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Abstract
Collimated illumination with a light-emitting diode (LED) is widely used in the residential, commercial, and industrial inspection fields. A slim LED illuminator composed of multi-parabolic surfaces to combine perpendicular collimated illuminations is proposed here. The multi-parabolic illuminator can produce a collimated rectangular illumination where light angle distributions for perpendicular axes can independently be controlled by different parabolic surfaces. This independence of the control makes the design of the illuminator simple. A prototype of the multi-parabolic illuminator is fabricated with an LED chip size of 3×3 mm and an opening aperture size of 20×60 mm. A maximum width of the prototype is 20 mm. The prototype demonstrates a production of a highly collimated rectangular illumination with half-intensity angles of about 7 degrees and 34 degrees for respective perpendicular axes.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Replacement of traditional incandescent lamps by light-emitting diodes (LEDs) has been progressing owing to the many advantages of LEDs such as high efficiency, eco-friendliness, and long lifetime [1–8]. For example, LED spotlights have been widely used in the residential, commercial, and industrial inspection fields [9–16]. Narrow-angle light distributions of the LED spotlights are suitable for accenting target areas with high light intensity or from a considerable distance.
In many manufacturing processes, a material surface is inspected for quality control by means of a camera. A micro-defect on the material surface is often needed to be detected in the inspection process. It is, however, difficult to capture an image of the micro-defect under an ambient illumination. The dark-field method using an LED spotlight that emits collimated light rays is therefore often used for detecting the micro-defects. A low-angle illumination of the collimated light rays is often used for the dark-field method where the micro-defects diffuse the light rays. The diffused light rays provide an image that highlights the micro-defects. For example, a low-angle LED ring array is used for the dark-field method to highlight surface defects, scratches, and engraving [17]. As like this example, incidence light rays in an incidence plane should be highly collimated whereas light rays in a plane orthogonal to the incidence plane need not be highly collimated. A collimated rectangular illumination that has different illuminance distributions in respective perpendicular axes is therefore suitable for the dark-field method.
There are several free-form optics that can emit rectangular illuminations [18–23]. When an illumination device is installed on a manufacturing line, a width of the device should often be slim enough to fit an empty space in the manufacturing line. A width of an industrial camera has then recently been downsized to a few tens of millimeters. According to it, a width of an illuminator is also sometimes desired to be that of the industrial camera, namely, a few tens of millimeters. A half-intensity angle of a light angle distribution of the illuminator in the incidence plane is also desired to be less than 10 degrees. It is, however, well known that an opening aperture of an illuminator should be larger if the light angle distribution is narrower owing to the etendue conservation law [24,25]. If the illuminator is axisymmetric or almost axisymmetric, the width of the illuminator is then often forced to be larger than that of the industrial camera. A light angle distribution of the illuminator for the dark-field method in a plane orthogonal to the incidence plane, however, needs not to be narrow. This enables a one-directional length of the opening aperture to be small, which indicates that the illuminator can be slim. It is often complicated, however, to design different light angle distributions in respective perpendicular planes with the free-form optics.
A slim LED illuminator composed of multi-parabolic surfaces to combine perpendicular collimated illuminations is thus proposed here. The multi-parabolic illuminator produces a collimated rectangular illumination where light angle distributions for respective perpendicular axes can independently be controlled by different parabolic surfaces. This independence of the control makes a design of the illuminator simple, which enables that the illuminator can be easily customized for various applications. A prototype of the multi-parabolic illuminator is fabricated with an LED chip size of 3×3 mm and an opening aperture size of 20×60 mm. A maximum width of the prototype is 20 mm. The prototype demonstrates a production of a highly collimated rectangular illumination with half-intensity angles of about 7 degrees and 34 degrees for respective perpendicular axes. The remainder of this paper is organized as follows. Section 2 describes the basic concept of the multi-parabolic illuminator. Section 3 describes a design of the multi-parabolic surfaces of the illuminator based on the etendue conservation theory. Section 4 describes a prototype of the illuminator and reports on the measurement of illumination angle distribution. Discussions are given in section 5, and conclusions are given in section 6.
2. Concept of the multi-parabolic illuminator
Figure 1 shows a schematic perspective view of the multi-parabolic illuminator with an LED source. The illuminator is made of a transparent material with a refractive index of n. A Cartesian coordinate system of (x, y, z) is taken with an origin O. The illuminator has a mirror surface and two total internal reflection (TIR) surfaces. These surfaces are either parabolic or quasi-parabolic.
Fig. 1. Schematic perspective view of multi-parabolic illuminator with an LED source. The illuminator is made of a transparent material with a refractive index of n. A Cartesian coordinate system of (x, y, z) is taken with an origin O. The illuminator has a mirror surface and two total internal reflection (TIR) surfaces. These surfaces are either parabolic or quasi-parabolic.
