Patent Publication Number: US-11042082-B2

Title: Projection optical system including movable lens groups, a non-movable lens group, and a curved mirror

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of U.S. application Ser. No. 15/707,565 filed Sep. 18, 2017, which is a continuation of U.S. application Ser. No. 14/971,671 filed Dec. 16, 2015 (now U.S. Pat. No. 9,766,438 issued Sep. 19, 2017), which is a continuation of U.S. application Ser. No. 14/272,838 filed May 8, 2014 (now U.S. Pat. No. 9,261,767 issued Feb. 16, 2016), and claims the benefit of priority under 35 U.S.C. § 119 from Japanese Patent Application No. 2013-105851 filed May 20, 2013, the entire contents of each of which are incorporated herein by reference. 
    
    
     BACKGROUND 
     Technical Field 
     The present invention relates to a projection optical system, and an image display apparatus provided with the projection optical system. 
     Description of the Related Art 
     The image display apparatuses, such as projectors, are usually provided with a projection optical system that magnifies and projects an image on a projection plane such as a screen. Recently, a demand for a projector of which projection distance is extremely short and which can greatly magnify a display size of an image on a screen (can implement a large screen display), that is, a front projection type projector with an ultra-short projection distance has been increased. In addition, a request for miniaturization of the projector has also been increased. 
     SUMMARY 
     Example embodiments of the present invention include a projection optical system including: in order from a reduction side to a magnification side, an image forming unit configured to form an image thereon; a refraction optical system including a plurality of lenses, a first reflecting surface and a second reflecting surface. When, an optical axis shared by the largest number of lenses among optical axes of the plurality of the lenses of the refraction optical system is defined by an optical axis of the projection optical system, when in arrangement where a distance between an intersection of a magnification-side surface of a lens which is arranged to be closest to the magnification side of the refraction optical system and the optical axis and an intersection of the first reflecting surface and the optical axis has a minimum value, the distance between the intersections is denoted by L, when a focal length of the refraction optical system is denoted by f, when a direction parallel to the optical axis is defined by a Z axis direction, when an arrangement direction of the first reflecting surface and the second reflecting surface is defined by a Y axis direction, when a maximum value of a distance between the optical axis and an end portion of the image forming unit in the Y axis direction is denoted by Y max, when in a YZ plane which is a plane parallel to the Y axis direction and the Z axis direction, a maximum value D 1  of a distance between an intersection of a light beam path from the image forming unit and the magnification-side surface of the lens which is arranged to be closest to the magnification side of the refraction optical system and the optical axis, when a sag amount ds 1  which is a sag amount of the magnification-side surface of the lens which is arranged to be closest to the magnification side of the refraction optical system at the D 1  and of which positive direction is defined by the direction from the reduction side in the Z axis toward the magnification side, when a point H of which distance from the optical axis has a maximum value among the intersections of the light beam and the first reflecting surface, when a point F of which distance from the optical axis has a minimum value among the intersections of the light beam and the second reflecting surface, an angle θ1 between a line connecting the H and the F and the optical axis satisfies the condition 1: 0&lt;Y max/f−1/tan θ1; and condition 2: −0.1&lt;(L−D 1 −ds 1 )/(L+D 1 −ds 1 )−1/tan θ1. 
     The above-described projection optical system may be applicable to any desired apparatus such as an image display apparatus. 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
         FIG. 1  is an optical layout diagram illustrating an image display apparatus according to an embodiment of the present invention; 
         FIG. 2  is a plan view illustrating an image forming unit included in the image display apparatus of  FIG. 1  as seen from an optical axis direction; 
         FIG. 3  is an optical layout diagram illustrating an example of a projection optical system included in the image display apparatus of  FIG. 1 ; 
         FIG. 4  is an optical layout diagram illustrating an example of a projection optical system included in the image display apparatus of  FIG. 1 ; 
         FIG. 5  is an optical layout diagram illustrating a moving locus of each of lens units constituting the projection optical system of  FIG. 3  during focusing; 
         FIG. 6  is a diagram illustrating an example of an image on a screen in a long distance projection period of the projection optical system of  FIG. 3 ; 
         FIG. 7A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 6 ; 
         FIG. 7B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 6 ; 
         FIG. 7C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 6 ; 
         FIG. 8  is a diagram illustrating an example of an image on a screen in a reference distance projection period of the projection optical system of  FIG. 3 ; 
         FIG. 9A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 8 ; 
         FIG. 9B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 8 ; 
         FIG. 9C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 8 ; 
         FIG. 10  is a diagram illustrating an example of an image on a screen in a close range projection period of the projection optical system of  FIG. 3 ; 
         FIG. 11A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 10 ; 
         FIG. 11B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 10 ; 
         FIG. 11C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 10 ; 
         FIG. 12  is a spot diagram in the long distance projection period of the projection optical system of  FIG. 3 ; 
         FIG. 13  is a spot diagram in the reference distance projection period of the projection optical system of  FIG. 3 ; 
         FIG. 14  is a spot diagram in the close range projection period of the projection optical system of  FIG. 3 ; 
         FIG. 15  is an optical layout diagram illustrating an image display apparatus according to another embodiment of the present invention; 
         FIG. 16  is an optical layout diagram illustrating a moving locus of each of lens units constituting a projection optical system of  FIG. 15  during focusing; 
         FIG. 17  is a diagram illustrating an example of an image on a screen in a long distance projection period of the projection optical system of  FIG. 15 ; 
         FIG. 18A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 17 ; 
         FIG. 18B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 17 ; 
         FIG. 18C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 17 ; 
         FIG. 19  is a diagram illustrating an example of an image on a screen in a reference distance projection period of the projection optical system of  FIG. 15 ; 
         FIG. 20A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 19 ; 
         FIG. 20B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 19 ; 
         FIG. 20C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 19 ; 
         FIG. 21  is a diagram illustrating an example of an image on a screen in a close range projection period of the projection optical system of  FIG. 15 ; 
         FIG. 22A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 21 ; 
         FIG. 22B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 21 ; 
         FIG. 22C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 21 ; 
         FIG. 23  is a spot diagram in the long distance projection period of the projection optical system of  FIG. 15 ; 
         FIG. 24  is a spot diagram in the reference distance projection period of the projection optical system of  FIG. 15 ; 
         FIG. 25  is a spot diagram in the close range projection period of the projection optical system of  FIG. 15 ; 
         FIG. 26  is an optical layout diagram illustrating an image display apparatus according to still another embodiment of the present invention; 
         FIG. 27  is an optical layout diagram illustrating a moving locus of each of lens units constituting the projection optical system of  FIG. 26  during focusing; 
         FIG. 28  is a diagram illustrating an example of an image on a screen in a long distance projection period of the projection optical system of  FIG. 26 ; 
         FIG. 29A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 28 ; 
         FIG. 29B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 28 ; 
         FIG. 29C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 28 ; 
         FIG. 30  is a diagram illustrating an example of an image on a screen in a reference distance projection period of the projection optical system of  FIG. 26 ; 
         FIG. 31A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 30 ; 
         FIG. 31B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 30 ; 
         FIG. 31C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 30 ; 
         FIG. 32  is a diagram illustrating an example of an image on a screen in a close range projection period of the projection optical system of  FIG. 26 ; 
         FIG. 33A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 32 ; 
         FIG. 33B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 32 ; 
         FIG. 33C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 32 ; 
         FIG. 34  is a spot diagram in the long distance projection period of the projection optical system of  FIG. 26 ; 
         FIG. 35  is a spot diagram in the reference distance projection period of the projection optical system of  FIG. 26 ; 
         FIG. 36  is a spot diagram in the close range projection period of the projection optical system of  FIG. 26 ; 
         FIG. 37  is an optical layout diagram illustrating an image display apparatus according to further still another embodiment of the present invention; 
         FIG. 38  is an optical layout diagram illustrating a moving locus of each of lens units constituting the projection optical system of  FIG. 37  during focusing; 
         FIG. 39  is a diagram illustrating an example of an image on a screen in a long distance projection period of the projection optical system of  FIG. 37 ; 
         FIG. 40A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 39 ; 
         FIG. 40B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 39 ; 
         FIG. 40C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 39 ; 
         FIG. 41  is a diagram illustrating an example of an image on a screen in a reference distance projection period of the projection optical system of  FIG. 37 ; 
         FIG. 42A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 41 ; 
         FIG. 42B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 41 ; 
         FIG. 42C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 41 ; 
         FIG. 43  is a diagram illustrating an example of an image on a screen in a close range projection period of the projection optical system of  FIG. 37 ; 
         FIG. 44A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 43 ; 
         FIG. 44B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 43 ; 
         FIG. 44C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 43 ; 
         FIG. 45  is a spot diagram in the long distance projection period of the projection optical system of  FIG. 37 ; 
