JP5578420B2 - Optical system for image projection device and image projection device - Google Patents

Description

  The present invention relates to an image projection apparatus that enlarges and projects an image or the like of an image display element.

  A plurality of methods have been devised and put into practical use for image projection apparatuses (so-called projectors) that project an enlarged image of an image display element. For example, after light from a light source is transmitted through a liquid crystal display element, an image (light) obtained through the liquid crystal display element is screened (projection surface) via a projection optical system including a plurality of lenses. An image projection apparatus that projects an image on an enlarged scale is known (see, for example, Patent Document 1).

JP 2008-134350 A

  However, such a video projection device is required to be downsized and to reduce distortion (for example, trapezoidal distortion) of an image projected on a screen.

  The present invention has been made in view of such problems, and has an optical system for a video projection device capable of satisfactorily correcting trapezoidal distortion of a projected image, and a video projection provided with the same, while having a compact configuration. An object is to provide an apparatus.

In order to achieve such an object, according to an aspect of the present invention, an optical system for an image projection apparatus that enlarges an image displayed on an image display element and projects the image on a predetermined projection surface from an oblique direction, A rotationally symmetric lens group composed of rotationally symmetric lenses formed in a rotationally symmetric manner with respect to the central axis and arranged in order from the image display element side along the optical axis, and a plurality of non-rotationally symmetric lenses formed with respect to the central axis A free-form surface lens group including a free-form surface lens and a free-form surface mirror having a reflection surface formed in a non-rotationally symmetric manner with respect to a central axis, and the free-form surface lens group is closest to the free-form surface mirror The cross-sectional shape of the mirror-side free-form surface lens is perpendicular to the first cross-section passing through the optical axis between the free-form surface lens group and the free-form surface mirror and the optical axis between the free-form surface mirror and the projection surface. the second cross section Oite has a meniscus shape with a concave surface on the image display element side,
In the second cross section, the coordinate axis of the optical axis direction in which the direction from the image display element toward the free-form curved mirror is positive is the z-axis, and the coordinate axis perpendicular to the z-axis is the x-axis, Define a local coordinate system (x, z) with the origin at the intersection of the lens surface and the optical axis,
When i = 1 for the lens surface of the free-form surface lens on the image display element side and i = 2 for the lens surface of the free-form surface lens on the free-form surface mirror side, the free-form surface lens The maximum point coordinate within the effective range is (xoi, zoi), and the sag amount of the free-form lens is zfi (x).
zfci (x) = zfi (x + xoi) −zoi
Define
The air conversion center distance from the diaphragm to the lens surface on the image display element side of the free-form surface lens is L1, and the following formula
rx1 = -L1 / 2.5
Define
The air conversion center distance from the diaphragm to the lens surface of the free-form curved lens side of the free-form curved lens is L2, and the following formula
rx2 = −L2 / 0.1
Define
Where rx1 and rx2 are rxi,
zsi (x) = rxi + [(rxi) ² −x ² ] ^1/2
When defining
When the absolute value of the length in the x direction from the origin to the end of the effective range is | radxi |
In the range − | radxi | <x <| radxi |
zfci (x) <zs2 (x)
While satisfying the conditions of
When | rx1 | <| radxi |, the range is − | rx1 | <x <| rx1 |. When | radxi | <| rx1 |, the range is − | radxi | <x <| radxi |
zs1 (x) <zfci (x)
An optical system for an image projection apparatus is provided that satisfies the above condition .

  Further, according to an aspect of the present invention, the image is projected from an oblique direction on a surface that is used on a predetermined installation surface and is substantially the same as the installation surface or substantially parallel to the installation surface. There is provided an image projection apparatus, wherein the optical system constituting the image projection apparatus is the optical system for an image projection apparatus according to the present invention.

  According to the present invention, it is possible to satisfactorily correct the trapezoidal distortion of a projected image while having a compact configuration.

It is a sectional side view of the optical system for video projectors concerning the 1st example. It is a plane sectional view of the optical system for image projection devices concerning the 1st example. It is a graph which shows the amount of sag in the xz cross section of the 3rd free-form surface lens which concerns on 1st Example. (A) is a graph which shows the sag amount in the yz cross section of the 2nd free-form surface lens which concerns on 1st Example, (b) is a graph which shows the sag amount in an xz cross section. It is a spot diagram of the optical system for image projection apparatuses concerning the 1st example. It is a figure which shows the distortion of the optical system for image projection apparatuses which concerns on 1st Example. It is a sectional side view of the optical system for image projection apparatuses which concerns on 2nd Example. It is a plane sectional view of the optical system for image projection devices concerning the 2nd example. It is a graph which shows the amount of sag in the xz cross section of the 3rd free-form surface lens concerning 2nd Example. (A) is a graph which shows the sag amount in the yz cross section of the 2nd free-form surface lens which concerns on 2nd Example, (b) is a graph which shows the sag amount in an xz cross section. It is a spot diagram of the optical system for video projectors concerning the 2nd example. It is a figure which shows the distortion of the optical system for image projection apparatuses which concerns on 2nd Example. It is a sectional side view of the optical system for image projection apparatuses which concerns on 3rd Example. It is a plane sectional view of the optical system for image projection devices concerning the 3rd example. It is a graph which shows the amount of sag in the xz cross section of the 2nd free-form surface lens concerning 3rd Example. It is a spot diagram of the optical system for video projectors concerning the 3rd example. It is a figure which shows the distortion of the optical system for image projection apparatuses which concerns on 3rd Example. It is a sectional side view of the optical system for image projection apparatuses which concerns on 4th Example. It is a plane sectional view of the optical system for video projection devices concerning the 4th example. It is a graph which shows the amount of sag in the xz cross section of the 2nd free-form surface lens which concerns on 4th Example. It is a spot diagram of the optical system for video projectors concerning the 4th example. It is a figure which shows the distortion of the optical system for image projection apparatuses which concerns on 4th Example. It is a perspective view of a video projection device. It is sectional drawing of an image projection apparatus. It is explanatory drawing which shows an example of a local coordinate system.

  Hereinafter, preferred embodiments of the present invention will be described with reference to the drawings. An image projection apparatus PRJ of this embodiment is shown in FIGS. FIG. 23 is a perspective view of the image projection device PRJ, and FIG. 24 is a cross-sectional view of the image projection device PRJ.

  23 and 24 includes a cylindrical box-shaped housing BD having a window W on the front surface, an image display element DS accommodated in each of the housings BD, and image projection. And an apparatus optical system PL. Such an image projection device PRJ is used in a state where it is installed on a predetermined installation surface Q such as the upper surface of a table (or desk) or a wall surface in the vicinity of a whiteboard, and light from a light source (not shown). Is transmitted through the image display element DS, and the image (light) obtained through the image display element DS is projected through the image projection apparatus optical system PL and the window W, which will be described in detail later. The apparatus PRJ (housing BD) is configured to be enlarged and projected from an oblique direction onto a projection plane R set on the same plane as the installation plane Q.

  As the light source (not shown), for example, a lamp that generates high-intensity white light such as a mercury lamp or a halogen lamp is used. Further, as the video display element DS, for example, a liquid crystal display element that displays video (or images) input from an external input device (such as a personal computer or a storage device) is used.

  The optical system PL for image projection apparatus includes a rotationally symmetric lens group G1 composed of a plurality of rotationally symmetric lenses formed in a rotationally symmetric manner with respect to the central axis, arranged in order from the video display element DS side along the optical axis, A free-form surface lens group G2 composed of a plurality of free-form surface lenses formed non-rotationally symmetric with respect to the axis, and a free-form surface mirror M having a reflection surface formed non-rotationally symmetric with respect to the central axis Composed. The free-form surface lens group G2 and the free-form surface mirror M may be collectively referred to as a free-form surface group. In such an image projection apparatus optical system PL, the light emitted from the image display element DS is transmitted through the rotationally symmetric lens group G1 and the free-form surface lens group G2, reflected by the free-form surface mirror M in an oblique direction, and projected. Projected onto the surface R.

  In the present embodiment, at least one of the lens surfaces on both sides of the free-form surface lens closest to the free-form surface mirror M in the free-form surface lens group G2 (for example, the third free-form surface lens L23 in the example) is the free-form surface lens group G2. On the image display element DS side in the cross section of the free-form surface lens perpendicular to the optical axis between the free-form surface mirror M and the plane passing through the optical axis between the free-form surface mirror M and the projection surface R (image surface I). It has a shape with a concave surface facing the surface. At this time, at least one of the lens surfaces on both sides of the free-form surface lens has only one local maximum point within the effective diameter in the xz cross section described later in detail.

  The optical system of the present embodiment corrects trapezoidal distortion that occurs in oblique projection by using a free-form surface group. When correcting a rotationally asymmetric trapezoidal distortion on a free-form surface, the free-form surface itself has a rotationally asymmetric shape, and rotationally asymmetric aberration occurs on this surface. Therefore, the free-form surface group must suppress the generation of rotationally asymmetric aberration as much as possible while correcting the trapezoidal distortion. The above configuration is for satisfying this.

  Since the optical system of the present embodiment is an enlarged projection, the principal ray emitted from each point of the image display element DS becomes a divergent ray after passing (transmitting) the rotationally symmetric lens group G1. At this time, if the lens surface of the free-form surface lens closest to the free-form surface mirror M has a shape in which the concave surface is directed to the image display element DS as described above, the light incident angle becomes small and the light beam can be easily transmitted. It can be bent.

  In the free-form surface lens group G2, the mirror-side free-form surface lens closest to the free-form surface mirror M (for example, the third free-form surface lens L23 in the embodiment) and the element side arranged next to the mirror-side free-form surface lens Of the free-form surface lens (for example, the second free-form surface lens L22 in the embodiment), at least one of the cross-sectional shapes is an optical axis between the free-form surface lens group G2 and the free-form surface mirror M, and the free-form surface mirror M and the projection surface. In a first cross section (y-z cross section described later) passing through the optical axis between R (image plane I) or a second cross section (x-z cross section described later) perpendicular to the first cross section. It is preferable to have a meniscus shape with a concave surface facing the image display element DS side. As described above, the meniscus shape allows the rotationally asymmetric aberration generated on the surface on the image display element DS side to be canceled immediately on the image side surface, and is efficient while suppressing the occurrence of rotationally asymmetric aberration. Thus, it becomes possible to correct trapezoidal distortion.

  In addition, the free-form lens has a meniscus shape having a concave surface facing the image display element DS in both the first cross section (yz cross section) and the second cross section (xz cross section). You may do it. Even if it does in this way, the same effect can be acquired.

  Here, a local coordinate system in a free-form surface lens or the like will be described. In this embodiment, the local coordinate system of the free-form surface lens is an (x, y, z) coordinate system with the origin at the intersection of each lens surface of the free-form surface lens and the optical axis. At this time, the coordinate axis in the optical axis direction in which the direction from the image display element DS toward the free-form surface mirror M is positive is the z-axis, the coordinate axis perpendicular to the z-axis in the first cross section is the y-axis, The coordinate axis perpendicular to the z-axis (and y-axis) in the two cross sections is taken as the x-axis.

In such a local coordinate system, the case where the cross-sectional shape of the free-form surface lens has a meniscus shape with the concave surface facing the image display element DS side in the first cross-section, that is, the yz cross-section will be described. Since x = 0 in the yz section, the (y, z) coordinate system can be defined as the local coordinate system. At this time, each of the lens surfaces on both sides of the free-form surface lens has only one maximum point within the effective diameter, and the maximum point in the yz section of the lens surface on the image display element DS side in the free-form surface lens is expressed as (y _o1, z _YO1) and then, can be a maximum point in y-z cross section of the lens surface on the image side (y _o2, z _yo2).

In the local coordinate system (y ₁ , z ₁ ) on the lens surface on the video display element DS side, when z ₁ = f ₁ (y) and | y ₁ + y _o1 | ≠ 0, the following conditional expression (1) Satisfy the conditions expressed. Here, f ′ (y) is the first derivative of f (y) with respect to y.

(y ₁ + y _o1 ) / | y ₁ + y _o1 | × f ₁ ′ (y ₁ + y _o1 ) <0 (1)

In addition, in the local coordinate system (y ₂ , z ₂ ) on the image side lens surface, when z ₂ = f ₂ (y) and | y ₂ + y _o2 | ≠ 0, the following conditional expression (2) Satisfy the conditions.

(y ₂ + y _o2 ) / | y ₂ + y _o2 | × f ₂ ′ (y ₂ + y _o2 ) <0 (2)

On the other hand, the case where the cross-sectional shape of the free-form surface lens has a meniscus shape with the concave surface facing the image display element DS side in the second cross-section, that is, the xz cross-section will be described. Since x = 0 in the xz section, the (x, z) coordinate system can be defined as the local coordinate system. At this time, the maximum point in the xz section of the lens surface on the image display element DS side in the free-form surface lens is (x _o1 , z _xo1 ), and the maximum point in the xz section of the image side lens surface is ( x _o2 , z _xo2 ).

