Microarray type diffuser

Abstract

According to the present invention, lenses are disposed in a lattice on an array surface (32) of a micro lens array (31). A lens (30) has a cylindrical convex surface through cross-sections which are parallel to the lattice direction of the lens (30) and perpendicular to the array surface (32). On each of the lines of longitude on each cross-section parallel to the lattice direction of the lens (30), a sag amount is corrected so that the inclination of skirt increases. A lens according to another aspect has a cylindrical concave surface through cross-sections which are parallel to the lattice direction of the lens and perpendicular to an array surface. On each of the line of longitude on each cross-section parallel to the lattice direction of the lens, a sag amount is corrected so that the inclination of a rim increases.

Description

[Technical Field]

[0001] The present invention relates to a microarray type diffuser plate, and more particularly to a transmission type microlens array and a reflection type microconvex mirror array and a microconcave mirror array. [Background technology]

[0002] Non-Patent Document 1 discloses a curved surface structure of a microlens array. A technology has been proposed in which a diffuser using such a microlens array is used as a screen for head-up displays, laser projectors, and the like. The use of a microlens array has the advantage of suppressing speckle noise compared to the use of diffusers such as milky white or frosted glass. Speckle noise refers to bright areas that occur accidentally on these diffusers, which depend on the randomness of their microstructure and arrangement.

[0003] For example, Patent Document 1 proposes an image forming device having a laser projector that uses laser light as a light source to project an image formed by an array of multiple pixels, and a diffusion plate that uses a microlens array in which multiple microlenses are arranged. When a microlens array is used, it is possible to appropriately diffuse incident light and also to freely design the required diffusion angle.

[0004] Paragraph

[0023] of Patent Document 2 discloses a structured screen surface that enables control of both the elementary structures that define the surface, i.e., the microstructure, and the relative distribution of the elementary structures within the device surface. The control of the surface shape and relative spatial arrangement is completely deterministic, in contrast to prior art that relies on randomness in the microstructure and arrangement. [Prior art documents] [Patent documents]

[0005] [Patent Document 1] Japanese Patent Application Laid-Open No. 2010-145745 [Patent Document 2] Special Publication No. 2004-505306 [Non-patent literature]

[0006] [Non-Patent Document 1] Ushio Inc., "Ushio's Microfabrication Case Studies", [online], Ushio Inc., [Retrieved September 23, 2020], Internet<URL:https: / / www.ushio.co.jp / jp / feature / functional_device / part / > [Non-patent document 2] Kyoji Matsushima, "Wave Field Tools", [online], March 11, 2020, Kansai University, Faculty of Systems Science and Engineering, Optical Information Systems Laboratory, [Retrieved April 20, 2020], Internet,<URL:http: / / www.laser.ee.kansai-u.ac.jp / WaveFieldTools / > Summary of the Invention

[0007] The inventors have discovered that when a microlens array is used as a screen, brightness unevenness occurs in the image projected onto the microlenses.The inventors have also discovered that similar brightness unevenness occurs when a microconvex mirror array or a microconcave mirror array is used as a screen.An object of the present invention is to provide a means for reducing brightness unevenness in these microarray-type diffusers.

[0008] <1> A microlens array in which lenses are arranged in a grid on the array surface, the lens has a cylindrical convex surface through each cross section parallel to the lattice direction of the lens and perpendicular to the array surface; the amount of sag is corrected so that the inclination of the skirt increases in each meridian on each cross section of the lens parallel to the grating direction; Microlens array. <2> A microlens array in which lenses are arranged vertically and horizontally on an array surface, When the vertical and horizontal directions of the lens array are defined as the vertical and horizontal directions of the lens itself, the lens has a cross-cylindrical convex lens surface formed by combining a cylindrical convex surface that is parallel to the vertical direction and extends horizontally through each cross section perpendicular to the array surface, and a cylindrical convex surface that is parallel to the horizontal direction and extends vertically through each cross section perpendicular to the array surface, the amount of sag is corrected so that the inclination of the skirt increases along each meridian on each cross section parallel to the longitudinal and lateral directions; Microlens array. <3> the meridians in each cross section parallel to the longitudinal and lateral directions are formed by conic sections corrected for the amount of sag; <2> The microlens array according to claim 1. <4> In the following range of horizontal coordinate x centered on the axis of symmetry of the meridian,

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[0009] According to the present invention, it is possible to reduce uneven brightness in an image projected onto a microarray type diffuser. [Brief explanation of the drawings]

[0010] [Figure 1] FIG. [Figure 2] FIG. [Figure 3] Cross section and luminance distribution of the reference lens. [Figure 4] Cross section and luminance distribution of the correction lens. [Figure 5] A close-up of the skirt. [Figure 6] A close-up of the brightness curve. [Figure 7] A close-up of the brightness curve. [Figure 8] Brightness distribution of emitted light. [Figure 9] A graph of sag and compensation amount. [Figure 10] A graph of the amount of sag and correction near the skirt. [Figure 11] Vertical luminance distribution. [Figure 12] Horizontal luminance distribution. [Figure 13] Simulation of observed image. [Figure 14] Vertical luminance distribution. [Figure 15] Horizontal luminance distribution. [Figure 16] Simulation of observed image. [Figure 17] Vertical luminance distribution. [Figure 18] Horizontal luminance distribution. [Figure 19] Simulation of observed image. [Figure 20] Brightness distribution and observed image. [Figure 21] Brightness distribution and observed image. [Figure 22] Brightness distribution and observed image. [Figure 23] Cross section and luminance distribution of the correction lens. [Figure 24] Cross section of a concave mirror and brightness distribution. [Figure 25] A close-up of the skirt. [Figure 26] Brightness distribution and observed image. [Figure 27] FIG. 1 is a plan view of a microlens array. [Figure 28] Brightness distribution and observed image. [Figure 29] Brightness distribution and observed image. [Figure 30] Brightness distribution and observed image. [Figure 31] Brightness distribution and observed image. DETAILED DESCRIPTION OF THE INVENTION

[0011] <Microlens array>

[0012] FIG. 1 shows a microlens array 31 equipped with lenses 30. For convenience, the y-axis direction in the figure is defined as the vertical direction, and the x-axis direction is defined as the horizontal direction. The direction in which the sag of the lenses 30 increases is defined as the z-axis direction. The microlens array 31 can be suitably used in a transmission screen. Furthermore, the transmission screen can be suitably used in a head-up display.

[0013] 1, a microlens array 31 has an array surface 32. Lenses 30 are repeatedly arranged vertically and horizontally on the array surface 32. The lenses 30 are rectangular lenses. The lenses 30 may also be square lenses.

[0014] 1, the vertical and horizontal directions of the rectangle of the lens 30 are aligned with the vertical and horizontal directions of the lens array. The cross section Sy of the lens 30 is parallel to the vertical direction of the lens 30. The cross section Sx of the lens 30 is parallel to the horizontal direction of the lens 30.

[0015] 1, the pitch of the lenses 30 in the vertical direction is represented by Py. The pitch of the lenses 30 in the horizontal direction is represented by Px. The pitch Py may be equal to the vertical length of the lenses. The pitch Px may be equal to the horizontal length of the lenses.

[0016] FIG. 2 shows a lens 30, which is one microlens in a microlens array. The lens 30 has a cross-cylindrical convex lens surface 34. The convex lens surface 34 is both horizontally cylindrically convex and vertically cylindrically convex. In one embodiment, the lens 30 is a plano-convex lens with only one convex lens. More specifically, the convex lens surface 34 is horizontally cylindrical through each cross section, such as Sy, that is parallel to the vertical direction of the lens 30 and perpendicular to the array surface 32. Furthermore, the convex lens surface 34 is vertically cylindrical through each cross section, such as Sx, that is parallel to the horizontal direction of the lens 30 and perpendicular to the array surface 32. In one embodiment, the term cross-cylindrical may or may not include an elliptical paraboloid. In one embodiment, the term elliptical paraboloid may or may not include a paraboloid of revolution.

