9.8.1: Image Formation by Lenses

Learning Objectives

• List the rules for ray tracking for thin lenses.
• Illustrate the formation of images using the technique of ray tracking.
• Determine the power of a lens given the focal length.

Lenses are found in a huge array of optical instruments, ranging from a simple magnifying glass to the eye to a camera’s zoom lens. In this section, we will use the law of refraction to explore the properties of lenses and how they form images.

The word lens derives from the Latin word for a lentil bean, the shape of which is similar to the convex lens in Figure \(\PageIndex{1}\). The convex lens shown has been shaped so that all light rays that enter it parallel to its axis cross one another at a single point on the opposite side of the lens. (The axis is defined to be a line normal to the lens at its center, as shown in Figure \(\PageIndex{1}\).) Such a lens is called a converging (or convex) lens for the converging effect it has on light rays. An expanded view of the path of one ray through the lens is shown, to illustrate how the ray changes direction both as it enters and as it leaves the lens. Since the index of refraction of the lens is greater than that of air, the ray moves towards the perpendicular as it enters and away from the perpendicular as it leaves. (This is in accordance with the law of refraction.) Due to the shape of the lens, light is thus bent toward the axis at both surfaces. The point at which the rays cross is defined to be the focal point F of the lens. The distance from the center of the lens to its focal point is defined to be the focal length \(f\) of the lens. Figure \(\PageIndex{2}\) shows how a converging lens, such as that in a magnifying glass, can converge the nearly parallel light rays from the sun to a small spot.

Figure \(\PageIndex{1}\): Rays of light entering a converging lens parallel to its axis converge at its focal point F. (Ray 2 lies on the axis of the lens.) The distance from the center of the lens to the focal point is the focal length, f, of the lens. An expanded view of the path taken by ray 1 shows the perpendiculars and the angles of incidence and refraction at both surfaces.

Definition: CONVERGING OR CONVEX LENS

The lens in which light rays that enter it parallel to its axis cross one another at a single point on the opposite side with a converging effect is called converging lens.

Definition: FOCAL POINT F

The point at which the light rays cross is called the focal point F of the lens.

Definition: FOCAL LENGTH \(f\)

The distance from the center of the lens to its focal point is called focal length, \(f\).

Figure \(\PageIndex{2}\): Sunlight focused by a converging magnifying glass can burn paper. Light rays from the sun are nearly parallel and cross at the focal point of the lens. The more powerful the lens, the closer to the lens the rays will cross.

The greater effect a lens has on light rays, the more powerful it is said to be. For example, a powerful converging lens will focus parallel light rays closer to itself and will have a smaller focal length than a weak lens. The light will also focus into a smaller and more intense spot for a more powerful lens. The power \(P\) of a lens is defined to be the inverse of its focal length. In equation form, this is

\[P=\frac{1}{f}. \nonumber \]

Definition: POWER \(P\)

The power \(P\) of a lens is defined to be the inverse of its focal length. In equation form, this is

\[P=\frac{1}{f}. \nonumber\],

where \(f\) is the focal length of the lens, which must be given in meters (and not cm or mm). The power of a lens P has the unit diopters (D), provided that the focal length is given in meters. That is, \(1 ~\mathrm{D}=1 / \mathrm{m}\), or \(1 \mathrm{~m}^{-1}\). (Note that this power (optical power, actually) is not the same as power in watts. It is a concept related to the effect of optical devices on light.) Optometrists prescribe common spectacles and contact lenses in units of diopters.

Figure \(\PageIndex{3}\) shows a concave lens and the effect it has on rays of light that enter it parallel to its axis (the path taken by ray 2 in the figure is the axis of the lens). The concave lens is a diverging lens, because it causes the light rays to bend away (diverge) from its axis. In this case, the lens has been shaped so that all light rays entering it parallel to its axis appear to originate from the same point, \(F\), defined to be the focal point of a diverging lens. The distance from the center of the lens to the focal point is again called the focal length \(f\) of the lens. Note that the focal length and power of a diverging lens are defined to be negative. For example, if the distance to \(F\) in Figure \(\PageIndex{3}\) is 5.00 cm, then the focal length is \(f=-5.00 \mathrm{~cm}\) and the power of the lens is \(P=-20 ~\mathrm{D}\). An expanded view of the path of one ray through the lens is shown in the figure to illustrate how the shape of the lens, together with the law of refraction, causes the ray to follow its particular path and be diverged.

