Patent Publication Number: US-11042083-B2

Title: Projection system and projection-type image display apparatus

Description:
The present application is based on, and claims priority from JP Application Serial Number 2018-141034, filed Jul. 27, 2018 and 2019-105945, filed Jun. 6 2019, the disclosures of which are hereby incorporated by reference herein in its entirety. 
     BACKGROUND 
     1. Technical Field 
     The present disclosure relates to a projection system having a concave reflective surface on the magnifying side of an intermediate image, and to a projection-type image display apparatus including the projection system. 
     2. Related Art 
     JP-A-2010-20344 describes a projection-type image display apparatus that enlarges a projection image formed by an image formation section and projects the enlarged projection image via a projection system. The projection system described in JP-A-2010-20344 is formed of a first optical system and a second optical system sequentially arranged from the demagnifying side toward the magnifying side. The first optical system includes a refractive optical system. The second optical system is formed of a reflection mirror having a concave reflective surface. The image formation section includes a light source and a light valve. The image formation section forms a projection image in the demagnifying-side image formation plane of the projection system. The projection system forms an intermediate image in a position between the first optical system and the reflective surface and projects a final image on a screen disposed in the magnifying-side image formation plane of the projection system. 
     JP-A-2010-20344 is an embodiment of the related art. 
     In the projection system described in JP-A-2010-20344, shorting the projection distance causes the intermediate image located at the demagnifying side of the reflective surface to incline toward the direction along the optical axis of the first optical system. 
     The size of the intermediate image increases as the intermediate image inclines. When the size of the intermediate image increases, it is necessary to increase the size of the reflective surface located at the magnifying side of the intermediate image. Therefore, in a projection system having only a concave reflective surface on the magnifying side of the intermediate image, shorting the projection distance causes an increase in the size of the reflective surface. 
     SUMMARY 
     An advantage of some aspects of the present disclosure is to provide a projection system capable of suppressing an increase in size of a concave reflective surface disposed on the magnifying side of an intermediate image even in the case of a short projection distance. Another advantage of some aspects of the present disclosure is to provide a projection-type image display apparatus including the projection system. 
     An aspect of the present disclosure is directed to a projection system including a first optical system and a second optical system sequentially arranged from a demagnifying side toward a magnifying side, the projection system forming an intermediate image in a position between a demagnifying-side image formation plane and a magnifying-side image formation plane of the projection system. The second optical system is a lens. The lens has a first transmissive surface, a reflective surface, and a second transmissive surface sequentially arranged from the demagnifying side toward the magnifying side. Three axes perpendicular to one another are called axes X, Y, and Z, with an axis-Z direction being a direction in which the first transmissive surface and the reflective surface are arranged, an upper side being one side of the axis Y, a lower side being another side of the axis Y, and a plane YZ being a plane perpendicular to the axis X and containing the axes Y and Z. The first transmissive surface and the reflective surface are located at the lower side of an imaginary axis extending in the axis-Z direction. The second transmissive surface is located at the upper side of the imaginary axis. The reflective surface has a concave shape. The second transmissive surface has a convex shape protruding toward the magnifying side. An imaginary line that connects an upper intersection to a lower intersection inclines with respect to an imaginary vertical line perpendicular to the imaginary axis in the plane YZ, the upper intersection being an intersection where an upper peripheral light ray of an upper-end light flux that is a light ray passing through an axis-Y-direction upper end of an effective range of the second transmissive surface and an upper peripheral light ray of a lower-end light flux that is a light ray passing through an axis-Y-direction lower end of the effective range intersect with each other in the plane YZ, and the lower intersection being an intersection where a lower peripheral light ray of the upper-end light flux and a lower peripheral light ray of the lower-end light flux intersect with each other in the plane YZ. The intermediate image is located in the lens between the first transmissive surface and the reflective surface. 
     In the aspect of the present disclosure, the first optical system may be a refractive optical system. 
     In the aspect of the present disclosure, the imaginary axis may coincide with an optical axis of the first optical system. 
     In the aspect of the present disclosure, principal rays between the first optical system and the second optical system may approach each other toward the second optical system. 
     In the aspect of the present disclosure, any of the first transmissive surface, the reflective surface, and the second transmissive surface may be an aspheric surface. 
     In the aspect of the present disclosure, the first transmissive surface may be an aspheric surface. 
     In the aspect of the present disclosure, the intermediate image may have a shape that allows reduction in trapezoidal distortion of a final image formed in the magnifying-side image formation plane. 
     In the aspect of the present disclosure, the first transmissive surface, the reflective surface, and the second transmissive surface may form a coaxial optical system having surfaces rotationally symmetric with respect to the imaginary axis, and the imaginary axis may be a design reference axis. 
     In the aspect of the present disclosure, the projection system may satisfy a conditional expression below,
 
0°&lt;θ&lt;90°+γ  (1)
 
where θ is an inclination angle over which an end of the imaginary line facing the upper intersection rotates counterclockwise relative to the imaginary vertical line around an intersection of the imaginary vertical line and the imaginary line, and γ is an angle from the imaginary axis to the lower peripheral light ray of the lower-end light flux and intersects the imaginary axis.
 
     In the aspect of the present disclosure, the projection system may satisfy a conditional expression (2) below,
 
90°&lt;θ  (2)
 
     A projection-type image display apparatus according to another aspect of the present disclosure includes any of the projection systems described above and a display that displays a projection image in the demagnifying-side image formation plane of the projection system. 
     In the aspect of the present disclosure, the display may form the projection image on the upper side of the optical axis of the first optical system. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic configuration diagram of a projection-type image display apparatus including a projection system according to an embodiment of the present disclosure. 
         FIG. 2  is a light ray diagram of a projection system according to Embodiment 1. 
         FIG. 3  is a partially enlarged view of a portion A in  FIG. 2 . 
         FIG. 4  is a partially enlarged view of a portion B in  FIG. 2 . 
         FIG. 5  is a light ray diagram of the projection system according to Embodiment 1 enlarged. 
         FIG. 6  is a light ray diagram with a portion including a second optical system of the projection system according to Embodiment 1 enlarged. 
         FIG. 7  describes the magnification of the projection system in a case where the second optical system has only a reflective surface. 
         FIG. 8  describes the magnification of the projection system in a case where the second optical system has the reflective surface and a second transmissive surface which is disposed on the magnifying side of the reflective surface. 
         FIG. 9  shows an MTF of the projection system according to Embodiment 1 on the magnifying side. 
         FIG. 10  is a spot diagram showing spots produced by the projection system according to Embodiment 1. 
         FIG. 11  is a light ray diagram of a projection system according to Comparative Embodiment. 
         FIG. 12  is a light ray diagram of the projection system according to Comparative Embodiment enlarged. 
         FIG. 13  shows an MTF of the projection system according to Comparative Embodiment on the magnifying side. 
         FIG. 14  is a spot diagram showing spots produced by the projection system according to Comparative Embodiment. 
         FIG. 15  is a light ray diagram of a projection system according to Embodiment 2. 
         FIG. 16  is a light ray diagram of the projection system according to Embodiment 2. 
         FIG. 17  is a light ray diagram with a portion including a second optical system of the projection system according to Embodiment 2 enlarged. 
         FIG. 18  shows the MTF of the projection system according to Embodiment 2 on the magnifying side. 
         FIG. 19  is a spot diagram showing spots produced by the projection system according to Embodiment 2. 
         FIG. 20  is a light ray diagram of a projection system according to Embodiment 3. 
         FIG. 21  is a light ray diagram of the projection system according to Embodiment 3 enlarged. 
         FIG. 22  is a light ray diagram with a portion including a second optical system of the projection system according to Embodiment 3 enlarged. 
         FIG. 23  shows an MTF of the projection system according to Embodiment 3 on the magnifying side. 
         FIG. 24  is a spot diagram showing spots produced by the projection system according to Embodiment 3. 
         FIG. 25  is a light ray diagram of a projection system according to Embodiment 4. 
         FIG. 26  is a light ray diagram of the projection system according to Embodiment 4 enlarged. 
         FIG. 27  is a light ray diagram with a portion including a second optical system of the projection system according to Embodiment 4 enlarged. 
         FIG. 28  shows an MTF of the projection system according to Embodiment 4 on the magnifying side. 
         FIG. 29  is a spot diagram showing spots produced by the projection system according to Embodiment 4. 
     
    
    