         FIG. 46  is a spot diagram in the reference distance projection period of the projection optical system of  FIG. 37 ; 
         FIG. 47  is a spot diagram in the close range projection period of the projection optical system of  FIG. 37 ; 
         FIG. 48  is a diagram illustrating field positions corresponding to angles of view of the image forming unit according to an embodiment of the present invention; 
         FIG. 49  is example mathematical formula representing an aspherical surface and a free-form curved surface; 
         FIG. 50  is a table  1  illustrating examples of optical elements of the projection optical system of  FIG. 3 ; 
         FIG. 51  is a table  2  illustrating examples of lens intervals during focusing in the projection optical system of  FIG. 3 ; 
         FIG. 52  is a table  3  illustrating examples of numerical values of aspherical coefficients in the projection optical system of  FIG. 3 ; 
         FIG. 53  is a table  4  illustrating examples of numerical values of free-form curved surface coefficients in the projection optical system of  FIG. 3 ; 
         FIG. 54  is a table  5  illustrating characteristics of an image forming unit of the projection optical system of  FIG. 3 ; 
         FIG. 55  is a table  6  illustrating examples of position coordinates and rotation angles of mirrors in the refraction optical system of  FIG. 3 ; 
         FIGS. 56A and 56B  are a table  7  illustrating examples of optical elements of the projection optical system of  FIG. 15 ; 
         FIG. 57  is a table  8  illustrating examples of lens intervals during focusing in the projection optical system of  FIG. 15 ; 
         FIG. 58  is a table  9  illustrating examples of numerical values of aspherical coefficients in the projection optical system of  FIG. 15 ; 
         FIG. 59  is a table  10  illustrating examples of numerical values of free-form curved surface coefficients in the projection optical system of  FIG. 15 ; 
         FIG. 60  is a table  11  illustrating characteristics of an image forming unit of the projection optical system of  FIG. 15 ; 
         FIG. 61  is a table  12  illustrating examples of position coordinates and rotation angles of mirrors in the refraction optical system of  FIG. 15 ; 
         FIGS. 62A and 62B  are a table  13  illustrating examples of optical elements of the projection optical system of  FIG. 26 ; 
         FIG. 63  is a table  14  illustrating examples of lens intervals during focusing in the projection optical system of  FIG. 26 ; 
         FIG. 64  is a table  15  illustrating examples of numerical values of aspherical coefficients in the projection optical system of  FIG. 26 ; 
         FIG. 65  is a table  16  illustrating examples of numerical values of free-form curved surface coefficients in the projection optical system of  FIG. 26 ; 
         FIG. 66  is a table  17  illustrating characteristics of an image forming unit of the projection optical system of  FIG. 26 ; 
         FIG. 67  is a table  18  illustrating examples of position coordinates and rotation angles of mirrors in the refraction optical system of  FIG. 26 ; 
         FIGS. 68A and 68B  are a table  19  illustrating examples of optical elements of the projection optical system of  FIG. 37 ; 
         FIG. 69  is a table  20  illustrating examples of lens intervals during focusing in the projection optical system of  FIG. 37 ; 
         FIG. 70  is a table  21  illustrating examples of numerical values of aspherical coefficients in the projection optical system of  FIG. 37 ; 
         FIG. 71  is a table  22  illustrating examples of numerical values of free-form curved surface coefficients in the projection optical system of  FIG. 37 ; 
         FIG. 72  is a table  23  illustrating characteristics of an image forming unit of the projection optical system of  FIG. 37 ; 
         FIG. 73  is a table  24  illustrating examples of position coordinates and rotation angles of mirrors in the refraction optical system of  FIG. 37 ; 
         FIG. 74  is a table  25  illustrating example numerical values representing the condition in each one of the above-described examples; and 
         FIG. 75  is a table  26  illustrating example values representing each condition in each of the above-described examples. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Hereinafter, a projection optical system and an image display apparatus according to embodiments of the present invention will be described with reference to the drawings. In the following, the image display apparatus is provided with the projection optical system, which projects an image formed at an image forming unit on a projection plane. 
     First Embodiment of Image Display Apparatus 
       FIG. 1  is an optical layout diagram illustrating a projector  1 , which is one example of image display apparatus. The projector  1  includes an image forming unit  10 , a parallel plate  40 , a projection optical system  100 , an illumination optical system  20  including a light source that illuminates the image forming unit  10  with illumination light, and other members used for image formation, which may be accommodated in a housing  30 . 
     The image forming unit  10  may be implemented by any device capable of forming a to-be-projected image thereon, such as, a digital micromirror device (DMD), a transmission type liquid crystal panel, a reflection type liquid crystal panel, etc. 
     The parallel plate  40  is a cover glass (seal glass), which is arranged in the vicinity of the image forming unit  10  to protect the image forming unit  10 . 
     The projection optical system  100  includes a refraction optical system  101 , a plane mirror  102  functioning as a first reflecting surface, and a curved mirror  103  functioning as a second reflecting surface. As illustrated in  FIG. 1 , in this example, the plane mirror  102  is arranged such that the normal line of the plane mirror  102  is rotated by 45 degrees from the Z axis toward the Y axis direction on the YZ plane. The curved mirror  103  may be a concave mirror or a free-form curved surface mirror of which reflecting surface has a shape of a free-form curved surface. Details of the projection optical system  100  will be described later. 
     The illumination optical system  20  includes, for example, a rod integrator, a flyeye integrator, or the like in order to efficiently perform uniform illumination on the image forming unit  10 . In addition, the illumination optical system  20  is provided with a light source. As the light source, a white light source such as an ultra-high pressure mercury lamp, a xenon lamp, a halogen lamp, and a light-emitting diode (LED) or a monochromatic light source such as a monochromatic light-emitting LED and a monochromatic light-emitting LD may be used. 
     In the below description, the image forming unit  10  is assumed to be an “image forming unit having no light-emission function” such as a DMD. However, the image forming unit applicable to this embodiment of the present invention is not limited thereto, but a “self-emission type image forming unit having a light-emission function of emitting light on a generated image” may be used. 
     The image forming unit  10  which is a DMD is illuminated with illumination light of the illumination optical system  20  and reflects the illumination light. Image information is formed by the reflected light. In other words, the image information generated by the DMD is a flux of light which is two-dimensionally intensity-modulated. The flux of light becomes a flux of projection light as object light. The image formed on the image forming unit  10  is magnified and projected on a screen. 
     The screen may be arranged to be perpendicular to the image forming unit  10 . In other words, the normal line of an image formation plane of the image forming unit  10  is perpendicular to the normal line of the screen as a projection plane. 
     An intermediate image which is conjugate with the image information formed in the image forming unit  10  is formed by the light beam passing through the refraction optical system  101 . The intermediate image is formed as a spatial image in the side closer to the image forming unit  10  than the plane mirror  102 . In addition, the intermediate image is not necessarily formed as a plane image, but the intermediate image may be formed as a curved image. 
     The image is displayed on the screen by magnifying and projecting the intermediate image by using the curved mirror  103  which is arranged to be closest to the magnification side in the projection optical system  100 . Although the intermediate image has a curvature of field or distortion, the reflecting surface of the curved mirror  103  is configured to have a shape of a free-form curved surface, so that it is possible to correct the curvature of field and the distortion. Accordingly, since a burden of aberration correction on the refraction optical system  101  is reduced, a degree of freedom in the design of the projection optical system  100  is increased, so that it is advantageous to miniaturization. 
     The refraction optical system  101  is configured so that the first lens unit  11  having a group of positive lenses, the plane mirror  102 , and the curved mirror  103  are fixed with respect to the image forming unit  10  during focusing from a long distance side to a close range side. The second lens unit  12  having a group of positive lenses and the third lens unit  13  having a group of negative lenses are moved to the magnification side at one time and, after that, are moved to the image forming unit  10  side. The fourth lens unit  14  having a group of positive lenses, is moved to the magnification side during focusing from a long distance side to a close range side. In other words, the projection optical system  100  can control a curvature of field or distortion aberration at a high accuracy by performing floating focusing. 
     In the refraction optical system  101 , an aspherical lens is arranged in the lens unit which is moved during focusing. With this configuration, the effect of the correction is further improved. 
     The projector  1  illustrated in  FIG. 1  is an image display apparatus according to Example 1 described below.  FIG. 1  is also an optical path diagram illustrating a case of 48-inch projector where the front lens elements provided to the projection optical system  100  are drawn out to the extreme degree. As clarified from  FIG. 1 , a sufficient interval between each lens and each light beam is maintained, so that interference between each light beam and each lens or a lens barrel (not illustrated) can be avoided in this state. 
     Next, the image forming unit included in the projection optical system according to an example embodiment of the present invention will be described.  FIG. 2  is a plan view illustrating the image forming unit  10  according to the embodiment as seen from the image formation plane side. A plane in the image forming unit  10  on which the image information is formed is defined by an image formation plane. In  FIG. 2 , the image forming unit  10  is shifted in the Y axis direction with respect to the optical axis Lx described below. As illustrated in  FIG. 2 , the intersection of the Y direction axis line passing through a center C of the image forming unit  10  and the optical axis Lx is indicated by BO. The intersection of the Y direction axis line passing through the center C of the image forming unit  10  and the end portion of the image forming unit  10  is indicated by L 0 . The maximum value of the distance between the intersection BO and the intersection L 0  is denoted by a symbol “Y max”. The distance Y max is denoted by a maximum angle of view in the Y axis direction. 
