25.6 Image Formation by Lenses

Image Formation by Lenses

Lenses bend light to form images, and they show up everywhere: eyeglasses, cameras, microscopes, projectors. Understanding how a lens produces an image ties together the core ideas of refraction and geometric optics.

Ray tracing is the main technique you'll use to predict where an image forms, how big it is, and whether it's upright or inverted. By following a few simple rules for drawing light rays through a lens, you can figure out all of these characteristics without any equations.

Image Formation by Lenses

Rules for thin lens ray tracing

A thin lens is one whose thickness is small enough (compared to its diameter and focal length) that you can treat it as a flat plane for ray-tracing purposes. Three key rays, all drawn from the top of the object, are enough to locate the image:

• Parallel ray: Travels parallel to the optical axis, then refracts through the lens and passes through the focal point on the opposite side.
• Central ray: Passes straight through the center of the lens without bending at all.
• Focal ray: Passes through the focal point on the object's side of the lens, then refracts and exits parallel to the optical axis.

The image of the top of the object forms where these three rays converge (or appear to converge) on the other side of the lens. In practice, two rays are enough to find the intersection, but drawing all three serves as a built-in check.

Image formation through ray tracing

Here's the step-by-step process:

1. Draw a horizontal optical axis with the thin lens as a vertical line at the center. Mark the focal points (F) on both sides of the lens.

2. Place the object (usually drawn as an upright arrow) on the left side of the lens.

3. From the tip of the object, draw the three key rays described above:

   • The parallel ray goes horizontally to the lens, then bends to pass through F on the image side.
   • The central ray goes straight through the center of the lens, undeviated.
   • The focal ray passes through F on the object side, hits the lens, and exits parallel to the axis.

4. Find the point where the rays intersect on the image side. That point is the tip of the image.

5. Drop a perpendicular line from that intersection point down to the optical axis. This gives you the full image: its position, height, and orientation.

The image you get depends on how far the object sits from the lens. For a converging (convex) lens, the key cases are:

Notice the pattern: as the object moves closer to the focal point, the image gets larger. Once the object is inside the focal length, the rays diverge on the other side of the lens and never actually meet. Instead, they appear to come from a point on the same side as the object, producing a virtual, upright, magnified image. This is exactly what happens when you hold a magnifying glass close to a page.

Magnification ($m$) is the ratio of image height to object height. A magnification greater than 1 means the image is larger; less than 1 means it's smaller. A negative magnification means the image is inverted.

Lens power from focal length

Lens power ($P$) measures how strongly a lens converges or diverges light. It's defined as the reciprocal of the focal length in meters:

$P = \frac{1}{f}$

The unit of lens power is the diopter (D), which equals $\text{m}^{-1}$.

• A shorter focal length means a higher power and a stronger bending effect. A magnifying glass with $f = 0.10 \text{ m}$ has $P = +10 \text{ D}$.
• A longer focal length means a lower power and a gentler bending effect.

Converging lenses (convex) have positive focal lengths and positive power. Diverging lenses (concave) have negative focal lengths and negative power.

A couple of worked examples:

• A converging lens with $f = 0.25 \text{ m}$: $P = \frac{1}{0.25} = +4 \text{ D}$
• A diverging lens with $f = -0.50 \text{ m}$: $P = \frac{1}{-0.50} = -2 \text{ D}$

Optometrists use diopters to write eyeglass prescriptions. A prescription of $-3.0 \text{ D}$ means a diverging lens that corrects nearsightedness, while $+2.0 \text{ D}$ means a converging lens for farsightedness.

Lens Characteristics and Image Formation

A few additional properties affect how real lenses behave:

• Refraction is the bending of light at each surface of the lens. How much the light bends depends on the index of refraction of the lens material and the curvature of the surfaces.
• Aberration refers to imperfections in image formation. Spherical aberration occurs because rays hitting the edges of a lens focus at a slightly different point than rays near the center, causing a blurry image.
• Chromatic dispersion happens because the index of refraction varies slightly with wavelength. Different colors of light bend by different amounts, so a single lens can produce color fringing around the edges of an image. Camera lenses and telescopes use multi-element lens designs to minimize this effect.