Light rays emitted from the LED will be reflected by the mirror surface. The reflected light rays in the cross-sectional x - z plane are collimated at this time. On the other hand, the reflected light rays from the mirror surface in a cross section parallel to the y - z plane are guided by the illuminator to arrive at z = 0 plane. In other words, the light rays reach the z = 0 plane owing to the total internal reflection of the illuminator, and are almost spatial randomly distributed over the entire cross-sectional z = 0 plane of the illuminator. Note that all light rays that reach the cross-sectional z = 0 plane can be considered as a secondary light source. The light rays emitted from the secondary light source will then be collimated by the TIR surfaces. All the light rays will thus be emitted from an opening aperture. A light diffuser to make light distribution smooth can be attached to the opening aperture.
Figure 2 shows schematic cross-sectional views of the multi-parabolic illuminator in (a) x – z plane and (b) y – z plane. In the x – z plane, the LED is placed to face a flat plane including an incidence surface. A light ray emitted from the LED is incident on the flat incidence surface and is refracted by the surface. Mirror surface is parabolic surface with a focal point corresponding to the coordinate origin O, which makes light rays emitted from the origin collimated. Sizes of the opening aperture and the LED in the x – z plane are set to A1 and L1, respectively. On the other hand, in the y – z plane, the two TIR surfaces facing each other construct the same structure as the compound parabolic concentrator (CPC) [24]. The light rays reflected from the mirror surface will be guided by the illuminator, and reach the cross-sectional z = 0 plane of the illuminator. The light rays are almost spatial randomly distributed over this cross section, which can be considered as the secondary light source. The light rays emitted from the secondary light source will then be collimated by the TIR surfaces owing to the total internal reflection. Sizes of the opening aperture and the LED are set to A2 and L2, respectively. A width of the mirror surface is set to B.
Fig. 2. Schematic cross-sectional views of multi-parabolic illuminator in (a) x – z plane and (b) y – z plane. In the x – z plane, an LED is placed to face a flat plane including an incidence surface. Mirror surface is parabolic surface with a focal point corresponding to the coordinate origin O, which makes light rays emitted from LED collimated. Sizes of the opening aperture and the LED in the x – z plane are set to A1 and L1, respectively. Sizes of the opening aperture and the LED in the y – z plane are set to A2 and L2, respectively. A width of the mirror surface is set to B.
In an optical system, there is a conserved quantity, namely, an etendue. In three-dimensional space, the etendue can be written as a product of an area of a cross section where light rays pass through and a maximum solid angle of the light rays on the cross section. In two-dimensional space, assuming that an infinitesimal line element of dl that is immersed in a medium of refractive index n is crossed by the light rays confined to an angle dΘ with an incident angle Θ, the infinitesimal etendue, dG, can be written [24,25] as
In the cross section parallel to the x - z plane, a light ray emitted from the LED is incident on the flat plane and is refracted by the transparent material with the refractive index of n. A maximum angle of the refracted light ray in the material corresponds to a critical angle Θc of the material. Assuming that an ambient refractive index is 1.0, the critical angle Θc can be written as
The following relationship can thus be derived using etendue values for the incidence surface and opening aperture with their respective sizes of L1 and A1 as
where an angle Θo, 1 denotes a maximum divergence angle of the light rays emitted from the opening aperture. The divergence angle Θo, 1 can then be written using Eqs. (2) and (3) asEquation (4) indicates that the larger the opening aperture is, the smaller the divergence angle becomes.
In the cross section parallel to the y - z plane, a divergence angle Θo, 2 that denotes a maximum divergence angle of the light rays emitted from the opening aperture can be derived in the same way as mentioned above. Using etendue values for the flat incidence surface and an entrance plane of the CPC (i.e., the cross-sectional z = 0 plane) with their respective sizes of L2 and B, the following relationship can be derived with a divergence angle Θm at the cross-sectional z = 0 plane as
Note that the light rays at the cross-sectional z = 0 plane of the illuminator can be considered as the secondary light source. If the TIR surfaces construct the ideal CPC, etendue values for the entrance plane and the opening aperture with their respective sizes of B and A2 are also conserved. Using the divergence angle Θo, 2 at the opening aperture, the relationship of the etendue values can also be written as
Using Eqs. (2), (5), and (6), the divergence angle Θo, 2 can be written as
Equation (7) indicates that the larger the opening aperture is, the smaller the divergence angle becomes.
