Lenses are among the most important optical devices used in science, medicine, and everyday life. They are found in eyeglasses, contact lenses, cameras, microscopes, telescopes, endoscopes, smartphones, and even in the human eye. By bending light through refraction, lenses allow us to focus images, magnify tiny structures, and correct vision problems. For students pursuing careers in the health sciences, understanding how lenses work is especially important. Medical professionals rely on optical instruments every day—from surgical loupes and operating microscopes to ophthalmoscopes used to examine the retina. Understanding the basic physics of lenses provides the foundation for understanding how these instruments produce clear images. The word lens comes from the Latin word for the lentil bean because the shape of a common convex lens resembles that of a lentil.

A converging (convex) lens is thicker in the middle than at the edges. As shown in Figure 69.1, rays of light entering the lens parallel to its principal axis are refracted toward the axis and meet at a single point on the opposite side of the lens. The principal axis is an imaginary line passing through the center of the lens and perpendicular to its surfaces. The point where parallel rays meet after passing through the lens is called the focal point, labeled F. The distance from the center of the lens to this point is known as the focal length, represented by f.

A familiar example of a converging lens is a magnifying glass. When sunlight passes through a magnifying glass, the nearly parallel rays from the Sun are focused into a very small spot, as shown in Figure 69.2. Because the light energy is concentrated into such a small area, the temperature at the focal point can become high enough to ignite paper or other combustible materials.

The ability of a lens to bend light is described by its optical power. A lens that bends light strongly brings parallel rays to a focus over a short distance and therefore has a short focal length. Conversely, a weaker lens bends light less and has a longer focal length. The optical power P of a lens is defined as the reciprocal of its focal length:

where the focal length f must be expressed in meters. The SI unit of optical power is the diopter (D):

Optical power is different from power measured in watts. Instead of describing the rate of energy transfer, diopters measure how strongly a lens converges or diverges light. Optometrists and ophthalmologists prescribe eyeglasses and contact lenses using diopters. For example, a reading glass labeled +2.00 D has a focal length of 0.50 m, while a lens labeled −3.00 D is a diverging lens used to correct nearsightedness.

Diverging Lenses

A diverging lens, also called a concave lens, is thinner in the middle than at the edges. Instead of bringing parallel rays together, it bends them away from the principal axis. As shown in Figure 69.3, rays entering a diverging lens parallel to the principal axis spread apart after passing through the lens. When the outgoing rays are extended backward, they appear to originate from a common point on the same side of the lens as the incoming light. This point is the focal point of the diverging lens. Because the rays do not actually pass through the focal point, the focal point is described as virtual. By convention, both the focal length and optical power of a diverging lens are negative. For example, if a diverging lens has a focal length of

then its optical power is

Light paths through a lens are reversible. This means that if a ray can travel through a lens in one direction, it can follow the same path in reverse. For example, parallel rays entering a converging lens meet at the focal point, as shown in Figure 69.1. Reversing the path, light emitted from a small source placed at the focal point emerges from the lens as a set of parallel rays.

Ray Tracing and Thin Lenses

Ray tracing is a graphical method used to determine the paths followed by light rays and locate the images formed by optical systems. For lenses, ray tracing provides a simple way to predict whether an image will be upright or inverted, enlarged or reduced, and real or virtual. Detailed lens systems, such as those found in cameras, microscopes, and medical imaging instruments, may contain many curved surfaces and require computer modeling. However, the behavior of a single lens can often be understood using the thin-lens approximation. A thin lens is an idealized lens whose thickness is small compared with the distances involved in forming an image. Although light is refracted at both surfaces of the lens, we treat the total change in direction as if it occurs at a single plane passing through the center of the lens. This approximation allows us to ignore small effects associated with lens thickness, including some aberrations and dispersion.

A symmetrical thin lens has two focal points, one on each side of the lens. Both focal points are the same distance from the lens, so the focal length has the same magnitude regardless of the direction in which light travels. This symmetry follows from the reversibility of light paths.

Another useful property of a thin lens is that a ray passing through its center is deflected by only a very small amount. In ray diagrams, this ray is treated as continuing in a straight line. Together with the behavior of rays parallel to the axis and rays passing through a focal point, this provides a simple set of rules for locating images.

Ray tracing can be performed using a pencil, straightedge, and a carefully drawn diagram. Although light leaves every point on an object in many directions, only two rays are needed to locate the corresponding point on the image. A third ray can be added to check the construction. The most useful rays are those whose paths can be predicted using the following rules.

When constructing a ray diagram, begin at a specific point on the object, usually the top of an arrow representing the object. Trace at least two of the principal rays through the lens. The image of that point is located where the refracted rays meet or where their backward extensions appear to meet.

• If the refracted rays actually meet, the lens forms a real image.
• If the refracted rays remain separated but appear to come from a common point when extended backward, the lens forms a virtual image.

The complete image is formed by repeating this reasoning for every point on the object. In practice, tracing rays from the top of the object is sufficient to determine the image's position, height, and orientation.

Image Formation by Thin Lenses

One of the most important applications of lenses is their ability to form images. Cameras, microscopes, telescopes, projectors, eyeglasses, and the human eye all rely on lenses to produce images that are either projected onto a surface or viewed directly. To determine where an image forms, we use ray tracing. Instead of following every light ray leaving an object, we trace just a few principal rays whose paths are easy to predict. The point where those rays intersect (or appear to intersect) determines the location of the image. Figure 69.7 illustrates image formation by a converging lens. Three principal rays are drawn from the top of the object:

1. A ray parallel to the principal axis passes through the focal point after leaving the lens.
2. A ray passing through the center of the lens continues in essentially a straight line.
3. A ray passing through the focal point before reaching the lens emerges parallel to the principal axis.

