Microarray-type diffusion plate

Abstract

On an array surface (32) of a microlens array (31), lenses are arranged in a lattice shape. The lenses (30) have convex surfaces which are cylindrical through cross sections parallel to a lattice direction of the lenses (30) and orthogonal to the array surface (32). In each meridian on each cross section of the lenses (30) parallel to the lattice direction, the sag is corrected so that the inclination of the skirt becomes large. In other modes, the lenses have concave surfaces which are cylindrical through cross sections parallel to a lattice direction of the lenses and orthogonal to the array surface. In each meridian on each cross section of the lenses parallel to the lattice direction, the sag is corrected so that the inclination of the edge becomes large.

Description

TECHNICAL FIELD

[0001] The present application relates to a microarray type diffusion plate, and particularly to a transmissive micro-lens array and a reflective micro-lens array. BACKGROUND

[0002] Non-Patent Literature 1 discloses a curved surface configuration of a micro-lens array. A technology is proposed in which a diffusion plate using such a micro-lens array is applied to a head-up display, a laser projector, or the like as a screen. In the case of using a micro-lens array, compared to the case of using a diffusion plate of a milky translucent board (Japanese: milk half board), ground glass, or the like, there is an advantage that speckle noise can be suppressed. Speckle noise refers to a bright portion that is occasionally generated in the diffusion plate depending on the randomness of the microstructure and the arrangement.

[0003] For example, in Patent Literature 1, an image forming apparatus is proposed which has a diffusion plate that uses a laser projector that projects an image formed of a plurality of pixels using laser light as a light source, and a micro-lens array in which a plurality of micro-lenses are arranged. In the case of using a micro-lens array, it is possible to appropriately diffuse the incident light, and it is possible to freely design a desired diffusion angle.

[0004] Paragraph

[0023] of Patent Literature 2 discloses a structured screen surface that can simultaneously control the underlying structure, i.e., the microstructure, of the defining surface and the relative distribution of the underlying structure within the device surface. In contrast to the prior art that depends on the randomness of the microstructure and the arrangement, the control of the surface shape and the relative arrangement of the space is completely deterministic.

[0005] Prior Art Documents

[0006] Patent Literature

[0007] Patent Literature 1: Japanese Patent Application Publication No. 2010-145745

[0008] Patent Literature 2: Japanese Patent Application Publication No. 2004-505306

[0009] Non-Patent Literature

[0010] Non-Patent Literature 1: Ushio Denki Co., Ltd., “Introduction of Ushio’s Microfabrication Examples”, [online], Ushio Denki Co., Ltd., [retrieved on September 23, 2020], the Internet <URL: https: / / www.ushio.co.jp / jp / feature / functional_device / part / >

[0011] Non-Patent Literature 2: Matsushima, Takashi, "Wave Field Tools", [online], March 11, 2021, Department of Electrical and System Engineering, Kansai University, [searched on April 20, 2021], Internet, <URL: http: / / www.laser.ee.kansai-u.ac.jp / WaveFieldTools / > SUMMARY

[0012] PROBLEMS TO BE SOLVED BY THE INVENTION

[0013] The inventors have found that, in the case where a microlens array is used as a screen, unevenness in brightness occurs in an image projected on the microlens. In addition, the inventors have found that, in the case where a microconvex mirror array or a microconcave mirror array is used as a screen, the same unevenness in brightness occurs. The object of the present invention is to provide a scheme for reducing the unevenness in brightness in the microarray-type diffusion plate.

[0014] SCHEME FOR SOLVING THE PROBLEM

[0015] <1> A microlens array in which lenses are arranged in a lattice shape on an array surface, wherein

[0016] the lenses have a convex surface that is cylindrical in cross sections parallel to a lattice direction of the lenses and orthogonal to the array surface, and in each meridian on each cross section parallel to the lattice direction of the lenses, sag amount is corrected so as to make the inclination of a skirt larger.

[0017] <2> A microlens array in which lenses are arranged in a longitudinal direction and a lateral direction on an array surface, wherein

[0018] in the case where the longitudinal direction and the lateral direction in which the lenses are arranged are set as a longitudinal direction and a lateral direction of the lenses individually, the lenses have a convex lens surface that is a cross-cylindrical convex surface combined from a convex surface that is cylindrical toward the lateral direction in cross sections parallel to the longitudinal direction of the lenses and orthogonal to the array surface, and a convex surface that is cylindrical toward the longitudinal direction in cross sections parallel to the lateral direction of the lenses and orthogonal to the array surface, and in each meridian on each cross section parallel to the longitudinal direction and each cross section parallel to the lateral direction, sag amount is corrected so as to make the inclination of a skirt larger.

[0019] <3> The microlens array according to <2>, further comprising

[0020] in each cross section parallel to the longitudinal direction and each cross section parallel to the lateral direction, the meridian is constituted by a conic curve that is corrected with respect to the sag amount.

[0021] <4> The microlens array according to <3>, further comprising

[0022] In the following range of the horizontal coordinate x centered on the axis of symmetry of the meridian line,

[0023] [Math. 1]

[0024]

[0025] The sag z is expressed as a conic curve corrected as shown in the following equation,

[0026] [Math. 2]

[0027]

[0028] where Δx and Δz are as shown below,

[0029] [Math. 3]

[0030]

[0031] [Math. 4]

[0032]

[0033] L is the width of the meridian line, r is the radius of curvature of the conic curve, k is a conic constant, a is a real number of 0.5 or more and 2 or less, β is a real number of 0.15 or more and 0.6 or less, γ is a real number of 1 or more and 10 or less, λ is the wavelength of a visible light ray, and n is the absolute refractive index of the lens.

[0034] <5> The microlens array according to <4>, wherein

[0035] a is a real number of 0.9 or more and 1.1 or less, β is a real number of 0.25 or more and 0.35 or less, and γ is a real number of 2 or more and 10 or less.

[0036] <6> The microlens array according to <4> or <5>, wherein

[0037] λ is 650 nm.

[0038] <7> The microlens array according to <4> or <5>, wherein

[0039] λ is 530 nm.

[0040] <8> The microlens array according to any one of <4> to <7>, wherein

[0041] In the following range of the horizontal coordinate x centered on the axis of symmetry of the meridian line,

[0042] [Math. 5]

[0043]

[0044] The sag z is expressed as an uncorrected conic curve as shown in the following equation,

[0045] [Equation 6]

[0046]

[0047] <9> The microlens array according to any one of <2> to <8>, wherein

[0048] The lens is a rectangular lens, and the longitudinal direction and the lateral direction of the lens coincide with the longitudinal direction and the lateral direction in which the lens is arranged.

[0049] <10> The microlens array according to any one of <2> to <9>, wherein

[0050] The diffusion angle of the convex lens surface in the longitudinal direction is different from the diffusion angle in the lateral direction.

[0051] <11> A microlens array in which lenses are arranged in a lattice pattern on an array surface, wherein

[0052] The lens has a concave surface that is cylindrical by each cross section parallel to a lattice direction of the lens and orthogonal to the array surface,

[0053] In each meridian on each cross section parallel to the lattice direction of the lens, a sag is corrected so that the inclination of the edge becomes larger.

[0054] <12> A microlens array in which lenses are arranged in longitudinal and lateral directions on an array surface, wherein

[0055] In a case where the longitudinal direction and the lateral direction in which the lenses are arranged are set as the longitudinal direction and the lateral direction of the lens individual, the lens has a concave lens surface that is a cross-cylindrical concave lens surface that is combined by a concave surface that is cylindrical toward the lateral direction by each cross section parallel to the longitudinal direction of the lens and orthogonal to the array surface, and a concave surface that is cylindrical toward the longitudinal direction by each cross section parallel to the lateral direction of the lens and orthogonal to the array surface,

[0056] In each meridian on each cross section parallel to the longitudinal direction and each meridian on each cross section parallel to the lateral direction, a sag is corrected so that the inclination of the edge becomes larger.

[0057] <13> A transmissive screen, wherein

[0058] The transmissive screen includes the microlens array according to any one of <1> to <12>.

[0059] <14> A head-up display, wherein

[0060] The head-up display has the transmissive screen described in <13>.

[0061] <15> A micro-concave mirror array in which concave mirrors are arranged in a lattice shape on an array surface, wherein

[0062] The concave mirrors have a concave surface that is a cylindrical surface in each cross section parallel to a lattice direction of the concave mirrors and orthogonal to the array surface,

[0063] In each meridian line on each cross section of the concave mirrors parallel to the lattice direction, a sag amount is corrected so that a gradient of an edge becomes larger.

[0064] <16> A micro-concave mirror array in which concave mirrors are arranged in a longitudinal direction and a lateral direction on an array surface, wherein

[0065] In a case where the longitudinal direction and the lateral direction in which the concave mirrors are arranged are set as a longitudinal direction and a lateral direction of the concave mirror units, the concave mirrors have a concave surface that is a cross-cylindrical surface that is combined from a concave surface that is a cylindrical surface in each cross section parallel to the longitudinal direction of the concave mirrors and orthogonal to the array surface and oriented in the lateral direction, and a concave surface that is a cylindrical surface in each cross section parallel to the lateral direction of the concave mirrors and orthogonal to the array surface and oriented in the longitudinal direction,

[0066] In each meridian line on each cross section parallel to the longitudinal direction and each meridian line on each cross section parallel to the lateral direction, a sag amount is corrected so that a gradient of an edge becomes larger.

[0067] <17> The micro-concave mirror array described in <16>, wherein

[0068] In each cross section parallel to the longitudinal direction and each cross section parallel to the lateral direction, the meridian line is constituted by a conic curve that is corrected with respect to the sag amount.