     Next, the projection optical system  100  according to the embodiment will be described in more detail.  FIGS. 3 and 4  are optical layout diagrams of the projection optical system  100 . As illustrated in  FIGS. 3 and 4 , the projection optical system  100  includes the image forming unit  10 , the refraction optical system  101 , the plane mirror  102  which is a first reflecting surface, and the curved mirror  103  which is a second reflecting surface. 
     Next, symbols for describing relations between optical elements in the present disclosure will be described with reference to  FIGS. 3 and 4 .  FIGS. 3 and 4  are optical path diagrams illustrating optical paths of light beams projected from the image forming unit  10  to the screen. 
     As illustrated in  FIGS. 3 and 4 , an axis shared by the largest number of lenses among a plurality of the lenses constituting the refraction optical system  101  is defined by the optical axis Lx. The direction parallel to the optical axis Lx is defined by the Z axis direction; and the arrangement direction of the plane mirror  102  and the curved mirror  103  is defined by the Y axis direction. In other words, the direction perpendicular to the optical axis Lx on the plane including the light beam path passing through the center C (refer to  FIG. 2 ) of the image forming unit  10 , the center of a stop (not illustrated) included in the refraction optical system  101 , and the center of the screen (not illustrated) is defined by the Y axis direction. In addition, the direction perpendicular to the Y axis direction and the Z axis direction is defined by the X axis direction. 
       FIGS. 3 and 4  are also cross-sectional views of the projection optical system  100  on the YZ plane.  FIGS. 3 and 4  illustrate only the light beam paths parallel to the YZ plane among the light beams from the image forming unit  10 . In addition, in  FIGS. 3 and 4 , the rotation from the +Z axis direction to the +Y axis direction on the YZ plane is defined by +a rotation. 
     As illustrated in  FIG. 3 , among lens surfaces of the lens which is arranged to be closest to the magnification side among the lenses included in the refraction optical system  101 , the lens surface closer to the magnification side is denoted by “S 1 ”. In focusing, a minimum value of the distance between the intersection of the lens surface S 1  and the optical axis Lx and the intersection of the plane mirror  102  and the optical axis Lx is denoted by “L”. When the L has a minimum value, among the intersections of the plane mirror  102  and the light beam paths parallel to the YZ cross section, the point on the plane mirror  102  of which distance from the optical axis Lx has a maximum value is denoted by “H”. Among the intersections of the curved mirror  103  and the light beam paths parallel to the YZ cross section, the point on the curved mirror  103  of which distance from the optical axis Lx has a minimum value is denoted by “F”. In addition, an angle between the segment connecting the point H and the point F and the optical axis Lx is denoted by θ 1 . 
     Among the intersections of the light beam paths parallel to the YZ cross section and the surface S 1 , the distance between the point on the surface S 1  of which distance from the optical axis Lx has a maximum value and the optical axis Lx is denoted by “D 1 ”. Further, a sag amount of the surface S 1  at the distance D 1  from the top of the lens surface S 1  is denoted by “ds 1 ”. The positive direction of the sag amount ds 1  is defined by the direction from the reduction side to the magnification side. In other words, in this disclosure, the sag amount of the surface S 1  may be referred to as a surface sag indicating the height of the surface S 1 . 
     Among the light beams illustrated in  FIG. 4 , the light beam indicated by the thick solid black line is an upper light beam at the maximum angle of view in the Y axis direction. The distance between the intersection of the upper light beam and the surface S 1  and the optical axis Lx is denoted by “D 2 ”. The sag amount of the surface S 1  at the distance D 2  is denoted by “ds 2 ”. Further, an angle between the light beam emitted from the refraction optical system  101  for the upper light beam and the optical axis Lx is denoted by θ 2 . 
     Among the symbols described above, the meanings of the symbols representing the positional relation between the optical elements constituting the projector  1  are the same in each example described below. 
     Next, specific numerical examples of the projection optical system  100  will be described. First, meanings of symbols used in each example will be described. The meanings of symbols are as follows. 
     f: focal distance of the entire system of the projection optical system  100   
     NA: aperture efficiency 
     ω: half angle of view (deg) 
     R: radius of curvature (paraxial radius of curvature of an aspheric surface) 
     D: surface interval 
     Nd: refractive index 
     νd (Vd): Abbe number 
     K: conic constant of an aspheric surface 
     Ai: i-th order aspherical constant 
     Cj: free-form curved surface coefficient 
     C: reciprocal of paraxial radius of curvature (paraxial curvature) 
     H: height from optical axis 
     K: conic constant 
     A shape of an aspherical surface is represented as an aspherical amount X in the optical axis direction by the Mathematical Formula 1 (Equation 1) illustrated in  FIG. 49  using the paraxial curvature C, the height H from the optical axis, the conic constant K, and the i-th order aspherical constants Ai. 
     The shape of an aspherical surface is specified by applying the paraxial curvature C, the conic constant K, and the aspherical constants Ai to the aforementioned Mathematical Formula 1. 
     A shape of a free-form curved surface is expressed as a free-form curved surface amount X in the optical axis direction by the Mathematical Formula 2 (Equation 2) illustrated in  FIG. 49  using the paraxial curvature C, the height H from an optical axis, the conic constant K, and the free-form curved surface coefficients Cj. 
     Herein, j is represented by the Mathematical Formula 3 (Equation 3) illustrated in  FIG. 49 . 
     The shape of a free-form curved surface is specified by applying the paraxial curvature C, the conic constant K, and the free-form curved surface coefficients Cj to the aforementioned Mathematical Formula 2. 
     Example 1 
       FIG. 5  is an optical layout diagram illustrating the refraction optical system  101  according to this example. As illustrated in  FIG. 5 , the refraction optical system  101  includes a first lens unit  11 , a second lens unit  12 , a third lens unit  13 , and a fourth lens unit  14 . 
     In  FIG. 5 , each solid line represents a moving locus of each of the lens units constituting the refraction optical system  101  during focusing from a long distance side (far distance) to a close range side (near distance). In addition, the long distance side is defined by the case where an image size projected on a screen is 80 inches; and the close range side is defined by the case where the image size is 48 inches. 
     The first lens unit  11  is configured to include, in order from the image forming unit  10  side, a both-side aspherical biconvex lens having a stronger convex surface toward the image forming unit  10  side, a negative meniscus lens having a convex surface toward the image forming unit  10  side, a cemented lens of a biconvex lens having a stronger convex surface toward the magnification side and a negative meniscus lens having a convex surface toward the magnification side, an aperture stop (not illustrated), a biconvex lens having a stronger convex surface toward the magnification side, a biconcave lens having a stronger concave surface at the magnification side, a cemented lens of a positive meniscus lens having a convex surface toward the magnification side and a negative meniscus lens having a convex surface toward the magnification side, and a biconvex lens having a stronger convex surface toward the magnification side. 
     The second lens unit  12  is configured with a positive meniscus lens having a convex surface toward the image forming unit  10  side. 
     The third lens unit  13  is configured to include a biconcave lens B having a stronger concave surface toward the image forming unit  10  side and a both-side aspherical biconcave lens A having a stronger concave surface toward the image forming unit  10  side and having a shape that has a negative power on the axis and a positive power in the periphery. 
     The fourth lens unit  14  is configured to include a both-side aspherical negative meniscus lens having a convex surface toward the image forming unit  10  side and having a shape that has a negative power on the axis and a positive power in the periphery and a both-side aspherical biconvex lens having a stronger convex surface toward the magnification side and having a shape that has a positive power on the axis and a negative power in the periphery. 
     Table  1  illustrated in  FIG. 50  lists data representing examples of optical elements included in the projection optical system  100  according to Example 1. 
     In the table, S denotes each lens surface of the refraction optical system  101  as indicated by the numeral in  FIG. 5 . Further, a light beam distance, or an optical length, denotes a distance between a lower light beam at the maximum angle of view in the Y axis direction on each surface and the optical axis Lx. 
     Table  2  illustrated in  FIG. 51  represents a specific example of lens intervals during focusing in the projection optical system  100  according to this example. 
     Table  3  illustrated in  FIG. 52  represents a specific example of numerical values of aspherical coefficients in the projection optical system  100  according to this example. The aspherical surface is expressed by the above-described Mathematical Formula 1 (Equation 1) illustrated in  FIG. 49 . 
     Table  4  illustrated in  FIG. 53  represents a specific example of numerical values of free-form curved surface coefficients in the projection optical system  100  according to this example. The free-form curved surface is expressed by the above-described Mathematical Formula 2 (Equation 2) illustrated in  FIG. 49 . 
     Table  5  illustrated in  FIG. 54  represents a specific example of a DMD used as the image forming unit  10  in the projection optical system  100  according to this example. 
