Head-mounted display optical system and head-mounted display

Abstract

Provided is a head-mounted display optical system (LS) having: an optical deflection element (M1); a first lens group (G1); a second lens group (G2); and an optical reflection element (M2). The head-mounted display optical system is configured such that light from a light source, which is reflected on a reflection surface (M2r) of the optical reflection element (M2) and reaches a drawing surface (I) assumed to be located on a retina when a user wears the head-mounted display, moves on the drawing surface (I) according to the change of a travelling direction of the light caused by the optical deflection element (M1), and an image is drawn on the drawing surface (I). The first lens group (G1) includes a free-shaped surface lens having a free-shaped surface which is rotationally asymmetrical with respect to a reference axis, and the reflection surface (M2r) of the optical reflection element (M2) is formed to be rotationally asymmetrical with respect to a reference axis.

Claims

1. A head-mounted display optical system, comprising: an optical deflection element that changes a travelling direction of light from a light source; a first lens group that has positive refractive power and collects light entered via the optical deflection element; a second lens group that is disposed near a position of intermediate image forming by the first lens group; and an optical reflection element that has a reflection surface to reflect light transmitted through the second lens group, and allows light, entered from an opposite surface to the reflection surface, to transmit therethrough, the head-mounted display optical system being configured such that light from the light source, which is reflected on the reflection surface and reaches a drawing surface assumed to be located on a retina when a user wears the head-mounted display, moves on the drawing surface in accordance with the change of a travelling direction of the light caused by the optical deflection element, and an image is drawn on the drawing surface, the first lens group including a free-shaped surface lens having a free-shaped surface which is rotationally asymmetrical with respect to a reference axis, the reflection surface of the optical reflection element being formed to be rotationally asymmetrical with respect to a reference axis, and the following conditional expressions being satisfied:
0.20<m<3.00
20<|f2/f1|<3000 where m denotes an afocal magnification of the head-mounted display optical system, f1 denotes a focal length of the first lens group, and f2 denotes a focal length of the second lens group.

2. The head-mounted display optical system according to claim 1, wherein the following conditional expression is satisfied:
0.000015[mm.sup.1]<|1/f2 |<0.005[mm.sup.1].

3. The head-mounted display optical system according to claim 1, wherein the following conditional expression is satisfied:
15<d<45 where d denotes a maximum value of a difference of Abbe numbers between a lens having positive refractive power and a lens having negative refractive power in a the first lens group.

4. The head-mounted display optical system according to claim 1, wherein the first lens group is constituted by resin lenses, the second lens group further includes a lens having a diffuse transmission surface, and a deflection angle of light that is reflected on the reflection surface of the optical reflection element is 45 or more.

5. The head-mounted display optical system according to claim 1, wherein the first lens group includes the free-shaped surface lens having positive refractive power and a negative lens, and the second lens group includes a free-shaped surface lens having a free-shaped surface which is rotationally asymmetrical with respect to a reference axis.

6. The head-mounted display optical system according to claim 5, wherein the second lens group has negative refractive power near the reference axis.

7. The head-mounted display optical system according to claim 5, wherein the free-shaped surface lens of the first lens group has the free-shaped lens surface and a lens surface located on the opposite side of this lens surface and rotationally symmetrical with respect to a reference axis, and is formed in a meniscus shape, the negative lens has a lens surface rotationally asymmetrical with respect to a reference axis, and is formed in a meniscus shape, and the free-shaped surface lens of the second lens group has positive refractive power and is formed in a meniscus shape.

8. The head-mounted display optical system according to claim 7, wherein when (x, y) are the coordinates that pass through an intersection point of the free-shaped lens surface and a reference axis and are on a plane perpendicular to the reference axis, and a sag of the free-shaped lens surface is expressed by a polynomial of x and y in the free-shaped surface lens of the first lens group, the polynomial of x and y includes a term, of which degree is at least 8, and when h is a distance from the reference axis and a sag of the aspherical lens surface is expressed by a polynomial of h in the free-shaped surface lens of the first lens group, the polynomial of h includes a term, of which degree is at least 8.

9. The head-mounted display optical system according to claim 7, wherein the second lens group further includes a lens having a diffuse transmission surface, and when (x, y) are coordinates that pass through an intersection point of the free-shaped lens surface and a reference axis and are on a plane perpendicular to the reference axis, and a sag of the free-shaped lens surface is expressed by a polynomial of x and y in the free-shaped surface lens of the second lens group, the polynomial of x and y includes a term, of which degree is at least 8.

10. The head-mounted display optical system according to claim 1, wherein the first lens group includes a lens having a diffraction optical surface, and the following conditional expression is satisfied:
0.01<fd/f1<10.00 where fd denotes a focal length of the lens having the diffraction optical surface, and f1 denotes a focal length of the first lens group.

11. The head-mounted display optical system according to claim 1, wherein the image that is drawn is superposed on an image formed by the light that has transmitted through the optical reflection element and reached the drawing surface.

12. A head-mounted display including the head-mounted display optical system according to claim 1.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is an optical path diagram of a head-mounted display optical system according to Example 1;

(2) FIG. 2 is a spot diagram of a head-mounted display optical system according to Example 1;

(3) FIG. 3 is an optical path diagram of a head-mounted display optical system according to Example 2;

(4) FIG. 4 is a spot diagram of a head-mounted display optical system according to Example 2;

(5) FIG. 5 is an optical path diagram of a head-mounted display optical system according to Example 3;

(6) FIG. 6 is a spot diagram of a head-mounted display optical system according to Example 3;

(7) FIG. 7A is a schematic diagram depicting an example of a cross section of a separated dual layer type diffraction optical element, and FIG. 7B is a schematic diagram depicting an example of a cross section of a contacted dual layer type diffraction optical element;

(8) FIG. 8 is a schematic diagram depicting a head-mounted display; and

(9) FIG. 9 is a diagram depicting an example of a local coordinate system.

DESCRIPTION OF THE EMBODIMENTS

(10) An embodiment of the present invention will now be described with reference to drawings. For example, a head-mounted display optical system LS according to this embodiment has, in order from a light source: an optical deflection element M1, a first lens group G1 having positive refractive power, a second lens group G2, and an optical reflection element M2 as shown in FIG. 1. The optical deflection element M1 changes the travelling direction of light (e.g. approximately parallel light, such as laser light and LED light) from the light source. For the optical deflection element M1 of this embodiment, a galvano-mirror, for example, is used. Turning the galvano-mirror changes the reflection angle (that is, the travelling direction of the light from the light source) by a degree in accordance with the drive voltage that is inputted from a drive circuit (not illustrated).

(11) The first lens group G1 collects light entered from the light source via the optical deflection element M1. The second lens group G2 is disposed near the position of light collection by the first lens group G1. The optical reflection element M2 has a reflection surface M2r that reflects light transmitted through the second lens group G2. The optical reflection element M2 is also configured to transmit light entered from a surface M2s on the opposite side of this reflection surface M2r. For the optical reflection element M2 of this embodiment, a half mirror, for example, is used.

(12) In this head-mounted display optical system LS, light from the light source, which is reflected on the reflection surface M2r of the optical reflection element M2 and reaches a surface assumed to be located on a retina when a user wears the head-mounted display (hereafter called drawing surface I), moves on the drawing surface I as if scanning in a two-dimensional direction in accordance with the change of the traveling direction of the light by the optical deflection element M1. In this case, the image that is drawn is superposed on an image formed by the light that transmitted through the optical reflection element M2 and reached the drawing surface I (retina). Thereby the user can visually recognize the image superposed on an image (external view) formed by the light that transmitted through the optical reflection element M2, when the user wears the head mounted display. The first lens group G1 collects light from the light source, hence an image equivalent to the image drawn on the drawing surface I is formed as an intermediate image at the collection position of the light that transmitted through the first lens group G1, in accordance with the change of the travelling direction of the light from the light source caused by the optical deflection element M1. Hereafter, the position of the intermediate image formed by the first lens group G1 is called intermediate image forming position. In other words, the second lens group G2 is disposed near the intermediate image forming position by the first lens group G1.

(13) Conventionally, in various optical systems, an aspherical lens has implemented specifications, aberration, performance and compactness, which a spherical lens cannot implement. However if an aspherical lens, which is rotationally symmetrical with respect to a reference axis, is used, the aberration correction effect for an eccentric optical system is insufficient, or an uncorrectable aberration will remain. To solve this problem, lately a free-shaped surface lens, which is rotationally asymmetrical with respect to a reference axis, has started to be used as processing techniques advance.

(14) In this embodiment, the first lens group G1 includes a free-shaped surface lens having a free-shaped surface which is rotationally asymmetrical with respect to a reference axis. Further, the reflection surface Mgr of the optical reflection element M2 is formed to be rotationally asymmetrical with respect to a reference axis. Here the reference axis refers to an axis obtained by tracing (ray tracing) the light that passes through the center of the optical deflection element M1 and reaches the center of the drawing surface I. By using a free-shaped surface which is rotationally asymmetrical with respect to the reference axis, trapezoidal distortion, that is generated by diagonal reflection on the optical reflection element M2, can be effectively corrected. Further, by using the free-shaped surface, the spread of a point image, when the angle of view is large, can be decreased, and resolution can be improved.

(15) A local coordinate system in the free-shaped surface lens will be described here. In this embodiment, the local coordinate system of the free-shaped surface lens is assumed to be an (x, y, z) coordinate system (right handed system) of which origin is an intersection of the lens surface of the free-shaped surface lens and the reference axis, for example, as shown in FIG. 9. In this case, the z axis is assumed to be the coordinate axis in the reference axis direction where the direction from the optical deflection element M1 to the optical reflection element M2 is positive. The y axis is a coordinate axis that is perpendicular to the z axis in the cross section of the free-shaped surface lens that passes through the reference axis between the optical deflection element M1 and the optical reflection element M2, and the reference axis between the optical reflection element M2 and the drawing surface I. The x axis is a coordinate axis that is perpendicular to the z axis and the y axis. In this case, the (x, y) coordinate system becomes a coordinate system that passes through the intersection between the lens surface of the free-shaped surface lens and the reference axis, and is on the plane that is perpendicular to the reference axis.

(16) In the head-mounted display optical system LS of this embodiment, it is preferable that the following conditional expression (1) is satisfied.
0.20<m<3.00(1)
where m denotes an afocal magnification of the head-mounted display optical system LS.

(17) The conditional expression (1) specifies an appropriate range of an afocal magnification. In this embodiment, the afocal magnification is an absolute value of a ratio of the focal length between an optical system on the front side (light source side) and an optical system on the rear side (drawing surface I side) of the intermediate image forming position. After the parallel light passes through the optical systems before and after the intermediate image forming position, the optical system LS need not be perfectly afocal (the object point is ). Taking the adjusting capability of human eyes and optical devices into account, the optical system LS can be regarded as approximately afocal, if the object image formed by the luminous flux that enters the pupils is further than 2 [m] (0.5 [m.sup.1]). If the upper limit value of the conditional expression (1) is exceeded, distortion and chromatic aberration in the peripheral area of the screen increase, and image quality drops. Further, curvature of image field in the peripheral area of the screen increases, and sharpness tends to drop. If the lower limit value of the conditional expression (1) is not reached, the afocal magnification becomes too small, hence the image superposed on the external view becomes too small and becomes difficult to see. To secure sufficient pixels and resolution, a large deflection angle of the optical deflection element must be taken, which is not advantageous for the system configuration.

(18) To demonstrate the effect of this embodiment with certainty, it is preferable that the upper limit value of the conditional expression (1) is 2.00. To demonstrate the effect of this embodiment with certainty, it is preferable that the lower limit value of the conditional expression (1) is 0.35.

(19) In the head-mounted display optical system LS of this embodiment, it is preferable that the following conditional expression (2) is satisfied.
20<|f2/f1|<3000(2)
where f1 denotes a focal length of the first lens group G1, and f2 denotes a focal length of the second lens group G2.

(20) The conditional expression (2) specifies an appropriate ratio of refractive power between the first lens group G1 and the second lens group G2. If the upper limit value of the conditional expression (2) is exceeded, the refractive power of the diffraction optical surface becomes too small when the diffraction optical surface is used, hence correction of chromatic aberration for short wavelengths tends to be insufficient. Even if the diffraction optical surface is not used, generation of chromatic aberration increases, which diminishes the image quality. In particular, the coloring of the edge of an image increases. If the lower limit value of the conditional expression (2) is not reached, the refractive power of the diffraction optical surface becomes too high when the diffraction optical surface is used, hence correction of chromatic aberration for short wavelengths tends to be insufficient. Even if the diffraction optical surface is not used, generation of chromatic aberration increases, which diminishes the image quality. In particular, the coloring of an edge of an image increases. The focal length and refractive power are values acquired by tracing the rays of micro-luminous flux around the optical axis or the reference axis. The values acquired by tracing the rays of micro-luminous flux around the reference axis are a calculated value corresponding to the paraxial ray tracing.

(21) To demonstrate the effect of this embodiment with certainty, it is preferable that the upper limit value of the conditional expression (2) is 2000. To demonstrate the effect of this embodiment with certainty, it is preferable that the lower limit value of the conditional expression (2) is 35.