Figure \(\PageIndex{3}\): Rays of light entering a diverging lens parallel to its axis are diverged, and all appear to originate at its focal point \(F\). The dashed lines are not rays—they indicate the directions from which the rays appear to come. The focal length \(f\) of a diverging lens is negative. An expanded view of the path taken by ray 1 shows the perpendiculars and the angles of incidence and refraction at both surfaces.

Definition: DIVERGING LENS

A lens that causes the light rays to bend away from its axis is called a diverging lens.

As noted in the initial discussion of the law of refraction, the paths of light rays are exactly reversible. This means that the direction of the arrows could be reversed for all of the rays in Figure \(\PageIndex{1}\) and Figure \(\PageIndex{3}\). For example, if a point light source is placed at the focal point of a convex lens, as shown in Figure \(\PageIndex{4}\), parallel light rays emerge from the other side.

Figure \(\PageIndex{4}\): A small light source, like a light bulb filament, placed at the focal point of a convex lens, results in parallel rays of light emerging from the other side. The paths are exactly the reverse of those shown in Figure \(\PageIndex{1}\). This technique is used in lighthouses and sometimes in traffic lights to produce a directional beam of light from a source that emits light in all directions.

Ray Tracing and Thin Lenses

Ray tracing is the technique of determining or following (tracing) the paths that light rays take. For rays passing through matter, the law of refraction is used to trace the paths. Here we use ray tracing to help us understand the action of lenses in situations ranging from forming images on film to magnifying small print to correcting nearsightedness. While ray tracing for complicated lenses, such as those found in sophisticated cameras, may require computer techniques, there is a set of simple rules for tracing rays through thin lenses. A thin lens is defined to be one whose thickness allows rays to refract, as illustrated in Figure \(\PageIndex{1}\), but does not allow properties such as dispersion and aberrations. An ideal thin lens has two refracting surfaces but the lens is thin enough to assume that light rays bend only once. A thin symmetrical lens has two focal points, one on either side and both at the same distance from the lens. (See Figure \(\PageIndex{5}\).) Another important characteristic of a thin lens is that light rays through its center are deflected by a negligible amount, as seen in Figure \(\PageIndex{6}\).

Definition: THIN LENS

A thin lens is defined to be one whose thickness allows rays to refract but does not allow properties such as dispersion and aberrations.

TAKE-HOME EXPERIMENT: A VISIT TO THE OPTICIAN

Look through your eyeglasses (or those of a friend) backward and forward and comment on whether they act like thin lenses.

Figure \(\PageIndex{5}\): Thin lenses have the same focal length on either side. (a) Parallel light rays entering a converging lens from the right cross at its focal point on the left. (b) Parallel light rays entering a diverging lens from the right seem to come from the focal point on the right.

Figure \(\PageIndex{6}\): The light ray through the center of a thin lens is deflected by a negligible amount and is assumed to emerge parallel to its original path (shown as a shaded line).

Using paper, pencil, and a straight edge, ray tracing can accurately describe the operation of a lens. The rules for ray tracing for thin lenses are based on the illustrations already discussed:

• Principal Ray 1: a ray entering a converging lens parallel to its axis passes through the focal point F of the lens on the other side (see rays 1 and 3 in Figure \(\PageIndex{1}\)); a ray entering a diverging lens parallel to its axis seems to come from the focal point F (see rays 1 and 3 in Figure \(\PageIndex{3}\))
• Principal Ray 2: a ray passing through the center of either a converging or a diverging lens does not change direction (see Figure \(\PageIndex{6}\), and see ray 2 in Figure \(\PageIndex{1}\) and Figure \(\PageIndex{3}\))
• Principal Ray 3: a ray entering a converging lens through its focal point exits parallel to its axis (the reverse of rays 1 and 3 in Figure \(\PageIndex{1}\)); a ray that enters a diverging lens by heading toward the focal point on the opposite side exits parallel to the axis (the reverse of rays 1 and 3 in Figure \(\PageIndex{3}\)). The third principal ray is optional and may be used to verify the accuracy of image location.