[0069] <18> The micro-concave mirror array described in <16> or <17>, wherein

[0070] In a range of a horizontal coordinate x centered on a symmetry axis of the meridian line,

[0071] [Math. 7]

[0072]

[0073] A sag amount z is expressed as a conic curve that is corrected as follows,

[0074] [Math. 8]

[0075]

[0076] where Δx and Δz are shown below,

[0077] [Equation 9]

[0078]

[0079] [Equation 10]

[0080]

[0081] L is the width of the meridian, a is a real number of 0.5 or more and 2 or less, β is a real number of 0.15 or more and 0.6 or less, γ is a real number of 1 or more and 10 or less, r is the radius of curvature of the conic section, and λ is the wavelength of a visible light ray.

[0082] <19> The micro-concave mirror array of <18> is further characterized in that,

[0083] a is a real number of 0.9 or more and 1.1 or less, β is a real number of 0.25 or more and 0.35 or less, and γ is a real number of 2 or more and 10 or less.

[0084] <20> A micro-lens array is a micro-convex mirror array in which convex mirrors are arranged in a lattice pattern on an array surface, wherein

[0085] the convex mirrors have convex surfaces that are cylindrical in shape in each cross section parallel to a lattice direction of the convex mirrors and orthogonal to the array surface,

[0086] in each meridian on each cross section of the convex mirrors parallel to the lattice direction, the sag is corrected so that the inclination of the skirt becomes larger.

[0087] <21> A micro-convex mirror array is an array in which convex mirrors are arranged in a lattice pattern on an array surface, wherein

[0088] in a case where a longitudinal direction and a lateral direction in which the convex mirrors are arranged are set as a longitudinal direction and a lateral direction of the convex mirror unit, the convex mirrors have convex surfaces that are cross-cylindrical, the cross-cylindrical convex surfaces being combined from convex surfaces that are cylindrical in shape toward the lateral direction in each cross section parallel to the longitudinal direction of the convex mirrors and orthogonal to the array surface, and convex surfaces that are cylindrical in shape toward the longitudinal direction in each cross section parallel to the lateral direction of the convex mirrors and orthogonal to the array surface,

[0089] in each meridian on each cross section parallel to the longitudinal direction and each meridian on each cross section parallel to the lateral direction, the sag is corrected so that the inclination of the skirt becomes larger.

[0090] <22> A reflection-type screen is a screen in which

[0091] The reflection type screen has either one of the micro-concave mirror arrays of <15> to <19> and the micro-convex mirror arrays of <20> to <21>.

[0092] Effects of the Invention

[0093] By the present invention, it is possible to reduce the unevenness of brightness in the image projected on the microarray type diffusion plate. BRIEF DESCRIPTION OF DRAWINGS

[0094] Figure 1 is a perspective view of a microlens array.

[0095] Figure 2 is a perspective view of a microlens.

[0096] Figure 3 is a cross section and brightness distribution of a reference lens.

[0097] Figure 4 is a cross section and brightness distribution of a correction lens.

[0098] Figure 5 is an enlarged view of a skirt.

[0099] Figure 6 is an enlarged view of a brightness curve.

[0100] Figure 7 is an enlarged view of a brightness curve.

[0101] Figure 8 is a brightness distribution of the emergent light.

[0102] Figure 9 is a graph of sag and correction.

[0103] Figure 10 is a graph of sag and correction near the skirt.

[0104] Figure 11 is a longitudinal brightness distribution.

[0105] Figure 12 is a transverse brightness distribution.

[0106] Figure 13 is a simulation of the observed image.

[0107] Figure 14 is a longitudinal brightness distribution.

[0108] Figure 15 is a transverse brightness distribution.

[0109] Figure 16 is a simulation of the observed image.

[0110] Figure 17is a longitudinal luminance distribution.

[0111] Figure 18 is a lateral luminance distribution.

[0112] Figure 19 is a simulation of the observed image.

[0113] Figure 20 is a luminance distribution and an observed image.

[0114] Figure 21 is a luminance distribution and an observed image.

[0115] Figure 22 is a luminance distribution and an observed image.

[0116] Figure 23 is a cross section of a corrective lens and a luminance distribution.

[0117] Figure 24 is a cross section of a concave mirror and a luminance distribution.

[0118] Figure 25 is an enlarged view of a skirt.

[0119] Figure 26 is a luminance distribution and an observed image.

[0120] Figure 27 is a plan view of a microlens array.

[0121] Figure 28 is a luminance distribution and an observed image.

[0122] Figure 29 is a luminance distribution and an observed image.

[0123] Figure 30 is a luminance distribution and an observed image.

[0124] Figure 31 is a luminance distribution and an observed image. DETAILED DESCRIPTION

[0125] <MICROLENS ARRAY>

[0126] Figure 1 A microlens array 31 provided with lenses 30 is shown. For convenience, the y-axis direction in the drawing is taken as the longitudinal direction, and the x-axis direction is taken as the lateral direction. The direction in which the sag of the lenses 30 increases is taken as the z-axis direction. The microlens array 31 can preferably be used for a transmissive screen. In addition, the transmissive screen can preferably be used for a head-up display.

[0127] In Figure 1In the array, the microlens array 31 has an array surface 32. Lenses 30 are repeatedly arranged on the array surface 32 along the longitudinal and transverse directions. Lenses 30 are rectangular lenses. Lenses 30 can also be square lenses.

[0128] exist Figure 1 In the image, the longitudinal and transverse directions of the rectangle of lens 30 are consistent with the longitudinal and transverse directions of the arrangement of lenses. The cross-section Sy of lens 30 is parallel to the longitudinal direction of lens 30. The cross-section Sx of lens 30 is parallel to the transverse direction of lens 30.

[0129] exist Figure 1 In this diagram, Py represents the longitudinal spacing of lens 30, and Px represents the transverse spacing of lens 30. The spacing Py can be equal to the longitudinal length of the lens, and the spacing Px can be equal to the transverse length of the lens.

[0130] Figure 2 Lens 30 is shown as one of the microlenses in a microlens array. Lens 30 has a cross-cylindrical convex lens surface 34. The convex lens surface 34 is a cylindrical convex surface in the transverse direction and in the longitudinal direction. In one embodiment, lens 30 is a plano-convex lens that is a convex lens only on one side. More specifically, the convex lens surface 34 is cylindrical in the transverse direction through sections other than section Sy, which is parallel to the longitudinal direction of lens 30 and orthogonal to array surface 32. Furthermore, the convex lens surface 34 is cylindrical in the longitudinal direction through sections other than section Sx, which is parallel to the transverse direction of lens 30 and orthogonal to array surface 32. In one embodiment, the term cross-cylindrical may or may not include an elliptic paraboloid. In one embodiment, the term elliptic paraboloid may or may not include a paraboloid of revolution.

[0131] exist Figure 2 In one example shown, the diffusion angles in the longitudinal and transverse directions are different in the convex lens surface 34. A beam 35 orthogonal to the array surface 32 is incident on the convex lens surface 34 and refracted. The beam 35 converges to the focal point Fx at section Sx. After convergence, the taper of the diffused beam 35 is 2θx.

[0132] like Figure 2 As shown, beam 35 converges to focus Fy at section Sy. The taper of the diverging beam 35 after convergence is 2θy. In one configuration illustrated, 2θx is greater than 2θy. In other configurations, 2θx is less than 2θy. In still other configurations, 2θx and 2θy are equal.

[0133] <Correction of Uneven Brightness and Skirt>

[0134] Figure 3A cross section Sx and a luminance distribution of a lens are shown. In the present embodiment, the luminance distribution is a radiant luminance distribution unless otherwise specified. A coordinate axis x represents a horizontal coordinate centered on the axis of symmetry of the meridian of the cross section Sx. Light is set to be incident from the -z direction in the figure. The meridian Mo on the cross section Sx before correction is a conic curve. The conic curve in the present embodiment includes an ellipse, a parabola, and a hyperbola. The ellipse includes a circle. The conic curve does not include a double straight line. The terms of the conic curve are explained as follows. A lens having the meridian Mo before correction is sometimes referred to as a reference lens.

[0135] In Figure 3 , the light beam 35 is incident with respect to the lens. The light beam 35 refracted at the convex lens face 34 is diffused. The angle θ of the light ray after refraction is determined by the x coordinate of the meridian. In the figure, the angle θ is the opening angle of the light ray after refraction with respect to the axis of symmetry of the meridian on the cross section Sx. In the figure, the angle θ is expressed with a positive value and a negative value for convenience. The half width at half maximum (HWHM) of the luminance distribution of the light beam 35 after refraction is a diffusion angle θ0 of the reference lens. In the figure, the diffusion angle θ0 is 10 degrees. The taper of the diffused light beam 35 is 2θ0 = 20 degrees. This taper is different from 2θx shown in Figure 2 .

[0136] In Figure 3 , there is a shoulder of the luminance curve Lo around the angle θ = ±10 degrees. In the vicinity, a ripple of the luminance curve that causes luminance unevenness can be seen.

[0137] Figure 4 A meridian Mc on the cross section Sx after further correction is shown. A lens having the meridian Mc after correction is sometimes referred to as a corrected lens. In one aspect, the center of the meridian Mc is represented by a conic curve. The skirt of the meridian Mc is represented by a corrected conic curve. Here, the skirt refers to the vicinity of the rim of the convex lens shape.

[0138] In Figure 4 , the half width at half maximum of the luminance distribution of the light beam 35 through the corrected lens is a diffusion angle θc. In the figure, θc is 10 degrees. The taper of the diffused light beam 35 is 2θc = 20 degrees.

[0139] Figure 5 is an enlarged view of the skirt of the meridian. The skirt of the meridian Mc is slightly sagging in the +z direction compared with the skirt of the meridian Mo. The size of the range in the horizontal direction in which the sag is corrected is set to Δx. The amount of correction of the sag is set to Δz. Thus, in the meridian on the cross section Sx, the sag is corrected so that the inclination of the skirt is made larger.