(22) In the head-mounted display optical system LS of this embodiment, it is preferable that the following conditional expression (3) is satisfied.
0.000015 [mm.sup.1]<|1/f2|<0.005 [mm.sup.1](3)

(23) The conditional expression (3) specifies an appropriate range of the refractive power of the second lens group G2 disposed near the intermediate image forming position. If the upper limit value of the conditional expression (3) is exceeded, the refractive power of the second lens group G2 becomes too high, hence distortion in a peripheral area of the screen tends to be generated and coma aberration is also generated, which drops image quality. If the lower limit value of the conditional expression (3) is not reached, the refractive power of the second lens group G2 becomes too low, hence capability to correct various aberrations (particularly flatness of the image plane) in the peripheral area of the screen becomes insufficient. Further, the refractive power of the second lens group G2 becomes too low, hence it becomes difficult to dispose the pupil position at an optimum location, and correction of coma aberration also becomes difficult.

(24) To demonstrate the effect of this embodiment with certainty, it is preferable that the upper limit value of the conditional expression (3) is 0.003 [mm.sup.1]. To demonstrate sufficiently the effect of this embodiment, it is preferable that the lower limit value of the conditional expression (3) is 0.00003 [mm.sup.1]. In order to sufficiently separate the pupil position from the semi-transparent reflection mirror (optical reflection element M2) in the configuration, it is preferable that the second lens group G2 has negative refractive power near the reference axis.

(25) In the head-mounted display optical system LS, it is preferable that the following conditional expression (4) is satisfied.
15<d<45(4)
where d denotes a maximum value of the difference of Abbe numbers between a lens having positive refractive power and a lens having negative refractive power in the first lens group G1.

(26) The conditional expression (4) specifies an appropriate range of Abbe numbers of a positive lens and a negative lens in the first lens group G1. If the upper limit value of the conditional expression (4) is exceeded, a material having a high refractive index must be used for the negative lens side, which easily diminishes flatness of the image plane. Moreover, a material having a high refractive index tends to have high specific gravity, hence the weight of the entire optical system tends to increase. If the lower limit value of the conditional expression (4) is not reached, the refractive power of each lens becomes too high, and high order aberrations tend to be generated. Further, the curvature of field of the lens surface increases, which makes fabrication difficult.

(27) To demonstrate the effect of this embodiment with certainty, it is preferable that the upper limit value of the conditional expression (4) is 40. To demonstrate the effect of this embodiment with certainty, it is preferable that the lower limit value of the conditional expression (4) is 20.

(28) In the head-mounted display optical system LS of this embodiment, it is preferable that the first lens group G1 is constituted by resin lenses, the second lens group G2 further includes a lens having a diffuse transmission surface, and the deflection angle of light that is reflected on the reflection surface M2r of the optical reflection element M2 (angle formed by the light that enters the reflection surface M2r of the optical reflection element M2 and the light that is reflected by the reflection surface M2r) is 45 or more. The specific gravity of the resin is less than that of glass, so using resin lenses is effective in decreasing the weight of the optical system. Moreover, if the diffuse transmission surface for diffusing the transmitting light is disposed on any one of the lens surfaces of the second lens group G2, the luminous flux diameter after diffusion can be increased, hence the pupil diameter can be increased. Thereby the eye motion area (area which is not eclipsed even when eyes move) can be increased, which is effective in using an HMD (Head-Mounted Display) apparatus as a mobile unit. Micro-bumps to diffuse the transmitting light are formed, for example, on the diffuse transmission surface of the lens having a diffuse transmission surface. Alternatively a diffraction grating that can diffuse the transmitting light, for example, may be formed on the diffuse transmission surface of the lens. To decrease size and weight of the optical system LS, it is preferable that the specific gravity of the resin material is 2.0 or less. To demonstrate the effect even more so, it is preferable that the specific gravity of the resin material is 1.6 or less.

(29) In the head-mounted display optical system LS of this embodiment, it is preferable that the first lens group G1 includes a free-shaped surface lens having positive refractive power and a negative lens, and the second lens group G2 includes a free-shaped surface lens having a free-shaped surface which is rotationally asymmetrical with respect to a reference axis. According to this configuration, complicated distortion that is rotationally asymmetrical with respect to the reference axis can be effectively corrected by the free-shaped surface lens. Further, chromatic aberration can also be corrected by the combination of the positive lens and the negative lens in the first lens group G1.

(30) It is preferable that the free-shaped surface lens of the first lens group G1 has a free-shaped lens surface and an aspherical lens surface, which is located on the opposite side of this lens surface and is rotationally symmetric with respect to the reference axis, and is formed in a meniscus shape, and the negative lens has an aspherical lens surface which is rotationally symmetrical with respect to a reference axis, and is formed in a meniscus shape. Further, the free-shaped surface lens of the second lens group G2 has positive refractive power and is formed in a meniscus shape. By this configuration, an aberration component that is rotationally symmetrical with respect to the reference axis can be effectively corrected.

(31) When (x, y) are the coordinates that pass through the intersection point of the free-shaped lens surface and reference axis, and are on a plane perpendicular to the reference axis, and a sag of the free-shaped lens surface is expressed by a polynomial of x and y in the free-shaped surface lens of the first lens group G1, it is preferable that the polynomial of x and y includes a term of which degree is at least 8. When h is a distance from the reference axis and a sag of the aspherical lens surface is expressed by a polynomial of h in the free-shaped surface lens of the first lens group G1, it is preferable that the polynomial of h includes a term of which degree is at least 8. By this configuration, distortion that is rotationally asymmetrical with respect to the reference axis and an aberration component that is rotationally symmetrical with respect to the reference axis can be effectively corrected.

(32) It is preferable that the second lens group G2 further includes a lens having a diffuse transmission surface, and when (x, y) are coordinates that pass through the intersection point of the free-shaped lens surface and a reference axis, and are on a plane perpendicular to the reference axis, and a sag of the free-shaped lens surface is expressed by a polynomial of x and y in the free-shaped surface lens of the second lens group G2, the polynomial of x and y includes a term of which degree is at least 8. By this configuration, distortion that is rotationally asymmetrical with respect to the reference axis can be effectively corrected. Further, as mentioned above, the luminous flux diameter after diffusion can be increased, hence the pupil diameter can be increased, which is effective in using an HMD apparatus as a mobile unit.

(33) In a refraction optical system and a reflection optical system, various attempts to integrate the diffraction optical surface into the optical system has been made to increase performance and decrease size in a way that conventional methods cannot implement. An example is a pickup lens for an optical disk. However, in the case of a single layer diffraction optical element, more flares are generated in wavelengths that deviated from the design wavelength, which diminishes image quality and image forming performance. The use of a single layer diffraction optical element has been limited to a single wavelength or narrow wavelength region of a laser light source or the like.

(34) Recently however, in a diffraction optical element having such a diffraction optical surface, a diffraction optical element called either a dual layer type or a stacked type has been proposed. This type of diffraction optical element is constituted by a plurality of diffraction element members having a serrated surface, which are stacked in a separated or contacted state, and has characteristics where high diffraction efficiency is maintained in almost all of a desired wide wavelength region (e.g. visible light region), in other words, wavelength characteristics are good.

(35) Generally dual layer type diffraction optical element is configured by a first optical element member 111 constituted by a first material and a second optical element member 112 constituted by a second material of which refractive index and dispersion value are different from the first material, and serrated diffraction gratings 111a and 112a are formed on the surface of the respective optical element members which face each other, as shown in FIG. 7A and FIG. 7B for example. Here in order to satisfy achromatic conditions for two specific wavelengths, the grating height (height of grooves) h1 of the first optical element member 111 is determined to a predetermined value, and the grating height h2 of the second optical element member 112 is determined to another predetermined value. Thereby the diffraction efficiency is 1.0 for the two specific wavelengths, and very high diffraction efficiency can be obtained even for other wavelengths. By using the dual layer type for the diffraction optical element, the diffraction optical element can be applied to almost all the wavelengths, and can also be easily used even for an imaging lens of a photo camera that uses white light. The diffraction efficiency (diffraction efficiency of the first order diffracted light: first order diffracted light is used in this embodiment) is defined as (=I.sub.1/I.sub.0), that is, a ratio of the intensity I.sub.0 of the light that enters the diffraction optical element and the intensity I.sub.1 of the first order diffracted light included in the light transmitted through the diffraction optical element in the case of the transmission type diffraction optical element.

(36) By satisfying a predetermined condition, a contacted dual layer type diffraction optical element, where the grating height h1 of the first optical element member 111 and the grating height h2 of the second optical element member 112 are matched as shown in FIG. 7B, can be implemented. Compared with the separated dual layer type diffraction optical element shown in FIG. 7A, the contacted dual layer type diffraction optical element has merits in that the error sensitivity (tolerance) of the grating height is relaxed, and the error sensitivity (tolerance) of the surface roughness of the grating surface is also relaxed, which makes fabrication of the diffraction optical element easier, increases mass producibility, and has the advantage of reducing cost of the optical product.

(37) The diffraction optical surface will be further described. Refraction and reflection are known as methods for deflecting a ray, but diffraction is also known as a third method. The diffraction optical element is an optical element that utilizes the diffractive phenomenon of light, and presents behavior that is different from refraction and reflection. For example, a diffraction grating and a Fresnel zone plate are known. Even for a natural light or a white light, an obvious diffraction phenomenon can be generated by the interference of light waves if a structure in a wavelength order is created, since the coherence length is normally several . A surface having such a diffraction effect is called a diffraction optical surface. A known characteristic of the diffraction optical surface is that if the diffraction optical surface has positive power, the diffraction optical surface presents negative dispersion values, which is very effective to correct chromatic aberration. If the diffraction optical surface has negative power, the diffraction optical surface presents positive dispersion values, which is also very effective to correct chromatic aberration. Therefore chromatic aberration can be corrected very well, which cannot be implemented by ordinary glass, and can be implemented only by expensive special low dispersion glass.

(38) In this embodiment, a surface having a function to deflect a ray, such as a diffraction grating and Fresnel zone plate, is formed on the surface of an optical member constituted by glass or plastic, by applying the diffraction phenomenon, so that good optical performance is obtained by this function. A surface that has a function to deflect a ray using the diffraction phenomenon like this is called a diffraction optical surface, and an optical element having such a surface is normally called a diffraction optical element. For details on the diffraction optical element, see Diffraction Optical Element Primer (compiled and supervised by the Optical Society of Japan (Japan Society of Applied Physics), expanded and revised edition, 2007).

(39) Generally the smaller the angle of a ray that passes through the diffraction optical surface of an optical system the better. Because if a ray angle increases, flares are more easily generated from the walls or the like of the grating of the diffraction optical surface, which diminishes the image quality. In order to obtain a good image in this optical system with little influence of flares, it is preferable that the ray angle is 30 or less. As long as this condition is satisfied, the diffraction optical surface may be disposed anywhere in the optical system.

(40) By forming the diffraction optical surface, in addition to the free-shaped surface described above, a head-mounted display optical system that is compact and light and has excellent optical performance can be implemented. The refractive power of the first optical element member 111 and the second optical element member 112 may be either positive or negative. The configuration can be designed to be convenient for implementing the specifications and correcting aberrations considering the design requirements. For the diffraction optical element as a whole to have positive refractive power, both the first optical element member 111 and the second optical element member 112 are designed to have positive refractive power, so that the diffraction optical surface has positive refractive power, then the diffraction optical element can have negative dispersion and good achromatism can be implemented in the diffraction optical element as a whole. For the diffraction optical element as a whole to have negative refractive power, on the other hand, both the first optical element member 111 and the second optical element member 112 are designed to have negative refractive power, so that the diffraction optical surface has negative refractive power, then the diffraction optical element can have positive dispersion, and good achromatism can be implemented in the diffraction optical element as a whole.

(41) If the first lens group G1 includes a lens having a diffraction optical surface, it is preferable that the following conditional expression (5) is satisfied.
0.01<fd/f1<10.00(5)
where fd denotes a focal length of the lens having the diffraction optical surface, and f1 denotes a focal length of the first lens group G1.

(42) The conditional expression (5) specifies an appropriate ratio between the focal length of the first lens group G1 and the focal length of the lens having a diffraction optical surface. If the upper limit value of the conditional expression (5) is exceeded, fd becomes too large, hence a good aberration balance throughout the screen tends to be lost. Therefore blurring of a point image easily occurs throughout the screen, and correction of chromatic aberration tends to be insufficient. Moreover, refractive power of the other convex lenses in the first lens group G1 becomes too high, which makes it difficult to fabricate the head-mounted display optical system. If the lower limit value of the conditional expression (5) is not reached, fd becomes too small, hence aberration balance is easily lost. Therefore the image forming performance of luminous flux drops, particularly when the angle of view is large, and blurs and color shifts of a point image spread on the image forming surface, and the view near the visual field worsens. Furthermore, if fd becomes small, the radius of curvature of the surface tends to be small, which makes it more difficult to process the diffraction optical surface.

(43) To demonstrate the effect of this embodiment with certainty, it is preferable that the upper limit value of the conditional expression (5) is 6.00. To demonstrate the effect of this embodiment with certainty, it is preferable that the lower limit value of the conditional expression (5) is 0.20.

(44) For an application to enlarge and observe an image generated using a compact display and objective lens, it is preferable that the following conditional expression (6) is satisfied.
0.005 [mm.sup.1]<dm/(fm).sup.2<1.00 [mm.sup.1](6)
where dm denotes a distance from the second lens group G2 to the optical reflection element M2, and fm denotes a focal length of the optical reflection element M2.