Using Eqs. (4) and (7), the divergence angles in respective perpendicular planes can be estimated. These equations indicate that the emitting surface of the LED should be as small as possible in comparison with the opening aperture to make the divergence angles small. The CPC length (i.e., a length of the TIR surfaces), however, is known to tend to be too long in the z-direction to make the divergence angle small. In other words, it is difficult to make the opening aperture size of A2 large with keeping the length of the CPC small. The opening aperture size of the A2 is therefore set to smaller than that of the A1, which enables the multi-parabolic illuminator to be more compact than that with these sizes set in reverse.
3. Design of multi-parabolic surfaces
3.1 Mirror surface
Figure 3 shows a schematic cross-sectional view of the multi-parabolic illuminator in the x – z plane. The LED is placed to face the flat plane including the incidence surface. A distance between the LED emitting surface and the incidence plane is assumed to be 0.5 mm with 0.2 mm tolerance. A light angle distribution of the LED is assumed to be Lambertian. A heat dissipator of any size can be easily attached to the LED. Mirror surface is parabolic surface with a focal point corresponding to the coordinate origin O. A local cylindrical coordinate system of (ρ, θ) is also taken with the coordinate origin of O. The azimuth angle θ is defined as an angle from the negative direction of z-axis. A light ray will be reflected internally by the mirror at a position vector of q having an azimuth angle of θ and then travels toward the opening aperture. The angle Θc denotes the critical angle of the transparent material of the illuminator. A position vector of q0 having an azimuth angle of θ0 is parallel to a normal vector of the incidence flat plane.
Fig. 3. Schematic cross-sectional view of multi-parabolic illuminator in the x – z plane. The LED is placed to face the flat plane including the incidence surface. Mirror surface is parabolic surface with a focal point corresponding to the coordinate origin O. A local cylindrical coordinate system of (ρ, θ) is also taken with the coordinate origin of O. The azimuth angle θ is defined as an angle from the negative direction of z-axis. The angle Θc denotes the critical angle of the transparent material of the illuminator. A position vector of q0 having an azimuth angle of θ0 is parallel to a normal vector of the incidence flat plane.
A relationship of coordinate components of ρ and θ of the position vector q can be written with the focal length f of the parabolic mirror as
from which the position vector q of (x, z) can be derived as,Because light rays emitted from the LED are incident on the flat surface, a divergence angle of the light ray can be limited in a range of 2Θc where the Θc denotes the critical angle. In this work, the material is acrylic with refractive index of 1.49. The critical angle Θc can then be calculated using Eq. (2) as
It is preferable to reduce light energy loss caused by absorption of light rays returned to the LED. Additionally, it is also preferable to make the multi-parabolic illuminator compact. An azimuth angle θ0 of the position vector q0 is thus set to 45 degrees. A range of the azimuth angle θ of a light ray can then be confined to from about 3 degrees to 87 degrees, which prevents the light ray from getting out of the illuminator without reaching the mirror. This also prevents the light rays from returning back to the LED.
The emitting surface area of the LED is set to 3×3 mm, which means that the L1 and L2 are 3 mm and 3 mm, respectively. The aperture size of A1 is set to 60 mm. A full-width divergence angle of 2Θo, 1 can thus be calculated using Eq. (4) to be 5.7 degrees.
3.2 TIR surfaces
Figure 4 shows a schematic projection view of multi-parabolic illuminator in the y – z plane. Light rays emitted from the LED are guided to reach the cross-sectional z = 0 plane. The light rays at the cross section can be considered as the secondary light source. The two TIR surfaces are arranged in mirror symmetry with respect to the x = 0 plane. A focal point of one TIR surface is on the other TIR surface in the z = 0 plane. The two TIR surfaces facing each other can construct the ideal CPC that conserves the etendue values at the entrance plane and the opening aperture, which is represented by Eqs. (5) - (7). In this work, however, a non-ideal truncated CPC is adopted instead of the ideal CPC in order to make the illuminator compact. The aperture size of A2 and the mirror surface width of B are set to 20 mm and 6 mm, respectively. The length of the CPC along the z-direction is set to 60 mm. A full-width divergence angle of 2Θo,2 is then calculated using the ray-tracing simulation [26] to be about 34 degrees.
Fig. 4. Schematic projection view of multi-parabolic illuminator in the y – z plane. Light rays emitted from the LED are guided to reach the cross-sectional z = 0 plane. The light rays at the cross section can be considered as a secondary light source. Two TIR surfaces are arranged in mirror symmetry with respect to the x = 0 plane. A focal point of one TIR surface is on the other TIR surface in the z = 0 plane.