[0017] In the example shown in Figure 2, the convex lens surface 34 has different divergence angles in the vertical and horizontal directions. A beam 35 perpendicular to the array surface 32 is incident on the convex lens surface 34 and refracted. The beam 35 converges at a focal point Fx on the cross section Sx. The taper of the beam 35, which diverges after converging, is 2θx.

[0018] As shown in Figure 2, the beam 35 further converges at a focal point Fy on the cross section Sy. The taper of the beam 35 that diverges after converging is 2θy. In one embodiment shown in the figure, 2θx is greater than 2θy. In another embodiment, 2θx is less than 2θy. In another embodiment, 2θx and 2θy are equal.

[0019] <Correction of uneven brightness and skirt>

[0020] FIG. 3 shows the cross section Sx of the lens and its luminance distribution. Unless otherwise specified, in this embodiment, the luminance distribution is a radiance distribution. The coordinate axis x indicates the horizontal coordinate centered on the meridian symmetry axis of the cross section Sx. Light is assumed to be incident from the -z direction in the figure. The meridian Mo on the cross section Sx before correction is a conic section. In this embodiment, conic sections include ellipses, parabolas, and hyperbolas. Ellipses include perfect circles. Conic sections do not include two lines. The term conic section will be interpreted in the same way below. An uncorrected lens having meridian Mo may be referred to as a reference lens.

[0021] In Figure 3, a beam of light 35 is incident on a lens. The beam 35 is refracted by the convex lens surface 34 and diverges. The angle θ of the refracted light ray is determined by the x-coordinate of the meridian. In the figure, the angle θ is the divergence angle of the refracted light ray with respect to the axis of symmetry of the meridian on the cross section Sx. For convenience, the angle θ is expressed as positive and negative values ​​in the figure. The half width at half maximum (HWHM) of the luminance distribution of the refracted beam 35 is the divergence angle θo of the reference lens. In the figure, the divergence angle θo is 10 degrees. The taper of the diverging beam 35 is 2θo = 20 degrees. This taper is different from 2θx shown in Figure 2.

[0022] In Figure 3, there is a shoulder on the luminance curve Lo near the angle θ = ±10 degrees. Ripples in the luminance curve that cause uneven luminance can be seen in this vicinity.

[0023] FIG. 4 further shows the meridian Mc on the cross section Sx after correction. A corrected lens having the meridian Mc may be referred to as a corrected lens. In one embodiment, the center of the meridian Mc is represented by a conic section. The skirt of the meridian Mc is represented by a corrected conic section. Here, the skirt refers to the vicinity of the edge of the convex lens shape.

[0024] In Figure 4, the half-width at half maximum of the brightness distribution of the beam 35 passing through the correction lens is the divergence angle θc. In the figure, θc is 10 degrees. The taper of the diverging beam 35 is 2θc = 20 degrees.

[0025] Figure 5 is an enlarged view of the skirts of the meridians. The skirt of meridian Mc sags slightly in the +z direction compared to the skirt of meridian Mo. The size of the horizontal range over which the sag is corrected is Δx. The amount of sag correction is Δz. In this way, the sag is corrected so that the slope of the skirt increases at the meridian on cross section Sx.

[0026] Returning to Figure 4, this figure shows the change in luminance distribution before and after correction. In this figure, there is a shoulder on the luminance curve Lc near the angle θ = ±10 degrees. Ripples in the luminance curve, which cause luminance unevenness, can be seen in this vicinity. However, the luminance fluctuation, or ripple, is reduced in the luminance curve Lc compared to the luminance curve Lo.

[0027] Figure 6 shows an enlarged view of the luminance curve Lo. For each of the red, green, and blue colors, the ripple increases with each x-coordinate. Therefore, when an observer looks at the microlens, they will notice uneven luminance. Furthermore, the peaks differ for each color, indicating that this lens has chromatic aberration.

[0028] Figure 7 shows an enlarged view of the luminance curve Lc. For each of the red, green, and blue colors, ripples are reduced by correcting the amount of sag. Therefore, even when an observer looks at the microlens, it is difficult to find uneven luminance. Chromatic aberration is also reduced by correcting the amount of sag.

[0029] The sag amount is corrected in each cross section parallel to the cross section Sx as shown in FIG. 2. Therefore, brightness unevenness is reduced throughout the entire vertical direction, i.e., the y-axis direction. The sag amount is also corrected in each cross section parallel to the cross section Sy as shown in FIG. 2. Therefore, brightness unevenness is reduced throughout the entire horizontal direction, i.e., the x-axis direction. In this embodiment, each meridian in each cross section parallel to the vertical and horizontal directions consists of a conic section that has been corrected for the sag amount.

[0030] <Ripple Analysis>

[0031] The correction method is described below. First, we will explain the ripple that we want to reduce through correction. Figure 8 shows the directional characteristics of the luminance distribution of the emitted light. The vertical axis is luminance. It is upside down compared to the graph in Figure 4. The horizontal axis is sinθ with respect to the angle θ. The dashed line is the center of the luminance fluctuation range. X m is the x-coordinate on the lens that produces the outermost ripple peak. p is the x-coordinate of the lens edge. m From X p Distance to X mp is expressed as the following formula:

[0032]

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[0033] When visible light is projected onto a microlens, in one embodiment, the wavelength λ ranges from 400 (nm) to 700 (nm). Ideally, the conditions shown below using several mathematical formulas are satisfied throughout the entire visible light range. Furthermore, conveniently selecting a representative wavelength to analyze ripples is effective in reducing brightness unevenness throughout the entire visible light range. In one embodiment, ripples are analyzed using a green wavelength with high luminosity, for example, 530 nm, as λ. In another embodiment, ripples are analyzed using a red wavelength, for example, 650 nm, at which brightness unevenness due to diffraction is easily noticeable. This embodiment is effective when brightness unevenness is not noticeable in other visible light ranges. In another embodiment, ripples are analyzed using a yellow wavelength, for example, 590 nm, which is intermediate between the green and red wavelengths, as λ.

[0034] <Reference lens>

[0035] Next, we will explain the correction of sag using Figures 9 and 10. Figure 9 is an overall graph of sag and correction amount. Figure 10 is a graph of sag and correction amount near the skirt. First, we will explain the reference lens in the graph. The sag amount z of the reference lens is expressed as an uncorrected conic section as shown in the following formula. k is the conic constant. r is the radius of curvature of the conic section.

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[0036] <Correction method>

[0037] Next, the correction method will be explained based on Figures 9 and 10. First, the center of the lens, which is not subjected to correction, will be explained. In the following range of the horizontal x coordinate, the sag amount z is expressed as a conic section that is not subjected to correction, as shown in the above formula.

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[0038] On the other hand, the correction performed on the skirt in FIGS. 9 and 10 is preferably performed within the following range of the x coordinate in the horizontal direction centered on the axis of symmetry of the meridian.

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[0039] L is the width of the meridian, i.e., the lens diameter. In one embodiment, the lens diameter L is equal to the lens pitch. In Figures 9 and 10, the size of the horizontal range Δx, for which the sag amount is corrected, is expressed as follows:

[0040]

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[0041] α is a correction width coefficient, preferably 0<α<2, preferably 0.5<α<1.5, preferably 0.7<α<1.3, preferably 0.9<α<1.1, and preferably α=1.0.

[0042] γ is a real number representing the correction order. Preferably, 1≦γ≦10, preferably 2<γ, preferably 3<γ. When γ=1, the skirt becomes a straight line.

[0043] λ is the wavelength of the light.

[0044] r is the radius of curvature of the conic section.

[0045] n is the absolute refractive index of the lens. n is approximated by the relative refractive index of the lens to air.

[0046] The square root of the following is X shown in Figure 7 mp is equivalent to

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[0047] The amount of correction Δz of the sag amount shown in FIGS.

[0048]

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[0049] β is a correction coefficient for converting the optical path difference in wavelength units, and is preferably 0<β<0.6, preferably 0.15<β<0.45, preferably 0.2<β<0.4, preferably 0.25<β<0.35, and preferably β=0.3.

