Microarray type diffuser

Abstract

Lenses are arranged in a grid on the array surface (32) of a micro-lens array (31). The lens (30) has a cylindrical convex surface through each cross-section that is parallel to the grid direction of the lens (30) and orthogonal to the array surface (32). The amount of sag is corrected so that the slope of the skirt increases at each meridian on each cross-section parallel to the grid direction of the lens (30). In another embodiment, the lens has a cylindrical concave surface through each cross-section that is parallel to the grid direction of the lens and orthogonal to the array surface. The amount of sag is corrected so that the slope of the rim increases at each meridian on each cross-section parallel to the grid direction of the lens.

Description

Technology Field

[0001] The present invention relates to a microarray type diffuser, and in particular to a transmissive type micro lens array and a reflective type micro convex mirror array and a micro concave mirror array. Background Technology

[0002] Non-patent document 1 discloses a curved structure of a micro-lens array. A technology has been proposed to apply a diffuser plate using such a micro-lens array as a screen in head-up displays or laser projectors. When using a micro-lens array, there is an advantage in that speckle noise can be suppressed compared to when using diffusers such as opaque plates or opaque glass. Speckle noise refers to bright areas that occur accidentally in these diffusers, which depend on the randomness of the microstructure and arrangement.

[0003] For example, Patent Document 1 proposes an image forming device having a laser projector that projects an image formed by an array of multiple pixels using laser light as a light source, and a diffuser plate using a micro-lens array in which multiple micro-lenses are arranged. When a micro-lens array is used, the incident light can be appropriately diffused, and the required diffusion angle can be freely designed.

[0004] Paragraph

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

[0005] Japanese Published Patent Application No. 2010-145745, Japanese Published Patent Application No. 2004-505306

[0006] Ushio Electric Co., Ltd., "Introduction to Ushio's Micro-machining Cases", [online], Ushio Electric Co., Ltd., [Accessed September 23, 2020], Internet<URL : https: / / www.ushio.co.jp / jp / feature / functional_device / part / > Kyoji Matsushima, "Wave Field Tools", [online], March 11, 2020, Laboratory of Optical Information Systems, Faculty of Systems Science and Engineering, Kansai University, [Accessed April 20, 2020], Internet,<URL : http: / / www.laser.ee.kansai-u.ac.jp / WaveFieldTools / > The problem to be solved

[0007] The inventors have discovered that when a micro-lens array is used as a screen, luminance non-uniformity occurs in the image projected onto the micro-lenses. The inventors have also discovered that the same luminance non-uniformity occurs when a micro-convex mirror array or a micro-concave mirror array is used as a screen. The present invention aims to provide a means for reducing luminance non-uniformity in these micro-array type diffusers. means of solving the problem

[0008] <1> As a microlens array in which lenses are grid-arranged on an array plane,

[0009] The lens has a cylindrical convex surface through each cross-section that is parallel to the grid direction of the lens and orthogonal to the array plane, and

[0010] In each meridian on each cross-section parallel to the grid direction of the above lens, the sag amount is corrected so that the slope of the skirt increases.

[0011] Microlens array.

[0012] <2> As a micro-lens array in which lenses are arranged longitudinally and transversely on an array plane,

[0013] When the longitudinal and transverse directions of the array of the above lenses are the longitudinal and transverse directions of the lens group, the lens has a cross-cylindrical convex lens surface formed by the merging of a convex surface that is transversely cylindrical through each cross-section parallel to the longitudinal direction and perpendicular to the array plane, and a convex surface that is longitudinally cylindrical through each cross-section parallel to the transverse direction and perpendicular to the array plane.

[0014] In each meridian on each cross-section parallel to the above longitudinal and transverse directions, the sag amount is corrected so that the slope of the skirt increases,

[0015] Microlens array.

[0016] <3> In each cross-section parallel to the above longitudinal and transverse directions, the meridian is formed by a conic section corrected for the sag amount,

[0017] <2> Microlens array described in

[0018] <4> In the following range of horizontal coordinate x centered on the axis of symmetry of the above meridian,

[0019]

[0020] The sag amount z is represented as a corrected conic section as shown in the following equation, and

[0021]

[0022] However, Δx and Δz are represented as follows, and

[0023]

[0024]

[0025] L is the width of the above meridian, r is the radius of curvature of the above conic section, k is a conic constant, α is a real number between 0.5 and 2 inclusive, β is a real number between 0.15 and 0.6 inclusive, γ is a real number between 1 and 10 inclusive, λ is the wavelength of visible light, and n is the absolute refractive index of the lens,

[0026] <3> Microlens array described in

[0027] <5> 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.

[0028] <4> Microlens array described in

[0029] <6> λ is 650 nm,

[0030] <4> or <5> Microlens array described in

[0031] <7> λ is 530 nm,

[0032] <4> or <5> Microlens array described in

[0033] <8> In the following range of the above horizontal coordinate x,

[0034]

[0035] The sag amount z is represented as an uncorrected conic section as shown in the following equation,

[0036]

[0037] <4> ~ <7> Micro lens array described in any one of the following.

[0038] <9> The above lens is a rectangular lens, and its longitudinal and transverse directions are aligned with the longitudinal and transverse directions of the arrangement of the lens,

[0039] <2> ~ <8> Micro lens array described in any one of the following.

[0040] <10> The above convex lens surface has different diffusion angles in the longitudinal and transverse directions,

[0041] <2> ~ <9> Micro lens array described in any one of the following.

[0042] <11> As a microlens array in which lenses are grid-arranged on an array plane,

[0043] The lens has a cylindrical concave surface through each cross-section that is parallel to the grid direction of the lens and orthogonal to the array plane, and

[0044] In each meridian on each cross-section parallel to the grid direction of the lens, the amount of sag is corrected so that the inclination of the rim increases.

[0045] Microlens array.

[0046] <12> As a micro-lens array in which lenses are arranged longitudinally and transversely on an array plane,

[0047] When the longitudinal and transverse directions of the array of lenses are set as the longitudinal and transverse directions of the lens group, the lens has a cross-cylindrical concave lens surface formed by the merging of a transversely cylindrical concave surface through cross-sections that are parallel to the longitudinal direction and perpendicular to the array plane, and a longitudinally cylindrical concave surface through cross-sections that are parallel to the transverse direction and perpendicular to the array plane.

[0048] In each meridian on each cross-section parallel to the above longitudinal and transverse directions, the sag amount is corrected so that the inclination of the rim increases,

[0049] Microlens array.

[0050] <13> <1> ~ <12> A transmissive screen having a micro-lens array as described in any one of the above.

[0051] <14> <13> A head-up display having a transparent screen as described in

[0052] <15> As a micro-concave mirror array in which concave mirrors are grid-arranged on an array surface,

[0053] The above concave mirror has a cylindrical concave surface through each cross-section that is parallel to the grid direction of the above concave mirror and orthogonal to the array plane, and

[0054] In each meridian line on each cross-section parallel to the grid direction of the above-mentioned concave mirror, the sag amount is corrected so that the inclination of the rim increases.

[0055] Microlens array.

[0056] <16> As a micro-concave mirror array in which concave mirrors are arranged longitudinally and transversely on an array surface,

[0057] When the longitudinal and transverse directions of the above-mentioned concave mirror are defined as the longitudinal and transverse directions of the above-mentioned concave mirror unit, the above-mentioned concave mirror has a cross-cylindrical concave surface formed by merging a transversely cylindrical concave surface through cross-sections that are parallel to the longitudinal direction and perpendicular to the array plane, and a longitudinally cylindrical concave surface through cross-sections that are parallel to the transverse direction and perpendicular to the array plane.

[0058] In each meridian on each cross-section parallel to the above longitudinal and transverse directions, the sag amount is corrected so that the inclination of the rim increases,

[0059] Micro-concave mirror array.

[0060] <17> In each cross-section parallel to the above longitudinal and transverse directions, the meridian is formed by a conic section corrected for the sag amount,

[0061] <16> Micro-concave mirror array described in

[0062] <18> In the following range of horizontal coordinate x centered on the axis of symmetry of the above meridian,

[0063]

[0064] The sag amount z is represented as a corrected conic section as shown in the following equation, and

[0065]

[0066] However, Δx and Δz are represented as follows, and

[0067]

[0068]

[0069] L is the width of the above meridian, α is a real number between 0.5 and 2, β is a real number between 0.15 and 0.6, γ is a real number between 1 and 10, r is the radius of curvature of the above conic section, k is a conic integer, and λ is the wavelength of visible light,

[0070] <16> or <17> Micro-concave mirror array described in

[0071] <19>

[0072] 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.