Image Formation by Thin Lenses

In some circumstances, a lens forms an obvious image, such as when a movie projector casts an image onto a screen. In other cases, the image is less obvious. Where, for example, is the image formed by eyeglasses? We will use ray tracing for thin lenses to illustrate how they form images, and we will develop equations to describe the image formation quantitatively.

Consider an object some distance away from a converging lens, as shown in Figure \(\PageIndex{7}\). To find the location and size of the image formed, we trace the paths of principal rays originating from one point on the object, in this case the top of the person’s head. The figure shows the three principal rays from the top of the object as described above. (Note that there are many light rays leaving this point going in many directions, but we concentrate on the three principal rays which can be traced by simple rules.) The first ray is one that enters the lens parallel to its axis and passes through the focal point on the other side (Principal Ray 1). The second ray passes through the center of the lens without changing direction (Principal Ray 2). The third ray passes through the nearer focal point on its way into the lens and leaves the lens parallel to its axis (Principal Ray 3). If correctly drawn, all three rays cross at the same point on the other side of the lens. The image of the top of the person’s head is located at this point. All rays, including those that are not principal rays, that come from the same point on the top of the person’s head are refracted in such a way as to cross at the point shown; the principal rays are what we use to find this point. Rays from another point on the object, such as her belt buckle, will also cross at another common point, forming a complete image, as shown. Although three rays are traced in Figure \(\PageIndex{7}\), only two are necessary to locate the image. Before applying ray tracing to other situations, let us consider the example shown in Figure \(\PageIndex{7}\) in more detail.

Figure \(\PageIndex{7}\): Ray tracing is used to locate the image formed by a lens. Rays originating from the same point on the object are traced—the three chosen rays each follow one of the rules for ray tracing, so that their paths are easy to determine. The image is located at the point where the rays cross. In this case, a real image—one that can be projected on a screen—is formed.

The image formed in Figure \(\PageIndex{7}\) is a real image, meaning that it can be projected. That is, light rays from one point on the object actually cross at the location of the image and can be projected onto a screen, a piece of film, or the retina of an eye, for example. Figure \(\PageIndex{8}\) shows how such an image would be projected onto film by a camera lens. This figure also shows how a real image is projected onto the retina by the lens of an eye. Note that the image is there whether it is projected onto a screen or not.

Definition: REAL IMAGE

The image in which light rays from one point on the object actually cross at the location of the image and can be projected onto a screen, a piece of film, or the retina of an eye is called a real image.

Figure \(\PageIndex{8}\): Real images can be projected. (a) A real image of the person is projected onto film. (b) The converging nature of the multiple surfaces that make up the eye result in the projection of a real image on the retina.

Several important distances appear in Figure \(\PageIndex{7}\). We define \(d_{\mathrm{o}}\) to be the object distance, the distance of an object from the center of a lens. Image distance \(d_{\mathrm{i}}\) is defined to be the distance of the image from the center of a lens. The height of the object and height of the image are given the symbols \(h_{\mathrm{o}}\) and \(h_{\mathrm{i}}\), respectively. Images that appear upright relative to the object have heights that are positive and those that are inverted have negative heights. Using the rules of ray tracing and making a scale drawing with paper and pencil, like that in Figure \(\PageIndex{8}\), we can accurately describe the location and size of an image. But the real benefit of ray tracing is in visualizing how images are formed in a variety of situations. To obtain numerical information, we use a pair of equations that can be derived from a geometric analysis of ray tracing for thin lenses. The thin lens equations are

\[\frac{1}{d_{\mathrm{o}}}+\frac{1}{d_{\mathrm{i}}}=\frac{1}{f} \nonumber \]

and

\[\frac{h_{\mathrm{i}}}{h_{\mathrm{o}}}=-\frac{d_{\mathrm{i}}}{d_{\mathrm{o}}}=m. \nonumber \]

We define the ratio of image height to object height (\(h_{\mathrm{i}} / h_{\mathrm{o}}\)) to be the magnification \(m\). (The minus sign in the equation above will be discussed shortly.) The thin lens equations are broadly applicable to all situations involving thin lenses (and “thin” mirrors, as we will see later). We will explore many features of image formation in the following worked examples.