[0050] In FIGS. 9 and 10, the sag amount z is expressed as a corrected conic section as follows:

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[0051] <Example 1: Transmission screen, microlens array>

[0052] The microlens array 31 shown in Fig. 1 was designed on a computer, and its optical properties were investigated by simulation through calculations. The simulation was carried out using the Wave Field Library, a wave optics calculation support toolkit. See Non-Patent Document 2.

[0053] <Example 1-1>

[0054] To study the correction width coefficient α described above, a lens 30 as shown in FIG. 2 was designed. The pitches Px and Py were each set to 30 μm. In a cross section Sy parallel to the vertical direction of the lens 30, the divergence angle θo of the beam 35 for the lens 30 before correction was set to 10 degrees. In a cross section Sx parallel to the horizontal direction of the lens 30, the divergence angle θo of the beam 35 for the lens 30 before correction was set to 20 degrees. The lens 30 diverges the beam more widely in the horizontal direction than in the vertical direction.

[0055] FIG. 11 shows a graph of the vertical cross section Sy shown in FIGS. 2 and 3, with the luminance on the vertical axis and the angle of light after refraction on the horizontal axis. The vertical axis is luminance (au). The horizontal axis is angle (deg) representing the angle θ. As shown in FIG. 3, angle θ is the opening angle of the light beam after refraction. The same applies hereinafter unless otherwise stated.

[0056] The graph in the upper left of Figure 11 is for the reference lens. The sag amount z of the reference lens is expressed as an uncorrected conic section as shown in the following formula. The same applies hereinafter unless otherwise stated.

[0057]

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[0058] The radius of curvature is r = 40 μm. The conic constant is k = -1.0. The wavelength is λ = 630 nm.

[0059] In Figure 11, the other eight graphs are for the corrective lens. Correction by the skirt will be explained with reference to Figure 9 again. The x-axis in Figure 9 will be replaced with the y-axis for consideration. In the following range, which corresponds to the skirt of the corrective lens,

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[0060] The sag amount z is expressed as a corrected conic section as shown in the following formula.

[0061]

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[0062] However, Δy and Δz are expressed as follows:

[0063]

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[0064]

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[0065] The lens diameter L is equal to the pitch Py shown in Figure 2, and its value is 30 μm. The correction order γ mentioned above is fixed at 4. The correction width coefficient α is between 0.5 and 2.0. The absolute refractive index n is 1.5. The correction coefficient β mentioned above is fixed at 0.3. The sag amount z in the uncorrected portion is expressed as a conic section, the same as the reference lens.

[0066] 11, as the correction width coefficient α increases, the ripple decreases, but once α reaches approximately 1.0, the ripple increases again.

[0067] FIG. 12 shows a graph in which the vertical axis represents luminance and the horizontal axis represents the angle of light after refraction for the cross section Sx in the vertical direction shown in FIGS.

[0068] The graph in the upper left of Figure 12 is for the reference lens. The sag amount z of the reference lens is expressed as an uncorrected conic section as follows:

[0069]

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[0070] The radius of curvature is r = 20 μm. The conic constant is k = -1.0. The wavelength is λ = 630 nm.

[0071] In Figure 12, the other eight graphs are for corrective lenses. Correction by the skirt will be explained with reference to Figure 9 again. In the following range, which corresponds to the skirt of the corrective lens,

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[0072] The sag amount z is expressed as a corrected conic section as shown in the following formula.

[0073]

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[0074] However, Δx and Δz are expressed as follows:

[0075]

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[0076]

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[0077] The lens diameter L is equal to the pitch Px shown in Figure 2, and its value is 30 μm. The correction order γ mentioned above is fixed at 4. The correction width coefficient α is between 0.5 and 2.0. The absolute refractive index n is 1.5. The correction coefficient β mentioned above is fixed at 0.3. The sag amount z in the uncorrected portion is expressed as a conic section, the same as the reference lens.

[0078] 12, as the correction width coefficient α increases, the ripple decreases, but once α reaches approximately 1.0, the ripple increases again.

[0079] Figure 13 shows a simulation of the observed image when the microlens array is viewed in plan view. It can be seen that the brightness unevenness is reduced in the vicinity of α = 1.0 in both the vertical and horizontal directions, which have different diffusion angles.

[0080] <Example 1-2>

[0081] In order to examine the above-mentioned correction coefficient β, a lens was designed in the same manner as in Example 1-1.

[0082] FIG. 14 shows a graph in which the vertical axis represents luminance and the horizontal axis represents the angle of light after refraction for the vertical cross section Sy shown in FIGS.

[0083] The graph in the upper left of Figure 14 is for the reference lens. This reference lens is the same as the reference lens described in the explanation of Figure 11. The radius of curvature r, conic constant k, and wavelength λ are all as described in the explanation of Figure 11.

[0084] In Figure 14, the other eight graphs are for the corrective lens. Correction by the skirt will be explained with reference to Figure 9 again. The x-axis in Figure 9 will be replaced with the y-axis for consideration. In the following range, which corresponds to the skirt of the corrective lens,

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[0085] The sag amount z is expressed as a corrected conic section as shown in the following formula.

[0086]

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[0087] However, Δy and Δz are expressed as follows:

[0088]

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[0089]

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[0090] The lens diameter L is equal to the pitch Py shown in Figure 2, and its value is 30 μm. The correction order γ mentioned above is fixed at 4. The correction width coefficient α mentioned above is fixed at 1. The absolute refractive index n is 1.5. The correction coefficient β is between 0.15 and 0.6. The sag amount z in the uncorrected portion is expressed as a conic section, the same as that of the reference lens.

[0091] 14, as the correction coefficient β increases, the ripple decreases, but once β reaches approximately 0.3, the ripple increases again.

[0092] FIG. 15 shows a graph in which the vertical axis represents luminance and the horizontal axis represents the angle of light after refraction for the cross section Sx in the vertical direction shown in FIGS.

[0093] The graph in the upper left of Figure 15 is for the reference lens. This reference lens is the same as the reference lens described in the description of Figure 12. The radius of curvature r, conic constant k, and wavelength λ are all as described in the description of Figure 12.

[0094] In Figure 15, the other eight graphs are for corrective lenses. Correction by the skirt will be explained with reference to Figure 9 again. In the following range, which corresponds to the skirt of the corrective lens,

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[0095] The sag amount z is expressed as a corrected conic section as shown in the following formula.

[0096]

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[0097] However, Δx and Δz are expressed as follows:

[0098]

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[0099]

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[0100] The lens diameter L is equal to the pitch Px shown in Figure 2, and its value is 30 μm. The correction order γ mentioned above is fixed at 4. The correction width coefficient α mentioned above is fixed at 1. The absolute refractive index n is 1.5. The correction coefficient β is between 0.15 and 0.6. The sag amount z in the uncorrected portion is expressed as a conic section, the same as that of the reference lens.

[0101] 15, as the correction coefficient β increases, the ripple decreases, but once the correction coefficient β reaches approximately 0.3, the ripple increases again.

[0102] Figure 16 shows a simulation of the observed image when the microlens array is viewed in plan view. It can be seen that the brightness unevenness is reduced in the vicinity of β = 0.3 in both the vertical and horizontal directions, which have different diffusion angles.

[0103] <Examples 1-3>

[0104] In order to examine the above-mentioned correction order γ, a lens was designed in the same manner as in Example 1-1.

[0105] FIG. 17 shows a graph in which the vertical axis represents luminance and the horizontal axis represents the angle of light after refraction for the vertical cross section Sy shown in FIGS.

[0106] The graph in the upper left of Figure 17 is for the reference lens. This reference lens is the same as the reference lens described in the description of Figure 11. The radius of curvature r, conic constant k, and wavelength λ are all as described in the description of Figure 11.

