Fresnel Lenses
George created this article series as a primer for anyone interested in tackling a project using optical sensors. In this article he presents Fresnel lenses, lens power, and lens speed.
To conclude this series, let’s begin by talking about Fresnel lenses, which mentioned in Part 2. The lens was invented in the 19th century by a French physicist Augustin-Jean Fresnel for use in light houses where the limited space did not allow for lenses with large aperture and short focal length of conventional design.
FRESNEL
Fresnel lenses are light and compact and thus use very little material. Figure 1 depicts how a conventional lens design is transformed into a Fresnel lens.
Transformation of a conventional to a Fresnel lens
Imagine you slice a conventional plano-convex lens and as a result you have onion-like rings placed in a single plane. Fresnel lenses can be found everywhere: reflectors, automobile lights, magnifiers, projectors, and passive infrared (PIR) detectors for intrusion alarms or light control. They can be made from glass or injection molded from plastics.
There are two types of Fresnel lenses: imaging and non-imaging. Imaging lenses act as equivalent plano-convex lenses. They produce a sharp, focused image, although not as clear as their equivalent plano-convex lenses because of light diffraction at the edges of the ridges. Non-imaging lenses can be found, for instance, in solar panels where they can greatly increase the sunlight concentration onto the solar cell. With such an increase in efficiency, a solar panel size can be significantly reduced.
Figure 2 shows the principle of a reflector with a Fresnel lens. The ridges are clearly visible in the lens cross-section. Lenses used in PIR detectors deserve a special mention. First, the lenses are usually molded from Polyethylene (PE) because, unlike glass which stops it, PE is transparent to far infrared radiation (heat) emitted by warm objects. Second, for intrusion detection or for switching lights, it is desirable for many detectors to have a wide field of view. This is achieved using pyroelectric sensor(s) located in the focal plane and an array of lenses (see Photo 1). Lenses with 270° horizontal vision (two sensors are needed) are commercially available.
Transformation of a conventional to a Fresnel lens
Actual PIR lens with 15 horizontal and three vertical bands of lenses
DIOPTERS
When specifying lenses, opticians talk about “diopters.” A diopter is a unit of measure of the refractive power of a lens. It is the reciprocal of the lens’s focal length expressed in meters. Thus, a lens with focal length of 0.1 m (100 mm or 3.94″) has a power of 10 diopters.
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The total power of a lens is the sum of diopters of its front and back surfaces. In other words, D = D1 + D2. For example, if the power of the front surface of a lens D1 is 7.0 and the power of the back surface D2 is –3, the total power is 4 (i.e., 7 – 3) diopters. Notice that the power of the back surface is negative for convex lenses because their radii lay on opposite sides on the principal axis. The refracting index of the lens material must also be plugged into the Lensmaker’s Equation:
D is the total diopters. f is the focal length in meters. n is the lens material refraction index (typically around 1.5). R1 is the radius of curvature of the front surface. R2 is the radius of curvature of the back surface. For a convex lens, as mentioned above, R2 is negative. The Lensmaker’s Equation is an approximation usable for thin lenses only. Thick lenses are treated as two thin plano-convex lenses with a block of glass between them. In general, a lens is considered thin when its thickness at the principal axis is negligible when compared with both R1 and R2. This means usually smaller than 1/10 to 1/100 of the radii.
APERTURE
Another term frequently encountered with lenses is an aperture. It is an opening that determines the cone angle through which light rays come to the focus in the image plane. The aperture determines how collimated the incoming light rays are. With a narrow aperture, the light rays are highly collimated, which creates a well-focused, sharp image within a certain distance (the depth of field) of the plane. A wide aperture admits uncollimated rays, and this results in a sharp image within much smaller depth of field. The aperture also controls how much light is passed through the lens. In photography a smaller aperture requires a longer exposure.