     DESCRIPTION OF EXEMPLARY EMBODIMENTS 
     A projection system according to an embodiment of the present disclosure and a projection-type image display apparatus including the projection system will be described below in detail with reference to the drawings. 
     Projection-Type Image Display Apparatus 
       FIG. 1  is a schematic configuration diagram of the projection-type image display apparatus including the projection system according to the embodiment of the present disclosure. A projection-type image display apparatus  1  includes an image formation section  2 , which generates image light to be projected on a screen S, a projection system  3 , which enlarges and projects the image light, and a controller  4 , which controls the action of the image formation section  2 , as shown in  FIG. 1 . 
     Image Light Generation System and Controller 
     The image formation section  2  includes a light source  10 , a first optical integration lens  11 , a second optical integration lens  12 , a polarization converter  13 , and a superimposing lens  14 . The light source  10  is formed, for embodiment, of an ultrahigh-pressure mercury lamp, a solid-state light source, or any other light source. The first optical integration lens  11  and the second optical integration lens  12  each include a plurality of lens elements arranged in an array. The first optical integration lens  11  divides a light flux from the light source  10  into a plurality of light fluxes. The lens elements of the first optical integration lens  11  focus the light flux from the light source  10  in the vicinity of the lens elements of the second optical integration lens  12 . 
     The polarization converter  13  converts the light from the second optical integration lens  12  into predetermined linearly polarized light. The superimposing lens  14  superimposes images of each of the lens elements of the first optical integration lens  11  on one another on a display area of each of liquid crystal panels  18 R,  18 G, and  18 B, which will be described later, via the second optical integration lens  12 . 
     The image formation section  2  further includes a first dichroic mirror  15 , a reflection mirror  16 , a field lens  17 R, and the liquid crystal panel  18 R. The first dichroic mirror  15  reflects R light, which is part of light rays incident via the superimposing lens  14 , and transmits G light and B light, which are part of the light rays incident via the superimposing lens  14 . The R light reflected off the first dichroic mirror  15  travels via the reflection mirror  16  and the field lens  17 R and is incident on the liquid crystal panel  18 R. The liquid crystal panel  18 R is an image display device. The liquid crystal panel  18 R modulates the R light in accordance with an image signal to form a red projection image. One of the liquid crystal panels  18 R,  18 G, and  18 B is also called a display. 
     The image formation section  2  further includes a second dichroic mirror  21 , a field lens  17 G, and the liquid crystal panel  18 G. The second dichroic mirror  21  reflects the G light, which is part of the light rays from the first dichroic mirror  15 , and transmits B light, which is part of the light rays from the first dichroic mirror  15 . The G light reflected off the second dichroic mirror  21  passes through the field lens  17 G and is incident on the liquid crystal panel  18 G. The liquid crystal panel  18 G is an image display device. The liquid crystal panel  18 G modulates the G light in accordance with an image signal to form a green projection image. 
     The image formation section  2  further includes a relay lens  22 , a reflection mirror  23 , a relay lens  24 , a reflection mirror  25 , a field lens  17 B, and the liquid crystal panel  18 B. The B light having passed through the second dichroic mirror  21  travels via the relay lens  22 , the reflection mirror  23 , the relay lens  24 , the reflection mirror  25 , and the field lens  17 B and is incident on the liquid crystal panel  18 B. The liquid crystal panel  18 B is an image display device. The liquid crystal panel  18 B modulates the B light in accordance with an image signal to form a blue projection image. 
     The liquid crystal panels  18 R,  18 G, and  18 B surround a cross dichroic prism  19  in such a way that the liquid crystal panels  18 R,  18 G, and  18 B face three sides of the cross dichroic prism  19 . The cross dichroic prism  19 , which is a prism for light combination, combines the light modulated by the liquid crystal panel  18 R, the light modulated by the liquid crystal panel  18 G, and the light modulated by the liquid crystal panel  18 B with one another into image light. 
     The cross dichroic prism  19  forms part of the projection system  3 . The projection system  3  enlarges and projects the combined image light from the cross dichroic prism  19  (projection images formed by liquid crystal panels  18 R,  18 G, and  18 B) on the screen S. 
     The controller  4  includes an image processor  6 , to which an external image signal, such as a video signal, is inputted, and a display driver  7 , which drives the liquid crystal panels  18 R,  18 G, and  18 B based on image signals outputted from the image processor  6 . 
     The image processor  6  converts the image signal inputted from an external apparatus into image signals each containing grayscales and other factors on a color basis. The display driver  7  operates the liquid crystal panels  18 R,  18 G, and  18 B based on the color projection image signals outputted from the image processor  6 . The image processor  6  thus displays projection images corresponding to the image signals on the liquid crystal panels  18 R,  18 G, and  18 B. 
     Projection System 
     The projection system  3  will next be described. In the following sections, Embodiments 1 to 4 will be described as configuration embodiments of the projection system  3  incorporated in the projection-type image display apparatus  1 . 
     Embodiment 1 
       FIG. 2  is a light ray diagram of a projection system according to Embodiment 1 of the present disclosure.  FIG. 3  is a partially enlarged view of a portion A in  FIG. 2 .  FIG. 4  is a partially enlarged view of a portion B in  FIG. 2 .  FIG. 5  is a light ray diagram of the projection system according to Embodiment 1 enlarged.  FIG. 6  is a light ray diagram with a portion including a second optical system of the projection system according to Embodiment 1 enlarged.  FIG. 2  diagrammatically shows 11 light fluxes F 1  to F 11 , which exit out of a projection system  3 A and reach the screen S. The light flux F 1  is a light flux that reaches a lowest image height position. The light flux F 11  is a light flux that reaches a highest image height position. The light fluxes F 2  to F 10  are light fluxes that reach height positions between the height position that the light flux F 1  reaches and the height position that the light flux F 11  reaches. 
     The projection system  3 A according to the present embodiment is formed of a first optical system  31  and a second optical system  32  sequentially arranged from the demagnifying side toward the magnifying side, as shown in  FIG. 2 . The projection system  3 A forms an intermediate image  33  in a position between the demagnifying-side image formation plane and the magnifying-side image formation plane of the projection system  3 A, as shown in  FIG. 5 . 
     The first optical system  31  is a refractive optical system including a plurality of lenses. In the present embodiment, the first optical system  31  includes 15 lenses. The second optical system  32  is formed of a lens  35 . The liquid crystal panels  18 R,  18 G, and  18 B of the image formation section  2  are disposed in the demagnifying-side image formation plane.  FIGS. 2 and 5  show the liquid crystal panel  18 G, which is one of the three liquid crystal panels  18 R,  18 G, and  18 B. The liquid crystal panels  18 R,  18 G, and  18 B form projection images on one side of an optical axis N of the first optical system  31  in the demagnifying-side image formation plane. The screen S is disposed in the magnifying-side image formation plane. 
     The intermediate image  33  is formed in the second optical system  32 , that is, the lens  35 . The intermediate image  33  is formed on the other side of the optical axis N of the first optical system  31 . A final image projected on the screen S has an oblong shape elongated in the lateral direction. In the present embodiment, the final image has an aspect ratio of 16:10. 
     The first optical system  31  includes the cross dichroic prism  19  and a first lens L 1  to a fifteenth lens L 15 , which form 15 lenses, as shown in  FIG. 5 . The first lens L 1  to the fifteenth lens L 15  are arranged in the presented order from the demagnifying side toward the magnifying side. In the present embodiment, the second lens L 2  and the third lens L 3  are bonded to each other to forma first doublet L 21 . The fourth lens L 4  and the fifth lens L 5  are bonded to each other to form a second doublet L 22 . The eleventh lens L 11  and the twelfth lens L 12  are bonded to each other to form a third doublet L 23 . The thirteenth lens L 13  and the fourteenth lens L 14  are bonded to each other to forma fourth doublet L 24 . A stop O is disposed between the seventh lens L 7  and the eighth lens L 8 . In the present embodiment, the sixth lens L 6 , the ninth lens L 9 , and the fifteenth lens L 15  are three lenses that are each an aspheric lens having aspheric surfaces on both surfaces. 
     In the first optical system  31 , the fifteenth lens L 15 , which is located in a position closest to the magnifying side, has positive power. In the first optical system  31 , a magnifying-side lens group LG 25 , which is located in a position closest to the second optical system  32 , has positive power. The magnifying-side lens group LG 25  is formed of three lenses arranged from the side facing the second optical system  32 . That is, the thirteenth lens L 13 , the fourteenth lens L 14 , and the fifteenth lens L 15  form the magnifying-side lens group LG 25 . Since the magnifying-side lens group LG 25  has positive power in the first optical system  31 , the principal rays between the first optical system  31  and the second optical system  32  approach each other toward the second optical system  32 , as shown in  FIG. 5 . 
     The lens  35 , which forms the second optical system  32 , is made of resin. The lens  35  has a first transmissive surface  41 , a reflective surface  42 , and a second transmissive surface  43  sequentially arranged from the demagnifying side toward the magnifying side, as shown in  FIG. 6 . In the case where the lens  35  is made of resin, the lens  35  can be manufactured in injection molding. A lens  35  having a complicated shape is therefore readily manufactured. 
     In the following description, three axes perpendicular to one another are called axes X, Y, and Z for convenience. A first direction in which the first transmissive surface  41  and the reflective surface  42  are arranged is called an axis-Z direction. A second direction, which is called an axis-y direction, coincides with the vertical direction of the screen S. One side of the axis Y is called an upper side Y 1 , and the other side of the axis Y is called a lower side Y 2 . A first plane perpendicular to the axis X and containing the axes Y and Z is called a plane YZ.  FIGS. 1 to 6  therefore each show the plane YZ. The optical axis N of the first optical system  31  extends in the axis-Z direction. The image formation section  2  forms a projection image on the upper side Y 1  of the optical axis N of the first optical system  31 . The intermediate image  33  is formed on the lower side Y 2  of the optical axis N of the first optical system  31 . The lateral direction of the screen S coincides a third direction, which is called with the axis-X direction. In the following description, an imaginary axis M extending in the axis-Z direction is set in the plane YZ. The imaginary axis M is a reference axis used in the design of the lens  35 . The imaginary axis M is perpendicular to the screen S, which is the magnifying-side image formation plane. The imaginary axis M is substantially perpendicular to the screen S in some cases. 
     The first transmissive surface  41  and the reflective surface  42  are located at the lower side Y 2  of the imaginary axis M. The second transmissive surface  43  is located at the upper side Y 1  of the imaginary axis M. The reflective surface  42  has a concave shape. The reflective surface  42  therefore has positive power. The reflective surface  42  is provided by externally forming a reflective coating on the lens  35 . The second transmissive surface  43  has a convex shape protruding toward the magnifying side. The second transmissive surface  43  therefore has positive power. The first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43  form a coaxial optical system having surfaces rotationally symmetric with respect to the imaginary axis M. Therefore, the imaginary axis M is the reference axis used in the design of the lens  35  and is the optical axis of the lens  35 . In the present embodiment, the imaginary axis M coincides with the optical axis N of the first optical system  31 . The imaginary axis M does not necessarily coincide with the optical axis N of the first optical system  31 . 
     The upper and lower halves of the lens  35  are each configured to be rotationally symmetric with respect to the imaginary axis M. That is, the first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43  are so shaped that the cross-sectional shape in the plane YZ shown in  FIG. 6  is rotated around the imaginary axis M over an angular range of 90° toward one side and the other side of the axis-X direction. In the present embodiment, the first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43  are each an aspheric surface. 
     An imaginary line P can be specified in the lens  35  of the second optical system  32 , as shown in  FIG. 6 . The imaginary line P connects an upper intersection  53  to a lower intersection  54 , the upper intersection  53  being an intersection where an upper peripheral light ray  51   a  of an upper-end light flux  51 , where the upper-end light flux  51  is the light flux passing through the axis-Y-direction upper end of an effective range  50  of the second transmissive surface  43 , and an upper peripheral light ray  52   a  of a lower-end light flux  52 , where the lower-end light flux  52  is the light flux passing through the axis-Y-direction lower end of the effective range  50 , intersect with each other in the plane YZ, and the lower intersection  54  being an intersection where a lower peripheral light ray  51   b  of the upper-end light flux  51  and a lower peripheral light ray  52   b  of the lower-end light flux  52  intersect with each other in the plane YZ. The imaginary line P inclines by 98.77° with respect to an imaginary vertical line V, which is perpendicular to the imaginary axis M in the plane YZ. Further, in the lens  35  of the second optical system  32 , the upper peripheral light ray  51   a  of the upper-end light flux  51  travels from the second transmissive surface  43  toward the reflective surface  42 , and the point where the upper peripheral light ray  51   a  intersects another light ray reflected off the reflective surface  42  for the first time is called a first intersection. Similarly, the lower peripheral light ray  52   b  of the lower-end light flux  52  travels from the second transmissive surface  43  toward the reflective surface  42 , and the point where the lower peripheral light ray  52   b  intersects another light ray reflected off the reflective surface  42  for the first time is called a second intersection. Under the above definition, the line that connects the first intersection to the second intersection is named as a pupil  44 . In the present embodiment, the pupil  44  inclines by 103.315° with respect to the imaginary vertical line V in the plane YZ. 
     The intermediate image  33  is an inverted final image turned upside down. The intermediate image  33  is an image so distorted that an oblong final image is projected on the screen S, which is the magnifying-side image formation plane. More specifically, the intermediate image  33  has a shape that allows reduction in trapezoidal distortion of the final image formed on the screen S. That is, the intermediate image  33  has distortion opposite the trapezoidal distortion of the final image. The intermediate image  33  is therefore so formed that the shortest edge thereof is the edge having the highest image height on the screen S. 
     Lens Data 
     Data on the lenses of the projection system  3 A are as follows: The surfaces of the lenses are numbered sequentially from the demagnifying side toward the magnifying side. A surface having a surface number with * is an aspheric surface. The surface number  1  represents the demagnifying-side surface of the cross dichroic prism  19 , and the surface number  2  represents the magnifying-side surface thereof. The fields labeled with the surface number  20  show dummy data. The row of the reference characters represents the reference characters of the lenses thereof. The reference characters given in the second optical system  32  are the reference characters of the first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43 . That is, the surface number  31  represents the first transmissive surface  41  of the lens  35 . The surface number  33  represents the reflective surface  42  of the lens  35 . The surface number  35  represents the second transmissive surface  43  of the lens  35 . Reference character r denotes the radius of curvature in millimeters. Reference character d denotes the on-axis inter-surface distance in millimeters. Reference character nd denotes the refractive index. Reference character νd denotes the Abbe number. Reference character Y denotes the effective radius in the axis-Y direction. Reference character X denotes the effective radius in the axis-X direction. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
               
                 Surface 
                 Reference 
                   
                   
                   
                   
                   
                   
               
               
                 number 
                 character 
                 r 
                 d 
                 nd 
                 vd 
                 Y 
                 X 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Object 
                   
                 0 
                 9.39 
                   
                   
                   
                   
               
               
                 plane 
                   
                   
                   
                   
                   
                   
                   
               
               
                  1 
                   
                 0 
                 25.91 
                 1.516331 
                 64.14 
                 13.27 
                 13.27 
               
               
                  2 
                   
                 0 
                 0.1 
                   
                   
                 16.289 
                 16.289 
               
               
                  3 
                 L1 
                 56.80565 
                 6.965816 
                 1.646732 
                 58.36 
                 16.758 
                 16.758 
               
               
                  4 
                   
                 −42.68536 
                 0.1995 
                   
                   
                 16.795 
                 16.795 
               
               
                  5 
                 L2 
                 32.84132 
                 9.903502 
                 1.483259 
                 59.69 
                 15.087 
                 15.087 
               
               
                  6 
                 L3 
                 −33.73055 
                 1.197 
                 1.841678 
                 29.18 
                 14.07 
                 14.07 
               
               
                  7 
                   
                 599.36572 
                 0.1995 
                   
                   
                 13.431 
                 13.431 
               
               
                  8 
                 L4 
                 25.79278 
                 10.192649 
                 1.516331 
                 64.14 
                 12.589 
                 12.589 
               
               
                  9 
                 L5 
                 −20.37894 
                 1.197 
                 1.903658 
                 31.32 
                 11.379 
                 11.379 
               
               
                  10 
                   
                 21.51184 
                 1.012845 
                   
                   
                 10.637 
                 0.637 
               
               
                 *11 
                 L6 
                 14.98812 
                 1.3965 
                 1.437002 
                 95.1 
                 10.831 
                 10.831 
               
               
                 *12 
                   
                 15.82309 
                 0.370963 
                   
                   
                 10.927 
                 10.927 
               
               
                  13 
                 L7 
                 21.34706 
                 5.192064 
                 1.458241 
                 77.43 
                 11.381 
                 11.381 
               
               
                  14 
                   
                 −203.44571 
                 2.49375 
                   
                   
                 11.28 
                 11.28 
               
               
                 Stop 
                   
                 0 
                 3.49125 
                   
                   
                 12 
                 12 
               
               
                 plane 
                   
                   
                   
                   
                   
                   
                   
               
               
                  16 
                 L8 
                 27.16508 
                 4.1895 
                 1.846663 
                 23.78 
                 10.379 
                 10.379 
               
               
                  17 
                   
                 −90.48264 
                 3.3915 
                   
                   
                 10 
                 10 
               
               
                 *18 
                 L9 
                 127.94794 
                 1.7955 
                 1.787436 
                 38.46 
                 9.476 
                 9.476 
               
               
                 *19 
                   
                 24.80943 
                 2.9925 
                   
                   
                 9.581 
                 9.581 
               
               
                  20 
                 dummy 
                 0 
                 1.662166 
                   
                   
                 9.975 
                 9.975 
               
               
                  21 
                 L10 
                 −30.9257 
                 11 
                 1.846663 
                 23.78 
                 9.978 
                 9.978 
               
               
                  22 
                   
                 −37.46179 
                 8.254402 
                   
                   
                 13.744 
                 13.744 
               
               
                  23 
                 L11 
                 −96.87787 
                 10 
                 1.665537 
                 43.64 
                 17.097 
                 17.097 
               
               
                  24 
                 L12 
                 −20.35957 
                 11 
                 1.834105 
                 30.43 
                 17.528 
                 17.528 
               
               
                  25 
                   
                 −43.79383 
                 14.751624 
                   
                   
                 23.012 
                 23.012 
               
               
                  26 
                 L13 
                 39.90785 
                 16.825248 
                 1.604992 
                 61.79 
                 30 
                 30 
               
               
                  27 
                 L14 
                 −209.92296 
                 1.995 
                 1.844769 
                 25.57 
                 29.686 
                 29.686 
               
               
                  28 
                   
                 85.46293 
                 25.235494 
                   
                   
                 27.781 
                 27.781 
               
               
                 *29 
                 L15 
                 −94.94674 
                 11 
                 1.531132 
                 55.75 
                 25.435 
                 25.435 
               
               
                 *30 
                   
                 52.35571 
                 2.93364 
                   
                   
                 24.657 
                 24.657 
               
               
                 *31 
                 41 
                 −48.1898 
                 33.25 
                 1.531132 
                 55.75 
                 24.603 
                 24.603 
               
               
                  32 
                   
                 0 
                 0 
                 1.531132 
                 55.75 
                 22.06 
                 22.06 
               
               
                 *33 
                 42 
                 −26.026 
                 0 
                 1.531132 
                 55.75 
                 21.22 
                 21.22 
               
               
                  34 
                   
                 0 
                 −33.25 
                 1.531132 
                 55.75 
                 41.157 
                 41.157 
               
               
                 *35 
                 43 
                 20.45512 
                 −466.75 
                   
                   
                 20.295 
                 20.295 
               
               
                 Image 
                   
                 0 
                 0 
                   
                   
                 1451.182 
                 1451.182 
               
               
                 plane 
               
               
                   
               
            
           
         
       
     
     As shown in the lens data, the refractive index nd of the lens  35  is 1.531132, and the Abbe number νd of the lens  35  is 55.75. The field of the on-axis inter-surface distance d labeled with the surface number  35  shows the distance between the screen S and the second transmissive surface  43  of the lens  35 . The field of the on-axis inter-surface distance d labeled with the surface number  35  therefore shows the projection distance f of the projection system  3 A. In the present embodiment, f=466.75 mm. In the present embodiment, the effective radius of the reflective surface  42  is 21.22 mm in the axis-Y direction and 21.22 mm in the axis-X direction. 
     Aspheric data of the surface number  11  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                 −4.291852E−01 
               
               
                   
                 Fourth-order coefficient 
                 −2.147243E−04 
               
               
                   
                 Sixth-order coefficient 
                 6.917248E−07 
               
               
                   
                 Eighth-order coefficient 
                 −6.424421E−10 
               
               
                   
                 Tenth-order coefficient 
                 0 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  12  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                 −2.985186E−01 
               
               
                   
                 Fourth-order coefficient 
                 −2.429582E−04 
               
               
                   
                 Sixth-order coefficient 
                  6.907024E−07 
               
               
                   
                 Eighth-order coefficient 
                  −1.68077E−09 
               
               
                   
                 Tenth-order coefficient 
                  5.472176E−13 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  18  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                     −1E+00 
               
               
                   
                 Fourth-order coefficient 
                 −9.067839E−05 
               
               
                   
                 Sixth-order coefficient 
                 0 
               
               
                   
                 Eighth-order coefficient 
                 0 
               
               
                   
                 Tenth-order coefficient 
                 0 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  19  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                    −8.8E−01 
               
               
                   
                 Fourth-order coefficient 
                 −5.124337E−05 
               
               
                   
                 Sixth-order coefficient 
                  8.250618E−08 
               
               
                   
                 Eighth-order coefficient 
                  −8.04982E−11 
               
               
                   
                 Tenth-order coefficient 
                 0 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  29  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                 −7.854261E+01 
               
               
                   
                 Fourth-order coefficient 
                 1.117938E−05 
               
               
                   
                 Sixth-order coefficient 
                 −4.392964E−09 
               
               
                   
                 Eighth-order coefficient 
                 −1.552816E−11 
               
               
                   
                 Tenth-order coefficient 
                 −1.508818E−15 
               
               
                   
                 Twelfth-order 
                 2.210216E−17 
               
               
                   
                 coefficient 
                   
               
               
                   
                 Fourteenth-order 
                 −8.140301E−21 
               
               
                   
                 coefficient 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  30  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                  2.047967E+00 
               
