Focal Length
Part of Optics
Understanding focal length — the fundamental parameter governing lens behavior — and how it determines magnification, image distance, field of view, and practical design decisions in optical instruments.
Why This Matters
Focal length is the single most important number describing a lens. It determines how much a lens bends light, how far away it forms images, how much it magnifies, and what field of view it captures. Understanding focal length is the entry point to all lens design and optical instrument construction.
A lens maker who does not understand focal length cannot determine what curve to grind into a lens blank or why. An instrument builder who does not understand focal length cannot calculate the correct spacing between components or predict the performance of a device before building it. This is foundational knowledge.
The concept is simple in principle but rich in consequences: once you understand focal length, the design of magnifying glasses, telescopes, microscopes, and spectacles follows logically.
What Focal Length Is
When parallel light rays (such as sunlight from a very distant source) pass through a converging lens, they are bent toward a common point. This point of convergence is the focal point. The distance from the lens (specifically from its principal plane) to the focal point is the focal length, denoted f.
Short focal length: The lens bends light strongly, converging it quickly. The focal point is close to the lens. Short focal length lenses have higher curvature and are optically “stronger.”
Long focal length: The lens bends light weakly, converging it at a greater distance. Long focal length lenses have flatter curves and are optically “weaker.”
Diverging lenses have a negative focal length — they bend parallel rays outward rather than toward a focal point. The focal point for a diverging lens is virtual (on the same side as the incoming light).
Measuring Focal Length
The simplest method requires only a sunny day:
- Hold the lens in sunlight
- Hold a piece of paper or white card behind the lens
- Move the paper toward and away from the lens until the sun’s image on the paper is at its smallest and sharpest
- Measure the distance from the paper to the lens
That distance is the focal length. This works because sunlight arrives as essentially parallel rays (the sun is far enough away to approximate infinity for this purpose).
More precise method: Use a distant but controlled source — a candle flame or lamp 10+ meters away. Image it on a card while measuring the image distance. Use the thin lens equation:
1/f = 1/d_object + 1/d_image
With a very distant object (d_object >> f), d_image ≈ f, confirming the sunlight method.
Testing a diverging lens: Diverging lenses do not form real focused images of distant objects. Their focal length must be measured by combining them with a known converging lens and measuring the combined focal length, then calculating.
The Thin Lens Equation
For a thin lens (lens thickness small compared to focal length):
1/f = 1/do + 1/di
Where:
- f = focal length
- do = object distance (from lens to object)
- di = image distance (from lens to image)
Conventions: distances are positive when on the expected side (object in front, real image behind), negative for virtual images.
Magnification of an image: m = -di/do (negative indicates image inversion)
If m = -5, the image is 5 times larger than the object and inverted.
Example: An object 200 mm from a lens with 50 mm focal length: 1/50 = 1/200 + 1/di 1/di = 1/50 - 1/200 = 4/200 - 1/200 = 3/200 di = 66.7 mm m = -66.7/200 = -0.33 (image is 1/3 the size of the object, inverted)
Focal Length and Magnification
Magnifying glass (simple lens used as loupe): When an object is held inside the focal length, the lens acts as a magnifying glass. The apparent magnification depends on the ratio of the near point (the closest comfortable viewing distance, approximately 250 mm for a normal eye) to the focal length:
M = 250/f + 1 (approximately)
So a lens with 25 mm focal length provides approximately 11x magnification; a 50 mm lens provides approximately 6x.
Telescope magnification: M = f_objective / f_eyepiece
An objective with 1000 mm focal length and an eyepiece with 20 mm focal length: M = 50x.
Microscope magnification: The objective creates a magnified intermediate image. The magnification of the objective in a standard 160 mm tube length microscope: M_objective ≈ 160/f_objective
For a 4 mm focal length objective: M = 40x. Combined with a 10x eyepiece: total = 400x.
Field of View and Focal Length
Field of view (the angular width of the scene visible through an instrument) is inversely related to magnification — and therefore directly related to focal length ratio.
Short focal length objective = high magnification = narrow field of view Long focal length objective = low magnification = wide field of view
For a telescope: Field of view (degrees) ≈ eyepiece field stop diameter (mm) / f_objective × 57.3
This trade-off is fundamental: you cannot have both high magnification and wide field of view simultaneously. High-power observations require first finding the object at low power (wide field), then switching to higher power (narrow field).
Depth of Field and Focal Length
Depth of field (the range of distances over which objects appear acceptably sharp) decreases as focal length decreases (for the same aperture/f-ratio) and as magnification increases.
At low magnification (long focal length), a lens can focus on objects over a wide range of distances simultaneously. At high magnification (short focal length), focus is extremely critical — small changes in object or image distance cause blurring. This is why fine-focus mechanisms are essential on microscopes but less critical on simple magnifying glasses.
f-Number and Light Gathering
The f-number (focal ratio) = f/D where D is the lens aperture (diameter).
- f/4 lens: aperture is 1/4 of focal length; collects 4x more light than f/8 for the same focal length
- Lower f-number = “faster” lens = more light = brighter image = shorter exposure for cameras
- Lower f-number = larger aperture = less depth of field = more prominent aberrations at the edges
For visual instruments (telescopes, microscopes), aperture determines resolution (ability to see fine detail). For cameras and projection systems, aperture determines exposure and depth of field.
Practical Implications for Instrument Design
| Instrument | Objective f | Eyepiece f | f-ratio | Notes |
|---|---|---|---|---|
| Magnifying glass | 25-50 mm | — | — | Used at object-to-focal-length distance |
| Refracting telescope | 500-2000 mm | 10-25 mm | f/8-f/15 | Long f reduces chromatic aberration |
| Reflecting telescope | 500-2000 mm | 10-25 mm | f/6-f/12 | No chromatic aberration; faster OK |
| Low-power microscope | 25-40 mm | 25 mm | — | 100-200 mm tube |
| High-power microscope | 4-8 mm | 25 mm | — | 160 mm tube standard |
| Spectacles | 100-500 mm | — | — | Depends on prescription needed |
Lens Maker’s Equation
For a lens maker grinding glass to a specific focal length:
1/f = (n-1) × [1/R1 - 1/R2]
Where:
- n = refractive index of the glass (typically 1.5 for crown glass)
- R1 = radius of curvature of front surface (positive if center of curvature is to the right)
- R2 = radius of curvature of rear surface
For a plano-convex lens (one flat surface, one curved):
- R2 = infinity (flat) → 1/R2 = 0
- 1/f = (n-1)/R1
- R1 = f × (n-1) = f × 0.5 (for n=1.5)
So a 50 mm focal length plano-convex lens in glass with n=1.5 needs a front surface radius of curvature of 25 mm. This is the key relationship that converts “desired focal length” into “grinding radius for the surface.” A spherometer or radius gauge calibrated to this value tells the grinder when they have achieved the correct curve.
Understanding focal length, combined with the lens maker’s equation, transforms lens grinding from an art of trial and error into a calculable engineering process.