Image Formation by Lenses

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 this figure. 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 this figure.) 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 lens’s shape, 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. This figure 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.

$The figure on the right shows a convex lens. Three rays heading from left to right, 1, 2, and 3, are considered. Ray 2 falls on the axis and rays 1 and 3 are parallel to the axis. The distance from the center of the lens to the focal point F is small f on the right side of the lens. Rays 1 and 3 after refraction converge at F on the axis. Ray 2 on the axis goes undeviated. The figure on the left shows an expanded view of refraction for ray 1. The angle of incidence is theta 1 and angle of refraction theta 2 and a dotted line is the perpendicular drawn to the surface of the lens at the point of incidence. The ray after the refraction at the second surface emerges with an angle equal to theta 1 prime with the perpendicular drawn at that point. The perpendiculars are shown as dotted lines.$

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 lens’s focal length $f$. An expanded view of the path taken by ray 1 shows the perpendiculars and the angles of incidence and refraction at both surfaces.

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.

Focal Point F

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

Focal Length $f$

The distance from the center of the lens to its focal point is called focal length $f$.

A person’s hand is holding a magnifying glass to focus the sunlight to a point. The magnifying glass focuses the sunlight to burn paper.

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

$P=\cfrac{1}{f}.$

Power $P$

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

$P=\cfrac{1}{f}\text{.}$

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, $\text{1 D}=1/\text{m}$, or ${\text{1 m}}^{-1}$. (Note that this power (optical power, actually) is not the same as power in watts defined in Work, Energy, and Energy Resources. It is a concept related to the effect of optical devices on light.) Optometrists prescribe common spectacles and contact lenses in units of diopters.

Example: What is the Power of a Common Magnifying Glass?

Suppose you take a magnifying glass out on a sunny day and you find that it concentrates sunlight to a small spot 8.00 cm away from the lens. What are the focal length and power of the lens?

Strategy

The situation here is the same as those shown in this figure and this figure. The Sun is so far away that the Sun’s rays are nearly parallel when they reach Earth. The magnifying glass is a convex (or converging) lens, focusing the nearly parallel rays of sunlight. Thus the focal length of the lens is the distance from the lens to the spot, and its power is the inverse of this distance (in m).

Solution

The focal length of the lens is the distance from the center of the lens to the spot, given to be 8.00 cm. Thus,

$f=8.00 cm.$

To find the power of the lens, we must first convert the focal length to meters; then, we substitute this value into the equation for power. This gives

$P=\cfrac{1}{f}=\cfrac{1}{0\text{.}\text{0800 m}}=\text{12}\text{.}5 D.$

Discussion

This is a relatively powerful lens. The power of a lens in diopters should not be confused with the familiar concept of power in watts. It is an unfortunate fact that the word “power” is used for two completely different concepts. If you examine a prescription for eyeglasses, you will note lens powers given in diopters. If you examine the label on a motor, you will note energy consumption rate given as a power in watts.

This figure 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, $\text{F}$, 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 this figure is 5.00 cm, then the focal length is $f=–5.00 cm$ and the power of the lens is $P=\text{–20 D}$. 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.

$The figure on the top shows an expanded view of refraction for ray 1 falling on a concave lens. The angle of incidence is theta 1 and angle of refraction theta 2. The ray after the refraction at the second surface emerges with an angle equal to theta 1 prime with the perpendicular drawn at that point. Perpendiculars are shown as dotted lines. The figure at the bottom shows a concave lens. Three rays, 1, 2, and 3, are considered. Ray 2 falls on the axis and rays 1 and 3 are parallel to the axis. Rays 1 and 3 after refraction appear to come from a point F on the axis. The distance from the center of the lens to F is small f and is measured from the same side as the incident rays. Ray 2 on the axis goes undeviated.$

Rays of light entering a diverging lens parallel to its axis are diverged, and all appear to originate at its focal point $\text{F}$. 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.

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 in 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 this figure and this figure. For example, if a point light source is placed at the focal point of a convex lens, as shown in this figure, parallel light rays emerge from the other side.

$Three light rays coming from a light bulb filament are incident on a convex lens and the rays after refraction are rendered parallel.$

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 this figure. 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.

Image Formation by Lenses