Patent Publication Number: US-11378877-B2

Title: Homogenizer, illuminating optical system, and illuminator

Description:
TECHNICAL FIELD 
     The present invention relates to a homogenizer which converts an incident light having an uneven light-intensity spatial distribution into a light having an even light-intensity distribution on an irradiation plane, and to an illuminating optical system and an illuminator each including the homogenizer. 
     BACKGROUND ART 
     A homogenizer including a pair of lens arrays called integrator lenses or fly-eye lenses is a common technique for enabling illuminating optical systems, such as liquid-crystal projectors, which are for enlarging and projecting images produced by liquid-crystal display elements, and exposure devices, to attain an even light-intensity distribution on irradiation planes. 
       FIG. 23A  and  FIG. 23B  are cross-sectional views showing one example of illuminating optical systems used in liquid-crystal projectors. The illuminating optical system  200  shown in  FIG. 23A  and  FIG. 23B  includes a discharge lamp  50 , a parabolic mirror  51 , a homogenizer  52  including a pair of convex-lens arrays ( 52   a  and  52   b ), and a field lens  53 .  FIG. 23A  and  FIG. 23B  further show a liquid-crystal display element  54 , which is an irradiation plane, and a projection lens  55 , as components of a liquid-crystal projector.  FIG. 23A  is a cross-sectional view showing an example of the arrangement of the major components of the illuminating optical system  200 .  FIG. 23B  is a cross-sectional view in which an example of optical paths in the illuminating optical system  200  has been added. 
     In this example, the light emission point of the discharge lamp  50  has been disposed at the focal point of the parabolic mirror  51 . A visible light emitted by the light emission point is reflected by the parabolic mirror  51  to become approximately parallel light and enters the homogenizer  52 . The approximately parallel visible light which has entered the homogenizer  52  is condensed and caused to strike on the surface of the convex lenses of the convex-lens array  52   b  by the convex lenses of the convex-lens array  52   a , the convex lenses of the convex-lens array  52   b  being paired with the convex lenses of the convex-lens array  52   a  and being disposed in the vicinity of the focal points of the convex lenses of the convex-lens array  52   a . The light is emitted, by the convex lenses of the convex-lens array  52   b , as divergent light which is superimposed at the irradiation plane (display surface of the liquid-crystal display element  54 ). 
     The field lens  53  has been disposed in order to cause the optical axes of the individual convex lenses ( 521   a  and  521   b ) of the homogenizer  52  to meet each other at the center of the irradiation plane (display surface of the liquid-crystal display element  54 ). The light emitted by the individual lenses of the convex-lens array  52   b  is caused to strike on the display surface of the liquid-crystal display element  54  by the field lens  53  to become light for projecting images produced by the liquid-crystal display element  54 , on a screen which is not shown. In this example, the images produced by the liquid-crystal display element  54  are enlarged and projected by the projection lens  55 . 
     In this illuminating optical system  200 , fluxes of incident light which differ in light-intensity distribution on the surface of the convex-lens array  52   a  are emitted from the convex lenses  521   a  of the convex-lens array  52   a  and the convex lenses  521   b  of the convex-lens array  52   b  and superimposed at the irradiation plane. Thus, the light-intensity distributions of the light which has entered the individual convex lenses are averaged in accordance with the number of arrayed convex lenses and an even light-intensity distribution is obtained. 
       FIG. 24A  and  FIG. 24B  are views which illustrate a more detailed example of the configuration of the homogenizer  52 .  FIG. 24A  is a cross-sectional view of the homogenizer  52 , while  FIG. 24B  is a plan view of one convex-lens array ( 52   a  or  52   b ) of the homogenizer. The concentric dotted-line circles within each lens surface are contour lines each connecting points equal in lens depth (sag value) (hereinafter the same applies in other plan views). 
     In the homogenizer  52 , in each of the convex-lens arrays  52   a  and  52   b , lenses having the same convex-lens shape are closely disposed in an array arrangement on one surface of a light-transmitting substrate. The convex-lens arrays  52   a  and  52   b  in this example are each a convex-lens array in which lenses having an X-axis-direction width of Wx and a Y-axis-direction width of Wy are disposed in an array arrangement such that the X-axis-direction number of lenses is Nx and the Y-axis-direction number of lenses is Ny and there is no flat boundary. Hereinafter, the individual convex lenses of the convex-lens array  52   a  are often referred to as convex lenses  521   a  and the individual convex lenses of the convex-lens array  52   b  are often referred to as convex lenses  521   b . In the case where the convex-lens arrays  52   a  and  52   b  are inclusively mentioned without distinction, the convex lenses of either or both of the arrays  52   a  and  52   b  are often referred to as convex lenses  521 . Hereinafter, the same applies to other homogenizers. 
     The incident light which has entered the homogenizer  52  becomes divergent light having a Z-axis-direction maximum incidence angle α, depending on the light-emission length (on the order of millimeter) of the discharge lamp  50 . The maximum incidence angle α corresponds to a maximum diffusion angle (half angle) of fluxes of the light emitted by the discharge lamp  50 . The maximum diffusion angle (half angle) is also called a maximum emission angle (half angle). 
     As shown in  FIG. 24A , parallel-light components of the light which has entered the convex-lens array  52   a  are condensed on the axes on the focal plane of each of the convex lenses  521   a , and are caused to pass through the top flat portions of the convex lenses  521   b  disposed in the vicinity of the focal plane. The light then becomes a divergent light having a maximum diffusion angle (half angle) β to illuminate the liquid-crystal display element  54 . In the case where each of the convex lenses has the lens width Wx being different from the lens width Wy, that is, where each of the convex lenses has a rectangular outer shape (in periphery shape), the X-axis-direction maximum diffusion angle (half angle) βx differs from the Y-axis-direction maximum diffusion angle (half angle) βy and the relationship therebetween is as follows: sin(βx)/sin(βy)=Wx/Wy. Consequently, the X-axis-direction maximum diffusion angle β of the light to be emitted from the homogenizer  52  and the Y-axis-direction maximum diffusion angle β thereof can be independently regulated. 
     Meanwhile, divergent-light components of the light which has entered the convex-lens array  52   a  are condensed outside the axes on the focal plane of the convex lenses  521   a . In the case where the positions where the condensation occurs are within the surface of the convex lenses  521   b , which are paired with the convex lenses  521   a , the light that has been refracted by the convex lenses  521   b  and emitted therefrom illuminates the liquid-crystal display element  54 . That is, the divergent light having a maximum diffusion angle α which has entered the convex-lens array  52   a  including the convex lenses  521   a  having lens widths of Wx×Wy is converted, by the convex lenses  521   b  of the convex-lens array  52   b , into divergent light having an outer shape analogous to Wx×Wy and having a maximum diffusion angle β, and is enlarged and projected on the irradiation plane. 
       FIG. 25A  and  FIG. 25B  are cross-sectional views showing one example of illuminating optical systems used in ultraviolet exposure devices. The illuminating optical system  210  shown in  FIG. 25A  and  FIG. 25B  includes a discharge lamp  60 , an ellipsoidal mirror  61 , a dichroic mirror  66 , a homogenizer  62  including a pair or convex-lens arrays ( 62   a  and  62   b ), and a field lens  63 .  FIG. 25A  and  FIG. 25B  further show a condenser lens  65  as a component of an ultraviolet exposure device. Although the irradiation plane in this example is not shown, an effective region of the entrance surface of the condenser lens  65  corresponds to a simulated irradiation plane.  FIG. 25A  is across-sectional view showing an example of the arrangement of the major components of the illuminating optical system  210 , while  FIG. 25B  is a cross-sectional view in which an example of optical paths in the illuminating optical system  210  has been added. 
     In this example, the light emission point of the discharge lamp  60  has been disposed at the first focal point of the ellipsoidal mirror  61 . Ultraviolet light emitted by the light emission point is reflected by the ellipsoidal mirror  61  and the dichroic mirror  66 , and condensed so as to strike on the homogenizer  62 , which has been disposed at a second focal point of the ellipsoidal mirror  61 . The light which has entered the homogenizer  62  is condensed by the convex lenses of the convex-lens array  62   a  so as to strike on the apertures of the convex lenses of the convex-lens array  62   b , which is paired with the convex-lens array  62   a  and has been disposed in the vicinity of the focal points of the convex lenses of the convex-lens array  62   a . The light is emitted by the convex lenses of the convex-lens array  62   b , as divergent light which is superimposed on an irradiation plane (not shown). 
     The field lens  63  has been disposed in order to cause the optical axes of the individual convex lenses ( 621   a  and  621   b ) of the homogenizer  62  to meet each other at the center of the irradiation plane. The condenser lens  65  is a lens for converting the divergent light emitted from the field lens  63  into approximately parallel light. Thus, the light emitted by the homogenizer  62  passes through the field lens  63 , which causes the optical axes of the convex lenses of the homogenizer  62  to meet each other at the center of the irradiation plane, is converted to approximately parallel light by the condenser lens  65 , and reaches the irradiation plane (not shown). 
     Also in the illuminating optical system  210 , fluxes of incident light which differ in light intensity distribution on the surface of the convex-lens array  62   a  are emitted from the convex lenses  621   a  of the convex-lens array  62   a  and the convex lenses  621   b  of the convex-lens array  62   b  and superimposed at the irradiation plane. Thus, the light-intensity distributions of the light which has entered the individual convex lenses are averaged in accordance with the number of arrayed convex lenses and an even light-intensity distribution is obtained. The term “even light-intensity distribution” herein means a light-intensity distribution of, for example, 85% or higher. 
       FIG. 26A  and  FIG. 26B  are views which illustrate a more detailed example of the configuration of the homogenizer  62 .  FIG. 26A  is a cross-sectional view of the homogenizer  62 .  FIG. 26B  is a plan view of one convex-lens array ( 62   a  or  62   b ) of the homogenizer. 
     The homogenizer  62  is a both-side convex-lens array including an entrance surface and an emission surface which have been processed respectively into a convex-lens array  62   a  and a convex-lens array  62   b . In each of the two convex-lens arrays ( 62   a  and  62   b ), two or more convex lenses are disposed in an array arrangement such that the boundaries therebetween include neither a flat surface nor a gap. In the case of ultraviolet exposure devices, the homogenizer  62  typically employs synthetic quartz, which shows little absorption in an ultraviolet wavelength range. Synthetic quartz has a softening temperature as high as 1,000° C. or higher and it is difficult to produce a lens array therefrom by die forming. Because of this, the following configuration has been frequently used: synthetic-quartz blocks having a prismatic outer shape are ground to form a convex lens in each of the upper and lower faces and the resultant both-side convex lenses  621  are disposed in an array arrangement. This configuration is poor in mass productivity. 
       FIG. 26B  shows an example of a convex-lens array ( 62   a  or  62   b ) including columnar lenses each having convex-lens surfaces having the shape of a regular hexagon, which are disposed in a fly-eye arrangement at the same intervals as the lens width W. However, the convex-lens array ( 62   a  or  62   b ) may be one obtained by disposing columnar lenses each having convex-lens surfaces of a quadrilateral shape, in an array arrangement. Although the homogenizer  62  in this example differs in configuration from the homogenizer  52 , the principle of even illumination is common therebetween. 
       FIG. 27  is a view (YZ cross-sectional view) illustrating a relationship between the entrance surface and emission surface of a pair of convex lenses (i.e., a pair of convex lenses  621   a  and  621   b ) in the homogenizer  62  and the irradiation plane. In the illuminating optical system  210  in this example, divergent light having a maximum diffusion angle α which strikes on the convex lens  621   a  is refracted at the surface of the convex lens  621   a  and is condensed so as to strike on the focal plane thereof. If some of the divergent light which has entered the aperture of the convex lens  621   a  reaches the outside of the convex lens  621   b , which is paired with the convex lens  621   a , the some of the divergent light become stray light, which is not condensed so as to strike on the desired irradiation plane, resulting in a poor illumination-plane intensity distribution and a decrease in light utilization efficiency. In order that the divergent light which has entered the aperture of the convex lens  621   a  is condensed so as to strike on the aperture of the convex lens  621   b , which is paired with the convex lens  621   a , it is necessary that the convex-lens array  62   a  should have a convex-lens surface having a large numerical aperture. In addition, since spherical convex lenses having a large numerical aperture have off-axial aberration and poor condensing properties, it is preferred to employ an aspherical shape. 
     The distance between the entrance surface of the convex lens  621   a  and the principal point of the convex lens  621   b , which is paired with the convex lens  621   a , is expressed by S 1 , and the distance between the principal point of the convex lens  621   b  and the irradiation plane (denoted by numeral  65  in the figure) is expressed by D (D=S 2 ). Furthermore, the focal distance of the convex lenses  621   b  is expressed by f. Then, the S 1 , S 2 , and f are correlated with each other by a paraxial approximate expression for convex lenses, and an entrance pupil A (more specifically, the width W of the aperture shape) which has entered the aperture of the convex lens  621   a  is caused, by the convex lens  621   b , to form an image as an emission pupil B on the given irradiation plane (numeral  65 ). In the case where the S 2  is sufficiently larger than the S 1 , the aperture B (maximum width of the irradiation plane) of the emission pupil B is approximated by B=2·S 2 ·tan(β), from the maximum diffusion angle β of the light emitted by homogenizer  62  upon reception of incident light parallel with the optical axis. 
     Thus, by using a homogenizer including a pair of convex-lens arrays including convex-lens pairs each having a common symmetry axis which are disposed in an array arrangement, the light-intensity distributions on the individual convex-lens entrance surfaces are superimposed and averaged, thereby illuminating the irradiation plane at an even light intensity. The larger the number of convex lenses (number of arrayed convex lenses), the more the evenness improves. The number of arrayed convex lenses in the same surface is, for example, preferably 16 or more, more preferably 25 or more, still more preferably 50 or more. There is no particular upper limit on the number of arrayed convex lenses. An upper limit of the number of arrayed convex lenses per mm 2  may be 10,000. 
     Meanwhile, LEDs and semiconductor lasers (LDs) have come to be practically used as high-intensity light sources in place of the discharge lamps, and are spreading increasingly because these light sources have the feature of being small and having a high luminescent efficiency. In the illuminating optical system  210  shown above, light is emitted over a wide light distribution angle by the discharge lamp having a light-emission length on the order of millimeter and the light is condensed by the condenser mirror (ellipsoidal mirror  61 ) so as to strike on the homogenizer. Thus, the illuminating optical system  210  is difficult to be reduced in size. In the case where an LD light source having a light-emission length on the order of micrometer and efficiently emitting light having high directivity (that is, the emitted light has a narrow light distribution angle) is used, a size reduction in illuminating optical systems can be realized. Consequently, LDs are expected to be utilized in a wide range of illumination applications as light sources in place of the discharge lamps. 
     Patent Literatures 1 and 2 describe examples of illuminators or illuminating optical systems in which laser light sources are used as light sources for illumination. For example, Patent Literature 1 describes a laser illuminator including a laser light source and a homogenizer including a first lens and a second lens each of which includes a plurality of minute lens elements. Patent Literature 2, for example, describes an illuminating optical system for use in semiconductor exposure devices employing laser light as a light source for illumination, the illuminating optical system having a configuration in which a fly-eye integrator is used as a homogenizer. 
     CITATION LIST 
     Patent Literature 
     Patent Literature 1: Japanese Patent No. 4880746 
     Patent Literature 2: JP-A-H05-251309 
     SUMMARY OF THE INVENTION 
     Technical Problems 
     As described above, use of a laser light source attains a size reduction in the light-source part including a condensing system. For example, an LD light source having a light-emission length on the order of micrometer efficiently gives illuminating light having high directivity. Furthermore, a vertical-cavity surface-emitting laser (VCSEL) which produces laser light having a wavelength in the near infrared range of 780 nm to 1,300 nm has an advantage in that a structure including laser light emission points two-dimensionally arrayed at intervals of several tens of micrometers in a semiconductor wafer can be easily produced and that a high-power laser light source having a power of several tens of watts to several hundreds of watts is obtained by increasing the number of light emission points. 
       FIG. 28A  and  FIG. 28B  is respectively a cross-sectional view and a plan view which show an example of laser light sources. The laser light source  11  shown in  FIG. 28  includes a semiconductor substrate  11   a  and surface emitting laser light emission points (VCSEL light-emitting layers)  11   b  disposed in an array arrangement on the semiconductor substrate  11   a . The laser light source  11  having such configuration emits divergent light having a light-intensity distribution of emitted light which can be approximated to a Gaussian distribution. For example, in the case where the laser light source  11  includes laser light emission points in an X-axis direction at intervals of “a”, the number of the light emission points being Na, and includes laser light emission points in a Y-axis direction at intervals of “b”, the number of the light emission points being Nb, on the surface of the substrate, this laser light source  11  as a whole has a light intensity of P=p·Na·Nb, where p is the light intensity of the light emitted by each light emission point. For example, in the case where a=b=30 μm, Na=Nb=100, and P=10 mW, a laser output of P=100 W is obtained from an emission surface of 3 mm×3 mm. 
     In the case where such a VCSEL array is used as a light source, a laser light source having an increased output can be obtained by increasing the number of arrayed laser light emission points. However, the light emitted by each laser light emission point has a narrow emission angle (half angle) δ, which is the maximum emission angle (half angle) of the light emitted by the laser array light source (see  FIG. 28A ). Because of this, for illuminating a wide area, it is necessary to elongate the distance L to an irradiation plane (the entrance surface of the homogenizer in the aforementioned illuminating optical system). This results in an illumination-plane light-intensity distribution which is an uneven Gaussian distribution. 
     If the emission angle δ of the light emitted by each unit laser light emission point in an array-based laser light source is defined as an angle at which a light-intensity ratio determined by normalizing the light-intensity distribution with Gaussian-distribution center intensity (emission angle: 0°) is e −2 , a light-intensity distribution I(θ) for the laser light source alone at any laser light emission angle θ is expressed by formula (1).
 