All three rays intersect at the same location, showing where the image of the top of the object forms. Repeating this process for every point on the object produces the complete image.

The image shown in Figure 69.7 is a real image. A real image forms when light rays actually converge at the image location. Because the rays physically meet, the image can be projected onto a screen, camera sensor, photographic film, or the retina of the eye.

Figure 69.8 shows two familiar examples. A camera lens projects a real image onto the image sensor (or film), while the lens of the eye projects a real image onto the retina, where specialized photoreceptor cells convert light into electrical signals that are sent to the brain.

Healthcare Connection: Vision and the Retina

The image formed on the retina is real, inverted, and usually smaller than the object being viewed. Although the retinal image is upside down, the brain automatically interprets the visual information so that we perceive the world as upright. Many vision disorders occur because the eye cannot focus this real image precisely on the retina. Corrective lenses change the paths of incoming rays so that the image forms at the proper location.

Several distances and dimensions are used to describe image formation:

• Object distance ([latex]d_o[/latex]): the distance from the object to the center of the lens.
• Image distance ([latex]d_i[/latex]): the distance from the center of the lens to the image.
• Object height ([latex]h_o[/latex]): the height of the object.
• Image height ([latex]h_i[/latex]): the height of the image.

By convention, upright images have positive heights, while inverted images have negative heights. Although carefully drawn ray diagrams provide an intuitive understanding of image formation, quantitative calculations are performed using the thin lens equation and the magnification equation.

Thin Lens Equation

[latex]\frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{f}[/latex]

Magnification Equation

[latex]m=\frac{h_i}{h_o}=-\frac{d_i}{d_o}[/latex]

The magnification describes how much larger or smaller the image is compared with the object. A positive magnification corresponds to an upright image, while a negative magnification corresponds to an inverted image.

These two equations apply to all ideal thin lenses and allow us to predict the position, orientation, and size of images formed by cameras, microscopes, telescopes, magnifying glasses, and the human eye.

Example 69.2: Finding the Image of a Light Bulb Using Ray Tracing and the Thin Lens Equation

A clear light bulb is placed [latex]0.750~\text{m}[/latex] in front of a converging lens with a focal length of [latex]0.500~\text{m}[/latex]. Use ray tracing to estimate the image location. Then calculate:

1. the image distance
2. the magnification using the thin lens equations.

Strategy

Because the object is farther from the lens than its focal length ([latex]d_o>f[/latex]), a real image will form on the opposite side of the lens. Ray tracing provides an estimate, while the thin lens equation gives the exact location. Given:

[latex]d_o=0.750~\text{m} \qquad f=0.500~\text{m}[/latex]

Solution

The principal rays intersect approximately [latex]1.50~\text{m}[/latex] from the lens. The image is inverted and about twice as tall as the object. Using the thin lens equation:

[latex]\frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{f}[/latex]

Solving for [latex]d_i[/latex],

[latex]\frac{1}{d_i}=\frac{1}{f}-\frac{1}{d_o}[/latex]

[latex]\frac{1}{d_i}=\frac{1}{0.500}-\frac{1}{0.750}=2.00-1.33=0.667~\text{m}^{-1}[/latex]

[latex]d_i=\frac{1}{0.667}=1.50~\text{m}[/latex]

The magnification is

[latex]m=-\frac{d_i}{d_o}=-\frac{1.50}{0.750}=-2.00[/latex]

Discussion

The negative magnification indicates that the image is inverted. The magnitude of the magnification shows that the image is twice as tall as the object. The agreement between ray tracing and the thin lens equation illustrates how the graphical and mathematical approaches complement one another.

A real image such as this is produced whenever a converging lens has a positive focal length and the object is located farther from the lens than the focal length ([latex]d_o>f[/latex]). Cameras, projectors, and the human eye all operate under these conditions. A different situation occurs when the object is moved closer to the lens than its focal length. This is how a magnifying glass works. Instead of producing a real image on the opposite side of the lens, the refracted rays continue to diverge. However, when extended backward, they appear to originate from a point on the same side of the lens as the object.

This second case is illustrated in Figure 69.11. The refracted rays never actually meet, but they appear to come from a common point on the object's side of the lens.

Images Formed by Diverging Lenses

A diverging lens forms a virtual, upright, and reduced image for any position of a real object. Looking through eyeglasses designed to correct nearsightedness provides a familiar example. Objects viewed through these lenses appear smaller but remain upright, as shown in Figure 69.12.

The ray tracing for a diverging lens is illustrated in Figure 69.13. A ray entering parallel to the principal axis is refracted so that it appears to originate from the focal point on the object's side of the lens. A second ray passes through the center of the lens with negligible deflection. The refracted rays do not actually meet. However, their backward extensions intersect on the same side of the lens as the object, locating the virtual image. Because the image is upright and smaller than the object, its magnification is positive but less than 1.

Images Formed by Thin Lenses

Table 69.1 summarizes the three types of images produced by thin lenses.

Only real images can be projected onto a screen because the light rays physically converge at the image location. Virtual images cannot be projected directly; they can be seen only by looking through the lens or by using an additional optical system that forms a real image.

In the next section, we will see that mirrors produce the same three categories of images as thin lenses. The same sign conventions and many of the same mathematical relationships apply to spherical mirrors.