Design of a plastic aspheric Fresnel lens with a spherical shape
Optical Engineering and Manufacturing
This paper provides a detailed description of the design of a plastic, aspheric Fresnel lens upon a spherical surface. The design technique is based on calculating solutions to Snell's law along the lens surface. These lenses were intended for the wavelength range of 8 to 12 microns. However, his method easily applies to lenses designed for other wavelengths. The procedure used to manufacture the lenses will be to construct a diamond turned master surface, followed by injection molding with high density polyethylene.
Keywords: Fresnel Lenses, Diffractive Optics, Plastic Lenses
Infrared motion sensors are installed in many homes and businesses as part of a security system or a light control system. I began this project with the goal of allowing product designers to have more creative options to the boxes that we see in many homes and businesses today. My method allows for the design of a fresnel lens on a spherical surface. Such a lens would allow a designer to design a product with curved lines, with the lens blending smoothly into the housing, while maintaining a high performance optical system. This lens design is different from a dome lens because the radius of the outer surface does not need to match the focal length of the lens.
The method specifies the input to be parallel light rays and the output as a set of rays that all converge at the focal point. The design procedure solves Snell's law to determine the profile of the surface that must lie between the input and output. This method will create a Fresnel lens with constant groove width and variable groove depth. The grooves will be fairly flat and shallow near the center of the lens and deep with steep angles near the outside. I selected the constant groove width method because this method establishes the exact location of each groove peak and simplifies the mathematics.
The design proceeds according to the following procedure. The lens designer specifies the focal length, spherical radius groove width, and index of refraction. The grooves will have large dimensions to avoid diffraction effects. A computer program handles all the necessary calculations. The program breaks the lens into a set of concentric grooves and solves Snell's law for each groove. At this point we will have a collection of flat grooves. Each flat groove will have some variance in focal length along its cross section. We will eliminate most of this variance by fitting the flat grooves to the aspheric lens equation (1). The program assembles the lens surface so it includes the material thickness and places each groove peak along a spherical radius. Finally the program tests the lens surface for focal length and transmittance.
Fig 1. Spherical Lens with Constant Groove Width
Calculation of Groove Angle
The design of a curved Fresnel lens is more complex than the design of a flat lens. With a curved lens, the outer surface will refract in incident ray. We must match the grooved surface to the outer surface so the two combine to produce the correct focal length. This analysis will calculate the angle of each groove, based on the location of each groove peak.
Figure 2. Spherical Fresnel Lens Geometry
Figure 3. Detail of Groove Geometry
Figures 2 and 3 illustrate a ray of light striking the outer surface of a spherical Fresnel lens. The location where this ray strikes the outer surface is the incidence point (x'). The location where the ray exits the lens is the exit point (x). By using Snell's law, we can establish a relationship between the incidence point (x'), exit point (x), and the groove angle (d). Angles a and b are functions of the incidence point (x'), while angle e is a function of the exit point (x), or groove peak. We can solve (5) for d, the groove angle. Equation (6) defines the groove angle that is necessary to refract an incoming ray onto the focal point and is the most important result of this analysis.
In order to use (6), we must know the incidence point (x') and the groove peak (x). However, we only know the locations of the groove peaks (x). Figure 4 illustrates the relationship between the groove peak and the incidence point. The refraction angle (b) and (x-x') are very small throughout the lens, therefore t » t'. We need to find values for the x' term in (6). The simplest method is to solve (7) and (8) for x as in (9) and use a numerical method to solve for x'. After performing this numerical calculation, we can use (6) to calculate the required angle for each groove.
Figure 4. Relationship between incidence point (x') and exit point (x).
The procedure above calculates the groove angles on the basis of the locations of the groove peaks. However, I would prefer to base the lens design on the groove centers. Because the lens has variable groove depth, we cannot predict the coordinates of each groove center. To approximate the angle at the groove centers I calculated the average angle at the preceding and following groove peak. A focal length test later verifies this approximation.
We can improve the focus of the lens by fitting the groove angles to a continuous, aspheric curve. To accomplish this we must first calculate the derivative of (1) with respect to x. Equation (10), describes the slope of the aspheric curve at any point. We can use the Levenberg-Marquardt method to fit the groove angles and their x coordinates to (10).
Constructing the Final Lens Surface
The final lens surface will consist of aspheric grooves with the peaks arranged so they lie along a sphere. Equation (11) calculates the depth of each groove and will be used to break the aspheric curve into a series of grooves with the peaks placed along a flat line. Equation (12) calculates the radius of the inner lens surface and will align the peaks along that radius. Equation (13) combines (1), (11) and (12) to produce the final grooved lens surface.
We can use (13) to diamond turn a master tool that will contain the Fresnel lens surface. Motion sensors will employ lens arrays that may include twenty or more individual elements. We will produce electroform copies from the master tool and machine them to size to create each lens element. The assembled array is placed into an injection molding tool and is used to mold lenses that will be used in products.