 I (θ)=exp{−2(θ/δ) 2 }  (1)
 
     The light-intensity distribution of a laser light source including an assembly of individual laser light emission points changes depending on the Z-axis-direction distance L from the light emission points. When a=b and Na=Nb, the diameter φ of an irradiation plane where the light-intensity ratio for the light emitted by a laser light emission point is e −2  is expressed by φ=2·L·tan(δ). Because of this, where the distance L satisfies φ≥a, the light emitted by adjoining light emission points is superimposed. Where the distance L satisfies φ≥√2·a, the light-intensity ratio for the light emitted from the overall array area (Na×a)×(Nb×b) is e −2  or more. Furthermore, where the distance L satisfies φ&gt;√2·a·Na, the irradiation plane has an increased area and the light-intensity ratio for the light emitted from the overall array area is close to the Gaussian light-intensity distribution I(θ) for a laser light emission point. 
     The laser array light source shown above as an example is a VCSEL array light source obtained by two-dimensionally arraying VCSELs producing laser light having a wavelength in the near infrared range of 780 nm to 1,300 nm. It is, however, possible to convert the near infrared light emitted by the VCSEL array light source into visible laser light through second harmonic generation (SHG) using a nonlinear optical crystal such as LiNbO 3 . 
     In the case of a Fabry-Pérot laser, which includes a laser emission active layer having a waveguide structure, laser light having wavelengths in the ultraviolet to the near infrared range is obtained. By one-dimensionally arraying light emission points of such a laser or by enlarging the width of the waveguide of the laser emission active layer, a high-output laser light source can be obtained. However, the emitted laser light has a Gaussian intensity distribution, and in the case of the laser light source operated in a multi-mode in which the laser emission active layer has a larger width, the emitted light has a more uneven intensity distribution. 
     Use of such a laser light source attains a remarkable size reduction in light-source parts including a condensing system. However, a widely diffusing homogenizer is necessary for causing the divergent light having an uneven light-intensity distribution emitted by a laser light source to strike on an irradiation plane evenly (for example, a light-intensity distribution on the irradiation plane of 85% or higher), while a high condensing efficiency is maintained. 
     In the conventional illuminating optical system in which a discharge lamp is used as a light source, the light having a wide distribution angle emitted by the light source is condensed with a condensing mirror so as to strike on the entrance surface of the homogenizer. Because of this, the light-intensity distribution on the entrance surface is uneven and the maximum diffusion angle α of the incident light is regulated to 10° or less such that the number of the apertures of the convex-lens arrays in the entrance surface and emission surface is small. As a result, the outer shape of the condensing mirror becomes large, and the effective diameter φ of the entrance surface of the homogenizer is also enlarged. Consequently, a homogenizer including an entrance surface and an emission surface constituted of convex-lens arrays including spherical convex lenses or aspherical convex lenses having a close shape to spherical ones has been able to be used to attain even illumination even under such illumination conditions that the maximum diffusion angel β is relatively small. However, in the case of a homogenizer configured to convert divergent light which has been emitted at a narrow angle by a laser light source and which is a light flux having a maximum diameter φ on the entrance surface of the homogenizer into light having an even light-intensity distribution on a given irradiation plane, it is necessary that the convex lenses constituting each of the convex-lens pairs in the pair of convex-lens arrays of the homogenizer should have an XY-surface width W of φ/4 or less (in the case where the number of arrayed convex lenses is 4×4=16). In the case where the X-direction width differs from the Y-direction width, the XY-surface width W of each convex lens is defined as the larger one of the widths along the two directions. The width W is more preferably φ/5 or less (assuming the case where the number of arrayed convex lenses is 5×5=25), still more preferably φ/7 or less (assuming the case where the number of arrayed convex lenses is 7×7=49). 
     For example, in the case where the homogenizer has an effective diameter of 2.0 mm, i.e., φ=2.0 mm, it is required that W≤0.5 mm, or W≤0.4 mm, or W≤0.3 mm. Furthermore, in the case where the homogenizer has an effective diameter of 1.0 mm, i.e., φ=1.0 mm, it is required that W≤0.25 mm, or W≤0.2 mm, or W≤0.15 mm. 
     As already explained above, in an illuminating optical system, the light emitted from the homogenizer is enlarged at a maximum diffusion angle β to strike on an irradiation plane having a width B. Consequently, the convex-lens array ( 52   b  or  62   b ) lying on the emission side in the homogenizer corresponds to diffusing micro light sources. The intensity of light emitted from such diffusing micro light sources, on an irradiation plane, is restricted by the “cosine fourth power law” for illuminating optical systems, which is based on the “inverse square law of distance” and the “cosine characteristics of oblique incident light” both regarding illumination. 
       FIG. 29A  and  FIG. 29B  are respectively a schematic view and a graph which show that a light flux emitted by diffusing micro surface light sources has light intensities according to the cosine fourth power law on an irradiation plane facing the emission surface of the light sources.  FIG. 29B  shows a light-intensity ratio (E θ /E 0 ), which is the ratio of a light intensity (E θ ) as measured at any angle θ in the diffusion angle β direction to a light intensity (directly-under-light-source intensity E 0 ) as measured at a perpendicularly opposed position. As  FIG. 29A  schematically shows, in the case where a plane opposite to the emission surface of the diffusing micro surface light sources is illuminated, the light intensity E θ  as measured at a point on the irradiation plane which lies at an angle θ is cos 4 θ times the light intensity (directly-under-light-source intensity E 0 ) as measured on a surface perpendicularly opposite to the light sources.  FIG. 29B  is a graph showing calculated values of light-intensity ratio cos 4 θ for an area in the irradiation plane lying in the angle range of θ=0 to 50°. 
     As  FIG. 29B  shows, the light-intensity ratio on the irradiation plane has values larger than 0.9 in the case where θ≤10°, but decreases to less than 0.8 in the case where θ=20° and to less than 0.6 in the case where θ=30°. 
     Because of this, in the case where the light emitted from a homogenizer is made to have a larger maximum diffusion angle β (e.g., β≥12°) than in conventional configurations by a method in which the light fluxes emitted from the individual convex lenses of the emission-side convex-lens array are made to have a larger diffusion angle, for example, by merely regulating the lens shape of the convex lenses, the resultant light-intensity distribution on the irradiation plane is uneven due to restrictions by the cosine fourth power law, etc. That is, the method in which the convex lenses of the emission-side convex-lens array in a conventional homogenizer are merely made to emit light fluxes at a wider angle cannot necessarily attain an even light-intensity distribution of 0.9 or higher in the case where the maximum diffusion angle β≥12°, due to restrictions by the cosine fourth power law, etc. 
     Although Patent Literature 1 shows a laser illuminator including a laser light source, the unevenness of light-intensity distribution on an irradiation plane due to the use of the laser light source is not considered. Patent Literature 1 contains no specific disclosure regarding the diffusion of the light emitted from the homogenizer or the light-intensity distribution on irradiation planes. The same applies to Patent Literature 2. 
     An object of the present invention is to provide a homogenizer which is small and has a satisfactory utilization efficiency and which can emit light showing high evenness on irradiation planes, and to provide an illuminating optical system and an illuminator. 
     Another object of the present invention is to provide a homogenizer by which divergent light having an uneven light-intensity distribution emitted by a laser light source can be projected at a maximum diffusion angle of 12° or larger on an irradiation plane so as to result in a light-intensity distribution as even as 85% or more while a high condensing efficiency is maintained, and to provide an illuminating optical system and an illuminator employing the homogenizer. In particular, the object is to provide a homogenizer showing such properties for divergent light having a Gaussian intensity distribution emitted by a semiconductor laser light source, and to provide an illuminating optical system and an illuminator employing the homogenizer. 
     Solution to Problem 
     A homogenizer in the present invention includes a convex-lens array pair including a first convex-lens array disposed on a light entrance side and a second convex-lens array disposed on a light emission side, 
     in which the first convex-lens array and the second convex-lens array are disposed so as to face each other such that each of the convex-lens arrays has a lens surface opposed to each other outward or inward, 
     in which the first convex-lens array includes a plurality of first convex lenses having a same shape which are disposed in an array arrangement on one surface, 
     in which the second convex-lens array includes a plurality of second convex lenses having a same shape which are disposed in an array arrangement on one surface, 
     in which the first convex lenses and the second convex lenses form convex-lens pairs in each of which the first convex lens and the second convex lens face each other and have a common symmetry axis, 
     in which the first convex lens, in a lens cross-section including the symmetry axis, has an average internal transmission angle for incident light entering a lens-surface center region in the lens cross-section and being in parallel with the symmetry axis, the average internal transmission angle being equal to or more than 1.3 times an average internal transmission angle in a lens-surface center region of a spherical convex lens. 
     In addition, another homogenizer in the present invention includes two convex-lens array pairs each including a first convex-lens array disposed on a light entrance side and a second convex-lens array disposed on a light emission side, 
     in which in each of the convex-lens array pairs, the first convex-lens array and the second convex-lens array are disposed so as to face each other such that each of the convex-lens arrays has a lens surface opposed to each other outward or inward, 
     in which the first convex-lens array includes a plurality of first convex lenses having a same shape which are disposed in an array arrangement on one surface, each of the first convex lenses being a convex cylindrical lens, the plurality of first convex lenses being disposed such that lens-function axes of the first convex lenses are parallel with each other, 
     in which the second convex-lens array includes a plurality of second convex lenses having a same shape which are disposed in an array arrangement on one surface, each of the second convex lenses being a convex cylindrical lens, the plurality of second convex lenses being disposed such that lens-function axes of the second convex lenses are parallel with each other, 
     in which the first convex lenses and the second convex lenses in each of the convex-lens array pairs form convex-lens pairs in each of which the first convex lens and the second convex lens face each other and have a common symmetry axis, 
     in which the two convex-lens array pairs are serially disposed along an optical-axis direction, which is a traveling direction of incident light, such that the two convex-lens array pairs differ from each other in lens-function axis direction by 90°, 
     in which in each of the convex-lens array pairs, when a cross-section of each convex lens which is perpendicular to a base-line direction of the convex lens is referred to as a lens cross-section and a position of a symmetry plane in the lens cross-section is referred to as symmetry axis, the first convex lens has an average internal transmission angle for incident light entering a lens-surface center region in the lens cross-section and being in parallel with the symmetry axis, the average internal transmission angle being equal to or more than 1.3 times an average internal transmission angle in a lens-surface center region of a spherical convex lens. 
     An illuminating optical system in the present invention includes a laser light source configured to emit a divergent light having an uneven light-intensity distribution and any one of the homogenizer described above, 
     in which the divergent light emitted by the laser light source enters the homogenizer, is emitted as more widely diffused divergent light from the homogenizer, and is expanded and projected on a given irradiation plane with an even light-intensity distribution. 
     An illuminator in the present invention includes the illuminating optical system described above. 
     Advantageous Effects of Invention 
     The present invention can provide a homogenizer which is small and has a satisfactory utilization efficiency and which can emit light showing high evenness on irradiation planes, and an illuminating optical system and an illuminator. 
     Furthermore, the present invention can provide a homogenizer by which divergent light having an uneven light-intensity distribution emitted by a laser light source can be projected at a maximum diffusion angle of 120 or larger on an irradiation plane so as to result in a light-intensity distribution as even as 85% or more while a high condensing efficiency is maintained, and an illuminating optical system and an illuminator employing the homogenizer. In particular, the present invention can provide a homogenizer showing such properties for divergent light having a Gaussian intensity distribution emitted by a semiconductor laser light source, and an illuminating optical system and an illuminator employing the homogenizer. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  and  FIG. 1B  are respectively a cross-sectional view and a plan view which show an example of a homogenizer according to a first embodiment. 
         FIG. 2A  and  FIG. 2B  are respectively a cross-sectional view and a plan view of a convex lens  121   a.    
         FIG. 3  is a view schematically illustrating the function of a pair of convex lenses (a convex lens  121   a  and a convex lens  121   b  paired therewith) whereby incident light parallel with the symmetry axis is transmitted. 
         FIG. 4A ,  FIG. 4B  and  FIG. 4C  are cross-sectional views showing a lens function of a pair of convex lenses. 
         FIG. 5A  and  FIG. 5B  are respectively a schematic cross-sectional view and a schematic plan view which show an example of convex lenses  221   a  of a homogenizer  22  according to a second embodiment. 
         FIG. 6  is a view schematically illustrating the function of a pair of convex lenses (a convex lens  221   a  and a convex lens  221   b  paired therewith) whereby incident light parallel with the symmetry axis is transmitted. 
         FIG. 7  is a slant view of a homogenizer  32  according to a third embodiment. 
         FIG. 8A  and  FIG. 8B  are respectively a cross-sectional view and a plan view which show an example of a homogenizer  32   x.    
         FIG. 9A  and  FIG. 9B  are respectively a cross-sectional view and a plan view which show an example of a homogenizer  32   y.    
         FIG. 10  is a cross-sectional view showing an example of an illuminator  100  according to a fourth embodiment. 
         FIG. 11  is a cross-sectional view showing an example of an illuminator  110  according to a fifth embodiment. 
         FIG. 12  is a presentation for explaining the lens shapes of convex lenses  121   a  of the homogenizers  12  of Examples 1-1 to 1-10. 
         FIG. 13  is graphs showing the results of calculating the angle (θ 1 -θ 2 ) of traveling direction of light transmitted in the homogenizers  12  according to the first Examples. 
         FIG. 14A ,  FIG. 14B ,  FIG. 14C  and  FIG. 14D  are graphs (1) showing the results of calculating light-intensity distributions on an irradiation plane  17  which were obtained with illuminators  100  employing homogenizers  12  according to first Examples. 
         FIG. 15A ,  FIG. 15B ,  FIG. 15C  and  FIG. 15D  are graphs (2) showing the results of calculating light-intensity distributions on an irradiation plane  17  which were obtained with illuminators  100  employing homogenizers  12  according to first Examples. 
         FIG. 16A ,  FIG. 16B , and  FIG. 16C  are graphs (3) showing the results of calculating light-intensity distributions on an irradiation plane  17  which were obtained with illuminators  100  employing homogenizers  12  according to first Examples. 
         FIG. 17A ,  FIG. 17B ,  FIG. 17C  and  FIG. 17D  are graphs showing the results of calculating light-intensity distributions on an irradiation plane  17  which were obtained with illuminators  100  employing the homogenizers of Reference Examples 1 to 4. 
         FIG. 18A  and  FIG. 18B  are views schematically illustrating, by gradation, light-intensity distributions (normalized) on an irradiation plane which were obtained with an illuminator  100  employing the homogenizer of Example 1-5. 
         FIG. 19A  and  FIG. 19B  are views schematically illustrating, by gradation, light-intensity distributions (normalized) on an irradiation plane which were obtained with an illuminator  100  employing the homogenizer of Comparative Example 1. 
         FIG. 20A ,  FIG. 20B ,  FIG. 20C ,  FIG. 20D  and  FIG. 20E  are views illustrating an example of methods for producing one cylindrical-lens array of the homogenizer  32  according to the third Example. 
         FIG. 21A  and  FIG. 21B  are views schematically illustrating, by gradation, light-intensity distributions (normalized) on an irradiation plane which were obtained with an illuminator  100  employing the homogenizer according to the third Example. 
         FIG. 22A ,  FIG. 22B ,  FIG. 22C ,  FIG. 22D  and  FIG. 22E  are views illustrating an example of methods for producing one cylindrical-lens array of the homogenizer  32  according to the fourth Example. 
         FIG. 23A  and  FIG. 23B  are cross-sectional views showing one example of illuminating optical systems for use in liquid-crystal projectors. 
         FIG. 24A  and  FIG. 24B  are views which illustrate a more detailed example of the configuration of a homogenizer  52 . 
         FIG. 25A  and  FIG. 25B  are cross-sectional views showing one example of illuminating optical systems for use in ultraviolet exposure devices. 
         FIG. 26A  and  FIG. 26B  are views which illustrate a more detailed example of the configuration of a homogenizer  62 . 
         FIG. 27  is a view illustrating a relationship between the entrance and emission surfaces of a pair of convex lenses (pair of convex lenses  621   a  and  621   b ) in the homogenizer  62  and an irradiation plane. 
         FIG. 28A  and  FIG. 28B  are respectively a cross-sectional view and a plan view which show an example of laser array light sources. 
         FIG. 29A  and  FIG. 29B  are respectively a schematic view and a graph which show that a light flux emitted by diffusing micro surface light sources has light intensities according to the cosine fourth power law on an irradiation plane facing the emission surface of the light sources. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     Embodiment 1 
     Examples of embodiments of the present invention are explained below by reference to drawings.  FIG. 1A  and  FIG. 1B  are respectively a cross-sectional view and a plan view which show an example of a homogenizer according to a first embodiment. As  FIG. 1A  shows, the homogenizer  12  according to this embodiment includes a light-transmitting substrate  10  having a flat-plate shape and a first convex-lens array  12   a  and a second convex-lens array  12   b , which are disposed respectively on two surfaces (first surface and second surface) of the light-transmitting substrate  10 . 
     In the first convex-lens array  12   a  and the second convex-lens array  12   b , convex lenses  121   a  constituting the first convex-lens array are respectively paired with convex lenses  121   b  constituting the second convex-lens array. More specifically, the first convex-lens array  12   a  and the second convex-lens array  12   b  are each a gap-less lens array in which flat portions between adjoining convex lenses are small, and each of the convex lenses  121   a  constituting the first convex-lens array  12   a  has a lens symmetry axis coinciding with that of the convex lens  121   b  paired therewith and has the same XY-plane widths W (Wx and Wy) as the paired convex lens  121   b . The term “symmetry axis” for a lens has the same meaning as an optical axis, and means the center axis of the lens or a symmetry axis having at least two-fold symmetry. 
     The offset in symmetry axis between the convex lens  121   a  and the convex lens  121   b  is preferably 5% or less, more preferably 2% or less, with respect to lens width. In the convex-lens arrays  12   a  and  12   b , the width G (see  FIG. 3 , which will be described later) of a boundary portion which lies between adjoining convex lenses and which has a shape deviated from a desired convex-lens shape is preferably 5% or less with respect to the lens width (that is, 2.5% or less per convex lens (G/2/W×100=5/2=2.5), more preferably 2% or less (1% or less per convex lens). For example, in the case where W=100 μm, G≤5 μm is preferably satisfied. 
     In the homogenizer  12  according to this embodiment, in the case where light from a light source strikes at a maximum incidence angle α (which corresponds to the maximum emission angle δ in the case of a laser light source) on the convex lenses  121   a  of the first convex-lens array  12   a , the light is refracted at the surfaces of the convex lenses  121   a , is transmitted through the light-transmitting substrate  10 , and reaches the surfaces of the convex lenses  121   b  of the second convex-lens array  12   b . The light is then refracted at the surfaces of the convex lenses  121   b  and is finally emitted as divergent light having a maximum diffusion angle β. 
     Here, the lens surface shape of the convex lenses  121   a  and the thickness of the substrate (distance between the first convex-lens array  12   a  and the second convex-lens array  12   b ) are set such that the incident light which has entered the convex lenses  121   a  reaches effective surfaces of the convex lenses  121   b  which are paired with the convex lenses  121   a , as in the homogenizers  52  and  62  described above. 
       FIG. 2A  and  FIG. 2B  are respectively a cross-sectional view and a plan view of a convex lens  121   a  included in the first convex-lens array  12   a . As  FIG. 2A  shows, the homogenizer  12  according to this embodiment differs from conventional homogenizers in the cross-sectional shape of the convex lenses  121   a  of the first convex-lens array  12   a , which are disposed on the entrance side. More specifically, as compared with the cross-sectional shape of a conventional convex lens having approximately the same maximum diffusion angle β, that is, the cross-sectional shape of a spherical convex lens  921  having approximately the same inclination in the peripheral portion, in the cross-sectional shape of the convex lenses  121   a , the surface flat portion in the vicinity of the symmetry axis is smaller, and the surface in the vicinity of the symmetry axis other than the flat portion is more inclined. Namely, the convex lenses  121   a  each have a cross-sectional shape which is close to those of conical lenses having conical shapes and which includes a curved surface having a small radius of curvature near the top. 
     In  FIG. 2A  and  FIG. 2B , the lens-surface depth (sag value) distribution of the convex lens  121   a , based on the symmetry axis, is shown by contour lines (dotted lines). In  FIG. 2A , the cross-sectional shape of the spherical convex lens  921  is indicated by a broken line for comparison. 
     The surface shapes of a convex lens  121   a  and a convex lens  121   b  can be expressed using the formula (2), which is an aspherical-lens formula indicating a sag value Z at a radial-direction radius r as a radial distance from the symmetry axis. 
     