               
                   
                 Fourth-order coefficient 
                 −3.292408E−05 
               
               
                   
                 Sixth-order coefficient 
                  3.055446E−08 
               
               
                   
                 Eighth-order coefficient 
                 −2.077281E−11 
               
               
                   
                 Tenth-order coefficient 
                  2.14935E−14 
               
               
                   
                 Twelfth-order 
                 −2.539946E−17 
               
               
                   
                 coefficient 
                   
               
               
                   
                 Fourteenth-order 
                  2.403648E−20 
               
               
                   
                 coefficient 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  31  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                 0 
               
               
                   
                 Fourth-order coefficient 
                 1.258824E−05 
               
               
                   
                 Sixth-order coefficient 
                 3.935005E−09 
               
               
                   
                 Eighth-order coefficient 
                 −1.154515E−11 
               
               
                   
                 Tenth-order coefficient 
                 2.202788E−14 
               
               
                   
                 Twelfth-order 
                 0 
               
               
                   
                 coefficient 
                   
               
               
                   
                 Fourteenth-order 
                 0 
               
               
                   
                 coefficient 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  33  are as follows. 
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 Conic constant 
                     −1E+00 
               
               
                   
                 Fourth-order coefficient 
                 4.920678E−06 
               
               
                   
                 Sixth-order coefficient 
                 −5.820738E−09  
               
               
                   
                 Eighth-order coefficient 
                 6.336457E−12 
               
               
                   
                 Tenth-order coefficient 
                 1.339523E−14 
               
               
                   
                 Twelfth-order 
                 −3.33883E−17 
               
               
                   
                 coefficient 
               
               
                   
                 Fourteenth-order 
                 2.827146E−20 
               
               
                   
                 coefficient 
               
               
                   
                   
               
            
           
         
       
     
     Aspheric data of the surface number  35  are as follows. 
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 Conic constant 
                 0 
               
               
                   
                 Fourth-order coefficient 
                 −1.223012E−06 
               
               
                   
                 Sixth-order coefficient 
                 −3.111316E−08 
               
               
                   
                 Eighth-order coefficient 
                 9.115488E−11 
               
               
                   
                 Tenth-order coefficient 
                 −8.917847E−14 
               
               
                   
                 Twelfth-order 
                 0 
               
               
                   
                 coefficient 
               
               
                   
                 Fourteenth-order 
                 0 
               
               
                   
                 coefficient 
               
               
                   
                   
               
            
           
         
       
     
     In the projection system  3 A according to the present embodiment, the lens  35 , which forms the second optical system  32 , has the reflective surface  42  with concave shape and the second transmissive surface  43  with a convex shape protruding toward the magnifying side. The light fluxes reflected off the reflective surface  42  are refracted by the second transmissive surface  43 . Therefore, the projection distance of the projection system  3 A is readily shortened, as compared with a case where the second optical system  32  has only the reflective surface  42 . 
     Further, in the present embodiment, an increase in the size of the reflective surface  42 , which is disposed on the magnifying side of the intermediate image  33 , can be suppressed even in the case of a short projection distance. 
     The effects described above will be described in detail with reference to  FIGS. 7 and 8 .  FIG. 7  describes the magnification of the projection system in the case where the second optical system  32  has only the reflective surface  42  on the magnifying side of the intermediate image  33 .  FIG. 8  describes the magnification of the projection system in the case where the second optical system  32  has the reflective surface  42  and the second transmissive surface  43  with the convex shape protruding toward the magnifying side on the magnifying side of the intermediate image  33 . 
     In the case where the second optical system  32  has only the reflective surface  42  on the magnifying side of the intermediate image  33 , the magnification Q of the projection system is represented by the ratio of T to R, where T is a distance from the reflective surface  42  to the screen S, and R is a distance between the intermediate image  33  and the reflective surface  42  along the optical path of a specific part of the light ray from the intermediate image  33  to the screen S, as shown in  FIG. 7 . That is, Q=T/R. The intermediate image  33 , which is conjugate with the screen S which corresponds to the magnifying-side image formation plane, greatly inclines so that the magnification Q of each of light fluxes is equal and has field curvature due to the inclination. The size of the intermediate image  33  increases as the intermediate image  33  inclines. When the size of the intermediate image  33  increases, it is necessary to increase the size of the reflective surface  42  located at the magnifying side of the intermediate image  33 . Therefore, in the projection system having only the concave reflective surface  42  on the magnifying side of the intermediate image  33 , shorting the projection distance causes an increase in the size of the reflective surface  42 . The size of the intermediate image  33  increases as the intermediate image  33  inclines, as described above. This means that a wider distance between the first optical system  31  and the second optical system  32  is necessary, resulting in an increase in the overall length of the projection system  3 A. 
     In contrast, in the present embodiment, in which the second optical system  32  has the second transmissive surface  43  with the convex shape protruding toward the magnifying side on the magnifying side of the reflective surface  42 , an increase in the size of the intermediate image  33  can be suppressed. That is, in the present embodiment, the magnification Q of the projection system  3 A is represented by the ratio of T′ to (R 1 +R 2 ), where T′ is a distance between the second transmissive surface  43  and the screen S, R 1  is a distance between the intermediate image  33  and the reflective surface  42  along the optical path of a specific part of the light ray from the intermediate image  33  to the screen S, and R 2  is a distance between the reflective surface  42  and the second transmissive surface  43 , as shown in  FIG. 8 . That is, Q=T′/(R 1 +R 2 ). The intermediate image  33 , which is conjugate with the screen S which corresponds to the magnifying-side image formation plane, therefore does not greatly incline and has reduced field curvature. As a result, an increase in the size of the intermediate image  33  can be suppressed. An increase in the size of the reflective surface  42 , which is located at the magnifying side of the intermediate image  33 , can therefore be suppressed. Further, if the upper-end light flux  51  can be refracted inward when passing through the first transmissive surface  41 , the size of the reflective surface  42  can be further reduced. Moreover, the second transmissive surface  43 , which has positive power, causes the light flux to be convergent, whereby an increase in the size of the reflective surface  42  can be suppressed, as compared with the case where no second transmissive surface  43  is provided. 
     Further, in the present embodiment, the magnifying-side lens group LG 25 , which is located in a position closest to the magnifying side in the first optical system  31 , has positive power, as shown in  FIG. 5 . The principal rays between the first optical system  31  and the second optical system  32  therefore approach each other toward the second optical system  32 . Therefore, the intermediate image  33  is readily formed, and the size of the intermediate image  33  can be reduced. The size of the reflective surface  42 , which is located at the magnifying side of the intermediate image  33 , can therefore be further reduced. 
     Further, in the present embodiment, the intermediate image  33  is located in the lens  35  between the first transmissive surface  41  and the reflective surface  42 . The first optical system  31  can be closer to the lens  35  than in a case where the intermediate image  33  is formed on the demagnifying side of the first transmissive surface  41 . The overall length of the projection system  3 A can therefore be shortened. 
     In the present embodiment, the first optical system  31  includes three aspheric lenses. Further, the first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43  of the second optical system  32  are each an aspheric surface. Aberrations produced by the projection system  3 A according to the present embodiment can therefore be suppressed. 
     Further, in the present embodiment, since the first transmissive surface  41 , which is adjacent to the intermediate image  33  on the demagnifying side, is an aspheric surface, aberrations produced in the position of the intermediate image  33  can be suppressed. Moreover, in the present embodiment, since the second optical system  32  has the second transmissive surface  43  with the convex shape protruding toward the magnifying side on the magnifying side of the reflective surface  42 , the intermediate image  33  does not greatly incline along the imaginary axis M. In other words, in the present embodiment, the intermediate image  33  extends in the direction perpendicular to the imaginary axis M. Therefore, according to the present embodiment, the first transmissive surface  41  and the intermediate image  33  can readily approach each other in the axis-Z direction, whereby the aspheric surface can be disposed in a position close to the intermediate image  33 . Aberrations produced in the position of the intermediate image  33  can therefore be efficiently corrected. 
     In the present embodiment, the lens  35 , which forms the second optical system  32 , satisfies the following conditional expressions (1) to (4), as shown in  FIG. 6 .
 
0°&lt;θ&lt;90°+γ  (1)
 
90°&lt;θ  (2)
 
0°&lt;η&lt;90°+γ  (3)
 
90°&lt;η  (4)
 
     θ: Inclination angle over which an end of the imaginary line P facing the upper intersection  53  rotates counterclockwise relative to the imaginary vertical line V around the intersection of the imaginary vertical line V and the imaginary line P 
     η: Inclination angle over which an end of the pupil  44  facing the first intersection rotates counterclockwise relative to the imaginary vertical line V around the intersection of the imaginary vertical line V and the pupil  44   
     γ: Angle from the imaginary axis M to the lower peripheral light ray  52   b  passing through the effective range  50  of the second transmissive surface  43  and intersects the imaginary axis M 
     That is, in the present embodiment, the imaginary line P inclines by 98.77° with respect to the imaginary vertical line V. Therefore, θ=98.77°, which satisfies the conditional expressions (1) and (2) are satisfied. Further, in the present embodiment, the pupil  44  of the lens inclines by 103.315° with respect to the imaginary vertical line V. Therefore, η=103.315°, which satisfies the conditional expressions (3) and (4). 
     In the present embodiment, in which the conditional expression (1) is satisfied, the imaginary line P inclines with respect to the imaginary vertical line V. In a case where θ=0°, the imaginary line P is perpendicular to the imaginary axis M. That is, the imaginary line P is perpendicular to the design reference axis. Further, in the present embodiment, in which the conditional expression (3) is satisfied, the pupil  44  inclines with respect to the imaginary vertical line V. In a case where η=0°, the pupil  44  is perpendicular to the imaginary axis M. In a case where at least one of θ in the conditional expression (1) and η in the conditional expression (3) is greater than the upper limit, the lower-end light flux  52  is blocked. In the present embodiment, in which the conditional expressions (1) and (3) are satisfied, the lower-end light flux  52  passing through the lower end of the effective range  50  of the second transmissive surface  43  is not blocked but reaches the screen S. 
     Further, since the lens  35  according to the present embodiment satisfies the conditional expression (1), a decrease of brightness at the upper periphery of the screen S can be suppressed, as compared with a case where the imaginary line P is parallel to the imaginary vertical line V. 
     That is, in the case where the imaginary line P is perpendicular to the imaginary axis M, that is, θ=0°, a divergence angle θ 0  of a light flux that reaches an upper portion of the screen S decreases as the angle of view on the magnifying side increases. The divergence angle θ 0  is shown in  FIGS. 3 and 4 . Further, the difference in the divergence angle θ 0  between the light flux that reaches the upper portion of the screen S and a light flux that reaches a lower portion of the screen S increases. As a result, the amount of light at the upper periphery of the screen S is smaller than the amount of light at the lower periphery of the screen S. 
     On the other hand, in the present embodiment, in which the conditional expression (1) is satisfied, the imaginary line P inclines with respect to the imaginary vertical line V. As a result, the divergence angle θ 0  of the light flux that reaches the upper portion of the screen S increases. The amount of light that reaches the upper portion of the screen S therefore increases. Further, when the divergence angle θ 0  of the light flux that reaches the upper portion of the screen S increases, the difference in the divergence angle θ 0  between the light flux that reaches the upper portion of the image formation plane and the light flux that reaches the lower portion of the image formation plane decreases. The situation in which the amount of light at the upper periphery of the screen S is smaller than the amount of light at the lower periphery of the screen S can therefore be suppressed. 
     In addition to the above, since the lens  35  according to the present embodiment satisfies the conditional expression (2), the divergence angle θ 0  of the light flux that reaches the lower portion of the screen S decreases. The difference in the divergence angle θ 0  between the light flux that reaches the upper portion of the screen S and the light flux that reaches the lower portion of the screen S therefore decreases, whereby the difference in the amount of light between the upper portion and the lower portion of the screen S can be suppressed. 
       FIG. 9  shows an MTF of the projection system  3 A on the magnifying side. The MTF was calculated under the following conditions: The image formation planes were divided along the axis Y; and the resultant halves were each divided into 11 areas. Light rays used in the calculation of the MTF are so weighted that the weighting ratio among light rays having a wavelength of 620 nm, light rays having a wavelength of 550 nm, and light rays having a wavelength of 470 nm is 2:7:1. The horizontal axis of  FIG. 9 , which shows the MTF, represents the spatial frequency. A spatial frequency of 0.24 cycles corresponds to a resolution of 16.7 μm. The vertical axis of  FIG. 9  represents a contrast reproduction ratio. In the present embodiment, a decrease in resolution is suppressed, as shown in  FIG. 9 .  FIG. 10  is a spot diagram showing spots produced by the projection system  3 A. In the present embodiment, variation in the size of the spots is suppressed, as shown in  FIG. 10 . 
     Comparative Embodiment 
     The fact that the projection system  3 A can provide the effect of suppressing an increase in the size of the concave reflective surface  42  disposed on the magnifying side of the intermediate image  33  even in the case of a short projection distance will be shown below as compared with Comparative Embodiment. 
     Comparative Embodiment relates to a projection system in which the second optical system is formed only of a reflection mirror having a reflective surface.  FIG. 11  is a light ray diagram of the projection system according to Comparative Embodiment.  FIG. 12  is a light ray diagram of the projection system according to Comparative Embodiment enlarged.  FIG. 11  diagrammatically shows 11 light fluxes F 1  to F 11 , which exit out of a projection system  100  according to Comparative Embodiment and reach the screen S. The light flux F 1  is a light flux that reaches the lowest image height position. The light flux F 11  is a light flux that reaches the highest image height position. The light fluxes F 2  to F 10  are light fluxes that reach height positions between the height position that the light flux F 1  reaches and the height position that the light flux F 11  reaches. The projection system  100  according to Comparative Embodiment has a configuration corresponding to the projection system described in JP-A-2010-20344 having been described as related art. The projection system  100  according to Comparative Embodiment further has a configuration corresponding to the projection system  3 A in the embodiment described above, and the corresponding components have the same reference characters in the following description. 
     The projection system  100  according to the present embodiment is formed of the first optical system  31  and the second optical system  32  sequentially arranged from the demagnifying side toward the magnifying side, as shown in  FIG. 11 . The projection system  100  forms the intermediate image  33  in a position between the demagnifying-side image formation plane and the magnifying-side image formation plane, as shown in  FIG. 12 . The first optical system  31  is a refractive optical system including a plurality of lenses. The second optical system  32  is a reflection mirror  101  having a reflective surface. The liquid crystal panels  18 R,  18 G, and  18 B of the image formation section  2  are disposed in the demagnifying-side image formation plane.  FIGS. 11 and 12  show the liquid crystal panel  18 G, which is one of the three liquid crystal panels  18 R,  18 G, and  18 B. The liquid crystal panels  18 R,  18 G, and  18 B form projection images on one side of the optical axis N of the first optical system  31  in the demagnifying-side image formation plane. The screen S is disposed in the magnifying-side image formation plane. The intermediate image  33  is formed on the other side of the optical axis N of the first optical system  31 . A final image projected on the screen S has an oblong shape elongated in the lateral direction. In the present embodiment, the final image has the aspect ratio of 16:10. 
     The first optical system  31  includes the cross dichroic prism  19  and the first lens L 1  to the fifteenth lens L 15 , which form 15 lenses, as shown in  FIG. 12 . The first lens L 1  to the fifteenth lens L 15  are arranged in the presented order from the demagnifying side toward the magnifying side. In the present embodiment, the second lens L 2  and the third lens L 3  are bonded to each other to form the first doublet L 21 . The fourth lens L 4  and the fifth lens L 5  are bonded to each other to form the second doublet L 22 . The eleventh lens L 11  and the twelfth lens L 12  are bonded to each other to form the third doublet L 23 . The stop O is disposed between the seventh lens L 7  and the eighth lens L 8 . In the present embodiment, the sixth lens L 6 , the ninth lens L 9 , and the fifteenth lens L 15  are three lenses that are each an aspheric lens having aspheric surfaces on both surfaces. 
     In the first optical system  31 , the fifteenth lens L 15 , which is located in a position closest to the magnifying side, has negative power. The principal rays between the first optical system  31  and the second optical system  32  separate away from each other toward the second optical system  32 . 
     The second optical system  32  is formed of the reflection mirror  101  having the reflective surface  42 . The reflective surface  42  is located at the lower side Y 2  of the optical axis N of the first optical system  31 . The reflective surface  42  is so shaped that the cross-sectional shape in the plane YZ shown in  FIG. 12  is rotated around the optical axis N over the angular range of 90° toward one side and the other side of the axis-X direction. The reflective surface  42  is an aspheric surface. 
     The intermediate image  33  is formed in a position between the fifteenth lens L 15  of the first optical system  31  and the reflection mirror  101  of the second optical system  32 . The intermediate image  33 , which is conjugate with the screen S which corresponds to the magnifying-side image formation plane, greatly inclines so that the magnification Q of each of light fluxes is equal, as shown in  FIG. 12 . The intermediate image  33  is a conjugate image that is an inverted final image turned upside down. The intermediate image  33  is so distorted that an oblong final image is projected on the screen S, which is the magnifying-side image formation plane. More specifically, the intermediate image  33  has a shape that allows reduction in trapezoidal distortion of the final image formed on the screen S. That is, the intermediate image  33  has distortion opposite the trapezoidal distortion of the final image. The intermediate image  33  is therefore so formed that the shortest edge thereof is the edge having the highest image height on the screen S. 
     Lens Data 
     Data on the lenses of the projection system  100  are as follows: The surfaces of the lenses are numbered sequentially from the demagnifying side toward the magnifying side. A surface having a surface number with * is an aspheric surface. The surface number  1  represents the demagnifying-side surface of the cross dichroic prism  19 , and the surface number  2  represents the magnifying-side surface thereof. The fields labeled with the surface number  20  show dummy data. The reference characters given in the first optical system  31  are the reference characters of the lenses thereof. The row of the reference character represents the reference character of the reflective surface  42 . That is, the surface number  32  represents the reflective surface  42  of the reflection mirror  101 . Reference character r denotes the radius of curvature in millimeters. Reference character d denotes the on-axis inter-surface distance in millimeters. Reference character nd denotes the refractive index. Reference character νd denotes the Abbe number. Reference character Y denotes the effective radius in the axis-Y direction. Reference character X denotes the effective radius in the axis-X direction. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
               
