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Fundamental Optics – Part 1: Curved Mirrors

Written by George Novacek

Understanding the fundamentals of working with optical sensors will help you with future robotics engineering projects. There are two primary types of optics: reflective and refractive. George introduces reflective optics in this article.

  • How to understand the fundamentals of optics
  • How do concave mirrors work?
  • How do convex mirrors work?
  • What’s the difference between reflective and refractive optics?
  • Optical sensors
  • Concave mirrors
  • Convex mirrors

Designers of embedded and robotic systems are increasingly faced with the use of optical sensors. In most instances, these need additional optics for their operation. The design of optical systems is an engineering discipline in its own right, requiring years of study and experience. Present-day optical systems are designed with the help of computers, but many of those systems can be analyzed and understood by simple manual ray tracing. This can be accomplished the old-fashioned way with a piece of paper, a straight edge, and a pencil. As electrical engineers we do not have to be able to design optics, even though some optical rules apply in electrical engineering. Dish and Yagi antennas are two examples. This three-part series will not make you an optical expert, but it will explain the fundamentals, which will be helpful when you’re working with optical sensors.

PRIMARY OPTICS
There are two primary types of optics: reflective and refractive. It is not unusual to find both types combined in a single system, such as in a telescope. With some exceptions, most optical systems consist of numerous elements to achieve the desired characteristics. Understanding the characteristics of the individual elements is the first step to understanding the characteristics of the entire system.

Reflective elements are mirrors, usually called curved mirrors. They work by reflecting light rays in a predetermined way. Their curvature is often spherical, meaning the mirror is a portion of a sphere. As we learned in descriptive geometry, different curves can be obtained by slicing a cone—namely, a circle, parabola, ellipsis, and hyperbola. Thus, parabolic and elliptical mirrors can also be found alongside the spherical ones. The reasons for selecting something other than a spherical mirror are beyond the scope of this column, but one reason is mentioned later in this article. Flat mirrors are used to change the direction of the light rays without any other effect.

CONCAVE MIRRORS
Depending on which surface of the spherical mirror is coated with a reflective surface, curved mirrors are either concave or convex. Figure 1 shows the concave mirror which has the inner spherical surface reflective.

Figure 1
Image forming by a concave mirror

The axis passing through the center of a spherical mirror perpendicular to its surface is called the principal axis. Two important points lie on it: Point C, which is called the center of curvature, is the center of the sphere making the mirror. Point F is called the focal point. In case of the spherical mirror, the focal point F lies halfway between point C and the reflective surface of the mirror. Locations of point C and the focal point for parabolic and elliptical mirrors are different.

We use ray tracing to see how the image is created. In Figure 1, the blue traces are for the incident rays. The red traces are for the reflected rays. The rules are simple: incident rays moving parallel with the principal axis are reflected to pass through the focal point. Incident rays passing through the focal point are reflected in parallel to the principal axis. The image will be created where the reflected rays intersect and will be inverted with respect to the object. Additional optical elements would have to be used to turn the image right side up.

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When the object is located beyond the center of curvature C, the image will be located between the center of curvature and the focal point, inverted, and the image dimensions will be smaller than the object dimensions. With the object located at C, the image will also be located at C and will be of the same size as the object, but inverted. The object located between points C and F will cause the image to be seen beyond the center of curvature, and it will be inverted and larger than the object. In all these cases the images will be real, with the light rays converging at the image location, where a sensor, for example, can be placed. You can verify this with a piece of paper, a straight edge, and a pencil.

When the object is located at the focal point, no image is formed but when the object is located between the focal point and the reflecting surface, a virtual image, right side up, magnified, that is larger than the object, will be formed somewhere beyond the mirror. This effect can be seen, with magnifying bathroom mirrors, for instance.

The other way around, when light rays emanating from the focal point of a concave mirror hit the mirror’s surface, they are reflected parallel to the principal axis. This is the principle we can see used in some stage reflectors, light houses, flashlights, halogen light bulbs and other applications where a focused light beam is required.

Concave mirrors are commonly used in astronomical telescopes, but have also been used in the early passive infrared (PIR) intrusion detectors. They are still used in specialized, long range, narrow field of view PIRs such as those used to protect a site perimeter. Photo 1 shows several PIR lenses.

Photo 1
Concave mirrors for PIRs

It should also be noted that depending on the reflective surface material, mirrors can be used to reflect wavelengths other than light. Concave mirrors form the heart of dish antennas, be it for communications or radars, heat (infrared) waves can be focused in a specific direction in PIRs and so forth. Even just a small part of a concave mirror can increase the gain of an antenna by reflecting the wave to the dipole as we can see with Yagi antenna reflector elements.

CONVEX MIRRORS
Convex mirrors use the outer surface of the sphere as is shown in Figure 2. Similar to the concave mirror, we have the principal axis with the center of the curvature measured from the reflective surface and the focal point half way in between (in case of the spherical mirror). To trace the rays, the same rules apply. The incident rays are shown in blue and the reflected rays in red. Notice that unlike the concave mirror, the center of the curvature and the focal point of the convex mirror are located behind it.

Figure 2
Convex diverging mirror

Because an incident light from a single point will reflect off the mirror and diverge, never to intersect at the object side of the mirror, the convex mirror is often referred to as a diverging mirror. And, therefore, the convex mirror can produce a virtual image located behind the mirror only. While the virtual image appears to an observer, the light rays do not pass through the image location and cannot act on a sensor.

To trace the light rays to determine the image location, we use the same process as for the concave mirror. The rays traveling parallel with the principal axis reflect to the focal point while the rays traveling through the focal point reflect in parallel with the principal axis. The rays cannot pass through the mirror to reach the focal point behind it, so it is the extensions of these rays that must be considered. In Figure 2 the extensions are shown as broken lines.

Unlike concave mirrors, the image created by convex mirrors is not affected by the location of the object with respect to the mirror. It will always be a virtual image behind the mirror, and will be smaller than the object. This effect is used to observe a wider angle of view when compared with an ordinary flat mirror. Because of this characteristic, convex mirrors are frequently used as rear view mirrors in cars, in supermarkets, stores, or traffic intersections where a normal, limited view could lead to a collision.

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Due to its light divergent characteristic convex mirrors are also used in some street lamps and lamps with point light sources, such as LEDs, to spread the light over a wider area. As convex mirrors provide a virtual image only as the light rays do not converge, I am not aware of any application of convex mirrors alone in electronic sensors, but would welcome any feedback from the readers. Convex mirrors are also used for entertainment (see Photo 2).

Photo 2
A convex mirror in Chicago

Even perfectly made concave spherical mirrors suffer from aberration, meaning that they are not generating image exactly as they are expected to do. These mirrors are unable to focus all the incident light from the same location to the same point. It is most noticeable with the light rays reflected from the edges of the mirrors. All light rays striking the edge of the mirror are subject to this aberration, causing some blurring of the image. This situation can be corrected by covering the edges of the spherical mirror with a mask or, better, by the use of a parabolic mirror.

I hope you enjoyed this introduction to fundamental optics. In Part 2, we’ll take a closer look at refractive optics—that is, lenses. 

Read Part 2 Here

PUBLISHED IN CIRCUIT CELLAR MAGAZINE • DECEMBER 2015 #305 – 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.

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Fundamental Optics – Part 1: Curved Mirrors

by George Novacek time to read: 6 min