[0140] Returning to Figure 4This figure illustrates the change in luminance distribution before and after correction. In this figure, a shoulder peak exists on the luminance curve Lc near angle θ = ±10 degrees. Fluctuations in the luminance curve, leading to uneven luminance, can be observed in this area. However, compared to the luminance curve Lo, the amplitude, i.e., the fluctuation, of the luminance in the luminance curve Lc is reduced.

[0141] Figure 6 The brightness curve Lo is magnified. For each color (red, green, and blue), the fluctuation of each x-coordinate increases. Therefore, an observer will perceive uneven brightness when viewing the microlens. Furthermore, since the peak value also differs for each color, it can be concluded that the lens exhibits chromatic aberration.

[0142] Figure 7 The brightness curve Lc is magnified. For each color—red, green, and blue—the fluctuations are reduced through sag correction. Therefore, even when observing the microlens, it is difficult for an observer to detect brightness unevenness. Furthermore, chromatic aberration is also reduced through sag correction.

[0143] The correction of the amount of sag is in relation to Figure 2 The correction is performed on sections parallel to section Sx shown. Therefore, brightness is unevenly reduced along the longitudinal direction, i.e., the entire y-axis. The correction for sag is also performed in conjunction with... Figure 2 The process is carried out in sections parallel to Sy. Therefore, in the transverse direction, i.e., the entire x-axis direction, brightness is unevenly reduced. In this embodiment, in sections parallel to both the longitudinal and transverse directions, each meridian is composed of a conic section corrected for sag.

[0144] <Analysis of Fluctuations>

[0145] The following describes the correction method. First, let's explain the fluctuations that we hope to reduce through correction. Figure 8 Its directional characteristics are represented by the brightness distribution of the emitted light. The vertical axis represents brightness. Figure 4 Compared to the previous chart, this one is upside down. The horizontal axis is sinθ relative to the angle θ. The dashed line is the center of the brightness amplitude. X m It is the x-coordinate on the lens that produces the peak of the outermost wave. X p Let X be the x-coordinate of the lens end. m To X p Distance X mp It is represented by the following formula.

[0146] [Mathematical Expression 11]

[0147]

[0148] In the formula, λ is the wavelength of the incident light. In the formula, R is the focal length of the lens.

[0149] In the case of projecting visible light toward the microlenses, in one aspect, the wavelength λ is in the range of 400 (nm) to 700 (nm). Desirably, the condition expressed by the following mathematical expression is satisfied in the entire visible light region. In addition, it is effective for reducing the unevenness in brightness in the entire visible light region to analyze the fluctuation with a representative wavelength selected for convenience. In one aspect, a green wavelength, for example, 530 nm, in which the visual sensitivity is high, is set as λ and the fluctuation is analyzed. In another aspect, a red wavelength, for example, 650 nm, in which the unevenness in brightness caused by the diffraction phenomenon is more noticeable, is set as λ and the fluctuation is analyzed. This aspect is effective for the case in which the unevenness in brightness is not noticeable in other visible light regions. In still another aspect, a yellow wavelength, for example, 590 nm, which is intermediate between the green wavelength and the red wavelength, is set as λ and the fluctuation is analyzed.

[0150] <Reference Lens>

[0151] Next, the correction of the sag amount will be described using Figure 9 and Figure 10 . Figure 9 is a graph of the entire sag amount and the correction amount. Figure 10 is a graph of the sag amount and the correction amount near the skirt. First, the reference lens in the graph will be described. The sag amount z of the reference lens is expressed as a conic curve that is not corrected as in the following equation. k is a conic constant. r is the radius of curvature of the conic curve.

[0152] [Equation 12]

[0153]

[0154] <Correction Method>

[0155] The correction method will be described next based on Figure 9 and Figure 10 . First, the center of the lens that is not corrected will be described. In the following range of the x coordinate in the horizontal direction, the sag amount z is expressed as a conic curve that is not corrected as in the above equation.

[0156] [Equation 13]

[0157]

[0158] On the other hand, in Figure 9 and Figure 10 , the correction performed at the skirt is preferably performed in the following range of the x coordinate in the horizontal direction centered on the axis of symmetry of the meridian.

[0159] [Equation 14]

[0160]

[0161] L is the width of the meridian, i.e., the lens diameter. In one configuration, the lens diameter L is equal to the distance between the lenses. Figure 9 and Figure 10 The magnitude of the horizontal range Δx for correcting sag is shown below.

[0162] [Mathematical Expression 15]

[0163]

[0164] α is the 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.

[0165] γ is a real number representing the number of corrections. Preferably, 1 ≤ γ ≤ 10, more preferably 2 < γ, and more preferably 3 < γ. When γ = 1, the skirt is a straight line.

[0166] λ is the wavelength of the light.

[0167] r is the radius of curvature of the conic section.

[0168] n is the absolute refractive index of the lens. n can be approximated by the relative refractive index of the lens relative to air.

[0169] In addition, the following square root is equivalent to Figure 7 The X shown mp .

[0170] [Mathematical Expression 16]

[0171]

[0172] Figure 9 and Figure 10 The correction amount Δz for the sag is shown below.

[0173] [Mathematical Expression 17]

[0174]

[0175] β is a correction coefficient used to convert optical path difference using wavelength units. Preferably, 0 < β < 0.6, more preferably 0.15 < β < 0.45, more preferably 0.2 < β < 0.4, more preferably 0.25 < β < 0.35, and more preferably β = 0.3.

[0176] exist Figure 9 and Figure 10 In this equation, the sag z is expressed as a corrected conic section as follows.

[0177] [Mathematical Expression 18]

[0178]

[0179] <Example 1: Transmissive Screen, Microlens Array>

[0180] Design on a computer Figure 1 The optical properties of the microlens array 31 shown were studied through computational simulation. The Wave Field Library, a toolkit supporting wave optics computation, was used in this simulation. See Non-Patent Literature 2.

[0181] <Example 1-1>

[0182] To study the aforementioned correction width factor α, a design was developed. Figure 2 Lens 30 is shown. Spacing Px and spacing Py are both set to 30 μm. In section Sy, parallel to the longitudinal direction of lens 30, the diffusion angle θo of beam 35 is set to 10 degrees for lens 30 before correction. In section Sx, parallel to the transverse direction of lens 30, the diffusion angle θo of beam 35 is set to 20 degrees for lens 30 before correction. Lens 30 causes the beam to diffuse more widely in the transverse direction relative to the longitudinal direction.

[0183] Figure 11 It is aimed at Figure 2 and Figure 3 The vertical cross-section shown is a graph with luminance as the vertical axis and the angle of refracted light as the horizontal axis. The vertical axis is set to luminance, i.e., Luminance (au). The horizontal axis is set to Angle (deg), representing angle θ. Figure 3 As shown, angle θ is the opening angle of the refracted light rays. Unless otherwise specified, the same applies below.

[0184] Figure 11 The graph in the upper left corner is the graph of the reference lens. The sag z of the reference lens is expressed as an uncorrected conic section as follows. Unless otherwise specified, the same applies below.

[0185] [Mathematical Expression 19]

[0186]

[0187] Radius of curvature r = 40 μm. Conic constant k = -1.0. Wavelength λ = 630 nm.

[0188] exist Figure 11 The other eight charts are for correcting lenses. Refer again... Figure 9 The correction of the skirt hem is explained. Figure 9 Replace the x-axis with the y-axis and conduct the study. Within the following range corresponding to the skirt of the corrective lens,

[0189] [Mathematical Expression 20]

[0190]

[0191] The sag z is expressed as a conic curve after correction as shown in the following equation.

[0192] [Equation 21]

[0193]

[0194] where Δy and Δz are shown below.

[0195] [Equation 22]

[0196]

[0197] [Equation 23]

[0198]

[0199] The lens diameter L is equal to the pitch Py shown in Figure 2 The correction number γ is fixed at 4. The correction width coefficient α is 0.5 to 2.0. The absolute refractive index n is 1.5. The correction coefficient β is fixed at 0.3. The sag z of the portion not subjected to correction is expressed as a conic curve identical to that of the reference lens.

[0200] In Figure 11 , as the correction width coefficient α increases, the fluctuation decreases. However, the fluctuation again increases around α = 1.0.

[0201] Figure 12 is shown for the longitudinal cross section Sx shown in Figure 2 and Figure 3 is shown for the longitudinal cross section Sx shown in

[0202] Figure 12 The chart on the upper left of

[0203] [Equation 24]

[0204]

[0205] The radius of curvature r = 20 μm. The conic constant k = -1.0. The wavelength λ = 630 nm.

[0206] In Figure 12 , the other eight charts are charts of the correction lenses. Reference is again made to Figure 9 The correction of the skirt is described. In the following range corresponding to the skirt of the correction lens,

[0207] [Math. 25]

[0208]

[0209] The sag z is expressed as a conic curve after correction as shown in the following equation.

[0210] [Math. 26]

[0211]

[0212] where Δx and Δz are shown below.

[0213] [Math. 27]

[0214]

[0215] [Math. 28]

[0216]

[0217] The lens diameter L is equal to the pitch Px shown in the figure, and has a value of 30 μm. The correction number γ is fixed at 4. The correction width coefficient α is 0.5 to 2.0. The absolute refractive index n is 1.5. The correction coefficient β is fixed at 0.3. The sag z of the portion not subjected to correction is expressed as a conic curve identical to that of the reference lens. Figure 2

[0218] In the figure, as the correction width coefficient α increases, the fluctuation decreases. However, the fluctuation again increases around α = 1.0. Figure 12

[0219] A simulation of the observed image when the microlens array is observed from above is shown. It is known that in the longitudinal direction and the lateral direction in which the diffusion angle is different, the unevenness of the brightness decreases around α = 1.0. Figure 13 <Example 1-2>

[0220] In order to investigate the correction coefficient β, a lens was designed in the same manner as in <Example 1-1>.

