9.5 Lenses

Lenses are optical elements that bend light to form images. They show up everywhere, from your eyes to cameras to telescopes, and understanding how they work gives you the tools to predict where images form, how big they'll be, and whether they're real or virtual.

This guide covers lens types, key properties, ray diagrams, the thin lens equation, aberrations, lens combinations, and applications.

Types of Lenses

Converging vs. Diverging Lenses

These are the two fundamental categories, and the distinction comes down to what happens when parallel light rays enter the lens.

• Converging lenses bend parallel rays inward so they meet at a single point (the focal point). They have a positive focal length. Common shapes include biconvex and plano-convex.
• Diverging lenses spread parallel rays outward so they appear to come from a focal point behind the lens. They have a negative focal length. Common shapes include biconcave and plano-concave.

The shape of the lens surfaces determines which type it is. If the lens is thicker in the middle, it converges. If it's thinner in the middle, it diverges.

Spherical vs. Aspherical Lenses

• Spherical lenses have surfaces that are sections of a sphere (constant radius of curvature). They're easier and cheaper to manufacture, but they suffer from spherical aberration, where rays hitting the edge of the lens don't focus at the same point as rays near the center.
• Aspherical lenses have non-spherical surface profiles designed to reduce these aberrations. They produce sharper images and are used in high-performance systems like camera lenses and telescopes.

Simple vs. Compound Lenses

A simple lens is a single piece of glass or plastic. It works fine for basic tasks, but it introduces various aberrations that degrade image quality.

A compound lens combines multiple simple lenses to cancel out those aberrations. Microscope objectives, camera zoom lenses, and telescope eyepieces are all compound lens systems.

Lens Properties

Focal Length

The focal length ($f$) is the distance from the center of the lens to the focal point, where parallel rays converge (for a converging lens) or appear to diverge from (for a diverging lens). It's measured in meters.

A shorter focal length means a stronger lens that bends light more sharply. Lens strength is often expressed as power, defined as:

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

Power is measured in diopters (D). A lens with $f = 0.5 \text{ m}$ has a power of $+2 \text{ D}$.

Optical Center

The optical center is the point on the lens where light passes straight through without changing direction. For a symmetrical lens, it sits right at the geometric center. It serves as the reference point for all distance measurements in lens calculations.

Principal Axis

The principal axis is an imaginary line that passes through the optical center and is perpendicular to the lens surface. All the key reference points (focal points, center of curvature) lie along this line. Rays parallel to the principal axis converge at the focal point after passing through the lens.

Radius of Curvature

The radius of curvature describes how sharply each lens surface is curved. A smaller radius means a more steeply curved surface, which produces a shorter focal length and a stronger lens.

The relationship between surface curvature and focal length is given by the lensmaker's equation:

$\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$

• $n$ is the refractive index of the lens material
• $R_1$ and $R_2$ are the radii of curvature of the front and back surfaces

This equation shows that both the material (through $n$) and the shape (through $R_1$ and $R_2$) determine the focal length.

Ray Diagrams

Ray diagrams are your main visual tool for figuring out where an image forms and what it looks like. You trace specific rays from the top of the object through the lens, and where they intersect tells you the image location.

Rules for Ray Tracing

For any thin lens, draw at least two of these three principal rays:

1. Parallel ray: Travels parallel to the principal axis, then refracts through the focal point on the other side (for converging) or appears to come from the focal point on the same side (for diverging).
2. Central ray: Passes straight through the optical center with no bending.
3. Focal ray: Passes through the focal point on the object's side, then emerges parallel to the principal axis.

Where any two of these rays intersect, that's where the image forms. Use arrowheads on your rays to show the direction light travels.

Real vs. Virtual Images

• Real images form where light rays actually converge. You can project them onto a screen. Converging lenses produce real images when the object is beyond the focal point.
• Virtual images form where rays only appear to diverge from. You can't project them onto a screen. Diverging lenses always produce virtual images. Converging lenses also produce virtual images when the object is inside the focal point (this is how a magnifying glass works).

Upright vs. Inverted Images

• Upright images have the same orientation as the object. Diverging lenses always produce upright images, and converging lenses do too when the object is inside the focal point.
• Inverted images are flipped upside-down. Converging lenses produce inverted images when the object is beyond the focal point.

You can determine orientation directly from your ray diagram or from the sign of the magnification (covered below).

Thin Lens Equation

This is the core equation for quantitative lens problems. It connects three quantities: focal length, object distance, and image distance.

The Equation and How to Use It

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

• $f$ = focal length
• $d_o$ = object distance (from the lens to the object)
• $d_i$ = image distance (from the lens to the image)

If you know any two of these, you can solve for the third. This equation works for both converging and diverging lenses, as long as you follow the sign conventions.

Example: An object is placed 30 cm from a converging lens with $f = 10$ cm. Where does the image form?

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

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

$d_i = 15 \text{ cm}$

The positive value tells you it's a real image, formed 15 cm on the other side of the lens.

Sign Conventions

Getting signs right is critical. Here are the rules:

• Focal length: Positive for converging lenses, negative for diverging lenses
• Object distance ($d_o$): Always positive (for single-lens problems with real objects)
• Image distance ($d_i$): Positive for real images (opposite side from object), negative for virtual images (same side as object)
• Heights: Positive above the principal axis, negative below

Magnification Formula

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

• $M$ = magnification
• $h_i$ = image height
• $h_o$ = object height

What the value of $M$ tells you:

• Negative $M$ → inverted image
• Positive $M$ → upright image
• $|M| > 1$ → image is larger than the object (enlarged)
• $|M| < 1$ → image is smaller than the object (reduced)

Lens Aberrations

Real lenses don't produce perfect images. Aberrations are the imperfections that cause blurring, distortion, or color fringing.