[0107] In Figure 17, the other four graphs are for the corrective lens. Correction by the skirt will be explained with reference to Figure 9 again. The x-axis in Figure 9 will be replaced with the y-axis for consideration. In the following range, which corresponds to the skirt of the corrective lens,

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[0108] The sag amount z is expressed as a corrected conic section as shown in the following formula.

[0109]

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[0110] However, Δy and Δz are expressed as follows:

[0111]

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[0112]

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[0113] The lens diameter L is equal to the pitch Py shown in Figure 2, and its value is 30 μm. The correction order γ is between 1 and 4. The correction width coefficient α mentioned above is fixed at 1. The absolute refractive index n is 1.5. The correction coefficient β mentioned above is fixed at 0.3. The sag amount z in the uncorrected portion is expressed as a conic section, the same as the reference lens.

[0114] In FIG. 17, as the correction order γ increases, the ripple decreases.

[0115] FIG. 18 shows a graph in which the vertical axis represents luminance and the horizontal axis represents the angle of light after refraction for the cross section Sx in the vertical direction shown in FIGS.

[0116] The graph in the upper left of Figure 18 is for the reference lens. This reference lens is the same as the reference lens described in the description of Figure 12. The radius of curvature r, conic constant k, and wavelength λ are all as described in the description of Figure 12.

[0117] In Figure 18, the other four graphs are for the corrective lens. Referring again to Figure 9, correction by the skirt will be explained. In the following range, which corresponds to the skirt of the corrective lens,

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[0118] The sag amount z is expressed as a corrected conic section as shown in the following formula.

[0119]

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[0120] However, Δx and Δz are expressed as follows:

[0121]

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[0122]

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[0123] The lens diameter L is equal to the pitch Px shown in Figure 2, and its value is 30 μm. The correction order γ is between 1 and 4. The correction width coefficient α mentioned above is fixed at 1. The absolute refractive index n is 1.5. The correction coefficient β mentioned above is fixed at 0.3. The sag amount z in the uncorrected portion is expressed as a conic section, the same as the reference lens.

[0124] In FIG. 18, as the correction order γ increases, the ripple decreases.

[0125] Figure 19 shows a simulation of the observed image when the microlens array is viewed in plan view. It can be seen that the luminance unevenness is reduced as the correction order γ approaches 4 in both the vertical and horizontal directions, which have different diffusion angles.

[0126] <Examples 1-4>

[0127] In order to examine the above-mentioned lens diameter L, that is, the lens pitch, lenses were designed in the same manner as in Example 1-1.

[0128] Figure 20 shows graphs of the vertical cross section Sy and horizontal cross section Sx shown in Figures 2 and 3, with the vertical axis representing brightness and the horizontal axis representing the angle of light after refraction. It also shows a simulation of the observed image when the microlens array is viewed in plan view. The upper graphs are both for the reference lens. The lower graphs are both for the correction lens. The graph on the left is for the vertical cross section Sy. The graph on the right is for the horizontal cross section Sx.

[0129] Referring again to FIG. 9, correction of the vertical skirt will be explained. The x-axis in FIG. 9 will be replaced with the y-axis for consideration. In the following range, which corresponds to the skirt of the correction lens,

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[0130] The sag amount z is expressed as a corrected conic section as shown in the following formula.

[0131]

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[0132] However, Δy and Δz are expressed as follows:

[0133]

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[0134]

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[0135] In the vertical direction, the lens diameter L is equal to the pitch Py shown in Figure 2, and its value is 60 μm. The radius of curvature r = 80 μm. The conic constant k = -1.0. The wavelength λ = 630 nm. The above-mentioned correction order γ is fixed at 4. The above-mentioned correction width coefficient α is fixed at 1. The absolute refractive index n is 1.5. The above-mentioned correction coefficient β is fixed at 0.3. The sag amount z in the uncorrected portion is expressed as a conic section, the same as for the reference lens.

[0136] Referring again to FIG. 9, correction by the skirt in the horizontal direction will be explained. In the following range, which corresponds to the skirt of the correction lens,

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[0137] The sag amount z is expressed as a corrected conic section as shown in the following formula.

[0138]

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[0139] However, Δx and Δz are expressed as follows:

[0140]

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[0141]

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[0142] In the horizontal direction, the lens diameter L is equal to the pitch Px shown in Figure 2, and its value is 60 μm. The radius of curvature r = 40 μm. The conic constant k is -1.0. The wavelength λ is the same as the wavelength λ in the vertical direction. The correction order γ mentioned above is fixed at 4. The correction width coefficient α mentioned above is fixed at 1. The absolute refractive index n is the same as the absolute refractive index n in the vertical direction. The correction coefficient β mentioned above is fixed at 0.3. The sag amount z in the uncorrected portion is expressed as a conic section, the same as that of the reference lens.

[0143] As shown in the observation image in Figure 20, even when the vertical and horizontal lens pitch was 60 μm, ripples were reduced by skirt correction.

[0144] <Examples 1-5>

[0145] To study the lens diameter L, i.e., the lens pitch, lenses were designed in the same manner as in Example 1-4. The graphs and observed images shown in Figure 21 are interpreted in the same way as in Figure 20. Corrections for the skirt in the vertical and horizontal directions were made in the same manner as in Example 1-4.

[0146] In the vertical direction, the lens diameter L is equal to the pitch Py shown in Figure 2, and its value is 100 μm. The radius of curvature r = 133.3 μm. The conic constant k is -1.0. The wavelength λ is 630 nm. The above-mentioned correction order γ is fixed at 4. The above-mentioned correction width coefficient α is fixed at 1. The absolute refractive index n is 1.5. The above-mentioned correction coefficient β is fixed at 0.3. The sag amount z in the uncorrected portion is expressed as a conic section, the same as that of the reference lens.

[0147] In the horizontal direction, the lens diameter L is equal to the pitch Px shown in Figure 2, and its value is 100 μm. The radius of curvature r = 66.7 μm. The conic constant k is -1.0. The wavelength λ is the same as the wavelength λ in the vertical direction. The correction order γ mentioned above is fixed at 4. The correction width coefficient α mentioned above is fixed at 1. The absolute refractive index n is the same as the absolute refractive index n in the vertical direction. The correction coefficient β mentioned above is fixed at 0.3. The sag amount z in the uncorrected portion is expressed as a conic section, the same as that of the reference lens.

[0148] As shown in the observation image in Figure 21, even when the vertical and horizontal lens pitch was 100 μm, ripples were reduced by skirt correction.

[0149] <Examples 1-6>

[0150] To study the lens diameter L, i.e., the lens pitch, lenses were designed in the same manner as in Example 1-4. The graphs and observed images shown in Figure 22 are interpreted in the same way as in Figure 20. Corrections for the skirt in the vertical and horizontal directions were made in the same manner as in Example 1-4.

[0151] In the vertical direction, the lens diameter L is equal to the pitch Py shown in Figure 2, and its value is 150 μm. The radius of curvature r = 200 μm. The conic constant k is -1.0. The wavelength λ = 630 nm. The above-mentioned correction order γ is fixed at 4. The above-mentioned correction width coefficient α is fixed at 1. The absolute refractive index n is 1.5. The above-mentioned correction coefficient β is fixed at 0.3. The sag amount z in the uncorrected portion is expressed as a conic section, the same as that of the reference lens.

[0152] In the horizontal direction, the lens diameter L is equal to the pitch Px shown in Figure 2, and its value is 150 μm. The radius of curvature r = 100 μm. The conic constant k is -1.0. The wavelength λ is the same as the wavelength λ in the vertical direction. The correction order γ mentioned above is fixed at 4. The correction width coefficient α mentioned above is fixed at 1. The absolute refractive index n is the same as the absolute refractive index n in the vertical direction. The correction coefficient β mentioned above is fixed at 0.3. The sag amount z in the uncorrected portion is expressed as a conic section, the same as that of the reference lens.

[0153] As shown in the observation image in Figure 22, even when the vertical and horizontal lens pitch was 150 μm, ripples were reduced by skirt correction.

[0154] <Modification 1: Microlens array of concave lenses>

[0155] The above-mentioned microlens array is a microarray of convex lenses. A microarray of concave lenses may also be designed as follows.