(45) The conditional expression (6) specifies an appropriate relationship between eye relief and the focal length when this optical system is applied to an observation optical system. Taking sufficient eye relief is important to construct an observation optical system. If the upper limit value of the conditional expression (6) is exceeded, dm becomes too long, hence the size of the optical system tends to increase. In particular, the optical reflection element M2 becomes too large. Moreover, the curvature of field increases, which makes it impossible to implement good image quality. If the lower limit value of the conditional expression (6) is not reached, dm becomes too short, which results in a decrease in the eye relief. This makes observation difficult when the spectacles are mounted, which makes observation difficult in practical terms. In this embodiment, eye relief refers to a point where the principal ray that passes through the very lower right corner of the mirror of the optical system intersects with the optical axis in the optical path diagram. The shape of a pupil need not be circular, but may be rectangular or elliptical depending on the application and design specification. The shape of a pupil can be created by considering the shape of the lenses and the shape of the diaphragm.

(46) To demonstrate the effect of this embodiment with certainty, it is preferable that the upper limit value of the conditional expression (6) is 0.10 [mm.sup.1]. To demonstrate the effect of this embodiment with certainty, it is preferable that the lower limit value of the conditional expression (6) is 0.01 [mm.sup.1].

(47) If the first lens group G1 includes a lens having a diffraction optical surface, it is preferable that the following conditional expression (7) is satisfied.
0.01<NdhNdl<0.45(7)
where Ndh denotes a refractive index of a layer of which refractive index is relatively high in the dual layer type diffraction optical element, and Ndl denotes a refractive index of a layer of which refractive index is relatively low in the dual layer type diffraction optical element.

(48) The conditional expression (7) specifies an appropriate range of the difference of refractive indexes before and after the diffraction optical surface. In the conditional expression (7), a case of blazing spectral lines to which human eyes are highly sensitive, such as the d-line and e-line, by a first order diffracted light, is considered. If the upper limit value of the conditional expression (7) is exceeded, the difference of the refractive indexes become too large, which increases noise light reflected by the interface (diffraction optical surface) and deteriorates image quality. Moreover, the error sensitivity with respect to the grating height increases, which makes processing difficult. If the lower limit value of the conditional expression (7) is not reached, the grating height becomes too high, hence high diffraction efficiency cannot be implemented if the light deviates from vertical entry, and the generation of flares increases, and as a result good image quality cannot be implemented. Processing also becomes difficult.

(49) To demonstrate the effect of this embodiment with certainty, it is preferable that the upper limit value of the conditional expression (7) is 0.20. To demonstrate the effect of this embodiment with certainty, it is preferable that the lower limit value of the conditional expression (7) is 0.02.

(50) If the first lens group G1 includes a lens having a diffraction optical surface, it is preferable that the following conditional expression (8) is satisfied.
0.001<(d1+d2)/f1<0.25(8)
where d1 denotes thickness (on the optical axis) of a layer of which refractive index is relatively high in the dual layer type diffraction optical element, d2 denotes thickness (on the optical axis) of a layer of which refractive index is relatively low in the dual layer type diffraction optical element, and f1 denotes focal length of the first lens group G1.

(51) The conditional expression (8) specifies an appropriate range when the sum of d1 and d2 is normalized by f1. If the upper limit value of the conditional expression (8) is exceeded, d1 becomes too large, hence the light absorption on the short wavelength side increases, and transmittance of the entire optical system on the short wavelength side tends to deteriorate. If the lower limit value of the conditional expression (8) is not reached, d1 becomes too small, and molding of the diffraction optical element becomes difficult.

(52) To demonstrate the effect of this embodiment with certainty, it is preferable that the upper limit value of the conditional expression (8) is 0.10. To demonstrate the effect of this embodiment with certainty, it is preferable that the lower limit value of the conditional expression (8) is 0.005.

(53) Thus according to this embodiment, compactness, light weight and excellent optical performance can be implemented.

(54) When the optical system is actually constructed, it is preferable that the following requirements are further satisfied.

(55) The optical member having the free-shaped surface may be constituted by either resin or glass, but is preferably fabricated by a molding method.

(56) In order to maintain good moldability of the contacted dual layer type diffraction optical element and ensure excellent mass producibility, the viscosity (viscosity of uncured material) of material constituting at least one of the first and second diffraction element members is at least 40 [mPa.circle-solid.s]. If 40 [mPa.circle-solid.s] or less, resin easily flows during molding, which makes it difficult to mold a precise shape. The viscosity of material constituting the other diffraction element member, on the other hand, is preferably at least 2000 [mPa.circle-solid.s].

(57) The material of the first and second diffraction element members constituting the diffraction optical element is preferably UV curable resin. This increases production efficiency, which is desirable. In this case, labor time can also be reduced, which leads to cost reduction. To decrease size and weight, it is preferable that the specific gravity of any resin material is 2.0 or less. Resin, of which specific gravity is less compared with glass, is very effective to decrease weight of the optical system. To demonstrate the effect even more, it is preferable that the specific gravity of the resin material is 1.6 or less. Furthermore, it is preferable that the first and second diffraction element members have a refraction surface having positive refractive power at the interface with air, and this refraction surface is aspherical. Suitable resin materials are acrylic, polycarbonate, olefine, acryl-styrene co-polymer, polyester or the like.

(58) It is preferable that the incident angle of all the rays with respect to the diffraction optical surface is 15 or less. If the incident angle increases, generation of flares increases, and image quality drops.

(59) It is also possible to provide a color filter effect by mixing dye with any of the resins constituting each optical member. For example, an infrared cut-off filter may be constructed by this method, whereby a compact imaging optical system is constructed.

(60) The diaphragm may be freely disposed in the optical system, but is preferably constructed such that unwanted rays are cut off and only rays useful for forming an image are transmitted. For example, the lens frame itself may be used as the aperture stop, or a diaphragm may be constructed by a mechanical member at a position distant from the lens. The diaphragm constructed by a mechanical member may be disposed between the lens and the image plane, for example. The shape of the diaphragm is not limited to a circle, but may be an ellipse or rectangle, according to the design specification.

(61) An optical system constituted by a plurality of composing elements incorporating the optical element of this embodiment is within the scope of the present invention. Further, an optical system incorporating a gradient index lens, a crystal material lens or the like is also within the scope of the present invention.

(62) A head-mounted display according to this embodiment will be described next. FIG. 8 shows the head-mounted display DSP according to this embodiment. The head-mounted display DSP includes a spectacle type frame FL, a light source unit SU and a projection unit PU. The spectacle type frame FL holds the light source unit SU and the projection unit PU, and is mounted on the head of the user, along with the light source unit SU and the projection unit PU. The power supply unit SU generates an image signal based on image information inputted from an external input device (not illustrated), and outputs a laser light having intensity in accordance with the image signal (hereafter called image light) to the projection unit PU.

(63) The projection unit PU two-dimensionally scans the image light emitted from the light source unit SU, and projects the light onto the eyes E of the user. Thereby when the user wears the head-mounted display DSP, the image light is two-dimensionally scanned and an image is projected onto the retina of the eyes E of the user, and the user can visually recognize an image corresponding to the image signal. In the projection unit PU, a half mirror HM is disposed in a location corresponding to the eyes E of the user. An external light La transmits through the half mirror HM and enters the eyes E of the user, and the image light Lb emitted from the projection unit PU is reflected by the half mirror HM and enters the eyes E of the user. As a result, the user can visually recognize the image formed by the image light Lb in a state of being superposed on an outside view formed by the natural light La.

(64) The projection unit PU includes the head-mounted display optical system LS according to the above mentioned embodiment. Therefore compactness, light weight and excellent optical performance can be implemented, as described above. In the head-mounted display DSP according to this embodiment, the half mirror HM corresponds to the optical reflection element M2 of the head-mounted display optical system LS.

EXAMPLES

Example 1

(65) Each example of the present invention will now be described with reference to the accompanying drawings. Example 1 of the present invention will be described first with reference to FIG. 1, FIG. 2 and Table 1. FIG. 1 is an optical path diagram of a head-mounted display optical system LS (LS1) according to Example 1. The head-mounted display optical system LS1 according to Example 1 has, in order from a light source, an optical deflection element M1, a first lens group G1 having positive refractive power, a second lens group G2 having negative refractive power near the reference axis, and an optical reflection element M2. The optical deflection element M1 changes the traveling direction of a light from the light source (e.g. approximately parallel light, such as laser light and LED light). In FIG. 1, the state of change of the traveling direction of light in the optical deflection element M1 is indicated by the light that transmits through a diaphragm (this diaphragm is indicated as the optical deflection element M1) at a plurality of types of incident angles, to simplify description.

(66) The first lens group G1 collects light entered from the light source via the optical deflection element M1. The first lens group G1 includes, in order from the light source, a negative lens L11, a first free-shaped surface lens L12 having positive refractive power, and a second free-shaped surface lens L13 having positive refractive power. The negative lens L11 is formed in a meniscus shape having a convex surface facing the light source. An aspherical surface, which is rotationally symmetrical with respect to the reference axis, is formed on both lens surfaces of the negative lens L11. The first free-shaped surface lens L12 is formed in a meniscus shape having a concave surface facing the light source. An aspherical surface, which is rotationally symmetrical with respect to the reference axis, is formed on the light source side lens surface of the first free-shaped surface lens L12, and a free-shaped surface, which is rotationally asymmetrical with respect to the reference axis, is formed on the drawing surface I side lens surface of the first free-shaped surface lens L12. The second free-shaped surface lens L13 is formed in a meniscus shape having a convex surface facing the light source. An aspherical surface, which is rotationally symmetrical with respect to the reference axis, is formed on the light source side lens surface of the second free-shaped surface lens L13, and a free-shaped surface, which is rotationally asymmetrical with respect to the reference axis, is formed on the drawing surface I side lens surface of the second free-shaped surface lens L13.

(67) The second lens group G2 is disposed near an intermediate image forming position (collecting position) by the first lens group G1. The second lens group G2 includes, in order from the light source, a diffusion lens L21, and a third free-shaped surface lens L22 having positive refractive power. The diffusion lens L21 is formed in a meniscus shape having a convex surface facing the light source. A diffuse transmission surface is formed on the drawing surface I side lens surface of the diffusion lens L21. The third free-shaped surface lens L22 is formed in a meniscus shape having a convex surface facing the light source. A free-shaped surface, which is rotationally asymmetrical with respect to the reference axis, is formed on the light source side lens surface of the third free-shaped surface lens L22, and an aspherical surface, which is rotationally symmetrical with respect to the reference axis, is formed on the drawing surface I side lens surface of the third free-shaped surface lens L22.

(68) The optical reflection element M2 has a reflection surface M2r that reflects light transmitted through the second lens group G2. The optical reflection element M2 is also configured to transmit the light entered from the surface M2s on the opposite side of the reflection surface M2r. The reflection surface M2r of the optical reflection element M2 is formed to be rotationally asymmetrical with respect to the reference axis.

(69) This head-mounted display optical system LS1 is configured such that light from the light source, which is reflected on the reflection surface M2r of the optical reflection element M2 and reached a surface assumed to be located on a retina when the user wears the head-mounted display (drawing surface I), moves on the drawing surface I in a manner of two-dimensionally scanning in accordance with the change of a travelling direction of the light caused by the optical deflection element M1, and an image is drawn on the drawing surface I. In this case, the image that is drawn is superposed on an image formed by the light that is transmitted through the optical reflection element M2 and reached the drawing surface I (retina). In Example 1, it is assumed that an aplanatic lens (not illustrated), of which focal length f=17 [mm], is disposed between the optical reflection element M2 and the drawing surface I. Thereby the light reflected on the reflection surface Mgr of the optical reflection element M2 transmits through the aplanatic lens (not illustrated) and is collected on the drawing surface I (retina).

(70) In each example, the phase difference of the diffraction optical surface is calculated using a phase function method.

(71) A phase polynomial to determine a shape of the diffraction optical surface is given by the following Expression (A).
[Math. 1]
Z=C.sub.jx.sup.my.sup.n (A)

(72) In Expression (A), the relationship given by the following Expression (B) is established among j, m and n.

(73) $\begin{array}{cc}\left[\mathrm{Math}.2\right]& \\ j=\frac{{\left(m+n\right)}^2+m+3n}{2}& \left(B\right)\end{array}$

(74) The free-shaped surface is defined by the following expression (C). In Expression (C), Z denotes a sag of a surface in parallel with the reference axis, c denotes a curvature at a vertex of the surface (origin), k denotes a conic constant, h denotes a distance from an origin on the reference axis on a plane that intersects vertically with the reference axis at the origin, and C.sub.j denotes a coefficient of an xy polynomial (polynomial of x and y).

(75) $\begin{array}{cc}\left[\mathrm{Math}.3\right]& \\ Z=\frac{{\mathrm{ch}}^2}{1+\sqrt{1-\left(1+k\right)c^2{h}^2}}+.Math.C_j{x}^m{y}^n& \left(C\right)\end{array}$

(76) Here the relationships given by the following Expressions (D) and (E) are established among j, m and n of Expression (C).

(77) $\begin{array}{cc}\left[\mathrm{Math}.4\right]& \\ j=\frac{{\left(m+n\right)}^2+m+3n}{2}+1& \left(D\right)\\ \left[\mathrm{Math}.5\right]& \\ m+n10& \left(E\right)\end{array}$

(78) The aspherical surface is defined by the following Expression (F). In Expression (F), Z denotes a sag of a surface in parallel with the reference axis (optical axis), c denotes a curvature at a vertex of the surface (on the reference axis (optical axis)), k denotes a conic constant, and h denotes a distance from the reference axis (optical axis) on a plane that intersects vertically with the reference axis (optical axis), and A4 to A12 are coefficients according to each power series term of h.