The lens speed is defined by its focal length divided by its aperture. An f = 50 mm lens with 50-mm aperture diameter would have the maximum speed of 1. Most camera lenses start around f/1.4. The lens’s speed needs to be often modified, so the aperture size is controlled by an iris, also known as a diaphragm, usually an integral part of the lens (see Figure 3). The iris increases or decreases the size of the aperture, known as the f-number or f-stop, marked on lenses as f/2, f/2.8, f/4, and so forth. This is the ratio of the focal length to the effective aperture diameter. Typically, f-stops are separated by √2 (1.41) ratio. The ability to modify the amount of light reaching the sensor (or film) to allow as much light as necessary for detection while avoiding sensor’s saturation is crucial. As I mentioned, the aperture size also modifies the depth of field, describing the extent to which the subject lying closer or farther from the plane of focus appears to be in focus.
Construction of an iris
In digital photography or when definition of the aperture is needed for electronic sensors, an “equivalent aperture” number is often specified. It is the actual aperture number adjusted for the equivalent value of a 35-mm camera.
The collimated light mentioned above means that its rays travel in parallel and will spread minimally as they propagate. A perfectly collimated light is said to be focused at infinity. An example of an almost perfectly collimated light is a laser beam. Optics, such as curved mirrors and lenses, can produce collimated light, but they are too large to concentrate enough light to achieve the power of lasers. A simple optical collimator often used with optical microscopes is in Figure 4.
Figure 4
Optical collimator
Collimators are not used with visible light exclusively. There are particle collimators for neutron particles, X-rays, and Gamma rays, to name just a few. In embedded control systems, an optical collimator such as the one in Figure 4 could be an LED with an integral lens illuminating a volume of air for detection of particles (e.g., smoke) by a photodiode, whose lens is focused on the illuminated volume.
ASTIGMATISM
Astigmatism is a form of aberration we may encounter when working with optics. Astigmatism causes the rays propagating in two perpendicular planes to focus at two different distances from the lens. Astigmatism occurs in two different forms. The Third-order Astigmatism affects light rays traveling from any point on an object in two different planes. The tangential plane is defined by the point on the object and the axis of symmetry. If this includes the principal axis it’s called a meridian plane. The second, sagittal plane is defined by the point on the object and is orthogonal to the tangential plane. Because it affects every single wavelength of light, the Third-order Astigmatism is also called “monochromatic aberration”.
The second type of astigmatism occurs in lenses which are not rotationally symmetrical. This can be due to a manufacturing flaw or design. This type of astigmatism is not necessarily a bad thing. It is sometimes intentionally designed into lenses. Some telescopes or compact disc players use astigmatic lenses to their advantage. A CD-reading lens turns the round dots (pits) on the CD into ovals. The oval indicates which axis is in better focus and, therefore, which direction the read lens needs to move. With a square array of four such lenses the read lens can be brought to a perfect focus.
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There are other optical devices often used in conjunction with electronic sensors. Among them are filters, such as dichroic filters, analogous to electronic band-pass filters. Flat mirrors are used to divert light rays to a different direction. Prisms split the light into its fundamental wavelengths, while reticles in gun sights and measuring magnifying glasses, for example, add cross-hairs or a measuring scale.
Diffraction grating is a piece of glass with lines scribed into it to split and diffract light into beams traveling in different directions. Those directions depend on the spacing of the grating and the wavelengths of the light. The effect is similar to that of a prism, but it is much more versatile. Gratings are commonly used in monochromators and spectrometers, as they are also much sharper than filters.
THE BASICS
As I wrote in Part 1, optical engineering is a science in its own right. My intent in writing this series was merely to explain the basics which could be useful to electronics engineers faced with a project using optical sensors. 
REFERENCES
EIFA Industry Co., “PIR Lens,” www.chinaeifa.com/sdp/460657/4/pd-2535294/3485466-1448758/PIR_lens.html.
Wikipedia, “Fresnel Lens,” http://en.wikipedia.org/wiki/Fresnel_lens.
PUBLISHED IN CIRCUIT CELLAR MAGAZINE • FEBRUARY 2016 #307 – Get a PDF of the issue
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George Novacek was a retired president of an aerospace company. He was a professional engineer with degrees in Automation and Cybernetics. George’s dissertation project was a design of a portable ECG (electrocardiograph) with wireless interface. George has contributed articles to Circuit Cellar since 1999, penning over 120 articles over the years. George passed away in January 2019. But we are grateful to be able to share with you several articles he left with us to be published.