[0221]

[0222] The graph shown in the figure is a graph in which the brightness is taken as the vertical axis and the angle of the refracted light is taken as the horizontal axis, with respect to the longitudinal cross section Sy shown in the figure. Figure 14 Figure 2 Figure 3 The graph in the upper left of the figure is a graph of the reference lens. This reference lens is identical to the reference lens shown in the description of

[0223] Figure 14 The graph in the upper left of the figure is a graph of the reference lens. This reference lens is identical to the reference lens shown in the description of Figure 11 Figure 11 ​​​​The values shown in the description are the same.

[0224] In Figure 14 , the other eight graphs are graphs of the correction lens. Referring again to Figure 9 The correction of the skirt is described. Figure 9 The x axis in

[0225] [Equation 29]

[0226]

[0227] The sag z is expressed as a conic curve after correction as shown in the following equation.

[0228] [Equation 30]

[0229]

[0230] where Δy and Δz are shown below.

[0231] [Equation 31]

[0232]

[0233] [Equation 32]

[0234]

[0235] The lens diameter L is equal to the interval Py shown in Figure 2 , which has a value of 30 μm. The correction number γ is fixed at 4. The correction width coefficient α is fixed at 1. The absolute refractive index n is 1.5. The correction coefficient β is 0.15 to 0.6. The sag z of the portion not subjected to correction is expressed as a conic curve identical to that of the reference lens.

[0236] In Figure 14 , as the correction coefficient β increases, the fluctuation decreases. However, the fluctuation again increases around β = 0.3.

[0237] Figure 15 is a graph showing the brightness as the vertical axis and the angle of the refracted light as the horizontal axis with respect to the longitudinal cross section Sx shown in Figure 2 and Figure 3 .

[0238] Figure 15 The graph in the upper left of Figure 12 is a graph of the reference lens. This reference lens is identical to the reference lens shown in the description of Figure 12 . The radius of curvature r, the conic constant k, and the wavelength λ are identical to the values shown in the description of

[0239] exist Figure 15 The other eight charts are for correcting lenses. Refer again... Figure 9 The correction of the skirt edge is explained. Within the following range corresponding to the skirt edge of the correction lens,

[0240] [Mathematical Expression 33]

[0241]

[0242] The sag z is expressed as a corrected conic section as follows.

[0243] [Mathematical Expression 34]

[0244]

[0245] The values ​​of Δx and Δz are shown below.

[0246] [Mathematical Expression 35]

[0247]

[0248] [Mathematical Expression 36]

[0249]

[0250] Lens diameter L and Figure 2 The spacing Px shown is equal, with a value of 30 μm. The correction order γ is fixed at 4. The correction width coefficient α is fixed at 1. The absolute refractive index n is 1.5. The correction coefficient β is 0.15 to 0.6. The sag z of the uncorrected portion is represented by the same conic section as the reference lens.

[0251] exist Figure 15 In the middle, as the correction coefficient β increases, the fluctuation decreases. However, around the correction coefficient β = 0.3, the fluctuation increases again.

[0252] Figure 16 The diagram shows a simulation of the image observed from above when viewing the microlens array. It can be seen that the unevenness in brightness decreases around β = 0.3 in both the longitudinal and lateral directions at different diffusion angles.

[0253] <Examples 1-3>

[0254] To study the aforementioned correction number γ, a lens was designed in the same manner as in <Example 1-1>.

[0255] Figure 17 It is aimed at Figure 2 and Figure 3 The longitudinal section Sy is shown as a graph with brightness as the vertical axis and the angle of refracted light as the horizontal axis.

[0256] Figure 17 The top left diagram is a diagram of the reference lens. This reference lens is related to... Figure 11 The reference lens shown in the description is the same. The radius of curvature r, conic constant k, and wavelength λ are all the same as those of the reference lens. Figure 11 The values ​​shown in the description are the same.

[0257] exist Figure 17 The other four charts are for correcting lenses. Refer again... Figure 9 The correction of the skirt hem is explained. Figure 9 Replace the x-axis with the y-axis and conduct the study. Within the following range corresponding to the skirt of the corrective lens,

[0258] [Mathematical Expression 37]

[0259]

[0260] The sag z is expressed as a corrected conic section as follows.

[0261] [Mathematical Expression 38]

[0262]

[0263] The values ​​of Δy and Δz are shown below.

[0264] [Mathematical Expression 39]

[0265]

[0266] [Mathematical Expression 40]

[0267]

[0268] Lens diameter L and Figure 2 The spacing Py shown is equal, with a value of 30 μm. The correction order γ is 1 to 4. The correction width coefficient α is fixed at 1. The absolute refractive index n is 1.5. The correction coefficient β is fixed at 0.3. The sag z of the uncorrected portion is represented by the same conic section as the reference lens.

[0269] exist Figure 17 In the middle, as the number of corrections γ increases, the fluctuation decreases.

[0270] Figure 18 It is aimed at Figure 2 and Figure 3 The longitudinal section Sx is shown as a graph with brightness as the vertical axis and the angle of refracted light as the horizontal axis.

[0271] Figure 18 The top left diagram is a diagram of the reference lens. This reference lens is related to... Figure 12The reference lens shown in the description is the same. The radius of curvature r, conic constant k, and wavelength λ are all the same as those of the reference lens. Figure 12 The values ​​shown in the description are the same.

[0272] exist Figure 18 The other four charts are for correcting lenses. Refer again... Figure 9 The correction of the skirt edge is explained. Within the following range corresponding to the skirt edge of the correction lens,

[0273] [Mathematical Expression 41]

[0274]

[0275] The sag z is expressed as a corrected conic section as follows.

[0276] [Mathematical Expression 42]

[0277]

[0278] The values ​​of Δx and Δz are shown below.

[0279] [Mathematical Expression 43]

[0280]

[0281] [Mathematical Expression 44]

[0282]

[0283] Lens diameter L and Figure 2 The spacing Px shown is equal, with a value of 30 μm. The correction order γ is 1 to 4. The correction width coefficient α is fixed at 1. The absolute refractive index n is 1.5. The correction coefficient β is fixed at 0.3. The sag z of the uncorrected portion is represented by the same conic section as the reference lens.

[0284] exist Figure 18 In the middle, as the number of corrections γ increases, the fluctuation decreases.

[0285] Figure 19 The diagram shows a simulation of the image observed from above when viewing the microlens array. It can be seen that, in both the longitudinal and lateral directions with different diffusion angles, the unevenness of brightness decreases as the number of corrections γ approaches 4.

[0286] <Examples 1-4>

[0287] In order to study the lens diameter L, i.e. the lens spacing, the lens was designed in the same way as in <Example 1-1>.

[0288] Figure 20 It is aimed at Figure 2 and Figure 3The diagram shows a graph with brightness as the vertical axis and the angle of refracted light as the horizontal axis, using the longitudinal section Sy and the transverse section Sx as the horizontal axis. A simulation of the observation image when viewing the microlens array from above is also shown. The graphs above are for the reference lens. The graphs below are for the correction lens. The graph on the left is the longitudinal section Sy. The graph on the right is the transverse section Sx.

[0289] Refer again Figure 9 The correction of the skirt hem in the longitudinal direction is explained. Figure 9 Replace the x-axis with the y-axis and conduct the study. Within the following range corresponding to the skirt of the corrective lens,

[0290] [Mathematical Expression 45]

[0291]

[0292] The sag z is expressed as a corrected conic section as follows.

[0293] [Mathematical Expression 46]

[0294]

[0295] The values ​​of Δy and Δz are shown below.

[0296] [Mathematical Expression 47]

[0297]

[0298] [Mathematical Expression 48]

[0299]

[0300] In the longitudinal direction, the lens diameter L and... Figure 2 The spacing Py shown is equal and has a value of 60 μm. The radius of curvature r = 80 μm. The conic constant k = -1.0. The wavelength λ = 630 nm. The correction order γ is fixed at 4. The correction width coefficient α is fixed at 1. The absolute refractive index n is 1.5. The correction coefficient β is fixed at 0.3. The sag z of the uncorrected portion is represented by the same conic section as the reference lens.

[0301] Refer again Figure 9 The correction of the skirt in the lateral direction is explained. Within the following range corresponding to the skirt of the correction lens,

[0302] [Mathematical Expression 49]

[0303]

[0304] The sag z is expressed as a corrected conic section as follows.

[0305] [Mathematical Expression 50]

[0306]

[0307] The values ​​of Δx and Δz are shown below.

[0308] [Mathematical Expression 51]

[0309]

[0310] [Mathematical Expression 52]

[0311]

[0312] In the horizontal direction, the lens diameter L and... Figure 2 The spacing Px shown is equal, with a value of 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 longitudinal direction. The correction order γ is fixed at 4. The correction width coefficient α is fixed at 1. The absolute refractive index n is the same as the absolute refractive index n in the longitudinal direction. The correction coefficient β is fixed at 0.3. The sag z of the uncorrected portion is represented by the same conic section as the reference lens.

[0313] like Figure 20 As shown in the observed images, when the spacing between the horizontal and vertical lenses is both 60 μm, the fluctuation is also reduced by the skirt correction.

[0314] <Examples 1-5>

[0315] In order to study the lens diameter L, i.e. the lens spacing, the lens was designed in the same way as in <Examples 1-4>. Figure 21 The observation methods of the charts and images shown are the same as those of the others. Figure 20 Same. The correction of the skirt hem in both the longitudinal and transverse directions follows the example in <Examples 1-4>.