In the local coordinate system (x ₁ , z ₁ ) on the lens surface on the image display element DS side, when z ₁ = f ₁ (x) and | x ₁ + x _o1 | ≠ 0, the following conditional expression (3) Satisfy the conditions expressed. Here, f ′ (x) is the first derivative of f (x) with respect to x.

(x ₁ + x _o1 ) / | x ₁ + x _o1 | × f ₁ ′ (x ₁ + x _o1 ) <0 (3)

In addition, in the local coordinate system (x ₂ , z ₂ ) on the image side lens surface, when z ₂ = f ₂ (x) and | x ₂ + x _o2 | ≠ 0, the following conditional expression (4) Satisfy the conditions.

(x ₂ + x _o2 ) / | x ₂ + x _o2 | × f ₂ ′ (x ₂ + x _o2 ) <0 (4)

  In addition, when the lens surface of the free-form surface lens has a shape facing the concave surface toward the image display element DS in the y-z cross section, the maximum point coordinates within the effective range of the lens surface (free-form surface) of the free-form surface lens are Again (yoi, zoi). Here, i = 1 for the lens surface on the image display element DS side of the free-form surface lens, i = 2 for the lens surface on the image side (free-form surface mirror M side) of the free-form surface lens, and so on. Express. In addition, the effective range of a free-form surface is the range which the light ray of each free-form surface passes.

  Assuming that the sag amount of the free-form surface lens is zfi (y), for example, as shown in FIG. 25, the above-mentioned maximum point coordinates are translated to the origin of the (y, z) coordinate system (local coordinate system). The sag amount zfci (y) can be expressed as the following equation (5). The sag amount zfi (y) of the lens surface of the free-form surface lens is the same as the sag amount Z in the equation (20) described in an example described later.

  zfci (y) = zfi (y + yoi) −zoi (5)

  Further, the radius of curvature ry1 is defined by the following equation (6), where L1 is an air-converted center distance from the aperture stop S to the lens surface on the image display element DS side of the free-form surface lens.

  ry1 = -L1 / 3.5 (6)

  Similarly, the radius of curvature ry2 is defined as the following equation (7), where L2 is the air-converted center distance from the stop S to the lens surface on the image side of the free-form surface lens (the free-form surface mirror M).

  ry2 = −L2 (7)

  The radii of curvature ry1 and ry2 defined by Equation (6) and Equation (7) are the virtual spherical surface (at the surface vertex) that is set to determine the range of the sag amount zfci (y) of the lens surface of the free-form surface lens. ) The radius of curvature. The curvature radii ry1 and ry2 are combined (that is, i = 1 for the lens surface on the image display element DS side of the free-form surface lens, and the lens surface on the image side of the free-form surface lens (the free-form surface mirror M side)) Ryi (assuming i = 2), the sag amount zsi (y) of this virtual spherical surface can be expressed by the following equation (8).

zsi (y) = ryi + [(ryi) ² −y ² ] ^1/2 (8)

  When the spherical sag amount zsi (y) is defined in this way, the length in the y direction from the origin of the (y, z) coordinate system (local coordinate system) to the end of the effective range of the lens surface (free-form surface) of the free-form surface lens When the absolute value of | radyi | is satisfied, the condition expressed by the following conditional expression (9) is satisfied in the range of − | radyi | <y <| radyi | and | ry1 | <| radyi | Sometimes in the range of − | ry1 | <y <| ry1 |, in the case of | radyi | <| ry1 |, in the range of − | radyi | <y <| radyi | Is preferably satisfied.

zfci (y) <zs2 (y) (9)
zs1 (y) <zfci (y) (10)

  In this way, if the free-form surface lens has a meniscus shape that satisfies the conditional expressions (9) and (10), the light incident angle becomes small and the light can be bent without difficulty. Further, by adopting a meniscus shape that satisfies the conditional expressions (9) and (10), the rotationally asymmetric aberration generated on the surface on the image display element DS side can be immediately canceled on the surface on the image side, and the rotationally asymmetrical shape. It is possible to efficiently correct trapezoidal distortion while suppressing the occurrence of various aberrations. When the condition is less than the lower limit of the conditional expression (10), a large amount of rotationally asymmetric aberration occurs when correcting trapezoidal distortion on a free-form surface, which is particularly short when the optical system is downsized. It becomes very difficult when all aberrations must be corrected well at distance. In addition, the incident angle of the chief ray is increased, making it difficult to process. On the other hand, when the condition exceeds the upper limit of conditional expression (9), the rotationally asymmetric trapezoidal distortion is insufficiently corrected.

  Similarly, when the lens surface of the free-form surface lens has a shape facing a concave surface toward the image display element DS in the xz section, the maximum point coordinates within the effective range of the lens surface (free-form surface) of the free-form surface lens Is again (xoi, zoi). If the sag amount of the free-form surface lens is zfi (x), the sag amount zfci (x) obtained by translating the above maximal point coordinates to the origin of the (x, z) coordinate system (local coordinate system) is It can be expressed as the following formula (11). Note that the sag amount zfi (x) of the lens surface of the free-form surface lens is the same as the sag amount Z in the equation (20) described in an example described later.

  zfci (x) = zfi (x + xoi) −zoi (11)

  Further, the radius of curvature rx1 is defined as the following equation (12), where L1 is the air-converted center distance from the stop S to the lens surface on the image display element DS side of the free-form surface lens.

  rx1 = -L1 / 2.5 (12)

  Similarly, the radius of curvature rx2 is defined by the following equation (13), where L2 is an air-converted center distance from the stop S to the lens surface on the image side of the free-form surface lens (the free-form surface mirror M).

  rx2 = -L2 / 0.1 (13)

  The radii of curvature rx1 and rx2 defined by Equation (12) and Equation (13) are the virtual spherical surface (at the vertex of the surface) set to determine the range of the sag amount zfci (x) of the lens surface of the free-form surface lens. ) The radius of curvature. When the curvature radii rx1 and rx2 are collectively represented as rxi, the sag amount zsi (x) of the virtual spherical surface can be expressed as the following equation (14).

zsi (x) = rxi + [(rxi) ² −x ² ] ^1/2 (14)

  When the spherical sag zsi (x) is defined in this way, the x-direction length from the origin of the (x, z) coordinate system (local coordinate system) to the end of the effective range of the lens surface (free-form surface) of the free-form surface lens When the absolute value of | radxi | is satisfied, the condition represented by the following conditional expression (15) is satisfied in the range of − | radxi | <x <| radxi | and | rx1 | <| radxi | Sometimes in the range of − | rx1 | <x <| rx1 |, in the case of | radxi | <| rx1 |, in the range of − | radxi | <x <| radxi |, the condition expressed by the following conditional expression (16) Is preferably satisfied.

zfci (x) <zs2 (x) (15)
zs1 (x) <zfci (x) (16)

  Thus, if the free-form surface lens has a meniscus shape that satisfies the above conditional expressions (15) and (16), the same effect as when the lens surface of the free-form surface lens has a meniscus shape in the y-z cross section is obtained. Can be obtained. In addition, since a free-form surface lens is a rotationally asymmetric lens, there are cases where there are two types of values of | radyi | and | radxi |. However, in this case, if it is properly used depending on the shape of the free-form surface lens Good.

  The free-form surface lens group G2 is preferably composed of three free-form surface lenses. In this way, the trapezoidal distortion can be corrected satisfactorily. Further, the burden of correcting trapezoidal distortion on each free-form surface (lens surface) can be reduced, and a free-form surface shape suitable for mass production can be obtained.

  Further, it is preferable that the condition represented by the following conditional expression (17) is satisfied, where f is the focal length of the rotationally symmetric lens group G1 and TL is the total length of the optical system PL for image projection apparatus.

  15 ≦ | TL / f | ≦ 30 (17)

  Note that the total length TL of the optical system PL for the image projection apparatus is an absolute value of the entire distance from the display surface (object surface) to the image surface I of the image display element DS.

  Conditional expression (17) is for optimizing the balance between the size of the optical system and the power of the rotationally symmetric lens group G1. When the condition is less than the lower limit value of the conditional expression (17), the focal length of the rotationally symmetric lens group G1 becomes long, that is, the rotationally symmetric lens group G1 becomes a telephoto type, and the maximum field angle of the principal ray becomes small. Although it is advantageous for making the entire system compact in the xy cross section, the entire back focus is extended due to insufficient power of the rotationally symmetric lens group G1. In order to reduce this, the free-form surface group must have a large positive power, and correction of an extraneous asymmetric aberration caused thereby is performed by the free-form surface group itself (in the rotationally symmetric lens group G1, The asymmetric aberration cannot be corrected by the angle of view), and the burden increases. For this reason, it is difficult to simultaneously perform trapezoidal distortion correction and aberration correction, which is not preferable. On the other hand, when the condition exceeds the upper limit value of the conditional expression (17), the focal length of the rotationally symmetric lens group G1 becomes short, that is, the rotationally symmetric lens group G1 becomes a wide-angle type, and the maximum field angle of the principal ray increases. This is not preferable because the xy cross-sectional area of the optical system increases and the size of the apparatus increases. In addition, since the back focus becomes too short, the free-form surface group must have a large negative power in order to extend it, and it becomes difficult to simultaneously perform trapezoidal distortion correction and aberration correction as described above. Should be avoided.

  If the following conditional expression (17-1) is satisfied instead of the conditional expression (17), the balance between the size of the optical system and the power of the rotationally symmetric lens group G1 is made more optimal. Therefore, it is possible to more compactly and correct aberrations more efficiently.

  18 ≦ | TL / f | ≦ 20 (17-1)

  Further, as shown in FIG. 1, the inclination angle of the reflecting surface of the free-form surface mirror M with respect to a surface perpendicular to the optical axis between the free-form surface lens group G2 and the free-form surface mirror M is θm, and the free-form surface mirror M and the projection are projected. When the inclination angle of the projection plane R with respect to a plane perpendicular to the optical axis between the plane R (image plane I) is θs, it is preferable that the condition represented by the following conditional expression (18) is satisfied.

  0.5 ≦ | θm / θs | ≦ 0.7 (18)

  Conditional expression (18) is for optimizing the size of the optical system. When the condition is lower than the lower limit value of the conditional expression (18), the lens positioned in front of the free-form surface mirror M and the light beam bent by the free-form surface mirror M interfere with each other, which is not desirable. In addition, the trapezoidal distortion increases. On the other hand, when the condition exceeds the upper limit value of conditional expression (18), the angle formed by the optical axis of the projection lens and the screen becomes small, leading to an increase in the size of the apparatus. Further, the distance between the image display element DS and the projection surface R (screen) becomes long, and projection at a short distance becomes impossible.

  If the following conditional expression (18-1) is satisfied instead of the above conditional expression (18), the size of the optical system can be further optimized.

  0.55 ≦ | θm / θs | ≦ 0.6 (18-1)

  As described above, according to the present embodiment, the optical system PL for a video projection apparatus that can satisfactorily correct the trapezoidal distortion of a projected image, and the video projection apparatus PRJ including the same, with a compact configuration. Can be obtained.

  In the video projection apparatus PRJ of the present embodiment, an image processing unit (not shown) that performs predetermined image processing on the video displayed on the video display element DS and corrects trapezoidal distortion of the video projected from an oblique direction. ). In this way, the trapezoidal distortion in the direction to cancel the trapezoidal distortion of the image that occurs when performing oblique projection is generated in the input image in advance, thereby reducing the burden of trapezoidal distortion correction on the free-form surface group. Thus, the shape of the free-form surface group can be changed to a free-form surface shape suitable for mass production.

  Embodiments of the present application will be described below with reference to the accompanying drawings. In each embodiment, an aspheric surface that is rotationally symmetric with respect to the optical axis (center axis) and a free-form surface that is not rotationally symmetric with respect to the central axis are used. Therefore, before describing each embodiment, these defining formulas will be described.

  First, an aspherical surface that is rotationally symmetric with respect to the optical axis (center axis) is defined by the following equation (19). In the following equation (19), Z is the sag amount of the surface parallel to the optical axis, c is the curvature at the surface vertex (on the optical axis), k is the conic coefficient, and h is the optical axis. Is a distance perpendicular to this, and A to J are coefficients for each power series term of h.

  Next, a free-form surface that is not rotationally symmetric with respect to the central axis is defined by the following equation (20). In the following equation (20), Z is the sag amount of the surface parallel to the central axis, c is the curvature at the surface vertex (origin), k is the conic constant, and h is the central axis. This is the distance from the origin in the plane perpendicular to this at the origin, and Cj is the coefficient of the xy polynomial.