(79) $\begin{array}{cc}\left[\mathrm{Math}.6\right]& \\ Z=\frac{{\mathrm{ch}}^2}{1+\sqrt{1-\left(1+k\right)c^2{h}^2}}+A4h^4+A6h^6+A8h^8+A10h^{10}+A12h^{12}& \left(F\right)\end{array}$

(80) In each example, each spectral line of C-line, d-line, F-line and g-line is used to calculate the aberration characteristics. The wavelength of each spectral line is as follows.

(81) Wavelength (nm) C-line 656.273 d-line 587.562 F-line 486.133 g-line 435.835

(82) Table 1 to Table 3 shown below are tables listing various data of the head-mounted display optical system LS according to Examples 1 to 3 respectively. In [Lens Data] in each table, the first column N shows a sequential number of the lens surface counted from the object side, the second column R shows a radius of curvature of the lens surface, the third column D shows the distance between lenses, the fourth column nC shows a refractive index with respect to the C-line, the fifth column nd shows a refractive index with respect to the d-line, the sixth column nF shows a refractive index with respect to the F-line, and the seventh column ng shows a refractive index with respect to the g-line. *a attached to the right of the first column indicates that this lens surface is aspherical, *b attached to the right of the first column indicates that this lens surface is a free-shaped surface, and *c attached to the right of the first column indicates that this lens surface is a diffraction optical surface. of the radius of curvature indicates a plane, and the refractive index of air is omitted.

(83) In [Eccentricity Data], XDE, YDE and ZDE indicate shift in the x direction, the y direction and the z direction respectively, and ADE, BDE and CDE indicate an inclination (unit: degrees) around the x axis, y axis and z axis respectively. In [Aspherical Data], [Free-shaped surface data] and [Diffraction optical surface data], En means 10.sup.n. In all the data values herein below, mm is normally used as the unit of radius of curvature R, surface distance D and other lengths. However, unit is not limited to mm, since an equivalent optical performance is obtained even if the optical system is proportionally expanded or proportionally reduced. The same symbols as in this example are also used for various data values of Example 2 and Example 3, which are described later.

(84) Table 1 shows each data value of Example 1. The radius of curvature R of surface 1 to surface 14 in Table 1 correspond to symbols R1 to R14 attached to surface 1 to surface 1 in FIG. 1 (excluding surface 10 and surface 13, which are virtual surfaces).

(85) TABLE-US-00001 TABLE 1 [Lens Data] N R D nC nd nF ng Object surface 1 7.0737 (aperture stop) 2*a 26.1079 3.5000 1.6074 1.6142 1.6314 1.6463 3*a 6.0261 1.2000 4*a 139.6362 3.2500 1.5283 1.53113 1.53783 1.54319 5*b 6.6413 1.1000 6*a 6.0884 2.5000 1.5283 1.53113 1.53783 1.54319 7*b 7.1934 20.0000 8 36.0000 1.0000 1.4883 1.4908 1.4969 1.5016 9 36.0000 1.0000 10 0.0000 11*b 98.4222 1.8000 1.4883 1.4908 1.4969 1.5016 12*a 82.9218 0.0000 13 31.1429 14*b 66.0000 45.0000 (reflec- tion) Image 0.0000 plane (drawing surface) [Eccentricity Data] Surface Surface Surface Surface Surface Eccentricity 8 10 11 13 14 XDE 0.0 0.0 0.0 0.0 0.0 YDE 0.0 0.0 0.0 0.0 0.0 ZDE 0.0 0.0 0.0 0.0 0.0 ADE 6.0 6.0 6.0 6.0 25.0 BDE 0.0 0.0 0.0 0.0 0.0 CDE 0.0 0.0 0.0 0.0 0.0 [Aspherical Data] Surface 2 coefficient k = 121.7194695202487 A4 = 0.001204087810046505 A6 = 0.0001388073049376515 A8 = 0.3272747753555764E5 A10 = 0.8651482588206862E7 A12 = 0.4577336492603486E8 Surface 3 coefficient k = 3.899985247951888 A4 = 0.000692579625568129 A6 = 0.736953824073804E6 A8 = 0.5744863840903909E7 A10 = 0.1089711828346197E8 A12 = 0.42714750385595E10 Surface 4 coefficient k = 500.0 A4 = 0.0001885783788227973 A6 = 0.7290580702481259E6 A8 = 0.1687839086919377E6 A10 = 0.3866962063772566E8 A12 = 0.1672547615430344E9 Surface 6 coefficient k = 4.103478542047234 A4 = 0.0002580575996569321 A6 = 0.1247180061433354E4 A8 = 0.1934054239498101E6 A10 = 0.1851585581105413E8 A12 = 0.2038231252720517E11 Surface 12 coefficient k = 0.00001 A4 = 0.2609431778754561E4 A6 = 0.1879205705223268E7 A8 = 0.1737233337684116E10 A10 = 0.4084729769386872E13 A12 = 0.3965647731154538E14 [Free-shaped surface data 1] Term Surface 5 coefficient Surface 7 coefficient C1(k) 0.3017745850875778 6.289974218996122 C4(x{circumflex over ()}2) 0.003466799143357255 0.001298452145258934 C5(x*y) 0.001115603104928361 0.0003822737406034326 C6(y{circumflex over ()}2) 0.0070562666856616 0.01357172844453788 C7(x{circumflex over ()}3) 0.0002593761539094165 0.0003967924865306014 C8(x{circumflex over ()}2*y) 0.000213250191978018 0.001371988100285954 C9(x*y{circumflex over ()}2) 0.0006191442572110238 0.0006519318827385893 C10(y{circumflex over ()}3) 0.0004718785005503118 0.0007140652552868351 C11(x{circumflex over ()}4) 0.0003098822638427462 0.001333147444714297 C12(x{circumflex over ()}3*y{circumflex over ()}1) 0.3407768810806382E4 0.3702664184744909E4 C13(x{circumflex over ()}2*y{circumflex over ()}2) 0.0006208558182873208 0.0001486993056817897 C14(x{circumflex over ()}1*y{circumflex over ()}3) 0.0001792129244483547 0.8202377408038175E4 C15(y{circumflex over ()}4) 0.0003580494710146699 0.1805720322028896E5 C16(x{circumflex over ()}5) 0.2663050945651949E4 0.3564660331163073E4 C17(x{circumflex over ()}4*y) 0.0001474696025030011 0.0002908357809224226 C18(x{circumflex over ()}3*y{circumflex over ()}2) 0.3826719889708307E4 0.5278851232820081E4 C19(x{circumflex over ()}2*y{circumflex over ()}3) 0.1026510876791286E4 0.1129260388688707E4 C20(x*y{circumflex over ()}4) 0.89755244405221E5 0.3255869135725364E5 C21(y{circumflex over ()}5) 0.881872878453439E5 0.1565166610057392E4 C22(x{circumflex over ()}6) 0.2469668814869958E4 0.0001044613639817373 C23(x{circumflex over ()}5*y) 0.9577511038872208E5 0.485644559192588E5 C24(x{circumflex over ()}4*y{circumflex over ()}2) 0.5046560131654857E4 0.4070365124589857E5 C25(x{circumflex over ()}3*y{circumflex over ()}3) 0.3118417520710068E5 0.1206368572452636E6 C26(x{circumflex over ()}2*y{circumflex over ()}4) 0.2390039944775742E4 0.8706445825913175E5 C27(x*y{circumflex over ()}5) 0.2021899900909428E5 0.7102202017053954E6 C28(y{circumflex over ()}6) 0.1121912351982155E4 0.3920146580746637E5 C29(x{circumflex over ()}7) 0.3119742444603493E5 0.1022712596352382E5 C30(x{circumflex over ()}6*y) 0.1560107102046973E4 0.2061553131129934E4 C31(x{circumflex over ()}5*y{circumflex over ()}2) 0.1022165397653761E6 0.2524689087473794E5 C32(x{circumflex over ()}4*y{circumflex over ()}3) 0.6256105666354908E6 0.1076326368048734E5 C33(x{circumflex over ()}3*y{circumflex over ()}4) 0.2345864653335014E6 0.6829314149181599E6 C34(x{circumflex over ()}2*y{circumflex over ()}5) 0.2093443610038782E5 0.817752813267785E6 C35(x{circumflex over ()}1*y{circumflex over ()}6) 0.483868092159451E6 0.1440162325040492E6 C36(y{circumflex over ()}7) 0.7510694585638476E6 0.146433761705865E6 C37(x{circumflex over ()}8) 0.2467932549368314E6 0.440537048345761E5 C38(x{circumflex over ()}7*y{circumflex over ()}1) 0.174350235824826E7 0.100773324541987E7 C39(x{circumflex over ()}6*y{circumflex over ()}2) 0.3030380767038765E5 0.6710754426117793E6 C40(x{circumflex over ()}5*y{circumflex over ()}3) 0.862388498644196E6 0.9535259048086168E7 C41(x{circumflex over ()}4*y{circumflex over ()}4) 0.1114429145219428E5 0.1957221945793596E6 C42(x{circumflex over ()}3*y{circumflex over ()}5) 0.1311338944233999E7 0.3649699847740186E7 C43(x{circumflex over ()}2*y{circumflex over ()}6) 0.7673913237103807E6 0.7638710741140231E7 C44(x{circumflex over ()}1*y{circumflex over ()}7) 0.1022816718413322E6 0.1531614658352089E7 C45(y{circumflex over ()}8) 0.1038023304987815E6 0.4631307996723954E8 C46(x{circumflex over ()}9) 0.2077668320265781E6 0.7659178291049076E7 C47(x{circumflex over ()}8*y) 0.8811215797461278E6 0.4286648225719628E6 C48(x{circumflex over ()}7*y{circumflex over ()}2) 0.1017463461555473E6 0.4690214684721334E7 C49(x{circumflex over ()}6*y{circumflex over ()}3) 0.7048883610168791E6 0.9082440518878004E7 C50(x{circumflex over ()}5*y{circumflex over ()}4) 0.4252785573753692E7 0.16753421062955E7 C51(x{circumflex over ()}4*y{circumflex over ()}5) 0.1144701685012133E6 0.2319990072526336E7 C52(x{circumflex over ()}3*y{circumflex over ()}6) 0.3692806162779624E7 0.3301471336612236E9 C53(x{circumflex over ()}2*y{circumflex over ()}7) 0.5502278825410581E7 0.1239194838045183E7 C54(x*y{circumflex over ()}8) 0.8886779900973679E9 0.181425272749395E8 C55(y{circumflex over ()}9) 0.1400634098335568E7 0.4289199154548536E10 C56(x{circumflex over ()}10) 0.6394953871821582E7 0.9682846367662571E7 C57(x{circumflex over ()}9*y) 0.2540103437709849E7 0.342555825602362E8 C58(x{circumflex over ()}8*y{circumflex over ()}2) 0.978580552741408E7 0.9438044604115687E9 C59(x{circumflex over ()}7*y{circumflex over ()}3) 0.1703644781583354E7 0.698764528549666E11 C60(x{circumflex over ()}6*y{circumflex over ()}4) 0.463820833904626E7 0.1325497883224603E8 C61(x{circumflex over ()}5*y{circumflex over ()}5) 0.3049088350270744E7 0.9226805612114128E9 C62(x{circumflex over ()}4*y{circumflex over ()}6) 0.4055561730231605E7 0.446228256709647E8 C63(x{circumflex over ()}3*y{circumflex over ()}7) 0.2411367852597294E9 0.9391879112780659E9 C64(x{circumflex over ()}2*y{circumflex over ()}8) 0.1071492894406766E7 0.21236224132437E8 C65(x{circumflex over ()}9*y) 0.1805437181357719E8 0.1460283831938418E9 C66(y{circumflex over ()}10) 0.1848835057183689E8 0.7361519164041878E10 [Free-shaped surfaced data 2] Term Surface 17 coefficient C1(k) 81.71254939167278 C4(x{circumflex over ()}2) 0.00791570619172745 C5(x*y) 0.001123378770402062 C6(y{circumflex over ()}2) 0.009679528636615819 C7(x{circumflex over ()}3) 0.226322034107961E4 C8(x{circumflex over ()}2*y) 0.5360693330540352E4 C9(x*y{circumflex over ()}2) 0.1377957183120677E4 C10(y{circumflex over ()}3) 0.0003519578519071908 C11(x{circumflex over ()}4) 0.0001075753092625291 C12(x{circumflex over ()}3*y{circumflex over ()}1) 0.4241041725681437E6 C13(x{circumflex over ()}2*y{circumflex over ()}2) 0.7789536633500974E4 C14(x{circumflex over ()}1*y{circumflex over ()}3) 0.5062402485802235E5 C15(y{circumflex over ()}4) 0.2409472911968187E4 C16(x{circumflex over ()}5) 0.762063479810465E6 C17(x{circumflex over ()}4*y) 0.7480814903816188E5 C18(x{circumflex over ()}3*y{circumflex over ()}2) 0.1438089929671677E5 C19(x{circumflex over ()}2*y{circumflex over ()}3) 0.556591328611352E5 C20(x*y{circumflex over ()}4) 0.7945076353643055E6 C21(y{circumflex over ()}5) 0.1338230016045893E5 C22(x{circumflex over ()}6) 0.2878012125528234E6 C23(x{circumflex over ()}5*y) 0.624022839236557E6 C24(x{circumflex over ()}4*y{circumflex over ()}2) 0.4599179052765449E6 C25(x{circumflex over ()}3*y{circumflex over ()}3) 0.2644884073907662E6 C26(x{circumflex over ()}2*y{circumflex over ()}4) 0.9578696536819849E8 C27(x*y{circumflex over ()}5) 0.2720859549328555E6 C28(y{circumflex over ()}6) 0.849145886566742E7 C29(x{circumflex over ()}7) 0.2734043214198553E7 C30(x{circumflex over ()}6*y) 0.4912163351311643E8 C31(x{circumflex over ()}5*y{circumflex over ()}2) 0.2018147279015479E6 C32(x{circumflex over ()}4*y{circumflex over ()}3) 0.5266506284409001E7 C33(x{circumflex over ()}3*y{circumflex over ()}4) 0.9551172420924838E7 C34(x{circumflex over ()}2*y{circumflex over ()}5) 0.7127478731636541E7 C35(x{circumflex over ()}1*y{circumflex over ()}6) 0.1719394408507042E7 C36(y{circumflex over ()}7) 0.445746144917605E7 C37(x{circumflex over ()}8) 0.3220832302879691E7 C38(x{circumflex over ()}7*y{circumflex over ()}1) 0.3084425750501727E7 C39(x{circumflex over ()}6*y{circumflex over ()}2) 0.8346726588083648E7 C40(x{circumflex over ()}5*y{circumflex over ()}3) 0.1867881673338012E7 C41(x{circumflex over ()}4*y{circumflex over ()}4) 0.7908158309130722E7 C42(x{circumflex over ()}3*y{circumflex over ()}5) 0.2537263125823934E8 C43(x{circumflex over ()}2*y{circumflex over ()}6) 0.7681488722460555E9 C44(x{circumflex over ()}1*y{circumflex over ()}7) 0.1596511474214415E8 C45(y{circumflex over ()}8) 0.6698019179202518E9 [Free-shaped surface data 3] Term Surface 14 coefficient C1(k) 3.962964493781687 C4(x{circumflex over ()}2) 0.001028153765764826 C5(x*y) 0.175569482042806E4 C6(y{circumflex over ()}2) 0.001259624894796678 C7(x{circumflex over ()}3) 0.5221346423219113E5 C8(x{circumflex over ()}2*y) 0.1861546693528138E4 C9(x*y{circumflex over ()}2) 0.1520365201410679E4 C10(y{circumflex over ()}3) 0.1845501708191367E4 C11(x{circumflex over ()}4) 0.4772526614248559E4 C12(x{circumflex over ()}3*y{circumflex over ()}1) 0.1044451265574271E5 C13(x{circumflex over ()}2*y{circumflex over ()}2) 0.122787191899436E4 C14(x{circumflex over ()}1*y{circumflex over ()}3) 0.1600166184381832E5 C15(y{circumflex over ()}4) 0.7664122368760205E5 C16(x{circumflex over ()}5) 0.1834252350968806E6 C17(x{circumflex over ()}4*y) 0.3122096431780654E5 C18(x{circumflex over ()}3*y{circumflex over ()}2) 0.2768680465570283E6 C19(x{circumflex over ()}2*y{circumflex over ()}3) 0.1321919572017405E5 C20(x*y{circumflex over ()}4) 0.147599246363685E6 C21(y{circumflex over ()}5) 0.8156383807089381E7 C22(x{circumflex over ()}6) 0.2611234595310471E5 C23(x{circumflex over ()}5*y) 0.2970252998342986E7 C24(x{circumflex over ()}4*y{circumflex over ()}2) 0.6158655020171089E6 C25(x{circumflex over ()}3*y{circumflex over ()}3) 0.7105247483583933E8 C26(x{circumflex over ()}2*y{circumflex over ()}4) 0.2454274046766187E8 C27(x*y{circumflex over ()}5) 0.9744965415921332E8 C28(y{circumflex over ()}6) 0.1124077125995596E7 C29(x{circumflex over ()}7) 0.1163511506909664E7 C30(x{circumflex over ()}6*y) 0.3028808052213724E6 C31(x{circumflex over ()}5*y{circumflex over ()}2) 0.1198316702193867E7 C32(x{circumflex over ()}4*y{circumflex over ()}3) 0.193432647259742E7 C33(x{circumflex over ()}3*y{circumflex over ()}4) 0.2456697279766292E8 C34(x{circumflex over ()}2*y{circumflex over ()}5) 0.323741057368371E7 C35(x{circumflex over ()}1*y{circumflex over ()}6) 0.3355065242253185E9 C36(y{circumflex over ()}7) 0.1488724197468286E8 C37(x{circumflex over ()}8) 0.7147963609945736E7 C38(x{circumflex over ()}7*y{circumflex over ()}1) 0.3441028882735257E8 C39(x{circumflex over ()}6*y{circumflex over ()}2) 0.1636702703538912E7 C40(x{circumflex over ()}5*y{circumflex over ()}3) 0.1892078012001264E8 C41(x{circumflex over ()}4*y{circumflex over ()}4) 0.8673389748241515E8 C42(x{circumflex over ()}3*y{circumflex over ()}5) 0.9327542722958507E10 C43(x{circumflex over ()}2*y{circumflex over ()}6) 0.1808477328812757E9 C44(x{circumflex over ()}1*y{circumflex over ()}7) 0.1230400754690463E9 C45(y{circumflex over ()}8) 0.7695547891324175E9 C46(x{circumflex over ()}9) 0.2538572458767442E9 C47(x{circumflex over ()}8*y) 0.5596033598779898E8 C48(x{circumflex over ()}7*y{circumflex over ()}2) 0.8299884465237318E10 C49(x{circumflex over ()}6*y{circumflex over ()}3) 0.1408407485040278E8 C50(x{circumflex over ()}5*y{circumflex over ()}4) 0.1026572981410108E9 C51(x{circumflex over ()}4*y{circumflex over ()}5) 0.2842618728614499E9 C52(x{circumflex over ()}3*y{circumflex over ()}6) 0.9551906470507542E10 C53(x{circumflex over ()}2*y{circumflex over ()}7) 0.4050425393925585E9 C54(x*y{circumflex over ()}8) 0.3804279681288634E10 C55(y{circumflex over ()}9) 0.3697621660603705E10 C56(x{circumflex over ()}10) 0.8177789522569431E9 C57(x{circumflex over ()}9*y) 0.812957348690838E10 C58(x{circumflex over ()}8*y{circumflex over ()}2) 0.3084476996537277E9 C59(x{circumflex over ()}7*y{circumflex over ()}3) 0.5631426025373504E10 C60(x{circumflex over ()}6*y{circumflex over ()}4) 0.5305140315017303E10 C61(x{circumflex over ()}5*y{circumflex over ()}5) 0.5476471825119294E11 C62(x{circumflex over ()}4*y{circumflex over ()}6) 0.8166445259833216E10 C63(x{circumflex over ()}3*y{circumflex over ()}7) 0.6976224960109545E11 C64(x{circumflex over ()}2*y{circumflex over ()}8) 0.3150044437407047E10 C65(x{circumflex over ()}9*y) 0.5285680302196486E11 C66(y{circumflex over ()}10) 0.199790881969388E11