                 Surface 
                 Reference 
                   
                   
                   
                   
                   
                   
               
               
                 number 
                 character 
                 r 
                 d 
                 nd 
                 vd 
                 Y 
                 X 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Object 
                   
                 0 
                 9.5 
                   
                   
                   
                   
               
               
                 plane 
                   
                   
                   
                   
                   
                   
                   
               
               
                  1 
                   
                 0 
                 25.91 
                 1.516331 
                 64.14 
                 13.149 
                 13.149 
               
               
                  2 
                   
                 0 
                 0 
                   
                   
                 15.92 
                 15.92 
               
               
                  3 
                 L1 
                 28.94031 
                 9.6 
                 1.497 
                 81.55 
                 16.8 
                 16.8 
               
               
                  4 
                   
                 −72.35054 
                 0.2 
                   
                   
                 16.597 
                 16.597 
               
               
                  5 
                 L2 
                 28.00787 
                 7.6 
                 1.497 
                 81.55 
                 14.95 
                 14.95 
               
               
                  6 
                 L3 
                 −80.72822 
                 1.2 
                 1.805181 
                 25.43 
                 14.201 
                 14.201 
               
               
                  7 
                   
                 84.18233 
                 0.2 
                   
                   
                 13.279 
                 13.279 
               
               
                  8 
                 L4 
                 23.1621 
                 10.5 
                 1.516331 
                 64.14 
                 12.332 
                 12.332 
               
               
                  9 
                 L5 
                 −18.39049 
                 1.2 
                 1.903658 
                 31.32 
                 10.829 
                 10.829 
               
               
                  10 
                   
                 49.86134 
                 0.2 
                   
                   
                 10.088 
                 10.088 
               
               
                 *11 
                 L6 
                 22.23222 
                 1.4 
                 1.589131 
                 61.15 
                 10.088 
                 10.088 
               
               
                 *12 
                   
                 14.51177 
                 0.5 
                   
                   
                 9.871 
                 9.871 
               
               
                  13 
                 L7 
                 18.70105 
                 4 
                 1.487491 
                 70.24 
                 9.984 
                 9.984 
               
               
                  14 
                   
                 85.77957 
                 2.5 
                   
                   
                 9.744 
                 9.744 
               
               
                 Stop 
                   
                 0 
                 2.657363 
                   
                   
                 9.434 
                 9.434 
               
               
                 plane 
                   
                   
                   
                   
                   
                   
                   
               
               
                  16 
                 L8 
                 26.25511 
                 4.2 
                 1.84666 
                 23.78 
                 9.973 
                 9.973 
               
               
                  17 
                   
                 −72.32644 
                 2.804685 
                   
                   
                 9.8 
                 9.8 
               
               
                 *18 
                 L9 
                 −285.96073 
                 1.8 
                 1.743198 
                 49.3 
                 9.312 
                 9.312 
               
               
                 *19 
                   
                 20.09443 
                 3 
                   
                   
                 9.563 
                 9.563 
               
               
                  20 
                 dummy 
                 0 
                 11.16296 
                   
                   
                 10 
                 10 
               
               
                  21 
                 L10 
                 294.94797 
                 3.8 
                 1.761821 
                 26.52 
                 16.702 
                 16.702 
               
               
                  22 
                   
                 −185.55163 
                 0.2 
                   
                   
                 17.394 
                 17.394 
               
               
                  23 
                 L11 
                 88.77106 
                 11.7 
                 1.654115 
                 39.68 
                 18.691 
                 18.691 
               
               
                  24 
                 L12 
                 −32.58663 
                 2 
                 1.805181 
                 25.43 
                 19.198 
                 19.198 
               
               
                  25 
                   
                 −207.71339 
                 7.258462 
                   
                   
                 20.588 
                 20.588 
               
               
                  26 
                 L13 
                 47.46195 
                 11 
                 1.581439 
                 40.75 
                 24.7 
                 24.7 
               
               
                  27 
                   
                 −286.50997 
                 1.90653 
                   
                   
                 24.527 
                 24.527 
               
               
                  28 
                 L14 
                 −162.79029 
                 2 
                 1.805181 
                 25.43 
                 24.367 
                 24.367 
               
               
                  29 
                   
                 86.54168 
                 13.837579 
                   
                   
                 24.298 
                 24.298 
               
               
                 *30 
                 L15 
                 278 
                 2.8 
                 1.531132 
                 55.75 
                 26.772 
                 26.772 
               
               
                 *31 
                   
                 32.7897 
                 112.362421 
                   
                   
                 28.077 
                 28.077 
               
               
                 *32 
                 42 
                 −53.40115 
                 −501 
                   
                   
                 59.723 
                 59.723 
               
               
                  33 
                   
                 0 
                 0 
                   
                   
                 1450.989 
                 1450.989 
               
               
                 Image 
                   
                 0 
                 0 
                   
                   
                 1450.989 
                 1450.989 
               
               
                 plane 
               
               
                   
               
            
           
         
       
     
     As shown in the lens data, the field of the on-axis inter-surface distance d labeled with the surface number  32  shows the distance between the screen S and the reflective surface  42 . The field of the on-axis inter-surface distance d labeled with the surface number  32  therefore shows the projection distance f of the projection system  100 . In the present embodiment, f=501 mm. In the present embodiment, the effective radius of the reflective surface  42  is 59.723 mm in the axis-Y direction and 59.723 mm in the axis-X direction. 
     Aspheric data of the surface number  11  are as follows. 
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 Conic constant 
                 1.891766E+00 
               
               
                   
                 Fourth-order coefficient 
                 −3.21888E−04 
               
               
                   
                 Sixth-order coefficient 
                 1.748382E−06 
               
               
                   
                 Eighth-order coefficient 
                 −4.334786E−09  
               
               
                   
                 Tenth-order coefficient 
                 0 
               
               
                   
                   
               
            
           
         
       
     
     Aspheric data of the surface number  12  are as follows. 
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 Conic constant 
                 −1.715462E+00 
               
               
                   
                 Fourth-order coefficient 
                 −2.456905E−04 
               
               
                   
                 Sixth-order coefficient 
                 2.077486E−06 
               
               
                   
                 Eighth-order coefficient 
                 −8.095391E−09 
               
               
                   
                 Tenth-order coefficient 
                 1.652912E−11 
               
               
                   
                   
               
            
           
         
       
     
     Aspheric data of the surface number  18  are as follows. 
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 Conic constant 
                 −1E+00 
               
               
                   
                 Fourth-order coefficient 
                 −9E−05 
               
               
                   
                 Sixth-order coefficient 
                 0 
               
               
                   
                 Eighth-order coefficient 
                 0 
               
               
                   
                 Tenth-order coefficient 
                 0 
               
               
                   
                   
               
            
           
         
       
     
     Aspheric data of the surface number  19  are as follows. 
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 Conic constant 
                 −5.793922E−01 
               
               
                   
                 Fourth-order coefficient 
                  −5.29339E−05 
               
               
                   
                 Sixth-order coefficient 
                  1.037748E−07 
               
               
                   
                 Eighth-order coefficient 
                 −1.806724E−10 
               
               
                   
                 Tenth-order coefficient 
                 0 
               
               
                   
                   
               
            
           
         
       
     
     Aspheric data of the surface number  30  are as follows. 
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 Conic constant 
                      9E+01 
               
               
                   
                 Fourth-order coefficient 
                 −6.930833E−06 
               
               
                   
                 Sixth-order coefficient 
                  2.131125E−08 
               
               
                   
                 Eighth-order coefficient 
                 −4.144247E−11 
               
               
                   
                 Tenth-order coefficient 
                  3.842719E−14 
               
               
                   
                 Twelfth-order 
                 −1.672012E−17 
               
               
                   
                 coefficient 
               
               
                   
                   
               
            
           
         
       
     
     Aspheric data of the surface number  31  are as follows. 
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 Conic constant 
                 0 
               
               
                   
                 Fourth-order coefficient 
                 −2.83341E−05 
               
               
                   
                 Sixth-order coefficient 
                  3.95412E−08 
               
               
                   
                 Eighth-order coefficient 
                 −6.268725E−11  
               
               
                   
                 Tenth-order coefficient 
                 5.533984E−14 
               
               
                   
                 Twelfth-order 
                 −2.65766E−17 
               
               
                   
                 coefficient 
               
               
                   
                 Fourteenth-order 
                 2.765817E−21 
               
               
                   
                 coefficient 
               
               
                   
                   
               
            
           
         
       
     
     Aspheric data of the surface number  32  are as follows. 
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 Conic constant 
                 −9.858117E−01 
               
               
                   
                 Fourth-order coefficient 
                 1.107991E−07 
               
               
                   
                 Sixth-order coefficient 
                 3.623153E−11 
               
               
                   
                 Eighth-order coefficient 
                 −6.165432E−14 
               
               
                   
                 Tenth-order coefficient 
                 2.288554E−17 
               
               
                   
                 Twelfth-order 
                 −4.232985E−21 
               
               
                   
                 coefficient 
               
               
                   
                 Fourteenth-order 
                 3.064401E−25 
               
               
                   
                 coefficient 
               
               
                   
                   
               
            
           
         
       
