What is Focal Length? Complete Equation & Calculator Guide

Effective focal length (EFL) is the distance from a lens's rear principal plane to its focal point, and it sets how much scene a sensor sees. For a quick far-conjugate estimate, engineers use EFL ≈ working distance × sensor width ÷ FOV width. The free Commonlands calculator on this page computes it instantly.

Instant FOV Calculations
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15 min Complete Guide

What is Focal Length of a Lens? Technical Definition

Focal length is the distance from the lens's rear principal plane to the rear focal point when the lens is focused at infinity, measured in air (refractive index n=1). Together with the sensor dimension, this fundamental optical parameter sets the angular field of view. Together with object distance, it sets magnification.

Technical Precision

For thick lens systems, focal length is specifically measured from the rear principal plane H' to the rear focal point F'. For thin lenses where thickness t << f, the principal planes coincide at the lens center, simplifying to the elementary definition.

The focal length of lens systems controls two inverse relationships:

  • Longer focal length → Narrower field of view + Higher magnification
  • Shorter focal length → Wider field of view + Lower magnification

The Focal Length Equation: Gaussian Optics

The fundamental focal length equation relates object distance, image distance, and focal length through the Gaussian lens formula:

1/f = 1/u + 1/v Gaussian Lens Equation (Sign Convention: Real is Positive)
Where: f = focal length, u = object distance, v = image distance
All distances measured from principal planes

For optical engineers, the focal length equation more commonly appears in its magnification form:

m = v/u = f/(u - f) Lateral magnification (Real is Positive convention)
Gives the magnification magnitude. The real image is inverted
Valid for paraxial rays only

What is Focal Length's Role in Field of View?

Focal length and sensor size determine the ideal field of view at a distant focus. For rectilinear lenses, the angular field of view follows:

θ = 2 × arctan(d/(2f)) Field of View for Rectilinear Projection
θ = full angle FOV, d = sensor dimension, f = effective focal length
Ideal rectilinear projection at a distant focus. Distortion and finite focus distance change the result

Focal Length Calculator – Find Required EFL

Required Focal Length: 5.54 mm

EFL vs BFL vs FFL: Which Focal Length Is Which?

Three lengths share the name and cause most datasheet confusion:

Effective focal length (EFL) is the distance from the rear principal plane H′ to the rear focal point with the lens focused at infinity. It is the optical quantity that sets field of view and magnification, and the principal plane it starts from can sit in front of, inside, or behind the physical barrel. Commonlands specs every lens by its EFL.

Back focal length (BFL) is the distance from the vertex of the last glass surface to the rear focal point. It is an optical distance that helps assess clearance. Usable mechanical space also depends on the rear barrel, focus distance, and filter or sensor stack. Telephoto designs have BFL < EFL. Retrofocus designs have BFL > EFL.

Front focal length (FFL) is the distance from the vertex of the first glass surface to the front focal point. Together with EFL, BFL, and the defined vertex references, it can locate the principal planes in the same surrounding medium.

Related but distinct: flange focal distance is a mount standard (17.526mm for C-mount), not a property of the lens design.

Worked Example: CIL059 on an OX08B40 Sensor

The Commonlands CIL059 has a 5.9mm EFL, and the Omnivision OX08B40 on its compatibility list has an 8.064mm-wide active area. At a 2m working distance:

Scene width ≈ WD × w / f = 2000 × 8.064 / 5.9 ≈ 2733mm Far-conjugate estimate: about 2.7m of scene width
Angular check: θ = 2 × arctan(8.064 / (2 × 5.9)) ≈ 69° horizontal
Measured on this pairing: 71° horizontal, 79° diagonal – the lens's −4% barrel distortion maps slightly more scene onto the sensor than the zero-distortion estimate predicts

Focal-Length Ratio Calculator

This tool reports f2/f1, the ratio of angular image scales for distant objects. It approximates the magnification ratio at the same distant object distance. At finite distance, compare m = f/(u-f) for each refocused thin lens instead.

Distant-object image-scale ratio (f2/f1): 3.1×

Sensor Format Corrections

Format Diagonal (mm) Width × Height (mm) Crop Factor
1/3" 6.0 4.8 × 3.6 7.2×
1/2.3" 7.7 6.2 × 4.6 5.6×
1/2" 8.0 6.4 × 4.8 5.4×
1/1.8" 8.9 7.2 × 5.3 4.9×
2/3" 11.0 8.8 × 6.6 3.9×
1" 16.0 12.8 × 9.6 2.7×

For sensor dimensions, pixel size, and lens compatibility beyond this table, see the full Image Sensors list.