  Here, a relationship represented by the following expressions (21) and (22) is established between j, m, and n in the expression (20).

(First embodiment)
1st Example of this application is described using FIGS. 1-6 and Table 1-Table 4. FIG. 1 is a side sectional view (y-z sectional view) of an optical system for an image projection apparatus according to a first embodiment, and FIG. 2 is a plan sectional view (x of an optical system for an image projection apparatus according to the first embodiment). -Z sectional view). The optical system PL for image projection apparatus according to the first example is composed of a plurality of rotationally symmetric lenses L11 to L14 that are arranged in order from the image display element DS side along the optical axis and are formed rotationally symmetrically with respect to the central axis. A rotationally symmetric lens group G1, a free-form surface lens group G2 composed of a plurality of free-form surface lenses L21 to L23 formed in a non-rotational symmetry with respect to the central axis, and a reflection formed in a non-rotational symmetry with respect to the central axis. And a free-form surface mirror M having a surface. Two parallel flat plates are arranged between the display surface (element surface) of the video display element DS and the rotationally symmetric lens group G1, and this parallel flat plate has a video display element on the video display element DS side. A DS cover glass CV is a prism P (PBS: polarization beam splitter) that reflects S-polarized light and transmits P-polarized light on the image side. A stop S is disposed between the rotationally symmetric lens group G1 and the free-form surface lens group G2.

  The rotationally symmetric lens group G1 is arranged in order from the image display element DS side along the optical axis, and includes a biconvex first rotationally symmetric lens L11, a biconvex second rotationally symmetric lens L12, and a biconcave shape. The lens includes a third rotationally symmetric lens L13 and a fourth rotationally symmetric lens L14 that is a meniscus lens having a concave surface facing the element side. The element-side lens surface (seventh surface) in the second rotationally symmetric lens L12, the image-side lens surface (ninth surface) in the third rotationally symmetric lens L13, and the image-side lens surface in the fourth rotationally symmetric lens L14. The lens surface (the 12th surface) is an aspherical surface. Further, the second rotationally symmetric lens L12 and the third rotationally symmetric lens L13 are bonded lenses.

  The free-form surface lens group G2 includes a first free-form surface lens L21, a second free-form surface lens L22, and a third free-form surface lens L23 arranged in order from the image display element DS side. The cross-sectional shape of the third free-form surface lens L23 has a meniscus shape with the concave surface facing the image display element DS side in the xz cross section. The cross-sectional shape of the second free-form surface lens L22 has a meniscus shape with a concave surface facing the image display element DS side in the yz cross section and the xz cross section. Regarding the free-form surface mirror M, the z-axis of the local coordinate system of the free-form surface mirror M is inclined by +35 degrees in the yz plane with the coordinate origin as the center with respect to the optical axis of the free-form surface lens group G2. In each of the following embodiments, the local coordinate system of each lens and mirror is a right-handed system (xyz coordinate system), the z-axis is an optical axis, and the sign is positive in the light traveling direction. However, after mirror reflection, the light traveling direction is negative. The tilt of each lens, mirror, etc. indicates the tilt in the y-z plane (plane parallel to the cross section of FIG. 1), and the sign is counterclockwise when viewed in the positive direction of the x-axis. Define as positive.

  In order to alleviate the focus shift between the upper and lower parts of the screen due to oblique projection, a virtual surface (fourth surface) is provided on the incident surface of the rotationally symmetric lens group G1, and this virtual surface is rotated around the x axis (local coordinate system). And the rotationally symmetric lens group G1 is tilted (tilted). Further, by providing virtual surfaces (the 13th surface and the 15th surface) before and after the stop S, the 13th surface is shifted in the y direction, and the 15th surface is rotated around the x axis, so that the image display area The light beam emitted from the center along the optical axis generally passes through the center of the stop and travels along the optical axis after passing. A virtual surface (tenth surface) is also provided between the biconcave third rotationally symmetric lens L13 and the fourth rotationally symmetric lens L14.

  Table 1 below shows various data of the optical system PL for the image projection apparatus according to the first example. In the table (numerical data of the optical system) shown in each of the following examples, “* a” described therein represents that the surface is a rotationally symmetric aspheric surface, and “* f” represents This indicates that the surface is a non-rotationally symmetric free-form surface.

(Table 1)
(Overall specifications)
Object side numerical aperture 0.21739
Projection size 284.48mm x 213.36mm
Magnification 37.8x Rotationally symmetric lens group focal length f 14.817mm

(Lens data)
Surface number Curvature radius Surface spacing Refractive index / νd
Element plane Plane 1.00
1 plane 0.70 1.51680 / 64.2
2 Plane 13.00 1.51680 / 64.2
3 Plane 3.51
4 plane 0.00
5 20.42000 6.00 1.49700 / 81.6
6 -20.42000 2.00
7 * a 12.96154 5.00 1.61305 / 59.0
8 -19.92000 2.00 1.72250 / 29.2
9 * a 44.52031 2.50
10 plane 0.00
11 -12.23000 2.00 1.84400 / 24.8
12 * a -15.05107 1.00
13 plane 0.00
14 (Aperture) Plane 0.00
15 plane 1.00
16 * f plane 3.00 1.53113 / 55.7
17 * f plane 9.70
18 * f plane 4.00 1.53113 / 55.7
19 * f plane 5.40
20 * f plane 5.00 1.53113 / 55.7
21 * f plane 24.00
22 * f Plane -196.64 Reflection Image plane Plane 0.00

  In the lens data of Table 1, the fourth surface to the twelfth surface are lens surfaces of the rotationally symmetric lens group G1, and among them, the seventh surface, the ninth surface, and the twelfth surface are rotationally symmetric aspheric surfaces. . Table 2 below shows the aspherical coefficients of the seventh surface, the ninth surface, and the twelfth surface, respectively.

(Table 2)
(Aspheric data)
Term 7th surface coefficient 9th surface coefficient 12th surface coefficient c 0.07715 0.02246 -0.06644
k -0.311292 0 0
A (4th order) -1.3400918E-04 -3.2982930E-04 1.9974879E-04
B (6th) -5.3237190E-07 -2.3536542E-06 -2.4346185E-06
C (8th) -6.7510945E-08 -1.5816492E-08 3.4025478E-08
D (10th order) 1.6899482E-09 -1.5838434E-09 4.6050232E-09
E (12th order) -2.7610274E-11 0 0
(Since all coefficients after F are 0, they are omitted)

  In the lens data in Table 1, the 16th to 22nd surfaces are non-rotationally symmetric free-form surfaces. In the present embodiment, the 22nd surface is the reflecting surface of the free-form surface mirror M. Table 3 below shows each term coefficient of these free-form surfaces.

(Table 3)
(Free curved surface data 1)
Term 16th surface coefficient 17th surface coefficient 18th surface coefficient c 0 0 0
C1 (k) 0 0 0
C3 (y) -2.7164989E-01 -2.9945917E-01 3.3015954E-01
C4 (x ^ 2) -2.7371870E-03 -2.2986215E-02 -9.0285606E-02
C6 (y ^ 2) 3.2852532E-02 1.3992659E-02 -1.4697519E-01
C8 (x ^ 2 * y) -4.0397838E-03 -2.6111557E-03 3.0777626E-03
C10 (y ^ 3) -1.3811026E-03 3.9109077E-04 1.1235385E-02
C11 (x ^ 4) -8.5753499E-05 -2.2309354E-04 -2.9732017E-04
C13 (x ^ 2 * y ^ 2) 8.0887249E-04 6.9120615E-04 -7.7221879E-04
C15 (y ^ 4) 8.2475225E-04 8.4245035E-04 -1.0374485E-03
C17 (x ^ 4 * y) 2.6565573E-05 5.8438271E-05 6.0941586E-05
C19 (x ^ 2 * y ^ 3) 6.1193547E-05 6.4895466E-05 6.8301523E-05
C21 (y ^ 5) 4.0706773E-05 6.7982959E-05 1.3403154E-04
C22 (x ^ 6) -1.2849015E-05 -7.2690008E-06 -1.5980850E-06
C24 (x ^ 4 * y ^ 2) -2.1577674E-05 -1.5848149E-05 -3.0007477E-05
C26 (x ^ 2 * y ^ 4) -1.9130400E-06 9.2196998E-06 -3.4089703E-05
C28 (y ^ 6) -1.7362079E-06 3.3034439E-06 -3.6587100E-05
C30 (x ^ 6 * y) 1.9194423E-06 1.3329724E-06 3.5671502E-06
C32 (x ^ 4 * y ^ 3) 2.4643407E-06 3.2668438E-06 1.2944243E-05
C34 (x ^ 2 * y ^ 5) -3.9669435E-07 2.0179026E-06 1.5037660E-05
C36 (y ^ 7) -1.2391018E-06 -5.5418285E-07 1.0099056E-05
C37 (x ^ 8) 4.3597774E-07 2.6450808E-07 1.7727747E-09
C39 (x ^ 6 * y ^ 2) 2.0287590E-06 1.6187944E-06 -4.4838348E-07
C41 (x ^ 4 * y ^ 4) 2.6898904E-06 2.7761672E-06 -1.2230089E-06
C43 (x ^ 2 * y ^ 6) 1.3906298E-06 2.1496462E-06 -1.2566581E-06
C45 (y ^ 8) 5.4403332E-07 1.1781426E-06 -7.2636295E-07
(Since all coefficients after C46 are 0, they are omitted.)

(Free curved surface data 2)
Term 19th surface coefficient 20th surface coefficient 21st surface coefficient c 0 0 0
C1 (k) 0 0 0
C3 (y) 3.1928411E-01 3.5684097E-01 2.1412290E-01
C4 (x ^ 2) -7.2597729E-02 -3.9708065E-02 -3.2895026E-02
C6 (y ^ 2) -9.1747765E-02 -1.6558219E-02 -1.1925637E-02
C8 (x ^ 2 * y) -3.2004372E-03 -1.5693633E-03 1.0716634E-03
C10 (y ^ 3) 8.5947565E-05 1.3115829E-03 3.4613120E-03
C11 (x ^ 4) 1.6210408E-04 2.4664734E-04 1.3367935E-05
C13 (x ^ 2 * y ^ 2) 6.6926503E-05 -5.4739437E-05 -2.4939498E-05
C15 (y ^ 4) 8.0767480E-06 1.2034245E-04 1.6444545E-04
C17 (x ^ 4 * y) -3.4868190E-05 -9.1628408E-06 -1.3775191E-05
C19 (x ^ 2 * y ^ 3) -6.0665856E-05 7.7908870E-06 -9.6385703E-06
C21 (y ^ 5) -2.2403749E-05 1.3951857E-05 5.3980110E-06
C22 (x ^ 6) 2.6143604E-07 -2.3863466E-06 -1.9406514E-07
C24 (x ^ 4 * y ^ 2) 2.3874868E-06 -9.8668211E-07 -1.7687467E-06
C26 (x ^ 2 * y ^ 4) 8.8907512E-06 1.8430607E-06 -1.8067207E-06
C28 (y ^ 6) 2.8897175E-06 -3.3632489E-07 -4.8877080E-07
C30 (x ^ 6 * y) 4.3265630E-07 -7.9222888E-09 4.9827665E-08
C32 (x ^ 4 * y ^ 3) 3.3844674E-07 -2.7566738E-07 -2.9934843E-08
C34 (x ^ 2 * y ^ 5) -1.8853146E-07 -3.1562484E-07 -1.0162105E-07
C36 (y ^ 7) 1.0494137E-07 -7.2497867E-08 -1.2359180E-08
C37 (x ^ 8) -4.8221827E-10 3.4660156E-09 -2.2262943E-09
C39 (x ^ 6 * y ^ 2) -2.9046452E-08 7.5635782E-09 1.0112620E-08
C41 (x ^ 4 * y ^ 4) -4.8452367E-08 -3.5037115E-08 4.7040778E-09
C43 (x ^ 2 * y ^ 6) 5.7980725E-08 -1.3607590E-08 -3.4451643E-10
C45 (y ^ 8) 2.9086309E-08 -3.2759598E-10 1.1769751E-10
(Since all coefficients after C46 are 0, they are omitted.)

