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Fundamental Optics – Part 2: Lenses and Refraction

Written by George Novacek

In part 1, George familiarized you with curved mirrors and their uses. Here, in part 2, he covers lenses and important topics such as chromatic aberration.

  • How to understand the fundamentals of optics
  • What is chromatic aberration?
  • Concave lenses
  • Convex lenses
  • Converging lenses 
  • Diverging lenses

Last month we discussed curved mirrors, their characteristics and their applications such as in passive infrared (PIR) intrusion detectors. Now it’s time to talk about lenses. Rather than reflecting, lenses work by refracting—that is, bending—light rays. Just as with curved mirrors we can use ray tracing techniques to understand how lenses form an image.

There are two principal types of lenses: positive (also called converging or convex lenses) and negative (also called diverging or concave lenses). Each positive and negative lens exists in three basic types. Positive lenses are bi-convex (also called dual convex), plano-convex, and positive meniscus (see Figure 1). Similarly, negative or concave lenses are bi-concave or dual concave, plano concave, and negative meniscus (see Figure 2).

Figure 1
Positive lenses: (a) bi-convex, (b) plano-convex, and (c) positive meniscus
Figurer 2 
Negative lenses: (a) bi-concave, (b) plano-concave, and (c) negative meniscus

Converging lenses produce both real and virtual images. Ray tracing rules are similar to ray tracing of curved mirrors. Figure 3 shows ray tracing of a converging bi-convex lens. For the moment let’s consider lenses to be “thin.” (More about it in Part 3 of this article series.) Any ray traveling in parallel to the principal axis (red traces) is refracted, i.e. bent, to reach the focal point at the other side of the lens. Rays traveling through the focal point towards the lens (blue traces) will refract through the lens and continue parallel to the principal axis. And, finally, rays traveling via the center of the lens (green traces) will continue in the same direction.

Figure 3 
Double convex lens ray tracing

As you can see in Figure 3a, the lens bends the rays twice, once on its each surface. The resulting refraction is the sum of the two bends. As long as we know the location of the focal points—which, by the way, do not have to be the same distance from the center of the lens—and the location of the center of the lens, the tracing rules listed above apply. We don’t have to draw the lens, the object and the image. Mere vertical lines in their particular locations will suffice, as illustrated by Figure 3b.

Figure 3 shows the object located more than two focal lengths (2F) away from the lens. In case of a spherical lens, 2F distance coincides with the center of the curvature (C) as is the case with spherical mirrors described in Part 1 of this article series. The resulting image will be inverted, smaller than the object and located between the focal point (F) and 2F on the opposite side of the lens. When the object is located far enough beyond 2F and the image thus located very close to F, the object is considered to be in infinity and everything beyond that point will be in focus. The image will be real. The light rays will converge at the location of the image and can be projected on a screen, film, or sensor.

With the object located at 2F distance from the lens, the image will appear at the 2F distance from the opposite size of the lens. The image will be real, inverted, and the same size as the object. In other words, the lens magnification will be 1.


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With the object located between 2F and F points, the lens will form an image beyond the 2F point on the other side of the lens. Once again, the image will be real and inverted, but the image dimensions will be larger than those of the object. Therefore, the magnification will be greater than 1.

An object located at the focal point F will create no image as the refracted rays will neither converge nor diverge. You can take a straight edge, a pencil, and a piece of paper and verify it for yourselves.

Finally, when the object is located between the focal point F and the lens, the image will be located behind the object between points F and 2F on the side of the lens where the object is located. It will be larger than the object—that is, with magnification greater than 1, upright and virtual. That means that upon refraction the light rays will diverge. Tracing the rays using the rules listed above you will find that they indeed intersect beyond the object only. This is the case of a magnifying glass, also called a loupe.

Light rays emanating from the focal point are bent by the convex lens such that they are projected parallel to the principal axis. Similar to curved mirrors, this principle is used where a collimated light beam is desired. Often the light source is positioned in the focal point of the lens as well as of the focal point of a concave mirror behind it. Typical applications would be stage reflectors, automobile head and back lights and so forth. It should be noted that Fresnel lenses are usually used for this purpose. We shall address the Fresnel lens next month in the last part of this series. Many LEDs have a lens molded as an integral part of their package to provide focused and/or dispersed light.

Now, let’s consider a diverging lens shown in Figure 4. The tracing can also be simplified by using vertical lines as shown in Figure 4b. The rules for ray tracing of negative lenses are essentially the same as for the positive lenses except that the focal point locations are inverted. Let’s summarize them. An incident ray traveling parallel to the principal axis will refract such that its extension shown in broken line will pass through the focal point F on the object side of the lens (red trace). The incident ray passing from the object through the lens toward the focal point at the opposite side of the lens (blue trace) will be refracted to continue from the lens in parallel with the principal axis. And the ray passing through the center of the lens (green trace) will not be bent and will continue in the same direction. Similar to the convex mirror, concave lenses can spread point light source over a wide area.

Figure 4 
Double concave lens ray tracing

Simple optics found in reading glasses, magnifying glasses and many sensors use just a single element lens. Complicated optical systems—such as camera lenses, microscopes, projectors, and so forth—use lenses comprising numerous elements to compensate for aberrations and to provide required features such as focusing and aperture control. Some elements are movable for focusing and zooming, others fixed, some elements are even glued together. Figure 5 is an example of a camera lens.

Figure 5 
Camera lens

Lenses, like mirrors, unfortunately, do not always work the way we expect them to because they suffer from aberrations, which is a deviation from a norm. Chromatic aberration—also known as color fringing, achromatism, and by other similar names—is a failure of a lens to focus all colors to the same convergence point. It is caused by different refractive indexes of the lens material for different color wavelengths. It manifests itself by images having fringes of colors around object boundaries. There are two types of chromatic aberration. Axial (or longitudinal aberration) occurs when different wavelengths converge at different points on the principal axis (see Figure 6a). Transverse (or lateral) aberration occurs when different wavelengths focus at different positions at the focal plane (see Figure 6b).

Figure 6 
These are chromatic aberrations: axial aberration (a) and transverse (lateral) aberration (b)

Chromatic aberration can be reduced by increasing the focal length of the lens. With the present-day drive for miniaturization, this is not a favored option in most cases. Therefore, lenses are made achromatic by being assembled from several materials with different dispersions. Most common materials are crown and flint glass, forming so-called achromatic doublets. Such lenses reduce the chromatic aberration to an acceptable degree, although they are not perfect. Better results are achieved with three layers of glass, each focusing one basic color (i.e., blue, green, and red). Such apochromatic lenses produce a very small focusing error.

Another method is the combination of converging lens with a diverging lens. Since both lenses exhibit chromatic aberration but in opposite directions, the chromatic aberrations cancel each other out. Lenses used with electrical sensors where the signal is digitized—such as in video cameras—can have transverse aberration corrected by digital processing. Axial aberration cannot be thus corrected.


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Barrel and pincushion distortions are deviations from rectilinear projection of the lens, generally radially symmetric. These are caused by the lens magnification increase (pincushion) or decrease (barrel) with the incident ray’s distance from the lens’s optical axis. They are not an issue with most optical sensors other than those processing photo and video images. Both distortions can be corrected by lens design or digital processing. 

Read Part 1 Here

Cambridge in Colour, “Understanding Camera Lenses,” .
The Physics Classroom, “Refraction and the Ray Model of Light,” Lesson 5,


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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 2: Lenses and Refraction

by George Novacek time to read: 6 min