     
       FIG. 13  shows an MTF of the projection system  100  according to Comparative Embodiment on the magnifying side.  FIG. 14  is a spot diagram showing spots produced by the projection system  100  according to Comparative Embodiment. 
     The projection distance of the projection system  100  according to Comparative Embodiment is f=501 mm, as shown in the lens data. In the projection system  100  according to Comparative Embodiment, the effective radius of the reflective surface  42  is 59.723 mm in the axis-Y direction and 59.723 mm in the axis-X direction. In contrast, the projection distance of the projection system  3 A according to Embodiment 1 is f=466.75 mm. On the other hand, in the projection system  3 A according to Embodiment 1, the effective radius of the reflective surface  42  is 21.22 mm in the axis-X direction and 21.22 mm in the axis-Y direction. The projection system  3 A according to Embodiment 1 therefore allows suppression of an increase in the size of the reflective surface  42  even in the case of a short projection distance. 
     Embodiment 2 
       FIG. 15  is a light ray diagram of a projection system according to Embodiment 2 of the present disclosure.  FIG. 16  is a light ray diagram of the projection system according to Embodiment 2 enlarged.  FIG. 17  is a light ray diagram with a portion including a second optical system of the projection system according to Embodiment 1 enlarged.  FIG. 15  diagrammatically shows the 11 light fluxes F 1  to F 11 , which exit out of a projection system  3 B and reach the screen S. The light flux F 1  is a light flux that reaches the lowest image height position. The light flux F 11  is a light flux that reaches the highest image height position. The light fluxes F 2  to F 10  are light fluxes that reach height positions between the height position that the light flux F 1  reaches and the height position that the light flux F 11  reaches. The projection system  3 B according to the present embodiment has a configuration corresponding to the projection system  3 A in the embodiment described above, and the corresponding components therefore have the same reference characters in the description. 
     The projection system  3 B according to the present embodiment is formed of the first optical system  31  and the second optical system  32  sequentially arranged from the demagnifying side toward the magnifying side, as shown in  FIG. 15 . The projection system  3 B forms the intermediate image  33  in a position between the demagnifying-side image formation plane and the magnifying-side image formation plane of the projection system  3 B, as shown in  FIG. 16 . 
     The first optical system  31  is a refractive optical system including a plurality of lenses. The second optical system  32  is a lens  35 . The liquid crystal panels  18 R,  18 G, and  18 B of the image formation section  2  are disposed in the demagnifying-side image formation plane.  FIGS. 15 and 16  show the liquid crystal panel  18 G, which is one of the three liquid crystal panels  18 R,  18 G, and  18 B. The liquid crystal panels  18 R,  18 G, and  18 B form projection images on one side of the optical axis N of the first optical system  31  in the demagnifying-side image formation plane. The screen S is disposed in the magnifying-side image formation plane. 
     The intermediate image  33  is formed in the second optical system  32 , that is, the lens  35 . The intermediate image  33  is formed on the other side of the optical axis N of the first optical system  31 . A final image projected on the screen S has an oblong shape elongated in the lateral direction. In the present embodiment, the final image has the aspect ratio of 16:10. 
     The first optical system  31  includes the cross dichroic prism  19  and a first lens L 1  to a fifteenth lens L 15 , which form 15 lenses, as shown in  FIG. 16 . The first lens L 1  to the fifteenth lens L 15  are arranged in the presented order from the demagnifying side toward the magnifying side. In the present embodiment, the second lens L 2  and the third lens L 3  are bonded to each other to form the first doublet L 21 . The fourth lens L 4  and the fifth lens L 5  are bonded to each other to form the second doublet L 22 . The eleventh lens L 11  and the twelfth lens L 12  are bonded to each other to form the third doublet L 23 . The thirteenth lens L 13  and the fourteenth lens L 14  are bonded to each other to form the fourth doublet L 24 . In the present embodiment, the first lens L 1  to the fifteenth lens L 15 , which form the first optical system  31 , are each a spherical lens. 
     In the first optical system  31 , the fifteenth lens L 15 , which is located in a position closest to the magnifying side, has positive power. In the first optical system  31 , the magnifying-side lens group LG 25 , which is located in a position closest to the second optical system  32 , has positive power. The magnifying-side lens group LG 25  is formed of three lenses arranged from the side facing the second optical system  32 . That is, the thirteenth lens L 13 , the fourteenth lens L 14 , and the fifteenth lens L 15  form the magnifying-side lens group LG 25 . Since the magnifying-side lens group LG 25  has positive power in the first optical system  31 , the principal rays between the first optical system  31  and the second optical system  32  approach each other toward the second optical system  32 . 
     The lens  35 , which forms the second optical system  32 , is made of resin. The lens  35  has the first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43  sequentially arranged from the demagnifying side toward the magnifying side, as shown in  FIG. 17 . In the case where the lens  35  is made of resin, the lens  35  can be manufactured in injection molding. The lens  35  having a complicated shape is therefore readily manufactured. 
     Also, in the following description of the present embodiment, three axes perpendicular to one another are called the axes X, Y, and Z for convenience. A first direction in which the first transmissive surface  41  and the reflective surface  42  are arranged is called the axis-Z direction. A second direction, which is called an axis-y direction, coincides with the vertical direction of the screen S. One side of the axis Y is called the upper side Y 1 , and the other side of the axis Y is called the lower side Y 2 . A first plane perpendicular to the axis X and containing the axes Y and Z is called the plane YZ.  FIGS. 15 to 17  therefore each show the plane YZ. The optical axis N of the first optical system  31  extends in the axis-Z direction. The image formation section  2  forms a projection image on the upper side Y 1  of the optical axis N of the first optical system  31 . The intermediate image  33  is formed on the lower side Y 2  of the optical axis N of the first optical system  31 . The lateral direction of the screen S coincides with a third direction, which is called the axis-X direction. In the following description, the imaginary axis M extending in the axis-Z direction is set in the plane YZ. The imaginary axis M is the reference axis used in the design of the lens  35 . The imaginary axis M is perpendicular to the screen S, which is the magnifying-side image formation plane. The imaginary axis M is substantially perpendicular to the screen S in some cases. 
     The first transmissive surface  41  and the reflective surface  42  are located at the lower side Y 2  of the imaginary axis M. The second transmissive surface  43  is located at the upper side Y 1  of the imaginary axis M. The reflective surface  42  has a concave shape. The reflective surface  42  therefore has positive power. The reflective surface  42  is provided by externally forming a reflective coating on the lens  35 . The second transmissive surface  43  has a convex shape protruding toward the magnifying side. The second transmissive surface  43  therefore has positive power. The first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43  form a coaxial optical system having surfaces rotationally symmetric with respect to the imaginary axis M. Therefore, the imaginary axis M is the reference axis used in the design of the lens  35  and is the optical axis of the lens  35 . In the present embodiment, the imaginary axis M coincides with the optical axis N of the first optical system  31 . The imaginary axis M may not necessarily coincide with the optical axis N of the first optical system  31 . 
     The upper and lower halves of the lens  35  are each configured to be rotationally symmetric with respect to the imaginary axis M. That is, the first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43  are so shaped that the cross-sectional shape in the plane YZ shown in  FIG. 17  is rotated around the imaginary axis M over the angular range of 90° toward one side and the other side of the axis-X direction. In the present embodiment, the first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43  are each an aspheric surface. 
     The imaginary line P can be specified in the lens  35  of the second optical system  32 , as shown in  FIG. 17 . The imaginary line P connects the upper intersection  53  to the lower intersection  54 , the upper intersection  53  being an intersection where the upper peripheral light ray  51   a  of the upper-end light flux  51 , where the upper-end light flux  51  is the light flux passing through the axis-Y-direction upper end of the effective range  50  of the second transmissive surface  43 , and the upper peripheral light ray  52   a  of the lower-end light flux  52 , where the lower-end light flux  52  is the light flux passing through the axis-Y-direction lower end of the effective range  50 , intersect with each other in the plane YZ, and the lower intersection  54  being an intersection where the lower peripheral light ray  51   b  of the upper-end light flux  51  and the lower peripheral light ray  52   b  of the lower-end light flux  52  intersect with each other in the plane YZ. The imaginary line P inclines by 96.64° with respect to the imaginary vertical line V, which is perpendicular to the imaginary axis M in the plane YZ. Further, in the lens  35  of the second optical system  32 , the upper peripheral light ray  51   a  of the upper-end light flux  51  travels from the second transmissive surface  43  toward the reflective surface  42 , and the point where the upper peripheral light ray  51   a  intersects another light ray reflected off the reflective surface  42  for the first time is called the first intersection. Similarly, the lower peripheral light ray  52   b  of the lower-end light flux  52  travels from the second transmissive surface  43  toward the reflective surface  42 , and the point where the lower peripheral light ray  52   b  intersects another light ray reflected off the reflective surface  42  for the first time is called the second intersection. Under the above definition, the line that connects the first intersection to the second intersection is named as a pupil  44 . In the present embodiment, the pupil  44  inclines by 100.348° with respect to the imaginary vertical line V in the plane YZ. The imaginary axis M, which is the optical axis of the lens  35 , is not necessary to pass through the center of the imaginary line P. 
     The intermediate image  33  is an inverted final image turned upside down. The intermediate image  33  is an image so distorted that an oblong final image is projected on the screen S, which is the magnifying-side image formation plane. More specifically, the intermediate image  33  has a shape that allows reduction in trapezoidal distortion of the final image formed on the screen S. That is, the intermediate image  33  has distortion opposite the trapezoidal distortion of the final image. The intermediate image  33  is therefore so formed that the shortest edge thereof is the edge having the highest image height on the screen S. 
     Lens Data 
     Data on the lenses of the projection system  3 B are as follows: The surfaces of the lenses are numbered sequentially from the demagnifying side toward the magnifying side. A surface having a surface number with * is an aspheric surface. The surface number  1  represents the demagnifying-side surface of the cross dichroic prism  19 , and the surface number  2  represents the magnifying-side surface thereof. The fields labeled with the surface number  20  show dummy data. The intermediate image  33  is formed in a position between the surface number  31  and the surface number  33 . The row of the reference characters represents the reference characters of the lenses thereof. The reference characters given in the second optical system  32  are the reference characters of the first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43 . That is, the surface number  31  represents the first transmissive surface  41  of the lens  35 . The surface number  33  represents the reflective surface  42  of the lens  35 . The surface number  35  represents the second transmissive surface  43  of the lens  35 . Reference character r denotes the radius of curvature in millimeters. Reference character d denotes the on-axis inter-surface distance in millimeters. Reference character nd denotes the refractive index. Reference character νd denotes the Abbe number. Reference character Y denotes the effective radius in the axis-Y direction. Reference character X denotes the effective radius in the axis-X direction. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
               
                 Surface 
                 Reference 
                   
                   
                   
                   
                   
                   
               
               
                 number 
                 character 
                 r 
                 d 
                 nd 
                 vd 
                 Y 
                 X 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Object 
                   
                 0 
                 7.125 
                   
                   
                   
                   
               
               
                 plane 
                   
                   
                   
                   
                   
                   
                   
               
               
                  1 
                   
                 0 
                 19.4325 
                 1.516331  
                 64.14 
                 10.237 
                 10.237 
               
               
                  2 
                   
                 0 
                 0.29133 
                   
                   
                 12.974 
                 12.974 
               
               
                  3 
                 L1 
                 −291.41008 
                 3.665508 
                 1.700675  
                 28.54 
                 13 
                 13 
               
               
                  4 
                   
                 −30.58875 
                 0.15 
                   
                   
                 13.118 
                 13.118 
               
               
                  5 
                 L2 
                 87.97506 
                 6.762902 
                 1.437002 
                 95.1 
                 12.773 
                 12.773 
               
               
                  6 
                 L3 
                 −19.01953 
                 0.9 
                 1.846579  
                 23.86 
                 12.67 
                 12.67 
               
               
                  7 
                   
                 −56.62914 
                 0.15 
                   
                   
                 13.166 
                 13.166 
               
               
                  8 
                 L4 
                 64.3433 
                 6.854463 
                 1.516331 
                 64.14 
                 13.218 
                 13.218 
               
               
                  9 
                 L5 
                 −23.14892 
                 0.9 
                 1.903658 
                 31.32 
                 13.134 
                 13.134 
               
               
                  10 
                   
                 −673.61176 
                 0 
                   
                   
                 13.581 
                 13.581 
               
               
                  11 
                 L6 
                 68.20357 
                 7.449857 
                 1.437002 
                 95.1 
                 13.827 
                 13.827 
               
               
                  12 
                   
                 −32.37093 
                 32.969471 
                   
                   
                 13.955 
                 13.955 
               
               
                  13 
                 L7 
                 0 
                 6 
                   
                   
                 9 
                 9 
               
               
                  14 
                   
                 15.69631 
                 1.983635 
                 1.437135 
                 94.92 
                 10.047 
                 10.047 
               
               
                 Stop 
                   
                 17.29696 
                 7.423181 
                   
                   
                 9.814 
                 9.814 
               
               
                 plane 
                   
                   
                   
                   
                   
                   
                   
               
               
                  16 
                 L8 
                 22.66617 
                 3.7217 
                 1.825709 
                 24.28 
                 10.179 
                 10.179 
               
               
                  17 
                   
                 519.71825 
                 1.496478 
                   
                   
                 9.907 
                 9.907 
               
               
                  18 
                 L9 
                 30.75598 
                 1.4 
                 1.437002  
                 95.1 
                 8.944 
                 8.944 
               
               
                  19 
                   
                 14.84402 
                 2.91106 
                   
                   
                 8.107 
                 8.107 
               
               
                  20 
                 dummy 
                 0 
                 1.885816 
                   
                   
                 8 
                 8 
               
               
                  21 
                 L10 
                 −17.94171 
                 3.372831 
                 1.835309 
                 41.31 
                 8 
                 8 
               
               
                  22 
                   
                 −37.88337 
                 2.921753 
                   
                   
                 8.913 
                 8.913 
               
               
                  23 
                 L11 
                 44.95773 
                 6.413839 
                 1.5256 
                 71.65 
                 9.921 
                 9.921 
               
               
                  24 
                 L12 
                 −15.02119 
                 6.2879 
                 1.846663 
                 23.78 
                 9.99 
                 9.99 
               
               
                  25 
                   
                 −64.64749 
                 41.303086 
                   
                   
                 11.656 
                 11.656 
               
               
                  26 
                 L13 
                 30.13656 
                 10 
                 1.755001  
                 52.32 
                 23 
                 23 
               
               
                  27 
                 L14 
                 45.37156 
                 11.668738  
                 1.590317 
                 63.23 
                 20.45 
                 20.45 
               
               
                  28 
                   
                 −77.39811 
                 0.227404 
                   
                   
                 19.09 
                 19.09 
               
               
                  29 
                 L15 
                 −83.1086 
                 10 
                 1.846663  
                 23.78 
                 18.642 
                 18.642 
               
               
                  30 
                   
                 28.59819 
                 3.217563 
                   
                   
                 14.833 
                 14.833 
               
               
                 *31 
                 41 
                 −29.46454  
                 20 
                 1.531132  
                 55.75 
                 14.986 
                 14.986 
               
               
                  32 
                   
                 0 
                 0 
                 1.531132  
                 55.75 
                 12.839 
                 12.839 
               
               
                 *33 
                 42 
                 −16.13835 
                 0 
                 1.531132 
                 55.75 
                 12.678 
                 12.678 
               
               
                  34 
                   
                 0 
                 −20 
                 1.531132 
                 55.75 
                 24.667 
                 24.667 
               
               
                 *35 
                 43 
                 12.4031 
                 −290 
                   
                   
                 12.305 
                 12.305 
               
               
                 Image 
                   
                 0 
                 0 
                   
                   
                 869.934 
                 869.934 
               
               
                 plane 
               
               
                   
               
            
           
         
       
     
     As shown in the lens data, the refractive index nd of the lens  35  is 1.531132, and the Abbe number νd of the lens  35  is 55.75. The field of the on-axis inter-surface distance d labeled with the surface number  35  shows the distance between the screen S and the second transmissive surface  43  of the lens  35 . The field of the on-axis inter-surface distance d labeled with the surface number  35  therefore shows the projection distance f of the projection system  3 B. In the present embodiment, f=290 mm. 
     Aspheric data of the surface number  31  are as follows. 
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 Conic constant 
                 0 
               
               
                   
                 Fourth-order coefficient 
                  2.54685E−04 
               
               
                   
                 Sixth-order coefficient 
                 −8.580888E−07 
               
               
                   
                 Eighth-order coefficient 
                  2.19535E−09 
               
               
                   
                 Tenth-order coefficient 
                 −2.461316E−12 
               
               
                   
                 Twelfth-order 
                 0 
               
               
                   
                 coefficient 
               
               
                   
                 Fourteenth-order 
                 0 
               
               
                   
                 coefficient 
               
               
                   
                   
               
            
           
         
       
     
     Aspheric data of the surface number  33  are as follows. 
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 Conic constant 
                     −1E+00 
               
               
                   
                 Fourth-order coefficient 
                  2.969055E−05 
               
               
                   
                 Sixth-order coefficient 
                 −1.565091E−07 
               
               
                   
                 Eighth-order coefficient 
                  1.255633E−09 
               
               
                   
                 Tenth-order coefficient 
                 −5.109606E−12 
               
               
                   
                 Twelfth-order 
                  1.345092E−14 
               
               
                   
                 coefficient 
               
               
                   
                 Fourteenth-order 
                  −1.73251E−17 
               
               
                   
                 coefficient 
               
               
                   
                   
               
            
           
         
       
     
     Aspheric data of the surface number  35  are as follows. 
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 Conic constant 
                 0 
               
               
                   
                 Fourth-order coefficient 
                 −2.946854E−05 
               
               
                   
                 Sixth-order coefficient 
                 −2.297529E−07 
               
               
                   
                 Eighth-order coefficient 
                 2.645623E−09 
               
               
                   
                 Tenth-order coefficient 
                 −1.009952E−11 
               
               
                   
                 Twelfth-order 
                 0 
               
               
                   
                 coefficient 
               
               
                   
                 Fourteenth-order 
                 0 
               
               
                   
                 coefficient 
               
               
                   
                   
               
            
           
         
       
     
     The present embodiment can also provide the same effects as those provided by the embodiment described above. 
     In the present embodiment, the inclination angle θ by which the imaginary line P inclines with respect to the imaginary vertical line V is 96.64°. Therefore, θ=96.64°, which satisfies the conditional expressions (1) and (2) below. Further, the inclination angle η by which the pupil  44  inclines with respect to the imaginary vertical line V is 100.348°. Therefore, η=100.348°, which satisfies the conditional expressions (3) and (4) below. Therefore, the projection apparatus  3 B according to the present embodiment allows suppression of the difference in the amount of light between the upper portion and the lower portion of the screen S.
 