(Free curved surface data 3)
Term 22nd surface coefficient c 0
C1 (k) 0
C3 (y) 2.4201793E-02
C4 (x ^ 2) 1.4453640E-02
C6 (y ^ 2) 1.6891420E-03
C8 (x ^ 2 * y) 7.2876362E-04
C10 (y ^ 3) 1.4602557E-04
C11 (x ^ 4) -9.7793046E-06
C13 (x ^ 2 * y ^ 2) 2.1755522E-05
C15 (y ^ 4) 5.3084710E-06
C17 (x ^ 4 * y) -1.0393819E-06
C19 (x ^ 2 * y ^ 3) 3.3455725E-07
C21 (y ^ 5) 1.7384390E-07
C22 (x ^ 6) 4.8472761E-09
C24 (x ^ 4 * y ^ 2) -4.3839485E-08
C26 (x ^ 2 * y ^ 4) -3.1188580E-09
C28 (y ^ 6) 4.3253652E-09
C30 (x ^ 6 * y) 9.8767449E-10
C32 (x ^ 4 * y ^ 3) -9.1481892E-10
C34 (x ^ 2 * y ^ 5) -3.1904648E-10
C36 (y ^ 7) 5.3838608E-11
C37 (x ^ 8) 9.6498932E-12
C39 (x ^ 6 * y ^ 2) 3.9085698E-11
C41 (x ^ 4 * y ^ 4) -8.2300157E-12
C43 (x ^ 2 * y ^ 6) -7.7692101E-12
C45 (y ^ 8) -3.4775011E-13
C47 (x ^ 8 * y) 2.9808202E-13
C49 (x ^ 6 * y ^ 3) 4.2578716E-13
C51 (x ^ 4 * y ^ 5) 4.1876523E-15
C53 (x ^ 2 * y ^ 7) -7.7684026E-14
C55 (y ^ 9) -1.3376482E-14
(Since all coefficients after C56 are 0, they are omitted.)

  Furthermore, the eccentricity of the local coordinate system of the fourth surface, the thirteenth surface, the fifteenth surface, and the twenty-second surface in this example is shown in Table 4 below. In the eccentric data in Table 4, “ADE” is the magnitude of the inclination in a plane (y-z plane) parallel to the cross section of FIG. “YDE” is the magnitude of the shift, and the shift is set in a plane (y-z plane) parallel to the cross section of FIG. 1 and in a direction perpendicular to the optical axis (z-axis). Here, the sign rule such as inclination is as described above. The explanation of the eccentricity data is the same in the other examples below, and the explanation is omitted.

(Table 4)
(Eccentric data)
Surface number ADE (°) YDE (mm)
4 0.5 0
13 0.0 -0.13842
15 -0.555 0
22 35.0 0
Image plane -60.0 0

  The corresponding values for each conditional expression are shown below.

Conditional Expression (17) | TL / f | = | 290.4127 / 14.8168 | = 19.600
Conditional expression (18) | θm / θs | = | 35.0 / −60.0 | = 0.58
(Third free-form curved lens L23)
L1 = 20.665
L2 = 23.926
(x-z cross section)
Element side lens surface maximum point coordinates (x _o1 , z _xo1 ) = (0, 0)
Image side lens surface maximum point coordinates (x _o2 , z _xo2 ) = (0, 0)
Conditional expression (3) (x ₁ + x _o1 ) / | x ₁ + x _o1 | × f ₁ ′ (x ₁ + x _o1 ) <0
Conditional expression (4) (x ₂ + x _o2 ) / | x ₂ + x _o2 | × f ₂ ′ (x ₂ + x _o2 ) <0
Formula (12) rx1 = −L1 / 2.5 = −8.27
Formula (13) rx2 = −L2 / 0.1 = −239.26
Conditional expression (15) In the range of − | radxi | <x <| radxi |
zfc1 (x) <zs2 (x), zfc2 (x) <zs2 (x)
Conditional expression (16) | rx1 | <| radxi |, and in the range of − | rx1 |
zs1 (x) <zfc1 (x), zs1 (x) <zfc2 (x)
(Second free-form surface lens L22)
L1 = 12.656
L2 = 15.265
(y-z cross section)
Element-side lens surface maximum point coordinates _{(y o1, z yo1) =} (1.287803,0.202936)
Image-side lens surface maximum point coordinates _{(y o2, z yo2) =} (1.741274,0.278036)
Conditional expression (1) (y ₁ + y _o1 ) / | y ₁ + y _o1 | × f ₁ ′ (y ₁ + y _o1 ) <0
Conditional expression (2) (y ₂ + y _o2 ) / | y ₂ + y _o2 | × f ₂ ′ (y ₂ + y _o2 ) <0
Formula (6) ry1 = −L1 / 3.5 = −3.62
Formula (7) ry2 = −L2 = −15.27
Conditional expression (9) in the range of − | radyi | <y <| radyi |
zfc1 (y) <zs2 (y), zfc2 (y) <zs2 (y)
Conditional expression (10) | ry1 | <| radyi |, and in the range of − | ry1 | <y <| ry1 |
zs1 (y) <zfc1 (y), zs1 (y) <zfc2 (y)
(x-z cross section)
Element side lens surface maximum point coordinates (x _o1 , z _xo1 ) = (0, 0)
Image side lens surface maximum point coordinates (x _o2 , z _xo2 ) = (0, 0)
Conditional expression (3) (x ₁ + x _o1 ) / | x ₁ + x _o1 | × f ₁ ′ (x ₁ + x _o1 ) <0
Conditional expression (4) (x ₂ + x _o2 ) / | x ₂ + x _o2 | × f ₂ ′ (x ₂ + x _o2 ) <0
Formula (12) rx1 = −L1 / 2.5 = −5.06
Formula (13) rx2 = −L2 / 0.1 = −152.65
Conditional expression (15) In the range of − | radxi | <x <| radxi |
zfc1 (x) <zs2 (x), zfc2 (x) <zs2 (x)
Conditional expression (16) | rx1 | <| radxi |, and in the range of − | rx1 | <x <| rx1 |
zs1 (x) <zfc1 (x), zs1 (x) <zfc2 (x)

  Thus, it can be seen that conditional expression (1) and the like are satisfied in this embodiment. The sag amount in each cross section of the third free-form surface lens L23 and the second free-form surface lens L22 is shown in FIGS. 3 and 4, respectively.

  FIG. 5 is an e-line monochromatic spot diagram of the optical system PL for image projection apparatus according to the first embodiment. The length of the straight line displayed at the bottom of the spot diagram corresponds to 1 mm on the screen. Corresponding object point positions are (x coordinate, y coordinate) = (0.00, 0.00), (1.88, 0.00), (3.76, 0.00) in order from the bottom of the spot diagram. ), (0.00, 1.41), (1.88, 1.41), (3.76, 1.41), (0.00, -1.41), (1.88, -1 .41), (3.76, -1.41), (0.00, 2.82), (1.88, 2.82), (3.76, 2.82), (0.00, -2.82), (1.88, -2.82), (3.76, -2.82).

  FIG. 6 represents an image (distortion) when a grid is displayed over the entire display area. In the actual calculation, since the convolution integral of the object image and the point image intensity distribution of the optical system is calculated, not only the trapezoidal distortion state but also the resolving power is expressed. In FIG. 6, the lattice line width of the object is about 0.01 mm. It should be noted that the description of the distortion is the same in the following other embodiments, and the description thereof is omitted. 5 and 6, it can be seen that in the first example, the trapezoidal distortion is corrected well and the imaging performance is excellent.

(Second embodiment)
A second embodiment of the present application will be described with reference to FIGS. 7 to 12 and Tables 5 to 8. FIG. FIG. 7 is a side sectional view (y-z sectional view) of the optical system for the image projection apparatus according to the second embodiment, and FIG. 8 is a plan sectional view (x of the optical system for the image projection apparatus according to the second embodiment). -Z sectional view). The optical system for the image projection apparatus according to the second embodiment has the same configuration as that of the optical system for the image projection apparatus according to the first embodiment except for the configuration of the rotationally symmetric lens group. The same reference numerals as those in the case are attached and detailed description is omitted.

  The rotationally symmetric lens group G1 of the second example is a biconvex first rotationally symmetric lens L11 arranged in order from the image display element DS side along the optical axis, and a meniscus lens having a convex surface facing the element side. The lens includes a second rotationally symmetric lens L12, a third rotationally symmetric lens L13 that is a meniscus lens having a convex surface facing the element side, and a fourth rotationally symmetric lens L14 that is a meniscus lens having a convex surface facing the element side. The element-side lens surface (seventh surface) in the second rotationally symmetric lens L12, the image-side lens surface (ninth surface) in the third rotationally symmetric lens L13, and the image-side lens surface in the fourth rotationally symmetric lens L14. The lens surface (the 12th surface) is an aspherical surface. Further, the second rotationally symmetric lens L12 and the third rotationally symmetric lens L13 are bonded lenses.

  Table 5 below shows various data of the optical system PL for the image projection apparatus according to the second example.

(Table 5)
(Overall specifications)
Object side numerical aperture 0.20000
Projection size 284.48mm x 213.36mm
Magnification 37.0x Rotational symmetry lens group focal length f 14.260mm

(Lens data)
Surface number Curvature radius Surface spacing Refractive index / νd
Element plane Plane 1.00
1 plane 0.70 1.51680 / 64.2
2 plane 11.00 1.78800 / 47.4
3 Plane 10.93
4 plane 0.00
5 21.03444 7.00 1.49700 / 81.6
6 -21.03444 0.20
7 * a 10.90764 6.00 1.63246 / 63.8
8 70.72422 2.00 1.83917 / 23.9
9 * a 10.13481 4.00
10 plane 0.00
11 10.85532 2.00 2.14352 / 17.8
12 * a 10.71335 0.00
13 plane 0.50
14 (Aperture) Plane 0.00
15 plane 1.00
16 * f plane 3.00 1.53113 / 55.7
17 * f plane 9.50
18 * f plane 3.00 1.53113 / 55.7
19 * f plane 3.43
20 * f plane 5.00 1.53113 / 55.7
21 * f plane 21.00
22 * f Plane -199.61 Reflection Image plane Plane 0.00

  In the lens data in Table 5, the fourth surface to the twelfth surface are lens surfaces of the rotationally symmetric lens group G1, and among them, the seventh surface, the ninth surface, and the twelfth surface are rotationally symmetric aspheric surfaces. . Table 6 below shows the aspherical coefficients of the seventh surface, the ninth surface, and the twelfth surface, respectively.

(Table 6)
(Aspheric data)
Term 7th surface coefficient 9th surface coefficient 12th surface coefficient c 0.09168 0.09867 0.09334
k -2.90877 0 0
A (4th order) 1.9535635E-04 -7.7588897E-05 -3.2174401E-05
B (6th) -1.8450936E-06 -4.2829065E-06 1.4592384E-05
C (8th) -5.4203107E-09 -3.4566719E-08 -1.3440130E-06
D (10th order) 1.3623190E-10 -2.5784731E-09 6.4768446E-08
E (12th order) -3.9894819E-12 0 0
(Since all coefficients after F are 0, they are omitted)

  In the lens data of Table 5, the 16th to 22nd surfaces are non-rotationally symmetric free-form surfaces. In the present embodiment, the 22nd surface is the reflecting surface of the free-form surface mirror M. Table 7 below shows each term coefficient of these free-form surfaces.

(Table 7)
(Free curved surface data 1)
Term 16th surface coefficient 17th surface coefficient 18th surface coefficient c 0 0 0
C1 (k) 0 0 0
C3 (y) -2.6041813E-01 -3.3508872E-01 3.5978955E-01
C4 (x ^ 2) 1.1826274E-02 -5.8967222E-03 -7.0383083E-02
C6 (y ^ 2) 4.1661092E-02 2.1233688E-02 -1.3665859E-01
C8 (x ^ 2 * y) -3.9473639E-03 -2.7850136E-03 7.8857468E-04
C10 (y ^ 3) -1.1423394E-03 8.1331969E-04 1.5654117E-02
C11 (x ^ 4) -2.0340783E-04 -9.7625263E-05 -4.6365215E-04
C13 (x ^ 2 * y ^ 2) 2.8258800E-04 5.7103139E-04 -2.4407852E-03
C15 (y ^ 4) 6.4885200E-04 9.3071472E-04 -2.1261327E-03
C17 (x ^ 4 * y) 4.0327007E-05 7.1042274E-05 3.1254897E-04
C19 (x ^ 2 * y ^ 3) 2.7019582E-05 5.0337585E-05 7.8024735E-04
C21 (y ^ 5) 8.0589865E-06 3.2272800E-05 2.1096408E-04
C22 (x ^ 6) 3.8248095E-05 8.4371240E-06 5.3487013E-06
C24 (x ^ 4 * y ^ 2) 9.4081147E-05 2.2912845E-05 -1.6960606E-06
C26 (x ^ 2 * y ^ 4) 7.2352479E-05 1.7894704E-05 -3.6997135E-05
C28 (y ^ 6) 1.9260933E-05 4.1391628E-06 -5.6995135E-06
C30 (x ^ 6 * y) 1.5459067E-06 3.3649236E-06 5.9783348E-06
C32 (x ^ 4 * y ^ 3) 4.1144273E-08 3.7050894E-06 6.9738975E-06
C34 (x ^ 2 * y ^ 5) 4.7200657E-06 4.5974549E-06 1.0644933E-05
C36 (y ^ 7) 1.4674388E-06 9.3126428E-07 5.3075155E-07
C37 (x ^ 8) -3.7416310E-06 -5.3535501E-07 -2.6396075E-08
C39 (x ^ 6 * y ^ 2) -1.0276811E-05 -7.3029517E-07 -2.9183475E-06
C41 (x ^ 4 * y ^ 4) -8.4510780E-06 2.4603301E-06 -7.0051200E-06
C43 (x ^ 2 * y ^ 6) -7.0198445E-07 6.2811387E-06 -2.7315925E-06
C45 (y ^ 8) 5.1743326E-07 2.9867245E-06 -3.7276033E-07
C47 (x ^ 8 * y) -9.3177463E-08 -1.6770224E-07 -1.1637311E-08
C49 (x ^ 6 * y ^ 3) -1.1877155E-08 -3.4259129E-07 7.5237116E-07
C51 (x ^ 4 * y ^ 5) -3.2384158E-07 -5.5039427E-07 1.4251137E-06
C53 (x ^ 2 * y ^ 7) -6.5862869E-07 -6.1323078E-07 8.0560632E-08
C55 (y ^ 9) -1.8236665E-07 -2.4601735E-07 9.9509732E-08
C56 (y ^ 10) 1.7681756E-07 3.3688147E-08 -2.5093455E-10
C58 (x ^ 8 * y ^ 2) 6.1365036E-07 8.7698739E-08 -3.2341791E-09
C60 (x ^ 6 * y ^ 4) 7.9841144E-07 4.1824000E-08 -7.4344821E-08
C62 (X ^ 4 * y ^ 26) 3.7045248E-07 -5.8875173E-08 -7.8151512E-08
C64 (X ^ 2 * y ^ 28) -8.8716877E-08 -2.4475517E-07 3.9076794E-09
C66 (y ^ 10) -2.3966984E-08 -5.4473898E-08 -4.9980691E-09