       
         
           
             
               
                 
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     In the formula (2), c is the inverse of radius of curvature R (c=1/R), k is conic constant, and α 1  to α 4  are aspherical constants. The sag value Z corresponds to convex-lens depth, and setting the symmetry axis (r=0) as a base (Z=0), the sag value Z decreases (Z&lt;0) as the r increases. Here, the radius r, the radius of curvature R, and the sag value Z take length units. In this example, these numerals are expressed in m unit. The conic constant k is a dimensionless number. 
     By including the terms of higher orders of aspherical constants, various lens surfaces can be set. However, no aspherical constants are used here (all of α 1  to α 4  being 0), and an aspherical lens shape approximated with radius of curvature R and conic constant k is used as a prerequisite to show the parameters (radius of curvature and conic constant) for the convex lens  121   a  and the parameters for the convex lens  121   b  respectively as (R 1 , k 1 ) and (R 2 , k 2 ). The radius of curvature and conic constant of the convex lens  121   a  are R 1  and k 1  respectively, and the radius of curvature and conic constant of the convex lens  121   b  are R 2  and k 2  respectively. 
     The formula (2) indicates a convex-lens shape having rotational symmetry involving r=0 as asymmetry axis. However, the convex lenses  121   a  and  121   b  each may be a cylindrical lens in which only either the XZ cross-section or the YZ cross-section has a convex-lens shape and the other cross-section does not have a curved lens surface shape. In this case, the sag value Z is expressed by using X-axis coordinate x or Y-axis coordinate y where a convex lens shape is formed, in place of using the radial-direction radius r in the formula (2). 
     In the case where widely diffused divergent light such as one having a maximum diffusion angle α of 10° or larger strikes as incident light, for condensing the incident light by the convex lenses  121   a  of the entrance-side first convex-lens array  12   a  so as to strike on the apertures of the convex lenses  121   b  paired with the convex lenses  121   a , it is necessary that the convex lenses  121   a  should have a large numerical aperture. The numerical aperture NA 1  of convex lenses  121   a  is defined by the formula (3).
 
NA 1   =W /(2 f   1 )  (3)
 
     In order that the light which has struck on the apertures of the convex lenses  121   b  may be converted to widely diffused divergent light such as one having a maximum diffusion angle β of 100 or larger and be enlarged and projected on an irradiation plane, it is necessary the convex lenses  121   b  should have a large numerical aperture. The numerical aperture NA 2  of each convex lens  121   b  is defined by the formula (4).
 
NA 2   =W /(2 f   2 )  (4)
 
     In the formulae (3) and (4), W is the maximum width (maximum diameter) of the effective convex-lens surface, f 1  is the focal distance of the convex lens  121   a , and f 2  is the focal distance of the convex lens  121   b . When the entrance-side medium is air and the light enters, in parallel with the symmetry axis, the convex lens  121   a  having a refractive index n and is then condensed so as to enter the paired convex lens  121   b  having the refractive index n, the focal distances f 1  and f 2  in a paraxial region having a small value of W are approximated by the following formulae.
 
 f   1   ≈n·R   1 /( n− 1)  (5-1)
 
 f   2   ≈n·R   2 /( n− 1)  (5-2)
 
     As seen from the formulae (5-1) and (5-2), the focal distances f 1  and f 2  in paraxial regions having small values of W are not affected by the conic constants. 
     Consequently, in the case where aspherical convex lenses which have a large numerical aperture NA but have a conic constant k being a negative value are used, an improvement in condensing property can be attained as compared with spherical lenses, in which the lens-surface inclination angle abruptly increases in the peripheral portion of the lens to reduce the condensing properties. Specifically, the conic constant k preferably satisfies 0≥k≥−3, more preferably −0.5≥k≥−2.5. The case where k=−1 corresponds to paraboloid and k=−2 corresponds to hyperboloid. Those conditions of k are applicable to both k 1  and k 2  as conditions for aspherical lenses effective in improving the condensing properties of convex lenses having a large numerical aperture NA. Although those conditions of k are applicable also in the case where α&lt;10° or β&lt;10°, it is advantageous, especially in the case where α≥10°, to regulate the conic constant k 2  of the emission-side convex lenses  121   b  so as to satisfy those conditions. With respect to the conic constant k 1  of the convex lenses  121   a , the condition which will be described later is preferential. 
     The convex lenses  121   a  constituting the first convex-lens array  12   a  used in the homogenizer  12  according to this embodiment are convex lenses which, assuming the entrance of divergent light having a narrow light-intensity distribution such as one approximated to a Gaussian distribution, are for changing the light distribution angle of the divergent incident light to condense the incident light so as to enter the convex lenses  121   b  and finally strike evenly on an irradiation plane. Consequently, the convex lenses  121   a  differ in preferred shape from convex lenses for a mere improvement in the property of condensing light and causing the light to strike on the convex lenses  121   b.    
       FIG. 3  schematically shows the function of a pair of convex lenses (a convex lens  121   a  and a convex lens  121   b  paired therewith) in the homogenizer  12  according to this embodiment. The function is to transmit incident light parallel with the symmetry axis.  FIG. 3  schematically shows a cross-section of the pair of convex lenses  121   a  and  121   b  constituting a convex-lens pair in the homogenizer  12 . As  FIG. 3  shows, the inclination angle of the convex lens  121   a  at a radial distance r corresponds to the incidence angle θ 1  for the convex lens having a sag value Z(r). Consequently, incident light parallel with the symmetry axis and entering the convex lens  121   a  is light which enters at an incidence angle θ 1  in the position corresponding to the sag value Z(r), and is refracted at a refractive angle θ 2  satisfying Snell&#39;s refraction law sin(θ 1 )=n·sin(θ 2 ) in accordance with the refractive index n of the convex lens for the incident light and transmitted through the convex lens. The light is thereafter refracted at the surface of the subsequent convex lens  121   b  and emitted at an angle θ 3  with the symmetry axis. Since the light which is being transmitted through the inside of the homogenizer  12  having a refractive index n proceeds at an angle with Z axis of |θ 1 -θ 2 |, the distance Lz along Z-axis-direction (symmetry-axis-direction) between the entrance surface of the convex lens  121   a  and the intersection of the transmitted light and the symmetry axis is represented by Lz=r/tan|θ 1 -θ 2 |. In the case where the radial distance r is 0≤r≤(W−G/2)/2, the Lz is preferably within the range of 0.5 times or more and 2.0 times or less the thickness T of the homogenizer  12  (the maximum symmetry-axis-direction distance between the paired convex lenses), more preferably within the range of 0.7 times or more and 1.2 times or less. The above description about Lz holds even in the case where the intersection lies after the emission surface of the convex lens  121   b  (Lz≥T). 
     Furthermore, the convex lenses  121   a  each have an aspherical lens shape in which a lens-surface center region has an average inclination angle larger than the average inclination angle of a lens-surface region of a spherical lens and a lens-surface peripheral region has approximately the same average inclination angle as in the spherical lens. The term “lens-surface center region” means a region in a cross-section including the symmetry axis and having a maximum width in the XY plane of W max  (see  FIG. 2B ), in which the radial distance r from the symmetry axis is equal to or less than 50% of the maximum radial distance from the symmetry axis W max /2 (0≤r≤W max /4). The term “lens-surface peripheral region” means a region where the radial distance r is equal to or more than 50% of the maximum radial distance W max /2 (W max /4≤r≤W max /2). 
     A center region of the lens surface may be expressed by 0≤r/(W max /2)≤χ and a peripheral region of the lens surface may be expressed by (1−χ)≤r/(W max /2)≤1, using a value of χ satisfying 0&lt;χ≤0.5 to define each region. For example, χ may be 0.1, 0.2, 0.3, 0.4, or 0.5. Whichever set value of χ is used, the proportion Ar 1  of the average inclination angle of the lens-surface center region of the convex lens  121   a  to the average inclination angle of the lens-surface center region of the spherical lens is set at a value larger than 1, and the proportion Ar 2  of the average inclination angle of the lens-surface peripheral region of the convex lens  121   a  to the average inclination angle of the lens-surface peripheral region of the spherical lens is set at about 1. For example, in the case of a convex lens  121   a  having a refractive index n of 1.5 for the wavelength range of the incident light, it is preferable that Ar 1  is within the range of 1.3 to 3.2 and Ar 2  is within the range of 0.6 to 1.1. 
     As described above, in this embodiment, the proportion of the flat region near the symmetry axis is reduced by regulating the radius of curvature R 1  of the convex lens  121   a  to a value smaller than the radius of curvature R 2  of the convex lens  121   b  paired therewith. The ratio between the radii of curvature R 1  and R 2  (R 1 /R 2 ) is preferably within the range of 0.3 to 0.7, more preferably within the range of 0.4 to 0.6. 
     In this case, the radius of curvature R 2  of the convex lens  121   b  having a refractive index n≈1.5 is such that the ratio of R 2  to the XY-plane maximum width W max  of the convex lens  121   b  preferably satisfies R 2 /W max ≥0.5, more preferably R 2 /W max ≥0.9. From such ranges of R 2 /W max , using formulae (4) and (5-2) shown above, NA 2  preferably satisfies NA 2 ≤0.33, more preferably NA 2 ≤0.19. In this case, θ 3 =arcsin(NA 2 ) is 10° or larger. Incidentally, θ 3 ≈19° when NA 2 =0.33, and θ 3 ≈11° when NA 2 =0.19. In this case, the W max  may be within the range of 0.05 to 0.5 mm. In the case where a convex lens  121   b  having a refractive index n higher than 1.5 is used, NA 2  becomes large even in the case where R 2 /W max  is the same. 
     It is preferable that the conic constant k 1  of the convex lens  121   a , which relates to the inclination angle of the lens-surface peripheral region, is a negative value smaller than the conic constant k 2  of the convex lens  121   b  paired therewith. Specifically, it is preferable that the conic constant k 1  is set in accordance with the radius of curvature R 1  so as to result in an inclination angle smaller than the lens-surface inclination angle of the aperture (peripheral portion) of a spherical convex lens having k=0. In this case, the conic constant k 2  of the convex lens  121   b  may be within the range of −3 to 0 (−3≤k 2 ≤0). Furthermore, for example, in the case where the refractive index n of the convex lens  121   a  is 1.5, the conic constant k 1  may be within the range of −3 to −7, and is more preferably within the range of −3 to −4. 
     The refractive angle θ 2  of incident light parallel with the symmetry axis, in a portion of the convex lens  121   a  which has an inclination angle θ 1 , depends on the refractive index n, and the angle |θ 1 -θ 2 | of the transmitted light passing through the homogenizer also changes. Consequently, in order to realize the same optical function as a conventional homogenizer using a light-transmitting material having different refractive index from that having a refractive index n≈1.5 in the case where the thickness of the homogenizer  12  and the widths (Wx and Wy) of the convex lens  121   a  and convex lens  121   b  are the same as in the conventional homogenizer, the sag values Z(r) of the convex lens  121   a  and convex lens  121   b  may be regulated such that the angle |θ 1 -θ 2 | is the same as in the control homogenizer for comparison having a refractive index n≈1.5. 
     More specifically, the regulation of the sag values Z(r) of the convex lens  121   a  and convex lens  121   b  is attained by regulating the radii of curvature R 1  and R 2  and the conic constants k 1  and k 2  in the formula (2) in which all of α 1  to α 4  are 0. 
     The higher the refractive index n, the smaller the inclination angle of a portion of the convex-lens surface which has the same angle |θ 1 -θ 2 | and the more the maximum sag value Sa of the convex lens  121   a  and the maximum sag value Sb of the convex lens  121   b  (see  FIG. 3 ) can be reduced. Consequently, processability for producing the homogenizer  12  is improved. The refractive index n of the first convex-lens array  12   a  and second convex-lens array  12   b  may be, for example, 1.6 to 2.1. In this case, the light-transmitting substrate  10  on which the first convex-lens array  12   a  and the second convex-lens array  12   b  are to be formed may be a light-transmitting inorganic material differing in refractive index from the convex-lens arrays. For example, a light-transmitting dielectric having a refractive index n=2.1 may be disposed on both surfaces of a light-transmitting glass substrate having a refractive index n≈1.5 so as to form a first convex-lens array  12   a  having a maximum sag value Sa and a second convex-lens array  12   b  having a maximum sag value Sb. 
     Next, the function of one pair of convex lenses in a homogenizer  12  is explained by reference to  FIG. 4A ,  FIG. 4B  and  FIG. 4C .  FIG. 4A  to  FIG. 4C  are cross-sectional views showing the lens function of a pair of convex lenses (a convex lens  121   a  and a convex lens  121   b  paired therewith) of a homogenizer  12  according to this embodiment.  FIG. 4A  to  FIG. 4C  each show a noticeable optical path by hatching.  FIG. 4A  shows an example in which the convex-lens pair acts on an incident light flux which is approximately parallel with the symmetry axis and which strikes on the convex lens  121   a  at the position of the symmetry axis (lens-surface center).  FIG. 4B  and  FIG. 4C  each show an example in which the convex-lens pair acts on an incident light flux which is approximately parallel with the symmetry axis and which strikes on the convex lens  121   a  outside the symmetry axis. 
     As  FIG. 4A  shows, the light flux striking on the convex lens  121   a  at the position of the symmetry axis approximately in parallel with the symmetry axis is emitted from the paired convex lens  121   b  as divergent light having a narrow diffusion angle equal to those in conventional homogenizers, since the light flux enters the flat portion of the convex lens  121   a . Meanwhile, the light flux striking on an inclined surface of the convex lens  121   a  which lies slightly outside the position of the symmetry axis, approximately in parallel with the symmetry axis, proceeds in a more outward direction (i.e., a direction resulting in a larger angle) because of the intense refractive function of the convex lens  121   a , although the diffusion angle (degree of diffusion of the light flux) is the same, and enters a region in the paired convex lens  121   b  which lies more outside the symmetry axis, as shown in  FIG. 4B  and  FIG. 4C . As a result, the light is emitted as divergent light having a larger diffusion angle, by the action of the paired convex lens  121   b . Ina conventional homogenizer in which a flat surface occupies a larger area around the symmetry axis in the lens surface, an incident light flux striking on a region of the convex lens  121   a  which lies outside the symmetry axis but near the symmetry axis approximately in parallel with the symmetry axis is emitted as light having a narrow diffusion angle similar to that shown in  FIG. 4A . 
     As described above, in this embodiment, the convex lenses  121   a  each have a lens surface in which especially a region near the symmetry axis (lens-surface center region) has a larger inclination such that a light flux of incident diffused light which is approximately parallel with the symmetry axis of each convex lens  121   a  and which strikes on a lens surface near the symmetry axis is converted to (wider) divergent light having a larger diffusion angle than in conventional homogenizers and emitted from the paired convex lens  121   b.    
     As shown by hatching in  FIG. 4B  and  FIG. 4C , light-flux components which are linearly passing through regions near the symmetry axis in each convex lens  121   a  are converted to divergent light having a larger diffusion angle than conventional ones. This embodiment hence is effective in regulating a light-intensity distribution on an irradiation plane in the case of using an LD light source which emits light having a light-intensity distribution with a narrow light distribution angle, such as ones approximated to a Gaussian distribution in which the light intensity distribution is maximum at the center axis of a divergent light flux. Especially in the case where divergent light having a light-intensity distribution with a narrow light distribution angle is converted to wide-angle divergent light, by superimposing light emitted from a plurality of convex-lens pairs, to evenly illuminate a given irradiation plane, the light intensity of the light emitted from each convex lens  121   b  decreases in accordance with the cosine fourth power law as the radial-direction angle θ with the symmetry axis increases, as shown in  FIG. 29B . Hence, the light-intensity distribution of the light emitted from each convex lens  121   b  tends to be one in which the light intensity decreases toward the periphery. Meanwhile, in the case where a homogenizer  12  according to this embodiment is used, the light-intensity distribution of the incident light to enter each convex lens  121   b  in regard to the light distribution angle of the incident light to enter the homogenizer  12  can be regulated by changing the lens-surface shape (in particular, the inclination angle of the center region) of the convex lens  121   a  paired with the convex lens  121   b  or by regulating the refractive index n for the wavelength range of the incident light entering the homogenizer  12  (in particular, a light flux entering near the symmetry axis can be more widely diffused in radial directions). Consequently, evenness can be improved. 
     For example, the center-region inclination angle (e.g., the radius of curvature R 1  or the conic constant k 1 ) or the refractive index n of the convex lenses  121   a  in the homogenizer  12  may be regulated such that an average of the in-lens transmission angle |θ 1 -θ 2 | of incident light which enters the lens-surface center region of each convex lens  121   a  including the symmetry axis and which is parallel with the symmetry axis (hereinafter the average is referred to as “average in-lens transmission angle”) is equal to or more than 1.3 times the average in-lens transmission angle in spherical lenses. 
     As demonstrated by Examples mentioned later, the homogenizer  12  according to this embodiment can convert incident light which, for example, is light emitted by a laser light source and having a Gaussian distribution with a maximum diffusion angle α of about 12° into emitted divergent light which has a maximum diffusion angle β of 12° or larger and has a light-intensity distribution of 85% or higher or 90% or higher on a given irradiation plane. The term “light-intensity distribution” herein means the light intensity at a position in a given irradiation plane which is the lowest in light intensity, the light intensity at the center being taken as 100%. 
     For example, considering converting incident light entering a homogenizer  12  and having a large diffusion angle (e.g., α≥10°) and emitting as divergent light which has the same or a larger diffusion angle (e.g., β≥10°) and which strikes evenly on an irradiation plane, a conventional lens-array pairs including spherical lenses are ineffective in obtaining an even intensity distribution on the irradiation plane. Meanwhile, with the lens shapes according to this embodiment, even emission over a wide angle can be attained for incident light having such a large diffusion angle. 
     Divergent light components of light which enters the first convex-lens array  12   a  are condensed so as to strike on regions outside the axes in the focal plane of the convex lenses  121   a . However, the convex lenses  121   a  have a regulated lens surface shape such that the light is condensed so as to strike on the surfaces of the convex lenses  121   b  paired with the convex lenses  121   a . This regulation is the same as in conventional homogenizers. 
     Embodiment 2 
     Next, a second embodiment of the present invention is explained.  FIG. 5A  and  FIG. 5B  are respectively a schematic cross-sectional view and a schematic plan view which show an example of convex lenses  221   a  constituting a first convex-lens array  22   a  of a homogenizer  22  according to the second embodiment. As  FIG. 5A  and  FIG. 5B  show, the homogenizer  22  according to this embodiment differs from the homogenizer  12  according to the first embodiment in the configuration and function of the convex lenses  221   a  constituting the light-entrance-side first convex-lens array  22   a . The configurations and functions of a light-transmitting substrate  20  and convex lenses  221   b  constituting a light-emission-side second convex-lens array  22   b  are the same as those of the light-transmitting substrate  10  and the convex lenses  121   b  constituting the light-emission-side second convex-lens array  12   b  in the first embodiment. 
     The convex lens  221   a  according to this embodiment has a phase diffraction grating  23  formed in an approximately flat lens-surface region near the symmetry axis (e.g., a lens-surface center region). In  FIG. 5A , a lens-surface depth (sag value) distribution of the convex lens  221   a , based on the symmetry axis, is shown by contour lines (dotted lines). In  FIG. 5B , recesses of the phase diffraction grating  23  which are concentric grooves formed in the surface of the convex lens  221   a  are indicated by thick black lines. 
     As  FIG. 5A  shows, light striking on the lens-surface center region of the convex lens  221   a  in the Z-axis direction is diffracted in the directions of diffraction angles γ m  defined by the formula (6) in accordance with the period P of the recesses and protrusions of the phase diffraction grating  23  and with the wavelength λ of the incident light, and the diffracted light is transmitted through the light-transmitting substrate  20 .
 