[0073] <18> Micro-concave mirror array described in

[0074] <20> As a micro-convex mirror array in which convex mirrors are grid-arranged on an array surface,

[0075] The above convex mirror has a cylindrical convex surface through each cross-section that is parallel to the grid direction of the above convex mirror and orthogonal to the array plane, and

[0076] In each meridian on each cross-section parallel to the grid direction of the above convex mirror, the sag amount is corrected so that the slope of the skirt increases.

[0077] Microlens array.

[0078] <21> As a micro-convex mirror array in which convex mirrors are arranged longitudinally and transversely on an array surface,

[0079] When the longitudinal and transverse directions of the above convex mirror are defined as the longitudinal and transverse directions of the above convex mirror unit, the above convex mirror has a cross-cylindrical convex surface formed by merging a convex surface that is cylindrical in the transverse direction through each cross-section that is parallel to the longitudinal direction and perpendicular to the array plane, and a convex surface that is cylindrical in the longitudinal direction through each cross-section that is parallel to the transverse direction and perpendicular to the array plane.

[0080] In each meridian on each cross-section parallel to the above longitudinal and transverse directions, the sag amount is corrected so that the slope of the skirt increases,

[0081] Micro-convex mirror array.

[0082] <22> <15> ~ <19> The micro-concave mirror array described in and <20> ~ <21> A reflective screen having any of the micro-convex mirror arrays described in Effects of the invention

[0083] According to the present invention, luminance non-uniformity in an image projected onto a micro-array type diffuser can be reduced. Brief explanation of the drawing

[0084] FIG. 1 is a perspective view of a micro-lens array. Fig. 2 is a perspective view of a micro lens. Figure 3 shows the cross-section and luminance distribution of a reference lens. Figure 4 shows the cross-section and luminance distribution of the correction lens. Fig. 5 is an enlarged view of the skirt. Figure 6 is an enlarged view of the luminance curve. Fig. 7 is an enlarged view of the luminance curve. Fig. 8 shows the luminance distribution of the emitted light. Figure 9 is a graph of the sag amount and the correction amount. Figure 10 is a graph of the sag amount and correction amount near the skirt. Figure 11 shows the longitudinal luminance distribution. Fig. 12 shows the luminance distribution in the transverse direction. Fig. 13 is a simulation of the observed image. Fig. 14 shows the longitudinal luminance distribution. Fig. 15 shows the luminance distribution in the transverse direction. Fig. 16 is a simulation of the observed image. Fig. 17 shows the longitudinal luminance distribution. Fig. 18 shows the luminance distribution in the transverse direction. Fig. 19 is a simulation of the observed image. Figure 20 shows the luminance distribution and the observed image. Figure 21 shows the luminance distribution and the observed image. Figure 22 shows the luminance distribution and the observed image. Fig. 23 shows the cross-section and luminance distribution of the correction lens. Figure 24 shows the cross-section and luminance distribution of a concave mirror. Fig. 25 is an enlarged view of the skirt. Figure 26 shows the luminance distribution and the observed image. FIG. 27 is a top view of a microlens array. Figure 28 shows the luminance distribution and the observed image. Figure 29 shows the luminance distribution and the observed image. Figure 30 shows the luminance distribution and the observed image. Figure 31 shows the luminance distribution and the observed image. Specific details for implementing the invention

[0085] Microlens Array

[0086] FIG. 1 shows a micro-lens array (31) equipped with a lens (30). For convenience, the y-axis direction in the drawing is defined as the longitudinal direction, and the x-axis direction as the transverse direction. The direction in which the sag amount of the lens (30) increases is defined as the z-axis direction. The micro-lens array (31) can preferably be used in a transmissive screen. Also, the transmissive screen can preferably be used in a head-up display.

[0087] In FIG. 1, the micro-lens array (31) has an array surface (32). Lenses (30) are arranged repeatedly in a longitudinal and transverse manner on the array surface (32). The lenses (30) are rectangular lenses. The lenses (30) may be square lenses.

[0088] In FIG. 1, the longitudinal and transverse directions of the rectangular lens (30) are aligned with the longitudinal and transverse directions of the lens arrangement. The cross-section Sy of the lens (30) is parallel to the longitudinal direction of the lens (30). The cross-section Sx of the lens (30) is parallel to the transverse direction of the lens (30).

[0089] In FIG. 1, the pitch of the longitudinal lens (30) is denoted as Py. The pitch of the transverse lens (30) is denoted as Px. The pitch Py may be equal to the vertical length of the lens. The pitch Px may be equal to the horizontal length of the lens.

[0090] FIG. 2 shows a lens (30) which is a microlens of a microlens array. The lens (30) has a cross-cylindrical convex lens surface (34). The convex lens surface (34) is a convex surface that is cylindrical in the transverse direction and also a convex surface that is cylindrical in the longitudinal direction. In one embodiment, the lens (30) is a plano-convex lens in which only one side is a convex lens. More specifically, the convex lens surface (34) is cylindrical in the transverse direction through each cross-section other than the cross-section Sy which is parallel to the longitudinal direction of the lens (30) and perpendicular to the array surface (32). Additionally, the convex lens surface (34) is cylindrical in the longitudinal direction through each cross-section other than the cross-section Sx which is parallel to the transverse 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 rotational paraboloid.

[0091] In the example shown in FIG. 2, the divergence angles in the longitudinal and transverse directions are different at 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 at the focal point Fx on the cross section Sx. The taper of the beam (35) that diverges after convergence is 2θx.

[0092] As shown in FIG. 2, the beam (35) is also converged at the focal point Fy on cross-section Sy. The taper of the beam (35) after convergence and subsequent divergence is 2θy. In one embodiment shown in the figure, 2θx is greater than 2θy. In another embodiment, 2θx is smaller than 2θy. In another embodiment, 2θx and 2θy are equal.

[0093] <Luminance Non-uniformity and Skirt Correction>

[0094] FIG. 3 shows the cross-section Sx of the lens and the luminance distribution. Unless otherwise noted, the luminance distribution in this embodiment is a radiative luminance distribution. The coordinate axis x represents a horizontal coordinate centered on the axis of symmetry of the meridian of cross-section Sx. Light is assumed to be incident from the -z direction in the drawing. The meridian Mo on 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 straight lines. The term conic section will be interpreted in the same way below. A lens before correction having the meridian Mo may be referred to as a reference lens.

[0095] In FIG. 3, a beam of light (35) is incident on the lens. The beam (35) refracted by the convex lens surface (34) is diffused. The angle θ of the ray after refracting is determined by the x-coordinate of the meridian. In the drawing, the angle θ is the aperture angle of the ray after refracting with respect to the axis of symmetry of the meridian on cross-section Sx. In the drawing, the angle θ is conveniently shown as positive and negative values. The half width at half maximum (HWHM) of the luminance distribution of the beam (35) after refracting is the diffuse angle θo of the reference lens. In the drawing, the diffuse angle θo is 10 degrees. The taper of the diffused beam (35) is 2θo = 20 degrees. This taper is different from 2θx shown in FIG. 2.

[0096] In Fig. 3, there is a shoulder of the luminance curve Lo around the angle θ = ±10 degrees. In this area, ripples in the luminance curve, which cause luminance non-uniformity, are visible.

[0097] Figure 4 also shows the meridian Mc on the cross-section Sx after correction. A corrected lens having the meridian Mc is sometimes 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.

[0098] In FIG. 4, the half-width of the luminance distribution of the beam (35) passing through the correction lens is the diffusion angle θc. In the drawing, θc is 10 degrees. The taper of the diffused beam (35) is 2θc = 20 degrees.

[0099] Figure 5 is an enlarged view of the skirt of the meridian. The skirt of meridian Mc is slightly lower in the +z direction compared to the skirt of meridian Mo. The size of the horizontal range to which the sag amount is corrected is denoted as Δx. The amount of sag correction is denoted as Δz. In this way, the sag amount is corrected so that the slope of the skirt increases on the meridian on cross-section Sx.

[0100] Returning to Fig. 4, this figure shows the change in luminance distribution before and after correction. In this figure, there is a shoulder of the luminance curve Lc around the angle θ = ±10 degrees. In this area, ripples in the luminance curve, which cause luminance non-uniformity, are visible. However, compared to the luminance curve Lo, the range of this luminance deviation, i.e., the ripples, are reduced in the luminance curve Lc.