(Free curved surface data 2)
Term 19th surface coefficient 20th surface coefficient 21st surface coefficient c 0 0 0
C1 (k) 0 0 0
C3 (y) 3.3365563E-01 3.0685418E-01 2.4813533E-01
C4 (x ^ 2) -5.7532907E-02 -3.9396149E-02 -3.9798366E-02
C6 (y ^ 2) -4.7563856E-02 1.4470529E-02 -1.4712861E-02
C8 (x ^ 2 * y) -6.0052101E-03 2.1308948E-03 3.6417478E-03
C10 (y ^ 3) 7.7731371E-05 1.4409240E-03 3.5348542E-03
C11 (x ^ 4) 6.4145191E-05 1.7930250E-05 -8.3125745E-05
C13 (x ^ 2 * y ^ 2) -8.0340544E-05 5.4224552E-04 4.1739456E-04
C15 (y ^ 4) -1.4884064E-04 3.1351662E-04 4.2936576E-04
C17 (x ^ 4 * y) 6.8779530E-05 -7.7716810E-05 -4.3333795E-05
C19 (x ^ 2 * y ^ 3) 1.9806411E-04 1.6385645E-07 2.5845437E-05
C21 (y ^ 5) -8.6869718E-05 6.3297046E-06 3.4059708E-05
C22 (x ^ 6) -3.2867482E-08 -3.0406161E-06 -7.4305959E-07
C24 (x ^ 4 * y ^ 2) 6.2032943E-06 -2.5878450E-06 -8.5935792E-06
C26 (x ^ 2 * y ^ 4) 1.0075005E-05 -4.9807798E-06 -1.9596538E-06
C28 (y ^ 6) 1.5051609E-05 -1.2959658E-06 2.4796795E-06
C30 (x ^ 6 * y) 1.2945976E-06 1.9202003E-07 8.7618193E-08
C32 (x ^ 4 * y ^ 3) -2.0852128E-06 2.4247531E-07 -6.3635364E-07
C34 (x ^ 2 * y ^ 5) -2.0325580E-06 4.7425385E-07 -3.0162098E-07
C36 (y ^ 7) 4.4577925E-07 -1.1213258E-07 2.5052904E-08
C37 (x ^ 8) 7.8519053E-08 -1.6151932E-08 -6.4750976E-09
C39 (x ^ 6 * y ^ 2) -2.4844770E-07 -3.5322101E-08 6.5771445E-08
C41 (x ^ 4 * y ^ 4) -1.0185718E-07 -5.8353264E-09 5.1955449E-08
C43 (x ^ 2 * y ^ 6) -1.5961773E-07 4.5563395E-08 -3.9285141E-09
C45 (y ^ 8) -3.6739652E-07 3.4014020E-09 -3.1084567E-08
C47 (x ^ 8 * y) -4.6786334E-09 5.3747709E-09 1.5337922E-09
C49 (x ^ 6 * y ^ 3) -1.7275156E-08 -6.7034371E-09 2.8870106E-09
C51 (x ^ 4 * y ^ 5) 4.1882418E-08 -6.3748940E-10 1.7124915E-09
C53 (x ^ 2 * y ^ 7) -8.2685364E-09 -5.5034831E-09 8.2443453E-10
C55 (y ^ 9) 1.4271216E-08 3.6841661E-10 -6.7287913E-09
C56 (y ^ 10) -6.6959428E-10 3.7660422E-10 3.5422758E-11
C58 (x ^ 8 * y ^ 2) 4.3203371E-09 6.3872630E-10 -2.7250144E-10
C60 (x ^ 6 * y ^ 4) 5.9150691E-09 -2.7991354E-10 -3.7777723E-10
C62 (X ^ 4 * y ^ 26) 1.8390612E-09 -4.4279901E-11 -4.4085890E-10
C64 ((X ^ 2 * y ^ 28) 1.5641973E-09 -4.6766855E-10 -3.5877409E-11
C66 (y ^ 10) 3.5540724E-09 -1.1139770E-11 -3.4630667E-10

(Free curved surface data 3)
Term 22nd surface coefficient c 0
C1 (k) 0
C3 (y) 2.5554379E-02
C4 (x ^ 2) 1.6338720E-02
C6 (y ^ 2) 1.9654169E-03
C8 (x ^ 2 * y) 9.3479190E-04
C10 (y ^ 3) 2.2029628E-04
C11 (x ^ 4) -1.4539531E-05
C13 (x ^ 2 * y ^ 2) 3.3847778E-05
C15 (y ^ 4) 8.7220312E-06
C17 (x ^ 4 * y) -1.2598260E-06
C19 (x ^ 2 * y ^ 3) 7.6399847E-07
C21 (y ^ 5) 2.8112003E-07
C22 (x ^ 6) 2.3109057E-08
C24 (x ^ 4 * y ^ 2) -5.6065597E-08
C26 (x ^ 2 * y ^ 4) 1.3201618E-09
C28 (y ^ 6) 1.0733281E-08
C30 (x ^ 6 * y) 9.0361885E-10
C32 (x ^ 4 * y ^ 3) -1.6340322E-09
C34 (x ^ 2 * y ^ 5) -6.5153497E-10
C36 (y ^ 7) 4.1109091E-10
C37 (x ^ 8) -4.6209688E-11
C39 (x ^ 6 * y ^ 2) -3.0413647E-11
C41 (x ^ 4 * y ^ 4) -3.0822618E-11
C43 (x ^ 2 * y ^ 6) -1.9011268E-11
C45 (y ^ 8) 7.9009017E-12
C47 (x ^ 8 * y) -2.6946898E-12
C49 (x ^ 6 * y ^ 3) -2.4387396E-12
C51 (x ^ 4 * y ^ 5) -1.1331121E-13
C53 (x ^ 2 * y ^ 7) -8.2658852E-14
C55 (y ^ 9) -1.8480462E-14
C56 (y ^ 10) 1.5462337E-14
C58 (x ^ 8 * y ^ 2) -5.8626927E-14
C60 (x ^ 6 * y ^ 4) -3.2414498E-14
C62 (X ^ 4 * y ^ 26) 3.2916835E-15
C64 (X ^ 2 * y ^ 28) 2.7221317E-15
C66 (y ^ 10) -1.8842008E-15

  Furthermore, the eccentricity of the local coordinate system of the fourth surface, the thirteenth surface, the fifteenth surface, and the twenty-second surface in this example is shown in Table 8 below.

(Table 8)
(Eccentric data)
Surface number ADE (°) YDE (mm)
4 0.5 0.00000
13 0.0 -0.13842
15 -0.555 0
22 35.0 0
Image plane -60.0 0

  The corresponding values for each conditional expression are shown below.

Conditional Expression (17) | 290.8798 / 14.2598 | = 20.399
Conditional expression (18) | θm / θs | = | 35.0 / −60.0 | = 0.58
(Third free-form curved lens L23)
L1 = 17.846
L2 = 21.106
(x-z cross section)
Element side lens surface maximum point coordinates (x _o1 , z _xo1 ) = (0, 0)
Image side lens surface maximum point coordinates (x _o2 , z _xo2 ) = (0, 0)
Conditional expression (3) (x ₁ + x _o1 ) / | x ₁ + x _o1 | × f ₁ ′ (x ₁ + x _o1 ) <0
Conditional expression (4) (x ₂ + x _o2 ) / | x ₂ + x _o2 | × f ₂ ′ (x ₂ + x _o2 ) <0
Formula (12) rx1 = −L1 / 2.5 = −7.14
Formula (13) rx2 = −L2 / 0.1 = −211.06
Conditional expression (15) In the range of − | radxi | <x <| radxi |
zfc1 (x) <zs2 (x), zfc2 (x) <zs2 (x)
Conditional expression (16) | rx1 | <| radxi |, and in the range of − | rx1 | <x <| rx1 |
zs1 (x) <zfc1 (x), zs1 (x) <zfc2 (x)
(Second free-form surface lens L22)
L1 = 12.456
L2 = 14.413
(y-z cross section)
Element-side lens surface maximum point coordinates _{(y o1, z yo1) =} (1.684773,0.278734)
Image-side lens surface maximum point coordinates _{(y o2, z yo2) =} (3.144479,0.553191)
Conditional expression (1) (y ₁ + y _o1 ) / | y ₁ + y _o1 | × f ₁ ′ (y ₁ + y _o1 ) <0
Conditional expression (2) (y ₂ + y _o2 ) / | y ₂ + y _o2 | × f ₂ ′ (y ₂ + y _o2 ) <0
Formula (6) ry1 = −L1 / 3.5 = −3.56
Expression (7) ry2 = −L2 = −14.41
Conditional expression (9) in the range of − | radyi | <y <| radyi |
zfc1 (y) <zs2 (y), zfc2 (y) <zs2 (y)
Conditional expression (10) | ry1 | <| radyi |, and in the range of − | ry1 | <y <| ry1 |
zs1 (y) <zfc1 (y), zs1 (y) <zfc2 (y)
(x-z cross section)
Element side lens surface maximum point coordinates (x _o1 , z _xo1 ) = (0, 0)
Image side lens surface maximum point coordinates (x _o2 , z _xo2 ) = (0, 0)
Conditional expression (3) (x ₁ + x _o1 ) / | x ₁ + x _o1 | × f ₁ ′ (x ₁ + x _o1 ) <0
Conditional expression (4) (x ₂ + x _o2 ) / | x ₂ + x _o2 | × f ₂ ′ (x ₂ + x _o2 ) <0
Formula (12) rx1 = −L1 / 2.5 = −4.98
Formula (13) rx2 = −L2 / 0.1 = −144.13
Conditional expression (15) In the range of − | radxi | <x <| radxi |
zfc1 (x) <zs2 (x), zfc2 (x) <zs2 (x)
Conditional expression (16) | rx1 | <| radxi |, and in the range of − | rx1 | <x <| rx1 |
zs1 (x) <zfc1 (x), zs1 (x) <zfc2 (x)

  Thus, it can be seen that conditional expression (1) and the like are satisfied in this embodiment. The sag amount in each cross section of the third free-form surface lens L23 and the second free-form surface lens L22 is shown in FIGS. 9 and 10, respectively.

  FIG. 11 is an e-line monochromatic spot diagram of the optical system PL for an image projection apparatus according to the second embodiment. The length of the straight line displayed at the bottom of the spot diagram corresponds to 1 mm on the screen. Corresponding object point positions are (x coordinate, y coordinate) = (0.00, 0.00), (1.92, 0.00), (3.84, 0.00) in order from the bottom of the spot diagram. ), (0.00, 1.44), (1.92, 1.44), (3.84, 1.44), (0.00, -1.44), (1.92, -1 .44), (3.84, -1.44), (0.000, 2.88), (1.92, 2.88), (3.84, 2.88), (0.000, -2.88), (1.92, -2.88), (3.84, -2.88).