sin(γ m )= m ·λ/( n·P )  (6)
 
     In the formula (6), n is the refractive index of the convex lens  221   a  and  m  is the order of diffraction (integer). The formula (6) indicates that the diffraction angle γ m  of m-order diffracted light can be regulated with the period P of the phase diffraction grating  23  and the refractive index n. Furthermore, by changing the depth d of the recesses of the phase diffraction grating  23 , the phase difference 2π(n−n 0 )·d/λ between the light transmitted through the recesses and the light transmitted through the protrusions is changed, making it possible to regulate the diffraction efficiency of m-order diffracted light. Symbol n 0  is the refractive index of the recesses (n 0  is 1 in the case of air). 
     For example, in the case where the phase diffraction grating  23  is one in which the recesses and the protrusions have the same width and have a rectangular cross-sectional shape, this phase diffraction grating  23  gives diffracted light composed only of 0-order and odd-number-order diffracted light. In the case where the recess depth d is λ/{2(n−n 0 )}, which results in a phase difference of π, the 0-order transmitted light is minimal and the ±1-order diffraction efficiency is about 40% at the most. The efficiency for diffracted light produced by odd-number-order diffraction where |m|≥3 is a value obtained by multiplying the ±1-order diffraction efficiency by m −2 . By regulating the recess depth d so as to result in a phase difference of 0-π, the quantity of 0-order transmitted light and that of ±1-order diffracted light can be regulated. 
     As shown in  FIG. 6  by hatching, a light flux striking on a lens-surface region near the symmetry axis of the convex lens  221   a  in parallel with the symmetry axis is diffracted by the diffractive function of the phase diffraction grating  23  to proceed more outward directions (i.e., directions in a larger angle), and then strikes on the paired convex lens  221   b  in a region lying more outwardly apart from the symmetry axis. As a result, the light is emitted as divergent light having a larger diffusion angle, by the convex lens  221   b .  FIG. 6  shows a flux of ±1-order diffracted light emitted in the case where the phase diffraction grating  23  has a phase difference of π. By setting the phase difference of the phase diffraction grating  23  at a value within the range of 0 to π, a flux of 0-order transmitted light is also produced and the ratio between the quantities of the 0-order transmitted light and ±1-order diffracted light can be changed. Thus, it is possible to regulate the light-intensity distribution of light to strike on the lens surface of the convex lens  221   b  (that is, light-intensity distribution at the entrance position corresponding to a diffusion angle) and to regulate a light-intensity distribution on an irradiation plane. 
     That is, in the homogenizer  22  according to this embodiment, the light-intensity distribution of light which is to enter each convex lens  221   b  of the homogenizer  22  on which light having a light distribution angle strikes can be regulated with the phase diffraction grating  23 . Evenness can hence be improved. 
     In the example shown in  FIG. 5A  and  FIG. 5B , the recesses and protrusions of the phase diffraction grating  23  are concentric ones disposed at a certain period P, with the symmetry axis of the convex lens  221   a  as the center. However, it is possible to change the period P along the radial directions to regulate the diffraction angle. Furthermore, it is possible to heighten the diffraction efficiency for specific diffraction orders by employing a blaze diffraction grating including recesses having a serrate cross-sectional shape or a pseudo-blaze diffraction grating in which the serration are divided into steps. 
     For example, the phase diffraction grating  23  may be designed such that the optical function obtained by the homogenizer  12  according to the first embodiment, which employs the convex lenses  121   a , can be attained with the homogenizer  22  according to the second embodiment which employs the convex lenses  221   a.    
     As shown in  FIG. 3 , in the homogenizer  12  according to the first embodiment, light which has struck on the convex lens  121   a  at a position having radial distance r in parallel with the symmetry axis is transmitted through the homogenizer in a direction having an angle |θ 1 -θ 2 | with the symmetry axis, by the refractive function of the convex lens  121   a . Consequently, in order to obtain the same effect with the homogenizer  22  according to the second embodiment, when a center region of a convex lens  221   a  where the phase diffraction grating  23  is to be formed is expressed by 0≤r/(W max /2)≤χ (where 0&lt;χ≤0.5), the shape of the phase diffraction grating  23  and the average inclination angle of the lens-surface peripheral region are regulated such that the angle |θ 1 -θ 2 | at which light rays are transmitted through the homogenizer by the refractive and diffractive functions of the convex lens  221   a  is equal to the angle |θ 1 -θ 2 | in the homogenizer employing the convex lens  121   a.    
     A phase diffraction grating  23  produces diffracted light attributed to a plurality of diffraction orders m (m=0, ±1, ±2, . . . ) in accordance with the cross-sectional shape of the phase diffraction grating  23  and the depth of the grating. Hence, in the case where the phase diffraction grating  23  generates a plurality of angles |θ 1 -θ 2 | at which light rays are transmitted through the homogenizer, this phase diffraction grating  23  may be configured such that an average angle |θ 1 -θ 2 | determined by taking account of the diffraction efficiencies of the diffraction orders is equal to the angle |θ 1 -θ 2 | in the homogenizer employing the convex lens  121   a . The blaze phase diffraction grating or pseudo-blaze phase diffraction grating in which the diffraction efficiency for a specific diffraction order m (m≠0) is 80% or higher functions as the case where both a refracting lens and a diffracting lens are used. Since the phase diffraction grating  23  is employed for the purpose of reducing the proportion of rectilinear transmitted-light components in the center region of the convex lens  221   a , χ is more preferably 0.05≤χ≤0.3. 
     Besides the homogenizers according to the first and second embodiments, any homogenizer may have a similar evenly diffusing/illuminating function so long as the convex lenses constituting the light-entrance-side first convex-lens array each has a surface shape which converts some of rectilinear transmitted light striking on a surface region near the symmetry axis of the convex lens (in particular, an approximately flat region) to diffused light (that is, to light which strikes on the lens surface of each of the light-emission-side convex lenses paired with the light-entrance-side convex lenses, at a position outside the symmetry axis). 
     Third Embodiment 
     The homogenizers ( 12  and  22 ) according to the first and second embodiments are each configured of a first convex-lens array ( 12   a  or  22   a ) and a second convex-lens array ( 12   b  or  22   b ) each obtained by arraying convex lenses which each have a lens surface symmetric with respect to the axis and have an XY-plane outer shape that is quadrilateral (Wx×Wy). Because of this, the convex-lens surface has a maximum diameter and a maximum depth (maximum sag value) along diagonal directions of the quadrilateral shape. In the case where the outer shape is square (W=Wx=Wy), the maximum width (W max ) of each convex lens is expressed by the formula (7). Spherical lenses have a lens depth being twice as compared with the directions of the sides (X axis and Y axis) of the square.
 
[Math. 2]
 
 W   max =√{square root over (( Wx   2   +Wy   2 ))}=2· W   (7)
 