     Table  6  illustrated in  FIG. 55  represents a specific example of position coordinates and angles of α rotation of the plane mirror  102  and the curved mirror  103  from the vertex of the lens located to be closest to the plane mirror  102  side among the lenses included in the refraction optical system  101  in the focus state where the projected image has a maximum size. The rotation represents the angle between the normal line of the surface and the optical axis Lx. 
     Next, suppression of a deterioration in image quality at each projection distance in the projection optical system  100  according to the example will be described with reference to  FIGS. 6 to 11C .  FIGS. 6 to 11C  are diagrams of images illustrating positions of main light beams having a wavelength of 550 nm and diagrams illustrating distortion of an image at each angle of view when the image representing the positions of the main light beams is displayed on the screen with respect to each zoom position and each projection distance in the projection optical system  100  according to Example 1. 
       FIG. 6  illustrates an example of an image representing positions of spots having a wavelength of 550 nm displayed on the screen in the projection optical system  100  in a long distance projection period.  FIG. 7A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 6 .  FIG. 7B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 6 .  FIG. 7C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 6 . 
       FIG. 8  illustrates an example of an image representing positions of spots having a wavelength of 550 nm displayed on the screen in the projection optical system  100  in a reference distance projection period.  FIG. 9A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 8 .  FIG. 9B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 8 .  FIG. 9C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 8 . 
       FIG. 10  illustrates an example of an image representing positions of spots having a wavelength of 550 nm displayed on the screen in the projection optical system  100  in a close range projection period.  FIG. 11A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 10 .  FIG. 11B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 10 .  FIG. 11C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 10 . 
     Hereinbefore, as illustrated in  FIGS. 6 to 11C , in the projection optical system  100  according to this example, it is possible to project an image having small distortion with respect to each zoom position and each projection distance. 
     Next, it is described that a change in an image is suppressed during zooming at each angle of view by using spot diagrams in the projection optical system  100  according to the example with reference to  FIGS. 12 to 14 . The spots in each of the spot diagrams illustrated in  FIGS. 12 to 14  correspond to field positions indicated by F 1  to F 13  illustrated in  FIG. 48 . Each spot diagram illustrates image formation characteristics (mm) on the screen plane with respect to wavelengths of 625 nm (red), 550 nm (green), and 425 nm (blue). 
       FIG. 12  is a spot diagram in the long distance projection period.  FIG. 13  is a spot diagram in the reference distance projection period.  FIG. 14  is a spot diagram in the close range projection period. 
     As illustrated in  FIGS. 12 to 14 , according to the projection optical system  100  of this example, a variation in image quality at each zoom position and each projection distance is suppressed. 
     Second Embodiment of Image Display Apparatus 
     Next, another embodiment of the image display apparatus according to the present invention will be described. In the following description, the same components are denoted by the same reference numerals, and the detailed description thereof is not repeated. 
       FIG. 15  is an optical layout diagram illustrating a projector  1   a  according to an embodiment of the present invention. In  FIG. 15 , the projector  1   a  includes the image forming unit  10 , the parallel plate  40 , a projection optical system  100   a , the illumination optical system  20  including a light source which illuminates the image forming unit  10  with illumination light, and other members used for image formation, which may be accommodated in the housing  30 . 
     The projection optical system  100   a  includes a refraction optical system  101   a , a plane mirror  102   a  which is a first reflecting surface, and a curved mirror  103   a  which is a second reflecting surface. As illustrated in  FIG. 15 , the plane mirror  102   a  is arranged such that the normal line of the plane mirror  102   a  is rotated by 45 degrees from the Z axis towards the Y axis direction on the YZ plane. The curved mirror  103   a  may be a concave mirror or a free-form curved surface mirror of which reflecting surface has a shape of a free-form curved surface. 
     As described above in the case of First Embodiment, the image formed on the image forming unit  10  is magnified and projected on a screen (not illustrated). 
     The screen (not illustrated) is arranged to be perpendicular to the image forming unit  10 . In other words, the normal line of an image formation plane of the image forming unit  10  is perpendicular to the normal line of the screen as a projection plane. 
     An intermediate image which is conjugate with the image information formed in the image forming unit  10  is formed by the light beam passing through the refraction optical system  101   a . The intermediate image is formed as a spatial image in the side closer to the image forming unit  10  than the plane mirror  102   a . The intermediate image is not necessarily formed as a plane image, but the intermediate image may be formed as a curved image. 
     The image is displayed on the screen by magnifying and projecting the intermediate image by using the curved mirror  103   a  which is arranged to be closest to the magnification side in the projection optical system  100   a . Although the intermediate image has a curvature of field or distortion, the reflecting surface of the curved mirror  103   a  is configured to have a shape of a free-form curved surface, so that it is possible to correct the curvature of field and the distortion. Accordingly, since a burden of aberration correction on the refraction optical system  101   a  is reduced, a degree of freedom in the design of the projection optical system  100   a  is increased, so that it is advantageous in miniaturization. 
     The refraction optical system  101   a  is configured so that the first lens unit  11   a  having a group of lenses with a positive refractive power, the plane mirror  102   a , and the curved mirror  103   a  are fixed to the image forming unit  10  during focusing from a long distance side to a close range side. The second lens unit  12   a  which is a lens unit having a positive refractive power and the third lens unit  13   a  which is a lens unit having a negative refractive power are moved to the image forming unit  10  side. The fourth lens unit  14   a  which is a lens unit having a positive refractive power is moved to the magnification side. In other words, the projection optical system  100   a  can control a curvature of field or distortion aberration with high accuracy by performing floating focusing. 
     Further, since the refraction optical system  101   a  is configured so that an aspherical lens is arranged in the lens unit which is moved during focusing, it is possible to improve the effect of the correction. 
     The projector  1   a  illustrated in  FIG. 15  is an image display apparatus according to Example 2 described below.  FIG. 15  is also an optical path diagram illustrating a case of 48-inch projector where the front lens elements provided to the projection optical system  100   a  are drawn out to the extreme degree. In this disclosure, the front lens elements correspond to a lens which is arranged to be closest to the magnification side of the refraction optical system. As clarified from  FIG. 15 , a sufficient interval between each lens and each light beam is maintained, so that interference between each light beam and each lens or a lens barrel (not illustrated) can be avoided in this state. 
     Example 2 
       FIG. 16  is an optical layout diagram illustrating a refraction optical system  101   a  included in the projection optical system  100   a  according to this example. As illustrated in  FIG. 16 , the refraction optical system  101   a  includes, in order from the image forming unit  10  side, a first lens unit  11   a , a second lens unit  12   a , a third lens unit  13   a , and a fourth lens unit  14   a.    
     In  FIG. 16 , each solid line represents a moving locus of each of the lens units constituting the refraction optical system  101   a  during focusing from a long distance side (far distance) to a close range side (near distance). In addition, the long distance side is defined by the case where an image size projected on a screen is 80 inches; and the close range side is defined by the case where the image size is 48 inches. 
     The first lens unit  11   a  is configured to include, in order from the image forming unit  10  side, a both-side aspherical biconvex lens having a stronger convex surface toward the image forming unit  10  side, a negative meniscus lens having a convex surface toward the image forming unit  10  side, a cemented lens of a biconvex lens having a stronger convex surface toward the magnification side and a negative meniscus lens having a convex surface toward the magnification side, an aperture stop (not illustrated), a biconvex lens having a stronger convex surface toward the magnification side, a biconcave lens having a stronger concave surface at the magnification side, a cemented lens of a positive meniscus lens having a convex surface toward the magnification side and a negative meniscus lens having a convex surface toward the screen side, and a biconvex lens having a stronger convex surface toward the magnification side. 
     The second lens unit  12   a  is configured with a positive meniscus lens having a convex surface toward the image forming unit  10  side. 
     The third lens unit  13   a  is configured to include a negative meniscus lens Ba having a convex surface toward the magnification side and a both-side aspherical biconcave lens Aa having a stronger concave surface toward the image forming unit  10  side and having a shape that has a negative power on the optical axis Lx and a positive power in the periphery. 
     The fourth lens unit  14   a  is configured to include a both-side aspherical biconcave lens having a stronger concave surface toward the image forming unit  10  side and having a shape that has a negative power on the optical axis Lx and a positive power in the periphery and a both-side aspherical biconvex lens having a stronger convex surface toward the magnification side and having a shape that has a positive power on the optical axis Lx and a negative power in the periphery. 
     Table  7  (Tables  7 A and  7 B) illustrated in  FIGS. 56A and 56B  lists data representing examples of optical elements included in the projection optical system  100   a  of  FIG. 15  according to Example 2. In the table, S denotes each lens surface of the refraction optical system  101   a  as indicated by the numeral in  FIG. 16 . Further, a light beam distance, or an optical length, denotes a distance between a lower light beam at the maximum angle of view in the Y axis direction on each surface and the optical axis Lx. 
     Table  8  illustrated in  FIG. 57  represents a specific example of lens intervals during focusing in the projection optical system  100   a  according to this example. 
     Table  9  illustrated in  FIG. 58  represents a specific example of numerical values of aspherical coefficients in the projection optical system  100   a  according to this example. The aspherical surface is expressed by the above-described Mathematical Formula 1 (Equation 1) illustrated in  FIG. 49 . 