  FIG. 12 represents an image (distortion) when a grid is displayed over the entire display area. 11 and 12, it can be seen that in the second embodiment, the trapezoidal distortion is corrected well and the imaging performance is excellent.

(Third embodiment)
A third embodiment of the present application will be described with reference to FIGS. 13 to 17 and Tables 9 to 12. FIG. FIG. 13 is a side sectional view (y-z sectional view) of the optical system for the image projection apparatus according to the third embodiment, and FIG. 14 is a plan sectional view (x -Z sectional view). The optical system for the image projection apparatus of the third embodiment has the same configuration as the optical system for the image projection apparatus of the first embodiment, except for a part of the shape of the rotationally symmetric lens group and the free-form surface lens group. The same reference numerals as those in the first embodiment are assigned to the respective parts, and detailed description thereof is omitted.

  The rotationally symmetric lens group G1 of the third embodiment includes a first rotationally symmetric lens L11 that is a meniscus lens that is arranged in order from the image display element DS side along the optical axis and has a concave surface directed toward the element side, and a biconvex shape. The lens includes a second rotationally symmetric lens L12, a biconcave third rotationally symmetric lens L13, a biconcave fourth rotationally symmetric lens L14, and a biconvex fifth rotationally symmetric lens L15. The image-side lens surface (fifth surface) in the first rotationally symmetric lens L11, the element-side lens surface (sixth surface) in the second rotationally symmetric lens L12, and the image-side lens surface in the fifth rotationally symmetric lens L15. The lens surface (11th surface) is aspherical. Further, the second rotationally symmetric lens L12 and the third rotationally symmetric lens L13 are bonded lenses, and the fourth rotationally symmetric lens L14 and the fifth rotationally symmetric lens L15 are bonded lenses.

  In the free-form surface lens group G2 of the third example, the image-side lens surface of the third free-form surface lens L23 has a shape with a concave surface facing the image display element DS side in the xz section. The cross-sectional shape of the second free-form surface lens L22 has a meniscus shape with a concave surface facing the image display element DS side in the xz cross-section. Regarding the free-form surface mirror M of the third embodiment, the z-axis of the local coordinate system of the free-form surface mirror M is +30 degrees in the yz plane with the coordinate origin as the center with respect to the optical axis of the free-form surface lens group G2. Just leaning. Further, in the third embodiment, no virtual surface as in the first and second embodiments is provided.

  Table 9 below shows various data of the optical system PL for the image projection apparatus according to the third example.

(Table 9)
(Overall specifications)
Object side numerical aperture 0.17857
Projection size 284.48mm x 213.36mm
Magnification magnification 37.0 times Focal length f 14.897mm

(Lens data)
Surface number Curvature radius Surface spacing Refractive index / νd
Element plane Plane 1.00
1 plane 0.70 1.51680 / 64.2
2 plane 11.00 1.78800 / 47.4
3 Plane 2.14
4 -125.98570 3.50 1.62299 / 58.2
5 * a -10.97536 0.10
6 * a 8.20802 4.50 1.62299 / 58.2
7 -150.51597 3.00 1.72825 / 28.5
8 6.12558 2.34
9 -7.16940 1.00 1.60342 / 38.0
10 8.27052 4.00 1.74400 / 44.8
11 * a -8.05504 1.00
12 (aperture) plane 0.20
13 * f plane 3.00 1.53113 / 55.7
14 * f plane 10.66
15 * f plane 3.00 1.53113 / 55.7
16 * f plane 4.83
17 * f plane 5.00 1.53113 / 55.7
18 * f plane 34.03
19 * f Plane -199.60 Reflection Image plane Plane 0.00

  In the lens data of Table 9, the fourth surface to the eleventh surface are lens surfaces of the rotationally symmetric lens group G1, and among them, the fifth surface, the sixth surface, and the eleventh surface are rotationally symmetric aspheric surfaces. . Table 10 below shows the aspherical coefficients of the fifth surface, the sixth surface, and the eleventh surface, respectively.

(Table 10)
(Aspheric data)
Term 5th surface coefficient 6th surface coefficient 11th surface coefficient c -0.09111 0.12183 -0.12415
k -1.20030 -0.88844 -1.15656
A (4th order) 2.4210186E-05 1.4079225E-04 2.0253316E-05
B (6th) -9.5168770E-07 1.6343669E-06 7.8484153E-06
C (8th) -2.4608359E-09 -1.1931496E-07 -1.4005730E-06
D (10th order) 2.0094362E-10 1.3960379E-09 5.6569278E-08
(Since all coefficients after E are 0, they are omitted)

  In the lens data of Table 9, the thirteenth to nineteenth surfaces are non-rotationally symmetric free-form surfaces. In the present embodiment, the nineteenth surface is the reflecting surface of the free-form mirror M. Table 11 below shows each term coefficient of these free-form surfaces.

(Table 11)
(Free curved surface data 1)
Term 13th surface coefficient 14th surface coefficient 15th surface coefficient c 0 0 0
C1 (k) 0 0 0
C3 (y) 1.7250775E-01 1.3915417E-01 -3.6122881E-02
C4 (x ^ 2) -1.9002225E-02 -3.1345050E-02 -2.0610290E-02
C6 (y ^ 2) 2.6320783E-02 1.1968413E-03 -4.1030399E-02
C8 (x ^ 2 * y) 9.3272868E-04 2.5000954E-03 2.3504211E-03
C10 (y ^ 3) -3.1049318E-03 -3.3897704E-03 7.5443505E-03
C11 (x ^ 4) 4.2880010E-04 1.4680491E-04 1.4357239E-04
C13 (x ^ 2 * y ^ 2) 1.3455893E-03 9.6357894E-04 -8.5050443E-05
C15 (y ^ 4) 3.4710553E-04 7.8424715E-05 -1.3247950E-03
C17 (x ^ 4 * y) 1.1276481E-04 1.2088135E-04 1.1354916E-04
C19 (x ^ 2 * y ^ 3) 1.8644298E-04 2.2807626E-04 1.2753388E-04
C21 (y ^ 5) 3.2232856E-05 5.9697746E-05 7.3528783E-05
C22 (x ^ 6) -3.8622131E-06 -3.2757535E-06 -3.7704797E-06
C24 (x ^ 4 * y ^ 2) -1.4877007E-06 7.0837131E-06 -1.5472547E-05
C26 (x ^ 2 * y ^ 4) 9.5582625E-06 1.3951959E-05 -3.4580074E-06
C28 (y ^ 6) 6.4888253E-06 6.8246032E-06 -6.6046446E-06
C30 (x ^ 6 * y) -3.5899011E-07 -5.5069532E-08 -7.8152950E-07
C32 (x ^ 4 * y ^ 3) -1.2295113E-06 1.5413470E-06 1.7589525E-07
C34 (x ^ 2 * y ^ 5) -1.8078891E-06 2.5439966E-07 3.3071008E-06
C36 (y ^ 7) -7.6398257E-07 -1.1775534E-06 2.6033435E-06
C37 (x ^ 8) -2.8622400E-07 1.2517171E-07 3.7425363E-08
C39 (x ^ 6 * y ^ 2) -1.0667375E-06 4.1286056E-07 5.8740479E-08
C41 (x ^ 4 * y ^ 4) -8.2097091E-07 1.3777925E-06 -2.7534607E-07
C43 (x ^ 2 * y ^ 6) -5.4025084E-07 1.3354302E-06 -2.2125652E-07
C45 (y ^ 8) -2.2188808E-07 3.4488968E-07 1.2052859E-07
(Since all coefficients after C46 are 0, they are omitted.)

(Free curved surface data 2)
Term 16th surface coefficient 17th surface coefficient 18th surface coefficient c 0 0 0
C1 (k) 0 0 0
C3 (y) -5.5566831E-02 1.5134051E-01 1.8075196E-01
C4 (x ^ 2) -4.1031173E-03 6.3495806E-03 -4.5923987E-03
C6 (y ^ 2) 3.3772323E-02 6.4298933E-03 -2.2065420E-02
C8 (x ^ 2 * y) -9.0047104E-04 3.9471572E-03 3.8616721E-03
C10 (y ^ 3) 6.2899413E-03 3.6648525E-03 3.0600838E-03
C11 (x ^ 4) 1.2451268E-05 -5.6980632E-05 1.5490614E-05
C13 (x ^ 2 * y ^ 2) -6.7046368E-04 9.7541334E-05 3.6589232E-04
C15 (y ^ 4) -8.9111633E-04 2.3108981E-04 2.0411057E-04
C17 (x ^ 4 * y) 7.2067677E-05 -9.9652008E-06 1.4625001E-07
C19 (x ^ 2 * y ^ 3) -9.8714201E-06 3.2642804E-06 5.4924004E-05
C21 (y ^ 5) -1.1861145E-04 -1.0822109E-05 2.3936693E-05
C22 (x ^ 6) -2.3188121E-06 -1.0005172E-06 -7.5781234E-07
C24 (x ^ 4 * y ^ 2) -3.2890815E-06 -1.6061746E-06 -1.1787007E-06
C26 (x ^ 2 * y ^ 4) 6.6404026E-06 -4.8225836E-06 2.3107150E-06
C28 (y ^ 6) 3.6753129E-06 -3.3997840E-06 4.5925049E-07
C30 (x ^ 6 * y) -3.3777446E-07 3.2742773E-07 8.1162167E-08
C32 (x ^ 4 * y ^ 3) -6.8794920E-07 5.8420943E-07 -9.1013686E-08
C34 (x ^ 2 * y ^ 5) 1.9035179E-06 4.7607689E-07 -3.3860357E-08
C36 (y ^ 7) 1.9478057E-06 1.3555213E-07 -1.3543414E-07
C37 (x ^ 8) 2.2567124E-08 2.5710890E-09 2.2397413E-09
C39 (x ^ 6 * y ^ 2) -1.0144806E-08 -5.5596594E-08 -1.0068468E-08
C41 (x ^ 4 * y ^ 4) -6.0435431E-08 -4.2458541E-08 -4.0977315E-08
C43 (x ^ 2 * y ^ 6) 1.1712172E-07 7.4073704E-09 -4.8371751E-08
C45 (y ^ 8) 1.0492005E-07 1.1514768E-08 -1.5459613E-08
(Since all coefficients after C46 are 0, they are omitted.)

(Free curved surface data 3)
Term 19th surface coefficient c 0
C1 (k) 0
C3 (y) 8.2936434E-03
C4 (x ^ 2) 1.0440642E-02
C6 (y ^ 2) 9.3816839E-05
C8 (x ^ 2 * y) 3.9014555E-04
C10 (y ^ 3) 6.9731637E-05
C11 (x ^ 4) -5.6804450E-06
C13 (x ^ 2 * y ^ 2) 7.6417919E-06
C15 (y ^ 4) 1.9851881E-06
C17 (x ^ 4 * y) -5.1984756E-07
C19 (x ^ 2 * y ^ 3) -1.1335167E-07
C21 (y ^ 5) 2.9093770E-09
C22 (x ^ 6) 5.4936392E-09
C24 (x ^ 4 * y ^ 2) -1.8840571E-08
C26 (x ^ 2 * y ^ 4) -1.6626465E-08
C28 (y ^ 6) -9.3160115E-10
C30 (x ^ 6 * y) 5.3180021E-10
C32 (x ^ 4 * y ^ 3) -3.4441621E-10
C34 (x ^ 2 * y ^ 5) -5.6525887E-10
C36 (y ^ 7) 1.3960281E-11
C37 (x ^ 8) -1.2150278E-12
C39 (x ^ 6 * y ^ 2) 1.3560649E-11
C41 (x ^ 4 * y ^ 4) -3.8773757E-12
C43 (x ^ 2 * y ^ 6) -6.7802887E-12
C45 (y ^ 8) 7.1767824E-13
(Since all coefficients after C46 are 0, they are omitted.)

  Furthermore, the eccentricity of the local coordinate system of the 19th surface and the image surface in this example is shown in Table 12 below.

(Table 12)
(Eccentric data)
Surface number ADE (°) YDE (mm)
19 30.0 0
Image plane -60.0 0

  The corresponding values for each conditional expression are shown below.