(86) FIG. 2 is a spot diagram of the head-mounted display optical system LS1 according to Example 1. The length of the line indicated at the lower part of the spot diagram corresponds to 0.6 mm on the drawing surface. The ordinate of the spot diagram indicates the field position, and the abscissa of the spot diagram indicates the defocus amount. The description on the spot diagram is the same for other examples herein below, where description thereof is omitted. As FIG. 2 shows, according to Example 1, chromatic aberration is corrected satisfactorily, and excellent image forming performance is demonstrated.

Example 2

(87) Example 2 of the present invention will be described next with reference to FIG. 3, FIG. 4 and Table 2. FIG. 3 is an optical path diagram of a head-mounted display optical system LS (LS2) according to Example 2. The head-mounted display optical system LS2 according to Example 2 has, in order from a light source, an optical deflection element M1, a first lens group G1 having positive refractive power, a second lens group G2 having negative refractive power near the reference axis, and an optical reflection element M2. The optical deflection element M1 changes the traveling direction of a light from the light source (e.g. approximately parallel light, such as laser light and LED light). In FIG. 3, the state of change of the traveling direction of light in the optical deflection element M1 is indicated by the light that transmits through a diaphragm (this diaphragm is indicated as the optical deflection element M1) at a plurality of types of incident angles, to simplify description.

(88) The first lens group G1 collects light entered from the light source via the optical deflection element M1. The first lens group G1 includes, in order from the light source, a negative lens L11, a first free-shaped surface lens L12 having positive refractive power, and a second free-shaped surface lens L13 having positive refractive power. The negative lens L11 is formed in a meniscus shape having a convex surface facing the light source. An aspherical surface, which is rotationally symmetrical with respect to the reference axis, is formed on both lens surfaces of the negative lens L11. The first free-shaped surface lens L12 is formed in a meniscus shape having a concave surface facing the light source. An aspherical surface, which is rotationally symmetrical with respect to the reference axis, is formed on the light source side lens surface of the first free-shaped surface lens L12, and a free-shaped surface, which is rotationally asymmetrical with respect to the reference axis, is formed on the drawing surface I side lens surface of the first free-shaped surface lens L12. The second free-shaped surface lens L13 is formed in a meniscus shape having a convex surface facing the light source. A diffraction optical surface on an aspherical surface, which is rotationally symmetrical with respect to the reference axis, is formed on the light source side lens surface of the second free-shaped surface lens L13, and a free-shaped surface, which is rotationally asymmetrical with respect to the reference axis, is formed on the drawing surface I side lens surface of the second free-shaped surface lens L13.

(89) The second lens group G2 is disposed near an intermediate image forming position (collecting position) by the first lens group G1. The second lens group G2 includes, in order from the light source, a diffusion lens L21, and a third free-shaped surface lens L22 having positive refractive power. The diffusion lens L21 is formed in a meniscus shape having a convex surface facing the light source. A diffuse transmission surface is formed on the drawing surface I side lens surface of the diffusion lens L21. The third free-shaped surface lens L22 is formed in a meniscus shape having a convex surface facing the light source. A free-shaped surface, which is rotationally asymmetrical with respect to the reference axis, is formed on the light source side lens surface of the third free-shaped surface lens L22, and an aspherical surface, which is rotationally symmetrical with respect to the reference axis, is formed on the drawing surface I side lens surface of the third free-shaped surface lens L22.

(90) The optical reflection element M2 has a reflection surface M2r that reflects light transmitted through the second lens group G2. The optical reflection element M2 is also configured to transmit the light entered from the surface M2s on the opposite side of the reflection surface M2r. The reflection surface M2r of the optical reflection element M2 is formed to be rotationally asymmetrical with respect to the reference axis.

(91) This head-mounted display optical system LS is configured such that light from the light source, which is reflected on the reflection surface M2r of the optical reflection element M2 and reaches a surface assumed to be located on a retina when the user wears the head-mounted display (drawing surface I), moves on the drawing surface I in a manner of two-dimensionally scanning in accordance with the change of a travelling direction of the light caused by the optical deflection element M1, and an image is drawn on the drawing surface I. In this case, the image that is drawn is superposed on an image formed by the light that is transmitted through the optical reflection element M2 and reached the drawing surface I (retina). In Example 2, it is assumed that as a user's eye an aplanatic lens (not illustrated), of which focal length f=17 [mm], is disposed between the optical reflection element M2 and the drawing surface I. Thereby the light reflected on the reflection surface Mgr of the optical reflection element M2 transmits through the aplanatic lens (not illustrated) and is collected on the drawing surface I (retina).

(92) Table 2 shows each data value of Example 2. The radius of curvature R of surface 1 to surface 14 in Table 2 correspond to symbols R1 to R14 attached to surface 1 to surface 14 in FIG. 3 (excluding surface 10 and surface 13, which are virtual surfaces).