     Table  10  illustrated in  FIG. 59  represents a specific example of numerical values of free-form curved surface coefficients in the projection optical system  100   a  according to this example. The free-form curved surface is expressed by the above-described Mathematical Formula 2 (Equation 2) illustrated in  FIG. 49 . 
     Table  11  illustrated in  FIG. 60  represents a specific example of a DMD used as the image forming unit  10  in the projection optical system  100   a  according to this example. 
     Table  12  illustrated in  FIG. 61  represents a specific example of position coordinates and angles of α rotation of the plane mirror  102   a  and the curved mirror  103   a  from the vertex of the lens located to be closest to the plane mirror  102   a  side among the lenses included in the refraction optical system  101   a  in the focus state where the projected image has a maximum size. The rotation represents the angle between the normal line of the surface and the optical axis Lx. 
     Next, suppression of a deterioration in image quality at each projection distance in the projection optical system  100   a  according to this example will be described with reference to  FIGS. 17 to 22C .  FIGS. 17 to 22C  are diagrams illustrating positions of main light beams having a wavelength of 550 nm and diagrams illustrating distortion of an image at each angle of view when the image representing the positions of the main light beams is displayed on the screen with respect to each zoom position and each projection distance in the projection optical system  100   a  according to Example 2. 
       FIG. 17  illustrates an example of an image representing positions of spots having a wavelength of 550 nm displayed on the screen in the projection optical system  100   a  in a long distance projection period.  FIG. 18A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 17 .  FIG. 18B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 17 .  FIG. 18C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 17 . 
       FIG. 19  illustrates an example of an image representing positions of spots having a wavelength of 550 nm displayed on the screen in the projection optical system  100   a  in a reference distance projection period.  FIG. 20A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 19 .  FIG. 20B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 19 .  FIG. 20C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 19 . 
       FIG. 21  illustrates an example of an image representing positions of spots having a wavelength of 550 nm displayed on the screen in the projection optical system  100   a  in a close range projection period.  FIG. 22A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 21 .  FIG. 22B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 21 .  FIG. 22C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 21 . 
     Hereinbefore, as illustrated in  FIGS. 17 to 22C , according to the projection optical system  100   a  of Example 2, it is possible to project an image having small distortion with respect to each zoom position and each projection distance. 
     Next, suppression of a change in an image during zooming at each angle of view by using spot diagrams in the projection optical system  100   a  according to this example will be described. The spots in each of spot diagrams illustrated in  FIGS. 23 to 25  correspond to field positions indicated by F 1  to F 13  illustrated in  FIG. 48 . Each spot diagram illustrates image formation characteristics (mm) on the screen plane with respect to wavelengths of 625 nm (red), 550 nm (green), and 425 nm (blue). 
       FIG. 23  is a spot diagram in the long distance projection period.  FIG. 24  is a spot diagram in the reference distance projection period.  FIG. 25  is a spot diagram in the close range projection period. 
     As illustrated in  FIGS. 23 to 25 , according to the projection optical system  100   a  of this example, a variation in image quality at each zoom position and each projection distance is suppressed. 
     Third Embodiment of Image Display Apparatus 
     Next, still another embodiment of the image display apparatus according to the present invention will be described. In the following description, the same components are denoted by the same reference numerals, and the detailed description thereof is not repeated. 
       FIG. 26  is an optical layout diagram illustrating a projector  1   b  according to an embodiment of the present invention. In  FIG. 26 , the projector  1   b  includes the image forming unit  10 , th parallel plate  40 , a projection optical system  100   b , th illumination optical system  20  including a light source which illuminates the image forming unit  10  with illumination light, and other members used for image formation, which may be accommodated in the housing  30 . 
     The projection optical system  100   b  includes a refraction optical system  101   b , a plane mirror  102   b  which is a first reflecting surface, and a curved mirror  103   b  which is a second reflecting surface. As illustrated in  FIG. 26 , the plane mirror  102   b  is arranged such that the normal line of the plane mirror  102   b  is rotated by 45 degrees from the Z axis towards the Y axis direction on the YZ plane. The curved mirror  103   b  may be a concave mirror or a free-form curved surface mirror of which reflecting surface has a shape of a free-form curved surface. 
     As described above in the case of First Embodiment, the image formed on the image forming unit  10  is magnified and projected on a screen (not illustrated). 
     The screen (not illustrated) is arranged to be perpendicular to the image forming unit  10 . In other words, the normal line of an image formation plane of the image forming unit  10  is perpendicular to the normal line of the screen as a projection plane. 
     An intermediate image which is conjugate with the image information formed in the image forming unit  10  is formed by the light beam passing through the refraction optical system  101   b . The intermediate image is formed as a spatial image in the side closer to the image forming unit  10  than the plane mirror  102   b . The intermediate image is not necessarily formed as a plane image, but the intermediate image may be formed as a curved image. 
     The image is displayed on the screen by magnifying and projecting the intermediate image by using the curved mirror  103   b  which is arranged to be closest to the magnification side in the projection optical system  100   b . Although the intermediate image has a curvature of field or distortion, the reflecting surface of the curved mirror  103   b  is configured to have a shape of a free-form curved surface, so that it is possible to correct the curvature of field and the distortion. Accordingly, since a burden of aberration correction on the refraction optical system  101   b  is reduced, a degree of freedom in the design of the projection optical system  100   b  is increased, so that it is advantageous to miniaturization. 
     The refraction optical system  101   b  is configured so that the first lens unit  11   b  which is a lens unit having a positive refractive power, the plane mirror  102   b , and the curved mirror  103   b  are fixed with respect to the image forming unit  10  during focusing from a long distance side to a close range side. The second lens unit  12   b  which is a lens unit having a positive refractive power and the third lens unit  13   b  which is a lens unit having a negative refractive power are moved to the image forming unit  10  side. The fourth lens unit  14   b  which is a lens unit having a positive refractive power is moved to the magnification side. In other words, the projection optical system  100  can control a curvature of field or distortion aberration at a high accuracy by performing floating focusing. 
     Further, since the refraction optical system  101   b  is configured so that an aspherical lens is arranged in the lens unit which is moved during focusing, it is possible to improve the effect of the correction. 
     The projector  1   b  illustrated in  FIG. 26  is an image display apparatus according to Example 3 described below.  FIG. 26  is also an optical path diagram illustrating a case of 48-inch projector where the front lens elements provided to the projection optical system  100   b  are drawn out to the extreme degree. As clarified from  FIG. 26 , a sufficient interval between each lens and each light beam is maintained, so that interference between each light beam and each lens or a lens barrel (not illustrated) can be avoided in this state. 
     Example 3 
       FIG. 27  is an optical layout diagram illustrating a refraction optical system  101   b  included in the projection optical system  100   b  according to this example. As illustrated in  FIG. 27 , the refraction optical system  101   b  includes, in order from the image forming unit  10  side, a first lens unit  11   b , a second lens unit  12   b , a third lens unit  13   b , and a fourth lens unit  14   b.    
     In  FIG. 27 , each solid line represents a moving locus of each of the lens units constituting the refraction optical system  101   b  during focusing from a long distance side (far distance) to a close range side (near distance). In addition, the long distance side is defined by the case where an image size projected on a screen is 80 inches; and the close range side is defined by the case where the image size is 48 inches. 
     The first lens unit  11   b  is configured to include, in order from the image forming unit  10  side, a both-side aspherical biconvex lens having a stronger convex surface toward the image forming unit  10  side, a negative meniscus lens having a convex surface toward the image forming unit  10  side, a cemented lens of a negative meniscus lens having a convex surface toward the image forming unit  10  side and a biconvex lens having a stronger convex surface toward the image forming unit  10  side, an aperture stop (not illustrated), a both-side aspherical convex lens having a stronger convex surface toward the magnification side, a biconcave lens having a stronger concave surface at the magnification side, a cemented lens of a positive meniscus lens having a convex surface toward the magnification side and a negative meniscus lens having a convex surface toward the magnification side, and a biconvex lens having a stronger convex surface toward the magnification side. 
     The second lens unit  12   b  is configured with a positive meniscus lens having a convex surface toward the image forming unit  10  side. 
     The third lens unit  13   b  is configured to include a negative meniscus lens Bb having a convex surface toward the magnification side and a both-side aspherical biconcave lens Ab having a stronger concave surface toward the image forming unit  10  side and having a shape that has a negative power on the optical axis Lx and a positive power in the periphery. 
     The fourth lens unit  14   b  is configured to include a both-side aspherical negative meniscus lens having a convex surface toward the magnification side and having a shape that has a negative power on the optical axis Lx and a positive power in the periphery and a both-side aspherical positive meniscus lens having a convex surface toward the magnification side and having a shape that has a positive power on the optical axis Lx and a negative power in the periphery. 