     Furthermore, in each convex-lens array, quadrilateral convex lenses each having a maximum inclination angle at the corners are arranged such that four convex-lens surfaces discontinuously meet each other to form a structure having a maximum sag value. It is hence difficult to precisely perform processing for forming a lens shape, and the processing is prone to result in recessed surface regions where the inclination angle changes continuously. As a result, especially in the case where it is desired to evenly illuminate an irradiation plane with light having such a wide diffusion angle that maximum diffusion angle β≥15°, a homogenizer having the desired properties may not be obtained. In order to overcome such problem, a homogenizer  32  according to a third embodiment employs cylindrical-lens arrays having convex-lens cross-section shapes which are symmetric with respect to a plane (that is, arrays of convex cylindrical lenses). 
       FIG. 7  is a slant view of the homogenizer  32  according to the third embodiment. As  FIG. 7  shows, the homogenizer  32  according to this embodiment includes a pair of homogenizers including a homogenizer (convex-lens array pair)  32   x  for X axis, which has a lens power only in the X-axis direction, and a homogenizer (convex-lens array pair)  32   y  for Y axis, which has a lens power only in the Y-axis direction. The homogenizer  32   x  and the homogenizer  32   y  are disposed serially in the optical-axis direction, which is the light transmission direction, such that the lens-function axis directions of these differ from each other by 90°. The lens power is also called lens function. 
       FIG. 8A  and  FIG. 8B  are respectively a cross-sectional view and a plan view which show an example of the homogenizer  32   x .  FIG. 9A  and  FIG. 9B  are a cross-sectional view and a plan view which show an example of the homogenizer  32   y.    
     As  FIG. 8A  and  FIG. 8B  show, the homogenizer  32   x  includes a light-transmitting substrate  30   x  having a flat-plate shape, and a first cylindrical-lens array  32   xa  and a second cylindrical-lens array  32   xb , which each have convex-lens cross-sectional shapes that are symmetric with respect to a plane and which are disposed respectively on two surfaces (first surface and second surface) of the light-transmitting substrate  30   x . The first cylindrical-lens array  32   xa  and the second cylindrical-lens array  32   xb  have XZ cross-sections that are convex-lens array cross-sections similar to those of the first convex-lens array  12   a  and second convex-lens array  12   b  of the homogenizer  12  according to the first embodiment. However, the first and second cylindrical-lens arrays  32   xa  and  32   xb  have no lens power in the YZ cross-sections. 
     Meanwhile, as  FIG. 9A  and  FIG. 9B  show, the homogenizer  32   y  includes a light-transmitting substrate  30   y  having a flat-plate shape, and a first cylindrical-lens array  32   ya  and a second cylindrical-lens array  32   yb , which each have convex-lens cross-sectional shapes that are symmetric with respect to a plane and which are disposed respectively on two surfaces (first surface and second surface) of the light-transmitting substrate  30   y . The first cylindrical-lens array  32   ya  and the second cylindrical-lens array  32   yb  have YZ cross-sections that are convex-lens array cross-sections similar to those of the first convex-lens array  12   a  and second convex-lens array  12   b  of the homogenizer  12  according to the first embodiment. However, the first and second cylindrical-lens arrays  32   ya  and  32   yb  have no lens power in the XZ cross-sections. 
     Examples of a convex lens include a convex cylindrical lens in a broad sense. Hereinafter, an array of convex cylindrical lenses is often called a convex-lens array. In each of the homogenizers  32   x  and  32   y : the cylindrical lenses, which correspond to convex lenses of the first cylindrical-lens array disposed on the light entrance side, are often called first convex lenses; the cylindrical lenses, which correspond to convex lenses of the second cylindrical-lens array disposed on the light emission side, are often called second convex lenses; and a cross-section of each of these convex lenses which is perpendicular to the base-line direction is often called a lens cross-section and the position of a symmetry plane in the lens cross-section is often called a symmetry axis (in the lens cross-section). 
     In this embodiment, divergent light from a light source first enters the homogenizer  32   x , and this light is made, by the homogenizer  32   x , to have an even X-axis-direction light-intensity distribution on an irradiation plane, and then the light enters the homogenizer  32   y . The light which has entered the homogenizer  32   y  is made, by the homogenizer  32   y , to have an even Y-axis-direction light-intensity distribution on the irradiation plan, and then the light is emitted. As a result, an evenly diffusing/illuminating function similar to that of the homogenizer  12  of the first embodiment is obtained. 
     The sequence of Z-axis-direction arrangement of the homogenizers  32   x  and  32   y  is not particularly limited. Namely, either the homogenizer  32   x  or the homogenizer  32   y  may be on the light entrance side. The smaller the gap between the homogenizer  32   x  and the homogenizer  32   y , the more preferable, from the viewpoint of reducing the size of the homogenizer  32 . In the cylindrical lenses  321  (specifically,  321   xa ,  321   xb ,  321   ya , and  321   yb ) constituting the cylindrical-lens arrays of the homogenizers  32   x  and  32   y , the lens widths (Wx and Wy in the figures) are preferably set such that four or more cylindrical lenses, more preferably five or more cylindrical lenses, still more preferably seven or more cylindrical lenses, are included along the X direction or the Y direction in the plane where each cylindrical lens array is formed, in accordance with the beam diameter of the light to enter the homogenizers  32   x  and  32   y.    
     In this embodiment, the cylindrical lenses ( 321   xa ,  321   xb ,  321   ya , and  321   yb ) of the cylindrical-lens arrays ( 32   xa ,  32   xb ,  32   ya , and  32   yb ) can be made to have a reduced maximum depth (sag value) and a reduced lens-surface maximum inclination angle. In addition, since the lens boundaries, where convex-lens surfaces meet each other discontinuously, can be linear, a lens shape of optimal design can be precisely formed. Consequently, a homogenizer to attain the desired wide-angle diffusion and even illumination is easily obtained. 
     Another example of the configuration of the cylindrical-lens arrays  32   xa  and  32   ya , which are disposed on the light entrance side of the substrates, may be one in which an approximately flat surface region near the symmetry plane of each cylindrical lens ( 321   xa  or  321   ya ), for example a region corresponding to a lens-surface center region in a cross-section perpendicular to the base-line direction of the lens (hereinafter referred to simply as “lens-surface center region”), has a phase diffraction grating formed therein which includes recess grooves linearly extending in parallel with the symmetry plane. The lens-surface center region in the cylindrical lens may be a region where the radial distance r from the position of the symmetry plane in a cross-section perpendicular to the base-line direction of the lens is equal to or less than 50% of the maximum radial distance W/2 from the position (0≤r≤W/4), or may be a region which satisfies 0≤r/(W/2)≤χ where χ is within the range of 0&lt;χ≤0.5. 
     In other words, another example of the homogenizer  32   x  according to this embodiment may be one in which the first cylindrical-lens array  32   xa  and the second cylindrical-lens array  32   xb  are cylindrical-lens arrays which have XZ cross-sections that are lens cross-sections similar to those of the first convex-lens array  22   a  and second convex-lens array  22   b  of the homogenizer  22  according to the second embodiment but which have no lens power in the YZ cross-sections. Furthermore, another example of the homogenizer  32   y  may be one in which the first cylindrical-lens array  32   ya  and the second cylindrical-lens array  32   yb  are cylindrical-lens arrays which have YZ cross-sections that are lens cross-sections similar to those of the first convex-lens array  22   a  and second convex-lens array  22   b  of the homogenizer  22  according to the second embodiment but which have no lens power in the XZ cross-sections. 
     In the case where the homogenizers  32   x  and  32   y  are configured thus, an evenly diffusing/illuminating function similar to that of the second embodiment is obtained. 
     [Other Configurations of the Homogenizers] 
     It is preferable, in each of the homogenizers according to the embodiments described above, that in order to reduce Fresnel reflection, which occurs due to a difference in refractive index, an antireflection film (not shown) based on a common dielectric multilayer-film design in accordance with the wavelength of the incident light and the incidence angle range is formed on lens surfaces in contact with air (for example, the surfaces of the following lens arrays which are in contact with air: the first convex-lens array  12   a , first convex-lens array  22   a , first cylindrical-lens array  32   xa , first cylindrical-lens array  32   ya , second convex-lens array  12   b , second convex-lens array  22   b , second cylindrical-lens array  32   xb , and second cylindrical-lens array  32   yb ). 
     Furthermore, in each of the homogenizers according to the embodiments described above, the smaller the width of the boundary portion between adjoining convex lenses, the less the occurrence of stray light, which adversely affects the intensity distribution on irradiation planes. The width of the boundary portion is the distance G between inflection points in the boundary portion at which the area the change in lens-surface inclination angle with the radial distance increases shifts to the area in which the change decreases or vice versa. Specifically, the proportion of the G to the width W (Wx or Wy) of each lens, G/W, is preferably 10% or less, more preferably 5% or less, still more preferably 3% or less, especially preferably 2% or less. The distance G is preferably 10 μm or less. In particular, in the light-entrance-side first convex-lens arrays (the first convex-lens arrays  12   a  and  22   a  and the first cylindrical-lens arrays  32   xa  and  32   ya ), the distance G, which is the width of the boundary portion between adjoining convex lenses, is more preferably 5 μm or less. 
     In the homogenizers according to the embodiments described above, the widths W (Wx and Wy) of each convex lens are preferably within the range of 20 μm to 500 μm from the viewpoints of illuminator size reduction and even illumination, and are more preferably within the range of 50 μm to 200 μm. Furthermore, the distance T between the first convex-lens array and the second convex-lens array, which corresponds to the thickness of the homogenizer, is set approximately in proportion to the widths W of the convex lenses and is preferably 50 μm to 3,000 μm. From the viewpoint of stably forming a convex-lens array shape on each of both surfaces of a light-transmitting substrate, the T is more preferably within the range of 100 μm to 1,000 μm. 
     In each of the embodiments shown above, the homogenizer includes a pair of convex-lens arrays configured by disposing a convex-lens array on each of both surfaces of a light-transmitting substrate  10  having a refractive index n, from the viewpoint of ease of optical-axis alignment. However, the homogenizer may have, for example, a configuration including two light-transmitting substrates which each have a convex-lens array formed on one surface thereof and which are disposed such that the two lens surfaces face each other inward, such as that shown in  FIG. 24A . Even in such a case, the convex-lens array disposed on the light entrance side is referred to as a first convex-lens array and the convex-lens array disposed on the light emission side is referred to as a second convex-lens array. Although the two convex-lens arrays are apart from each other, this homogenizer is configured such that the first convex lenses, which are the convex lenses of the first convex-lens array, and the second convex lenses, which are the convex lenses of the second convex-lens array, face each other and share symmetry axes, and constitute convex-lens pairs. 
     However, in the configuration shown above, the average internal transmission angle |θ 1 -θ 2 | of the first convex lenses is defined as a relationship between the incidence angle θ 1  and the emission angle θ 2  on the lens surface of the first convex lens with respect to light entering the first light-transmitting substrate having a refractive index n, on which the first convex-lens array is formed, in parallel with the symmetry axis. Consequently, the “internal” in the term “average internal transmission angle” in this case indicates the inside of a pair of convex-lens arrays which is regarded as one optical member. 
     Embodiment 4 
     Next, an illuminator employing the homogenizers described above is explained as a fourth embodiment of the present invention.  FIG. 10  is a cross-sectional view showing an example of the illuminator  100  according to the fourth embodiment. As  FIG. 10  shows, the illuminator  100  according to this embodiment includes a package  13 , a laser light source  11  mounted and fixed in the package  13 , and a homogenizer  12  which is disposed on the emission side of the package and on which a light flux  15  emitted by the laser light source  11  strikes.  FIG. 10  shows the homogenizer  12  according to the first embodiment as an example of optical elements for converting the light flux  15  emitted by the laser light source  11  into more widely diffused divergent light  16 , which performs even illuminating on an irradiation plane  17 , and emitting the divergent light  16 . However, the optical element may be the homogenizer  22  according to the second embodiment or the homogenizer  32  according to the third embodiment. 
     From the viewpoint of enabling the laser light source  11  to work stably and retain long-term reliability, it is preferable that the illuminator  100  employs a package material and a homogenizer having high heat resistance and high heat dissipation property, and that the laser light source  11  is airtightly enclosed in the package  13 . It is more preferable in the illuminator  100  that the inside of the package  13  is filled with an inert gas  14 , e.g., dry nitrogen, according to need. In  FIG. 10 , a power source and electrical wiring for supplying electrical voltage/current to the laser light source  11 , a heat sink for cooling the laser light source  11  and the like are omitted. 
     In the illuminator  100 , the laser light source  11  emits a light flux  15  having a Gaussian light-intensity distribution with a maximum diffusion angle δ. The light flux  15  is transmitted, as such, as the light flux  15  having a maximum incidence angle α (=δ) through the homogenizer  12  to become divergent light  16  having a maximum diffusion angle β, and then strikes on an irradiation plane  17  to attain an even light-intensity distribution thereon. 
     Although  FIG. 10  shows an example of the illuminator  100  in which the homogenizer  12  is integrated with the package  13 , the illuminator  100  can be configured such that a package  13  and a homogenizer  12  are separately disposed. 
     As the laser light source  11 , an LD which emits light having a specific wavelength in the wavelength range of 300 nm to 2,000 nm is suitably used because an LD is small and has a high luminance. The laser light source  11  may have a single laser light emission point, or may be a laser-array light source including a plurality of light emission points disposed in an array arrangement. A surface-emitting laser (VCSEL), by which a plurality of laser light emission points can be highly densely integrated efficiently in producing the laser, is suitably used as the laser light source  11  of the illuminator  100  for large illumination areas because the intensity of the laser light to be emitted can be controlled to increase by increasing the number of emission points. Although typical surface-emitting lasers emit light having wavelengths in the near infrared range of 780 nm to 1,300 nm, it is possible to convert such a surface-emitting laser into a laser light source for emitting light in the visible light wavelength range of 400 nm to 700 nm, by using a nonlinear optical crystal to conduct a wavelength conversion. 
     As the laser light source  11 , a laser light source obtained by arraying semiconductor chips of a Fabry-Pérot LD, in which laser active layers having an optical waveguide structure are used to constitute an etalon resonator structure based on reflection at the ends of the waveguides, may be used. The Fabry-Pérot LD has an advantage in that a high light output is obtained by increasing the width of the optical waveguide structure of laser active layers and that a laser light source emitting light having a wavelength in an ultraviolet to green range is obtained therefrom by using a GaN crystal system. 
     The light emitted by LDs typically has an uneven distribution approximated to a Gaussian light-intensity distribution. In the case of a surface-emitting laser, the emitted light has an axially symmetrical emission-angle light-intensity distribution and has an FFP (total angle of emission where the light intensity ratio is e −2  of the maximum light intensity at the center of the symmetry axis) of about 9° to 20°. In the case of a Fabry-Pérot laser, the laser active layer has different cross-sectional shapes and the emitted light hence has a Gaussian light-intensity distribution in which the FFP along the horizontal direction (H) in the plane of the optical waveguide is different from that along the vertical direction (V); the FFP(H) is about 6° to 15° and the FFP(V) is about 20° to 40°. A relationship between the maximum emission angle δ and the FFP is FFP=2×δ. 
     The laser light source  11  to be used in the illuminator  100  of the present invention is preferably a surface-emitting laser having high directivity and a relatively small maximum emission angle δ. Moreover, since a laser light source including a single laser light emission point can only emit light having a low intensity, it is preferred to use a laser array light source including a plurality of surface-emitting laser light emission points. 
     The laser light source  11  may be, for example, a laser array light source obtained by arraying laser emission points  11   b  on a semiconductor substrate  11   a  such that laser light emission points are arranged in the X-axis direction at intervals of a, the number of the light emission points being Na, and laser light emission points are arranged in the Y-axis direction at intervals of b, the number of the light emission points being Nb, such as that shown in  FIG. 28A  and  FIG. 28B . In this case, the intervals a and bare preferably within the range of 20 μm to 100 μm, and the light flux emitted by each laser emission point  11   b , which has a Gaussian light-intensity distribution, preferably has an emission angle (half angle) δ of about 4° to 9°. 
     For example, when each laser emission point has an emission intensity of 20 mW and the intervals are a=b=50 μm and in the case where Na=Nb=10, i.e., 100 laser emission points are arrayed, a laser light source  11  having a total quantity of light of 2 W and an emission surface size of 0.5 mm×0.5 mm is obtained. Furthermore, in the case where Na=Nb=100, i.e., 10,000 laser emission points are arrayed, a laser light source  11  having a total quantity of light of 200 W and an emission surface size of 5 mm×5 mm is obtained. 
     Furthermore, when the light flux emitted by each laser emission point has an emission angle δ of 8° and in the case where the distance L from the laser emission points to the entrance surface of the homogenizer is 3.6 mm or larger, lights emitted by adjoining laser emission points are superimposed each other, and a light flux which has an uneven light-intensity distribution and has diffused to a size larger by 2·L·tan(δ) than the emission area of the laser light source  11  (Na×a in the X-axis direction; Nb×b in the Y-axis direction) enters the homogenizer  12 . 
     It is preferable in this case that in each of the first convex-lens array and second convex-lens array which constitute the homogenizer  12 , the number of arrayed convex lenses in each of the two directions (Nx and Ny) which lie in the light flux entering the homogenizer  12  is 4 or larger, that is, at least 16 arrayed convex lenses in total (Nx×Ny≥16) lie in the light flux. Furthermore, by regulating the number of arrayed convex lenses lying in the light flux to 5 or more in each of the X direction and Y direction (25 or more in total), or to 7 or more in each direction (49 or more in total), the light fluxes emitted respectively from the convex lenses of the light-entrance-side first convex-lens array are more superimposed at the irradiation plane  17  to improve the evenness of the light-intensity distribution. 
     As described above, according to this embodiment, it is possible to provide a compact illuminator  100  which employs a small high-luminance laser light source  11  to emit divergent light having a large maximum diffusion angle β and efficiently illuminate an irradiation plane located at a relatively short distance with the divergent light having an even intensity distribution. The laser light source  11  is not limited to VCSEL array light sources such as that described above, and a Fabry-Pérot laser array light source or a laser light source including a single emission surface can also be used to provide an illuminator  100  having a similar function. 
     Besides laser light sources, a conventional discharge emission lamp can be used to provide an illuminator which emits divergent light having a large maximum diffusion angle R and efficiently illuminates an irradiation plane located at a relatively short distance with the divergent light having an even intensity distribution, by using the homogenizer  12 ,  22 , or  32 . 
     Embodiment 5 
     Next, another example of illuminators employing the homogenizers described above is explained as a fifth embodiment of the present invention.  FIG. 11  is a cross-sectional view showing an example of the illuminator  110  according to the fifth embodiment. The illuminator  110  shown in  FIG. 11  differs from the illuminator  100  according to the fourth embodiment in that this illuminator  110  includes: a plurality of laser light sources  112 ; lenses  113  for condensing light emitted by the laser light sources  112 , and optical fibers  114 ; the light-emission-side ends of which are bundled and disposed in an array arrangement, which are for receiving the condensed light from the lenses  113  and through which the light is transmitted, and these are used in place of the laser light source  11  of the illuminator  100 , as a laser light source within the package  13 , namely as a laser light source (more specifically, laser emission point array)  111  for the homogenizer  12 . 
     In the laser light source  111 , the light-emission-side ends of the optical fibers  114  correspond to the plurality of laser emission points  11   b  shown in  FIG. 28A . The semiconductor substrate  11   a  is replaced by a fixing jig which bundles the light-emission-side ends of the optical fibers  114 . 
     Any laser light sources may be used as the plurality of laser light sources  112  as long as the laser light sources emit light that can be efficiently condensed by the lenses  113  so as to enter the optical fibers  114  and transmitted. As the optical fibers  114 , quartz-based multimode optical fibers which transmit LD light having wavelengths of 190 nm to 2,400 nm can be used. For example, the numerical aperture NA is 0.22 (emission angle δ=12.7°), the core diameter is 50 μm, and the clad diameter is 125 μm. Consequently, a plurality of such laser light sources  112  can be used as a laser light source  111  including a plurality of laser light emission points having an emission angle δ of 12.7° and emission-point intervals of about 125 μm. 
     The homogenizers ( 12 ,  22 , and  33 ) according to the embodiments cause no change in light-intensity distribution on irradiation planes due to the positional shifting of laser light emission points. Because of this, the function of the illuminator  110  can be maintained even in the case where the light-emission-side ends of the plurality of optical fibers  114  are disposed at different intervals. 
     In the illuminator  110  according to this embodiment, the laser light sources  112  and the laser light emission points can be disposed at any desired distance therebetween using the optical fibers  114 . Consequently, the design flexibility regarding laser light sources and cooling systems is increased to enable the illuminator  110  to be used in various illumination applications. 
     EXAMPLES 
     Example 1 
     Examples of the embodiments are shown below using specific numerical values. First, examples of the homogenizer  12  according to the first embodiment shown in  FIG. 1  and  FIG. 2  are shown as first Examples (more specifically, Examples 1-1 to 1-10). These homogenizers  12  are each produced by respectively forming a first convex-lens array  12   a  and a second convex-lens array  12   b  on the first and second surfaces of a light-transmitting substrate  10  which is a glass substrate having a refractive index n for use wavelength of 1.50 and a thickness T of 280 μm, by disposing convex-lens pairs (convex lenses  121   a  and  121   b ) having a rectangular shape with Wx=100 μm and Wy=80 μm and sharing symmetry axes, in an array arrangement in the XY plane such that Nx=10 and Ny=12, i.e., the total number of the convex-lens pairs is 120. The convex lenses  121   a  and the convex lenses  121   b  each have a cross-sectional shape that is symmetrical with respect to a rotation axis parallel with Z axis, and each convex lens  121   a  and the convex lens  121   b  paired therewith share a symmetry axis. 
       FIG. 12  is a diagram for illustrating the lens shapes of the convex lenses  121   a  in the homogenizer  12  according to the first embodiment. In this example, the lens surface shapes of the convex lenses  121   a  in the homogenizer  12  were aspherical lens shapes approximated with the radius of curvature R 1  and the conic constant k 1  by the formula (2) shown above in which α 1  to α 4  were each 0. 
     In  FIG. 12 , the lens surface shape indicated by a solid line is Comparative Example 1, (R 1 , k 1 )=(100, 0). The values of (R 1 , k 1 ) of the Examples 1-1 to 1-10 areas shown in Table 1. Besides the values of the Examples, parameters of the Comparative Example and Reference Examples are shown in Table 1.  FIG. 12  shows the results of calculations for determining aspherical lens shapes from the parameters of Comparative Example 1 and Examples 1-1 to 1-10 shown in Table 1. 
     
       
         
           
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 (R 1 , k 1 ) 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                   
                 Example 
                   
               
               
                   
                 1-1 
                 (60, −3) 
               
               
                   
                 1-2 
                 (60, −4) 
               
               
                   
                 1-3 
                 (60, −5) 
               
               
                   
                 1-4 
                 (50, −3) 
               
               
                   
                 1-5 
                 (50, −4) 
               
               
                   
                 1-6 
                 (50, −5) 
               
               
                   
                 1-7 
                 (50, −6) 
               
               
                   
                 1-8 
                 (40, −3) 
               
               
                   
                 1-9 
                 (40, −4) 
               
               
                   
                 1-10 
                 (30, −3) 
               
               
                   
                 Comparative Example 
               
               
                   
                 1 
                 (100, 0)  
               
               
                   
                 Reference Example 
               
               
                   
                 1 
                 (150, −4)  
               
               
                   
                 2 
                 (20, −4) 
               
               
                   
                 3 
                 (50, −1) 
               
               
                   
                 4 
                 (50, −2) 
               
               
                   
                   
               
            
           
         
       