[0101] Figure 6 shows an enlarged view of the luminance curve Lo. For each color of Red, Green, and Blue, the ripple at each x-coordinate increases. Therefore, when an observer observes the microlens, luminance non-uniformity is detected. Also, since the peaks differ for each color, it can be seen that this lens has chromatic aberration.

[0102] Figure 7 shows an enlarged view of the luminance curve Lc. For each of the colors Red, Green, and Blue, the ripple is reduced by the correction of the sag amount. Therefore, it is difficult for an observer to detect luminance non-uniformity even when observing the microlens. In addition, chromatic aberration is also reduced by the correction of the sag amount.

[0103] Correction of the sag amount is performed at each cross section parallel to cross section Sx as shown in FIG. 2. Accordingly, luminance non-uniformity is reduced throughout the entire longitudinal direction, i.e., the y-axis direction. Correction of the sag amount is also performed at each cross section parallel to cross section Sy as shown in FIG. 2. Accordingly, luminance non-uniformity is reduced throughout the entire transverse direction, i.e., the x-axis direction. In this embodiment, each meridian at each cross section parallel to the longitudinal and transverse directions is formed by a conic section that has undergone correction of the sag amount.

[0104] Ripple's Analysis

[0105] The correction method is described below. First, we will explain the ripple that is intended to be reduced by the correction. Figure 8 shows the directional characteristics of the luminance distribution of the emitted light. The vertical axis represents luminance. Compared to the graph in Figure 4, it is inverted vertically. The horizontal axis represents sinθ with respect to the angle θ. The dashed line represents the center of the luminance deviation range. X m is the x-coordinate on the lens that generates the outermost ripple peak. X p is the x-coordinate of the lens tip. X m From X p Distance to X mp It is represented as shown in the following formula.

[0106]

[0107] In the equation, λ is the wavelength of the incident light. In the equation, R is the focal length of the lens.

[0108] When visible light is projected onto a microlens, in one embodiment, the range of wavelength λ is 400 (nm) to 700 (nm). Ideally, the conditions expressed by the following formulas should be satisfied across the entire visible light region. Additionally, for convenience, selecting a representative wavelength to analyze the ripple is effective in reducing luminance non-uniformity across the entire visible light region. In one embodiment, the ripple is analyzed by setting λ to a green wavelength with high visual sensitivity, for example, 530 nm. In another embodiment, the ripple is analyzed by setting λ to a red wavelength, for example, 650 nm, where luminance non-uniformity due to diffraction phenomena is easily noticeable. This embodiment is effective when luminance non-uniformity is not noticeable in other visible light regions. In another embodiment, the ripple is analyzed by setting λ to a yellow wavelength, for example, 590 nm, which is intermediate between the green and red wavelengths.

[0109] <Reference Lens>

[0110] Next, the correction of the sag amount is explained using Figures 9 and 10. Figure 9 is the entire graph of the sag amount and the correction amount. Figure 10 is the graph of the sag amount and the correction amount near the skirt. First, the reference lens in the graph is explained. The sag amount z of the reference lens is represented as an uncorrected conic section as shown in the following equation. k is the conic constant. r is the radius of curvature of the conic section.

[0111]

[0112] Correction Method

[0113] Based on Figures 9 and 10, the method of correction is described below. First, the center of the lens without correction is described. In the following range of horizontal x-coordinates, the sag amount z is represented as an uncorrected conic section as shown in the above equation.

[0114]

[0115] Meanwhile, in FIGS. 9 and 10, the correction performed on the skirt is preferably performed within the following range of the horizontal x-coordinate centered on the axis of symmetry of the meridian.

[0116]

[0117] L is the width of the meridian, that is, the lens diameter. In one embodiment, the lens diameter L is equal to the pitch of the lens. In FIGS. 9 and 10, the size of the horizontal range Δx, to which the sag amount is corrected, is shown as follows.

[0118]

[0119] 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.

[0120] γ is a real number representing the correction order. Preferably, 1 ≤ γ ≤ 10, preferably 2 < γ, and preferably 3 < γ. When γ = 1, the skirt becomes a straight line.

[0121] λ is the wavelength of the light.

[0122] r is the radius of curvature of the conic section.

[0123] n is the absolute refractive index of the lens. n is approximated as the relative refractive index of the lens with respect to air.

[0124] Also, the following square root is X shown in Fig. 7 mp It is equivalent to.

[0125]

[0126] The correction amount Δz of the sag amount shown in FIGS. 9 and FIGS. 10 is expressed as follows.

[0127]

[0128] β is a correction factor for converting optical path difference into wavelength units. 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.

[0129] In Figures 9 and 10, the sag amount z is represented as a corrected conic section as shown in the following equation.

[0130]

[0131] <Example 1: Transmissive screen, microlens array>

[0132] The microlens array (31) shown in FIG. 1 was designed on a computer, and its optical properties were simulated by calculation. The Wave Field Library, a wave optical calculation support toolkit, was used for the simulation. See Non-patent Literature 2.

[0133] <Example 1-1>

[0134] To examine 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 the cross-section Sy parallel to the longitudinal direction of the lens (30), the diffusion angle θo of the beam (35) was set to 10 degrees relative to the lens (30) before correction. In the cross-section Sx parallel to the transverse direction of the lens (30), the diffusion angle θo of the beam (35) was set to 20 degrees relative to the lens (30) before correction. The lens (30) diffuses the beam more widely in the transverse direction than in the longitudinal direction.

[0135] FIG. 11 shows a graph with luminance on the vertical axis and the angle of light after refraction on the horizontal axis for the longitudinal cross-section Sy shown in FIG. 2 and FIG. 3. The vertical axis is denoted as Luminance (au). The horizontal axis is denoted as Angle (deg), representing the angle θ. As shown in FIG. 3, the angle θ is the aperture angle of the light ray after refraction. Unless otherwise noted, the same applies below.

[0136] The graph in the upper left corner of Fig. 11 is of the reference lens. The sag amount z of the reference lens is represented as an uncorrected conic section as shown in the following equation. Unless otherwise noted, the same applies below.

[0137]

[0138] The radius of curvature r = 40 μm. The cone constant k = -1.0. The wavelength λ = 630 nm.

[0139] In Fig. 11, the other eight graphs are of the correction lens. Referring again to Fig. 9, the correction at the skirt is explained. The x-axis in Fig. 9 is substituted with the y-axis for consideration. In the following range corresponding to the skirt of the correction lens,

[0140]

[0141] The sag amount z is represented as a corrected conic section as shown in the following equation.

[0142]

[0143] However, Δy and Δz are represented as follows.

[0144]

[0145]

[0146] The lens diameter L is the same as the pitch Py shown in FIG. 2, and its value is 30 μm. The correction order γ described 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 factor β described above is fixed at 0.3. The sag amount z in the uncorrected portion is represented as a conic section identical to that of the reference lens.

[0147] In Fig. 11, the ripple is reduced as the correction width coefficient α increases. However, the ripple increases again around the boundary of α = 1.0.

[0148] Figure 12 shows a graph in which the luminance is taken on the vertical axis and the angle of light after refraction is taken on the horizontal axis for the longitudinal cross-section Sx shown in Figures 2 and 3.

[0149] The graph in the upper left corner of Fig. 12 is of the reference lens. The sag amount z of the reference lens is represented as an uncorrected conic section as shown in the following equation.

[0150]

[0151] The radius of curvature r = 20 μm. The cone constant k = -1.0. The wavelength λ = 630 nm.

[0152] In Fig. 12, the other eight graphs are of the correction lens. Referring again to Fig. 9, the correction at the skirt is explained. In the following range corresponding to the skirt of the correction lens,

[0153]

[0154] The sag amount z is represented as a corrected conic section as shown in the following equation.

[0155]

[0156] However, Δx and Δz are represented as follows.

[0157]

[0158]

[0159] The lens diameter L is the same as the pitch Px shown in FIG. 2, and its value is 30 μm. The correction order γ described above is fixed at 4. The correction width factor α is between 0.5 and 2.0. The absolute refractive index n is 1.5. The correction factor β described above is fixed at 0.3. The sag amount z in the uncorrected portion is represented as a conic section identical to that of the reference lens.

[0160] In Fig. 12, the ripple is reduced as the correction width coefficient α increases. However, the ripple increases again around the boundary of α = 1.0.

[0161] Figure 13 shows a simulation of an observation image of a microlens array viewed from a plane. It can be seen that the non-uniformity of luminance is reduced at around α = 1.0 in both the longitudinal and transverse directions, where the diffusion angles are different.

[0162] <Example 1-2>

[0163] In order to examine the correction factor β described above, a lens was designed in the same way as in <Example 1-1>.