[0316] In the longitudinal direction, the lens diameter L and... Figure 2 The spacing Py shown is equal and has a value of 100 μm. The radius of curvature r = 133.3 μm. The conic constant k is -1.0. The wavelength λ = 630 nm. The correction order γ is fixed at 4. The correction width coefficient α is fixed at 1. The absolute refractive index n is 1.5. The correction coefficient β is fixed at 0.3. The sag z of the uncorrected portion is represented by the same conic section as the reference lens.

[0317] In the horizontal direction, the lens diameter L and... Figure 2The interval Px shown is equal to 100 μm. The radius of curvature r = 66.7 μm. The conic constant k is -1.0. The wavelength λ is the same as that in the longitudinal direction. The correction number γ is fixed at 4. The correction width coefficient α is fixed at 1. The absolute refractive index n is the same as that in the longitudinal direction. The correction coefficient β is fixed at 0.3. The sag z of the portion not subjected to correction is expressed as a conic curve identical to that of the reference lens.

[0318] As shown in the observation image of Figure 21 , when the interval in the lateral direction of the lens and the interval in the longitudinal direction of the lens are both 100 μm, the fluctuation is also reduced by the correction of the skirt.

[0319] Example 1-6

[0320] To investigate the lens diameter L, i.e., the interval of the lens, the lens was designed in the same manner as in Figure 22 . The observation method of the graphs and observation images shown is the same as in Figure 20 . The correction of the skirt in the longitudinal and lateral directions was imitated from

[0321] In the longitudinal direction, the lens diameter L is equal to the interval Py shown in Figure 2 , which is 150 μm. The radius of curvature r = 200 μm. The conic constant k is -1.0. The wavelength λ = 630 nm. The correction number γ is fixed at 4. The correction width coefficient α is fixed at 1. The absolute refractive index n is 1.5. The correction coefficient β is fixed at 0.3. The sag z of the portion not subjected to correction is expressed as a conic curve identical to that of the reference lens.

[0322] In the lateral direction, the lens diameter L is equal to the interval Px shown in Figure 2 , which is 150 μm. The radius of curvature r = 100 μm. The conic constant k is -1.0. The wavelength λ is the same as that in the longitudinal direction. The correction number γ is fixed at 4. The correction width coefficient α is fixed at 1. The absolute refractive index n is the same as that in the longitudinal direction. The correction coefficient β is fixed at 0.3. The sag z of the portion not subjected to correction is expressed as a conic curve identical to that of the reference lens.

[0323] As shown in the observation image of Figure 22 , when the interval in the lateral direction of the lens and the interval in the longitudinal direction of the lens are both 150 μm, the fluctuation is also reduced by the correction of the skirt.

[0324] Modified Example 1: Microlens array of concave lens

[0325] The microlens array described above is a microlens array of convex lens. A microlens array of concave lens can also be designed as follows.

[0326] Figure 23 A cross section Sx and a luminance distribution of a concave lens constituting a microarray are shown. The coordinate axis x shows a horizontal coordinate centered on the axis of symmetry of the meridian of the cross section Sx. Light is set to be incident from the +z direction in the figure. The meridian Mo on the cross section Sx before correction is a conic curve. The conic curve in the present embodiment includes an ellipse, a parabola, and a hyperbola. The ellipse includes a circle. The conic curve does not include a double straight line. The terms of the conic curve are explained as follows. The lens having the meridian Mo before correction is sometimes referred to as a reference lens.

[0327] Figure 23 A meridian Mc on the cross section Sx after correction is also shown. The lens having the meridian Mc after correction is sometimes referred to as a corrected lens. In one aspect, the center of the meridian Mc is represented by a conic curve. The edge of the meridian Mc is represented by a conic curve after correction. Here, the edge refers to the vicinity of the rim of the concave lens shape. The amount of sagging of the edge is corrected. The edge is slightly raised toward the +z direction.

[0328] The size of the range in the horizontal direction is set to Δx. The amount of correction of the sagging is set to Δz. In the following range of the x coordinate in the horizontal direction, the sagging z is represented by a conic curve that is not corrected as in the following equation. L is the lens width. In one aspect, the lens width L is equal to the pitch of the lens. k is a conic constant. r is the radius of curvature of the conic curve.

[0329] [Equation 53]

[0330]

[0331] [Equation 54]

[0332]

[0333] On the other hand, the correction performed at the edge is preferably performed in the following range of the x coordinate in the horizontal direction centered on the axis of symmetry of the meridian.

[0334] [Equation 55]

[0335]

[0336] The size Δx of the range in the horizontal direction that corrects the sagging is as follows.

[0337] [Equation 56]

[0338]

[0339] γ is a real number representing the number of corrections. It is preferably 1 ≤ γ ≤ 10, preferably 2 < γ, and preferably 3 < γ. When γ = 1, the skirt is a straight line.

[0340] λ is the wavelength of light.

[0341] r is a radius of curvature of the conic section.

[0342] n is an absolute refractive index of the lens. n can be approximated by a relative refractive index of the lens with respect to air.

[0343] The correction amount Δz of the sag amount is shown below.

[0344] [mathematical expression 57]

[0345]

[0346] The sag amount z is expressed as a conic section after correction as shown in the following equation.

[0347] [mathematical expression 58]

[0348]

[0349] In Figure 23 , the light beam 35 of light is incident with respect to the lens. The light beam 35 that is refracted at the concave lens face is diffused. 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 opening angle of the refracted light ray with respect to the axis of symmetry of the meridian on the cross section Sx. In the figure, the angle θ is expressed with positive and negative values for convenience. The half width at half maximum of the radiant intensity distribution of the refracted light beam 35 is the diffusion angle θc of the reference lens. In the figure, the corrected diffusion angle θc is 10 degrees. The diffusion angle θc is equal to the diffusion angle before correction. The cone angle of the diffused light beam 35 is 2θc = 20 degrees. The cone angle is equal to the cone angle before correction.

[0350] As Figure 23 shown, there is a shoulder peak of the luminance curve Lo before correction in the vicinity of the angle θ = ± 10 degrees. In the vicinity, a fluctuation of the luminance curve that causes luminance unevenness is visible. Similarly, there is a shoulder peak of the luminance curve Lc after correction in the vicinity of the angle θ = ± 10 degrees. In the vicinity, a fluctuation of the luminance curve that causes luminance unevenness is visible. However, in the luminance curve Lc, the amplitude of the luminance, that is, the fluctuation is reduced compared to the luminance curve Lo.

[0351] <Modified Example 2: Reflective Concave Mirror Array Type Diffusion Plate>

[0352] The above-described microlens array is a transmissive microlens array that is suitable for a transmissive screen. A reflective concave mirror array type diffusion plate that is suitable for a reflective screen is designed and fabricated as follows.

[0353] Figure 24A cross section of a metal film Mf constituting a reflecting micro-concave mirror array is shown. The concave mirror array is made by using the above-mentioned micro-lens array as a mold and vapor-depositing the metal film Mf on each convex surface thereof. The concave surface of the metal film Mf has a shape into which the convex surface of the convex lens shape shown is transferred. In one aspect, the metal film Mf is composed of aluminum. Figure 2

[0354] In the reflecting micro-concave mirror array, the concave mirrors are arranged in the longitudinal and lateral directions on the array surface. The concave mirrors have a shape into which the micro-lens array shown is transferred. Therefore, in the case where the longitudinal and lateral directions of the concave mirrors are set as the longitudinal and lateral directions of the concave mirror unit, the concave mirrors have a concave surface of a cross-cylinder shape, which is a combination of a concave surface that is a cylinder shape toward the lateral direction by each cross section parallel to the longitudinal direction of the concave mirror and orthogonal to the array surface, and a concave surface that is a cylinder shape toward the longitudinal direction by each cross section parallel to the lateral direction of the concave mirror and orthogonal to the array surface. Figure 1

[0355] In the reflecting micro-concave mirror array, the concave mirrors are arranged in the longitudinal and lateral directions on the array surface. The concave mirrors have a shape into which the micro-lens array shown is transferred. Therefore, in the case where the longitudinal and lateral directions of the concave mirrors are set as the longitudinal and lateral directions of the concave mirror unit, the concave mirrors have a concave surface of a cross-cylinder shape, which is a combination of a concave surface that is a cylinder shape toward the lateral direction by each cross section parallel to the longitudinal direction of the concave mirror and orthogonal to the array surface, and a concave surface that is a cylinder shape toward the longitudinal direction by each cross section parallel to the lateral direction of the concave mirror and orthogonal to the array surface. Figure 24 Figure 3 Figure 4

[0356] In the reflecting micro-concave mirror array, the concave mirrors are arranged in the longitudinal and lateral directions on the array surface. The concave mirrors have a shape into which the micro-lens array shown is transferred. Therefore, in the case where the longitudinal and lateral directions of the concave mirrors are set as the longitudinal and lateral directions of the concave mirror unit, the concave mirrors have a concave surface of a cross-cylinder shape, which is a combination of a concave surface that is a cylinder shape toward the lateral direction by each cross section parallel to the longitudinal direction of the concave mirror and orthogonal to the array surface, and a concave surface that is a cylinder shape toward the longitudinal direction by each cross section parallel to the lateral direction of the concave mirror and orthogonal to the array surface. Figure 24

[0357] Figure 25 is an enlarged view of the edge of the meridian. Here, the edge refers to the vicinity of the rim of the concave mirror shape. For comparison, a skirt of a convex lens for a transmissive diffusion plate is shown on the left. The edge of the concave mirror is shown on the right. In either case, the edge of the meridian Mc is slightly raised in the +z direction compared with the edge of the meridian Mo. In this way, in the meridian on the cross section, a correction amount Δz is added to the sag to make the inclination of the edge larger.