(93) TABLE-US-00002 TABLE 2 [Lens Data] N R D nC nd nF ng Object surface 1 (aperture 7.0831 stop) 2*a 26.1651 3.5000 1.6074 1.6142 1.6314 1.6463 3*a 6.0315 1.2000 4*a 143.3443 3.2500 1.5283 1.53113 1.53783 1.54319 5*b 6.6594 1.1000 6*a*c 6.3022 2.5000 1.5283 1.53113 1.53783 1.54319 7*b 7.5704 20.0000 8 36.0000 1.0000 1.4883 1.4908 1.4969 1.5016 9 36.0000 1.0000 10 0.0000 11*b 98.6116 1.8000 1.4883 1.4908 1.4969 1.5016 12*a 75.4656 0.0000 13 32.0547 14*b 66.0000 38.0560 (reflection) Image plane 0.0000 (drawing surface) [Eccentricity Data] Surface Surface Surface Surface Surface Eccentricity 8 10 11 13 14 XDE 0.0 0.0 0.0 0.0 0.0 YDE 0.0 0.0 0.0 0.0 0.0 ZDE 0.0 0.0 0.0 0.0 0.0 ADE 6.0 6.0 6.0 6.0 25.0 BDE 0.0 0.0 0.0 0.0 0.0 CDE 0.0 0.0 0.0 0.0 0.0 [Aspherical Data] Surface 2 coefficient k = 124.6212244263259 A4 = 0.001204200199273652 A6 = 0.0001386036941931049 A8 = 0.3276470962239764E5 A10 = 0.8461937659182494E7 A12 = 0.456637483894513E8 Surface 3 coefficient k = 3.910512067211907 A4 = 0.0006946049400910621 A6 = 0.7915311306384153E6 A8 = 0.563745700159402E7 A10 = 0.1090170748278784E8 A12 = 0.3938449383844765E10 Surface 4 coefficient k = 500.0 A4 = 0.0001884494623314058 A6 = 0.6684559289524473E6 A8 = 0.1661476652611027E6 A10 = 0.3827111832796201E8 A12 = 0.1635609710619695E9 Surface 6 coefficient k = 4.1416842368152 A4 = 0.0002575315743116521 A6 = 0.1246600508478533E4 A8 = 0.1934338103548995E6 A10 = 0.1853239196823036E8 A12 = 0.2025187349428312E11 Surface 12 coefficient k = 0.00001 A4 = 0.2839336271869116E4 A6 = 0.3408877404436407E8 A8 = 0.3706976730325451E10 A10 = 0.5323123873693355E12 A12 = 0.1644935738227222E13 [Free-shaped surface data 1] Term Surface 5 coefficient Surface 7 coefficient C1(k) 0.3044117239169427.sup. 6.210179649757263.sup. C4(x{circumflex over ()}2) 0.003713914513726664.sup. 0.001186163413066797.sup. C5(x*y) 0.001085562025148004.sup. .sup.0.0003514837217863681 C6(y{circumflex over ()}2) 0.007062231361900973.sup. 0.01343327779501104.sup. C7(x{circumflex over ()}3) 0.0002452754022695958.sup. .sup.0.0003949695507289595 C8(x{circumflex over ()}2*y) 0.0001651542812559088.sup. 0.001361146290953931.sup. C9(x*y{circumflex over ()}2) 0.0006135423357087519.sup. .sup.0.0006365921598598023 C10(y{circumflex over ()}3) 0.0004768304415626346.sup. .sup.0.0006970794286006876 C11(x{circumflex over ()}4) 0.0003037015705534265.sup. 0.001334625516756709.sup. C12(x{circumflex over ()}3*y{circumflex over ()}1) 0.3809856337282187E4 0.3504604790870343E4 C13(x{circumflex over ()}2*y{circumflex over ()}2) 0.0006140546431927162.sup. 0.0001453643454395092.sup. C14(x{circumflex over ()}1*y{circumflex over ()}3) 0.0001752632160152545.sup. 0.8191419681687872E4 C15(y{circumflex over ()}4) 0.0003727404398809901.sup. 0.1736454723033816E5 C16(x{circumflex over ()}5) 0.2719101080255722E4 0.3616353217176764E4 C17(x{circumflex over ()}4*y) 0.0001422956955870682.sup. .sup.0.0002886431884463202 C18(x{circumflex over ()}3*y{circumflex over ()}2) 0.3961105934359095E4 0.5288355376499949E4 C19(x{circumflex over ()}2*y{circumflex over ()}3) 0.8362038478789313E5 0.107536802370358E4 C20(x*y{circumflex over ()}4) 0.8595236888967398E5 0.3331184374513139E5 C21(y{circumflex over ()}5) 0.9120120637233973E5 0.1575072061576212E4 C22(x{circumflex over ()}6) 0.2521499675319464E4 .sup.0.0001047508559885795 C23(x{circumflex over ()}5*y) 0.9054552785237278E5 0.4795226503605732E5 C24(x{circumflex over ()}4*y{circumflex over ()}2) 0.4903665793867147E4 0.4201199000129014E5 C25(x{circumflex over ()}3*y{circumflex over ()}3) 0.2946360989471093E5 0.1306822845519472E6 C26(x{circumflex over ()}2*y{circumflex over ()}4) 0.242896687192317E4 0.8600226881862653E5 C27(x*y{circumflex over ()}5) 0.1794871126104695E5 0.720051619244834E6 C28(y{circumflex over ()}6) 0.1141762308719592E4 0.3955087197324593E5 C29(x{circumflex over ()}7) 0.3162220502466017E5 0.1079263896995276E5 C30(x{circumflex over ()}6*y) 0.1549116632015361E4 0.2064262131606276E4 C31(x{circumflex over ()}5*y{circumflex over ()}2) 0.6590678631915092E7 0.2530151828630695E5 C32(x{circumflex over ()}4*y{circumflex over ()}3) 0.1233082427792277E6 0.102244563477985E5 C33(x{circumflex over ()}3*y{circumflex over ()}4) 0.3152984712809762E6 0.6674948401506242E6 C34(x{circumflex over ()}2*y{circumflex over ()}5) 0.2104799942359669E5 0.7958365434359981E6 C35(x{circumflex over ()}1*y{circumflex over ()}6) 0.4928652007151956E6 0.139646220895246E6 C36(y{circumflex over ()}7) 0.733565522214735E6 0.1479076315725485E6 C37(x{circumflex over ()}8) 0.2310420623929306E6 0.437812709225447E5 C38(x{circumflex over ()}7*y{circumflex over ()}1) 0.4766461803638555E7 0.976503668600782E8 C39(x{circumflex over ()}6*y{circumflex over ()}2) 0.3070697697772037E5 0.6778004794915314E6 C40(x{circumflex over ()}5*y{circumflex over ()}3) 0.8736872537515536E6 0.9631003457531317E7 C41(x{circumflex over ()}4*y{circumflex over ()}4) 0.1112116510003865E5 0.1960482481917686E6 C42(x{circumflex over ()}3*y{circumflex over ()}5) 0.5315256549809658E8 0.3637806089075426E7 C43(x{circumflex over ()}2*y{circumflex over ()}6) 0.7520623414915488E6 0.7870088578336839E7 C44(x{circumflex over ()}1*y{circumflex over ()}7) 0.9210963166924476E7 0.1475987111596374E7 C45(y{circumflex over ()}8) 0.1046575213885323E6 0.4549120521731493E8 C46(x{circumflex over ()}9) 0.1984152022994444E6 0.7416742079181322E7 C47(x{circumflex over ()}8*y) 0.8128257292528562E6 0.4262633477758227E6 C48(x{circumflex over ()}7*y{circumflex over ()}2) 0.103301225933334E6 0.447235130671805E7 C49(x{circumflex over ()}6*y{circumflex over ()}3) 0.6360980963602419E6 0.9194039388537242E7 C50(x{circumflex over ()}5*y{circumflex over ()}4) 0.3748419276756708E7 0.1648682975516311E7 C51(x{circumflex over ()}4*y{circumflex over ()}5) 0.1183038351997398E6 0.2294812151740892E7 C52(x{circumflex over ()}3*y{circumflex over ()}6) 0.3907466626203889E7 0.1782889075113961E9 C53(x{circumflex over ()}2*y{circumflex over ()}7) 0.5355253376931649E7 0.1244118993844718E7 C54(x*y{circumflex over ()}8) 0.2336694636779812E9 0.1663627186711565E8 C55(y{circumflex over ()}9) 0.1393863018537273E7 0.9976994048176901E10 C56(x{circumflex over ()}10) 0.6831302926161432E7 0.9762883954596809E7 C57(x{circumflex over ()}9*y) 0.2258172909036116E7 0.3306134796185684E8 C58(x{circumflex over ()}8*y{circumflex over ()}2) 0.8968895216793801E7 0.4939235647611712E9 C59(x{circumflex over ()}7*y{circumflex over ()}3) 0.1441111681226747E7 0.5877766782239647E10 C60(x{circumflex over ()}6*y{circumflex over ()}4) 0.5511514414292549E7 0.1181052120051939E8 C61(x{circumflex over ()}5*y{circumflex over ()}5) 0.253556208676787E7 0.9139343620831635E9 C62(x{circumflex over ()}4*y{circumflex over ()}6) 0.4052925742840984E7 0.4377380782240374E8 C63(x{circumflex over ()}3*y{circumflex over ()}7) 0.5625299355916163E9 0.8988048665333446E9 C64(x{circumflex over ()}2*y{circumflex over ()}8) 0.1197441617119414E7 0.2051855291522901E8 C65(x{circumflex over ()}9*y) 0.185826615354817E8 0.1696631403403916E9 C66(y{circumflex over ()}10) 0.2038458080555793E8 0.6445655479796271E10 [Free-shaped surface data 2] Term Surface 11 coefficient C1(k) 79.46376069014747.sup. C4(x{circumflex over ()}2) 0.008781950775443515.sup. C5(x*y) .sup.0.0009308419560987458 C6(y{circumflex over ()}2) 0.01113704140423572.sup. C7(x{circumflex over ()}3) 0.1760347652094596E4 C8(x{circumflex over ()}2*y) .sup.0.0001460755227785545 C9(x*y{circumflex over ()}2) 0.3146182180201695E4 C10(y{circumflex over ()}3) 0.0002338112657631998.sup. C11(x{circumflex over ()}4) .sup.0.0001288763886808186 C12(x{circumflex over ()}3*y{circumflex over ()}1) 0.5588537747640256E5 C13(x{circumflex over ()}2*y{circumflex over ()}2) 0.7154476682496682E4 C14(x{circumflex over ()}1*y{circumflex over ()}3) 0.2233413996828774E5 C15(y{circumflex over ()}4) 0.2747947174614958E4 C16(x{circumflex over ()}5) 0.915849258267991E6 C17(x{circumflex over ()}4*y) 0.6148176140156556E5 C18(x{circumflex over ()}3*y{circumflex over ()}2) 0.1375538274399082E5 C19(x{circumflex over ()}2*y{circumflex over ()}3) 0.6233560560960316E5 C20(x*y{circumflex over ()}4) 0.1840270598725556E6 C21(y{circumflex over ()}5) 0.1237031224448769E5 C22(x{circumflex over ()}6) 0.5768411537875053E6 C23(x{circumflex over ()}5*y) 0.8034799634922124E6 C24(x{circumflex over ()}4*y{circumflex over ()}2) 0.4293992492596523E6 C25(x{circumflex over ()}3*y{circumflex over ()}3) 0.245074616484058E6 C26(x{circumflex over ()}2*y{circumflex over ()}4) 0.1089178317786131E6 C27(x*y{circumflex over ()}5) 0.2377423853387786E6 C28(y{circumflex over ()}6) 0.1294972263368165E6 C29(x{circumflex over ()}7) 0.6607708701685202E11 C30(x{circumflex over ()}6*y) 0.3189630748927253E7 C31(x{circumflex over ()}5*y{circumflex over ()}2) 0.2065713245155823E6 C32(x{circumflex over ()}4*y{circumflex over ()}3) 0.5940891834264707E7 C33(x{circumflex over ()}3*y{circumflex over ()}4) 0.8240175928329288E7 C34(x{circumflex over ()}2*y{circumflex over ()}5) 0.599382689524388E7 C35(x{circumflex over ()}1*y{circumflex over ()}6) 0.1943252964568572E7 C36(y{circumflex over ()}7) 0.3771743748042609E7 C37(x{circumflex over ()}8) 0.482034159810021E7 C38(x{circumflex over ()}7*y{circumflex over ()}1) 0.3199313441251986E7 C39(x{circumflex over ()}6*y{circumflex over ()}2) 0.8520322345961256E7 C40(x{circumflex over ()}5*y{circumflex over ()}3) 0.2216481770618415E7 C41(x{circumflex over ()}4*y{circumflex over ()}4) 0.7848660775257326E7 C42(x{circumflex over ()}3*y{circumflex over ()}5) 0.1306460670991727E8 C43(x{circumflex over ()}2*y{circumflex over ()}6) 0.6436629955747501E9 C44(x{circumflex over ()}1*y{circumflex over ()}7) 0.192000600061975E8 C45(y{circumflex over ()}8) 0.2171472696133448E9 [Free-shaped surface data 3] Term Surface 14 coefficient C1(k) 4.331113639069206.sup. C4(x{circumflex over ()}2) 0.0009066565006826442.sup. C5(x*y) 0.9422610759795473E5 C6(y{circumflex over ()}2) 0.00142035683124462.sup. C7(x{circumflex over ()}3) 0.5006378989079557E5 C8(x{circumflex over ()}2*y) 0.1279911190810272E4 C9(x*y{circumflex over ()}2) 0.1483714442744794E4 C10(y{circumflex over ()}3) 0.2263724302742891E4 C11(x{circumflex over ()}4) 0.4885153890283033E4 C12(x{circumflex over ()}3*y{circumflex over ()}1) 0.8907390616195318E6 C13(x{circumflex over ()}2*y{circumflex over ()}2) 0.1306776632445698E4 C14(x{circumflex over ()}1*y{circumflex over ()}3) 0.1678420737545771E5 C15(y{circumflex over ()}4) 0.8026685702591208E5 C16(x{circumflex over ()}5) 0.2442034237925676E6 C17(x{circumflex over ()}4*y) 0.3108842124348334E5 C18(x{circumflex over ()}3*y{circumflex over ()}2) 0.2726854812902458E6 C19(x{circumflex over ()}2*y{circumflex over ()}3) 0.1255872948855557E5 C20(x*y{circumflex over ()}4) 0.1493676576993912E6 C21(y{circumflex over ()}5) 0.1058325108957716E6 C22(x{circumflex over ()}6) 0.260984451952067E5 C23(x{circumflex over ()}5*y) 0.3725459143092297E7 C24(x{circumflex over ()}4*y{circumflex over ()}2) 0.6144438515964853E6 C25(x{circumflex over ()}3*y{circumflex over ()}3) 0.5427620822882202E8 C26(x{circumflex over ()}2*y{circumflex over ()}4) 0.2992722123534214E9 C27(x*y{circumflex over ()}5) 0.1088633717848592E7 C28(y{circumflex over ()}6) 0.1242491189436026E7 C29(x{circumflex over ()}7) 0.1095787164100878E7 C30(x{circumflex over ()}6*y) 0.2968894475677224E6 C31(x{circumflex over ()}5*y{circumflex over ()}2) 0.1194535303878235E7 C32(x{circumflex over ()}4*y{circumflex over ()}3) 0.1960863391147454E7 C33(x{circumflex over ()}3*y{circumflex over ()}4) 0.1157330225815343E8 C34(x{circumflex over ()}2*y{circumflex over ()}5) 0.3349277461157255E7 C35(x{circumflex over ()}1*y{circumflex over ()}6) 0.1107075191076766E9 C36(y{circumflex over ()}7) 0.2033353596333252E8 C37(x{circumflex over ()}8) 0.7057980834653433E7 C38(x{circumflex over ()}7*y{circumflex over ()}1) 0.3663006846702143E8 C39(x{circumflex over ()}6*y{circumflex over ()}2) 0.1601758454201476E7 C40(x{circumflex over ()}5*y{circumflex over ()}3) 0.1825926824770061E8 C41(x{circumflex over ()}4*y{circumflex over ()}4) 0.8690484051603E8 C42(x{circumflex over ()}3*y{circumflex over ()}5) 0.4154060781671823E10 C43(x{circumflex over ()}2*y{circumflex over ()}6) 0.1038979444268852E9 C44(x{circumflex over ()}1*y{circumflex over ()}7) 0.1652454713668999E9 C45(y{circumflex over ()}8) 0.7724559355566372E9 C46(x{circumflex over ()}9) 0.154408908057191E9 C47(x{circumflex over ()}8*y) 0.5699847320152124E8 C48(x{circumflex over ()}7*y{circumflex over ()}2) 0.4332746877460881E10 C49(x{circumflex over ()}6*y{circumflex over ()}3) 0.1277325470506736E8 C50(x{circumflex over ()}5*y{circumflex over ()}4) 0.1759993562231055E9 C51(x{circumflex over ()}4*y{circumflex over ()}5) 0.209641652630629E9 C52(x{circumflex over ()}3*y{circumflex over ()}6) 0.831181352914488E10 C53(x{circumflex over ()}2*y{circumflex over ()}7) 0.3870336258447297E9 C54(x*y{circumflex over ()}8) 0.3503243314912144E10 C55(y{circumflex over ()}9) 0.3238934302039904E10 C56(x{circumflex over ()}10) 0.8249139063163544E9 C57(x{circumflex over ()}9*y) 0.9829216038217835E10 C58(x{circumflex over ()}8*y{circumflex over ()}2) 0.2927583539831702E9 C59(x{circumflex over ()}7*y{circumflex over ()}3) 0.5842996838838534E10 C60(x{circumflex over ()}6*y{circumflex over ()}4) 0.5512513899861672E10 C61(x{circumflex over ()}5*y{circumflex over ()}5) 0.8905989254150218E11 C62(x{circumflex over ()}4*y{circumflex over ()}6) 0.8613607998148935E10 C63(x{circumflex over ()}3*y{circumflex over ()}7) 0.6496157888087305E11 C64(x{circumflex over ()}2*y{circumflex over ()}8) 0.3117619545237941E10 C65(x{circumflex over ()}9*y) 0.4381187587446189E11 C66(y{circumflex over ()}10) 0.2861260928939042E11 [Diffraction optical surface data] Term Surface 6 coefficient C3(x{circumflex over ()}2) 0.5078981020003178E4 C4(x*y) 0.1074272792154444E4 C5(y{circumflex over ()}2) 0.0001013950360598153.sup. C6(x{circumflex over ()}3) 0.7393093289878994E5 C7(x{circumflex over ()}2*y) 0.1082909716426623E4 C8(x*y{circumflex over ()}2) 0.1687515783535323E5 C9(y{circumflex over ()}3) 0.508045964678955E5 C10(x{circumflex over ()}4) 0.6002724002574973E5 C11(x{circumflex over ()}3*y{circumflex over ()}1) 0.1011785033518255E6 C12(x{circumflex over ()}2*y{circumflex over ()}2) 0.8967362674962745E5 C13(x{circumflex over ()}1*y{circumflex over ()}3) 0.7150890228122288E6 C14(y{circumflex over ()}4) 0.1884605645421943E5 C15(x{circumflex over ()}5) 0.7496012489836572E7 C16(x{circumflex over ()}4*y) 0.4478856492315672E6 C17(x{circumflex over ()}3*y{circumflex over ()}2) 0.6108732769621659E7 C18(x{circumflex over ()}2*y{circumflex over ()}3) 0.203867515925772E7 C19(x*y{circumflex over ()}4) 0.2137699921080692E7 C20(y{circumflex over ()}5) 0.9636626970058547E8 C21(x{circumflex over ()}6) 0.7390577981553802E7 C22(x{circumflex over ()}5*y) 0.1652236605672853E7 C23(x{circumflex over ()}4*y{circumflex over ()}2) 0.182216819275834E6 C24(x{circumflex over ()}3*y{circumflex over ()}3) 0.1350514661675573E7 C25(x{circumflex over ()}2*y{circumflex over ()}4) 0.1528045758514017E6 C26(x*y{circumflex over ()}5) 0.4953482058318769E8 C27(y{circumflex over ()}6) 0.1477315812746981E7 C28(x{circumflex over ()}7) 0.1967457563220266E7 C29(x{circumflex over ()}6*y) 0.536866450738596E7 C30(x{circumflex over ()}5*y{circumflex over ()}2) 0.2462938966361094E8 C31(x{circumflex over ()}4*y{circumflex over ()}3) 0.3064619953760897E7 C32(x{circumflex over ()}3*y{circumflex over ()}4) 0.3688442580226221E8 C33(x{circumflex over ()}2*y{circumflex over ()}5) 0.9203883957108073E9 C34(x{circumflex over ()}1*y{circumflex over ()}6) 0.1973056856945383E9 C35(y{circumflex over ()}7) 0.3414723247599195E9 C36(x{circumflex over ()}8) 0.7192992299042694E8 C37(x{circumflex over ()}7*y{circumflex over ()}1) 0.8111634641150032E9 C38(x{circumflex over ()}6*y{circumflex over ()}2) 0.703390767589344E8 C39(x{circumflex over ()}5*y{circumflex over ()}3) 0.5099463925355792E10 C40(x{circumflex over ()}4*y{circumflex over ()}4) 0.2606463011316836E9 C41(x{circumflex over ()}3*y{circumflex over ()}5) 0.4485956930629984E9 C42(x{circumflex over ()}2*y{circumflex over ()}6) 0.3351715768272032E9 C43(x{circumflex over ()}1*y{circumflex over ()}7) 0.7088549439987446E10 C44(y{circumflex over ()}8) 0.7474213547085642E10 C45(x{circumflex over ()}9) 0.7056671272072184E9 C46(x{circumflex over ()}8*y) 0.4176104169126524E8 C47(x{circumflex over ()}7*y{circumflex over ()}2) 0.6901507491686979E9 C48(x{circumflex over ()}6*y{circumflex over ()}3) 0.1707244701705906E8 C49(x{circumflex over ()}5*y{circumflex over ()}4) 0.1043681567188657E9 C50(x{circumflex over ()}4*y{circumflex over ()}5) 0.3868465931158265E9 C51(x{circumflex over ()}3*y{circumflex over ()}6) 0.1769864426701496E9 C52(x{circumflex over ()}2*y{circumflex over ()}7) 0.2904749839197674E9 C53(x*y{circumflex over ()}8) 0.5167553024223719E10 C54(y{circumflex over ()}9) 0.8669348988349122E11 C55(x{circumflex over ()}10) 0.3948617721101432E9 C56(x{circumflex over ()}9*y) 0.4701321313712419E10 C57(x{circumflex over ()}8*y{circumflex over ()}2) 0.5980309265501623E10 C58(x{circumflex over ()}7*y{circumflex over ()}3) 0.7984092689910972E10 C59(x{circumflex over ()}6*y{circumflex over ()}4) 0.2942608881005636E9 C60(x{circumflex over ()}5*y{circumflex over ()}5) 0.171946598084867E10 C61(x{circumflex over ()}4*y{circumflex over ()}6) 0.2126547520189481E9 C62(x{circumflex over ()}3*y{circumflex over ()}7) 0.3294818039868844E10 C63(x{circumflex over ()}2*y{circumflex over ()}8) 0.7937250427695667E10 C64(x{circumflex over ()}9*y) 0.9445603210003383E11 C65(y{circumflex over ()}10) 0.7447027309736033E11