[0164] Figure 14 shows a graph in which luminance is taken on the vertical axis and the angle of light after refraction is taken on the horizontal axis for the longitudinal cross-section Sy shown in Figures 2 and 3.

[0165] The graph in the upper left corner of Fig. 14 is of the reference lens. This reference lens is identical to the reference lens described in Fig. 11. The radius of curvature r, the cone constant k, and the wavelength λ are all as described in Fig. 11.

[0166] In Fig. 14, the other eight graphs are of the correction lens. Referring again to Fig. 9, the correction at the skirt is explained. The x-axis in Fig. 9 is substituted with the y-axis for consideration. In the following range corresponding to the skirt of the correction lens,

[0167]

[0168] The sag amount z is represented as a corrected conic section as shown in the following equation.

[0169]

[0170] However, Δy and Δz are represented as follows.

[0171]

[0172]

[0173] The lens diameter L is the same as the pitch Py shown in FIG. 2, and its value is 30 μm. The correction order γ described above is fixed at 4. The correction width coefficient α described above is fixed at 1. The absolute refractive index n is 1.5. The correction factor β is 0.15 to 0.6. The sag amount z in the uncorrected portion is represented as a conic section identical to that of the reference lens.

[0174] In Fig. 14, the ripple is reduced as the correction factor β increases. However, the ripple increases again around the boundary of β = 0.3.

[0175] Figure 15 shows a graph in which the luminance is taken on the vertical axis and the angle of light after refraction is taken on the horizontal axis for the longitudinal cross-section Sx shown in Figures 2 and 3.

[0176] The graph in the upper left corner of Fig. 15 is of the reference lens. This reference lens is identical to the reference lens described in Fig. 12. The radius of curvature r, the cone constant k, and the wavelength λ are all as described in Fig. 12.

[0177] In Fig. 15, the other eight graphs are of the correction lens. Referring again to Fig. 9, the correction at the skirt is explained. In the following range corresponding to the skirt of the correction lens,

[0178]

[0179] The sag amount z is represented as a corrected conic section as shown in the following equation.

[0180]

[0181] However, Δx and Δz are represented as follows.

[0182]

[0183]

[0184] The lens diameter L is the same as the pitch Px shown in FIG. 2, and its value is 30 μm. The correction order γ described above is fixed at 4. The correction width coefficient α described above is fixed at 1. The absolute refractive index n is 1.5. The correction factor β is 0.15 to 0.6. The sag amount z in the uncorrected portion is represented as a conic section identical to that of the reference lens.

[0185] In Fig. 15, the ripple is reduced as the correction factor β increases. However, the ripple increases again around the correction factor β = 0.3.

[0186] Figure 16 shows a simulation of an observation image of a microlens array viewed from a plane. It can be seen that the non-uniformity of luminance is reduced at around β = 0.3 in both the longitudinal and transverse directions, where the diffusion angles are different.

[0187] <Examples 1-3>

[0188] To examine the correction order γ described above, a lens was designed in the same way as in <Example 1-1>.

[0189] Figure 17 shows a graph in which luminance is taken on the vertical axis and the angle of light after refraction is taken on the horizontal axis for the longitudinal cross-section Sy shown in Figures 2 and 3.

[0190] The graph in the upper left corner of Fig. 17 is of the reference lens. This reference lens is identical to the reference lens described in Fig. 11. The radius of curvature r, the cone constant k, and the wavelength λ are all as described in Fig. 11.

[0191] In Fig. 17, the other four graphs are of the corrective lens. Referring again to Fig. 9, the correction at the skirt is explained. The x-axis in Fig. 9 is substituted with the y-axis for consideration. In the following range corresponding to the skirt of the corrective lens,

[0192]

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

[0194]

[0195] However, Δy and Δz are represented as follows.

[0196]

[0197]

[0198] The lens diameter L is the same as the pitch Py shown in FIG. 2, and its value is 30 μm. The correction order γ is 1 to 4. The correction width coefficient α described above is fixed at 1. The absolute refractive index n is 1.5. The correction coefficient β described above is fixed at 0.3. The sag amount z in the uncorrected portion is represented as a conic section identical to that of the reference lens.

[0199] In Fig. 17, as the correction order γ increases, the ripple is reduced.

[0200] Figure 18 shows a graph in which the luminance is taken on the vertical axis and the angle of light after refraction is taken on the horizontal axis for the longitudinal cross-section Sx shown in Figures 2 and 3.

[0201] The graph in the upper left corner of Fig. 18 is of the reference lens. This reference lens is identical to the reference lens described in Fig. 12. The radius of curvature r, the cone constant k, and the wavelength λ are all as described in Fig. 12.

[0202] In Fig. 18, the other four graphs are of the correction lens. Referring again to Fig. 9, the correction at the skirt is explained. In the following range corresponding to the skirt of the correction lens,

[0203]

[0204] The sag amount z is represented as a corrected conic section as shown in the following equation.

[0205]

[0206] However, Δx and Δz are represented as follows.

[0207]

[0208]

[0209] The lens diameter L is the same as the pitch Px shown in FIG. 2, and its value is 30 μm. The correction order γ is 1 to 4. The correction width coefficient α described above is fixed at 1. The absolute refractive index n is 1.5. The correction coefficient β described above is fixed at 0.3. The sag amount z in the uncorrected portion is represented as a conic section identical to that of the reference lens.

[0210] In Fig. 18, the ripple is reduced as the correction order γ increases.

[0211] Figure 19 shows a simulation of an observation image of a microlens array viewed from a plane. It can be seen that the non-uniformity of brightness is reduced as the correction order γ approaches 4 in both the longitudinal and transverse directions, where the diffusion angles are different.

[0212] <Examples 1-4>

[0213] To examine the lens diameter L, i.e., the lens pitch, described above, a lens was designed in the same way as in <Example 1-1>.

[0214] FIG. 20 shows a graph in which luminance is plotted on the vertical axis and the angle of light after refraction is plotted on the horizontal axis for the longitudinal cross-section Sy and the transverse cross-section Sx shown in FIG. 2 and 3. It also shows a simulation of the observed image when the microlens array is viewed in a plane. The upper graphs are all for the reference lens. The lower graphs are all for the correction lens. The graph on the left is for the longitudinal cross-section Sy. The graph on the right is for the transverse cross-section Sx.

[0215] Referring again to Fig. 9, the correction at the skirt in the longitudinal direction is explained. The x-axis in Fig. 9 is substituted with the y-axis for consideration. In the following range corresponding to the skirt of the correction lens,

[0216]

[0217] The sag amount z is represented as a corrected conic section as shown in the following equation.

[0218]

[0219] However, Δy and Δz are represented as follows.

[0220]

[0221]

[0222] In the longitudinal direction, the lens diameter L is the same as the pitch Py shown in Fig. 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 correction order γ described above is fixed at 4. The correction width coefficient α described above is fixed at 1. The absolute refractive index n is 1.5. The correction coefficient β described above is fixed at 0.3. The sag amount z in the uncorrected portion is represented as a conic section identical to that of the reference lens.

[0223] Referring again to Fig. 9, the correction in the skirt in the transverse direction is explained. In the following range corresponding to the skirt of the correction lens,

[0224]

[0225] The sag amount z is represented as a corrected conic section as shown in the following equation.

[0226]

[0227] However, Δx and Δz are represented as follows.

[0228]

[0229]

[0230] In the transverse direction, the lens diameter L is the same as the pitch Px shown in Fig. 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 longitudinal wavelength λ. The correction order γ described above is fixed at 4. The correction width coefficient α described above is fixed at 1. The absolute refractive index n is the same as the longitudinal absolute refractive index n. The correction coefficient β described above is fixed at 0.3. The sag amount z in the uncorrected portion is represented as a conic section identical to that of the reference lens.

[0231] As shown in the observation image of Fig. 20, ripple was reduced by skirt correction even when the pitch of the longitudinal and transverse lenses was 60 μm.

[0232] <Examples 1-5>

[0233] In order to examine the lens diameter L, i.e., the lens pitch, described above, a lens was designed in the same manner as in <Examples 1-4>. The views of each graph and observation image shown in Fig. 21 are the same as those in Fig. 20. The correction of the skirt in the longitudinal and transverse directions followed <Examples 1-4>.

[0234] In the longitudinal direction, the lens diameter L is the same as the pitch Py shown in Fig. 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 = 630 nm. The correction order γ described above is fixed at 4. The correction width coefficient α described above is fixed at 1. The absolute refractive index n is 1.5. The correction coefficient β described above is fixed at 0.3. The sag amount z in the uncorrected portion is represented as a conic section identical to that of the reference lens.

