 20.88 25 50.147 50.147 7.220 26* 56.182 56.182 4.200 1.68893 31.07 27* 32.777 32.777 14.813 28* 341.331 341.331 12.000 1.53775 74.70 29* 24.581 24.581 VARIABLE 30* 50.193 50.193 0.000 46.500 51.300 MIRROR 31 41.380 32 6.560 81.106 18.100 MIRROR 33 INFINITY 113.000 2.000 1.51680 64.20 33.200 ONLY SURFACE IS ECCENTRIC 34 INFINITY 115.000 VARIABLE 33.200 ONLY SURFACE IS ECCENTRIC 35 32.800 SCREEN

(100) Table 3 shows aspherical data.

(101) TABLE-US-00003 TABLE 3 FOURTH FIFTH TWENTY-SIXTH TWENTY-SEVENTH TWENTY-EIGHTH TWENTY-NINTH COEFFICIENT SURFACE SURFACE SURFACE SURFACE SURFACE SURFACE k 0.04170 0.00000 3.27897 0.47070 99.96900 0.09921 A 2.3514E05 2.4582E05 2.6382E06 3.4014E05 5.0467E05 2.2030E05 B 7.7306E08 8.2842E08 2.7334E07 3.2338E08 1.9327E07 6.4774E08 C 4.9027E10 2.8535E09 9.8427E10 5.9025E11 1.4604E09 1.9181E10 D 7.9495E13 6.7999E12 1.0277E12 1.1384E13 2.5041E12 7.8913E14 E 1.8289E14 1.2796E13 7.9603E17 2.7126E16 2.4112E17 3.7685E16 F 3.4998E17 9.4584E16 4.8660E19 5.5053E19 5.1603E19 1.2712E19 G 5.4231E19 1.6552E18 4.1088E22 9.6601E22 3.4709E21 9.1484E22

(102) Table 4 shows free-form surface data.

(103) TABLE-US-00004 TABLE 4 FREE CURVED SURFACE COEFFICIENT DEGREE MIRROR k 0 0.403199863 C5 X2 1.31862E03 C7 Y2 1.88251E02 C9 X2Y 1.84263E04 C11 Y3 6.09179E04 C12 X4 4.19506E06 C14 X2Y2 9.79777E06 C16 Y4 1.49872E05 C18 X4Y 7.94402E07 C20 X2Y3 5.26153E08 C22 Y5 1.50240E07 C23 X6 2.42504E09 C25 X4Y2 6.80124E08 C27 X2Y4 3.82936E09 C29 Y6 9.09269E11 C31 X6Y 5.59445E10 C33 X4Y3 2.82875E09 C35 X2Y5 1.71428E10 C37 Y7 9.87541E12 C38 X8 1.30229E11 C40 X6Y2 4.87074E11 C42 X4Y4 4.24071E11 C44 X2Y6 1.47586E11 C46 Y8 1.72149E14 C48 X8Y 5.54630E13 C50 X6Y3 1.56941E12 C52 X4Y5 4.76512E13 C54 X2Y7 2.67145E13 C56 Y9 1.02923E15 C57 X10 5.04121E14 C59 X8Y2 1.94831E14 C61 X6Y4 4.80831E15 C63 X4Y6 9.97033E15 C65 X2Y8 7.04488E16 C67 Y10 2.75018E17 C69 X10Y1 1.81743E15 C71 X8Y3 2.55302E15 C73 X6Y5 1.35471E15 C75 X4Y7 4.91298E16 C77 X2Y9 2.70828E17 C79 Y11 7.29563E19 C80 X12 1.58722E16 C82 X10Y2 8.81432E17 C84 X8Y4 8.60305E18 C86 X6Y6 2.24810E17 C88 X4Y8 7.08458E18 C90 X2Y10 1.18694E18 C92 Y12 1.75144E20 C94 X12Y1 6.84567E18 C96 X10Y3 4.10754E18 C98 X8Y5 6.51245E19 C100 X6Y7 4.86385E19 C102 X4Y9 2.79283E19 C104 X2Y11 1.51013E20 C106 Y13 1.45912E22 C107 X14 4.90231E20 C109 X12Y2 8.13046E20 C111 X10Y4 1.60892E19 C113 X8Y6 1.24813E20 C115 X6Y8 1.33666E20 C117 X4Y10 7.13494E21 C119 X2Y12 7.56101E22 C121 Y14 6.63380E24 C123 X14Y1 1.36470E21 C125 X12Y3 3.78806E22 C127 X10Y5 1.34835E21 C129 X8Y7 2.80183E22 C131 X6Y9 7.44194E23 C133 X4Y11 4.52702E23 C135 X2Y13 6.08254E24 C137 Y15 7.07601E26

(104) Table 5 shows zoom data.