Conditional expression (17) | TL / f | = | 294.6001 / 14.8972 | = 19.776
Conditional expression (18) | θm / θs | = | 30.0 / −60 | = 0.5
(Second free-form surface lens L22)
L1 = 12.819
L2 = 14.776
(x-z cross section)
Element side lens surface maximum point coordinates (x _o1 , z _xo1 ) = (0, 0)
Image side lens surface maximum point coordinates (x _o2 , z _xo2 ) = (0, 0)
Conditional expression (3) (x ₁ + x _o1 ) / | x ₁ + x _o1 | × f ₁ ′ (x ₁ + x _o1 ) <0
Conditional expression (4) (x ₂ + x _o2 ) / | x ₂ + x _o2 | × f ₂ ′ (x ₂ + x _o2 ) <0
Formula (12) rx1 = −L1 / 2.5 = −5.13
Formula (13) rx2 = −L2 / 0.1 = −147.76
Conditional expression (15) In the range of − | radxi | <x <| radxi |
zfc1 (x) <zs2 (x), zfc2 (x) <zs2 (x)
Conditional expression (16) | rx1 | <| radxi |, and in the range of − | rx1 | <x <| rx1 |
zs1 (x) <zfc1 (x), zs1 (x) <zfc2 (x)

  Thus, in this embodiment, it can be seen that conditional expression (3) and the like are satisfied. The sag amount in the xz section of the second free-form curved lens L22 is shown in FIG.

  FIG. 16 is an e-line monochromatic spot diagram of the optical system PL for image projection apparatus according to the third example. The length of the straight line displayed at the bottom of the spot diagram corresponds to 1 mm on the screen. The corresponding object point positions are (x coordinate, y coordinate) = (0.00, 0.00), (1.82, 0.00), (3.65, 0.00) in order from the bottom of the spot diagram. ), (0.00, 1.44), (1.78, 1.44), (3.55, 1.44), (0.00, -1.44), (1.87, -1 .44), (3.74, -1.44), (0.000, 2.88), (1.73, 2.88), (3.46, 2.88), (0.000, -2.88), (1.92, -2.88), (3.84, -2.88). In the third embodiment, the trapezoidal distortion is corrected by using image processing in the video display element DS by an image processing unit (not shown). Specifically, image processing is performed so that an input object whose short side is 5% shorter than the long side is projected onto a rectangle.

  FIG. 17 represents an image (distortion) when a grid is displayed over the entire display area. 16 and 17, it can be seen that in the third example, the trapezoidal distortion is corrected well and the imaging performance is excellent.

(Fourth embodiment)
A fourth embodiment of the present application will be described with reference to FIGS. 18 to 22 and Tables 13 to 16. 18 is a side sectional view (y-z sectional view) of the optical system for an image projection apparatus according to the fourth embodiment, and FIG. 19 is a plan sectional view (x of the optical system for an image projection apparatus according to the fourth embodiment). -Z sectional view). The optical system for the image projection apparatus of the fourth embodiment has the same configuration as that of the optical system for the image projection apparatus of the third embodiment, and each part is the same as that of the third embodiment (first embodiment). Reference numerals are assigned and detailed description is omitted. Regarding the free-form surface mirror M of the fourth embodiment, the z-axis of the local coordinate system of the free-form surface mirror M is in the yz plane with the coordinate origin as the center with respect to the optical axis of the free-form surface lens group G2. Tilt by +35 degrees.

  Table 13 below shows various data of the optical system PL for image projection apparatus according to the fourth example.

(Table 13)
(Overall specifications)
Object side numerical aperture 0.17857
Projection size 284.48mm x 213.36mm
Magnification magnification 37.0 times Focal length f 15.5059 mm

(Lens data)
Surface number Curvature radius Surface spacing Refractive index / νd
Element plane Plane 1.00
1 plane 0.70 1.51680 / 64.2
2 plane 11.00 1.78800 / 47.4
3 Plane 2.14
4 -130.44850 3.50 1.62299 / 58.2
5 * a -10.32502 0.10
6 * a 8.87430 4.50 1.62299 / 58.2
7 -47.02495 3.00 1.72825 / 28.5
8 6.02391 2.34
9 -7.34059 1.00 1.60342 / 38.0
10 10.98280 4.00 1.74400 / 44.8
11 * a -7.69123 1.00
12 (aperture) plane 1.00
13 * f plane 3.00 1.53113 / 55.7
14 * f plane 13.05
15 * f plane 3.00 1.53113 / 55.7
16 * f plane 7.61
17 * f plane 5.00 1.53113 / 55.7
18 * f plane 28.06
19 * f Plane -199.60 Reflection Image plane Plane 0.00

  In the lens data of Table 13, the fourth surface to the eleventh surface are lens surfaces of the rotationally symmetric lens group G1, and among them, the fifth surface, the sixth surface, and the eleventh surface are rotationally symmetric aspheric surfaces. . Table 14 below shows the aspheric coefficients of the fifth surface, the sixth surface, and the eleventh surface, respectively.

(Table 14)
(Aspheric data)
Term 5th surface coefficient 6th surface coefficient 11th surface coefficient c -0.09685 0.11269 -0.13002
k -1.20030 -0.88844 -1.15656
A (4th order) 2.4210186E-05 1.4079225E-04 -2.4286499E-04
B (6th) -9.5168770E-07 1.6343669E-06 1.3535281E-05
C (8th) -2.4608359E-09 -1.1931496E-07 -1.3667381E-06
D (10th order) 2.0094362E-10 1.3960379E-09 4.5797215E-08
(Since all coefficients after E are 0, they are omitted)

  In the lens data of Table 13, the thirteenth to nineteenth surfaces are non-rotationally symmetric free-form surfaces. In the present embodiment, the nineteenth surface is the reflecting surface of the free-form mirror M. Table 15 below shows each term coefficient of these free-form surfaces.

(Table 15)
(Free curved surface data 1)
Term 13th surface coefficient 14th surface coefficient 15th surface coefficient c 0 0 0
C1 (k) 0 0 0
C3 (y) 4.7652139E-02 7.3462603E-02 -7.8338236E-02
C4 (x ^ 2) 1.1552008E-02 -1.8268494E-03 -2.8671604E-02
C6 (y ^ 2) 2.0876597E-02 2.0926139E-03 -4.6825621E-02
C8 (x ^ 2 * y) 2.9017450E-03 4.4189750E-03 1.5176782E-03
C10 (y ^ 3) -2.2086246E-03 -2.4211969E-03 1.2580946E-03
C11 (x ^ 4) -1.4694978E-04 -2.4243297E-04 1.5898508E-04
C13 (x ^ 2 * y ^ 2) -1.0446856E-04 -2.6332921E-04 -4.5258924E-04
C15 (y ^ 4) -3.0221284E-05 -1.2027936E-04 -5.4928473E-04
C17 (x ^ 4 * y) 7.0039476E-05 8.2101878E-05 1.4621575E-04
C19 (x ^ 2 * y ^ 3) 6.9752373E-05 9.4262145E-05 1.1715665E-04
C21 (y ^ 5) 5.2796265E-06 1.9584930E-05 6.1730193E-05
C22 (x ^ 6) 1.2997090E-05 2.7395877E-06 -6.0932245E-06
C24 (x ^ 4 * y ^ 2) 4.3809389E-05 1.8619375E-05 -2.2012591E-06
C26 (x ^ 2 * y ^ 4) 4.2857223E-05 1.3190596E-05 -2.4548570E-06
C28 (y ^ 6) 1.3336728E-05 2.4605132E-06 1.0698771E-06
C30 (x ^ 6 * y) 1.5346809E-07 -1.6876819E-07 -9.0235022E-07
C32 (x ^ 4 * y ^ 3) 8.2601844E-07 8.7365154E-07 1.9022832E-06
C34 (x ^ 2 * y ^ 5) 4.4808179E-08 1.5804911E-07 1.0603083E-06
C36 (y ^ 7) -3.2177996E-07 -6.8377345E-07 -1.7360462E-07
C37 (x ^ 8) -6.1770269E-07 -8.7296908E-08 3.9272435E-08
C39 (x ^ 6 * y ^ 2) -2.8367265E-06 -7.7216410E-07 -1.6472144E-07
C41 (x ^ 4 * y ^ 4) -3.1258756E-06 -2.6171869E-07 -2.0881349E-07
C43 (x ^ 2 * y ^ 6) -2.2643212E-06 -1.8440039E-07 -2.6197597E-07
C45 (y ^ 8) -4.0446312E-07 1.8412799E-07 2.6523681E-08
(Since all coefficients after C46 are 0, they are omitted.)

(Free curved surface data 2)
Term 16th surface coefficient 17th surface coefficient 18th surface coefficient c 0 0 0
C1 (k) 0 0 0
C3 (y) -1.5268139E-01 1.2228328E-02 2.6862477E-02
C4 (x ^ 2) -1.2598700E-02 1.5016471E-03 -5.9162849E-03
C6 (y ^ 2) 9.1618962E-03 4.6044324E-03 -1.6095228E-02
C8 (x ^ 2 * y) -1.8078806E-03 2.3984151E-03 2.3453350E-03
C10 (y ^ 3) 9.3947510E-04 2.4847676E-03 1.7228463E-03
C11 (x ^ 4) 9.2039711E-05 -2.0503957E-04 -1.2961665E-04
C13 (x ^ 2 * y ^ 2) -4.2008388E-04 4.8584277E-05 1.6469446E-04
C15 (y ^ 4) -4.3207999E-04 -3.8172977E-05 1.0658944E-05
C17 (x ^ 4 * y) 9.6848266E-05 1.0094868E-06 -4.8011453E-06
C19 (x ^ 2 * y ^ 3) 4.7350800E-05 1.0802890E-05 2.5597458E-05
C21 (y ^ 5) -6.0758965E-06 -6.4088472E-06 4.1760857E-06
C22 (x ^ 6) -2.7308279E-06 -3.2026808E-07 -7.4932265E-08
C24 (x ^ 4 * y ^ 2) 6.3786092E-06 1.3677013E-06 -9.5156069E-07
C26 (x ^ 2 * y ^ 4) 5.2829942E-06 -6.9478239E-08 6.4520650E-07
C28 (y ^ 6) 5.3127809E-06 5.2364267E-07 1.9490525E-07
C30 (x ^ 6 * y) -9.4562262E-07 -1.5745306E-07 -2.1814435E-08
C32 (x ^ 4 * y ^ 3) -2.1787211E-07 -1.2511936E-08 -1.0080547E-07
C34 (x ^ 2 * y ^ 5) -5.0085211E-07 -4.7807859E-08 -5.3990595E-08
C36 (y ^ 7) 2.8574408E-08 1.4140473E-08 -9.3717598E-09
C37 (x ^ 8) 2.2019211E-08 4.9289145E-09 1.1568050E-09
C39 (x ^ 6 * y ^ 2) -5.8672253E-08 -5.2289355E-09 2.9419858E-09
C41 (x ^ 4 * y ^ 4) 3.0553300E-11 -3.4817972E-09 -2.5420698E-09
C43 (x ^ 2 * y ^ 6) -4.4148694E-08 1.0464664E-09 -5.3392775E-09
C45 (y ^ 8) -3.0243748E-08 -1.7367671E-09 -8.9563065E-10
(Since all coefficients after C46 are 0, they are omitted.)

(Free curved surface data 3)
Term 19th surface coefficient c 0
C1 (k) 0
C3 (y) 5.8255880E-03
C4 (x ^ 2) 9.6009201E-03
C6 (y ^ 2) 4.3331592E-05
C8 (x ^ 2 * y) 4.0340417E-04
C10 (y ^ 3) 7.9988288E-05
C11 (x ^ 4) -3.9342481E-06
C13 (x ^ 2 * y ^ 2) 1.0206092E-05
C15 (y ^ 4) 3.1649776E-06
C17 (x ^ 4 * y) -3.9478331E-07
C19 (x ^ 2 * y ^ 3) 3.7452432E-08
C21 (y ^ 5) 8.3350449E-08
C22 (x ^ 6) 2.5195773E-09
C24 (x ^ 4 * y ^ 2) -1.6553197E-08
C26 (x ^ 2 * y ^ 4) -8.8336537E-09
C28 (y ^ 6) 1.4433315E-09
C30 (x ^ 6 * y) 3.4563093E-10
C32 (x ^ 4 * y ^ 3) -2.9666608E-10
C34 (x ^ 2 * y ^ 5) -2.9482933E-10
C36 (y ^ 7) 1.4958837E-11
C37 (x ^ 8) -2.4395532E-12
C39 (x ^ 6 * y ^ 2) 1.4513725E-11
C41 (x ^ 4 * y ^ 4) -8.0939383E-13
C43 (x ^ 2 * y ^ 6) -2.8156119E-12
C45 (y ^ 8) -9.9923435E-14
C47 (x ^ 8 * y) -1.3613661E-13
C49 (x ^ 6 * y ^ 3) 2.4014824E-13
C51 (x ^ 4 * y ^ 5) 2.0470013E-14
C53 (x ^ 2 * y ^ 7) 5.3763726E-15
C55 (y ^ 9) -4.3152859E-15
(Since all coefficients after C55 are 0, they are omitted.)

  Further, the eccentricity of the local coordinate system of the nineteenth surface and the image surface in this example is shown in Table 16 below.

(Table 16)
(Eccentric data)
Surface number ADE (°) YDE (mm)
19 35.0 0
Image plane -60.0 0

  The corresponding values for each conditional expression are shown below.