     Table  13  (Tables  13 A and  13 B) of  FIGS. 62A and 62B  lists data representing examples of optical elements included in the projection optical system  100   b  according to Example 3. In the table, S denotes each lens surface of the refraction optical system  101   b  as indicated by the numeral in  FIG. 27 . Further, a light beam distance denotes a distance between a lower light beam at the maximum angle of view in the Y axis direction on each surface and the optical axis Lx. 
     Table  14  illustrated in  FIG. 63  represents a specific example of lens intervals during focusing in the projection optical system  100   b  according to this example. 
     Table  15  illustrated in  FIG. 64  represents a specific example of numerical values of aspherical coefficients in the projection optical system  100   b  according to this example. The aspherical surface is expressed by the above-described Mathematical Formula 1 (Equation 1) of  FIG. 49 . 
     Table  16  illustrated in  FIG. 65  represents a specific example of numerical values of free-form curved surface coefficients in the projection optical system  100   b  according to this example. The free-form curved surface is expressed by the above-described Mathematical Formula 2 (Equation 2) illustrated in  FIG. 49 . 
     Table  17  represents a specific example of a DMD used as the image forming unit  10  in the projection optical system  100   b  according to this example. 
     Table  18  represents a specific example of position coordinates and angles of α rotation of the plane mirror  102   b  and the curved mirror  103   b  from the vertex of the lens located to be closest to the plane mirror  102   b  side among the lenses included in the refraction optical system  101   b  in the focus state where the projected image has a maximum size. The rotation represents the angle between the normal line of the surface and the optical axis Lx. 
     Next, suppression of a deterioration in image quality at each zoom position and each projection distance in the projection optical system  100   b  according to this example will be described with reference to  FIGS. 28 to 33C .  FIGS. 28 to 33C  are diagrams illustrating positions of main light beams having a wavelength of 550 nm and diagrams illustrating distortion of an image at each angle of view when the image representing the positions of the main light beams is displayed on the screen with respect to each zoom position and each projection distance in the projection optical system  100   b  according to Example 3. 
       FIG. 28  illustrates an example of an image representing positions of spots having a wavelength of 550 nm displayed on the screen in the projection optical system  100   b  in a long distance projection period.  FIG. 29A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 28 .  FIG. 29B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 28 .  FIG. 29C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 28 . 
       FIG. 30  illustrates an example of an image representing positions of spots having a wavelength of 550 nm displayed on the screen in the projection optical system  100   b  in a reference distance projection period.  FIG. 31A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 30 .  FIG. 31B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 30 .  FIG. 31C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 30 . 
       FIG. 32  illustrates an example of an image representing positions of spots having a wavelength of 550 nm displayed on the screen in the projection optical system  100   b  in a close range projection period.  FIG. 33A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 32 .  FIG. 33B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 32 .  FIG. 33C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 32 . 
     Hereinbefore, as illustrated in  FIGS. 28 to 33C , according to the projection optical system  100   b  of this example, it is possible to project an image having small distortion with respect to each zoom position and each projection distance. 
     Next, suppression of a change in an image during zooming at each angle of view by using spot diagrams in the projection optical system  100   b  according to this example will be described. The spots in each of spot diagrams illustrated in  FIGS. 34 to 36  correspond to field positions indicated by F 1  to F 13  illustrated in  FIG. 48 . Each spot diagram illustrates image formation characteristics (mm) on the screen plane with respect to wavelengths of 625 nm (red), 550 nm (green), and 425 nm (blue). 
       FIG. 34  is a spot diagram in the long distance projection period.  FIG. 35  is a spot diagram in the reference distance projection period.  FIG. 36  is a spot diagram in the close range projection period. 
     As illustrated in  FIGS. 34 to 36 , according to the projection optical system  100   b  of this example, a variation in image quality at each zoom position and each projection distance is suppressed. 
     Fourth Embodiment of Image Display Apparatus 
     Next, further still another embodiment of the image display apparatus according to the present invention will be described. In the following description, the same components are denoted by the same reference numerals, and the detailed description thereof is not repeated. 
       FIG. 37  is an optical layout diagram illustrating a projector  1   c  according to an embodiment of the present invention. In  FIG. 37 , the projector  1   c  includes the image forming unit  10 , the parallel plate  40 , a projection optical system  100   c , the illumination optical system  20  including a light source which illuminates the image forming unit  10  with illumination light, and other members used for image formation, which may be accommodated in the housing  30 . 
     The projection optical system  100   c  is configured to include a refraction optical system  101   c , a plane mirror  102   c  which is a first reflecting surface, and a curved mirror  103   c  which is a second reflecting surface. As illustrated in  FIG. 37 , the plane mirror  102   c  is arranged such that the normal line of the plane mirror  102   c  is being rotated by 45 degrees from the Z axis towards the Y axis direction on the YZ plane. The curved mirror  103   c  may be a concave mirror or a free-form curved surface mirror of which reflecting surface has a shape of a free-form curved surface. 
     As described above in the case of Example 1, the image formed on the image forming unit  10  is magnified and projected on a screen (not illustrated). 
     The screen (not illustrated) is arranged to be perpendicular to the image forming unit  10 . In other words, the normal line of an image formation plane of the image forming unit  10  is perpendicular to the normal line of the screen as a projection plane. 
     An intermediate image which is conjugate with the image information formed in the image forming unit  10  is formed by the light beam passing through the refraction optical system  101   c . The intermediate image is formed as a spatial image in the side closer to the image forming unit  10  than the plane mirror  102   c . In addition, the intermediate image is not necessarily formed as a plane image, but the intermediate image may be formed as a curved image. 
     The image is displayed on the screen by magnifying and projecting the intermediate image by using the curved mirror  103   c  which is arranged to be closest to the magnification side in the projection optical system  100   c . Although the intermediate image has a curvature of field or distortion, the reflecting surface of the curved mirror  103   c  is configured to have a shape of a free-form curved surface, so that it is possible to correct the curvature of field and the distortion. Accordingly, since a burden of aberration correction on the refraction optical system  101   c  is reduced, a degree of freedom in the design of the projection optical system  100   c  is increased, so that it is advantageous to miniaturization. 
     The refraction optical system  101   c  is configured so that the first lens unit  11   c  which is a lens unit having a positive refractive power, the plane mirror  102   c , and the curved mirror  103   c  are fixed with respect to the image forming unit  10  during focusing from a long distance side to a close range side. The second lens unit  12   c  which is a lens unit having a positive refractive power and the third lens unit  13   c  which is a lens unit having a negative refractive power are moved to the image forming unit  10  side. The fourth lens unit  14   c  which is a lens unit having a positive refractive power is moved to the magnification side. In other words, the projection optical system  100   c  can control a curvature of field or distortion aberration at a high accuracy by performing floating focusing. 
     In addition, since the refraction optical system  101   c  is configured so that an aspherical lens is arranged in the lens unit which is moved during focusing, it is possible to improve the effect of the correction. 
     The projector  1   c  illustrated in  FIG. 37  is an image display apparatus according to Example 4 described below.  FIG. 37  is also an optical path diagram illustrating a case of 48-inch projector where the front lens elements provided to the projection optical system  100   c  are drawn out to the extreme degree. As clarified from  FIG. 37 , a sufficient interval between each lens and each light beam is maintained, so that interference between each light beam and each lens or a lens barrel (not illustrated) can be avoided in this state. 
     Example 4 
       FIG. 38  is an optical layout diagram illustrating a refraction optical system  101   c  included in the projection optical system  100   c  according to this embodiment. As illustrated in  FIG. 38 , the refraction optical system  101   c  is configured to include, in order from the image forming unit  10  side, a first lens unit  11   c , a second lens unit  12   c , a third lens unit  13   c , and a fourth lens unit  14   c.    
     In  FIG. 38 , each solid line represents a moving locus of each of the lens units constituting the refraction optical system  101   c  during focusing from a long distance side (far distance) to a close range side (near distance). In addition, the long distance side is defined by the case where an image size projected on a screen is 80 inches; and the close range side is defined by the case where the image size is 48 inches. 
     The first lens unit  11   c  is configured to include, in order from the image forming unit  10  side, a both-side aspherical biconvex lens having a stronger convex surface toward the image forming unit  10  side, a negative meniscus lens having a stronger convex surface toward the image forming unit  10  side, a cemented lens of a negative meniscus lens having a convex surface toward the image forming unit  10  side and a biconvex lens having a stronger convex surface toward the magnification side, an aperture stop (not illustrated), a both-side aspherical convex lens having a stronger convex surface toward the magnification side, a both-side aspherical biconvex lens having a stronger convex surface at the magnification side, a biconcave lens having a stronger concave surface at the magnification side, a cemented lens of a positive meniscus lens having a convex surface toward the magnification side and a negative meniscus lens having a convex surface toward the magnification side, and a biconvex lens having a stronger convex surface toward the magnification side. 
     The second lens unit  12   c  is configured with a positive meniscus lens having a convex surface toward the image forming unit  10  side. 
     The third lens unit  13   c  is configured to include a negative meniscus lens Bc having a convex surface toward the magnification side and a both-side aspherical biconcave lens Ac having a stronger concave surface toward the image forming unit  10  side and having a shape that has a negative power on the optical axis Lx and a positive power in the periphery. 