(105) TABLE-US-00005 TABLE 5 PROJECTION SIZE (INCH) 50 80 100 d5 2.953 2.826 2.787 d11 13.511 13.638 13.677 d22 6.163 6.850 7.050 d29 10.987 10.300 10.100 d34 239.141 377.779 457.905

(106) Table 6 shows single-lens data.

(107) TABLE-US-00006 TABLE 6 LENS FOCAL NUMBER DISTANCE L1 31.01 L2, L3, L4 35.70 L5 29.41 L6 88.59 L7 48.70 L8 137.50 L9 48.98 L10 111.45 L11, L12 88.20 L13 29.48 L14 48.61

(108) Table 7 shows lens group data.

(109) TABLE-US-00007 TABLE 7 LENS FOCAL GROUP DISTANCE G1 31.01 G2 225.44 G3 94.87 G4 124.47

Numerical Example 2

(110) Tables 8 to 13 show specific data of the transmission optical system of the numerical example 2. In the numerical example 2, the throw ratio is 0.176. The projection magnification ranges from 113.23 to 217.59. For a size of the image display element used, a long-side direction is 9.856 mm, and a short-side direction is 6.162 mm.

(111) Table 8 shows the surface data of each optical element of example 2.

(112) TABLE-US-00008 TABLE 8 SURFACE NUMBER DMD r (Y-AXIS r (X-AXIS EFFECTIVE ECCENTRICITY SURFACE DIRECTION) DIRECTION) d nd vd DIAMETER Y Tilt REMARK 1 INFINITY INFINITY 1.000 1.233 2 INFINITY INFINITY 15.000 1.51680 64.20 3 INFINITY INFINITY 11.706 4* 18.222 18.222 8.000 1.59349 67.00 5* 52516.060 52516.060 VARIABLE 6 16.584 16.584 0.900 1.95375 32.32 7 9.520 9.520 7.600 1.49700 81.61 8 16.480 16.480 1.500 1.90366 31.31 9 55.608 55.608 0.400 10 30.315 30.315 3.500 1.84666 23.78 11 133.595 133.595 VARIABLE 12 (STOP) INFINITY INFINITY 1.000 9.778 13 133.400 133.400 3.000 1.72825 28.32 14 45.067 45.067 26.215 15 14.844 14.844 0.800 1.71300 53.94 16 25.738 25.738 0.300 17 38.553 38.553 4.000 1.70154 41.15 18 62.660 62.660 2.749 19 53.548 53.548 6.600 1.75211 25.05 20 122.013 122.013 4.464 21 48.554 48.554 1.500 1.59349 67.00 22 159.561 159.561 VARIABLE 23 26.757 26.757 12.500 1.72916 54.67 24 96.740 96.740 1.500 1.92286 20.88 25 48.336 48.336 7.744 26* 53.360 53.360 4.300 1.68893 31.07 27* 32.603 32.603 14.048 28* 285.848 285.848 12.000 1.53775 74.70 29* 24.449 24.449 VARIABLE 30* 52.242 52.242 0.000 46.490 51.253 MIRROR 31 41.493 32 7.584 81.106 18.066 MIRROR 33 INFINITY 113.000 2.000 1.51680 64.20 33.188 ONLY SURFACE IS ECCENTRIC 34 INFINITY 115.000 VARIABLE 33.188 ONLY SURFACE IS ECCENTRIC 35 32.813 SCREEN

(113) Table 9 shows the aspherical data.