[0235] In the transverse direction, the lens diameter L is the same as the pitch Px shown in Fig. 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 longitudinal wavelength λ. The correction order γ described above is fixed at 4. The correction width coefficient α described above is fixed at 1. The absolute refractive index n is the same as the longitudinal absolute refractive index n. The correction coefficient β described above is fixed at 0.3. The sag amount z in the uncorrected portion is represented as a conic section identical to that of the reference lens.

[0236] As shown in the observation image of Fig. 21, ripple was reduced by the correction of the skirt even when the pitch of the vertical and horizontal lenses was 100 μm.

[0237] <Examples 1-6>

[0238] To examine the lens diameter L, i.e., the lens pitch, described above, a lens was designed in the same manner as in <Examples 1-4>. The views of each graph and observation image shown in Fig. 22 are the same as those in Fig. 20. The correction of the skirt in the longitudinal and transverse directions followed <Examples 1-4>.

[0239] In the longitudinal direction, the lens diameter L is the same as the pitch Py shown in Fig. 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 correction order γ described above is fixed at 4. The correction width coefficient α described above is fixed at 1. The absolute refractive index n is 1.5. The correction coefficient β described above is fixed at 0.3. The sag amount z in the uncorrected portion is represented as a conic section identical to that of the reference lens.

[0240] In the transverse direction, the lens diameter L is the same as the pitch Px shown in Fig. 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 longitudinal wavelength λ. The correction order γ described above is fixed at 4. The correction width coefficient α described above is fixed at 1. The absolute refractive index n is the same as the longitudinal absolute refractive index n. The correction coefficient β described above is fixed at 0.3. The sag amount z in the uncorrected portion is represented as a conic section identical to that of the reference lens.

[0241] As shown in the observation image of Fig. 22, ripple was reduced by the correction of the skirt even when the pitch of the longitudinal and transverse lenses was 150 μm.

[0242] <Variation Example 1: Microlens Array of Concave Lenses>

[0243] The micro-lens array described above is a micro-array of convex lenses. A micro-array of concave lenses may also be designed as follows.

[0244] FIG. 23 shows the cross-section Sx and luminance distribution of a concave lens constituting a microarray. The coordinate axis x represents a horizontal coordinate centered on the axis of symmetry of the meridian of cross-section Sx. Light is assumed to be incident from the +z direction in the figure. The meridian Mo on cross-section Sx before correction is a conic section. In this embodiment, conic sections include ellipses, parabolas, and hyperbolas. Ellipses include circles. Conic sections do not include straight lines. The term conic section will be interpreted in the same way below. A lens before correction having the meridian Mo may be referred to as a reference lens.

[0245] Figure 23 also shows the meridian Mc on the cross-section Sx after correction. A lens after correction having the meridian Mc is sometimes referred to as 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 amount of the rim is corrected. The rim is pushed up slightly in the +z direction.

[0246] Let Δx be the size of the range in the horizontal direction. Let Δz be the correction amount for the sag amount. In the range of the horizontal x-coordinates below, the sag amount z is represented as an uncorrected conic section as shown in the following formula. L is the lens width. In one embodiment, the lens width L is equal to the lens pitch. k is a conic constant. r is the radius of curvature of the conic section.

[0247]

[0248]

[0249] Meanwhile, it is preferable that the correction performed on the rim be carried out within the following range of the horizontal x-coordinate centered on the axis of symmetry of the meridian.

[0250]

[0251] The size of the horizontal range Δx, to which the sag amount is corrected, is expressed as follows.

[0252]

[0253] γ is a real number representing the correction order. Preferably, 1 ≤ γ ≤ 10, preferably 2 < γ, and preferably 3 < γ. When γ = 1, the skirt becomes a straight line.

[0254] λ is the wavelength of the light.

[0255] r is the radius of curvature of the conic section.

[0256] n is the absolute refractive index of the lens. n is approximated as the relative refractive index of the lens with respect to air.

[0257] The correction amount Δz of the sag amount is expressed as follows.

[0258]

[0259] The sag amount z is represented as a corrected conic section as shown in the following equation.

[0260]

[0261] In FIG. 23, a beam of light (35) is incident on the lens. The beam (35) refracted from the concave lens surface is diffused. The angle θ of the ray after refraction is determined by the x-coordinate of the meridian. In the drawing, the angle θ is the aperture angle of the ray after refraction with respect to the axis of symmetry of the meridian on cross-section Sx. In the drawing, the angle θ is shown as positive and negative values ​​for convenience. The half-width of the radiation luminance distribution of the beam (35) after refraction is the diffuse angle θc of the reference lens. In the drawing, the diffuse angle θc after correction is 10 degrees. The diffuse angle θc is the same as the diffuse angle before correction. The taper of the diffused beam (35) is 2θc = 20 degrees. This taper is the same as the taper before correction.

[0262] As shown in Fig. 23, there is a shoulder of the luminance curve Lo before correction around an angle θ = ±10 degrees. In this area, ripples in the luminance curve that cause luminance non-uniformity are visible. Likewise, there is a shoulder of the luminance curve Lc after correction around an angle θ = ±10 degrees. In this area, ripples in the luminance curve that cause luminance non-uniformity are visible. However, in the luminance curve Lc, compared to the luminance curve Lo, the range of luminance deviation, i.e., ripples, is reduced.

[0263] <Variation Example 2: Reflective Concave Mirror Array Type Diffusion Plate>

[0264] The micro-lens array described above is a transmissive micro-lens array preferred for a transmissive screen. A reflective concave mirror array type diffuser preferred for a reflective screen is designed and manufactured as follows.

[0265] FIG. 24 shows a cross-section of a metal film Mf that constitutes a reflective micro-concave mirror array. The concave mirror array is manufactured by using the shape of the micro-lens array described above as a mold and depositing a metal film Mf on 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 FIG. 2. In one embodiment, the metal film Mf is made of aluminum.

[0266] In a reflective micro-concave mirror array, concave mirrors are arranged longitudinally and transversely on the array surface. It has a shape that is a projection of the micro-lens array shown in Fig. 1. Therefore, when the longitudinal and transverse directions of the concave mirrors are defined as the longitudinal and transverse directions of the concave mirrors themselves, the concave mirrors have a cross-cylindrical concave surface formed by merging a concave surface that is transversely cylindrical through cross-sections that are parallel to the longitudinal direction and perpendicular to the array surface, and a concave surface that is longitudinally cylindrical through cross-sections that are parallel to the transverse direction and perpendicular to the array surface.

[0267] In FIG. 24, the concave surface of the metal film Mf has the same meridian as meridian Mc. The meridian Mc after correction is as described using FIG. 3 and 4. The meridian Mo before correction is also the same.

[0268] In FIG. 24, a beam of light traveling through air or vacuum is incident on a concave mirror array from the +z direction. The beam reflected from the concave surface is diffused. The angle θ of the light ray after reflection is determined by the x-coordinate of the meridian. In the drawing, the angle θ is the aperture angle of the light ray after reflection with respect to the axis of symmetry of the meridian on the cross-section. In the drawing, the angle θ is shown as positive and negative values ​​for convenience. The half-width of the radiation luminance distribution of the beam after reflection is the diffuse angle θc of the reference lens. In the drawing, the diffuse angle θc after correction is 10 degrees. The diffuse angle θc is the same as the diffuse angle before correction. The taper of the diffused beam (35) is 2θc = 20 degrees. This taper is the same as the taper before correction.

[0269] Figure 25 is an enlarged view of the rim of a meridian. Here, the rim refers to the vicinity of the edge of the concave mirror shape. For comparison, the skirt of a convex lens used in a transmissive diffuser is shown on the left. The rim of the concave mirror is shown on the right. In both cases, the rim of meridian Mc is pushed up 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 to increase the inclination of the rim on the meridian in the cross-section.

[0270] Returning to Fig. 24. There is a shoulder of the luminance curve Lo before correction around an angle θ = ±10 degrees. In this area, ripples in the luminance curve, which cause luminance non-uniformity, are visible. Likewise, there is a shoulder of the luminance curve Lc after correction around an angle θ = ±10 degrees. In this area, ripples in the luminance curve, which cause luminance non-uniformity, are visible. However, in the luminance curve Lc, compared to the luminance curve Lo, the range of luminance deviation, i.e., ripples, is reduced.