0°&lt;θ&lt;90°+γ  (1)
 
90°&lt;θ  (2)
 
0°&lt;η&lt;90°+γ  (3)
 
90°&lt;η  (4)
 
     θ: Inclination angle over which an end of the imaginary line P facing the upper intersection  53  rotates counterclockwise relative to the imaginary vertical line V around the intersection of the imaginary vertical line V and the imaginary line P 
     η: Inclination angle over which an end of the pupil  44  facing the first intersection rotates counterclockwise relative to the imaginary vertical line V around the intersection of the imaginary vertical line V and the pupil  44   
     γ: Angle from the imaginary axis M to the lower peripheral light ray  52   b  passing through the effective range  50  of the second transmissive surface  43  and intersects the imaginary axis M 
     The projection distance of the projection system  3 B according to the present embodiment is f=290 mm. In the projection system  3 B according to the present embodiment, the effective radius of the reflective surface  42  is 12.678 mm in the axis-X direction and 12.678 mm in the axis-Y direction. The projection system  3 B according to the present embodiment therefore allows suppression of an increase in the size of the reflective surface  42  even in the case of a short projection distance. 
       FIG. 18  shows an MTF of the projection system  3 B on the magnifying side. The MTF was calculated under the following conditions: The image formation planes were divided along the axis Y; and the resultant halves were each divided into 11 areas. Light rays used in the calculation of the MTF are so weighted that the weighting ratio among the light rays having the wavelength of 620 nm, the light rays having the wavelength of 550 nm, and the light rays having the wavelength of 470 nm is 2:7:1. The horizontal axis of  FIG. 18 , which shows the MTF, represents the spatial frequency. The spatial frequency of 0.24 cycles corresponds to the resolution of 16.7 μm. The vertical axis of  FIG. 18  represents the contrast reproduction ratio. In the present embodiment, a decrease in resolution is suppressed, as shown in  FIG. 18 .  FIG. 19  is a spot diagram showing spots produced by the projection system  3 B. In the present embodiment, variation in the size of the spots is suppressed, as shown in  FIG. 19 . 
     In the present embodiment, the first lens L 1  to the fifteenth lens L 15 , which form the first optical system  31 , are each a spherical lens but provide satisfactory optical characteristics. That is, in the present embodiment, aberrations produced by the projection system can be satisfactorily suppressed although the first optical system  31 , which is a refractive optical system, has no aspheric lens. 
     Embodiment 3 
       FIG. 20  is a light ray diagram of a projection system according to Embodiment 3 of the present disclosure.  FIG. 21  is a light ray diagram of the projection system according to Embodiment 3 enlarged.  FIG. 22  is a light ray diagram with a portion including a second optical system of the projection system according to Embodiment 3 enlarged.  FIG. 20  diagrammatically shows the 11 light fluxes F 1  to F 11 , which exit out of a projection system  3 C and reach the screen S. The light flux F 1  is a light flux that reaches the lowest image height position. The light flux F 11  is a light flux that reaches the highest image height position. The light fluxes F 2  to F 10  are light fluxes that reach height positions between the height position that the light flux F 1  reaches and the height position that the light flux F 11  reaches. The projection system  3 C according to the present embodiment has a configuration corresponding to those in the embodiments described above, and the corresponding components therefore have the same reference characters in the description. Further, the same configurations in the embodiments described above will not be described. 
     The projection system  3 C according to the present embodiment is formed of the first optical system  31  and the second optical system  32  sequentially arranged from the demagnifying side toward the magnifying side, as shown in  FIG. 20 . The projection system  3 C forms the intermediate image  33  in a position between the demagnifying-side image formation plane and the magnifying-side image formation plane of the projection system  3 C, as shown in  FIG. 21 . 
     The intermediate image  33  is formed in the second optical system  32 , that is, a lens  35 . The intermediate image  33  is formed on the other side of the optical axis N of the first optical system  31 . A final image projected on the screen S has an oblong shape elongated in the lateral direction. In the present embodiment, the final image has the aspect ratio of 16:10. 
     The first optical system  31  includes the cross dichroic prism  19  and the first lens L 1  to the fifteenth lens L 15 , which form 15 lenses, as shown in  FIG. 21 . The first lens L 1  to the fifteenth lens L 15  are arranged in the presented order from the demagnifying side toward the magnifying side. In the present embodiment, the second lens L 2  and the third lens L 3  are bonded to each other to form the first doublet L 21 . The fourth lens L 4  and the fifth lens L 5  are bonded to each other to form the second doublet L 22 . The tenth lens L 10  and the eleventh lens L 11  are bonded to each other to form the third doublet L 23 . The twelfth lens L 12  and the thirteenth lens L 13  are bonded to each other to form the fourth doublet L 24 . In the present embodiment, out of the lenses that form the first optical system  31 , the sixth lens L 6 , the ninth lens L 9 , and the fifteenth lens L 15  are each an aspheric lens, and the other lenses are each a spherical lens. The lenses that form the first optical system  31  may instead each be formed of a spherical lens. 
     In the first optical system  31 , the fifteenth lens L 15 , which is located in a position closest to the magnifying side, has positive power. In the first optical system  31 , the magnifying-side lens group LG 25 , which is located in a position closest to the second optical system  32 , has positive power. The magnifying-side lens group LG 25  is formed of the fourteenth lens L 14  and the fifteenth lens L 15 . Since the magnifying-side lens group LG 25  has positive power in the first optical system  31 , the principal rays between the first optical system  31  and the second optical system  32  approach each other toward the second optical system  32 . 
     The lens  35 , which forms the second optical system  32 , is made of resin. The lens  35  has the first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43  sequentially arranged from the demagnifying side toward the magnifying side, as shown in  FIG. 22 . In the case where the lens  35  is made of resin, the lens  35  can be manufactured in injection molding. The lens  35  having a complicated shape is therefore readily manufactured. 
     Also, in the present embodiment, the coordinate axes are set in the same manner as in the embodiments described above. Further, also in the following description, the imaginary axis M is the reference axis used in the design of the lens  35 . The imaginary axis M is perpendicular to the screen S, which is the magnifying-side image formation plane. The imaginary axis M is substantially perpendicular to the screen S in some cases. 
     The first transmissive surface  41  and the reflective surface  42  are located at the lower side Y 2  of the imaginary axis M. The second transmissive surface  43  is located at the upper side Y 1  of the imaginary axis M. The reflective surface  42  has a concave shape. The reflective surface  42  therefore has positive power. The reflective surface  42  is provided by externally forming a reflective coating on the lens  35 . The second transmissive surface  43  has a convex shape protruding toward the magnifying side. The second transmissive surface  43  therefore has positive power. The first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43  form a coaxial optical system having surfaces rotationally symmetric with respect to the imaginary axis M. Therefore, the imaginary axis M is the reference axis used in the design of the lens  35  and is the optical axis of the lens  35 . In the present embodiment, the imaginary axis M coincides with the optical axis N of the first optical system  31 . The imaginary axis M may not necessarily coincide with the optical axis N of the first optical system  31 . 
     The upper and lower halves of the lens  35  are each configured to be rotationally symmetric with respect to the imaginary axis M. That is, the first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43  are so shaped that the cross-sectional shape in the plane YZ shown in  FIG. 22  is rotated around the imaginary axis M over the angular range of 90° toward one side and the other side of the axis-X direction. In the present embodiment, the first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43  are each an aspheric surface. 
     The imaginary line P can be specified in the lens  35  of the second optical system  32 , as shown in  FIG. 22 . The imaginary line P connects the upper intersection  53  to the lower intersection  54 , the upper intersection  53  being an intersection where the upper peripheral light ray  51   a  of the upper-end light flux  51 , where the upper-end light flux  51  is the light flux passing through the axis-Y-direction upper end of the effective range  50  of the second transmissive surface  43 , and the upper peripheral light ray  52   a  of the lower-end light flux  52 , where the lower-end light flux  52  is the light flux passing through the axis-Y-direction lower end of the effective range  50 , intersect with each other in the plane YZ, and the lower intersection  54  being an intersection where the lower peripheral light ray  51   b  of the upper-end light flux  51  and the lower peripheral light ray  52   b  of the lower-end light flux  52  intersect with each other in the plane YZ. In the present embodiment, the imaginary line P inclines by 97.74° with respect to the imaginary vertical line V, which is perpendicular to the imaginary axis M in the plane YZ. Further, in the lens  35  of the second optical system  32 , the upper peripheral light ray  51   a  of the upper-end light flux  51  travels from the second transmissive surface  43  toward the reflective surface  42 , and the point where the upper peripheral light ray  51   a  intersects another light ray reflected off the reflective surface  42  for the first time is called the first intersection. Similarly, the lower peripheral light ray  52   b  of the lower-end light flux  52  travels from the second transmissive surface  43  toward the reflective surface  42 , and the point where the lower peripheral light ray  52   b  intersects another light ray reflected off the reflective surface  42  for the first time is called the second intersection. Under the above definition, the line that connects the first intersection to the second intersection is named as a pupil  44 . In the present embodiment, the pupil  44  inclines by 103.93° with respect to the imaginary vertical line V in the plane YZ. 
     The intermediate image  33  is an inverted final image turned upside down. The intermediate image  33  is an image so distorted that an oblong final image is projected on the screen S, which is the magnifying-side image formation plane. More specifically, the intermediate image  33  is so distorted as to cancel the distortion produced by the second optical system  32  so that the trapezoidal distortion of the final image formed on the screen S decreases. That is, the intermediate image  33  has distortion opposite the trapezoidal distortion of the final image. The intermediate image  33  is therefore so formed that the shortest edge thereof is the edge having the highest image height on the screen S. 
     Lens Data 
     Data on the lenses of the projection system  3 C are as follows: The surfaces of the lenses are numbered sequentially from the demagnifying side toward the magnifying side. A surface having a surface number with * is an aspheric surface. The surface number  1  represents the demagnifying-side surface of the cross dichroic prism  19 , and the surface number  2  represents the magnifying-side surface thereof. The row of the reference characters represents the reference characters of the lenses thereof. The reference characters given in the second optical system  32  are the reference characters of the first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43 . That is, the surface number  29  represents the first transmissive surface  41  of the lens  35 . The surface number  31  represents the reflective surface  42  of the lens  35 . The surface number  33  represents the second transmissive surface  43  of the lens  35 . Reference character r denotes the radius of curvature in millimeters. Reference character d denotes the on-axis inter-surface distance in millimeters. Reference character nd denotes the refractive index. Reference character νd denotes the Abbe number. Reference character Y denotes the effective radius in the axis-Y direction. Reference character X denotes the effective radius in the axis-X direction. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
               
                 Surface 
                 Reference 
                   
                   
                   
                   
                   
                   
               
               
                 number 
                 character 
                 r 
                 d 
                 nd 
                 vd 
                 Y 
                 X 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Object 
                   
                 0 
                 9.5 
                   
                   
                   
                   
               
               
                 plane 
                   
                   
                   
                   
                   
                   
                   
               
               
                  1 
                   
                 0 
                 25.91 
                 1.51633 
                 64.14 
                 13.806 
                 13.806 
               
               
                  2 
                   
                 0 
                 0 
                   
                   
                 17.538 
                 17.538 
               
               
                  3 
                 L1 
                 30.66626 
                 10.398215 
                 1.642947  
                 58.64 
                 19 
                 19 
               
               
                  4 
                   
                 −98.82479 
                 0.1 
                   
                   
                 18.725 
                 18.725 
               
               
                  5 
                 L2 
                 31.47401 
                 10.907936 
                 1.532449 
                 47.06 
                 16.268 
                 16.268 
               
               
                  6 
                 L3 
                 −29.14613 
                 0.9 
                 1.846551  
                 23.88 
                 15.353 
                 15.353 
               
               
                  7 
                   
                 −109.24868 
                 0.1 
                   
                   
                 14.457 
                 14.457 
               
               
                  8 
                 L4 
                 18.90506 
                 8.366757 
                 1.440532 
                 90.6 
                 11.564 
                 11.564 
               
               
                  9 
                 L5 
                 −23.45512 
                 2.394143 
                 1.841404  
                 29.55 
                 10.378 
                 10.378 
               
               
                  10 
                   
                 76.54712 
                 2.180012 
                   
                   
                 8.986 
                 8.986 
               
               
                 *11 
                 L6 
                 −28.52627 
                 8 
                 1.84133 
                 29.65 
                 8.869 
                 8.869 
               
               
                 *12 
                   
                 48.93629 
                 1.220916 
                   
                   
                 8.193 
                 8.193 
               
               
                 Stop 
                 L7 
                 26.28885 
                 5 
                 1.437001  
                 95.1 
                 7.716 
                 7.716 
               
               
                 plane 
                   
                   
                   
                   
                   
                   
                   
               
               
                  14 
                   
                 45.09823 
                 12.233959 
                   
                   
                 8.2 
                 8.2 
               
               
                  15 
                 L8 
                 −150.87431 
                 7.221167 
                 1.845819 
                 24.55 
                 17.096 
                 17.096 
               
               
                  16 
                   
                 −28.43438 
                 19.451476 
                   
                   
                 17.914 
                 17.914 
               
               
                 *17 
                 L9 
                 27.31547 
                 6 
                 1.531131  
                 55.75 
                 22.803 
                 22.803 
               
               
                 *18 
                   
                 23.24316 
                 3.756493 
                   
                   
                 25.218 
                 25.218 
               
               
                  19 
                 L10 
                 75.10348 
                 15 
                 1.487489  
                 70.44 
                 26.918 
                 26.918 
               
               
                  20 
                 L11 
                 −50.52584 
                 2 
                 1.846663  
                 23.78 
                 27.009 
                 27.009 
               
               
                  21 
                   
                 −115.09535 
                 0.1 
                   
                   
                 27.983 
                 27.983 
               
               
                  22 
                 L12 
                 89.00944 
                 12 
                 1.499021 
                 76.62 
                 28.568 
                 28.568 
               
               
                  23 
                 L13 
                 −84.73782 
                 8 
                 1.846663  
                 23.78 
                 28.411 
                 28.411 
               
               
                  24 
                   
                 137.31138 
                 5 
                   
                   
                 28.655 
                 28.655 
               
               
                  25 
                 L14 
                 85.69705 
                 12 
                 1.549983 
                 67.99 
                 30.314 
                 30.314 
               
               
                  26 
                   
                 −154.25872 
                 9.093339 
                   
                   
                 30.31 
                 30.31 
               
               
                 *27 
                 L15 
                 485.02001 
                 8 
                 1.531131 
                 55.75 
                 27.967 
                 27.967 
               
               
                 *28 
                   
                 31.34185 
                 7.83338 
                   
                   
                 25.384 
                 25.384 
               
               
                 *29 
                 41 
                 −92.52796 
                 26 
                 1.509398  
                 56.47 
                 22.114 
                 22.114 
               
               
                  30 
                   
                 0 
                 0 
                 1.509398  
                 56.47 
                 16.37 
                 16.37 
               
               
                 *31 
                 42 
                 −22.23699 
                 0 
                 1.509398  
                 56.47 
                 16.008 
                 16.008 
               