Spherical Aberration

Rays passing through the outer edges of a spherical lens focus at a slightly different point than rays near the center. The result is a blurred image. This effect is more pronounced with large-aperture lenses or short focal lengths. Fixes include using aspherical lens surfaces or combining multiple lenses.

Chromatic Aberration

Because the refractive index of glass varies with wavelength (dispersion), different colors of light focus at slightly different points. This produces color fringing, especially at high-contrast edges. It's corrected using achromatic doublets, which pair a converging lens made of crown glass with a diverging lens made of flint glass to bring two wavelengths to the same focus.

Astigmatism

This occurs when the lens curvature differs across different orientations (meridians), so vertical and horizontal lines can't both be in focus at the same time. Off-axis image points appear blurred or distorted. Cylindrical or aspherical lenses correct for this. In the human eye, astigmatism is one of the most common vision problems and is corrected with prescription lenses.

Lens Combinations

Lens Systems in Series

When multiple lenses are arranged along the same optical axis, the image formed by the first lens becomes the object for the second lens. You apply the thin lens equation to each lens in sequence. This approach gives you much greater control over magnification and aberration correction, and it's how compound microscopes, telescopes, and zoom lenses work.

Effective Focal Length

For two thin lenses in contact (touching each other), the combined focal length is:

$\frac{1}{f_{eff}} = \frac{1}{f_1} + \frac{1}{f_2}$

If the lenses are separated by a distance $d$, the formula becomes more complex, but for this course the in-contact case is the one to know.

Power of a Lens System

Since power is $P = \frac{1}{f}$, the total power of lenses in contact is simply the sum of their individual powers:

$P_{total} = P_1 + P_2$

This additive property makes power a convenient quantity in optometry, where prescriptions are written in diopters.

Lens Applications

The Human Eye

Your eye is a natural lens system. The cornea provides most of the refractive power (about 2/3), while the crystalline lens behind it fine-tunes the focus by changing shape, a process called accommodation.

Common vision problems:

• Myopia (nearsightedness): The image focuses in front of the retina. Corrected with a diverging lens.
• Hyperopia (farsightedness): The image focuses behind the retina. Corrected with a converging lens.
• Astigmatism: Uneven corneal curvature causes blurred vision. Corrected with cylindrical lenses.

Cameras

Cameras use lens systems to focus light onto a sensor or film. The aperture controls how much light enters and affects depth of field (how much of the scene is in focus). Zoom lenses adjust the effective focal length to change the field of view without moving the camera.

Microscopes

A compound microscope uses two lens stages. The objective lens (close to the specimen) forms a magnified real image inside the tube. The eyepiece then acts like a magnifying glass to further enlarge that image. Immersion oil is sometimes placed between the specimen and objective to increase the numerical aperture and improve resolution.

Telescopes

Refracting telescopes use a large objective lens to gather light and form a real image at its focal plane. The eyepiece magnifies that image for viewing. A larger objective lens collects more light, which is why bigger telescopes can reveal fainter objects.

Lens Manufacturing

Materials

• Optical glass (crown and flint types) is the standard for precision lenses. Crown glass has lower dispersion; flint glass has higher dispersion. Pairing them corrects chromatic aberration.
• Plastics are used for lightweight, low-cost lenses (eyeglasses, disposable cameras).
• Crystalline materials like quartz and fluorite serve specialized applications, such as UV optics.

Material choice depends on the required refractive index, dispersion, durability, and cost.

Grinding and Polishing

Lens fabrication follows a sequence:

1. Rough grinding shapes the blank to approximate curvature.
2. Fine grinding refines the surface closer to the final shape.
3. Polishing brings the surface to optical-quality smoothness.
4. Verification using interferometry confirms the surface accuracy meets specifications.

Computer-controlled machines handle most of this process for high-precision lenses.

Coating Processes

• Anti-reflection coatings reduce light loss and glare at each lens surface. They're applied through vacuum deposition.
• Single-layer coatings work well for a narrow wavelength range, while multi-layer coatings provide broadband performance across visible light.
• Hydrophobic coatings repel water and make cleaning easier.

Advanced Lens Concepts

Fresnel Lenses

A Fresnel lens collapses a thick curved lens into a thin, flat sheet with concentric grooves that mimic the curvature of a conventional lens. This dramatically reduces weight and material while maintaining the same optical power. Fresnel lenses are used in lighthouses, overhead projectors, and solar concentrators. The trade-off is reduced image quality compared to a conventional lens.

Gradient-Index (GRIN) Lenses

Instead of relying on curved surfaces, a GRIN lens bends light using a refractive index that varies continuously across the lens material. This allows for flat-surfaced lenses that still focus light, and it can reduce aberrations. GRIN lenses are found in fiber optic couplers, photocopiers, and some medical endoscopes.

Adaptive Optics

Adaptive optics systems correct for optical distortions in real time. They typically use a wavefront sensor to detect distortions and a deformable mirror (or liquid crystal element) to compensate. The most well-known application is in ground-based astronomy, where adaptive optics counteract the blurring caused by atmospheric turbulence. Emerging uses include retinal imaging and advanced microscopy.