[0271] In one embodiment shown in FIG. 24, the method of correction is as follows. First, the center of the concave mirror without correction is described. In the following range of horizontal x-coordinates, the sag amount z is represented as an uncorrected conic section, such as the equation described in the <Reference Lens> section. Meanwhile, the correction performed on the rim is carried out in the following range of horizontal x-coordinates centered on the axis of symmetry of the meridian.

[0272]

[0273] The sag amount z is represented as a corrected conic section as shown in the following equation.

[0274]

[0275] As shown in FIG. 25, in the transmissive diffuser described above, when calculating Δx, the difference in optical path length before and after correction is considered to be n-1 times. For details, refer to the chapter <Correction Method> described above.

[0276] In this regard, in the reflective diffuser plate related to the present variation example, as shown on the right side of FIG. 25, the light travels back and forth along the Δz section, so the difference in optical path length before and after correction becomes twofold.

[0277] Therefore, Δx is expressed as follows.

[0278]

[0279] By canceling out the coefficients inside the square root, Δx is expressed as follows.

[0280]

[0281] Also, Δz is expressed as follows.

[0282]

[0283] L is the width of the meridian, and α is a real number greater than or equal to 0.5 and less than or equal to 2. In one embodiment, α is 1. β is a real number greater than or equal to 0.15 and less than or equal to 0.6. In one embodiment, β is 0.3. 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. λ is the wavelength of visible light. In a preferred embodiment, 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. γ is a real number greater than or equal to 2 and less than or equal to 10.

[0284] Figure 26 shows a graph in which luminance is plotted on the vertical axis and the angle of light after refraction is plotted on the horizontal axis for the transverse (x-direction) cross-section shown in Figure 24 and the longitudinal (y-direction) cross-section perpendicular thereto. The graph on the left is for the longitudinal cross-section. The graph on the right is for the transverse cross-section. Also, the right side shows a simulation of the observation image when viewing the microlens array in a plane. The top image is for the reference concave surface. The bottom image is for the sag amount corrected. As shown in this simulation, ripple was reduced by rim correction even in the concave mirror.

[0285] <Variation Example 3: Reflective Convex Mirror Array Type Diffusion Plate>

[0286] Another embodiment of the micro-array type diffuser is a micro-convex mirror array in which convex mirrors are arranged longitudinally and transversely on an array surface. In one embodiment of the micro-convex mirror array, when the longitudinal and transverse directions of the convex mirrors are defined as the longitudinal and transverse directions of the individual convex mirrors, the convex mirror has a cross-cylindrical convex surface formed by merging a convex surface that is cylindrical in the transverse direction through each cross-section parallel to the longitudinal direction and perpendicular to the array surface, and a convex surface that is cylindrical in the longitudinal direction through each cross-section parallel to the transverse direction and perpendicular to the array surface. Additionally, the sag amount is corrected so that the slope of the skirt increases at each meridian on each cross-section parallel to the longitudinal and transverse directions. In a preferred embodiment, the meridian on each cross-section parallel to the longitudinal and transverse directions is formed by a conical curve that has undergone sag amount correction.

[0287] The method of correction is as follows. First, the center of the convex mirror without correction is described. In the following range of horizontal x-coordinates, the sag amount z is represented as an uncorrected conic section, such as the equation described above in the <Reference Lens> chapter. Meanwhile, the correction performed on the skirt is carried out in the following range of horizontal x-coordinates centered on the axis of symmetry of the meridian.

[0288]

[0289] The sag amount z is represented as a corrected conic section as shown in the following equation.

[0290]

[0291] Δx is expressed as follows.

[0292]

[0293] Also, Δz is expressed as follows.

[0294]

[0295] L is the width of the meridian, and α is a real number greater than or equal to 0.5 and less than or equal to 2. In one embodiment, α is 1. β is a real number greater than or equal to 0.15 and less than or equal to 0.6. In one embodiment, β is 0.3. 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. λ is the wavelength of visible light. In a preferred embodiment, 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. γ is a real number greater than or equal to 2 and less than or equal to 10.

[0296] <Variation Example 4: Microlens array composed of hexagonal microconvex lenses>

[0297] The top of FIG. 27 shows a micro-lens array (41) viewed in a plane. The micro-lens array (41) comprises a regular hexagonal lens (40) and a lens having a shape viewed in the same plane as the lens. Similar to the micro-lens array (31) shown in FIG. 1, the lens (40) and other lenses are arranged in a hexagonal grid on the array plane. In this example, the hexagonal grid is a regular hexagonal grid.

[0298] As shown in FIG. 27, the x-axis and y-axis are established with the center of the lens (40) as the origin. The x-axis is parallel to the two opposing sides of the lens (40) as viewed in a plane. The y-axis is orthogonal to the two opposing sides of the lens (40) as viewed in a plane. The y-axis is parallel to the grid direction. The x-axis is orthogonal to the said grid direction.

[0299] As shown in FIG. 27, when the lens (40) viewed in a plane is cut parallel to the xz-plane, the length in the x-axis direction is L x It shall be done as. L x is a function of y. L is the length in the y-axis direction when the lens (40) viewed in a plane is cut parallel to the yz-plane. y It shall be done as. L yis a function of x. The lens (40) has a cylindrical convex surface through each cross-section that is parallel to the x-axis direction and orthogonal to the array plane. The lens (40) has a cylindrical convex surface through each cross-section that is parallel to the y-axis direction and orthogonal to the array plane.

[0300] The bottom of FIG. 27 shows a cross-section when the lens (40) is cut by a plane parallel to the xz-plane. In either cross-section, the lens (40) has a meridian Mo with the same radius of curvature r. The meridian Mo is a conic section. Conic sections include ellipses, parabolas, and hyperbolas. Ellipses include circles. Conic sections do not include two straight lines. In the cross-section when the lens (40) is cut by a plane parallel to the yz-plane, the lens (40) also has a meridian formed by a radius of curvature determined in the same way. This is a conic section. Conic sections include ellipses, parabolas, and hyperbolas. Ellipses include circles. Conic sections do not include two straight lines.

[0301] For each meridian Mo on each cross-section parallel to the grid direction of the above lens (40), the sag amount is corrected so that the slope of the skirt increases. That is, the sag amount of the lens (40) increases in the +z direction. Also, a range in which the sag amount is corrected in the x-axis is defined. Each of these axes is hereinafter simply referred to as the x-axis or x-coordinate. The micro-lens array (41) can preferably be used in a transmissive screen. Also, the transmissive screen can preferably be used in a head-up display.

[0302] In FIG. 27, for the following range Δx of the horizontal x-coordinate on a cross-section parallel to the xz-plane, the sag amount z is represented as an uncorrected conic section as shown in the following equation. Let Δz be the correction amount for the sag amount. L x is the lens width. Lens width L xIt changes according to the y-coordinate of the cross-section.

[0303] In the following range of horizontal coordinate x,

[0304]

[0305] The sag amount z is represented as an uncorrected conic section as shown in the following equation, and

[0306]

[0307] In the following range of horizontal coordinate x,

[0308]

[0309] The sag amount z is represented as a corrected conic section as shown in the following equation.

[0310]

[0311] However, Δx and Δz are represented as follows.

[0312]

[0313]

[0314] r x k is the radius of curvature of the conic section. x is a cone integer. α x is a real number between 0.5 and 2 inclusive. β x is a real number between 0.15 and 0.6 inclusive. γ x ε is a real number between 1 and 10. λ is the wavelength of visible light. n is the absolute refractive index of the lens.

[0315] Likewise, for 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 equation. Let Δz be the correction amount for the sag amount. L y is the lens width. Lens width L y It changes depending on the x-coordinate of the cross-section.

[0316] In the following range of the horizontal coordinate y,

[0317]

[0318] The sag amount z is represented as an uncorrected conic section as shown in the following equation, and

[0319]

[0320] In the following range of the horizontal coordinate y,

[0321]

[0322] The sag amount z is represented as a corrected conic section as shown in the following equation.

[0323]

[0324] However, Δy and Δz are represented as follows.

[0325]

[0326]

[0327] r y k is the radius of curvature of the conic section. y is a cone integer. α y is a real number between 0.5 and 2 inclusive. β y is a real number between 0.15 and 0.6 inclusive. γ y ε is a real number between 1 and 10. λ is the wavelength of visible light. n is the absolute refractive index of the lens.

[0328] The luminance distribution and observed image in Fig. 28 are for when the lens pitch is 30 μm. k with correction x = k y = 4, α x = α y = 1, β x = β y = 0.3. The same applies below. Ripple was reduced by the correction of the skirt.