[0156] Figure 23 shows the cross section Sx and luminance distribution of a concave lens that makes up a microarray. The coordinate axis x indicates the horizontal coordinate centered on the axis of symmetry of the meridian of the cross section Sx. Light is assumed to be incident from the +z direction in the figure. The meridian Mo on the cross section Sx before correction is a conic section. In this embodiment, conic sections include ellipses, parabolas, and hyperbolas. Ellipses include perfect circles. Conic sections do not include two lines. The term conic section will be interpreted in the same way below. The lens before correction that has the meridian Mo may be referred to as the reference lens.

[0157] Figure 23 further shows the meridian Mc on the cross section Sx after correction. A corrected lens having the meridian Mc is sometimes called a corrected lens. In one embodiment, the center of the meridian Mc is represented by a conic section. The rim of the meridian Mc is represented by a corrected conic section. Here, the rim refers to the vicinity of the edge of the concave lens shape. The sag of the rim has been corrected. The rim rises slightly in the +z direction.

[0158] Let the size of the horizontal range be Δx. Let the amount of sag correction be Δz. In the following range of horizontal x coordinates, the sag z is expressed as an uncorrected conic section as follows: L is the lens width. In one embodiment, the lens width L is equal to the lens pitch. k is the conic constant. r is the radius of curvature of the conic section.

[0159]

number

[0160]

number

[0161] On the other hand, the correction performed on the rim is preferably performed in the following range of horizontal x coordinates centered on the meridian axis of symmetry:

number

[0162] The size of the horizontal range Δx where the sag amount is corrected is expressed as follows:

[0163]

number

[0164] γ is a real number representing the correction order. Preferably, 1≦γ≦10, preferably 2<γ, preferably 3<γ. When γ=1, the skirt becomes a straight line.

[0165] λ is the wavelength of the light.

[0166] r is the radius of curvature of the conic section.

[0167] n is the absolute refractive index of the lens. n is approximated by the relative refractive index of the lens to air.

[0168] The amount of sag correction Δz is expressed as follows:

[0169]

number

[0170] The sag amount z is expressed as a corrected conic section as shown in the following formula.

number

[0171] In Figure 23, a beam of light 35 is incident on a lens. The beam 35 is refracted by the concave lens surface and diverges. The angle θ of the refracted light ray is determined by the x-coordinate of the meridian. In the figure, the angle θ is the divergence angle of the refracted light ray with respect to the axis of symmetry of the meridian on the cross section Sx. For convenience, the angle θ is expressed as a positive and negative value in the figure. The half-width at half maximum of the radiance distribution of the refracted beam 35 is the divergence angle θc of the reference lens. In the figure, the divergence angle θc after correction is 10 degrees. The divergence angle θc is equal to the divergence angle before correction. The taper of the diverging beam 35 is 2θc = 20 degrees. This taper is equal to the taper before correction.

[0172] As shown in Figure 23, there is a shoulder in the brightness curve Lo before correction near the angle θ = ±10 degrees. Ripples in the brightness curve that cause brightness unevenness can be seen in this vicinity. Similarly, there is a shoulder in the brightness curve Lc after correction near the angle θ = ±10 degrees. Ripples in the brightness curve that cause brightness unevenness can be seen in this vicinity. However, the brightness fluctuation, or ripple, is reduced in the brightness curve Lc compared to the brightness curve Lo.

[0173] <Variation 2: Reflective concave mirror array type diffuser>

[0174] The above-mentioned microlens array is a transmission type microlens array suitable for a transmission type screen. A reflection type concave mirror array type diffuser suitable for a reflection type screen is designed and fabricated as follows.

[0175] Figure 24 shows a cross section of the metal film Mf that constitutes a reflective micro concave mirror array. The concave mirror array is created by using the shape of the microlens array described above as a mold and evaporating a metal film Mf onto each convex surface. The concave surface of the metal film Mf has a shape that is a transfer of the convex surface of the convex lens shape shown in Figure 2. In one embodiment, the metal film Mf is made of aluminum.

[0176] In a reflective micro concave mirror array, concave mirrors are arranged vertically and horizontally on the array surface. It has a shape that is a transfer of the microlens array shown in Figure 1. Therefore, if the vertical and horizontal directions of the concave mirror are considered to be the vertical and horizontal directions of the concave mirror itself, the concave mirror has a cross-cylindrical concave surface that is a combination of a horizontally cylindrical concave surface that is parallel to the vertical direction and across each cross section perpendicular to the array surface, and a vertically cylindrical concave surface that is parallel to the horizontal direction and across each cross section perpendicular to the array surface.

[0177] In Fig. 24, the concave surface of the metal film Mf has the same meridian as the meridian Mc. The meridian Mc after correction is as explained with reference to Figs. 3 and 4. The same is true for the meridian Mo before correction.

[0178] In Figure 24, a beam of light traveling through air or vacuum is incident on the concave mirror array from the +z direction. The beam reflected by the concave surface diverges. The angle θ of the reflected light ray is determined by the x-coordinate of the meridian. In the figure, angle θ is the divergence angle of the reflected light ray with respect to the meridian symmetry axis on the cross section. For convenience, angle θ is represented as a positive and negative value in the figure. The half-width at half maximum of the radiance distribution of the reflected beam is the divergence angle θc of the reference lens. In the figure, the divergence angle θc after correction is 10 degrees. The divergence angle θc is equal to the divergence angle before correction. The taper of the diverging beam 35 is 2θc = 20 degrees. This taper is equal to the taper before correction.

[0179] Figure 25 is an enlarged view of the rims of the meridians. Here, the rim refers to the area near the edge of the concave mirror shape. For comparison, the skirt of a convex lens used in a transmission diffuser is shown on the left. The rim of a concave mirror is shown on the right. In both cases, the rim of meridian Mc rises slightly in the +z direction compared to the rim of meridian Mo. In this way, a correction amount Δz is added to the sag amount so that the inclination of the rim increases on the meridian in the cross section.

[0180] Returning to Figure 24, there is a shoulder in the pre-correction luminance curve Lo near the angle θ = ±10 degrees. Ripples in the luminance curve that cause luminance unevenness can be seen in this vicinity. Similarly, there is a shoulder in the post-correction luminance curve Lc near the angle θ = ±10 degrees. Ripples in the luminance curve that cause luminance unevenness can be seen in this vicinity. However, the luminance fluctuation, or ripple, is reduced in the luminance curve Lc compared to the luminance curve Lo.

[0181] In one embodiment shown in Figure 24, the correction method is as follows. First, we will explain the center of the concave mirror, which does not receive correction. In the following range of horizontal x coordinate, the sag amount z is expressed as a conic section without correction, as explained in the <Reference Lens> chapter above. On the other hand, the correction performed at the rim is performed in the following range of horizontal x coordinate centered on the meridian symmetry axis:

number

[0182] The sag amount z is expressed as a corrected conic section as shown in the following formula.

number

[0183] As shown in Figure 25, when calculating Δx for the above-mentioned transmission-type diffuser, it is necessary to take into account that the difference in optical path length before and after correction is n-1 times. For details, see the above chapter on <Correction Methods>.

[0184] In contrast to this, in the reflective diffuser according to this modification, light travels back and forth within the section of Δz as shown on the right side of FIG. 25, so the difference in optical path length before and after correction is doubled.

[0185] Therefore, Δx is expressed as follows:

number

[0186] The coefficients in the root are reduced to express Δx as follows:

number

[0187] Also, Δz is expressed as follows:

number

[0188] L is the width of the meridian, and α is a real number of 0.5 to 2 inclusive. In one embodiment, α is 1. β is a real number of 0.15 to 0.6 inclusive. In one embodiment, β is 0.3. γ is a real number of 1 to 10 inclusive. r is the radius of curvature of the conic section. λ is the wavelength of visible light. In a preferred embodiment, α is a real number of 0.9 to 1.1 inclusive. β is a real number of 0.25 to 0.35 inclusive. γ is a real number of 2 to 10 inclusive.