Definition: IMAGE DISTANCE

The distance of the image from the center of the lens is called image distance.

THIN LENS EQUATIONS AND MAGNIFICATION

\[\frac{1}{d_{\mathrm{o}}}+\frac{1}{d_{\mathrm{i}}}=\frac{1}{f} \nonumber\]

\[\frac{h_{\mathrm{i}}}{h_{\mathrm{o}}}=-\frac{d_{\mathrm{i}}}{d_{\mathrm{o}}}=m \nonumber\]

Real images, such as the one considered in the previous example, are formed by converging lenses whenever an object is farther from the lens than its focal length. This is true for movie projectors, cameras, and the eye. We shall refer to these as case 1 images. A case 1 image is formed when \(d_{0}>f\) and \(f\) is positive, as in Figure \(\PageIndex{10}\)(a). (A summary of the three cases or types of image formation appears at the end of this section.)

A different type of image is formed when an object, such as a person's face, is held close to a convex lens. The image is upright and larger than the object, as seen in Figure \(\PageIndex{10}\)(b), and so the lens is called a magnifier. If you slowly pull the magnifier away from the face, you will see that the magnification steadily increases until the image begins to blur. Pulling the magnifier even farther away produces an inverted image as seen in Figure \(\PageIndex{10}\)(a). The distance at which the image blurs, and beyond which it inverts, is the focal length of the lens. To use a convex lens as a magnifier, the object must be closer to the converging lens than its focal length. This is called a case 2 image. A case 2 image is formed when \(d_{\mathrm{o}}<f\) and \(f\) is positive.

Figure \(\PageIndex{10}\): (a) When a converging lens is held farther away from the face than the lens’s focal length, an inverted image is formed. This is a case 1 image. Note that the image is in focus but the face is not, because the image is much closer to the camera taking this photograph than the face. (credit: DaMongMan, Flickr) (b) A magnified image of a face is produced by placing it closer to the converging lens than its focal length. This is a case 2 image. (credit: Casey Fleser, Flickr)

Figure \(\PageIndex{11}\) uses ray tracing to show how an image is formed when an object is held closer to a converging lens than its focal length. Rays coming from a common point on the object continue to diverge after passing through the lens, but all appear to originate from a point at the location of the image. The image is on the same side of the lens as the object and is farther away from the lens than the object. This image, like all case 2 images, cannot be projected and, hence, is called a virtual image. Light rays only appear to originate at a virtual image; they do not actually pass through that location in space. A screen placed at the location of a virtual image will receive only diffuse light from the object, not focused rays from the lens. Additionally, a screen placed on the opposite side of the lens will receive rays that are still diverging, and so no image will be projected on it. We can see the magnified image with our eyes, because the lens of the eye converges the rays into a real image projected on our retina. Finally, we note that a virtual image is upright and larger than the object, meaning that the magnification is positive and greater than 1.

Figure \(\PageIndex{11}\): Ray tracing predicts the image location and size for an object held closer to a converging lens than its focal length. Ray 1 enters parallel to the axis and exits through the focal point on the opposite side, while ray 2 passes through the center of the lens without changing path. The two rays continue to diverge on the other side of the lens, but both appear to come from a common point, locating the upright, magnified, virtual image. This is a case 2 image.

Definition: VIRTUAL IMAGE

An image that is on the same side of the lens as the object and cannot be projected on a screen is called a virtual image.