     The fourth lens unit  14   c  is configured to include a both-side aspherical negative meniscus lens having a convex surface toward the magnification side and having a shape that has a negative power on the optical axis Lx and a positive power in the periphery and a both-side aspherical positive meniscus lens having a convex surface toward the magnification side and having a shape that has a positive power on the optical axis Lx and a negative power in the periphery. 
     Table  19  (Tables  19 A and  19 B) illustrated in  FIGS. 68A and 68B  lists data representing examples of optical elements included in the projection optical system  100   c  according to this example. In the table, S denotes each lens surface of the refraction optical system  101   c  as indicated by the numeral in  FIG. 38 . Further, a light beam distance, or an optical length, denotes a distance between a lower light beam at the maximum angle of view in the Y axis direction on each surface and the optical axis Lx. 
     Table  20  illustrated in  FIG. 69  represents a specific example of lens intervals during focusing in the projection optical system  100   c  according to this example. 
     Table  21  illustrated in  FIG. 70  represents a specific example of numerical values of aspherical coefficients in the projection optical system  100   c  according to this example. The aspherical surface is expressed by the above-described Mathematical Formula 1 (Equation 1) illustrated in  FIG. 49 . 
     Table  22  illustrated in  FIG. 71  represents a specific example of numerical values of free-form curved surface coefficients in the projection optical system  100   c  according to this example. The free-form curved surface is expressed by the above-described Mathematical Formula 2 (Equation 2) illustrated in  FIG. 49 . 
     Table  23  illustrated in  FIG. 72  represents a specific example of a DMD used as the image forming unit  10  in the projection optical system  100   c  according to this example. 
     Table  24  illustrated in  FIG. 73  represents a specific example of position coordinates and angles of α rotation of the plane mirror  102   c  and the curved mirror  103   c  from the vertex of the lens located to be closest to the plane mirror  102   c  side among the lenses included in the refraction optical system  101   c  in the focus state where the projected image has a maximum size. The rotation represents the angle between the normal line of the surface and the optical axis Lx. 
     Next, suppression of a deterioration in image quality at each zoom position and each projection distance in the projection optical system  100   c  according to this example will be described with reference to  FIGS. 39 to 44C .  FIGS. 39 to 44C  are diagrams illustrating positions of main light beams having a wavelength of 550 nm and diagrams illustrating distortion of an image at each angle of view when the image representing the positions of the main light beams is displayed on the screen with respect to each zoom position and each projection distance in the projection optical system  100   c  according to Example 4. 
       FIG. 39  illustrates an example of an image representing positions of spots having a wavelength of 550 nm displayed on the screen in the projection optical system  100   c  in a long distance projection period.  FIG. 40A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 39 .  FIG. 40B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 39 .  FIG. 40C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 39 . 
       FIG. 41  illustrates an example of an image representing positions of spots having a wavelength of 550 nm displayed on the screen in the projection optical system  100   c  in a reference distance projection period.  FIG. 42A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 41 .  FIG. 42B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 41 .  FIG. 42C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 41 . 
       FIG. 43  illustrates an example of an image representing positions of spots having a wavelength of 550 nm displayed on the screen in the projection optical system  100   c  in a close range projection period.  FIG. 44A  is a diagram illustrating distortion in the upper side of the image exemplified in  FIG. 43 .  FIG. 44B  is a diagram illustrating distortion in the left end of the image exemplified in  FIG. 43 .  FIG. 44C  is a diagram illustrating distortion in the lower end of the image exemplified in  FIG. 43 . 
     Hereinbefore, as illustrated in  FIGS. 39 to 44C , according to the projection optical system  100   c  of this example, it is possible to project an image having small distortion with respect to each zoom position and each projection distance. 
     Next, suppression of a change in an image during zooming at each angle of view by using spot diagrams in the projection optical system  100   c  according to the example will be described. The spots in each of spot diagrams illustrated in  FIGS. 45 to 47  correspond to field positions indicated by F 1  to F 13  illustrated in  FIG. 48 . Each spot diagram illustrates image formation characteristics (mm) on the screen plane with respect to wavelengths of 625 nm (red), 550 nm (green), and 425 nm (blue). 
       FIG. 45  is a spot diagram in the long distance projection period.  FIG. 46  is a spot diagram in the reference distance projection period.  FIG. 47  is a spot diagram in the close range projection period. 
     As illustrated in  FIGS. 45 and 46 , according to the projection optical system  100   c  of this example, a variation in image quality at each zoom position and each projection distance is suppressed. 
     Now, main features of the above-described projection optical systems and image display apparatuses according to the present invention are as follows. 
     Feature 1 
     According to example embodiments of the present invention, there is provided a projection optical system including: in order from a reduction side to a magnification side, an image forming unit, a refraction optical system, and a first reflecting surface and a second reflecting surface. When an optical axis shared by the largest number of lenses among optical axes of a plurality of the lenses of the refraction optical system is defined by an optical axis of the projection optical system, when, in arrangement where a distance between an intersection of a magnification-side surface of a lens which is arranged to be closest to the magnification side of the refraction optical system and the optical axis and an intersection between the first reflecting surface and the optical axis has a minimum value, the distance between the intersections is denoted by L, when a focal length of the refraction optical system is denoted by f, when a direction parallel to the optical axis is defined by a Z axis direction, when an arrangement direction of the first reflecting surface and the second reflecting surface is defined by a Y axis direction, and when a maximum value of a distance between the optical axis and an end portion of the image forming unit in the Y axis direction is denoted by Y max, in a YZ plane which is a plane parallel to the Y axis direction and the Z axis direction, a maximum value D 1  of a distance between an intersection of a light beam from the image forming unit and the magnification-side surface of the lens which is arranged to be closest to the magnification side of the refraction optical system and the optical axis, a sag amount ds 1  which is a sag amount of the magnification-side surface of the lens which is arranged to be closest to the magnification side of the refraction optical system at the D 1  and of which positive direction is defined by the direction from the reduction side in the Z axis toward the magnification side, a point H of which distance from the optical axis has a maximum value among the intersections of the light beam and the first reflecting surface, a point F of which distance from the optical axis has a minimum value among the intersections of the light beam and the second reflecting surface, and an angle θ1 between a line connecting the H and the F and the optical axis satisfies the following conditions (1) and (2).
 
0&lt; Y  max/ f− 1/tan θ1  condition (1):
 
−0.1&lt;( L−D 1− ds 1)/( L+D 1− ds 1)−1/tan θ1  condition (2):
 
     In the projection optical system using mirrors, a method of inserting a folding mirror between the refraction optical system and the reflection surface is used in order to reduce a full-length direction size thereof. However, if conditions are not appropriate, interference occurs between the light beam and the lens or a barrel member. As a method of avoiding the interference (interference between the light beam and the member), there is a method of increasing a separation distance between the folding mirror and the refraction optical system or a method of reducing a diameter of the lens which is closest to the magnification side of the refraction optical system. However, the former method is contrary to the purpose of miniaturization in the full-length direction. In addition, in the latter method, since the role of the lens such as aberration correction, particularly, correction of a curvature of field or distortion aberration is decreased, the burden to the second reflecting surface is increased, so that the mirror size is increased. As a result, the size of the apparatus on which the projection optical system is mounted needs to be increased. In this manner, there is a problem in that, if the size in the full-length direction is tried to be reduced without any preparation, performance is easily deteriorated, and if the performance is tried to be secured, the size is increased. 
     With respect to the problem, according to the projection optical system satisfying the conditions (1) and (2), it is possible to provide a projection optical system having a small size, having no occurrence of interference with a light beam, and having high performance. Each of the aforementioned conditions (1) and (2) represents an appropriate range of an emission angle of the folded light beam from the first reflecting surface. 
     If the aforementioned value is less than the lower limit of the condition (1), interference easily occurs between the light emitted from the first reflecting surface and the lens, so that the emission angle of the light emitted from the refraction optical system is also increased. Accordingly, the size of the folding mirror becomes large, and thus, an intermediate image becomes large, so that the second reflecting surface needs to be large. As a result, the image display apparatus on which the projection optical system is mounted needs to be large. 
     In addition, in order to project an image at an appropriate position on the screen, a radius of curvature of the second reflecting surface needs to be small. Accordingly, in the correction of the curvature of field and the distortion aberration, it is difficult to maintain balance, so that a manufacturing error sensitivity of the second reflecting surface is increased or the performance is deteriorated. 
     In addition, if the aforementioned value is less than the lower limit of the condition (2), interference occurs between the lens and the light beam. 
     Therefore, when the conditions (1) and (2) are satisfied at the same time, although the distance between the refraction optical system and the first reflecting surface is shortened, the interference of the light beam does not occur, and the diameter of the lens which is closest to the magnification side of the refraction optical system can be increased. Accordingly, the burden to the lens, particularly, the role of the correction of the curvature of field is increased, so that the burden to the second reflecting surface can be reduced. 