(94) FIG. 4 is a spot diagram of the head-mounted display optical system LS2 according to Example 2. As FIG. 4 shows, according to Example 2, chromatic aberration is corrected satisfactorily, and excellent image forming performance is demonstrated.

Example 3

(95) Example 3 of the present invention will be described next with reference to FIG. 5, FIG. 6 and Table 3. FIG. 5 is an optical path diagram of a head-mounted display optical system LS (LS3) according to Example 3. The head-mounted display optical system LS3 according to Example 3 has, in order from a light source, an optical deflection element M1, a first lens group G1 having positive refractive power, a second lens group G2 having negative refractive power near the reference axis, and an optical reflection element M2. The optical deflection element M1 changes the traveling direction of a light from the light source (e.g. approximately parallel light, such as laser light and LED light). In FIG. 5, the state of change of the traveling direction of light in the optical deflection element M1 is indicated by the light that transmits through a diaphragm (this diaphragm is indicated as the optical deflection element M1) at a plurality of types of incident angles, to simplify description.

(96) The first lens group G1 collects light entered from the light source via the optical deflection element M1. The first lens group G1 includes, in order from the light source, a negative lens L11, a first free-shaped surface lens L12 having positive refractive power, and a second free-shaped surface lens L13 having positive refractive power. The negative lens L11 is formed in a biconcave. An aspherical surface, which is rotationally symmetrical with respect to the reference axis, is formed on both lens surfaces of the negative lens L11. The first free-shaped surface lens L12 is formed in a biconvex shape. An aspherical surface, which is rotationally symmetrical with respect to the reference axis, and a diffraction optical surface are formed on the light source side lens surface of the first free-shaped surface lens L12, and a free-shaped surface, which is rotationally asymmetrical with respect to the reference axis, is formed on the drawing surface I side lens surface of the first free-shaped surface lens L12. The second free-shaped surface lens L13 is formed in a meniscus shape having a convex surface facing the light source. An aspherical surface, which is rotationally symmetrical with respect to the reference axis, is formed on the light source side lens surface of the second free-shaped surface lens L13, and a free-shaped surface, which is rotationally asymmetrical with respect to the reference axis, is formed on the drawing surface I side lens surface of the second free-shaped surface lens L13.

(97) The second lens group G2 is disposed near an intermediate image forming position (collecting position) by the first lens group G1. The second lens group G2 includes a diffusion lens L21 having negative refractive power. The diffusion lens L21 is formed in a meniscus shape having a convex surface facing the light source. A diffuse transmission surface is formed on the light source side lens surface of the diffusion lens L21.

(98) The optical reflection element M2 has a reflection surface M2r that reflects light transmitted through the second lens group G2. The optical reflection element M2 is also configured to transmit the light entered from the surface M2s on the opposite side of the reflection surface M2r. The reflection surface M2r of the optical reflection element M2 is formed to be rotationally asymmetrical with respect to the reference axis.

(99) This head-mounted display optical system LS is configured such that light from the light source, which is reflected on the reflection surface M2r of the optical reflection element M2 and reaches a surface assumed to be located on a retina when the user wears the head-mounted display (drawing surface I), moves on the drawing surface I in a manner of two-dimensionally scanning in accordance with the change of a travelling direction of the light caused by the optical deflection element M1, and an image is drawn on the drawing surface I. In this case, the image that is drawn is superposed on an image formed by the light that is transmitted through the optical reflection element M2 and reached the drawing surface I (retina). In Example 3, it is assumed that as a user's eye an aplanatic lens (not illustrated), of which focal length f=17 [mm], is disposed between the optical reflection element M2 and the drawing surface I. Thereby the light reflected on the reflection surface Mgr of the optical reflection element M2 transmits through the aplanatic lens (not illustrated) and is collected on the drawing surface I (retina).

(100) Table 3 shows each data value of Example 3. The radius of curvature R of surface 1 to surface 12 in Table 3 correspond to symbols R1 to R12 attached to surface 1 to surface 12 in FIG. 5.