[0358] Figure 24 ​​​​​​​. A shoulder peak of the luminance curve Lo before correction exists in the vicinity of the angle θ = ± 10 degrees. In the vicinity, a fluctuation of the luminance curve that causes luminance unevenness can be seen. Likewise, a shoulder peak of the luminance curve Lc after correction exists in the vicinity of the angle θ = ± 10 degrees. In the vicinity, a fluctuation of the luminance curve that causes luminance unevenness can be seen. However, in the luminance curve Lc, the amplitude of the luminance, that is, the fluctuation is reduced compared to the luminance curve Lo.

[0359] In Figure 24 In the aspect shown in FIG. 6, the correction is performed as follows. First, the center of the concave mirror that is not subjected to correction is explained. In the following range of the horizontal direction x coordinate, the sag z is expressed as the above-mentioned formula as the conic curve that is not subjected to correction as explained in the chapter of <Reference Lens>. On the other hand, the correction at the edge is performed in the following range of the horizontal direction x coordinate centered on the axis of symmetry of the meridian.

[0360] [Equation 59]

[0361]

[0362] The sag z is expressed as the conic curve after correction as the following formula.

[0363] [Equation 60]

[0364]

[0365] As Figure 25 shown in FIG. 6, in the above-mentioned transmissive diffusion plate, when calculating Δx, the difference in the optical path length before and after correction is considered to be n-1 times. Details are referred to the above-mentioned chapter of <Manner of Correction>.

[0366] On the contrary, in the reflective diffusion plate of the present modified example, as shown on the right side of FIG. 7, the light reciprocates in the interval of Δz, and thus the difference in the optical path length before and after correction becomes twice. Figure 25

[0367] Therefore, Δx is as follows.

[0368] [Equation 61]

[0369]

[0370] The coefficient in the root is reduced and Δx is shown as follows.

[0371] [Equation 62]

[0372]

[0373] In addition, Δz is as follows.

[0374] [Equation 63]​

[0375]

[0376] L is the width of the meridian, and α is a real number between 0.5 and 2. In one embodiment, α is 1. β is a real number between 0.15 and 0.6. In another embodiment, β is 0.3. γ is a real number between 1 and 10. 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 between 0.9 and 1.1. β is a real number between 0.25 and 0.35. γ is a real number between 2 and 10.

[0377] Figure 26 It is aimed at Figure 24 The graphs shown are plotted with brightness as the vertical axis and the angle of refracted light as the horizontal axis, representing cross-sections in the transverse (x-direction) and longitudinal (y-direction) directions, respectively. The graph on the left is the longitudinal cross-section graph. The graph on the right is the transverse cross-section graph. Furthermore, a simulation of the observation image when viewing the microlens array from above is shown on the right. The graphs above are for the concave surface as a reference. The graphs below are after correction for sag. As the simulation shows, in the concave mirror, fluctuations are also reduced through edge correction.

[0378] <Modification Example 3: Reflective Convex Mirror Array Diffuser>

[0379] Other forms of micro-array diffusers are arrays of miniature convex mirrors arranged in the longitudinal and transverse directions on an array surface. In one form of the miniature convex mirror array, where the longitudinal and transverse directions of the convex mirrors are defined as the longitudinal and transverse directions of individual convex mirrors, the convex mirrors have intersecting cylindrical convex surfaces. These intersecting cylindrical convex surfaces are formed by combining cylindrical convex surfaces extending transversely through sections parallel to the longitudinal direction of the convex mirrors and orthogonal to the array surface, and cylindrical convex surfaces extending longitudinally through sections parallel to the transverse direction of the convex mirrors and orthogonal to the array surface. Furthermore, in each meridian on each section parallel to the longitudinal and transverse directions, the sag is corrected to increase the inclination of the skirt. In a preferred form, in each section parallel to the longitudinal and transverse directions, the meridians are formed by conic sections corrected for the sag.

[0380] The correction method is as follows. First, the center of the uncorrected convex mirror is described. Within the following range of the horizontal x-coordinate, the sag z is expressed as an uncorrected conic section, as explained in the chapter on "Reference Lenses". On the other hand, the correction performed at the skirt is performed within the following range of the horizontal x-coordinate centered on the axis of symmetry of the meridian.

[0381] [Mathematical Expression 64]

[0382]

[0383] The sag z is expressed as a conic curve after correction as shown in the following equation.

[0384] [Equation 65]

[0385]

[0386] Δx is shown below.

[0387] [Equation 66]

[0388]

[0389] In addition, Δz is shown below.

[0390] [Equation 67]

[0391]

[0392] L is the width of the meridian, and α is a real number of 0.5 or more and 2 or less. In one aspect, α is 1. β is a real number of 0.15 or more and 0.6 or less. In one aspect, β is 0.3. γ is a real number of 1 or more and 10 or less. r is the radius of curvature of the conic curve. λ is the wavelength of the visible light. In a preferred aspect, α is a real number of 0.9 or more and 1.1 or less. β is a real number of 0.25 or more and 0.35 or less. γ is a real number of 2 or more and 10 or less.

[0393] Example 4: Microlens array composed of hexagonal micro-lenticules

[0394] Figure 27 The microlens array 41 is shown from above. The microlens array 41 has a lens 40 that is a regular hexagon and a lens that has the same plan view shape as the lens 40. As with the microlens array 31 shown in Figure 1 The lens 40 and the other lens are arranged in a hexagonal lattice on the array surface, as with the microlens array 31 shown in

[0395] As shown in Figure 27 , an x-axis and a y-axis are set with the center of the lens 40 as the origin. The x-axis is parallel to the opposite sides of the lens 40 in plan view. The y-axis is orthogonal to the opposite sides of the lens 40 in plan view. The y-axis is parallel to the lattice direction. The x-axis is orthogonal to the lattice direction.

[0396] As shown in Figure 27 , the length in the x-axis direction when the lens 40 in plan view is cut parallel to the xz-plane is set as L x . L xis a function of y. A length in the y-axis direction when the lens 40 is cut in plan view parallel to the yz plane is set as L y . L y is a function of x. The lens 40 has a convex surface in a cylindrical shape by each cross section parallel to the x-axis direction and orthogonal to the array surface. The lens 40 has a convex surface in a cylindrical shape by each cross section parallel to the y-axis direction and orthogonal to the array surface.

[0397] Figure 27 below indicates a cross section when the lens 40 is cut with a plane parallel to the xz plane. In any of the cross sections, the lens 40 has a meridian Mo of the same radius of curvature r. The meridian Mo is a conic section. The conic section includes an ellipse, a parabola, and a hyperbola. The ellipse includes a circle. The conic section does not include a double straight line. In the cross section when the lens 40 is cut with a plane parallel to the yz plane, the lens 40 also has a meridian composed of a radius of curvature determined to be the same. It is the conic section. The conic section includes an ellipse, a parabola, and a hyperbola. The ellipse includes a circle. The conic section does not include a double straight line.

[0398] In each meridian Mo on each cross section parallel to the lattice direction of the lens 40, the sag is corrected so that the inclination of the skirt becomes large. That is, the sag of the lens 40 is increased in the +z direction. In addition, in the x-axis, the correction range of the sag is determined. Hereinafter, the axes are simply indicated as the x-axis or the x coordinate. The microlens array 41 can be preferably used for a transmissive screen. In addition, the transmissive screen can be preferably used for a head-up display.

[0399] In Figure 27 , in the following range Δx of the x coordinate in the horizontal direction on the cross section parallel to the xz plane, the sag z is expressed as a conic section not subjected to correction as in the following equation. The correction amount of the sag is set as Δz. L x is a lens width. The lens width L x varies in accordance with the y coordinate of the cross section.

[0400] In the following range of the horizontal coordinate x,

[0401] [Equation 68]

[0402]

[0403] The sag z is expressed as a conic section not subjected to correction as in the following equation,

[0404] [Equation 69]

[0405]

[0406] In the following range of the horizontal coordinate x,

[0407] [Equation 70]

[0408]

[0409] The sag z is expressed as a conic curve that is corrected as shown in the following equation.

[0410] [Equation 71]

[0411]

[0412] where Δx and Δz are as shown below.

[0413] [Equation 72]

[0414]

[0415] [Equation 73]

[0416]

[0417] r x is the radius of curvature of the conic curve. k x is the conic constant. a x is a real number of 0.5 or more and 2 or less. b x is a real number of 0.15 or more and 0.6 or less. g x is a real number of 1 or more and 10 or less. l is the wavelength of the visible light. n is the absolute refractive index of the lens.

[0418] Likewise, in the following range Ay of the y coordinate on the cross section parallel to the yz plane, the sag z is expressed as a conic curve that is not corrected as shown in the following equation. The correction amount of the sag is set to Δz. y is the lens width. The lens width L y varies in accordance with the x coordinate of the cross section.

[0419] In the following range of the horizontal coordinate y,

[0420] [Equation 74]

[0421]

[0422] The sag z is expressed as a conic curve that is not corrected as shown in the following equation,

[0423] [Equation 75]

[0424]

[0425] In the following range of the horizontal coordinate y,

[0426] [Equation 76]

[0427]

[0428] The sag z is expressed as a conic curve after correction as shown in the following equation.

[0429] [Equation 77]

[0430]

[0431] where Δy and Δz are shown below.

[0432] [Equation 78]

[0433]

[0434] [Equation 79]

[0435]

[0436] r y is the radius of curvature of the conic curve. k y is the conic constant. a y is a real number of 0.5 or more and 2 or less. b y is a real number of 0.15 or more and 0.6 or less. g y is a real number of 1 or more and 10 or less. l is the wavelength of the visible light. n is the absolute refractive index of the lens.

[0437] Figure 28 The luminance distribution and the observed image of FIG. 9 are the luminance distribution and the observed image when the pitch of the lens is 30 μm. In the case where correction is present, k x = k y = 4, a x = a y = 1, b x = b y = 0.3. The same applies below. Fluctuation is reduced by correction of the skirt.