[0189] Figure 26 shows graphs of the horizontal cross section (x direction) shown in Figure 24 and the vertical cross section (y direction) perpendicular to this, with brightness on the vertical axis and the angle of light after refraction on the horizontal axis. The graph on the left is for the vertical cross section. The graph on the right is for the horizontal cross section. Also on the right is a simulation of the observed image when the microlens array is viewed in plan. All of the images in the top row are for the reference concave surface. All of the images in the bottom row have been corrected for the amount of sag. As shown in this simulation, ripple was reduced even in concave mirrors by correcting the rim.

[0190] <Modification 3: Reflective convex mirror array type diffuser>

[0191] Another embodiment of the microarray type diffuser plate is a micro-convex mirror array in which convex mirrors are arranged vertically and horizontally on an array surface. In one embodiment of the micro-convex mirror array, when the vertical and horizontal directions of the convex mirror are the vertical and horizontal directions of a single convex mirror, the convex mirror has a cross-cylindrical convex surface that merges a horizontally cylindrical convex surface parallel to the vertical direction and across each cross section perpendicular to the array surface, and a vertically cylindrical convex surface parallel to the horizontal direction and across each cross section perpendicular to the array surface. Furthermore, the sag is corrected so that the slope of the skirt is increased along each meridian on each cross section parallel to the vertical and horizontal directions. In a preferred embodiment, the meridians on each cross section parallel to the vertical and horizontal directions are formed by conic sections that have been corrected for the sag.

[0192] The correction method is as follows. First, we will explain the center of the convex mirror, which does not receive correction. In the horizontal x-coordinate range shown below, the sag amount z is expressed as a conic section without correction, as explained in the <Reference Lens> chapter above. On the other hand, the correction performed on the skirt is performed in the horizontal x-coordinate range shown below, centered on the meridian symmetry axis.

number

[0193] The sag amount z is expressed as a corrected conic section as shown in the following formula.

number

[0194] Δx is expressed as follows:

number

[0195] Also, Δz is expressed as follows:

number

[0196] L is the width of the meridian, and α is a real number of 0.5 to 2 inclusive. In one embodiment, α is 1. β is a real number of 0.15 to 0.6 inclusive. In one embodiment, β is 0.3. γ is a real number of 1 to 10 inclusive. r is the radius of curvature of the conic section. λ is the wavelength of visible light. In a preferred embodiment, α is a real number of 0.9 to 1.1 inclusive. β is a real number of 0.25 to 0.35 inclusive. γ is a real number of 2 to 10 inclusive.

[0197] <Modification 4: Microlens array consisting of hexagonal microconvex lenses>

[0198] The top part of Figure 27 shows a microlens array 41 in plan view. The microlens array 41 includes regular hexagonal lenses 40 and lenses having the same shape in plan view. Similar to the microlens array 31 shown in Figure 1, the lenses 40 and other lenses are arranged in a hexagonal lattice on the array surface. In this example, the hexagonal lattice is a regular hexagonal lattice.

[0199] As shown in Figure 27, the x-axis and y-axis are set with the center of the lens 40 as the origin. The x-axis is parallel to two opposing sides of the lens 40 when viewed in a plan view. The y-axis is perpendicular to the two opposing sides of the lens 40 when viewed in a plan view. The y-axis is parallel to the lattice direction. The x-axis is perpendicular to the lattice direction.

[0200] When the lens 40 is cut parallel to the xz-plane in plan view as shown in FIG. 27, the length in the x-axis direction is L. x Let's say. L x is a function of y. When the lens 40 is cut parallel to the yz-plane in plan view, the length in the y-axis direction is L. y Let's say. L y is a function of x. Lens 40 has a cylindrical convex surface in each cross section parallel to the x-axis direction and perpendicular to the array surface. Lens 40 has a cylindrical convex surface in each cross section parallel to the y-axis direction and perpendicular to the array surface.

[0201] The lower part of Figure 27 shows cross sections of lens 40 cut by a plane parallel to the xz-plane. In all cross sections, lens 40 has meridians Mo with the same radius of curvature r. Meridian Mo is a conic section. Conic sections include ellipses, parabolas, and hyperbolas. Ellipses include perfect circles. Conic sections do not include bilinear lines. In cross sections of lens 40 cut by a plane parallel to the yz-plane, lens 40 also has meridians with the same determined radius of curvature. Such conic sections are such conic sections. Conic sections include ellipses, parabolas, and hyperbolas. Ellipses include perfect circles. Conic sections do not include bilinear lines.

[0202] The sag amount is corrected so that the inclination of the skirt increases along each meridian Mo on each cross section parallel to the lattice direction of the lens 40. That is, the sag amount of the lens 40 increases in the +z direction. The range within which the sag amount is corrected on the x-axis is also determined. These axes will be referred to simply as the x-axis or x-coordinate hereinafter. The microlens array 41 can be suitably used in a transmission screen. Furthermore, a transmission screen can be suitably used in a head-up display.

[0203] In FIG. 27, in the range Δx of the horizontal x coordinate on the cross section parallel to the xz-plane, the sag amount z is expressed as a conic section without correction as shown in the following formula. The correction amount of the sag amount is Δz. L x is the lens width. Lens width L x varies depending on the y-coordinate of the cross section.

[0204] In the following range of horizontal coordinate x,

number

[0205] The sag z is expressed as an uncorrected conic section as follows:

number

[0206] In the following range of horizontal coordinate x,

number

[0207] The sag amount z is expressed as a corrected conic section as shown in the following formula.

number

[0208] However, Δx and Δz are expressed as follows:

number

number

[0209] r x is the radius of curvature of the conic section. k x is the conic constant. α x is a real number between 0.5 and 2. x is a real number between 0.15 and 0.6. x is a real number between 1 and 10. λ is the wavelength of visible light. n is the absolute refractive index of the lens.

[0210] Similarly, in the following range Δy of the y coordinate on a cross section parallel to the yz-plane, the sag amount z is expressed as an uncorrected conic section as shown in the following formula. The correction amount for the sag amount is Δz. L y is the lens width. Lens width L y varies depending on the x-coordinate of the cross section.

[0211] In the following range of horizontal coordinate y,

number

[0212] The sag z is expressed as an uncorrected conic section as follows:

number

[0213] In the following range of horizontal coordinate y,

number

[0214] The sag amount z is expressed as a corrected conic section as shown in the following formula.

number

[0215] However, Δy and Δz are expressed as follows:

number

number

[0216] r y is the radius of curvature of the conic section. k y is the conic constant. α y is a real number between 0.5 and 2. y is a real number between 0.15 and 0.6. y is a real number between 1 and 10. λ is the wavelength of visible light. n is the absolute refractive index of the lens.

[0217] The brightness distribution and observed image in Figure 28 are for a lens pitch of 30 μm. x =k y = 4, α x =α y = 1, β x =β y =0.3. The same applies below. Ripple was reduced by correcting the skirt.

[0218] The brightness distribution and observed image in Figure 29 are for a lens pitch of 60 μm. Ripples were reduced by skirt correction.

[0219] The brightness distribution and observed image in Figure 30 are for a lens pitch of 100 μm. Ripples were reduced by skirt correction.

[0220] The brightness distribution and observed image in Figure 31 are for a lens pitch of 150 μm. Ripples were reduced by skirt correction.

[0221] <Modification 5: Microlens array consisting of hexagonal micro concave lenses>

[0222] Another embodiment of the microarray type diffuser plate is a microlens array in which regular hexagonal concave lenses are arranged in a lattice on the array surface. The concave lenses have a cylindrical concave surface through each cross section that is parallel to the two opposing sides of the concave lens in a planar view and perpendicular to the array surface. The concave lenses have a cylindrical concave surface through each cross section that is perpendicular to the two opposing sides and perpendicular to the array surface. In one embodiment, the lattice is a regular hexagonal lattice. The amount of sag is corrected so that the inclination of the rim increases on each meridian on each cross section parallel to the lattice direction of the concave lenses. The correction causes the rim to rise slightly.