Advanced Optical Considerations

Telecentricity in Machine Vision

Object-space and bi-telecentric lenses approximately preserve magnification as object distance changes, with residual telecentricity error. Image-space telecentricity alone does not provide that property. Object-space telecentricity places the entrance pupil at infinity, while image-space telecentricity places the exit pupil at infinity, ensuring chief rays are parallel to the optical axis.

F-Number and Numerical Aperture Relationship

f/# = f/D. Working f/# ≈ 1/(2×NA) for small image-space NA in air

Here D is the entrance-pupil diameter. At infinity focus, working f-number approaches nominal f-number. At finite magnification, use the working f-number, including the lens's pupil magnification, when relating aperture to image-space NA.

Where D = entrance pupil diameter, NA = numerical aperture
Determines depth of field and diffraction limit

What is Focal Length's Impact on System Design?

When choosing focal length, also check:

  • Depth of Field: DOF ≈ 2Nc(u²/f²) at low magnification and focus distances well below hyperfocal. Here N is f-number and c is the allowed geometric blur diameter. Use near and far limit equations near hyperfocal focus
  • Hyperfocal Distance: H = f²/(Nc) + f
  • Diffraction Limit: Airy disk diameter = 2.44λ(f/#)
  • Entrance/Exit Pupil: Determines vignetting and telecentricity

Distortion Effects on FOV

The standard focal length equation assumes zero distortion. Real lenses exhibit:

  • Barrel distortion: Actual FOV > calculated (typical in wide-angle)
  • Pincushion distortion: Actual FOV < calculated (typical in telephoto)
  • Fisheye projection: Requires alternative models (equidistant, stereographic)

Technical FAQ: Focal Length of Lens Systems

What is optical EFL?

Optical EFL, or effective focal length, is the distance from a lens's rear principal plane to its rear focal point at infinity focus. It sets field of view and magnification: for a far-conjugate estimate, EFL ≈ working distance × sensor width ÷ FOV width, which the free Commonlands focal length calculator computes from your sensor and coverage numbers.

How do I calculate the focal length I need?

Start from your sensor width and the scene you must cover: EFL ≈ working distance × sensor width ÷ FOV width, a far-conjugate estimate that works for low-distortion rectilinear lenses. The Commonlands calculator on this page runs the exact angular form θ = 2×arctan(d/2f). Verify the final pairing against the lens's measured FOV, since distortion shifts real coverage.

What is focal length in optical terms?

Focal length is the distance from a lens's rear principal plane to the rear focal point when focused at infinity, measured in air (n=1). It quantifies the lens's optical power: Power(diopters) = 1/f(meters). For compound lens systems, it's the equivalent focal length that would produce the same magnification as a single thin lens.

How does the focal length equation account for thick lenses?

For thick lenses, the Gaussian equation 1/f = 1/u + 1/v still applies, but distances are measured from the principal planes H and H', not the lens surfaces. The principal plane positions depend on lens thickness, surface curvatures, and refractive index, and are found from the thick lens equation or matrix ray tracing.

What determines the focal length of lens combinations?

For two thin lenses separated by distance d: 1/f_combined = 1/f₁ + 1/f₂ - d/(f₁×f₂). When d=0 (lenses in contact), this simplifies to adding optical powers. For thick lens combinations, use matrix ray tracing methods or cardinal point calculations.

How does temperature affect focal length?

Focal length changes with temperature through three mechanisms: refractive index variation in the lens material, thermal expansion of the lens elements, and expansion of the mount. The magnitude and sign of the shift depend on whether the elements are glass or plastic and on the housing material. Critical for precision applications.

What is the difference between EFL and back focal length?

Effective focal length (EFL) is an optical parameter measured from the rear principal plane to the focal point. It sets field of view. Back focal length (BFL) is the optical distance from the rear lens vertex to the rear focal point. It helps assess clearance, but usable space also depends on the barrel, focus setting, and sensor stack. In telephoto designs BFL is shorter than EFL. In retrofocus designs BFL is longer than EFL. Commonlands product pages spec each lens by EFL.