     
       FIG. 13  shows the results of calculations for determining the angles |θ 1 -θ 2 | of light rays (transmitted light) which are transmitted through the homogenizers of Comparative Example 1 and Examples 1-1 to 1-10. As  FIG. 12  shows, the convex lenses  121   a  in each of the homogenizers  12  of Examples 1-1 to 1-10 have a cross-sectional shape in which the lens-surface center region (r&lt;30 μm) has larger inclination angles and includes a narrower flat portion as compared with that in Comparative Example 1. Moreover, in Examples 1-1 to 1-10, the lens-surface peripheral region (r&gt;40 μm) has an inclination angle which is approximately constant, unlike that in Comparative Example 1, which increases monotonously. As a result, as  FIG. 13  shows, the angle |θ 1 -θ 2 | at which transmitted light proceeds through the homogenizer  12  depends on the cross-sectional shape of the convex lens  121   a  as follows. In Comparative Example 1, the angle |θ 1 -θ 2 | increases approximately linearly as the radial distance (radius) r increases. By contrast, in Examples 1-1 to 1-10, the angle |θ 1 -θ 2 | does not increase linearly as the radial distance (radius) r increases. More specifically, in each of the homogenizers  12  of Examples 1-1 to 1-10, angle |θ 1 -θ 2 | in the lens-surface center region (r&lt;30 μm) is kept larger than in Comparative Example 1-1 while the degree of change in angle |θ 1 -θ 2 | decreases, and the degree of change in angle |θ 1 -θ 2 | in the lens-surface peripheral region (r&gt;40 μm) considerably decreased. The portion where r&gt;55 μm in each Example has a smaller value of angle |θ 1 -θ 2 | than that in Comparative Example 1. 
     In each of Examples 1-1 to 1-10, the convex lenses  121   b  of the second convex-lens array  12   b  were spherical lenses in which (R 2 , k 2 )=(100, 0). 
       FIG. 14A  to  FIG. 14D ,  FIG. 15A  to  FIG. 15D ,  FIG. 16A  and  FIG. 16B  show the results of calculating light-intensity distributions on an irradiation plane  17  which are obtained with illuminators  100  shown in  FIG. 10  into which the homogenizers  12  of Examples 1-1 to 1-10 are incorporated. The homogenizer  12  of each Example was attached to the light emission window of the package  13  of the illuminator  100  shown in  FIG. 10 , the package  13  containing a laser light source  11  mounted therein. Further, D=100 mm. The laser light source  11  had a configuration obtained by disposing surface-emitting lasers having an emitted-light wavelength of 850 nm and a diffusion angle δ of 8° and having a Gaussian emitted-light-intensity distribution, such that Na=Nb=5 and thus  25  such surface-emitting lasers in total were disposed in an array arrangement at intervals of a=b=50 μm. The laser array emission surface had a size of 0.25 mm×0.25 mm, and the gap L between the laser array emission surface and the homogenizer  12  was 0.5 mm. 
       FIG. 16C  shows the results of calculating a light-intensity distribution on the irradiation plane  17  which is obtained with an illuminator  100  shown in  FIG. 10  into which the homogenizer of Comparative Example 1 is incorporated. 
       FIG. 17A  to  FIG. 17D  show the results of calculating light-intensity distributions on the irradiation plane  17  which are obtained with illuminators  100  shown in  FIG. 10  into which the homogenizers of Reference Examples 1 to 4 are incorporated. Reference Examples 1 to 4 respectively employed the following lens shapes of the convex lenses  121   a : (R 1 , k 1 )=(150, −4), (20, −4), (50, −1), and (50, −2). 
     In the figures, each light-intensity distribution is shown in terms of relative values in % obtained by normalization with a maximum light intensity in the irradiation plane of 80 mm×80 mm. 
     As  FIG. 14A  to  FIG. 17D  show, use of the homogenizers of Examples 1-1 to 1-10 gave light-intensity distributions in each of which the normalized light intensity was 90% or higher over an X-axis-direction width of about 50 mm and a Y-axis-direction width of about 40 mm, indicating that the light-intensity distributions on the irradiation plane had been more even than that obtained with the homogenizer of Comparative Example 1, which employed spherical convex lenses  921 . It can be seen that Reference Examples 1 to 3 gave light-intensity distributions poorer than that of Comparative Example 1. Reference Example 4, in which (R 1 , k 1 )=(50, −2), gave a light-intensity distribution in which the normalized light intensity was 85% or higher over an X-axis-direction width of about 50 mm and a Y-axis-direction width of about 40 mm, indicating that the light-intensity distribution on the irradiation plane had been more even than that obtained with the homogenizer of Comparative Example 1, which employed spherical convex lenses  921 . However, Reference Example 4 was inferior to Examples 1-1 to 1-10. 
     As demonstrated above, there is a range of optimal shapes for the convex lenses  121   a , depending on the laser light source and use conditions for the irradiation plane. These shapes can be determined on the basis of an optical design. 
     According to the Examples, on the irradiation plane being 100 mm distant from the homogenizer  12 , a light-intensity evenness of 90% or higher can be attained over a range having an X-axis-direction width of 50 mm on Y axis and a Y-axis-direction width of about 40 mm on X axis. The diffusion angles (half angles) β in X-axis direction and Y-axis direction obtained by converting this illumination range are βx=14.0° and βy=11.3°. The maximum diffusion angle βmax, which is an angle along a diagonal direction of the rectangular irradiation plane, is 17.8°. By changing the distance between the homogenizer and the irradiation plane, the area of the evenly illuminated range can be regulated. 
       FIG. 18A  and  FIG. 19A  show the results of calculations for illustrating, by gradation on XY plane, the light-intensity distributions (normalized) on the irradiation plane which were obtained with the illuminators  100  respectively employing Example 1-5 and Comparative Example 1. The gradational illustration shown in  FIG. 18A  is the calculation results for Example 1-5, while the gradational illustration shown in  FIG. 19A  is the calculation results for Comparative Example 1.  FIG. 18B  and  FIG. 19B  each show a black-and-white binarized image obtained by gradating the light-intensity distributions shown in  FIG. 18A  and  FIG. 19A  at intervals of 10% to obtain a gradational illustration and binarizing the gradational illustration, for an easier understanding. 
     As  FIG. 18A ,  FIG. 18B ,  FIG. 19A  and  FIG. 19B  show, use of the homogenizer of Comparative Example 1 gave a light-intensity distribution in which the light intensity in corner portions (maximum diffusion half angle βmax=17.8°) of the irradiation plane, which were the lowest in light intensity, had decreased to about 70%, while use of the homogenizer  12  of Example 1-5 gave a light-intensity distribution in which even the same corner portions retained a light intensity of about 80% or higher. 
     Table 2 shows calculated values of the average lens-surface inclination angle of each of lens-surface regions defined based on maximum width W max , with respect to the convex lenses (corresponding to  121   a ) of the light-entrance-side convex-lens array used in each of the homogenizers of Examples 1-1 to 1-10, Comparative Example 1, and Reference Examples 1 to 4. The convex lenses ( 121   a ) each had a rectangular shape with widths of Wx=100 μm and Wy=80 μm and, hence, W max =128 μm. The division into lens-surface regions was such that a lens-surface center region was expressed by 0≤r/(W max /2)≤χ and a lens-surface peripheral region was expressed by (1−χ)≤r/(W max /2)≤1 and that the center and peripheral regions were each divided into regions respectively corresponding to χ=0.125, 0.20, 0.30, 0.40, and 0.50. The refractive index n of each homogenizer was taken as 1.5. Table 2 also shows the results of normalization with the average lens-surface inclination angles of the lens-surface regions of the convex lens of Comparative Example 1, which was a spherical lens with R 1 =100 μm and k 1 =0. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                 TABLE 2 
               
               
                   
               
             
            
               
                   
                 Comparative 
                 Example 
                 Example 
                 Example 
                 Example 
                 Example 
                 Example 
               
               
                   
                 Example 1 
                 1-1 
                 1-2 
                 1-3 
                 1-4 
                 1-5 
                 1-6 
               
               
                   
               
               
                 (R 1 , k 1 ) 
                 (100, 0) 
                 (60, −3) 
                 (60, −4) 
                 (60, −5) 
                 (50, −3) 
                 (50, −4) 
                 (50, −5) 
               
            
           
           
               
               
            
               
                 Lens-surface region 
                 Average lens-surface inclination angle [°] 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 12.5% or less of outer shape 
                 2.3 
                 3.8 
                 3.8 
                 3.7 
                 4.5 
                 4.5 
                 4.4 
               
               
                   20% or less of outer shape 
                 3.7 
                 6.0 
                 5.9 
                 5.9 
                 7.1 
                 7.0 
                 6.9 
               
               
                   30% or less of outer shape 
                 5.5 
                 8.5 
                 8.3 
                 8.2 
                 10.0 
                 9.7 
                 9.4 
               
               
                   40% or less of outer shape 
                 7.5 
                 11.1 
                 10.8 
                 10.4 
                 12.8 
                 12.3 
                 11.8 
               
               
                   50% or less of outer shape 
                 9.3 
                 13.1 
                 12.5 
                 12.0 
                 14.9 
                 14.1 
                 13.5 
               
               
                   50% or more of outer shape 
                 28.9 
                 27.5 
                 24.8 
                 22.7 
                 29.3 
                 26.0 
                 23.7 
               
               
                   60% or more of outer shape 
                 30.8 
                 28.3 
                 25.3 
                 23.2 
                 29.9 
                 26.5 
                 24.1 
               
               
                   70% or more of outer shape 
                 33.1 
                 29.1 
                 25.9 
                 23.6 
                 30.6 
                 27.0 
                 24.4 
               
               
                   80% or more of outer shape 
                 35.1 
                 29.6 
                 26.3 
                 23.9 
                 31.0 
                 27.3 
                 24.6 
               
               
                 87.5% or more of outer shape 
                 36.9 
                 30.0 
                 26.6 
                 24.1 
                 31.3 
                 27.5 
                 24.8 
               
            
           
           
               
               
            
               
                 Lens-surface region 
                 Normalized average lens-surface inclination angle 
               
               
                   
                 (normalized with average lens-surface inclination angle of Comparative Example 1) 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 12.5% or less of outer shape 
                 1.0 
                 1.6 
                 1.6 
                 1.6 
                 2.0 
                 1.9 
                 1.9 
               
               
                   20% or less of outer shape 
                 1.0 
                 1.6 
                 1.6 
                 1.6 
                 1.9 
                 1.9 
                 1.9 
               
               
                   30% or less of outer shape 
                 1.0 
                 1.6 
                 1.5 
                 1.5 
                 1.8 
                 1.8 
                 1.7 
               
               
                   40% or less of outer shape 
                 1.0 
                 1.5 
                 1.4 
                 1.4 
                 1.7 
                 1.6 
                 1.6 
               
               
                   50% or less of outer shape 
                 1.0 
                 1.4 
                 1.4 
                 1.3 
                 1.6 
                 1.5 
                 1.5 
               
               
                   50% or more of outer shape 
                 1.0 
                 1.0 
                 0.9 
                 0.8 
                 1.0 
                 0.9 
                 0.8 
               
               
                   60% or more of outer shape 
                 1.0 
                 0.9 
                 0.8 
                 0.8 
                 1.0 
                 0.9 
                 0.8 
               
               
                   70% or more of outer shape 
                 1.0 
                 0.9 
                 0.8 
                 0.7 
                 0.9 
                 0.8 
                 0.7 
               
               
                   80% or more of outer shape 
                 1.0 
                 0.8 
                 0.7 
                 0.7 
                 0.9 
                 0.8 
                 0.7 
               
               
                 87.5% or more of outer shape 
                 1.0 
                 0.8 
                 0.7 
                 0.7 
                 0.8 
                 0.7 
                 0.7 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                   
                 Example 
                 Example 
                 Example 
                 Example 
                 Reference 
                 Reference 
                 Reference 
                 Reference 
               
               
                   
                 1-7 
                 1-8 
                 1-9 
                 1-10 
                 Example 1 
                 Example 2 
                 Example 3 
                 Example 4 
               
               
                   
               
               
                 (R 1 , k 1 ) 
                 (50, −6) 
                 (40, −3) 
                 (40, −4) 
                 (30, −3) 
                 (150, −4) 
                 (20, −4) 
                 (50, −1) 
                 (50, −2) 
               
            
           
           
               
               
            
               
                 Lens-surface region 
                 Average lens-surface inclination angle [°] 
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 12.5% or less of outer shape 
                 4.4 
                 5.6 
                 5.5 
                 7.3 
                 1.5 
                 10.0 
                 4.6 
                 4.5 
               
               
                   20% or less of outer shape 
                 6.8 
                 8.7 
                 8.5 
                 11.1 
                 2.5 
                 14.1 
                 7.4 
                 7.2 
               
               
                   30% or less of outer shape 
                 9.2 
                 12.0 
                 11.5 
                 14.8 
                 3.6 
                 17.5 
                 10.6 
                 10.3 
               
               
                   40% or less of outer shape 
                 11.4 
                 15.1 
                 14.2 
                 18.0 
                 4.8 
                 20.0 
                 14.3 
                 13.5 
               
               
                   50% or less of outer shape 
                 12.9 
                 17.2 
                 16.1 
                 20.2 
                 5.9 
                 21.5 
                 17.2 
                 15.9 
               
               
                   50% or more of outer shape 
                 21.9 
                 31.0 
                 27.3 
                 32.7 
                 15.5 
                 29.2 
                 43.3 
                 34.2 
               
               
                   60% or more of outer shape 
                 22.2 
                 31.6 
                 27.6 
                 33.0 
                 16.2 
                 29.3 
                 45.2 
                 35.2 
               
               
                   70% or more of outer shape 
                 22.4 
                 32.0 
                 27.9 
                 33.3 
                 17.0 
                 29.4 
                 47.3 
                 36.2 
               
               
                   80% or more of outer shape 
                 22.6 
                 32.4 
                 28.2 
                 33.5 
                 17.7 
                 29.5 
                 48.9 
                 37.0 
               
               
                 87.5% or more of outer shape 
                 22.7 
                 32.6 
                 28.3 
                 33.7 
                 18.2 
                 29.5 
                 50.2 
                 37.5 
               
            
           
           
               
               
            
               
                 Lens-surface region 
                 Normalized average lens-surface inclination angle 
               
               
                   
                 (normalized with average lens-surface inclination angle of Comparative Example 1) 
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 12.5% or less of outer shape 
                 1.9 
                 2.4 
                 2.4 
                 3.2 
                 0.7 
                 4.4 
                 2.0 
                 2.0 
               
               
                   20% or less of outer shape 
                 1.8 
                 2.3 
                 2.3 
                 3.0 
                 0.7 
                 3.8 
                 2.0 
                 1.9 
               
               
                   30% or less of outer shape 
                 1.7 
                 2.2 
                 2.1 
                 2.7 
                 0.7 
                 3.2 
                 1.9 
                 1.9 
               
               
                   40% or less of outer shape 
                 1.5 
                 2.0 
                 1.9 
                 2.4 
                 0.6 
                 2.7 
                 1.9 
                 1.8 
               
               
                   50% or less of outer shape 
                 1.4 
                 1.9 
                 1.7 
                 2.2 
                 0.6 
                 2.3 
                 1.9 
                 1.7 
               
               
                   50% or more of outer shape 
                 0.8 
                 1.1 
                 0.9 
                 1.1 
                 0.5 
                 1.0 
                 1.5 
                 1.2 
               
               
                   60% or more of outer shape 
                 0.7 
                 1.0 
                 0.9 
                 1.1 
                 0.5 
                 1.0 
                 1.5 
                 1.1 
               
               
                   70% or more of outer shape 
                 0.7 
                 1.0 
                 0.8 
                 1.0 
                 0.5 
                 0.9 
                 1.4 
                 1.1 
               
               
                   80% or more of outer shape 
                 0.6 
                 0.9 
                 0.8 
                 1.0 
                 0.5 
                 0.8 
                 1.4 
                 1.1 
               
               
                 87.5% or more of outer shape 
                 0.6 
                 0.9 
                 0.8 
                 0.9 
                 0.5 
                 0.8 
                 1.4 
                 1.0 
               
               
                   
               
            
           
         
       
     
     Table 2 shows the following. In each of the homogenizers  12  of Examples 1-1 to 1-10, in the lens-surface center region of each convex lens  121   a , the regions corresponding to the respective values of χ each had a normalized average inclination angle (angle obtained by normalizing the average inclination angle of the lens-surface region with the average inclination angle of the corresponding lens-surface region of the spherical lens of Comparative Example 1) of larger than 1 (1.3 to 3.2). The nearer to the center and the smaller the χ(χ=0.125 to 0.30), the larger the average lens-surface inclination angle as compared with that of Comparative Example 1. Meanwhile, in the lens-surface peripheral region, the normalized average inclination angles were within the range of 0.6 to 1.1, and the nearer to the periphery and the smaller the χ(χ=0.125 to 0.30), the smaller the average lens-surface inclination angle as compared with that of Comparative Example 1. 
     It was thus demonstrated that an illuminator  100  capable of causing light to strike on an irradiation plane  17  so as to have an even light-intensity distribution (more specifically, any of the light-intensity distributions shown in  FIG. 14A  to  FIG. 16B ) is obtained by using any of the homogenizers  12  of the Examples which each employ a first convex-lens array  12   a  including convex lenses  121   a  in which the lens-surface center region has normalized average inclination angles within the range of, for example, 1.3 to 3.2 and the lens-surface peripheral region has normalized average inclination angles within the range of, for example, 0.6 to 1.1. 
     In the homogenizers of Reference Examples 1 to 4, the convex lenses (corresponding to  121   a ) of the light-entrance-side convex-lens arrays had the following inclination angles. The normalized average inclination angles of the lens-surface center region and lens-surface peripheral region in Reference Example 1 were 0.5 to 0.7, which were smaller than 1, and those in Reference Example 3 were 1.4 to 2.0, which were larger than 1. In Reference Example 2, the normalized average inclination angles of the lens-surface peripheral region were 0.8 to 1.0, which were comparable to those in the Examples, but the normalized average inclination angles of the lens-surface center region were 2.3 to 4.4, which were larger than in the Examples. The results show that Reference Examples 1 to 3 were inferior to Examples 1-1 to 1-10 in the evenness of light-intensity distribution on the irradiation plane  17 . Reference Example 4 was somewhat inferior in the evenness of light-intensity distribution, because in Reference Example 4, the normalized average inclination angles of the lens-surface center region were 1.7 to 2.0, which were comparable to those in the Examples, but the normalized average inclination angles of the lens-surface peripheral region were 1.0 to 1.2, which were larger than in Examples 1-1 to 1-10. 
     Table 3 shows calculated values of the average internal transmission angle |θ 1 -θ 2 | of each of lens-surface regions defined with maximum width W max , with respect to the convex lenses (corresponding to  121   a ) of the light-entrance-side convex-lens array used in each of the homogenizers of Examples 1-1 to 1-10, Comparative Example 1, and Reference Examples 1 to 4, and the normalized average internal transmission angle (angle obtained by normalization with the average internal transmission angle of the corresponding lens-surface region of the convex lens of Comparative Example 1). 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                 TABLE 3 
               
               
                   
               
             
            
               
                   
                 Comparative 
                 Example 
                 Example 
                 Example 
                 Example 
                 Example 
                 Example 
               
               
                   
                 Example 1 
                 1-1 
                 1-2 
                 1-3 
                 1-4 
                 1-5 
                 1-6 
               
               
                   
               
               
                 (R 1 , k 1 ) 
                 (100, 0) 
                 (60, −3) 
                 (60, −4) 
                 (60, −5) 
                 (50, −3) 
                 (50, −4) 
                 (50, −5) 
               
            
           
           
               
               
            