[0223] The range in which the sag amount is corrected is determined in the direction perpendicular to the direction parallel to the two opposing sides, i.e., the x-axis and y-axis directions. The y-axis is parallel to the grid direction. The x-axis is perpendicular to the grid direction. The size of the horizontal range in which the sag amount is corrected in the x-axis direction is Δx. The amount of sag correction is Δz. In the following range of x-coordinate, the sag amount z is expressed as a conic section that is not corrected as shown in the following formula. L x is the lens width. Lens width L x varies depending on the y-coordinate of the cross section. x is the conic constant. r x is the radius of curvature of the conic section.

[0224]

number

[0225]

number

[0226] On the other hand, the correction performed on the rim is preferably performed in the following range of horizontal x coordinates centered on the meridian axis of symmetry:

number

[0227] The size of the horizontal range Δx where the sag amount is corrected is expressed as follows:

[0228]

number

[0229] gamma x is a real number representing the correction order. Preferably, 1≦γ x ≦10, preferably 2<γ x , preferably 3<γ x γ x When = 1, the skirt is a straight line. λ is the wavelength of the light. n is the absolute refractive index of the lens. n is approximated by the relative refractive index of the lens to air.

[0230] The amount of sag correction Δz is expressed as follows:

[0231]

number

[0232] The sag amount z is expressed as a corrected conic section as shown in the following formula.

number

[0233] The size of the horizontal range in which the sag amount is corrected in the y-axis direction is Δy. The amount of sag correction is Δz. In the following range of the y coordinate, the sag amount z is expressed as a conic section that is not corrected as shown in the following formula. L y is the lens width. Lens width L y varies depending on the x-coordinate of the cross section. y is the conic constant. r y is the radius of curvature of the conic section.

[0234]

number

[0235]

number

[0236] On the other hand, the correction performed on the rim is preferably performed in the following range of horizontal y coordinates centered on the meridian axis of symmetry:

number

[0237] The size Δy of the horizontal range in which the sag amount is corrected is expressed as follows:

[0238]

number

[0239] gamma y is a real number representing the correction order. Preferably, 1≦γ y ≦10, preferably 2<γ y , preferably 3<γ y γ y When = 1, the skirt is a straight line. λ is the wavelength of the light. n is the absolute refractive index of the lens. n is approximated by the relative refractive index of the lens to air.

[0240] The amount of sag correction Δz is expressed as follows:

[0241]

number

[0242] The sag amount z is expressed as a corrected conic section as shown in the following formula.

number

[0243] <Variation 6: Reflective hexagonal concave mirror array diffuser>

[0244] Another embodiment of the microarray type diffuser plate is a micro concave mirror array in which regular hexagonal concave mirrors are arranged in a lattice on the array surface. The concave mirror has a cylindrical concave surface through each cross section that is parallel to two opposing sides of the concave mirror in a plan view and perpendicular to the array surface. The concave mirror has a cylindrical concave surface through each cross section that is perpendicular to the two opposing sides and perpendicular to the array surface. In one embodiment, the lattice is a regular hexagonal lattice. The amount of sag is corrected so that the inclination of the rim increases on each meridian on each cross section parallel to the lattice direction of the concave mirror. The correction causes the rim to rise slightly.

[0245] The range in which the sag amount is corrected is determined in the direction perpendicular to the direction parallel to the two opposing sides, i.e., the x-axis and y-axis directions. The y-axis is parallel to the grid direction. The x-axis is perpendicular to the grid direction. The size of the horizontal range in which the sag amount is corrected in the x-axis direction is Δx. The amount of sag correction is Δz. In the following range of x-coordinate, the sag amount z is expressed as a conic section that is not corrected as shown in the following formula. L x is the lens width. Lens width L x varies depending on the y-coordinate of the cross section. x is the conic constant. r x is the radius of curvature of the conic section.

[0246]

number

[0247]

number

[0248] On the other hand, the correction performed on the rim is preferably performed in the following range of horizontal x coordinates centered on the meridian axis of symmetry:

number

[0249] The size of the horizontal range Δx where the sag amount is corrected is expressed as follows:

[0250]

number

[0251] gamma x is a real number representing the correction order. Preferably, 1≦γ x ≦10, preferably 2<γ x , preferably 3<γ x γ x When = 1, the skirt is a straight line. λ is the wavelength of the light. n is the absolute refractive index of the lens. n is approximated by the relative refractive index of the lens to air.

[0252] The amount of sag correction Δz is expressed as follows:

[0253]

number

[0254] The sag amount z is expressed as a corrected conic section as shown in the following formula.

number

[0255] The size of the horizontal range in which the sag amount is corrected in the y-axis direction is Δy. The amount of sag correction is Δz. In the following range of the y coordinate, the sag amount z is expressed as a conic section that is not corrected as shown in the following formula. L y is the lens width. Lens width L y varies depending on the x-coordinate of the cross section. y is the conic constant. r y is the radius of curvature of the conic section.

[0256]

number

[0257]

number

[0258] On the other hand, the correction performed on the rim is preferably performed in the following range of horizontal y coordinates centered on the meridian axis of symmetry:

number

[0259] The size Δy of the horizontal range in which the sag amount is corrected is expressed as follows:

[0260]

number

[0261] gamma y is a real number representing the correction order. Preferably, 1≦γ y ≦10, preferably 2<γ y , preferably 3<γ y γ y When = 1, the skirt is a straight line. λ is the wavelength of the light. n is the absolute refractive index of the lens. n is approximated by the relative refractive index of the lens to air.

[0262] The amount of sag correction Δz is expressed as follows:

[0263]

number

[0264] The sag amount z is expressed as a corrected conic section as shown in the following formula.

number

[0265] <Modification 7: Reflective hexagonal convex mirror array diffuser>

[0266] Another embodiment of the microarray type diffuser is a micro-convex mirror array in which regular hexagonal convex mirrors are arranged in a lattice on the array surface. The convex mirror has a cylindrical concave surface through each cross section that is parallel to two opposing sides of the convex mirror in a plan view and perpendicular to the array surface. The convex mirror has a cylindrical concave surface through each cross section that is perpendicular to the two opposing sides and perpendicular to the array surface. In one embodiment, the lattice is a regular hexagonal lattice. The amount of sag is corrected so that the slope of the skirt increases along each meridian on each cross section that is parallel to the lattice direction of the convex mirror. The correction causes the skirt to droop slightly.

[0267] The range in which the sag amount is corrected is determined in the direction perpendicular to the direction parallel to the two opposing sides, i.e., the x-axis and y-axis directions. The y-axis is parallel to the grid direction. The x-axis is perpendicular to the grid direction. The size of the horizontal range in which the sag amount is corrected in the x-axis direction is Δx. The amount of sag correction is Δz. In the following range of x-coordinate, the sag amount z is expressed as a conic section that is not corrected as shown in the following formula. L x is the lens width. Lens width L x varies depending on the y-coordinate of the cross section. x is the conic constant. r x is the radius of curvature of the conic section.

[0268]

number

[0269]

number

[0270] On the other hand, the correction performed on the skirt is preferably performed in the following range of the horizontal x coordinate centered on the meridian axis of symmetry:

number

[0271] The size of the horizontal range Δx where the sag amount is corrected is expressed as follows:

[0272]

number

[0273] gamma x is a real number representing the correction order. Preferably, 1≦γ x ≦10, preferably 2<γ x , preferably 3<γ x γ x When = 1, the skirt is a straight line. λ is the wavelength of the light. n is the absolute refractive index of the lens. n is approximated by the relative refractive index of the lens to air.

[0274] The amount of sag correction Δz is expressed as follows:

[0275]

number

[0276] The sag amount z is expressed as a corrected conic section as shown in the following formula.

number

[0277] The size of the horizontal range in which the sag amount is corrected in the y-axis direction is Δy. The amount of sag correction is Δz. In the following range of the y coordinate, the sag amount z is expressed as a conic section that is not corrected as shown in the following formula. L y is the lens width. Lens width L y varies depending on the x-coordinate of the cross section. y is the conic constant. r y is the radius of curvature of the conic section.

