16.3 Lenses

Lenses and Image Formation

Lenses bend (refract) light to form images. The two main types, convex (converging) and concave (diverging), behave differently and show up in different applications. Understanding how to trace rays through a lens and apply the thin-lens equation will let you predict where an image forms, how big it is, and whether it's real or virtual.

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Ray Diagrams for Lens Image Formation

Convex (Converging) Lenses

Convex lenses are thicker in the middle and bend incoming light rays inward toward a focal point. The type of image they produce depends on where the object sits relative to the focal point:

• Object beyond the focal point: Light rays actually converge on the opposite side of the lens, forming a real, inverted image. This is how projectors and cameras work.
• Object inside the focal point: Light rays diverge after passing through the lens, so your eye traces them back to a virtual, upright, magnified image on the same side as the object. This is how a magnifying glass works.

The closer the object gets to the focal point (from outside it), the larger the image becomes. Applications include magnifying glasses, telescopes, microscopes, and corrective lenses for farsightedness (hyperopia).

Concave (Diverging) Lenses

Concave lenses are thinner in the middle and spread light rays apart. No matter where you place the object, a concave lens always produces a virtual, upright, reduced image. The image appears on the same side of the lens as the object.

Primary applications include corrective lenses for nearsightedness (myopia) and as components in compound optical systems to control aberrations.

Drawing the Three Principal Rays

Ray diagrams are the graphical method for finding where a lens forms an image. For a thin lens, you draw three rays from the top of the object:

1. Parallel ray: Travels parallel to the optical axis, then refracts through the focal point on the opposite side of the lens (for convex) or appears to diverge from the focal point on the same side (for concave).
2. Central ray: Passes straight through the center of the lens with no deviation.
3. Focal ray: Passes through the focal point on the incident side, then refracts parallel to the optical axis on the opposite side.

The point where these rays (or their extensions) intersect is the image location. From that intersection you can read off the image's position, size, and orientation.

In thin-lens approximations, all refraction is treated as happening at a single principal plane at the center of the lens, which simplifies the diagram considerably.

Geometric Optics of the Human Eye

The human eye is itself a lens system. Here's how its parts work together:

• The cornea does most of the bending, providing roughly two-thirds of the eye's total focusing power.
• The lens (behind the iris) fine-tunes focus by changing shape, a process called accommodation. It flattens for distant objects and bulges for close ones.
• The pupil controls how much light enters, similar to an aperture on a camera.
• The retina at the back of the eye receives the focused image and converts it to electrical signals sent to the brain. The fovea, a small central region of the retina, has the highest concentration of cone cells and provides the sharpest vision.

Two reference distances matter for vision problems:

• Near point: The closest distance at which the eye can focus clearly, about 25 cm for a young adult with normal vision.
• Far point: The farthest distance for clear focus, which is effectively infinity for a normal eye.

Thin-Lens Equation Applications

The thin-lens equation connects three quantities: object distance ($d_o$), image distance ($d_i$), and focal length ($f$):

$\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}$

The magnification ($M$) tells you the image size relative to the object and whether the image is upright or inverted:

$M = \frac{h_i}{h_o} = -\frac{d_i}{d_o}$

• If $M$ is negative, the image is inverted.
• If $M$ is positive, the image is upright.
• If $|M| > 1$, the image is larger than the object; if $|M| < 1$, it's smaller.

Sign Conventions for Lenses

Getting signs right is half the battle on lens problems. Here are the rules:

• Focal length: Positive for convex lenses, negative for concave lenses.
• Object distance ($d_o$): Positive when the object is on the incoming-light side of the lens (this is almost always the case).
• Image distance ($d_i$): Positive when the image forms on the opposite side of the lens from the object (real image); negative when the image forms on the same side as the object (virtual image).

Solving Lens Problems Step by Step

1. Sketch the setup. Draw the lens, mark the focal points, and place the object.
2. List knowns and unknowns. Identify $d_o$, $d_i$, $f$, and which you need to find.
3. Assign signs using the conventions above.
4. Plug into the thin-lens equation and solve for the unknown.
5. Find magnification if the problem asks about image size or orientation.
6. Interpret your answer. A negative $d_i$ means virtual image; a negative $M$ means inverted.

Example: An object is placed 30 cm from a convex lens with $f = 10$ cm. Find the image distance and magnification.

$\frac{1}{30} + \frac{1}{d_i} = \frac{1}{10}$

$\frac{1}{d_i} = \frac{1}{10} - \frac{1}{30} = \frac{3-1}{30} = \frac{2}{30} = \frac{1}{15}$

$d_i = +15 \text{ cm}$

The positive value confirms a real image on the opposite side of the lens.

$M = -\frac{d_i}{d_o} = -\frac{15}{30} = -0.5$

The image is inverted (negative sign) and half the size of the object ($|M| = 0.5$).

Lens Characteristics and Measurements

• Index of refraction ($n$): A measure of how much a material slows light compared to vacuum. A higher $n$ means more bending. Glass typically has $n \approx 1.5$, which is why glass lenses refract light effectively.
• Diopters (D): The unit for lens power, defined as the reciprocal of the focal length in meters:

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

A lens with $f = 0.25$ m has a power of $+4$ D. Positive diopters indicate converging lenses; negative diopters indicate diverging lenses. Optometrists use diopters when writing prescriptions.

• Lens aberrations: Real lenses don't produce perfect images. Two common types are spherical aberration (rays far from the center focus at a slightly different point than rays near the center) and chromatic aberration (different wavelengths of light focus at slightly different points because $n$ varies with wavelength). Optical designers combine multiple lens elements to minimize these effects.
• The lensmaker's equation relates a lens's focal length to its index of refraction and the curvatures of its two surfaces. You likely won't need to derive it in this course, but it explains why changing the shape or material of a lens changes its focal length.