               
                  32 
                   
                 0 
                 −26 
                 1.509398  
                 56.47 
                 24.459 
                 24.459 
               
               
                 *33 
                 43 
                 20 
                 −444 
                   
                   
                 19.843 
                 19.843 
               
               
                 Image 
                   
                 0 
                 0 
                   
                   
                 1463.027 
                 1463.027 
               
               
                 plane 
               
               
                   
               
            
           
         
       
     
     As shown in the lens data, the refractive index nd of the lens  35  is 1.509398, and the Abbe number νd of the lens  35  is 56.47. The field of the on-axis inter-surface distance d labeled with the surface number  33  shows the distance between the screen S and the second transmissive surface  43  of the lens  35 . The field of the on-axis inter-surface distance d labeled with the surface number  33  therefore shows the projection distance f of the projection system  3 C. In the present embodiment, f=444 mm. Further, in the present embodiment, the effective radius of the reflective surface  42  is 16.008 mm in the axis-Y direction and 16.008 mm in the axis-X direction. 
     Aspheric data of the surface number  11  are as follows. 
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 Conic constant 
                   1.568E+00 
               
               
                   
                 Fourth-order coefficient 
                 −1.223449E−05 
               
               
                   
                 Sixth-order coefficient 
                  4.897519E−07 
               
               
                   
                 Eighth-order coefficient 
                 −9.903867E−10 
               
               
                   
                   
               
            
           
         
       
     
     Aspheric data of the surface number  12  are as follows. 
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 Conic constant 
                    −1.3E+00 
               
               
                   
                 Fourth-order coefficient 
                  1.916784E−05 
               
               
                   
                 Sixth-order coefficient 
                  2.58057E−07 
               
               
                   
                 Eighth-order coefficient 
                 −4.731664E−10 
               
               
                   
                   
               
            
           
         
       
     
     Aspheric data of the surface number  17  are as follows. 
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 Conic constant 
                     −1E+00 
               
               
                   
                 Fourth-order coefficient 
                 −2.948697E−05 
               
               
                   
                 Sixth-order coefficient 
                  2.641744E−09 
               
               
                   
                 Eighth-order coefficient 
                  2.077529E−13 
               
               
                   
                 Tenth-order coefficient 
                 −1.336104E−14 
               
               
                   
                   
               
            
           
         
       
     
     Aspheric data of the surface number  18  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                    −8.8E−01 
               
               
                   
                 Fourth-order coefficient 
                 −3.854004E−05 
               
               
                   
                 Sixth-order coefficient 
                  2.313839E−08 
               
               
                   
                 Eighth-order coefficient 
                 −1.744829E−11 
               
               
                   
                 Tenth-order coefficient 
                  7.544078E−15 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  27  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                      9E+01 
               
               
                   
                 Fourth-order coefficient 
                  1.180085E−05 
               
               
                   
                 Sixth-order coefficient 
                 −6.121714E−09 
               
               
                   
                 Eighth-order coefficient 
                  8.493511E−12 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  28  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                  0 
               
               
                   
                 Fourth-order coefficient 
                  −3.50673E−05 
               
               
                   
                 Sixth-order coefficient 
                  3.700765E−08 
               
               
                   
                 Eighth-order coefficient 
                 −2.816039E−12 
               
               
                   
                 Tenth-order coefficient 
                 −1.810892E−14 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  29  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                  0 
               
               
                   
                 Fourth-order coefficient 
                  4.37256E−05 
               
               
                   
                 Sixth-order coefficient 
                 −4.402619E−08 
               
               
                   
                 Eighth-order coefficient 
                  3.628378E−11 
               
               
                   
                 Tenth-order coefficient 
                  8.44907E−15 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  31  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                 −1.192647E+00 
               
               
                   
                 Fourth-order coefficient 
                  1.333695E−05 
               
               
                   
                 Sixth-order coefficient 
                 −3.524652E−08 
               
               
                   
                 Eighth-order coefficient 
                  1.148378E−10 
               
               
                   
                 Tenth-order coefficient 
                 −1.100817E−13 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  33  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                 0 
               
               
                   
                 Fourth-order coefficient 
                 −2.192121E−05 
               
               
                   
                 Sixth-order coefficient 
                  8.458547E−08 
               
               
                   
                 Eighth-order coefficient 
                 −1.888281E−10 
               
               
                   
                 Tenth-order coefficient 
                  1.917352E−13 
               
               
                   
               
            
           
         
       
     
     The present embodiment can also provide the same effects as those provided by the embodiments described above. 
     In the present embodiment, the inclination angle θ by which the imaginary line P inclines with respect to the imaginary vertical line V is 97.74°. Therefore, θ=97.74°, which satisfies the conditional expressions (1) and (2) below. Further, the inclination angle η by which the pupil  44  inclines with respect to the imaginary vertical line V is 103.93°. Therefore, η=103.93°, which satisfies the conditional expressions (3) and (4) below. Therefore, the projection apparatus  3 C according to the present embodiment allows suppression of the difference in the amount of light between the upper portion and the lower portion of the screen S.
 
0°&lt;θ&lt;90°+γ  (1)
 
90°&lt;θ  (2)
 
0°&lt;η&lt;90°+γ  (3)
 
90°&lt;η  (4)
 
     θ: Inclination angle over which an end of the imaginary line P facing the upper intersection  53  rotates counterclockwise relative to the imaginary vertical line V around the intersection of the imaginary vertical line V and the imaginary line P 
     η: Inclination angle over which an end of the pupil  44  facing the first intersection rotates counterclockwise relative to the imaginary vertical line V around the intersection of the imaginary vertical line V and the pupil  44   
     γ: Angle from the imaginary axis M to the lower peripheral light ray  52   b  passing through the effective range  50  of the second transmissive surface  43  and intersects the imaginary axis M 
     In the projection system  3 C according to the present embodiment, the effective radius of the reflective surface  42  is 16.008 mm in the axis-X direction and 16.008 mm in the axis-Y direction. The projection system  3 C according to the present embodiment therefore also allows suppression of an increase in the size of the reflective surface  42 . 
       FIG. 23  shows an MTF of the projection system  3 C on the magnifying side. The MTF was calculated under the following conditions: The image formation planes were divided along the axis Y; and the resultant halves were each divided into 11 areas. Light rays used in the calculation of the MTF are so weighted that the weighting ratio among the light rays having the wavelength of 620 nm, the light rays having the wavelength of 550 nm, and the light rays having the wavelength of 470 nm is 2:7:1. The horizontal axis of  FIG. 23 , which shows the MTF, represents the spatial frequency. The spatial frequency of 0.24 cycles corresponds to the resolution of 16.7 μm. The vertical axis of  FIG. 23  represents the contrast reproduction ratio. In the present embodiment, a decrease in resolution is suppressed, as shown in  FIG. 23 .  FIG. 24  is a spot diagram showing spots produced by the projection system  3 C. In the present embodiment, variation in the size of the spots is suppressed, as shown in  FIG. 24 . 
     Embodiment 4 
     Embodiment 4 of the present disclosure will be described below. Embodiment 4 differs from other embodiments described above in that the image formation section  2  includes a single light modulator  18 . The light modulator is also called a display. Embodiments of the light modulator  18  may include a digital mirror device (DMD) and a liquid crystal panel. In this case, the image formation section  2  includes a color wheel for causing the color light, that is, the red light, the green light, and the blue light, to sequentially emit or an optical system for collecting the color lights into spots at specific areas of each pixel of the liquid crystal panel. The color wheel or the optical system for collecting the color lights being disposed on the upstream of the light modulator  18 . In the present embodiment, it is unnecessary to combine the color projection images with one another, unlike the embodiments described above. Therefore, in the present embodiment, the projection system has no cross dichroic prism. 
       FIG. 25  is a light ray diagram of a projection system according to Embodiment 4 of the present disclosure.  FIG. 26  is a light ray diagram of the projection system according to Embodiment 4 enlarged.  FIG. 27  is a light ray diagram with a portion including a second optical system of the projection system according to Embodiment 4 enlarged.  FIG. 25  diagrammatically shows the 11 light fluxes F 1  to F 11 , which exit out of a projection system  3 D and reach the screen S. The light flux F 1  is a light flux that reaches the lowest image height position. The light flux F 11  is a light flux that reaches the highest image height position. The light fluxes F 2  to F 10  are light fluxes that reach height positions between the height position that the light flux F 1  reaches and the height position that the light flux F 11  reaches. The projection system  3 D according to the present embodiment has a configuration corresponding to those in the embodiments described above, and the corresponding components therefore have the same reference characters in the description. Further, the same configurations in the embodiments described above will not be described. 
     The projection system  3 D according to the present embodiment is formed of the first optical system  31  and the second optical system  32  sequentially arranged from the demagnifying side toward the magnifying side, as shown in  FIG. 25 . The projection system  3 D forms the intermediate image  33  in a position between the demagnifying-side image formation plane and the magnifying-side image formation plane of the projection system  3 D, as shown in  FIG. 26 . 
     The first optical system  31  is a refractive optical system including a plurality of lenses. The second optical system  32  is a lens  35 . The light modulator  18  of the image formation section  2  is disposed in the demagnifying-side image formation plane. The light modulator  18  forms the projection images on one side of the optical axis N of the first optical system  31  in the demagnifying-side image formation plane. The screen S is disposed in the magnifying-side image formation plane. 
     The intermediate image  33  is formed in the second optical system  32 , that is, the lens  35 . The intermediate image  33  is formed on the other side of the optical axis N of the first optical system  31 . A final image projected on the screen S has an oblong shape elongated in the lateral direction. In the present embodiment, the final image has the aspect ratio of 16:10. 
     The first optical system  31  includes the first lens L 1  to the fifteenth lens L 15 , which form 15 lenses, as shown in  FIG. 26 . In the present embodiment, the first optical system  31  includes no cross dichroic prism. The first lens L 1  to the fifteenth lens L 15  are arranged in the presented order from the demagnifying side toward the magnifying side. In the present embodiment, the second lens L 2  and the third lens L 3  are bonded to each other to form the first doublet L 21 . The fourth lens L 4  and the fifth lens L 5  are bonded to each other to form the second doublet L 22 . The tenth lens L 10  and the eleventh lens L 11  are bonded to each other to form the third doublet L 23 . The twelfth lens L 12  and the thirteenth lens L 13  are bonded to each other to form the fourth doublet L 24 . In the present embodiment, out of the lenses that form the first optical system  31 , the sixth lens L 6 , the ninth lens L 9 , and the fifteenth lens L 15  are each an aspheric lens, and the other lenses are each a spherical lens. The lenses that form the first optical system  31  may instead each be formed of a spherical lens. 
     In the first optical system  31 , the fifteenth lens L 15 , which is located in a position closest to the magnifying side, has positive power. In the first optical system  31 , the magnifying-side lens group LG 25 , which is located in a position closest to the second optical system  32 , has positive power. The magnifying-side lens group LG 25  is formed of the fourteenth lens L 14  and the fifteenth lens L 15 . Since the magnifying-side lens group LG 25  has positive power in the first optical system  31 , the principal rays between the first optical system  31  and the second optical system  32  approach each other toward the second optical system  32 . 
     The lens  35 , which forms the second optical system  32 , is made of resin. The lens  35  has the first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43  sequentially arranged from the demagnifying side toward the magnifying side, as shown in  FIG. 27 . In the case where the lens  35  is made of resin, the lens  35  can be manufactured in injection molding. The lens  35  having a complicated shape is therefore readily manufactured. 
     Also, in the following description of the present embodiment, three axes perpendicular to one another are called the axes X, Y, and Z for convenience. A first direction in which the first transmissive surface  41  and the reflective surface  42  are arranged is called the axis-Z direction. A second direction, which is called an axis-y direction, coincides with the vertical direction of the screen S. One side of the axis Y is called the upper side Y 1 , and the other side of the axis Y is called the lower side Y 2 . A first plane perpendicular to the axis X and containing the axes Y and Z is called the plane YZ.  FIGS. 25 to 27  therefore each show the plane YZ. The optical axis N of the first optical system  31  extends in the axis-Z direction. The image formation section  2  forms the projection images on the upper side Y 1  of the optical axis N of the first optical system  31 . The intermediate image  33  is formed on the lower side Y 2  of the optical axis N of the first optical system  31 . The lateral direction of the screen S coincides with a third direction, which is called the axis-X direction. In the following description, the imaginary axis M extending in the axis-Z direction is set in the plane YZ. The imaginary axis M is the reference axis used in the design of the lens  35 . The imaginary axis M is perpendicular to the screen S, which is the magnifying-side image formation plane. The imaginary axis M is substantially perpendicular to the screen S in some cases. 
     The first transmissive surface  41  and the reflective surface  42  are located at the lower side Y 2  of the imaginary axis M. The second transmissive surface  43  is located at the upper side Y 1  of the imaginary axis M. The reflective surface  42  has a concave shape. The reflective surface  42  therefore has positive power. The reflective surface  42  is provided by externally forming a reflective coating on the lens  35 . The second transmissive surface  43  has a convex shape protruding toward the magnifying side. The second transmissive surface  43  therefore has positive power. The first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43  form a coaxial optical system having surfaces rotationally symmetric with respect to the imaginary axis M. Therefore, the imaginary axis M is the reference axis used in the design of the lens  35  and is the optical axis of the lens  35 . In the present embodiment, the imaginary axis M coincides with the optical axis N of the first optical system  31 . The imaginary axis M may not necessarily coincide with the optical axis N of the first optical system  31 . 
     The upper and lower halves of the lens  35  are each configured to be rotationally symmetric with respect to the imaginary axis M. That is, the first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43  are so shaped that the cross-sectional shape in the plane YZ shown in  FIG. 27  is rotated around the imaginary axis M over the angular range of 90° toward one side and the other side of the axis-X direction. In the present embodiment, the first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43  are each an aspheric surface. 
     The imaginary line P can be specified in the lens  35  of the second optical system  32 , as shown in  FIG. 27 . The imaginary line P connects the upper intersection  53  to the lower intersection  54 , the upper intersection  53  being an intersection where the upper peripheral light ray  51   a  of the upper-end light flux  51 , where the upper-end light flux  51  is the light flux passing through the axis-Y-direction upper end of the effective range  50  of the second transmissive surface  43 , and the upper peripheral light ray  52   a  of the lower-end light flux  52 , where the lower-end light flux  52  is the light flux passing through the axis-Y-direction lower end of the effective range  50 , intersect with each other in the plane YZ, and the lower intersection  54  being an intersection where the lower peripheral light ray  51   b  of the upper-end light flux  51  and the lower peripheral light ray  52   b  of the lower-end light flux  52  intersect with each other in the plane YZ. In the present embodiment, the imaginary line P inclines by 100.03° with respect to the imaginary vertical line V, which is perpendicular to the imaginary axis M in the plane YZ. Further, in the lens  35  of the second optical system  32 , the upper peripheral light ray  51   a  of the upper-end light flux  51  travels from the second transmissive surface  43  toward the reflective surface  42 , and the point where the upper peripheral light ray  51   a  intersects another light ray reflected off the reflective surface  42  for the first time is called the first intersection. Similarly, the lower peripheral light ray  52   b  of the lower-end light flux  52  travels from the second transmissive surface  43  toward the reflective surface  42 , and the point where the lower peripheral light ray  52   b  intersects another light ray reflected off the reflective surface  42  for the first time is called the second intersection. Under the above definition, the line that connects the first intersection to the second intersection is named as a pupil  44 . In the present embodiment, the pupil  44  inclines by 104.34° with respect to the imaginary vertical line V in the plane YZ. 
     The intermediate image  33  is an inverted final image turned upside down. The intermediate image  33  is an image so distorted that an oblong final image is projected on the screen S, which is the magnifying-side image formation plane. More specifically, the intermediate image  33  is so distorted as to cancel the distortion produced by the second optical system  32  so that the trapezoidal distortion of the final image formed on the screen S decreases. That is, the intermediate image  33  has distortion opposite the trapezoidal distortion of the final image. The intermediate image  33  is therefore so formed that the shortest edge thereof is the edge having the highest image height on the screen S. 
     Lens Data 
     Data on the lenses of the projection system  3 D are as follows: The surfaces of the lenses are numbered sequentially from the demagnifying side toward the magnifying side. A surface having a surface number with * is an aspheric surface. The row of the reference characters represents the reference characters of the lenses thereof. The reference characters given in the second optical system  32  are the reference characters of the first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43 . That is, the surface number  27  represents the first transmissive surface  41  of the lens  35 . The surface number  29  represents the reflective surface  42  of the lens  35 . The surface number  31  represents the second transmissive surface  43  of the lens  35 . Reference character r denotes the radius of curvature in millimeters. Reference character d denotes the on-axis inter-surface distance in millimeters. Reference character nd denotes the refractive index. Reference character νd denotes the Abbe number. Reference character Y denotes the effective radius in the axis-Y direction. Reference character X denotes the effective radius in the axis-X direction. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
               