[0278]

number

[0279]

number

[0280] On the other hand, the correction performed on the skirt is preferably performed in the following range of the horizontal y coordinate centered on the meridian axis of symmetry:

number

[0281] The size Δy of the horizontal range in which the sag amount is corrected is expressed as follows:

[0282]

number

[0283] gamma y is a real number representing the correction order. Preferably, 1≦γ y ≦10, preferably 2<γ y , preferably 3<γ y γ y When = 1, the skirt is a straight line. λ is the wavelength of the light. n is the absolute refractive index of the lens. n is approximated by the relative refractive index of the lens to air.

[0284] The amount of sag correction Δz is expressed as follows:

[0285]

number

[0286] The sag amount z is expressed as a corrected conic section as shown in the following formula.

number

[0287] This application claims priority based on Japanese Patent Application No. 2020-190843, filed on November 17, 2020, the disclosure of which is incorporated herein in its entirety. [Explanation of symbols]

[0288] 30 lens, 31 microlens array, 32 array surface, 34 convex lens surface, 35 beam, Fx focus, Fy focus, Lc brightness curve, Lo brightness curve, Mc meridian, Mo meridian, Px pitch, Py pitch, Sx cross section, Sy cross section, θc divergence angle, θo divergence angle, 2θc taper, 2θo taper, 2θx taper, 2θy taper

Claims

1. A microlens array in which lenses are arranged vertically and horizontally on an array surface, When the vertical and horizontal directions of the lens array are defined as the vertical and horizontal directions of the lens itself, the lens has a cross-cylindrical convex lens surface formed by combining a cylindrical convex surface that is parallel to the vertical direction and extends horizontally through each cross section perpendicular to the array surface, and a cylindrical convex surface that is parallel to the horizontal direction and extends vertically through each cross section perpendicular to the array surface, the amount of sag is corrected so that the inclination of the skirt increases along each meridian on each cross section parallel to the longitudinal and lateral directions, the meridians in each cross section parallel to the longitudinal direction and the lateral direction are formed by conic sections corrected for the amount of sag, In the following range of horizontal coordinate x centered on the axis of symmetry of the meridian, [Equation 1] The sag amount z is expressed as a corrected conic section as follows: [Equation 2] However, Δx and Δz are expressed as follows: [Equation 3] [Equation 4] L is the width of the meridian, r is the radius of curvature of the conic section, k is a conic constant, α is a real number greater than or equal to 0.5 and less than or equal to 2, β is a real number greater than or equal to 0.15 and less than or equal to 0.6, γ is a real number greater than or equal to 1 and less than or equal to 10, λ is the wavelength of visible light, and n is the absolute refractive index of the lens. Microlens array.

2. α is a real number between 0.9 and 1.1, β is a real number between 0.25 and 0.35, and γ is a real number between 2 and 10. The microlens array of claim 1 .

3. λ is 650 nm; The microlens array according to claim 1 or 2.

4. λ is 530 nm; The microlens array according to claim 1 or 2.

5. In the following range of the horizontal coordinate x, [Equation 5] The sag z is expressed as an uncorrected conic section as follows: [Equation 6] The microlens array according to any one of claims 1 to 4.

6. The lens is a rectangular lens, and its vertical and horizontal directions are aligned with the vertical and horizontal directions of the lens array. The microlens array according to any one of claims 1 to 5.

7. The convex lens surface has different diffusion angles in the vertical direction and the horizontal direction. The microlens array according to any one of claims 1 to 6.

8. A microlens array in which lenses are arranged vertically and horizontally on an array surface, When the vertical and horizontal directions of the lens array are defined as the vertical and horizontal directions of the lens itself, the lens has a cross-cylindrical concave lens surface formed by combining a cylindrical concave surface that is parallel to the vertical direction and extends horizontally through each cross section perpendicular to the array surface, and a cylindrical concave surface that is parallel to the horizontal direction and extends vertically through each cross section perpendicular to the array surface, The amount of sag is corrected so that the inclination of the rim increases along each meridian on each cross section parallel to the longitudinal and lateral directions, the meridians in each cross section parallel to the longitudinal direction and the lateral direction are formed by conic sections corrected for the amount of sag, In the following range of horizontal coordinate x centered on the axis of symmetry of the meridian, [Equation 1] The sag amount z is expressed as a corrected conic section as follows: [Equation 2] However, Δx and Δz are expressed as follows: [Equation 3] [Equation 4] L is the width of the meridian, r is the radius of curvature of the conic section, and k is the conic constant; α is a real number greater than or equal to 0.5 and less than or equal to 2, β is a real number greater than or equal to 0.15 and less than or equal to 0.6, γ is a real number greater than or equal to 1 and less than or equal to 10, λ is the wavelength of visible light, and n is the absolute refractive index of the lens. Microlens array.

9. A transmission screen comprising the microlens array according to any one of claims 1 to 8.

10. A head-up display comprising the transmissive screen according to claim 9.

11. A micro concave mirror array in which concave mirrors are arranged vertically and horizontally on an array surface, When the vertical and horizontal directions of the concave mirror are defined as the vertical and horizontal directions of the concave mirror alone, the concave mirror has a cross-cylindrical concave surface formed by merging a horizontally cylindrical concave surface that is parallel to the vertical direction and across each cross section perpendicular to the array surface, and a vertically cylindrical concave surface that is parallel to the horizontal direction and across each cross section perpendicular to the array surface, The amount of sag is corrected so that the inclination of the rim increases along each meridian on each cross section parallel to the longitudinal and lateral directions, the meridians in each cross section parallel to the longitudinal direction and the lateral direction are formed by conic sections corrected for the amount of sag, In the following range of horizontal coordinate x centered on the axis of symmetry of the meridian, [Equation 7] The sag amount z is expressed as a corrected conic section as follows: [Equation 8] However, Δx and Δz are expressed as follows: [Equation 9] [Equation 10] L is the width of the meridian, α is a real number greater than or equal to 0.5 and less than or equal to 2, β is a real number greater than or equal to 0.15 and less than or equal to 0.6, γ is a real number greater than or equal to 1 and less than or equal to 10, r is the radius of curvature of the conic section, and λ is the wavelength of visible light. Micro concave mirror array.

12. α is a real number between 0.9 and 1.1, β is a real number between 0.25 and 0.35, and γ is a real number between 2 and 10. The micro concave mirror array according to claim 11.

13. A micro convex mirror array in which convex mirrors are arranged vertically and horizontally on an array surface, When the vertical and horizontal directions of the convex mirror are defined as the vertical and horizontal directions of the convex mirror alone, the convex mirror has a cross-cylindrical convex surface formed by combining a convex surface that is parallel to the vertical direction and is cylindrical in the horizontal direction in each cross section perpendicular to the array surface, and a convex surface that is parallel to the horizontal direction and is cylindrical in the vertical direction in each cross section perpendicular to the array surface, the amount of sag is corrected so that the inclination of the skirt increases along each meridian on each cross section parallel to the longitudinal and lateral directions, the meridians in each cross section parallel to the longitudinal direction and the lateral direction are formed by conic sections corrected for the amount of sag, In the following range of horizontal coordinate x centered on the axis of symmetry of the meridian, [Equation 7] The sag amount z is expressed as a corrected conic section as follows: [Equation 8] However, Δx and Δz are expressed as follows: [Equation 9] [Equation 10] L is the width of the meridian, α is a real number greater than or equal to 0.5 and less than or equal to 2, β is a real number greater than or equal to 0.15 and less than or equal to 0.6, γ is a real number greater than or equal to 1 and less than or equal to 10, r is the radius of curvature of the conic section, and λ is the wavelength of visible light. Micro-convex mirror array.

14. A reflective screen comprising either the micro-concave mirror array according to claim 11 or the micro-convex mirror array according to claim 13.

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