4. Light angle distribution measurement of the prototype
Figure 5 shows a perspective view of a fabricated prototype of the multi-parabolic illuminator composed of the mirror surface, the TIR surface, and the opening aperture. The design parameters used for the prototype are mentioned in the previous section. The mirror surface is made by aluminum deposition on acrylic. A light diffuser is attached on an opening aperture to make illuminance distribution smooth. When a narrow input beam is incident on the diffuser, an output beam is wider beam with a typical gaussian light angle distribution profile with about 2 degrees divergence angle defined at 1/e intensity.
Fig. 5. Perspective view of fabricated prototype of multi-parabolic illuminator composed of mirror surface, TIR surface, and opening aperture. The mirror surface is made by aluminum deposition on acrylic.
Figure 6 shows light angle distributions of the prototype for both simulation and experiment. The simulation is performed using the ray-tracing simulation [26]. An intensity of the light angle distribution is plotted with respect to an angle from the positive z-direction corresponding to 0 degrees. The x – z plane is vertical plane and y – z plane is horizontal plane. Solid line and dotted line indicate simulations in the vertical plane and the horizontal plane, respectively. Circle mark and diamond mark indicate experiments in the vertical plane and the horizontal plane, respectively. Light distribution with respect to angle from -180 degrees to 0 degrees is plotted on the left, and that with respect to angle from 0 degrees to +30 degrees is plotted on the right. The experimental results agree well with the simulation results. The prototype demonstrates a production of a highly collimated rectangular illumination with half-intensity angles of about 7 degrees and 34 degrees for respective perpendicular axes. A ratio of the half-intensity angle in the horizontal plane to that in the vertical plane is about 5.
Fig. 6. Light angle distributions of the prototype for both simulation and experiment. An intensity of the light angle distribution is plotted with respect to an angle from the positive z-direction corresponding to 0 degrees. The x – z plane is vertical plane and y – z plane is horizontal plane. Solid line and dotted line indicate simulations in the vertical plane and the horizontal plane, respectively. Circle mark and diamond mark indicate experiments in the vertical plane and the horizontal plane, respectively. Light distribution with respect to angle from -180 degrees to 0 degrees is plotted on the left, and that with respect to angle from 0 degrees to +30 degrees is plotted on the right.
5. Discussions
Other than the light angle distributions, an illuminance distribution is also often an important feature. In the far field, it can be calculated from the light angle distribution shown in Fig. 6. For the dark-filed method, however, the illuminance distribution is often not so critical because presence or absence of a diffused light ray is important for the inspection processes. Figure 7 shows an illuminance distribution of the prototype calculated using the ray-tracing simulation. The illuminance distribution in the x – y plane at distance of 500 mm from the prototype is contour-plotted with a grayscale intensity normalized by the maximum intensity value. Cross-sectional illuminance distributions are also plotted at x = 30 mm and y = 0 mm, respectively. A total luminaire efficiency that is the ratio of light output emitted by the prototype to the light output emitted by the LED is calculated to be 83% with the reflectance of the mirror of about 90%. The total luminaire efficiency of more than 80% can be considered sufficient for the dark-field method.
Fig. 7. Illuminance distribution of the prototype calculated using the ray-tracing simulation. The illuminance distribution is in the x – y plane at distance of 500 mm from the prototype, which is contour-plotted with a grayscale intensity normalized by the maximum intensity value. Cross-sectional illuminance distributions are also plotted at x = 30 mm and y = 0 mm, respectively.
Depending on the purpose of the illuminator, the illuminance distribution can be optimized by the design parameters of the multi-parabolic surfaces. The illuminance distribution could also be drastically controlled by attaching a free-form surface at the opening aperture.
The horizontal divergence angle can be narrower with longer length of the illuminator along the z-direction. Note that a width of the illuminator along y-axis can be kept slim, which makes the illuminator installed in a narrow space.
6. Conclusions
A slim LED illuminator composed of multi-parabolic surfaces to combine perpendicular collimated illuminations is proposed here. The multi-parabolic illuminator can produce a collimated rectangular illumination where light angle distributions for respective perpendicular axes can independently be controlled by different parabolic surfaces. This independence of the control makes a design of the illuminator simple. In the cross-sectional x – z plane of the illuminator, a parabolic mirror can collimate light rays emitted from an LED. On the other hand, in the cross-sectional y – z plane, a secondary light source can be generated over the cross-sectional z = 0 plane. The other two parabolic surfaces (i.e., the TIR surfaces) can collimate light rays emitted from the secondary light source. The maximum divergence angles can be estimated based on the etendue conservation law as represented by Eqs. (4) and (7). A prototype of the multi-parabolic illuminator is fabricated with an LED chip size of 3×3 mm and an opening aperture size of 20×60 mm. A maximum width of the prototype is 20 mm. The prototype demonstrates a production of a highly collimated rectangular illumination with half-intensity angles of about 7 degrees and 34 degrees for respective perpendicular axes.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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