(114) TABLE-US-00009 TABLE 9 FOURTH FIFTH TWENTY-SIXTH TWENTY-SEVENTH TWENTY-EIGHTH TWENTY-NINTH COEFFICIENT SURFACE SURFACE SURFACE SURFACE SURFACE SURFACE k 0.06294 100.00000 2.39088 0.39346 99.88494 0.11200 A 2.0455E05 2.3752E05 2.9453E06 3.4067E05 5.2625E05 2.3049E05 B 4.3767E08 8.4328E08 2.6673E07 3.5932E08 1.8860E07 6.2933E08 C 7.2302E10 2.6662E09 9.8397E10 5.0860E11 1.4665E09 1.8286E10 D 8.0267E13 6.1795E12 1.0393E12 1.2835E13 2.5209E12 7.6061E14 E 2.5214E14 1.2112E13 1.6879E16 2.3411E16 3.6438E17 3.6909E16 F 1.0676E16 7.9574E16 1.9109E19 5.7624E19 4.4936E19 1.1612E19 G 1.1776E18 6.3427E19 6.6748E22 1.0043E21 3.7229E21 9.2240E22

(115) Table 10 shows the free-form surface data.

(116) TABLE-US-00010 TABLE 10 FREE CURVED SURFACE COEFFICIENT DEGREE MIRROR k 0 0.339472892 C5 X2 1.68479E03 C7 Y2 1.81782E02 C9 X2Y 1.88023E04 C11 Y3 5.97931E04 C12 X4 3.28817E06 C14 X2Y2 1.03273E05 C16 Y4 1.49436E05 C18 X4Y 5.81747E07 C20 X2Y3 1.24404E07 C22 Y5 1.61047E07 C23 X6 5.44989E09 C25 X4Y2 3.97207E08 C27 X2Y4 1.64885E09 C29 Y6 5.43000E11 C31 X6Y 6.91988E10 C33 X4Y3 1.35166E09 C35 X2Y5 3.16738E10 C37 Y7 1.46824E11 C38 X8 4.41749E13 C40 X6Y2 3.56100E11 C42 X4Y4 2.35506E11 C44 X2Y6 1.49915E11 C46 Y8 6.48453E15 C48 X8Y 1.26815E14 C50 X6Y3 8.35959E13 C52 X4Y5 1.99455E13 C54 X2Y7 2.68628E13 C56 Y9 1.23732E15 C57 X10 2.21240E16 C59 X8Y2 3.98117E17 C61 X6Y4 7.74138E15 C63 X4Y6 9.51640E16 C65 X2Y8 1.82634E15 C67 Y10 4.47930E18 C69 X10Y1 3.97067E17 C71 X8Y3 1.41368E17 C73 X6Y5 1.18401E17 C75 X4Y7 2.38903E18 C77 X2Y9 6.51385E20 C79 Y11 1.03868E20 C80 X12 9.33180E19 C82 X10Y2 4.17633E19 C84 X8Y4 4.11424E20 C86 X6Y6 4.55254E20 C88 X4Y8 6.65658E20 C90 X2Y10 8.37395E21 C92 Y12 2.32703E22 C94 X12Y1 6.52974E20 C96 X10Y3 1.45458E20 C98 X8Y5 3.68586E21 C100 X6Y7 7.34041E21 C102 X4Y9 2.36112E21 C104 X2Y11 2.96099E22 C106 Y13 2.62847E24 C107 X14 5.01918E22 C109 X12Y2 1.26536E21 C111 X10Y4 2.76492E22 C113 X8Y6 1.86915E23 C115 X6Y8 8.62424E23 C117 X4Y10 2.28183E23 C119 X2Y12 2.00930E24 C121 Y14 3.24178E26 C123 X14Y1 5.18199E23 C125 X12Y3 1.02938E22 C127 X10Y5 5.78942E23 C129 X8Y7 1.54359E23 C131 X6Y9 4.52945E24 C133 X4Y11 1.08800E24 C135 X2Y13 1.45344E25 C137 Y15 9.03177E28

(117) Table 11 shows the zoom data.

(118) TABLE-US-00011 TABLE 11 PROJECTION SIZE (INCH) 50 80 100 d5 3.494 3.363 3.317 d11 12.988 13.119 13.165 d22 4.344 5.096 5.312 d29 12.286 11.535 11.319 d34 239.355 377.930 457.926

(119) Table 12 shows the single-lens data.

(120) TABLE-US-00012 TABLE 12 LENS FOCAL NUMBER DISTANCE L1 30.71 L2, L3, L4 37.11 L5 29.47 L6 92.14 L7 50.74 L8 133.70 L9 50.29 L10 118.19 L11, L12 93.51 L13 28.79 14 48.93

(121) Table 13 shows the lens group data.