[0438] Figure 29 The luminance distribution and the observed image of FIG. 10 are the luminance distribution and the observed image when the pitch of the lens is 60 μm. Fluctuation is reduced by correction of the skirt.

[0439] Figure 30 The luminance distribution and the observed image of FIG. 11 are the luminance distribution and the observed image when the pitch of the lens is 100 μm. Fluctuation is reduced by correction of the skirt.

[0440] Figure 31 The luminance distribution and the observed image of FIG. 12 are the luminance distribution and the observed image when the pitch of the lens is 150 μm. Fluctuation is reduced by correction of the skirt.

[0441] <Variant 5: Microlens array composed of hexagonal micro-concave lenses>

[0442] Another form of the microarray-type diffusion plate is a microlens array in which concave lenses of regular hexagons are arranged in a lattice shape on the array surface. The concave lenses have a concave surface in a cylindrical shape in each cross section parallel to opposite two sides of the concave lens when viewed from above and orthogonal to the array surface. The concave lenses have a concave surface in a cylindrical shape in each cross section orthogonal to the opposite two sides and orthogonal to the array surface. In one form, the lattice is a regular hexagonal lattice. In each meridian in each cross section parallel to the lattice direction of the concave lenses, the sag is corrected so that the inclination of the edge becomes large. By the correction, the edge slightly rises.

[0443] In the directions parallel and orthogonal to the opposite two sides, that is, the x-axis direction and the y-axis direction, the range in which the sag can be corrected is determined. The y-axis is parallel to the lattice direction. The x-axis is orthogonal to the lattice direction. In the x-axis direction, the size of the horizontal direction of the range in which the sag can be corrected is set to Δx. The amount of correction of the sag is set to Δz. In the following range of the x-coordinate, the sag z is expressed as a conic curve that is not corrected as in the following equation. L x L is the lens width. The lens width L x L varies accordingly according to the y-coordinate of the cross section. k x r is the conic constant. r x R is the radius of curvature of the conic curve.

[0444] [Equation 80]

[0445]

[0446] [Equation 81]

[0447]

[0448] On the other hand, the correction at the edge is preferably performed in the following range of the x-coordinate in the horizontal direction centered on the axis of symmetry of the meridian.

[0449] [Equation 82]

[0450]

[0451] The size Δx of the horizontal direction of the range in which the sag is corrected is as follows.

[0452] [Equation 83]

[0453]

[0454] γ x γ is a real number indicating the number of corrections. It is preferably 1 ≤ γ x≤ 10, preferably 2 < γ x , preferably 3 < γ x At γ x = 1, the skirt is a straight line. λ is the wavelength of light. n is the absolute refractive index of the lens. n can be approximated by the relative refractive index of the lens with respect to air.

[0455] The correction amount Δz of the sag amount is as follows.

[0456] [mathematical formula 84]

[0457]

[0458] The sag amount z is expressed as a conic curve that is corrected as follows.

[0459] [mathematical formula 85]

[0460]

[0461] In the y-axis direction, the size of the horizontal range of the corrected sag amount is set to Δy. The correction amount of the sag amount is set to Δz. In the following range of the y coordinate, the sag amount z is expressed as a conic curve that is not corrected as follows. L y is the lens width. The lens width L y varies in accordance with the x coordinate of the cross section. k y is the conic constant. r y is the radius of curvature of the conic curve.

[0462] [mathematical formula 86]

[0463]

[0464] [mathematical formula 87]

[0465]

[0466] On the other hand, the correction at the edge is preferably performed in the following range of the y coordinate in the horizontal direction with the axis of symmetry of the meridian as the center.

[0467] [mathematical formula 88]

[0468]

[0469] The size Δy of the horizontal range of the corrected sag amount is as follows.

[0470] [mathematical formula 89]

[0471]

[0472] γ yis a real number indicating the number of corrections. It is preferably 1 < γ y ≤ 10, preferably 2 < γ y , preferably 3 < γ y . When γ y = 1, the skirt is a straight line. λ is the wavelength of light. n is the absolute refractive index of the lens. n can be approximated by the relative refractive index of the lens with respect to air.

[0473] The correction amount Δz of the sag amount is as follows.

[0474] [mathematical formula 90]

[0475]

[0476] The sag amount z is expressed as a conic curve that is corrected as follows.

[0477] [mathematical formula 91]

[0478]

[0479] <Variant 6: Reflective Hexagonal Concave Mirror Array Type Diffusion Plate>

[0480] Another form of the microarray type diffusion plate is a micro concave mirror array in which regular hexagonal concave mirrors are arranged in a lattice shape on the array surface. The concave mirror has a concave surface that is in the shape of a cylinder by each cross section parallel to the opposite two sides of the concave mirror when viewed from above and orthogonal to the array surface. The concave mirror has a concave surface that is in the shape of a cylinder by each cross section orthogonal to the opposite two sides and orthogonal to the array surface. In one form, the lattice is a regular hexagonal lattice. In each meridian on each cross section of the concave mirror parallel to the lattice direction, the sag amount is corrected so that the inclination of the edge becomes large. By the correction, the edge slightly rises.

[0481] In the direction parallel to the opposite two sides and the direction orthogonal thereto, that is, the x-axis direction and the y-axis direction, the correction range of the sag amount is determined. The y-axis is parallel to the lattice direction. The x-axis is orthogonal to the lattice direction. In the x-axis direction, the size of the range in the horizontal direction of the correction of the sag amount is set to Δx. The correction amount of the sag amount is set to Δz. In the following range of the x coordinate, the sag amount z is expressed as a conic curve that is not corrected as follows. L x is the lens width. The lens width L x varies in accordance with the y coordinate of the cross section. k x is the conic constant. r x is the radius of curvature of the conic curve.

[0482] [mathematical formula 92]

[0483]

[0484] [Math. 93]

[0485]

[0486] On the other hand, the correction at the edge is preferably performed in the following range of the x coordinate in the horizontal direction centered on the axis of symmetry of the meridian.

[0487] [Math. 94]

[0488]

[0489] The size Δx of the range in the horizontal direction in which the sag is corrected is as follows.

[0490] [Math. 95]

[0491]

[0492] γ x is a real number indicating the number of times of correction. It is preferably 1 < γ x ≤ 10, preferably 2 < γ x , and preferably 3 < γ x . When γ x = 1, the skirt is a straight line. λ is the wavelength of light. n is the absolute refractive index of the lens. n can be approximated by the relative refractive index of the lens with respect to air.

[0493] The correction amount Δz of the sag is as follows.

[0494] [Math. 96]

[0495] Δz = 0.15 λ

[0496] The sag z is expressed as a conic curve after correction as follows.

[0497] [Math. 97]

[0498]

[0499] In the y-axis direction, the size of the range in the horizontal direction in which the sag is corrected is set to Δy. The correction amount of the sag is set to Δz. In the following range of the y coordinate, the sag z is expressed as a conic curve before correction as follows. L y is the lens width. The lens width L y varies accordingly depending on the x coordinate of the cross section. k y is the conic constant. r y is the radius of curvature of the conic curve.

[0500] [Math. 98]

[0501]

[0502] [Equation 99]

[0503]

[0504] On the other hand, the correction at the edge is preferably performed in the following range of the y coordinate in the horizontal direction centered on the axis of symmetry of the meridian.

[0505] [Equation 100]

[0506]

[0507] The size Δy of the range in the horizontal direction in which the sag is corrected is as follows.

[0508] [Equation 101]

[0509]

[0510] γ y is a real number indicating the number of times of correction. It is preferably 1 < γ y ≤ 10, and is preferably 2 < γ y , and is preferably 3 < γ y . When γ y = 1, the skirt is a straight line. λ is the wavelength of light. n is the absolute refractive index of the lens. n can be approximated by the relative refractive index of the lens with respect to air.

[0511] The amount of correction Δz of the sag is as follows.

[0512] [Equation 102]

[0513] Δz = 0.15 λ

[0514] The sag z is expressed as a conic curve after correction as follows.

[0515] [Equation 103]

[0516]

[0517] <Variant 7: Reflective Hexagonal Convex Mirror Array Type Diffusion Plate>

[0518] Another form of the microarray-type diffusion plate is a micro-lens array in which convex lenses of regular hexagonal shape are arranged in a lattice pattern on the array surface. The convex lenses have a concave surface in the shape of a cylinder defined by sections parallel to opposite sides of the convex lens when viewed from above and orthogonal to the array surface. The convex lenses have a concave surface in the shape of a cylinder defined by sections orthogonal to the opposite sides and orthogonal to the array surface. In one form, the lattice is a regular hexagonal lattice. In each meridian on a section parallel to the lattice direction of the convex lenses, the amount of sag is corrected so that the inclination of the skirt increases. By the correction, the skirt slightly sags.

[0519] The range of correction of the amount of sag is determined in the direction parallel to the opposite sides and the direction orthogonal thereto, i.e., the x-axis direction and the y-axis direction. The y-axis is parallel to the lattice direction. The x-axis is orthogonal to the lattice direction. In the x-axis direction, the size of the range in the horizontal direction in which the amount of sag is corrected is set to Δx. The amount of correction of the amount of sag is set to Δz. In the following range of the x-coordinate, the amount of sag z is expressed as a conic curve that has not been corrected as in the following equation. L x L is the lens width. The lens width L x varies in accordance with the y-coordinate of the section. k x is a conic constant. r x is the radius of curvature of the conic curve.

[0520] [Equation 104]

[0521]

[0522] [Equation 105]

[0523]

[0524] On the other hand, the correction performed on the skirt is preferably performed in the following range of the x-coordinate in the horizontal direction centered on the axis of symmetry of the meridian.

[0525] [Equation 106]

[0526]

[0527] The size Δx of the range in the horizontal direction in which the amount of sag is corrected is as follows.