A third type of image is formed by a diverging or concave lens. Try looking through eyeglasses meant to correct nearsightedness. (See Figure \(\PageIndex{12}\).) You will see an image that is upright but smaller than the object. This means that the magnification is positive but less than 1. The ray diagram in Figure \(\PageIndex{13}\) shows that the image is on the same side of the lens as the object and, hence, cannot be projected—it is a virtual image. Note that the image is closer to the lens than the object. This is a case 3 image, formed for any object by a negative focal length or diverging lens.

Figure \(\PageIndex{12}\): A car viewed through a concave or diverging lens looks upright. This is a case 3 image. (credit: Daniel Oines, Flickr)

Figure \(\PageIndex{13}\): Ray tracing predicts the image location and size for a concave or diverging lens. Ray 1 enters parallel to the axis and is bent so that it appears to originate from the focal point. Ray 2 passes through the center of the lens without changing path. The two rays appear to come from a common point, locating the upright image. This is a case 3 image, which is closer to the lens than the object and smaller in height.

Table \(\PageIndex{1}\) summarizes the three types of images formed by single thin lenses. These are referred to as case 1, 2, and 3 images. Convex (converging) lenses can form either real or virtual images (cases 1 and 2, respectively), whereas concave (diverging) lenses can form only virtual images (always case 3). Real images are always inverted, but they can be either larger or smaller than the object. For example, a slide projector forms an image larger than the slide, whereas a camera makes an image smaller than the object being photographed. Virtual images are always upright and cannot be projected. Virtual images are larger than the object only in case 2, where a convex lens is used. The virtual image produced by a concave lens is always smaller than the object—a case 3 image. We can see and photograph virtual images only by using an additional lens to form a real image.

Table \(\PageIndex{1}\): Three Types of Images Formed By Thin Lenses

Type	Formed when	Image type	di	m
Case 1	\(f\) positive, \(d_{o}>f\)	real	positive	negative
Case 2	\(f\) positive, \(d_{o}<f\)	virtual	negative	positive, \(m>1\)
Case 3	\(f\) negative	virtual	negative	positive, \(m<1\)

TAKE-HOME EXPERIMENT: CONCENTRATING SUNLIGHT

Find several lenses and determine whether they are converging or diverging. In general those that are thicker near the edges are diverging and those that are thicker near the center are converging. On a bright sunny day take the converging lenses outside and try focusing the sunlight onto a piece of paper. Determine the focal lengths of the lenses. Be careful because the paper may start to burn, depending on the type of lens you have selected.

Section Summary

• Light rays entering a converging lens parallel to its axis cross one another at a single point on the opposite side.
• For a converging lens, the focal point is the point at which converging light rays cross; for a diverging lens, the focal point is the point from which diverging light rays appear to originate.
• The distance from the center of the lens to its focal point is called the focal length \(f\).
• Power \(P\) of a lens is defined to be the inverse of its focal length, \(P=\frac{1}{f}\).
• A lens that causes the light rays to bend away from its axis is called a diverging lens.
• Ray tracing is the technique of graphically determining the paths that light rays take.
• The image in which light rays from one point on the object actually cross at the location of the image and can be projected onto a screen, a piece of film, or the retina of an eye is called a real image.
• Thin lens equations are \(\frac{1}{d_{\mathrm{o}}}+\frac{1}{d_{\mathrm{i}}}=\frac{1}{f}\) and \(\frac{h_{\mathrm{i}}}{h_{\mathrm{o}}}=-\frac{d_{\mathrm{i}}}{d_{\mathrm{o}}}=m\) (magnification).
• The distance of the image from the center of the lens is called image distance.
• An image that is on the same side of the lens as the object and cannot be projected on a screen is called a virtual image.

Glossary

converging lens

a convex lens in which light rays that enter it parallel to its axis converge at a single point on the opposite side

diverging lens

a concave lens in which light rays that enter it parallel to its axis bend away (diverge) from its axis

focal point

for a converging lens or mirror, the point at which converging light rays cross; for a diverging lens or mirror, the point from which diverging light rays appear to originate

focal length

distance from the center of a lens or curved mirror to its focal point

magnification

ratio of image height to object height

power

inverse of focal length

real image

image that can be projected

virtual image

image that cannot be projected