(122) TABLE-US-00013 TABLE 13 LENS FOCAL GROUP DISTANCE G1 30.71 G2 191.20 G3 92.40 G4 139.70

Numerical Example 3

(123) Tables 14 to 19 show specific data of the transmission optical system of the numerical example 3. In the numerical example 1, the throw ratio is 0.184. The projection magnification ranges from 112.85 to 217.12. For a size of the image display element used, a long-side direction is 9.856 mm, and a short-side direction is 6.162 mm.

(124) Table 14 shows the surface data of each optical element of example 3.

(125) TABLE-US-00014 TABLE 14 SURFACE NUMBER DMD r (Y-AXIS r (X-AXIS EFFECTIVE ECCENTRICITY SURFACE DIRECTION) DIRECTION) d nd vd DIAMETER Y Tilt REMARK 1 INFINITY INFINITY 1.000 1.237 2 INFINITY INFINITY 15.000 1.51680 64.20 3 INFINITY INFINITY 11.500 4* 18.654 18.654 8.710 1.59349 67.00 5* 234.636 234.636 VARIABLE 6 16.558 16.558 0.700 1.95375 32.32 7 9.942 9.942 9.107 1.49700 81.61 8 17.602 17.602 0.845 1.90366 31.31 9 52.918 52.918 0.593 10 30.721 30.721 4.705 1.84666 23.78 11 122.453 122.453 VARIABLE 12 1041.154 1041.154 0.700 13 117.875 117.875 0.187 1.69895 30.05 14 (STOP) INFINITY INFINITY 3.694 9.043 15 142.236 142.236 6.429 1.69895 30.05 16 40.484 40.484 21.565 17 15.927 15.927 0.707 1.72000 43.90 18 23.327 23.327 0.208 19 39.863 39.863 3.553 1.76182 26.61 20 58.936 58.936 2.697 21 60.058 60.058 6.434 1.68893 31.16 22 93.684 93.684 1.517 23 50.020 50.020 1.136 1.49700 81.61 24 515.019 515.019 VARIABLE 25 25.789 25.789 9.764 1.71300 53.94 26 84.288 84.288 0.700 1.94595 17.98 27 41.071 41.071 9.471 28* 47.124 47.124 3.491 1.72825 28.32 29* 33.877 33.877 13.516 30* 139.739 139.739 11.992 1.53775 74.70 31* 25.313 25.313 VARIABLE 32* 53.831 53.831 0.000 47.487 51.967 MIRROR 33 29.472 34 29.130 45.435 21.817 MIRROR 35 INFINITY 110.000 2.000 1.51680 64.17 30.150 ONLY SURFACE IS ECCENTRIC 36 INFINITY 112.000 VARIABLE 30.150 ONLY SURFACE IS ECCENTRIC 37 29.849 SCREEN

(126) Table 15 shows the aspherical data.

(127) TABLE-US-00015 TABLE 15 FOURTH FIFTH TWENTY-EIGHTH TWENTY-NINTH THIRTIETH THIRTY-FIRST COEFFICIENT SURFACE SURFACE SURFACE SURFACE SURFACE SURFACE k 0.11911 100.00000 1.49000 0.03343 100.00000 0.06990 A 1.7586E05 2.5727E05 5.5464E06 4.0062E05 5.9463E05 1.9718E05 B 6.1241E08 3.7358E08 2.6587E07 3.4773E08 1.8135E07 6.3309E08 C 3.7409E10 1.5158E09 9.6259E10 3.3119E11 1.4903E09 1.4495E10 D 8.4280E13 4.1544E12 1.0211E12 1.1247E13 2.5434E12 1.5352E13 E 1.8394E14 7.3391E14 5.0937E17 2.1517E16 9.2107E17 2.0710E16 F 8.2679E17 5.3758E16 7.6863E20 3.6714E19 4.1431E19 1.2043E19 G 4.8557E20 9.7649E19 1.6635E22 1.0047E21 3.6827E21 3.1127E22

(128) Table 16 shows the free-form surface data.