     In addition to the decrease in the manufacturing error sensitivity and the improvement of the performance, the miniaturization of the second reflecting surface and the miniaturization of the apparatus can be particularly effectively implemented. 
     Feature 2 
     In the projection optical system according to the present invention, in addition to Feature 1, when an angle between a light beam emitted from the refraction optical system for an upper light beam at the maximum angle of view in the Y axis direction and the optical axis is denoted by θ 2 , when a distance between an intersection of the light beam emitted from the refraction optical system for the upper light beam at the maximum angle of view in the Y axis direction and the closest-magnification-side surface of the refraction optical system and the optical axis is denoted by D 2 , and when a sag amount which is a sag amount of the magnification-side surface of the lens which is arranged to be closest to the magnification side of the refraction optical system at a height of the D 2  and of which positive direction is defined by the direction from the reduction side in the Z axis toward the magnification side is denoted by ds 2 , the angle θ 2  satisfies the following conditions (3) and (4).
 
0&lt; Y  max/ f− 1/tan θ2.  condition (3):
 
−0.05&lt;( L−D 2− ds 2)/( L+D 1− ds 2)−1/tan θ2.  condition (4):
 
     Each of the conditions (3) and (4) represents an appropriate range of the emission angle of the upper light beam at the maximum angle in the Y axis direction. 
     If the aforementioned value is less than the lower limit of the condition (3), the emission angle of the upper light beam is increased, and thus, the size of the folding mirror becomes large, and the intermediate image becomes large. If the intermediate image becomes large, the second reflecting surface becomes large, so that the projection optical system becomes large. In addition, since the angle between the light beam emitted from the first reflecting surface and the optical axis is decreased, the interference of the lens easily occurs. In addition, in order to project an image at an appropriate position on the screen, a radius of curvature of the second reflecting surface needs to be small. Accordingly, in the correction of the curvature of field and the distortion aberration, it is difficult to maintain balance, so that a manufacturing error sensitivity of the second reflecting surface is increased or the performance is deteriorated. 
     If the aforementioned value is less than the lower limit of the condition (4), interference occurs between the lens and the light beam. 
     Therefore, when the conditions (3) and (4) are satisfied at the same time, although the distance between the refraction optical system and the first reflecting surface is shortened, the interference of the light beam does not occur, and the diameter of the lens which is closest to the magnification side of the refraction optical system can be increased. Accordingly, the burden to the lens, particularly, the role of the correction of the curvature of field is increased, so that the burden to the second reflecting surface can be reduced. In addition, the manufacturing error sensitivity is decreased, the performance is improved, and the miniaturization of the second reflecting surface and the miniaturization of the apparatus can be particularly effectively implemented. 
     Feature 3 
     In the projection optical system according to example embodiments of the present invention, in addition to Features 1 and 2, a magnification-side surface of the lens which is arranged to be closest to the magnification side of the refraction optical system is a convex and aspherical surface. 
     According to the feature, since the closest-magnification-side lens is configured to have a convex surface, the effect of deflection of the main light beam is improved, so that the intermediate image can be small. Therefore, the first reflecting surface and the second reflecting surface can be miniaturized. In addition, the interference between the lens and the light beam can be easily avoided. In addition, since the closest-magnification-side lens is configured to have an aspherical surface, the effect of deflection of the main light beam and the effect of correction of the curvature of field can be improved. 
     Feature 4 
     In the projection optical system according to example embodiments of the present invention, in addition to Features 1 to 3, the projection optical system further includes a both-side aspherical biconcave lens A having a negative power on the optical axis and a positive power in the periphery. 
     Since the lens has a convex power in the periphery, an emission angle of the main light beam of off-axis light emitted from the refraction optical system can be reduced, the miniaturization and the high performance of the apparatus can be implemented. Preferably, the aspherical lens is different from the aspherical lens having a convex shape, which is described above referring to Feature 3. Alternatively, the aspherical lens may be the same as the aspherical lens having a convex shape. Since a plurality of aspherical surfaces are used, it is possible to control the distortion aberration and the curvature of field at a high accuracy, and it is possible to implement a high-performance projection optical system in combination with the effect of the concave mirror. 
     Feature 5 
     In the projection optical system according to example embodiments of the present invention, in addition to Features 1 to 4, the aspherical lens is arranged between a spherical lens of which distance between an intersection of a lower light beam at the maximum angle of view in the Y axis direction and a surface and the optical axis is at maximum, and the first reflecting surface. 
     According to the feature, it is possible to control distortion and a curvature of field at a high accuracy by using the aspherical lens as described above in a portion where light beams are sufficiently separated. 
     Feature 6 
     In the projection optical system according to example embodiments of the present invention, in addition to Features 1 to 5, the first reflecting surface is a plane mirror and is rotated by 45 degrees on the YZ plane. 
     Herein, the 45 degrees represent that the normal line of the plane mirror is rotated by 45 degrees (−45 degrees in the α direction) from the Z axis toward the Y axis direction. According to the feature, since the optical system can be folded and bended by 90 degrees without any change in performance, the miniaturization can be effectively implemented. 
     Feature 7 
     In the projection optical system according to example embodiments of the present invention, in addition to Features 1 to 6, the second reflecting surface is a concave mirror. 
     According to the feature, since the intermediate image of the refraction optical system is magnified and projected by the concave mirror, the miniaturization of the projection optical system can be implemented. 
     Feature 8 
     In the projection optical system according to example embodiment of the present invention, in addition to Features 1 to 7, in a focus state where the L is at minimum, when paraxial magnification of the refraction optical system is denoted by β, the following condition (5) is satisfied.
 
5&lt;β&lt;8.  condition (5):
 
     The condition (5) is a mathematical formula for specifying an appropriate range of the size of the intermediate image. If the aforementioned value is more than the upper limit of the condition (5), the power of the concave mirror can be decreased, and thus, the manufacturing error sensitivity is reduced. However, since the size of the concave mirror is increased, the miniaturization cannot be implemented. 
     In addition, if the aforementioned value is less than the lower limit of the condition (5), the miniaturization can be effectively implemented. However, the power of the concave mirror needs to be increased in order to obtain a desired size of the projection image, and thus, the manufacturing error sensitivity needs to be increased. In addition, more preferably, the following condition (5′) is satisfied.
 
6&lt;β&lt;7.  condition (5′):
 
     Feature 9 
     In the projection optical system according to example embodiments of the present invention, in addition to Features 1 to 8, the second reflecting surface has a shape of a free-form curved surface. 
     Since the second reflecting surface is configured to have a shape of a free-form curved surface, it is possible to correct the curvature of field and the distortion aberration at a high accuracy. 
     Feature 10 
     In the projection optical system according to example embodiments of the present invention, in addition to Features 1 to 9, the image forming unit does not intersect the optical axis. 
     Since an axial light beam is not used, it is possible to control the curvature of field and the distortion aberration at a high accuracy by using the aspherical lens and the free-form curved mirror. 
     Feature 11 
     In the projection optical system according to example embodiments of the present invention, in addition to Features 1 to 10, at least the lens which is closest to the magnification side of the refraction optical system is moved during focusing. 
     Accordingly, it is possible to control a curvature of field and distortion which may occur due to focusing at a high accuracy. More preferably, floating focusing is used. Since a difference between the incidence angles of the light beams of the upper and lower ends of the image plane to the screen are great, the variation in curvature of field is increased in the focusing according to the projection distance. Herein, since the floating focusing is used, it is possible to correct the variation in curvature of field according to the variation in the projection distance. 
     Feature 12 
     According to example embodiments of the present invention, there is provided an image display apparatus including: an illumination optical system which illuminates an image forming unit with light from a light source; and a projection optical system which projects an image formed in the image forming unit on a projection plane, in which the projection optical system is the projection optical system according to any one of Features 1 to 11. 
     According to the feature, it is possible to obtain an image display apparatus having a very small projection distance and a small size. 
     Specific Numerical Values of Examples 
     Next, Table  25  illustrated in  FIG. 74  lists examples of numerical values associated with the above-described conditions in the Examples 1 to 4. 
     Table  26  illustrated in  FIG. 75  lists values according to each condition in each Example. 
     As clarified from Tables  25  and  26 , in the projection optical systems according to Examples 1 to 4, the above-described values of the parameters associated with the conditions 1 to 5 are included within the range of each condition. 
     According the projection optical system specified by the above-described specific numerical examples, since an angle of a folded light beam, an effective diameter of a lens, a distance between a folding mirror and a lens, and a sag amount of an aspherical lens are set to appropriate values, it is possible to obtain a small-sized and high-performance image projection apparatus. 
     According to an embodiment of the present invention, it is possible to provide a projection optical system having an extremely short projection distance and a small size. 
     In addition, although appropriate specific examples of the present invention are exemplified in the above-described embodiments, the present invention is not limited thereto. 
     In particular, specific shapes and numerical values of components in Examples 1 to 4 are merely exemplified for implementing the present invention, and thus, the scope of the prevention invention should not be limited thereto. 
     The present invention is not limited to the description of the embodiments, but appropriate changes and modifications can be made without departing from the spirit of the present invention.