(101) TABLE-US-00003 TABLE 3 [Lens Data] N R D nC nd nF ng Object surface 1 (aperture 8.7664 stop) 2*a 55.2558 1.5000 1.6074 1.6142 1.6314 1.6463 3*a 7.4777 0.5000 4*a 79.0243 0.0500 1.5538 1.5571 1.565 1.5713 5*a*c 79.0243 0.0500 1.5233 1.5278 1.5391 1.5491 6*a 79.0243 3.5620 1.5283 1.53113 1.53783 1.54319 7*b 6.5189 1.5117 8*a 3.5334 2.9263 1.5283 1.53113 1.53783 1.54319 9*b 3.0785 15.0000 10 200.0000 1.0000 1.5283 1.53113 1.53783 1.54319 11 197.0000 31.5000 12*b 66.0000 45.0000 (reflection) Image plane 0.0000 (drawing surface) [Eccentricity Data] Eccentricity Surface 12 XDE 0.0 YDE 0.0 ZDE 0.0 ADE 25.0 BDE 0.0 CDE 0.0 [Aspherical Data] Surface 2 coefficient k = 61.53223377503105 A4 = 0.001117233798974202 A6 = 0.0001507028920918782 A8 = 0.2126725352871772E5 A10 = 0.91093073607483E7 Surface 3 coefficient k = 6.67720051575926 A4 = 0.0007382034101680365 A6 = 0.1602347044190949E6 A8 = 0.1185673617018457E6 A10 = 0.9483890402222131E9 Surface 4 coefficient k = 175.71268632111 A4 = 0.000381528250257577 A6 = 0.361449829130996E5 A8 = 0.107558931083213E6 A10 = 0.542103130739954E8 Surface 5 coefficient k = 175.71268632111 A4 = 0.000381528250257577 A6 = 0.361449829130996E5 A8 = 0.107558931083213E6 A10 = 0.542103130739954E8 Surface 6 coefficient k = 175.7126863211099 A4 = 0.0003815282502575771 A6 = 0.3614498291309964E5 A8 = 0.107558931083213E6 A10 = 0.542103130739954E8 Surface 8 coefficient k = 3.816528789284875 A4 = 0.0002997589906375087 A6 = 0.1152321916657979E4 A8 = 0.1969329274360649E6 A10 = 0.2182880862990949E8 [Free-shaped surface data 1] Term Surface 7 coefficient C1(k) 0.3364628619636353.sup. C4(x{circumflex over ()}2) 0.001111389809964225.sup. C5(x*y) 0.000345633023972219.sup. C6(y{circumflex over ()}2) 0.004957826540583992.sup. C7(x{circumflex over ()}3) 0.8403950712470563E4 C8(x{circumflex over ()}2*y) 0.0006334087154299026.sup. C9(x*y{circumflex over ()}2) 0.155374355054229E4 C10(y{circumflex over ()}3) 0.0005245756359215623.sup. C11(x{circumflex over ()}4) 0.000372141838784073.sup. C12(x{circumflex over ()}3*y{circumflex over ()}1) 0.1864744397607962E4 C13(x{circumflex over ()}2*y{circumflex over ()}2) 0.0006595063497386374.sup. C14(x{circumflex over ()}1*y{circumflex over ()}3) 0.2069965720960925E5 C15(y{circumflex over ()}4) 0.000379609866359328.sup. C16(x{circumflex over ()}5) 0.3837475529663155E5 C17(x{circumflex over ()}4*y) 0.199909940626342E4 C18(x{circumflex over ()}3*y{circumflex over ()}2) 0.4632702136050437E4 C19(x{circumflex over ()}2*y{circumflex over ()}3) 0.0001344231633986514.sup. C20(x*y{circumflex over ()}4) 0.1076425114314026E5 C21(y{circumflex over ()}5) 0.3079658423187322E5 C22(x{circumflex over ()}6) 0.4824012296215599E6 C23(x{circumflex over ()}5*y) 0.6811033610542694E5 C24(x{circumflex over ()}4*y{circumflex over ()}2) 0.3487930519274964E4 C25(x{circumflex over ()}3*y{circumflex over ()}3) 0.778306442825551E6 C26(x{circumflex over ()}2*y{circumflex over ()}4) 0.2776544560849299E4 C27(x*y{circumflex over ()}5) 0.9497638333389676E6 C28(y{circumflex over ()}6) 0.6709065015909173E5 C29(x{circumflex over ()}7) 0.6358093464635089E7 C30(x{circumflex over ()}6*y) 0.2246824033003574E5 C31(x{circumflex over ()}5*y{circumflex over ()}2) 0.2184062792257402E5 C32(x{circumflex over ()}4*y{circumflex over ()}3) 0.3159897962061326E5 C33(x{circumflex over ()}3*y{circumflex over ()}4) 0.1529397930536397E5 C34(x{circumflex over ()}2*y{circumflex over ()}5) 0.6616515291670437E5 C35(x{circumflex over ()}1*y{circumflex over ()}6) 0.5053653164983818E7 C36(y{circumflex over ()}7) 0.5100339662302451E7 C37(x{circumflex over ()}8) 0.3908927771969818E6 C38(x{circumflex over ()}7*y{circumflex over ()}41) 0.2295527350587795E6 C39(x{circumflex over ()}6*y{circumflex over ()}2) 0.1300101306420439E5 C40(x{circumflex over ()}5*y{circumflex over ()}3) 0.413292876878701E7 C41(x{circumflex over ()}4*y{circumflex over ()}4) 0.1311082815847791E5 C42(x{circumflex over ()}3*y{circumflex over ()}5) 0.2816190918246684E8 C43(x{circumflex over ()}2*y{circumflex over ()}6) 0.1210284265050765E5 C44(x{circumflex over ()}1*y{circumflex over ()}7) 0.2591395975591261E7 C45(y{circumflex over ()}8) 0.1606482713875591E6 [Free-shaped surface data 2] Term Surface 9 coefficient C1(k) 6.77657355626851.sup. C4(x{circumflex over ()}2) 0.002128358710400415.sup. C5(x*y) .sup.0.0006211529504435977 C6(y{circumflex over ()}2) 0.0146857471219256.sup. C7(x{circumflex over ()}3) .sup.0.0001607450550941153 C8(x{circumflex over ()}2*y) 0.001049920702896136.sup. C9(x*y{circumflex over ()}2) 0.5339830460557105E4 C10(y{circumflex over ()}3) .sup.0.0003155842783092288 C11(x{circumflex over ()}4) 0.001471833409422146.sup. C12(x{circumflex over ()}3*y{circumflex over ()}1) .sup.0.0001080632917351919 C13(x{circumflex over ()}2*y{circumflex over ()}2) 0.0001421361868672432.sup. C14(x{circumflex over ()}1*y{circumflex over ()}3) 0.1161179724972522E4 C15(y{circumflex over ()}4) 0.4298830280866484E5 C16(x{circumflex over ()}5) 0.3925938275569645E5 C17(x{circumflex over ()}4*y) .sup.0.0003800481494676694 C18(x{circumflex over ()}3*y{circumflex over ()}2) 0.2954850672310948E4 C19(x{circumflex over ()}2*y{circumflex over ()}3) 0.3788310072687263E4 C20(x*y{circumflex over ()}4) 0.5997479309307839E5 C21(y{circumflex over ()}5) 0.7168907711651808E5 C22(x{circumflex over ()}6) .sup.0.0001225000133339481 C23(x{circumflex over ()}5*y) 0.2113946080473899E5 C24(x{circumflex over ()}4*y{circumflex over ()}2) 0.8148380209039854E5 C25(x{circumflex over ()}3*y{circumflex over ()}3) 0.1095105430202548E5 C26(x{circumflex over ()}2*y{circumflex over ()}4) 0.256688443064381E4 C27(x*y{circumflex over ()}5) 0.5505210595164379E7 C28(y{circumflex over ()}6) 0.339586840784051E5 C29(x{circumflex over ()}7) 0.2820684838114497E6 C30(x{circumflex over ()}6*y) 0.2009020233590986E4 C31(x{circumflex over ()}5*y{circumflex over ()}2) 0.1227092916159645E5 C32(x{circumflex over ()}4*y{circumflex over ()}3) 0.4568725315420399E5 C33(x{circumflex over ()}3*y{circumflex over ()}4) 0.6426244168301948E6 C34(x{circumflex over ()}2*y{circumflex over ()}5) 0.156702271184445E5 C35(x{circumflex over ()}1*y{circumflex over ()}6) 0.7630426422037567E7 C36(y{circumflex over ()}7) 0.1355347911491596E6 C37(x{circumflex over ()}8) 0.3608217232904656E5 [Free-shaped surface data 3] Term Surface 12 coefficient C1(k) 4.0.sup. C4(x{circumflex over ()}2) 0.0020593819522202.sup. C7(x{circumflex over ()}3) 0.2511626120412692E5 C11(x{circumflex over ()}4) 0.4430092334334631E4 C16(x{circumflex over ()}5) 0.2499682762769764E6 C22(x{circumflex over ()}6) 0.2488002816824596E5 C29(x{circumflex over ()}7) 0.1183737952292583E7 C37(x{circumflex over ()}8) 0.6958708070498757E7 C46(x{circumflex over ()}9) 0.3912166160352333E9 C56(x{circumflex over ()}10) 0.9268050242165196E9 [Diffraction optical surface data] Term Surface 5 coefficient C1(x) 0.000186432913712339.sup. C2(y) 0.000334028384455803.sup. C3(x{circumflex over ()}2) 0.000400706036884263.sup. C4(x*y) 0.221926248104121E4 C5(y{circumflex over ()}2) 0.0005496459597546.sup. C6(x{circumflex over ()}3) 0.657292046059681E5 C7(x{circumflex over ()}2*y) 0.6278671042198401E5 C8(x*y{circumflex over ()}2) 0.8636794643805789E5 C9(y{circumflex over ()}3) 0.473536699269363E5 C10(x{circumflex over ()}4) 0.249196177587225E4 C11(x{circumflex over ()}3*y{circumflex over ()}1) 0.4470566188511271E5 C12(x{circumflex over ()}2*y{circumflex over ()}2) 0.5660848502605312E4 C13(x{circumflex over ()}1*y{circumflex over ()}3) 0.9879118046431838E6 C14(y{circumflex over ()}4) 0.22869493158194E4 C15(x{circumflex over ()}5) 0.282604520614315E5 C16(x{circumflex over ()}4*y) 0.138087340623321E4 C17(x{circumflex over ()}3*y{circumflex over ()}2) 0.357200792044669E5 C18(x{circumflex over ()}2*y{circumflex over ()}3) 0.4306011297920189E5 C19(x*y{circumflex over ()}4) 0.168874973117509E5 C20(y{circumflex over ()}5) 0.131812119121683E5 C21(x{circumflex over ()}6) 0.111836888819939E5 C22(x{circumflex over ()}5*y) 0.536585886344322E6 C23(x{circumflex over ()}4*y{circumflex over ()}2) 0.6205464353723851E5 C24(x{circumflex over ()}3*y{circumflex over ()}3) 0.171843584946866E6 C25(x{circumflex over ()}2*y{circumflex over ()}4) 0.240213149514409E5 C26(x*y{circumflex over ()}5) 0.597793502065084E7 C27(y{circumflex over ()}6) 0.444618149861892E7 C28(x{circumflex over ()}7) 0.898537554431373E9 C29(x{circumflex over ()}6*y) 0.270211057167523E5 C30(x{circumflex over ()}5*y{circumflex over ()}2) 0.227481948957823E6 C31(x{circumflex over ()}4*y{circumflex over ()}3) 0.181802535043457E5 C32(x{circumflex over ()}3*y{circumflex over ()}4) 0.200702230224553E7 C33(x{circumflex over ()}2*y{circumflex over ()}5) 0.663005133632598E6 C34(x{circumflex over ()}1*y{circumflex over ()}6) 0.6982200936000581E7 C35(y{circumflex over ()}7) 0.612245076412055E7 C36(x{circumflex over ()}8) 0.350655378073152E6 C37(x{circumflex over ()}7*y{circumflex over ()}1) 0.106574777334788E6 C38(x{circumflex over ()}6*y{circumflex over ()}2) 0.6632704708545499E6 C39(x{circumflex over ()}5*y{circumflex over ()}3) 0.111237522680437E8 C40(x{circumflex over ()}4*y{circumflex over ()}4) 0.188163478389313E6 C41(x{circumflex over ()}3*y{circumflex over ()}5) 0.821421390471722E7 C42(x{circumflex over ()}2*y{circumflex over ()}6) 0.100809875031883E6 C43(x{circumflex over ()}1*y{circumflex over ()}7) 0.111306164297601E9 C44(y{circumflex over ()}8) 0.220079301279993E7

(102) FIG. 6 is a spot diagram of the head-mounted display optical system LS3 according to Example 3. As FIG. 6 shows, according to Example 3, chromatic aberration is corrected satisfactorily, and excellent image forming performance is demonstrated.

(103) Table 4 shows conditional expression correspondence values in each example.

(104) TABLE-US-00004 TABLE 4 [Basic Data] Example 1 Example 2 Example 3 f1 20.2574 20.3887 17.797 f2 1311.24 741.839 27960.2 fd 41.8669 11.4242 d1 0.05 d2 0.05 fm 33.0 33.0 33.0 dm 31.1429 32.0547 31.5 Ndh 1.5571 Ndl 1.5278 [Conditional expression Example Example Example correspondence value] 1 2 3 Conditional m = 0.60355 0.60755 0.53834 expression (1) Conditional |f2/f1| = expression (2) (f2)/f1 = 64.729 36.385 1571.06 Conditional |1/f2| = expression (3) 1/(f2) = 0.000763 0.001348 0.000036 [mm.sup.1] Conditional d = 30.1 30.1 30.1 expression (4) Conditional fd/f1 = 2.0534 0.64192 expression (5) Conditional dm/(fm).sup.2 = 0.0286 0.0294 0.02893 expression (6) [mm.sup.1] Conditional Ndh Ndl = 0.0293 expression (7) Conditional (d1 + d2)/f1 = 0.00562 expression (8)

(105) As shown above, each of the conditional expressions is satisfied in each example. According to each example, compactness, light weight and excellent optical performance can be implemented.

EXPLANATION OF NUMERALS AND CHARACTERS

(106) DSP head-mounted display LS head-mounted display optical system G1 first lens group G2 second lens group M1 optical deflection element M2 optical reflection element (M2r reflection surface)