[0528] [Equation 107]

[0529]

[0530] γ x is a real number indicating the number of times of correction. It is preferably 1 ≤ γ x ≤ 10, preferably 2 < γ x , and preferably 3 < γx. In the case of γ x= 1, the skirt is a straight line. λ is the wavelength of light. n is the absolute refractive index of the lens. n can be approximated by the relative refractive index of the lens with respect to air.

[0531] The correction amount Δz of the sag amount is as follows.

[0532] [Math. 108]

[0533] Δz = 0.15λ

[0534] The sag amount z is expressed as a conic curve that is corrected as follows.

[0535] [Math. 109]

[0536]

[0537] In the y-axis direction, the size of the horizontal range of the corrected sag amount is set to Δy. The correction amount of the sag amount is set to Δz. In the following range of y coordinates, the sag amount z is expressed as a conic curve that is not corrected as follows. L y L is the lens width. The lens width L y varies in accordance with the x coordinate of the cross section. k y r is the conic constant. r y R is the radius of curvature of the conic curve.

[0538] [Math. 110]

[0539]

[0540] [Math. 111]

[0541]

[0542] On the other hand, the correction performed on the skirt is preferably performed in the following range of y coordinates in the horizontal direction with the axis of symmetry of the meridian as the center.

[0543] [Math. 112]

[0544]

[0545] The size Δy of the horizontal range of the corrected sag amount is as follows.

[0546] [Math. 113]

[0547]

[0548] γ y is a real number that represents the number of times of correction. It is preferably 1 ≤ γ y ≤ 10, preferably 2 < γ y , and preferably 3 < γy At γ y = 1, the skirt is a straight line. λ is the wavelength of the light. n is the absolute refractive index of the lens. n can be approximated by the relative refractive index of the lens with respect to air.

[0549] The correction amount Δz of the sag amount is as follows.

[0550] [mathematical formula 114]

[0551] Δz = 0.15λ

[0552] The sag amount z is expressed as a conic curve after correction as follows.

[0553] [mathematical formula 115]

[0554]

[0555] This invention claims priority to Japanese Application No. 2020-190843, filed November 17, 2020, the disclosure of which is incorporated herein in its entirety.

[0556] BRIEF DESCRIPTION OF DRAWINGS

[0557] 30, lens; 31, microlens array; 32, array surface; 34, convex lens surface; 35, light beam; Fx, focal point; Fy, focal point; Lc, luminance curve; Lo, luminance curve; Mc, meridian; Mo, meridian; Px, pitch; Py, pitch; Sx, cross section; Sy, cross section; θc, divergence angle; θo, divergence angle; 2θc, conicity; 2θo, conicity; 2θx, conicity; 2θy, conicity.

Claims

1. A microlens array, wherein lenses are arranged along the longitudinal and transverse directions on the array surface, wherein, When the longitudinal and transverse directions of the lens arrangement are defined as the longitudinal and transverse directions of the individual lens, the lens has a cross-cylindrical convex lens surface. This cross-cylindrical convex lens surface is formed by combining a cylindrical convex surface that is convex in the transverse direction through sections parallel to the longitudinal direction of the lens and orthogonal to the array plane, and a cylindrical convex surface that is convex in the longitudinal direction through sections parallel to the transverse direction of the lens and orthogonal to the array plane. In each warp line on each section parallel to the longitudinal direction and in each warp line on each section parallel to the transverse direction, the sag is adjusted to increase the inclination of the skirt hem. In each section parallel to the longitudinal direction and in each section parallel to the transverse direction, the meridian is formed by a conic section corrected for the sag. Within the following range of horizontal coordinate x centered on the axis of symmetry of the meridian, The sag z is expressed as a corrected conic section as follows: Wherein, Δx and Δz are shown below, L is the width of the meridian, r is the radius of curvature of the conic section, k is the conic constant, α is a real number greater than or equal to 0.5 and less than 2, β is a real number greater than or equal to 0.15 and less than 0.6, γ is a real number greater than or equal to 1 and less than 10, λ is the wavelength of visible light, and n is the absolute refractive index of the lens.

2. The microlens array according to claim 1, wherein, α is a real number greater than or equal to 0.9 and less than or equal to 1.1, β is a real number greater than or equal to 0.25 and less than or equal to 0.35, and γ is a real number greater than or equal to 2 and less than or equal to 10.

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

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

5. The microlens array according to claim 1 or 2, wherein, Within the following range of the horizontal coordinate x, The sag z is expressed as an uncorrected conic section as follows:

6. The microlens array according to claim 1 or 2, wherein, The lens is a rectangular lens, and the longitudinal and transverse directions of the lens are consistent with the longitudinal and transverse directions of the lens arrangement.

7. The microlens array according to claim 1 or 2, wherein, The diffusion angle of the convex lens surface in the longitudinal direction is different from that in the transverse direction.

8. A microlens array, wherein lenses are arranged in the longitudinal and transverse directions on the array surface, wherein, When the longitudinal and transverse directions of the lens arrangement are defined as the longitudinal and transverse directions of the individual lens, the lens has a cross-cylindrical concave lens surface. This cross-cylindrical concave lens surface is formed by combining a cylindrical concave surface extending transversely through sections parallel to the longitudinal direction of the lens and orthogonal to the array plane, and a cylindrical concave surface extending longitudinally through sections parallel to the transverse direction of the lens and orthogonal to the array plane. In each meridian on each section parallel to the longitudinal direction and in each meridian on each section parallel to the transverse direction, the sag is corrected to increase the inclination of the edge. In each section parallel to the longitudinal direction and in each section parallel to the transverse direction, the meridian is formed by a conic section corrected for the sag. Within the following range of horizontal coordinate x centered on the axis of symmetry of the meridian, The sag z is expressed as a corrected conic section as follows: Wherein, Δx and Δz are shown below, L is the width of the meridian, r is the radius of curvature of the conic section, k is the conic constant, α is a real number greater than or equal to 0.5 and less than 2, β is a real number greater than or equal to 0.15 and less than 0.6, γ is a real number greater than or equal to 1 and less than 10, λ is the wavelength of visible light, and n is the absolute refractive index of the lens.

9. The microlens array according to claim 8, wherein, α is a real number greater than or equal to 0.9 and less than or equal to 1.1, β is a real number greater than or equal to 0.25 and less than or equal to 0.35, and γ is a real number greater than or equal to 2 and less than or equal to 10.

10. A transmissive screen, wherein, The transmissive screen comprises a microlens array as described in any one of claims 1 to 9.

11. A head-up display, wherein, The head-up display has the transmissive screen as described in claim 10.

12. A miniature concave mirror array, wherein concave mirrors are arranged in the longitudinal and transverse directions on the array surface, wherein, When the longitudinal and transverse directions of the arrangement of the concave mirrors are defined as the longitudinal and transverse directions of a single concave mirror, the concave mirror has a cross-cylindrical concave surface. This cross-cylindrical concave surface is formed by combining a cylindrical concave surface extending transversely through sections parallel to the longitudinal direction of the concave mirror and orthogonal to the array plane, and a cylindrical concave surface extending longitudinally through sections parallel to the transverse direction of the concave mirror and orthogonal to the array plane. In each meridian on each section parallel to the longitudinal direction and in each meridian on each section parallel to the transverse direction, the sag is corrected to increase the inclination of the edge. In each section parallel to the longitudinal direction and in each section parallel to the transverse direction, the meridian is formed by a conic section corrected for the sag. Within the following range of horizontal coordinate x centered on the axis of symmetry of the meridian, The sag z is expressed as a corrected conic section as follows: Wherein, Δx and Δz are shown below, L is the width of the meridian, α is a real number greater than or equal to 0.5 and less than 2, β is a real number greater than or equal to 0.15 and less than 0.6, γ is a real number greater than or equal to 1 and less than 10, r is the radius of curvature of the conic section, and λ is the wavelength of visible light.

13. The micro-concave mirror array according to claim 12, wherein, α is a real number greater than or equal to 0.9 and less than or equal to 1.1, β is a real number greater than or equal to 0.25 and less than or equal to 0.35, and γ is a real number greater than or equal to 2 and less than or equal to 10.

14. A miniature convex mirror array, wherein convex mirrors are arranged in the longitudinal and transverse directions on the array surface, wherein, When the longitudinal and transverse directions of the arrangement of the convex mirrors are defined as the longitudinal and transverse directions of a single convex mirror, the convex mirror has a cross-cylindrical convex surface. This cross-cylindrical convex surface is formed by combining a cylindrical convex surface extending transversely through sections parallel to the longitudinal direction of the convex mirror and orthogonal to the array plane, and a cylindrical convex surface extending longitudinally through sections parallel to the transverse direction of the convex mirror and orthogonal to the array plane. In each warp line on each section parallel to the longitudinal direction and in each warp line on each section parallel to the transverse direction, the sag is adjusted to increase the inclination of the skirt hem. In each section parallel to the longitudinal direction and in each section parallel to the transverse direction, the meridian is formed by a conic section corrected for the sag. Within the following range of horizontal coordinate x centered on the axis of symmetry of the meridian, The sag z is expressed as a corrected conic section as follows: Wherein, Δx and Δz are shown below, L is the width of the meridian, α is a real number greater than or equal to 0.5 and less than 2, β is a real number greater than or equal to 0.15 and less than 0.6, γ is a real number greater than or equal to 1 and less than 10, r is the radius of curvature of the conic section, and λ is the wavelength of visible light.

15. The micro convex mirror array according to claim 14, wherein, α is a real number greater than or equal to 0.9 and less than or equal to 1.1, β is a real number greater than or equal to 0.25 and less than or equal to 0.35, and γ is a real number greater than or equal to 2 and less than or equal to 10.

16. A reflective screen, wherein, The reflective screen comprises either the micro-concave mirror array of claim 12 or the micro-convex mirror array of claim 14.

Images