[0329] The luminance distribution and observed image in Fig. 29 are for when the lens pitch is 60 μm. Ripple was reduced by the correction of the skirt.

[0330] The luminance distribution and observed image in Fig. 30 are for when the lens pitch is 100 μm. Ripple was reduced by the correction of the skirt.

[0331] The luminance distribution and observed image in Fig. 31 are for when the lens pitch is 150 μm. Ripple was reduced by the correction of the skirt.

[0332] <Variation Example 5: Microlens array composed of hexagonal microconcave lenses>

[0333] Another embodiment of the microarray type diffuser is a microlens array in which regular hexagonal concave lenses are grid-arranged on an array plane. The concave lenses have cylindrical concave surfaces through cross-sections that are parallel to the two opposing sides of the concave lenses viewed from a plane and are orthogonal to the array plane. The concave lenses have cylindrical concave surfaces through cross-sections that are orthogonal to the two opposing sides and are orthogonal to the array plane. In one embodiment, the grid is a regular hexagonal grid. The sag amount is corrected so that the inclination of the rim increases at each meridian on each cross-section parallel to the grid direction of the concave lenses. Due to the correction, the rim is pushed up slightly.

[0334] The range in which the sag amount is corrected is defined in the directions parallel to and orthogonal to the two opposing sides, namely the x-axis and y-axis directions. The y-axis is parallel to the grid direction. The x-axis is orthogonal to the corresponding grid direction. The magnitude of the horizontal range in which the sag amount is corrected in the x-axis direction is denoted as Δx. The amount of sag correction is denoted as Δz. In the following range of x-coordinates, the sag amount z is represented as an uncorrected conic section as shown in the following equation. L x is the lens width. Lens width L xchanges depending on the y-coordinate of the cross-section. k x r is a cone integer. x is the radius of curvature of the conic section.

[0335]

[0336]

[0337] Meanwhile, it is preferable that the correction performed on the rim be carried out within the following range of the horizontal x-coordinate centered on the axis of symmetry of the meridian.

[0338]

[0339] The size of the horizontal range Δx, to which the sag amount is corrected, is expressed as follows.

[0340]

[0341] γ x is a real number representing the correction order. Preferably, 1 ≤ γ x ≤ 10, preferably 2 < γ x , preferably 3 < γ x is. γ x When = 1, the skirt becomes a straight line. λ is the wavelength of the light ray. n is the absolute refractive index of the lens. n is approximated by the relative refractive index of the lens with respect to air.

[0342] The correction amount Δz of the sag amount is expressed as follows.

[0343]

[0344] The sag amount z is represented as a corrected conic section as shown in the following equation.

[0345]

[0346] Let Δy be the size of the horizontal range in which the sag amount is corrected in the y-axis direction. Let Δz be the correction amount for the sag amount. In the following range of the y-coordinate, the sag amount z is represented as an uncorrected conic section as shown in the following equation. Ly is the lens width. Lens width L y changes depending on the x-coordinate of the cross-section. k y r is a cone integer. y is the radius of curvature of the conic section.

[0347]

[0348]

[0349] Meanwhile, the correction performed on the rim is preferably carried out within the following range of the y-coordinate in the horizontal direction centered on the axis of symmetry of the meridian.

[0350]

[0351] The size of the horizontal range Δy, to which the sag amount is corrected, is expressed as follows.

[0352]

[0353] γ y is a real number representing the correction order. Preferably, 1 ≤ γ y ≤ 10, preferably 2 < γ y , preferably 3 < γ y is. γ y When = 1, the skirt becomes a straight line. λ is the wavelength of the light ray. n is the absolute refractive index of the lens. n is approximated by the relative refractive index of the lens with respect to air.

[0354] The correction amount Δz of the sag amount is expressed as follows.

[0355]

[0356] The sag amount z is represented as a corrected conic section as shown in the following equation.

[0357]

[0358] <Variation Example 6: Reflective Hexagonal Concave Mirror Array Type Diffusion Plate>

[0359] Another embodiment of the micro-array type diffuser is a micro-concave mirror array in which regular hexagonal concave mirrors are grid-arranged on an array plane. The concave mirrors have cylindrical concave surfaces through each cross-section that is parallel to the two opposing sides of the concave mirrors viewed from a plane and is orthogonal to the array plane. The concave mirrors have cylindrical concave surfaces through each cross-section that is orthogonal to the two opposing sides and is orthogonal to the array plane. In one embodiment, the grid is a regular hexagonal grid. The sag amount is corrected so that the inclination of the rim increases at each meridian on each cross-section parallel to the grid direction of the concave mirrors. Due to the correction, the rim is pushed up slightly.

[0360] The range in which the sag amount is corrected is defined in the directions parallel to and orthogonal to the two opposing sides, namely the x-axis and y-axis directions. The y-axis is parallel to the grid direction. The x-axis is orthogonal to the corresponding grid direction. The magnitude of the horizontal range in which the sag amount is corrected in the x-axis direction is denoted as Δx. The amount of sag correction is denoted as Δz. In the following range of x-coordinates, the sag amount z is represented as an uncorrected conic section as shown in the following equation. L x is the lens width. Lens width L x changes depending on the y-coordinate of the cross-section. k x r is a cone integer. x is the radius of curvature of the conic section.

[0361]

[0362]

[0363] Meanwhile, it is preferable that the correction performed on the rim be carried out within the following range of the horizontal x-coordinate centered on the axis of symmetry of the meridian.

[0364]

[0365] The size of the horizontal range Δx, to which the sag amount is corrected, is expressed as follows.

[0366]

[0367] γ x is a real number representing the correction order. Preferably, 1 ≤ γ x ≤ 10, preferably 2 < γ x , preferably 3 < γ x is. γ x When = 1, the skirt becomes a straight line. λ is the wavelength of the light ray. n is the absolute refractive index of the lens. n is approximated by the relative refractive index of the lens with respect to air.

[0368] The correction amount Δz of the sag amount is expressed as follows.

[0369]

[0370] The sag amount z is represented as a corrected conic section as shown in the following equation.

[0371]

[0372] Let Δy be the size of the horizontal range in which the sag amount is corrected in the y-axis direction. Let Δz be the correction amount for the sag amount. In the following range of the y-coordinate, the sag amount z is represented as an uncorrected conic section as shown in the following equation. L y is the lens width. Lens width L y changes depending on the x-coordinate of the cross-section. k y r is a cone integer. y is the radius of curvature of the conic section.

[0373]

[0374]

[0375] Meanwhile, the correction performed on the rim is preferably carried out within the following range of the y-coordinate in the horizontal direction centered on the axis of symmetry of the meridian.

[0376]

[0377] The size of the horizontal range Δy, to which the sag amount is corrected, is expressed as follows.

[0378]

[0379] γ y is a real number representing the correction order. Preferably, 1 ≤ γ y ≤ 10, preferably 2 < γ y , preferably 3 < γ y is. γ y When = 1, the skirt becomes a straight line. λ is the wavelength of the light ray. n is the absolute refractive index of the lens. n is approximated by the relative refractive index of the lens with respect to air.

[0380] The correction amount Δz of the sag amount is expressed as follows.

[0381]

[0382] The sag amount z is represented as a corrected conic section as shown in the following equation.

[0383]

[0384] <Variation Example 7: Reflective Hexagonal Convex Mirror Array Type Diffusion Plate>

[0385] Another embodiment of the microarray type diffuser is a microconvex mirror array in which regular hexagonal convex mirrors are grid-arranged on an array plane. The convex mirrors have cylindrical concave surfaces through each cross-section that is parallel to the two opposing sides of the convex mirrors viewed from a plane and is orthogonal to the array plane. The convex mirrors have cylindrical concave surfaces through each cross-section that is orthogonal to the two opposing sides and is orthogonal to the array plane. In one embodiment, the grid is a regular hexagonal grid. The sag amount is corrected so that the slope of the skirt increases at each meridian on each cross-section parallel to the grid direction of the convex mirrors. Due to the correction, the skirt sags slightly downward.

[0386] The range in which the sag amount is corrected is defined in the directions parallel to and orthogonal to the two opposing sides, namely the x-axis and y-axis directions. The y-axis is parallel to the grid direction. The x-axis is orthogonal to the corresponding grid direction. The magnitude of the horizontal range in which the sag amount is corrected in the x-axis direction is denoted as Δx. The amount of sag correction is denoted as Δz. In the following range of x-coordinates, the sag amount z is represented as an uncorrected conic section as shown in the following equation. L x is the lens width. Lens width L x changes depending on the y-coordinate of the cross-section. k x r is a cone integer. x is the radius of curvature of the conic section.