(129) TABLE-US-00016 TABLE 16 FREE CURVED SURFACE COEFFICIENT DEGREE MIRROR k 0 0.323338401 C5 X2 2.28863E03 C7 Y2 1.80948E02 C9 X2Y 3.37331E04 C11 Y3 6.50665E04 C12 X4 1.38434E06 C14 X2Y2 2.61702E05 C16 Y4 1.73183E05 C18 X4Y 2.36915E07 C20 X2Y3 9.73376E07 C22 Y5 2.04905E07 C23 X6 4.62630E09 C25 X4Y2 1.66324E08 C27 X2Y4 2.34240E08 C29 Y6 3.46805E11 C31 X6Y 5.21006E10 C33 X4Y3 6.80128E10 C35 X2Y5 4.69642E10 C37 Y7 2.32373E11 C38 X8 2.35093E12 C40 X6Y2 2.33734E11 C42 X4Y4 1.55433E11 C44 X2Y6 1.02562E11 C46 Y8 8.26876E15 C48 X8Y 1.64238E13 C50 X6Y3 5.33848E13 C52 X4Y5 1.82709E13 C54 X2Y7 1.60166E13 C56 Y9 3.31161E15 C57 X10 4.93041E16 C59 X8Y2 1.95319E15 C61 X6Y4 4.98663E15 C63 X4Y6 9.04991E16 C65 X2Y8 1.10711E15 C67 Y10 2.16733E17 C69 X10Y1 0.00000E+00 C71 X8Y3 0.00000E+00 C73 X6Y5 0.00000E+00 C75 X4Y7 0.00000E+00 C77 X2Y9 0.00000E+00 C79 Y11 0.00000E+00 C80 X12 0.00000E+00 C82 X10Y2 0.00000E+00 C84 X8Y4 0.00000E+00 C86 X6Y6 0.00000E+00 C88 X4Y8 0.00000E+00 C90 X2Y10 0.00000E+00 C92 Y12 0.00000E+00 C94 X12Y1 0.00000E+00 C96 X10Y3 0.00000E+00 C98 X8Y5 0.00000E+00 C100 X6Y7 0.00000E+00 C102 X4Y9 0.00000E+00 C104 X2Y11 0.00000E+00 C106 Y13 0.00000E+00 C107 X14 0.00000E+00 C109 X12Y2 0.00000E+00 C111 X10Y4 0.00000E+00 C113 X8Y6 0.00000E+00 C115 X6Y8 0.00000E+00 C117 X4Y10 0.00000E+00 C119 X2Y12 0.00000E+00 C121 Y14 0.00000E+00 C123 X14Y1 0.00000E+00 C125 X12Y3 0.00000E+00 C127 X10Y5 0.00000E+00 C129 X8Y7 0.00000E+00 C131 X6Y9 0.00000E+00 C133 X4Y11 0.00000E+00 C135 X2Y13 0.00000E+00 C137 Y15 0.00000E+00

(130) Table 17 shows the zoom data.

(131) TABLE-US-00017 TABLE 17 PROJECTION SIZE (INCH) 50 80 100 d5 3.802 3.622 3.556 d11 7.242 7.423 7.489 d24 9.023 9.622 9.782 d31 13.299 12.700 12.541 d36 218.379 369.918 457.250

(132) Table 18 shows the single-lens data.

(133) TABLE-US-00018 TABLE 18 LENS FOCAL NUMBER DISTANCE L1 29.49 L2, L3, L4 42.15 L5 29.42 L6 151.46 L7 78.91 L8 72.64 L9 149.64 L10 54.04 L11 111.56 L12, L13 92.40 L14 26.58 L15 55.45

(134) Table 19 shows the lens group data.

(135) TABLE-US-00019 TABLE 19 LENS FOCAL GROUP DISTANCE G1 29.49 G2 125.35 G3 105.82 G4 275.24

Other Exemplary Embodiments

(136) As described above, the exemplary embodiment has been described as an example of the technique disclosed in the present application. However, the technique in the present disclosure is not limited to the exemplary embodiment, and can also be applied to embodiments in which change, substitution, addition, omission, and the like are performed. A new exemplary embodiment can also be made by a combination of the components described in the above exemplary embodiment.

(137) The above exemplary embodiment is an illustration of the technique of the present disclosure. Therefore, various changes, replacements, additions, or omissions may be made to the exemplary embodiments within the scope of claims or their equivalents.

INDUSTRIAL APPLICABILITY

(138) The present disclosure can be applied to the projection that projects the image displayed on the image display element. Specifically, the present disclosure can be applied to a projector, a head-up display, and the like.

REFERENCE MARKS IN THE DRAWINGS

(139) 10: image projection device 100: projection optical system 110: transmission optical system 120: reflection optical system 121: first mirror 122: second mirror 130: image display element 140: transmission element A: aperture stop PB: prism SC: screen