I used the method described above to design a lens with a focal length of 22 mm and a radius of 30 mm. Figure 1 illustrates a cross section of this lens. The solid curve shows the Fresnel lens surface, while the dashed curve shows the smooth outer surface. The program tests the lens for focal length and transmittance.
The focal length test projects parallel rays onto the outer lens surface and performs a ray trace through the smooth outer surface, and inner grooved surface. To perform this ray trace we need to calculate the refraction angle at the outer surface and use numerical techniques to determine where the refracted ray intersects the fresnel lens surface. Figure 5 illustrates the results of this analysis. According to the test, the mean focal length will be 21.96 millimeters, with a standard deviation of 0.068 mm. Each groove is 0.2 mm wide and you can see that the focal length varies slightly throughout each groove. The aspheric curve works to minimize these variations. Figure 5 appears to have some abnormalities near the outer edge of the lens. This is caused by small inaccuracies in the numerical routine I used to perform the ray trace.
Figure 5. Focal Length Test of the lens in figure 1.
Figure 6. Transmittance of Lens
Figure 6 displays the expected transmittance of the lens. The transmittance model includes the following factors: losses through the material, Fresnel reflections, losses due to draft angle, and losses due to groove valley radii. The results shown in figure 6 appear to be typical when compared to previous flat lenses that I have designed. The transmittance drops near the outer part of the lens because the grooves become steeper towards the outside of the lens. From Fresnel's equations, steeper surfaces will reflect greater portions of the incident light. If we attempt to make the lens too wide, the grooves will become too steep and total internal reflection will occur.
The base material for this lens is high density polyethylene (HDPE) with an index of refraction of 1.54. We add several pigments to absorb white light and add a white color to the plastic. HDPE is fairly absorbent. Experimentally, I have determined the absorption coefficient of this blend to be 0.878 mm-1.
The transmittance model assumes that the lens includes draft angles and groove valley radii. The model assumes a draft angle of one degree and a groove valley radius of two microns. We add a draft to the back of each groove to improve how the surface performs during tool manufacture and injection molding. The draft angle insures that we will be able to separate electroform copies from the master. Later, during injection molding, the draft angle assists in separating the molded lens from the tool. The groove valley radius is a result of the radius of the diamond tool. At the foot of each groove, the lens surface will conform to the shape of the diamond stylus, rather than the aspheric curve.
Optimization of the Lens Design
The lens designer can design the lens to maximize transmittance. The designer accomplishes this by varying the groove width and examining the results of the transmittance model. A lens with narrow grooves will have shallow grooves. There will be only a slight variation in groove depth between the center and the outer part of the lens so it will be possible to mold the lens thinner. The transmittance of the lens will improve because there will be less material to absorb light. However, as the grooves become thinner, the number of grooves becomes larger. If there are too many grooves, the valleys and relief angles will begin to consume large portions of the lens and transmittance will drop.
The method described above allows a lens designer to place a Fresnel lens surface on any spherical surface. A ray trace verified that this method will produce the desired focal length. Calculations of the transmittance indicate transmittance will be lower for the outer portions of the lens. The method will produce an equation that will enable a diamond turning machine to transfer the surface onto a metal tool.
1. G. J. Zissis and W. L. Wolfe, The Infrared Handbook, ERIM, Ann Arbor, MI, 1989.
2. K. D. Moeller, Optics, University Science Books, Mill Valley, CA, 1988
3. H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan Publishing, New York, NY, 1961
4. W. H. Press, S. A. Teukolsky, W.T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran 2nd Ed., Cambridge University Press, 1992
5. K. Sakakibara, T. Asai, A. Nagakura, N. Kojima, and S. Igarashi, "Development of Smoothly Curved Fresnel Lens," 1991
6. B. Nabelek , M. Maly, and V.I. Jirka, "Linear Fresnel Lenses, Their Design and Use", Renewable Energy 3(4), 403-408 (1990)
7. C. L. Grendol, "Design and Development of Injection Molded Fresnel Lenses for Point-Focus Photovoltaic Systems," Sandia National Laboratories, 1-58 (1987)
8. G. Bradburn, "Design and Manufacture of High Quality Plastic Infrared Fresnel Lenses," SPIE Vol. 590 Infrared Technology and Applications, 87-92 (1985)
9. I. Powell, "Tracing Finite Rays Through a Fresnel Lens", Applied Optics 22(18), 2924-2926 (1983)
10. E. A. Boettner and N. E. Barnett, "Design and Construction of Fresnel Optics for Photoelectric Receivers", Journal of the Optical Society of America 46(11), 849-857 (1951).
15 East 26th Street, Suite 18C · New York, NY 10010, USA
Telephone: +1.916.834.9283 · Brasil: +55.11.3304.9840 · Hong Kong +852.2127.0512· Fax +1.917.472.7359