                 Surface 
                 Reference 
                   
                   
                   
                   
                   
                   
               
               
                 number 
                 character 
                 r 
                 d 
                 nd 
                 vd 
                 Y 
                 X 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Object 
                   
                 0 
                 35.41 
                   
                   
                   
                   
               
               
                 plane 
                   
                   
                   
                   
                   
                   
                   
               
               
                  1 
                 L1 
                 27.04015 
                 10.723313 
                 1.596189 
                 62.64 
                 19 
                 19 
               
               
                  2 
                   
                   
                 −184.03214 
                 0.1 
                   
                 18.587 
                 18.587 
               
               
                  3 
                 L2 
                 28.75428 
                 10.3193 
                 1.493446 
                 77.82 
                 16.057 
                 16.057 
               
               
                  4 
                 L3 
                 −35.15431 
                 0.9 
                 1.843135 
                 27.36 
                 15.03 
                 15.03 
               
               
                  5 
                   
                 −796.26153 
                 0.1 
                   
                   
                 13.976 
                 13.976 
               
               
                  6 
                 L4 
                 21.30955 
                 8.220652 
                 1.447755 
                 82.78 
                 11.913 
                 11.913 
               
               
                  7 
                 L5 
                 −24.50612 
                 3.781704 
                 1.838229 
                 34.67 
                 10.688 
                 10.688 
               
               
                  8 
                   
                 54.29446 
                 1.630421 
                   
                   
                 8.992 
                 8.992 
               
               
                  *9 
                 L6 
                 −324.9208 
                 8 
                 1.83624 
                 38.92 
                 8.898 
                 8.898 
               
               
                 *10 
                   
                 22.55933 
                 0.2 
                   
                   
                 8.21 
                 8.21 
               
               
                 Stop 
                 L7 
                 22.46308 
                 5 
                 1.440559 
                 90.57 
                 7.772 
                 7.772 
               
               
                 plane 
                   
                   
                   
                   
                   
                   
                   
               
               
                  12 
                   
                 43.18975 
                 10.0417 
                   
                   
                 8.2 
                 8.2 
               
               
                  13 
                 L8 
                 1027.43527 
                 7.142823 
                 1.724905 
                 27.48 
                 15.179 
                 15.179 
               
               
                  14 
                   
                 −27.759 
                 29.55954 
                   
                   
                 15.933 
                 15.933 
               
               
                 *15 
                 L9 
                 26.18195 
                 6 
                 1.531131 
                 55.75 
                 21.895 
                 21.895 
               
               
                 *16 
                   
                 24.46638 
                 3.251569 
                   
                   
                 24.544 
                 24.544 
               
               
                  17 
                 L10 
                 52.54104 
                 15 
                 1.487489 
                 70.44 
                 27.899 
                 27.899 
               
               
                  18 
                 L11 
                 −99.67582 
                 2 
                 1.846663 
                 23.78 
                 27.867 
                 27.867 
               
               
                  19 
                   
                 367.31767 
                 0.1 
                   
                   
                 28.209 
                 28.209 
               
               
                  20 
                 L12 
                 144.86083 
                 11.949038 
                 1.499714 
                 76.47 
                 28.501 
                 28.501 
               
               
                  21 
                 L13 
                 −70 
                 2 
                 1.846663 
                 23.78 
                 28.609 
                 28.609 
               
               
                  22 
                   
                 −2882.97645 
                 5 
                   
                   
                 29.502 
                 29.502 
               
               
                  23 
                 L14 
                 −143.56752 
                 12 
                 1.523218 
                 72.05 
                 29.907 
                 29.907 
               
               
                  24 
                   
                 −47.93833 
                 0.3 
                   
                   
                 30.731 
                 30.731 
               
               
                 *25 
                 L15 
                 −278.45036 
                 8 
                 1.531131 
                 55.75 
                 28.533 
                 28.533 
               
               
                 *26 
                   
                 32.5217 
                 6.620327 
                   
                   
                 25.549 
                 25.549 
               
               
                 *27 
                 41 
                 −91.46 
                 26 
                 1.509398 
                 56.47 
                 22.401 
                 22.401 
               
               
                  28 
                   
                 0 
                 0 
                 1.509398 
                 56.47 
                 16.349 
                 16.349 
               
               
                 *29 
                 42 
                 −23.52443 
                 0 
                 1.509398 
                 56.47 
                 16.243 
                 16.243 
               
               
                  30 
                   
                 0 
                 −26 
                 1.509398 
                 56.47 
                 24.783 
                 24.783 
               
               
                 *31 
                 43 
                 20 
                 −444 
                   
                   
                 19.834 
                 19.834 
               
               
                 Image 
                   
                 0 
                 0 
                   
                   
                 1478.174 
                 1478.174 
               
               
                 plane 
               
               
                   
               
            
           
         
       
     
     As shown in the lens data, the refractive index nd of the lens  35  is 1.509398, and the Abbe number νd of the lens  35  is 56.47. The field of the on-axis inter-surface distance d labeled with the surface number  31  shows the distance between the screen S and the second transmissive surface  43  of the lens  35 . The field of the on-axis inter-surface distance d labeled with the surface number  31  therefore shows the projection distance f of the projection system  3 D. In the present embodiment, f=444 mm. Further, in the present embodiment, the effective radius of the reflective surface  42  is 16.243 mm in the axis-Y direction and 16.243 mm in the axis-X direction. 
     Aspheric data of the surface number  9  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                   1.568E+00 
               
               
                   
                 Fourth-order coefficient 
                 −1.009992E−04 
               
               
                   
                 Sixth-order coefficient 
                  3.513449E−07 
               
               
                   
                 Eighth-order coefficient 
                  1.167873E−10 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  10  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                    −1.3E+00 
               
               
                   
                 Fourth-order coefficient 
                 −5.044473E−05 
               
               
                   
                 Sixth-order coefficient 
                  4.666818E−07 
               
               
                   
                 Eighth-order coefficient 
                 −8.772887E−10 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  15  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                     −1E+00 
               
               
                   
                 Fourth-order coefficient 
                 −2.682649E−05 
               
               
                   
                 Sixth-order coefficient 
                 −1.226436E−08 
               
               
                   
                 Eighth-order coefficient 
                 −5.685428E−12 
               
               
                   
                 Tenth-order coefficient 
                 −1.521922E−14 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  16  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                    −8.8E−01 
               
               
                   
                 Fourth-order coefficient 
                 −3.377957E−05 
               
               
                   
                 Sixth-order coefficient 
                  1.703745E−09 
               
               
                   
                 Eighth-order coefficient 
                 −3.308662E−12 
               
               
                   
                 Tenth-order coefficient 
                  1.385218E−14 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  25  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                      9E+01 
               
               
                   
                 Fourth-order coefficient 
                  1.231931E−05 
               
               
                   
                 Sixth-order coefficient 
                 −2.697087E−09 
               
               
                   
                 Eighth-order coefficient 
                  9.407508E−12 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  26  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                  0 
               
               
                   
                 Fourth-order coefficient 
                 −3.942288E−05 
               
               
                   
                 Sixth-order coefficient 
                  4.592455E−08 
               
               
                   
                 Eighth-order coefficient 
                  2.39597E−11 
               
               
                   
                 Tenth-order coefficient 
                 −5.310071E−14 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  27  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                  0 
               
               
                   
                 Fourth-order coefficient 
                  4.369855E−05 
               
               
                   
                 Sixth-order coefficient 
                 −3.679881E−08 
               
               
                   
                 Eighth-order coefficient 
                  4.239583E−11 
               
               
                   
                 Tenth-order coefficient 
                  5.34966E−15 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  29  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                 −1.029701E+00 
               
               
                   
                 Fourth-order coefficient 
                  1.198257E−05 
               
               
                   
                 Sixth-order coefficient 
                 −2.370324E−08 
               
               
                   
                 Eighth-order coefficient 
                  7.902733E−11 
               
               
                   
                 Tenth-order coefficient 
                 −6.397751E−14 
               
               
                   
               
            
           
         
       
     
     Aspheric data of the surface number  31  are as follows. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 Conic constant 
                  0 
               
               
                   
                 Fourth-order coefficient 
                 −2.293837E−05 
               
               
                   
                 Sixth-order coefficient 
                  1.062271E−07 
               
               
                   
                 Eighth-order coefficient 
                 −2.666016E−10 
               
               
                   
                 Tenth-order coefficient 
                  3.255388E−13 
               
               
                   
               
            
           
         
       
     
     The present embodiment can also provide the same effects as those provided by the embodiments described above. 
     In the present embodiment, the inclination angle θ by which the imaginary line P inclines with respect to the imaginary vertical line V is 100.03°. Therefore, θ=100.03°, which satisfies the conditional expressions (1) and (2) below. Further, the inclination angle η by which the pupil  44  inclines with respect to the imaginary vertical line V is 104.34°. Therefore, η=104.34°, which satisfies the conditional expressions (3) and (4) below. Therefore, the projection apparatus  3 D according to the present embodiment allows suppression of the difference in the amount of light between the upper portion and the lower portion of the screen S.
 
0°&lt;θ&lt;90°+γ  (1)
 
90°&lt;θ  (2)
 
0°&lt;η&lt;90°+γ  (3)
 
90°&lt;η  (4)
 
     θ: Inclination angle over which an end of the imaginary line P facing the upper intersection  53  rotates counterclockwise relative to the imaginary vertical line V around the intersection of the imaginary vertical line V and the imaginary line P 
     η: Inclination angle over which an end of the pupil  44  facing the first intersection rotates counterclockwise relative to the imaginary vertical line V around the intersection of the imaginary vertical line V and the pupil  44   
     γ: Angle from the imaginary axis M to the lower peripheral light ray  52   b  passing through the effective range  50  of the second transmissive surface  43  and intersects the imaginary axis M 
     In the projection system  3 D according to the present embodiment, the projection distance f is 444 mm. Therefore, in the projection system  3 D according to the present embodiment, the same projection distance of the projection system  3 C according to Embodiment 3 is achieved with no dichroic prism provided in the first optical system. The effective radius of the reflective surface  42  is 16.243 mm in the axis-X direction and 16.243 mm in the axis-Y direction. The projection system  3 D according to the present embodiment therefore also allows suppression of an increase in the size of the reflective surface  42 . 
       FIG. 28  shows an MTF of the projection system  3 D on the magnifying side. The MTF was calculated under the following conditions: The image formation planes were divided along the axis Y; and the resultant halves were each divided into 11 areas. Light rays used in the calculation of the MTF are so weighted that the weighting ratio among the light rays having the wavelength of 620 nm, the light rays having the wavelength of 550 nm, and the light rays having the wavelength of 470 nm is 2:7:1. The horizontal axis of  FIG. 28 , which shows the MTF, represents the spatial frequency. The spatial frequency of 0.24 cycles corresponds to the resolution of 16.7 μm. The vertical axis of  FIG. 28  represents the contrast reproduction ratio. In the present embodiment, a decrease in resolution is suppressed, as shown in  FIG. 28 .  FIG. 29  is a spot diagram showing spots produced by the projection system  3 D. In the present embodiment, variation in the size of the spots is suppressed, as shown in  FIG. 29 . 
     Other Embodiments 
     In Embodiments 1 to 4 described above, the lens  35  is made of resin. The lens  35  can instead be made of glass. In the case where the lens  35  is made of glass, the lens  35  can be processed with high precision as compared with the case where the lens  35  is made of resin. Further, in the case where the lens  35  is made of glass, the amount of expansion of the shape of the lens  35  can be suppressed when the temperature of the lens  35  rises due to internal absorption of light, as compared with the case where the lens  35  is made of resin. The lens  35  made of glass can therefore maintain the optical performance of the lens  35  even in the case where high luminance light enters the lens  35 , because deformation of the lens  35  due to the heat is suppressed. The lens  35  made of glass can further improve the reliability of the lens  35  because the heat resistance is improved as compared with the lens  35  made of resin. 
     Further, the lens  35  may be formed of a plurality of members. In this case, the plurality of members may be made of materials different from one another. For example, an area of the lens  35  that is an area containing the pupil  44  and the vicinity thereof, where the temperature is likely to increase, may be made of glass, and the remainder of the lens  35  may be made of resin. Since the first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43  are therefore made of resin, a complicated aspheric shape can be readily formed, whereby the optical performance of the lens  35  can be improved. Further, since the area where the temperature is likely to increase is made of glass, the heat resistance of the lens  35  can also be improved. A projection system including the lens  35  having higher optical performance and improved reliability can therefore be achieved. 
     In the embodiments described above, the first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43  of the lens  35  are each an aspheric surface. Instead, a configuration in which at least one of the surfaces is an aspheric surface allows suppression of aberrations that affect a final image projected on the screen S. 
     Further, in the embodiments described above, the inclination angle θ, by which the imaginary line P inclines with respect to the imaginary vertical line V, satisfies the conditional expressions (1) and (2). Instead, a configuration in which the inclination angle θ satisfies at least the conditional expression (1) allows suppression of the situation in which the amount of light at the upper periphery of the screen S is smaller than the amount of light at the lower periphery of the screen S. 
     In Embodiments 1 to 4 described above, the lens  35  may be provided with a light shield. In this case, the light shield blocks light rays that incline with respect to the imaginary line P and do not fall within the effective range  50 , through which the imaginary line P passes. Stray light produced in the lens  35  can thus be avoided. 
     Further, any of the first transmissive surface  41 , the reflective surface  42 , and the second transmissive surface  43  of the lens  35  may be formed of a free-form surface. The projection distance of the projection system is thus readily further shortened. 
     In addition, the projection system according to each of the embodiments described above can also be used as an imaging system for capturing an image formed by outside light incident via the second transmissive surface  43  of the lens  35  with an imaging device disposed in place of the demagnifying-side image formation plane where the light modulator  18  or each of the liquid crystal panels  18 R,  18 G, and  18 B is disposed.