               
                 Lens-surface region 
                 Average in-lens transmission angle |θ 1 -θ 2 | [°] 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 12.5% or less of outer shape 
                 0.8 
                 1.3 
                 1.3 
                 1.2 
                 1.5 
                 1.5 
                 1.5 
               
               
                   20% or less of outer shape 
                 1.2 
                 2.0 
                 2.0 
                 2.0 
                 2.4 
                 2.3 
                 2.3 
               
               
                   30% or less of outer shape 
                 1.8 
                 2.9 
                 2.8 
                 2.7 
                 3.4 
                 3.3 
                 3.2 
               
               
                   40% or less of outer shape 
                 2.5 
                 3.8 
                 3.6 
                 3.5 
                 4.4 
                 4.2 
                 4.0 
               
               
                   50% or less of outer shape 
                 3.1 
                 4.4 
                 4.2 
                 4.1 
                 5.1 
                 4.8 
                 4.6 
               
               
                   50% or more of outer shape 
                 10.2 
                 9.6 
                 8.6 
                 7.8 
                 10.3 
                 9.0 
                 8.2 
               
               
                   60% or more of outer shape 
                 10.9 
                 9.9 
                 8.8 
                 8.0 
                 10.5 
                 9.2 
                 8.3 
               
               
                   70% or more of outer shape 
                 11.8 
                 10.2 
                 9.0 
                 8.1 
                 10.8 
                 9.4 
                 8.4 
               
               
                   80% or more of outer shape 
                 12.6 
                 10.4 
                 9.1 
                 8.2 
                 10.9 
                 9.5 
                 8.5 
               
               
                 87.5% or more of outer shape 
                 13.3 
                 10.5 
                 9.2 
                 8.3 
                 11.1 
                 9.6 
                 8.5 
               
            
           
           
               
               
            
               
                 Lens-surface region 
                 Normalized average in-lens transmission angle |θ 1 -θ 2 | [°] 
               
               
                   
                 (normalized with average in-lens transmission angle of Comparative Example 1) 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 12.5% or less of outer shape 
                 1.0 
                 1.6 
                 1.6 
                 1.6 
                 2.0 
                 2.0 
                 1.9 
               
               
                   20% or less of outer shape 
                 1.0 
                 1.6 
                 1.6 
                 1.6 
                 1.9 
                 1.9 
                 1.9 
               
               
                   30% or less of outer shape 
                 1.0 
                 1.6 
                 1.5 
                 1.5 
                 1.8 
                 1.8 
                 1.7 
               
               
                   40% or less of outer shape 
                 1.0 
                 1.5 
                 1.4 
                 1.4 
                 1.7 
                 1.7 
                 1.6 
               
               
                   50% or less of outer shape 
                 1.0 
                 1.4 
                 1.4 
                 1.3 
                 1.6 
                 1.5 
                 1.5 
               
               
                   50% or more of outer shape 
                 1.0 
                 0.9 
                 0.8 
                 0.8 
                 1.0 
                 0.9 
                 0.8 
               
               
                   60% or more of outer shape 
                 1.0 
                 0.9 
                 0.8 
                 0.7 
                 1.0 
                 0.8 
                 0.8 
               
               
                   70% or more of outer shape 
                 1.0 
                 0.9 
                 0.8 
                 0.7 
                 0.9 
                 0.8 
                 0.7 
               
               
                   80% or more of outer shape 
                 1.0 
                 0.8 
                 0.7 
                 0.7 
                 0.9 
                 0.8 
                 0.7 
               
               
                 87.5% or more of outer shape 
                 1.0 
                 0.8 
                 0.7 
                 0.6 
                 0.8 
                 0.7 
                 0.6 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                   
                 Example 
                 Example 
                 Example 
                 Example 
                 Reference 
                 Reference 
                 Reference 
                 Reference 
               
               
                   
                 1-7 
                 1-8 
                 1-9 
                 1-10 
                 Example 1 
                 Example 2 
                 Example 3 
                 Example 4 
               
               
                   
               
               
                 (R 1 , k 1 ) 
                 (50, −6) 
                 (40, −3) 
                 (40, −4) 
                 (30, −3) 
                 (150, −4) 
                 (20, −4) 
                 (50, −1) 
                 (50, −2) 
               
            
           
           
               
               
            
               
                 Lens-surface region 
                 Average in-lens transmission angle |θ 1 -θ 2 | [°] 
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 12.5% or less of outer shape 
                 1.5 
                 1.9 
                 1.8 
                 2.4 
                 0.5 
                 3.4 
                 1.5 
                 1.5 
               
               
                   20% or less of outer shape 
                 2.3 
                 2.9 
                 2.9 
                 3.7 
                 0.8 
                 4.8 
                 2.5 
                 2.4 
               
               
                   30% or less of outer shape 
                 3.1 
                 4.0 
                 3.9 
                 5.0 
                 1.2 
                 60 
                 3.6 
                 3.5 
               
               
                   40% or less of outer shape 
                 3.9 
                 5.1 
                 4.8 
                 6.2 
                 1.6 
                 6.9 
                 4.9 
                 4.6 
               
               
                   50% or less of outer shape 
                 4.4 
                 5.9 
                 5.5 
                 7.0 
                 2.0 
                 7.4 
                 5.9 
                 5.4 
               
               
                   50% or more of outer shape 
                 7.5 
                 10.9 
                 9.5 
                 11.6 
                 5.2 
                 10.2 
                 16.2 
                 12.2 
               
               
                   60% or more of outer shape 
                 7.6 
                 11.1 
                 9.6 
                 11.7 
                 5.5 
                 10.3 
                 17.0 
                 12.6 
               
               
                   70% or more of outer shape 
                 7.7 
                 11.3 
                 9.7 
                 11.8 
                 5.8 
                 10.3 
                 18.0 
                 13.0 
               
               
                   80% or more of outer shape 
                 7.8 
                 11.5 
                 9.8 
                 11.9 
                 6.0 
                 10.3 
                 18.8 
                 13.3 
               
               
                 87.5% or more of outer shape 
                 7.8 
                 11.5 
                 9.9 
                 12.0 
                 6.2 
                 10.4 
                 19.4 
                 13.6 
               
            
           
           
               
               
            
               
                 Lens-surface region 
                 Normalized average in-lens transmission angle |θ 1 -θ 2 | [°] 
               
               
                   
                 (normalized with average in-lens transmission angle of Comparative Example 1) 
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 12.5% or less of outer shape 
                 1.9 
                 2.4 
                 2.4 
                 3.2 
                 0.7 
                 4.4 
                 2.0 
                 2.0 
               
               
                   20% or less of outer shape 
                 1.8 
                 2.3 
                 2.3 
                 3.0 
                 0.7 
                 3.9 
                 2.0 
                 1.9 
               
               
                   30% or less of outer shape 
                 1.7 
                 2.2 
                 2.1 
                 2.8 
                 0.7 
                 3.3 
                 2.0 
                 1.9 
               
               
                   40% or less of outer shape 
                 1.5 
                 2.0 
                 1.9 
                 2.5 
                 0.6 
                 2.7 
                 1.9 
                 1.8 
               
               
                   50% or less of outer shape 
                 1.4 
                 1.9 
                 1.8 
                 2.2 
                 0.6 
                 2.4 
                 1.9 
                 1.7 
               
               
                   50% or more of outer shape 
                 0.7 
                 1.1 
                 0.9 
                 1.1 
                 0.5 
                 1.0 
                 1.6 
                 1.2 
               
               
                   60% or more of outer shape 
                 0.7 
                 1.0 
                 0.9 
                 1.1 
                 0.5 
                 0.9 
                 1.6 
                 1.2 
               
               
                   70% or more of outer shape 
                 0.7 
                 1.0 
                 0.8 
                 1.0 
                 0.5 
                 0.9 
                 1.5 
                 1.1 
               
               
                   80% or more of outer shape 
                 0.6 
                 0.9 
                 0.8 
                 0.9 
                 0.5 
                 0.8 
                 1.5 
                 1.1 
               
               
                 87.5% or more of outer shape 
                 0.6 
                 0.9 
                 0.7 
                 0.9 
                 0.5 
                 0.8 
                 1.5 
                 1.0 
               
               
                   
               
            
           
         
       
     