[0387]

[0388]

[0389] Meanwhile, the correction performed on the skirt is preferably carried out within the following range of the horizontal x-coordinate centered on the axis of symmetry of the meridian.

[0390]

[0391] The size of the horizontal range Δx, to which the sag amount is corrected, is expressed as follows.

[0392]

[0393] γ x is a real number representing the correction order. Preferably, 1 ≤ γ x ≤ 10, preferably 2 < γ x , preferably 3 < γ x is. γ x When = 1, the skirt becomes a straight line. λ is the wavelength of the light ray. n is the absolute refractive index of the lens. n is approximated by the relative refractive index of the lens with respect to air.

[0394] The correction amount Δz of the sag amount is expressed as follows.

[0395]

[0396] The sag amount z is represented as a corrected conic section as shown in the following equation.

[0397]

[0398] Let Δy be the size of the horizontal range in which the sag amount is corrected in the y-axis direction. Let Δz be the correction amount for the sag amount. In the following range of the y-coordinate, the sag amount z is represented as an uncorrected conic section as shown in the following equation. L y is the lens width. Lens width L y changes depending on the x-coordinate of the cross-section. k y r is a cone integer. y is the radius of curvature of the conic section.

[0399]

[0400]

[0401] Meanwhile, it is preferable that the correction performed on the skirt be carried out within the following range of the y-coordinate in the horizontal direction centered on the axis of symmetry of the meridian.

[0402]

[0403] The size of the horizontal range Δy, to which the sag amount is corrected, is expressed as follows.

[0404]

[0405] γ y is a real number representing the correction order. Preferably, 1 ≤ γ y ≤ 10, preferably 2 < γ y , preferably 3 < γ y is. γ y When = 1, the skirt becomes a straight line. λ is the wavelength of the light ray. n is the absolute refractive index of the lens. n is approximated by the relative refractive index of the lens with respect to air.

[0406] The correction amount Δz of the sag amount is expressed as follows.

[0407]

[0408] The sag amount z is represented as a corrected conic section as shown in the following equation.

[0409]

[0410] This application claims priority based on Japanese patent application 2020-190843 filed on November 17, 2020, and incorporates the entire disclosure thereof herein. Explanation of the symbols

[0411] 30 : Lens 31: Microlens array 32 : Array surface 34: Convex lens surface 35 : Beam Fx : Focus Fy : Focus Lc: Luminance curve Lo: Luminance curve Mc: Primary Mo : Primary election Px : Pitch Py : Pitch Sx: Cross-section Sy : Section θc : Diffusion angle θo: Diffusion angle 2θc : Taper 2θo : Taper 2θx : Taper 2θy : Taper

Claims

Claim 1 A micro-lens array in which a plurality of lenses are arranged longitudinally and transversely on an array surface, wherein when the longitudinal and transverse directions of the arrangement of the plurality of lenses are defined as the longitudinal and transverse directions of any one of the plurality of lenses, the any one lens has a cross-cylindrical convex lens surface formed by merging a convex surface that is transversely cylindrical through each cross-section parallel to the longitudinal direction and perpendicular to the array surface, and a convex surface that is longitudinally cylindrical through each cross-section parallel to the transverse direction and perpendicular to the array surface, wherein the sag amount is corrected so that the slope of the skirt increases at each meridian on each cross-section parallel to the longitudinal and transverse directions, wherein the meridian on each cross-section parallel to the longitudinal and transverse directions is formed by a conical curve that has undergone the correction of the sag amount, and in the following range of a horizontal coordinate x centered on the axis of symmetry of the meridian, The sag amount z is represented as a corrected conic section as shown in the following equation, and However, Δx and Δz are represented as follows, and A microlens array, wherein L is the width of the above meridian, r is the radius of curvature of the above conic section, k is a conic integer, 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. Claim 2 A microlens array according to claim 1, wherein 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, and γ is a real number of 2 or more and 10 or less. Claim 3 A microlens array according to claim 1, wherein λ is 650 nm. Claim 4 A microlens array according to claim 1, wherein λ is 530 nm. Claim 5 In claim 1, within the following range of the horizontal coordinate x, The sag amount z is represented as an uncorrected conic section as shown in the following equation, Microlens array. Claim 6 A micro-lens array according to claim 1, wherein each of the plurality of lenses is a rectangular lens, and the longitudinal and transverse directions are aligned with the longitudinal and transverse directions of the arrangement of the plurality of lenses. Claim 7 In claim 1, the convex lens surface is a micro-lens array having different diffusion angles in the longitudinal and transverse directions. Claim 8 A micro-lens array in which a plurality of lenses are arranged longitudinally and transversely on an array surface, wherein when the longitudinal and transverse directions of the arrangement of the plurality of lenses are defined as the longitudinal and transverse directions of any one of the plurality of lenses, the any one lens has a cross-cylindrical concave lens surface formed by merging a transversely cylindrical concave surface through each cross-section parallel to the longitudinal direction and perpendicular to the array surface, and a longitudinally cylindrical concave surface through each cross-section parallel to the transverse direction and perpendicular to the array surface, wherein the sag amount is corrected so that the inclination of the rim increases at each meridian on each cross-section parallel to the longitudinal and transverse directions, wherein the meridian on each cross-section parallel to the longitudinal and transverse directions is formed by a conic section that has undergone the correction of the sag amount, and in the following range of a horizontal coordinate x centered on the axis of symmetry of the meridian, The sag amount z is represented as a corrected conic section as shown in the following equation, and However, Δx and Δz are represented as follows, and A microlens array, wherein L is the width of the above meridian, r is the radius of curvature of the above conic section, k is a conic integer, 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. Claim 9 A transmissive screen having a micro-lens array as described in any one of claims 1 to 8. Claim 10 A head-up display having a transparent screen as described in claim 9. Claim 11 A micro-concave mirror array in which a plurality of concave mirrors are arranged longitudinally and transversely on an array surface, wherein when the longitudinal and transverse directions of the arrangement of the plurality of concave mirrors are defined as the longitudinal and transverse directions of any one of the plurality of concave mirrors, the any one of the concave mirrors has a cross-cylindrical concave surface formed by merging a concave surface that is transversely cylindrical through each cross-section parallel to the longitudinal direction and perpendicular to the array surface, and a concave surface that is longitudinally cylindrical through each cross-section parallel to the transverse direction and perpendicular to the array surface, wherein the sag amount is corrected so that the inclination of the rim increases at each meridian on each cross-section parallel to the longitudinal and transverse directions, wherein the meridian on each cross-section parallel to the longitudinal and transverse directions is formed by a conic section that has undergone the correction of the sag amount, and in the following range of a horizontal coordinate x centered on the axis of symmetry of the meridian, The sag amount z is represented as a corrected conic section as shown in the following equation, and However, Δx and Δz are represented as follows, and A micro-concave mirror array, wherein L is the width of the above 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 above conic section, k is a conic integer, and λ is the wavelength of visible light. Claim 12 A micro-concave mirror array according to claim 11, 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. Claim 13 A micro-convex mirror array in which a plurality of convex mirrors are arranged longitudinally and transversely on an array surface, wherein when the longitudinal and transverse directions of the arrangement of the plurality of convex mirrors are defined as the longitudinal and transverse directions of any one of the plurality of convex mirrors, the one convex mirror has a cross-cylindrical convex surface formed by merging a convex surface that is cylindrical in the transverse direction through each cross-section parallel to the longitudinal direction and perpendicular to the array surface, and a convex surface that is cylindrical in the longitudinal direction through each cross-section parallel to the transverse direction and perpendicular to the array surface, wherein the sag amount is corrected so that the inclination of the skirt increases at each meridian on each cross-section parallel to the longitudinal and transverse directions, wherein the meridian on each cross-section parallel to the longitudinal and transverse directions is formed by a conical curve that has undergone the correction of the sag amount, and in the following range of a horizontal coordinate x centered on the axis of symmetry of the meridian, The sag amount z is represented as a corrected conic section as shown in the following equation, and However, Δx and Δz are represented as follows, and A microconvex mirror array, wherein L is the width of the above 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 above conic section, k is a conic integer, and λ is the wavelength of visible light. Claim 14 A reflective screen comprising either the micro-concave mirror array described in claim 11 or the micro-convex mirror array described in claim 13. Claim 15 delete Claim 16 delete Claim 17 delete Claim 18 delete Claim 19 delete Claim 20 delete Claim 21 delete Claim 22 delete

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