Conditional Expression (17) | TL / f | = | 294.5972 / 15.5059 | = 19.0
Conditional Expression (18) | θm / θs | = | 35.0 / −60 | = 0.58
(Second free-form surface lens L22)
L1 = 16.005
L2 = 17.961
(x-z cross section)
Element side lens surface maximum point coordinates (x _o1 , z _xo1 ) = (0, 0)
Image side lens surface maximum point coordinates (x _o2 , z _xo2 ) = (0, 0)
Conditional expression (3) (x ₁ + x _o1 ) / | x ₁ + x _o1 | × f ₁ ′ (x ₁ + x _o1 ) <0
Conditional expression (4) (x ₂ + x _o2 ) / | x ₂ + x _o2 | × f ₂ ′ (x ₂ + x _o2 ) <0
Formula (12) rx1 = −L1 / 2.5 = −6.40
Formula (13) rx2 = −L2 / 0.1 = −179.61
Conditional expression (15) In the range of − | radxi | <x <| radxi |
zfc1 (x) <zs2 (x), zfc2 (x) <zs2 (x)
Conditional expression (16) | radxi | <| rx1 |, and in the range of − | radxi | <x <| radxi |
zs1 (x) <zfc1 (x), zs1 (x) <zfc2 (x)

  Thus, in this embodiment, it can be seen that conditional expression (3) and the like are satisfied. The sag amount in the xz section of the second free-form curved lens L22 is shown in FIG.

  FIG. 21 is an e-line monochromatic spot diagram of the optical system PL for image projection apparatus according to the fourth example. The length of the straight line displayed at the bottom of the spot diagram corresponds to 1 mm on the screen. The corresponding object point positions are (x coordinate, y coordinate) = (0.00, 0.00), (1.82, 0.00), (3.65, 0.00) in order from the bottom of the spot diagram. ), (0.00, 1.44), (1.78, 1.44), (3.55, 1.44), (0.00, -1.44), (1.87, -1 .44), (3.74, -1.44), (0.000, 2.88), (1.73, 2.88), (3.46, 2.88), (0.000, -2.88), (1.92, -2.88), (3.84, -2.88). In the fourth embodiment, the trapezoidal distortion is corrected by using image processing in the video display element DS by an image processing unit (not shown). Specifically, image processing is performed so that an input object whose short side is 5% shorter than the long side is projected onto a rectangle.

  FIG. 22 represents an image (distortion) when a grid is displayed over the entire display area. 21 and 22, it can be seen that in the fourth example, the trapezoidal distortion is corrected well and the imaging performance is excellent.

  As described above, according to each of the embodiments, it is possible to realize the optical system PL for the image projection apparatus and the image projection apparatus PRJ including the image projection apparatus optical system PL that can satisfactorily correct the trapezoidal distortion of the projected image while having a compact configuration. it can.

  In the above-described embodiment, the projection plane R (image plane I) is set on the same plane as the installation plane Q of the video projection device PRJ (housing BD). However, the present invention is not limited to this. You may make it set on the surface parallel to the installation surface Q.

PRJ image projection device DS image display element PL optical system for image projection device G1 rotationally symmetric lens group L11 first rotationally symmetric lens L12 second rotationally symmetric lens L13 third rotationally symmetric lens L14 fourth rotationally symmetric lens L15 fifth rotationally symmetric lens G2 Free-form surface lens group L21 First free-form surface lens L22 Second free-form surface lens L23 Third free-form surface lens M Free-form surface mirror Q Installation surface

Claims (9)

An optical system for an image projection apparatus that enlarges an image displayed on an image display element and projects the image on a predetermined projection surface from an oblique direction,
A rotationally symmetric lens group composed of rotationally symmetric lenses formed in a rotationally symmetric manner with respect to the central axis and arranged in order from the image display element side along the optical axis, and a plurality of non-rotationally symmetric lenses formed with respect to the central axis A free-form surface lens group consisting of a free-form surface lens, and a free-form surface mirror having a reflection surface formed non-rotationally symmetric with respect to the central axis,
In the free-form surface lens group, the mirror-side free-form surface lens closest to the free-form surface mirror has an optical axis between the free-form surface lens group and the free-form surface mirror, the free-form surface mirror, and the projection surface. A second cross section perpendicular to the first cross section passing through the optical axis between the two , having a meniscus shape with a concave surface facing the image display element side ,
In the second cross section, the coordinate axis of the optical axis direction in which the direction from the image display element toward the free-form curved mirror is positive is the z-axis, and the coordinate axis perpendicular to the z-axis is the x-axis, Define a local coordinate system (x, z) with the origin at the intersection of the lens surface and the optical axis,
When i = 1 for the lens surface of the free-form surface lens on the image display element side and i = 2 for the lens surface of the free-form surface lens on the free-form surface mirror side, the free-form surface lens The maximum point coordinate within the effective range is (xoi, zoi), and the sag amount of the free-form lens is zfi (x).
zfci (x) = zfi (x + xoi) −zoi
Define
The air conversion center distance from the diaphragm to the lens surface on the image display element side of the free-form surface lens is L1, and the following formula
rx1 = -L1 / 2.5
Define
The air conversion center distance from the diaphragm to the lens surface of the free-form curved lens side of the free-form curved lens is L2, and the following formula
rx2 = −L2 / 0.1
Define
Where rx1 and rx2 are rxi,
zsi (x) = rxi + [(rxi) ² −x ² ] ^1/2
When defining
When the absolute value of the length in the x direction from the origin to the end of the effective range is | radxi |
In the range − | radxi | <x <| radxi |
zfci (x) <zs2 (x)
While satisfying the conditions of
When | rx1 | <| radxi |, the range is − | rx1 | <x <| rx1 |. When | radxi | <| rx1 |, the range is − | radxi | <x <| radxi |
zs1 (x) <zfci (x)
An optical system for an image projection apparatus, characterized by satisfying the following conditions . In the free-form surface lens group, at least one of the cross-sectional shape of the mirror-side free-form surface lens and the element-side free-form surface lens arranged next to the mirror-side free-form surface lens is the first cross-section , The optical system for an image projection apparatus according to claim 1, wherein the image display device has a meniscus shape with a concave surface facing the image display element side. In the first cross section , a local coordinate system (y, z) is defined with the origin of the intersection of the lens surface of the free-form surface lens and the optical axis, with the coordinate axis perpendicular to the z axis as the y axis,
When i = 1 for the lens surface of the free-form surface lens on the image display element side and i = 2 for the lens surface of the free-form surface lens on the free-form surface mirror side, the free-form surface lens The maximum point coordinate within the effective range is (yoi, zoi), and the sag amount of the lens surface of the free-form surface lens is zfi (y).
zfci (y) = zfi (y + yoi) −zoi
Define
The air conversion center distance from the diaphragm to the lens surface on the image display element side of the free-form surface lens is L1, and the following formula
ry1 = -L1 / 3.5
Define
The air conversion center distance from the diaphragm to the lens surface of the free-form curved lens side of the free-form curved lens is L2, and the following formula
ry2 = −L2
Define
Ry1 and ry2 as ryi,
zsi (y) = ryi + [(ryi) ² −y ² ] ^1/2
When defining
When the absolute value of the length in the y direction from the origin to the end of the effective range is | radyi |
In the range − | radyi | <y <| radyi |
zfci (y) <zs2 (y)
While satisfying the conditions of
When | ry1 | <| radyi |, the range is − | ry1 | <y <| ry1 |. When | radyi | <| ry1 |, the range is − | radyi | <y <| radyi |.
zs1 (y) <zfci (y)
The optical system for an image projection apparatus according to claim 2 , wherein the following condition is satisfied. An optical system for an image projection apparatus that enlarges an image displayed on an image display element and projects the image on a predetermined projection surface from an oblique direction,
A rotationally symmetric lens group composed of rotationally symmetric lenses formed in a rotationally symmetric manner with respect to the central axis and arranged in order from the image display element side along the optical axis, and a plurality of non-rotationally symmetric lenses formed with respect to the central axis A free-form surface lens group consisting of a free-form surface lens, and a free-form surface mirror having a reflection surface formed non-rotationally symmetric with respect to the central axis,
In the free-form surface lens group, at least one of the cross-sectional shape of the free-form surface lens closest to the free-form surface mirror and the element-side free-form surface lens arranged adjacent to the free-form mirror surface lens is the free-form lens. In the second cross section perpendicular to the first cross section passing through the optical axis between the curved lens group and the free curved mirror and the optical axis between the free curved mirror and the projection plane, It has a meniscus shape with a concave surface,
The cross-sectional shape in the first cross section of the element-side free-form surface lens has a meniscus shape with a concave surface facing the image display element side,
In the first cross section, a lens of a free-form surface lens with a coordinate axis in the optical axis direction that is positive in the direction from the image display element toward the free-form surface mirror being a z-axis, and a coordinate axis perpendicular to the z-axis is a y-axis Define a local coordinate system (y, z) with the origin at the intersection of the surface and the optical axis,
When i = 1 for the lens surface of the free-form surface lens on the image display element side and i = 2 for the lens surface of the free-form surface lens on the free-form surface mirror side, the free-form surface lens The maximum point coordinate within the effective range is (yoi, zoi), and the sag amount of the lens surface of the free-form surface lens is zfi (y).
zfci (y) = zfi (y + yoi) −zoi
Define
The air conversion center distance from the diaphragm to the lens surface on the image display element side of the free-form surface lens is L1, and the following formula
ry1 = -L1 / 3.5
Define
The air conversion center distance from the diaphragm to the lens surface of the free-form curved lens side of the free-form curved lens is L2, and the following formula
ry2 = −L2
Define
Ry1 and ry2 as ryi,
zsi (y) = ryi + [(ryi) ² −y ² ] ^1/2
When defining
When the absolute value of the length in the y direction from the origin to the end of the effective range is | radyi |
In the range − | radyi | <y <| radyi |
zfci (y) <zs2 (y)
While satisfying the conditions of
When | ry1 | <| radyi |, the range is − | ry1 | <y <| ry1 |.
zs1 (y) <zfci (y)
Satisfy the conditions of
In the second cross section, a local coordinate system (x, z) is defined with the origin of the intersection of the lens surface of the free-form surface lens and the optical axis, with the coordinate axis perpendicular to the z axis as the x axis,
When i = 1 for the lens surface of the free-form surface lens on the image display element side and i = 2 for the lens surface of the free-form surface lens on the free-form surface mirror side, the free-form surface lens The maximum point coordinate within the effective range is (xoi, zoi), and the sag amount of the free-form lens is zfi (x).
zfci (x) = zfi (x + xoi) −zoi
Define
The air conversion center distance from the diaphragm to the lens surface on the image display element side of the free-form surface lens is L1, and the following formula
rx1 = -L1 / 2.5
Define
The air conversion center distance from the diaphragm to the lens surface of the free-form curved lens side of the free-form curved lens is L2, and the following formula
rx2 = −L2 / 0.1
Define
Where rx1 and rx2 are rxi,
zsi (x) = rxi + [(rxi) ² −x ² ] ^1/2
When defining
When the absolute value of the length in the x direction from the origin to the end of the effective range is | radxi |
In the range − | radxi | <x <| radxi |
zfci (x) <zs2 (x)
While satisfying the conditions of
When | rx1 | <| radxi |, the range is − | rx1 | <x <| rx1 |.
zs1 (x) <zfci (x)
An optical system for an image projection apparatus , characterized by satisfying the following conditions .   The optical system for an image projection apparatus according to any one of claims 1 to 4, wherein the free-form surface lens group includes three free-form surface lenses. Assuming that the focal length of the rotationally symmetric lens group is f and the total length of the optical system for the image projection apparatus is TL, the following expression 15 ≦ | TL / f | ≦ 30
The optical system for a video projection apparatus according to claim 1, wherein the following condition is satisfied. An inclination angle of the reflecting surface with respect to a surface perpendicular to the optical axis between the free-form surface lens group and the free-form surface mirror is θm, and a surface perpendicular to the optical axis between the free-form surface mirror and the projection surface is used. When the inclination angle of the projection surface is θs, the following formula 0.5 ≦ | θm / θs | ≦ 0.7
The optical system for an image projection apparatus according to claim 1, wherein the following condition is satisfied.   An image projection device that is used in a state where it is installed on a predetermined installation surface and projects an image from an oblique direction on the same surface as the installation surface or a surface substantially parallel to the installation surface, the image projection device An image projection apparatus, wherein the optical system is the optical system for an image projection apparatus according to any one of claims 1 to 7.   9. The image projection according to claim 8, further comprising an image processing unit that performs predetermined image processing on the image displayed on the image display element to correct a trapezoidal distortion of the image projected from the oblique direction. apparatus.