     The average internal transmission angles shown in Table 3 are calculated values for the homogenizers  12  having the values of R 1  and k 1  shown for the Examples and having a refractive index n of 1.5. However, in the case of a homogenizer  12  having a different refractive index n, this homogenizer  12  is made to have improved evenness in light-intensity distribution like the Examples, by regulating the cross-sectional shape (sag value Z(r)) of the convex lenses ( 121   a ) so as to result in a value of average internal transmission angle |θ 1 -θ 2 | which is approximately the same as those of the homogenizers of the Examples, in which n=1.5. Here, the average internal transmission angle |θ 1 -θ 2 | is an average of in-lens transmission angles |θ 1 -θ 2 | for the individual lens-surface regions, the in-lens transmission angle |θ 1 -θ 2 | being an angle at which incident light which strikes, in parallel with the symmetry axis, on the convex lens  121   a  in a position lying at a radial distance r is refracted at the lens surface, as shown in  FIG. 3 . 
     A comparison between Table 2 and Table 3 shows that the average lens-surface inclination angles and average internal transmission angles which were normalized with the values of Comparative Example 1 (spherical lens) were nearly equal in numerical value and tendency in all the lens-surface regions. Consequently, the cross-sectional shape of convex lenses ( 121   a ) can be set such that the average inclination angle and the average internal transmission angle are within the numerical ranges corresponding to Examples 1-1 to 1-10 on the basis of the normalized average inclination angle and normalized average internal transmission angle which are values obtained by normalization with values for a spherical lens. 
     Next, a method for producing the homogenizer  12  of the Example is explained. For producing the homogenizer  12  of the Example, any processing method may be used so long as the convex-lens shapes of the first convex-lens array  12   a  and second convex-lens array  12   b  can be formed as designed. For example, in the case where dies for precision press forming are used for forming the convex-lens arrays, a diamond tool is used to cut die surfaces to impart thereto concave-lens array shapes which are the inversions of the convex-lens arrays, thereby producing dies for the first and second convex-lens arrays. A light-transmitting glass material is sandwiched between the produced two dies, and the dies and the glass material as a whole are heated to a temperature equal to or more than the softening temperature of the glass to transfer the die shapes to the glass surfaces and are then gradually cooled to room temperature. In this operation, the two dies are precisely positioned such that in the resultant first and second convex-lens arrays, each of the convex-lens pairs has no offset in symmetry axis. This die positioning is conducted so as to result in a symmetry-axis offset amount equal to or less than 1/10 of the convex-lens width Wx or Wy, preferably equal to or more than 1/20 thereof, more preferably equal to or less than 1/30 thereof. The die surfaces may be coated with an alloy film in order to improve the separability of the dies from the glass and the smoothness of the surfaces of the formed glass. 
     The glass material to be used for the die forming preferably has a lower softening point, because such glass materials can be formed at lower temperatures. Furthermore, the higher the refractive index, the higher the formability. This is because a glass material having a higher refractive index gives convex lenses having smaller surface inclination angles for obtaining the same lens power and hence having reduced sag values of Sa and Sb shown in  FIG. 1A . Specifically, on the assumption that light strikes on convex lenses having different refractive indexes of n A  and n B  in parallel with the symmetry axis at convex-lens-surface incidence angles of θ A  and θ B , respectively, and that the incident light is refracted at refractive angles of θ A ′ and θB′ such that the refracted light proceeds in the convex lenses at the same angle (internal transmission angle) with the lens-plane symmetry axis (that is, the two convex lenses have the same lens power), then the convex lenses having refractive indexes of n A  and n B  have inclination angles of θ A  and θ B , which are related with each other by: internal transmission angle θ A -θ A ′==θ B -θ B ′. For example, one convex lens has θ A =30° and n A =1.50, while use of a glass material having n B =2.0 results in θ B =21.4° and in an average lens-surface inclination angle reduced to about ⅔. 
     The convex lenses constituting the first convex-lens array  12   a  and second convex-lens array  12   b  have a gap-less configuration in which each convex lens has a rectangular shape having widths of Wx×Wy and the whole rectangular shape has a lens shape. Because of this, in the boundary portions of the convex lenses, in particular in corner portions of the rectangular shapes, the sag value and the inclination angle are maximum and the designed inclination angle changes discontinuously. This makes it difficult to produce dies and to form the lenses. 
     The convex lenses  121   a  constituting each of the homogenizers  12  of the Examples have an aspherical shape which has approximately the same sag values Z as in Comparative Example 1 (spherical convex lenses  921 ) but has a smaller maximum inclination angle, as shown in  FIG. 12  and  FIG. 13 . Hence, die production and lens forming are less difficult. In the case where a light-transmitting material having a high refractive index is used for the convex lenses, this results in a reduction in average lens-surface inclination angle, making the die production and lens forming still less difficult. 
     In the case where a light-transmitting resin which is a photosetting or thermosetting resin is used as a lens material and molded by die forming, limitations on the die material, such as heat resistance, are relieved because the light-transmitting resin is cured at low temperatures. In the case of using an ultraviolet-curing resin, it is preferred to use dies made of a glass material which transmits ultraviolet light, e.g., quartz. 
     Another method for producing a homogenizer  12  is to perform dry-etching processing for light-transmitting glass substrate directly, using a reactive gas. Continuous-tone photomasks (gray-scale masks) each having an ultraviolet-transmittance distribution corresponding to a lens-shape spatial distribution are used in order to form convex-lens arrays  12   a  and  12   b  each having a gap-less convex-lens shape. An ultraviolet-sensitive resist applied to a surface of the light-transmitting glass is exposed to ultraviolet light through the photomask. Thereafter, the photosensitive resist is developed to obtain a photosensitive-resist pattern having a gap-less convex-lens shape. Furthermore, a reactive-ion etching device used in semiconductor microfabrication is used to strike ions on the patterned resist surface, thereby transferring the convex-lens-shape resist pattern to the surface of the light-transmitting glass in accordance with the dry etching rates of the resist material and glass material. Thus, gap-less convex-lens arrays  12   a  and  12   b  are obtained. 
     From the viewpoint of shortening the processing time, it is preferred to select a glass material and a reactive gas which attain a high reactive-ion etching rate. For example, quartz, which is not used as a glass material for die forming because of the high glass softening point thereof, is applicable to dry etching because quartz has a high reactive-ion etching rate. The following method may also be used: an SiO 2  film is deposited on a substrate having a relatively low reactive-ion etching rate, such as a borosilicate glass substrate or a sapphire substrate, and the SiO 2  film only is etched to impart a convex-lens shape thereto. In place of the SiO 2  (n=1.45), a substance having a high refractive index may be used, such as TiO 2  (n=2.3), Ta 2 O (n=2.1), or Nb 2 O 5  (n=2.1). Convex lenses having a regulated refractive index may be produced by using SiN x O y , by controlling a ratio between x and y. In the reactive-ion etching of SiO 2 , CF 4  is typically used as the reactive gas. 
     In the case of processing a light-transmitting inorganic material by reactive-ion etching to produce convex lenses, it is preferred to employ a configuration of the homogenizer  12  in which each convex lens has a reduced maximum sag value Z, from the viewpoint of shortening the processing time. Specifically, in order to attain the same diffusing/homogenizing function with a smaller maximum sag value Z, the light-transmitting inorganic material may be processed so as to have a configuration in which the outer-shape size (Wx×Wy×T) of each convex lens is reduced at the same reduction ratio as the maximum sag value Z. 
     Example 2 
     Next, an example of the homogenizer  22  according to the second embodiment shown in  FIG. 5A  and  FIG. 5B  is shown as a second Example. This homogenizer  22  employs a first convex-lens array  22   a  including convex lenses  221   a  in which a phase diffraction grating  23  is formed, the phase diffraction grating  23  including circular recess grooves each having a rectangular cross-section, in a lens-surface region near the symmetry axis (lens-surface center region) of a spherical lens  921  having the shape of (R 1 , k 1 )=(100, 0). More specifically, each convex lens  221   a , which has a maximum width W max =128 μm, includes the phase diffraction grating  23  formed in a lens-surface center region defined by 0≤r/(W max /2)≤0.20. 
     In this Example, the phase diffraction grating  23  has a ratio between recess width and protrusion width of 1:1. Furthermore, the period P of the recesses and protrusions is set such that when incident light having a wavelength λ of 850 nm enters the convex lens in the Z-axis direction, the resultant ±1-order diffracted light is transmitted through the convex lens at a diffraction angle γ ±1  within the range of 0.2° to 8°, preferably in the range of 1° to 4°. For example, in the case where P=10 μm, then γ ±1 =3.3°. Furthermore, in the case where the recesses of the phase diffraction grating  23  have a depth d=0.5λ/(n−n 0 ) (in the case where n=1.5, d=850 nm), there is approximately no 0-order light, which proceeds rectilinear, and ±1-order diffracted light is produced in an amount of about 40% (81% in total). In the case where d=0.32λ/(n−n 0 ) (in the case where n=1.5, d=543 nm), the diffraction efficiency for 0-order light and that for 1-order diffracted light are equally about 29%, the total thereof being about 86%. 
     In the homogenizer of Comparative Example 1 (spherical lenses) having a refractive index n=1.5, light rays entering the region equal to or less than 20% of the outer shape have an average internal transmission angle |θ 1 -θ 2 | as small as 1.2° as shown in Table 3 in the first Examples. However, in the convex lens  221   a  of this Example, in which the phase diffraction grating  23  is formed, light rays entering the region having width equal to or less than 20% of the lens width W have an average internal transmission angle |θ 1 -θ 2 | of about 3.2° in the case where d=850 nm, the average internal transmission angel being more than two times. In the convex lens  221   a  of this Example, two average internal transmission angles are defined for the ±1-order diffracted light produced by the phase diffraction grating  23 . However, since average internal transmission angle is intended to indicate the function of diminishing rectilinear transmitted-light components proceeding in the Z-axis direction, the average internal transmission angle of the ±1-order diffracted light in this Example was calculated as an average of absolute values of |θ 1 -θ 2 |. 
     Thus, the homogenizer  22  of this Example can attain approximately the same average internal transmission angles as the convex lenses  121   a  of the homogenizers  12  of the first Examples. Consequently, by using the homogenizer  22  of this Example in the illuminator  100 , approximately the same evenness in light-intensity distribution as that obtained with the homogenizers  12  of the first Examples can be attained on an irradiation plane  17 . 
     The homogenizer  22  of this Example can be produced by die forming or reactive-ion etching like the homogenizers  12  of the first Examples. 
     Example 3 
     Next, an example of the homogenizer  32  according to the third embodiment shown in  FIG. 7  is shown as a third Example. This homogenizer  32  includes a pair of homogenizers  32   x  and  32   y , which are configured to respectively include cylindrical-lens array pairs ( 32   xa  and  32   xb ; and  32   ya  and  32   yb ) such that the cylindrical lenses ( 321   xa  and  321   xb ; and  321   ya  and  321   yb ) constituting the array pairs have cross-sectional shapes which are symmetric with respect to not an axis but a plane and have a lens power only either in the X-axis direction or Y-axis direction. 
     More specifically, the cylindrical lenses  321   xa  constituting the first cylindrical-lens array  32   xa  of the homogenizer  32   x  and the cylindrical lenses  321   xb  constituting the second cylindrical-lens array  32   xb  are each symmetric with respect to a YZ plane, and have XZ cross-sectional shapes which are the same as those of the convex lenses  121   a  and convex lenses  121   b  (in which W max =W x ) of any of the homogenizers  12  of the first Examples. Furthermore, the cylindrical lenses  321   ya  constituting the first cylindrical-lens array  32   ya  of the homogenizer  32   y  and the cylindrical lenses  321   yb  constituting the second cylindrical-lens array  32   yb  are each symmetric with respect to an XZ plane, and have YZ cross-sectional shapes which are the same as those of the convex lenses  121   a  and convex lenses  121   b  (in which W max =W y ) of any of the homogenizers  12  of the first Examples. 
     In this Example, synthetic quartz having a refractive index n of 1.45 and a thickness T of 280 μm is used as light-transmitting substrates  30   x  and  30   y  as a material for the homogenizer  32   x  and homogenizer  32   y . The homogenizer  32   x  of this Example is configured such that Wx=100 μm and Nx=12, while the homogenizer  32   y  of this Example is configured such that Wy=80 μm and Ny=15. These homogenizers  32   x  and  32   y  are serially disposed in the Z-axis direction so as to leave a 0.1-mm space therebetween. 
     This homogenizer  32  thus configured has the same optical function as any of the homogenizers  12  of the first Examples. Hence, the same effect as in the first Examples can be obtained. 
     Furthermore, in this homogenizer  32 , since the homogenizer  32   x  and the homogenizer  32   y  employ cylindrical-lens arrays, the lens boundaries where lens surfaces meet each other discontinuously are linear and the maximum sag value Z is reduced. As a result, a lens shape of optimal design can be precisely formed, enabling the homogenizer to stably emit light having desired properties. 
     This homogenizer  32  also can be produced by die forming or reactive-ion etching like the homogenizers  12  of the first Examples. 
       FIG. 20A  to  FIG. 20E  are views for illustrating an example of methods for producing one cylindrical-lens array (cylindrical-lens array  32   xa  and  32   xb ) of the homogenizer  32  of this Example. In the example shown in  FIG. 20A  to  FIG. 20E , a surface of a light-transmitting glass substrate is dry-etched with a reactive ion gas to form a cylindrical-lens array. 
     (a) First, in the case of producing the first cylindrical-lens array  32   xa  of the homogenizer  32   x , an ultraviolet-sensitive resist  324  is evenly applied in a given film thickness H to one surface (e.g., a first surface) of a quartz substrate as the light-transmitting substrate  30   x  and cured by heating. The film thickness H for finally obtaining a maximum sag value SR of the quartz convex lenses (cylindrical lenses) is set in accordance with the etching-rate ratio between the quartz and the resist. Also in the case of the cylindrical-lens arrays  32   ya  and  32   yb , a quartz substrate is used as the light-transmitting substrate  30   y.    
     (b) Next, the resist  324  is patterned such that the cylindrical lenses  321   xa  which are to constitute the cylindrical lens array  32   xa  each have a lens width of Wx and that linear resist portions lying on the lens boundaries and having a gap width G are removed. In the case of the cylindrical-lens array  32   xb , the resist  324  is patterned so as to form cylindrical lenses which share symmetry planes with the cylindrical lenses of the cylindrical-lens array  32   xa  and such that the cylindrical lenses  321   xb  each have a lens width of Wx and that linear resist portions lying on the lens boundaries and having a gap width G are removed. In the case of the cylindrical-lens arrays  32   ya  and  32   yb , the resist  324  is patterned such that these arrays  32   ya  and  32   yb  share symmetry planes and have a lens width of Wy and that linear resist portions lying on the lens boundaries and having a gap width G are removed. In  FIG. 20B , the patterned resist  324  is shown as resist  324 ′. 
     In this step, smaller values of gap G are preferred from the viewpoint of light utilization efficiency. Specifically, the gap G is preferably 5 μm or less, more preferably 2 μm or less, still more preferably 1 μm or less. 
     (c) Next, the patterned resist  324 ′ is liquefied by heating to a temperature at which the resist  324 ′ softens (e.g., 200° C. to 250° C.) to allow the resist surface to become spherical (reflow) by surface tension. Thereafter, the resist  324 ′ is cooled to room temperature to fix the spherical shape. As a result, a resist pattern  324 ″ having a cylindrical spherical lens shape having a sag value SP and a convex cross-section is formed. The radius of curvature R and sag value S P  of the resist pattern  324 ″ are determined by the volume of the rectangular-parallelepiped resist  324 ′, which is determined by the patterning width W (Wx or Wy) for the resist  324 ′ and the thickness H thereof. In order for the rectangular-parallelepiped resist  324 ′ to become the spherical resist pattern  324 ″ through the reflow heating, there is a range of patterning width W according to the thickness H of the resist  324 ′. More specifically, the width W is 10 μm to 300 μm, preferably 20 μm to 200 μm. 
     (d) The quartz substrate having the lens-shape resist pattern  324 ″ formed on the surface thereof is dry-etched with a reactive ion gas, thereby transferring the surface shape of the resist  324 ″ to the surface of the quartz substrate. The cross-sectional view given in  FIG. 20D  shows a state during the dry etching. 
     (e) Cylindrical lenses  321   xa  having a final shape are obtained after the removal of the resist  324 ″ by the dry etching. In the case where the quartz substrate and the resist  324 ″ are equal in etching rate, the surface shape of the resist  324 ″ is transferred as such to the surface of the quartz substrate (that is, [sag value S R  of the cylindrical lens]=[sag value S P  of the resist  324 ″]). In the case where the quartz substrate and the resist  324 ″ have different etching rates, the ratio between the sag value S R  of the cylindrical lens and the sag value S P  of the resist  324 ″ is different and the cylindrical lenses have a shape formed by compressing or extending the resist  324 ″ in the height direction. 
     The etching rate varies depending on the kind, pressure, and flow rate of the reactive gas. Hence, a cylindrical-lens array  32   xa  in which the cylindrical lenses each have a cross-sectional shape different from that of spherical lenses is obtained by changing the set conditions during the period from the start of the dry etching to the end thereof. By utilizing such a method for controlling the shape of convex lenses, a convex cylindrical-lens array having an aspherical cross-sectional shape can be produced. 
       FIG. 21A  and  FIG. 21B  schematically illustrate, by gradation, XY-plane light-intensity distributions on an irradiation plane  17  which were obtained with an illuminator  100  employing the homogenizer  32  of this Example. The light-intensity distributions shown in  FIG. 21A  and  FIG. 21B  are results obtained by using the homogenizer  32  of this Example in the illuminator  100  shown in the first Examples in place of the homogenizer  12  of the first Examples and calculating light-intensity distributions of the light striking on the irradiation plane  17  being 100 mm distant from the homogenizer  32 . In this Example, the gap between the homogenizer  32   x  and the homogenizer  32   y  was set at 0.1 mm, and the gap between the laser light emission points  11   b  of the laser light source  11  and the light-entrance-side homogenizer  32  (homogenizer  32   x ) was set at 2 mm. In the homogenizer  32  which gave the results shown in  FIG. 21A  and  FIG. 21B , the entrance-side cylindrical-lens array  32   xa  of the homogenizer  32   x  and the entrance-side cylindrical-lens array  32   ya  of the homogenizer  32   y  have the same cross-sectional lens shape as the convex lenses  121   a  of Example 1-5: (R 1 , k 1 )=(50, −4). 
     As  FIG. 21A  and  FIG. 21B  shows, it can be seen that the illuminator  100  employing the homogenizer of this Example gave a light-intensity distribution in which the light intensity in corner portions (maximum diffusion angle β=17.8°) of the irradiation plane, which were the lowest in light intensity, was as high as about 88%. 
     Example 4 
     In the fourth Example, convex cylindrical lenses having the same lens cross-section as the convex lenses  221   a  of the second Example are used in place of both the cylindrical lenses  321   xa  constituting the first cylindrical-lens array  32   xa  of the homogenizer  32   x  used in the homogenizer  32  of the third Example and the cylindrical lenses  321   ya  constituting the first cylindrical-lens array  32   ya  of the homogenizer  32   y  used in the homogenizer  32 . However, like the cylindrical lenses of the third Example, those cylindrical lenses have a lens power only either in the X-axis direction or in the Y-axis direction and have convex-lens surface shapes which are symmetric with respect to not an axis but a plane. Consequently, in this Example, the phase diffraction grating  23  formed in an approximately flat lens-surface region near the symmetry plane of each cylindrical lens, which has the same lens cross-section as a convex spherical lens, is not cyclic recess grooves which are axially symmetric but linear recess grooves extending in the direction in which the cylindrical lens has no lens power (i.e., in the direction parallel with the symmetry plane of the lens). 
     Next, referring to  FIG. 22A  to  FIG. 22E , a method for producing one convex spherical cylindrical-lens array of the homogenizer  32  of this Example (e.g., the first cylindrical-lens array  32   xa  of the homogenizer  32   x , which includes a plurality of cylindrical lenses  321   xa  each having the phase diffraction grating  23  formed therein) is explained. 
     (a) First, phase diffraction gratings  325   x  are formed in a first surface of a quartz substrate as a light-transmitting substrate  30   x , in areas corresponding to 50-μm-wide lens-surface regions including the symmetry planes of the cylindrical lenses  321   xa  as the centers, the phase diffraction gratings  325   x  each including a linear grating including rectangular recess grooves having a depth d b  of 1,000 nm and arranged at a period P of 10 μm, the recess width and the protrusion width each being 5 km. Specifically, five recess grooves having a rectangular cross-section and having a width of 5 μm and a depth d b  of 1,000 nm are formed, by processing, at a period of 10 μm over a length of 1 mm, in each of 50-μm-wide regions in the first surface of the quartz substrate which correspond to convex center portions of the cylindrical lenses  321   xa.    
     (b) Next, an ultraviolet-sensitive resist  324  is evenly applied in a given film thickness H to the first surface of the quartz substrate where the phase diffraction gratings  325   x  are formed, and is then cured by heating. Thus, the recess grooves of the phase diffraction gratings  325   x  are filled with the resist  324  and the resist has a flat surface. Although the recesses and protrusions of the phase diffraction gratings affect the flatness of the resist surface depending on the shape of the recess grooves of the phase diffraction gratings and the viscosity and curing temperature of the resist, any ruggedness of the resist surface just after the resist application is not problematic, owing to the heating at a reflow temperature which will be described later. 
     (c) Next, the resist  324  is patterned such that the cylindrical lenses  321   xa  which are to constitute the cylindrical lens array  32   xa  each have a lens width of Wx and that linear resist portions lying on the lens boundaries and having a gap width G are removed. In the case of the cylindrical-lens array  32   ya , Wy is used, in place of the Wx, as the lens width of the cylindrical lenses  321   ya . Also in this example, the patterned resist  324  is shown as resist  324 ′. 
     (d) Next, the patterned resist  324 ′ is liquefied by heating to a temperature at which the resist  324 ′ softens, thereby allowing the resist surface to become spherical (reflow) by surface tension. Thereafter, the resist  324 ′ is cooled to room temperature to fix the spherical shape. As a result, a resist pattern  324 ″ having a cylindrical spherical lens shape having a sag value S P  and a convex cross-section is formed. In this step, regardless of the surface shape of the quartz substrate which is finely processed, the surface shape of the resist  324 ′ becomes spherical due to the surface tension of the liquefied resist. 
     (e) The quartz substrate having the lens-shape resist pattern  324 ″ formed on the surface thereof is dry-etched with a reactive ion gas, thereby transferring the surface shape of the resist  324 ″ to the surface of the quartz substrate. Since the phase diffraction gratings  325   x  are formed in the surface of the quartz substrate, complete removal of the resist  324 ″ by the reactive-ion etching gives a cylindrical-lens array that includes a plurality of cylindrical lenses  321   xa  having a sag value S R  which are arranged in one direction and which each have a phase diffraction grating  23   x  formed in the lens-surface center region. 
     The recess groove depth d b  of the phase diffraction gratings  325   x  before the lens shape formation and the recess groove depth d R  of the phase diffraction gratings  23   x  after the lens shape formation depend on the ratio between the dry-etching rate r R  of the material of the protrusions of the phase diffraction gratings  23   x  and the dry-etching rate r P  of the resist  324 ″ filling the recesses of the phase diffraction gratings  325   x . Specifically, in the case where r R =r P , d R =0 and thus no phase diffraction gratings  23   x  are formed in the lens surfaces of the cylindrical lenses  321   xa . Meanwhile, in the case where r R &lt;r P , then phase diffraction gratings  23   x  having a recess depth d R &gt;0, which is determined by the ratio between the r R  and the r P , are formed. In the case where r R &gt;r P , phase diffraction gratings  23   x  having d R &lt;0 are formed, in which the recesses of the phase diffraction gratings  325   x  have become protrusions of the phase diffraction gratings  23   x  and the protrusions have a height determined by the ratio between the r R  and the r P . 
     For example, under the dry-etching conditions of r R =r P /2, phase diffraction gratings  23   x  are obtained in which d R =d b /2=500 nm. In the case where light having a wavelength k of 850 nm enters such a phase diffraction grating  23   x  in the direction perpendicular to the light-transmitting substrate  30 , the efficiency for 0-order transmitted light, which proceeds straight, is about 45% and the efficiency for 1-order diffracted light having a diffraction angle of 4.9° is about 22% (44% in total), resulting in an increase in the proportion of diffused transmitted-light components emitted from the lens-surface center region. Consequently, use of the homogenizer  32  of this Example makes it possible to obtain an illuminator  100  which attains improved evenness of the light-intensity distribution on an irradiation plane like the illuminator of the third Example. 
     Another method for producing the cylindrical lenses each having a phase diffraction grating in the lens-surface center region, in the homogenizer  32  of Example 4, may be one in which a corrosion-resistant layer of a material having a lower dry-etching rate than both the light-transmitting substrate  30  ( 30   x  or  30   y ) and the resist  324  is deposited in an even film thickness on a first surface of the light-transmitting substrate  30  and the portions of the corrosion-resistant layer which correspond to the recess groove pattern of the phase diffraction gratings are removed. 
     For example, an ultraviolet-sensitive resist is formed beforehand into a pattern of the phase diffraction gratings  325   x , thereafter an Al 2 O 3  film, which has a low dry-etching rate, is deposited, and then the resist is removed with a developing solution (lift-off process), thereby producing a patterned Al 2 O 3  film layer. The resultant light-transmitting substrate  30   x  having a grating pattern including the Al 2 O 3  film having a low etching rate is processed by the same procedure as that shown by  FIG. 20B  to  FIG. 20D . Thus, a cylindrical-lens array including a plurality of cylindrical lenses having phase diffraction gratings  23   x  formed therein is obtained. The recess groove depth d R  of the phase gratins  23   x  can be controlled by regulating the thickness of the Al 2 O 3  film. 
     While the present invention has been described in detail and with reference to specific embodiments thereof, it will be apparent to one skilled in the art that various changes and modifications can be made therein without departing from the spirit and scope thereof. This application is based on Japanese Patent Application No. 2018-052652 filed on Mar. 20, 2020, the entire subject matter of which is incorporated herein by reference. 
     INDUSTRIAL APPLICABILITY 
     The present invention is suitable for use in applications where an irradiation plane located at a relatively short distance is evenly illuminated. 
     REFERENCE SIGNS LIST 
     
         
           12 ,  22 ,  32  Homogenizer 
           10 ,  20 ,  30 ,  30   x ,  30   y ,  40  Light-transmitting substrate 
           12   a ,  12   b  Convex-lens array 
           22   a ,  22   b  Convex-lens array 
           121   a ,  121   b  Convex lens 
           221   a ,  221   b  Convex lens 
           23 ,  23   x ,  23   y  Phase diffraction grating 
           325   x  Phase diffraction grating 
           921  Spherical convex lens 
           32   x ,  32   y  Homogenizer 
           32   xa ,  32   xb ,  32   ya ,  32   yb  Cylindrical-lens array 
           321 ,  321   xa ,  321   xb ,  321   ya ,  321   yb  Cylindrical lens 
           100 ,  110  Illuminator 
           11  Laser light source 
           11   a  Semiconductor substrate 
           11   b  Laser light emission point 
           13  Package 
           14  Inert gas 
           15  Light flux 
           16  Divergent light 
           17  Irradiation plane 
           111  Laser light source 
           112  Laser light source 
           113  Lens 
           114  Optical fiber 
           324  Resist 
           200 ,  210  Illuminating optical system 
           52 ,  62  Homogenizer 
           52   a ,  62   a  Convex-lens array 
           52   b ,  62   b  Convex-lens array 
           521 ,  521   a ,  621 ,  621   a  Convex lens 
           521   b ,  621   b  Convex lens 
           50 ,  60  Discharge lamp 
           51  Parabolic mirror 
           61  Ellipsoidal mirror 
           53 ,  63  Field lens 
           54  Liquid-crystal display element 
           55  Projection lens 
           65  Condenser lens 
           66  Dichroic mirror