What is Focal Length? Equation & Calculator Guide

What is Focal Length? Complete Equation & Calculator Guide

Master the focal length equation and understand what the focal length of a lens means for optical design. Interactive calculators with technically accurate formulas for EFL, BFL, and FOV.

Instant FOV Calculations
2 Calculators Interactive Tools
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?

Understanding what is focal length requires examining its relationship to field of view. 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
Valid only for distortion < 2% and non-fisheye lenses

Focal Length Calculator – Find Required EFL

Required Focal Length: 5.54 mm

Critical Distinction: EFL vs BFL

Effective Focal Length (EFL): Optical distance from rear principal plane to focal plane (determines FOV)
Back Focal Length (BFL): Mechanical distance from rear lens vertex to focal plane (determines clearance)
Flange Focal Distance: Mount reference to sensor (17.526mm for C-mount, variable for M12)

Magnification Calculator

Magnification Ratio: 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

For measurement applications, telecentric lenses maintain constant magnification regardless of object distance within the depth of field. 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 = 1/(2×NA) for NA << 1 Where D = entrance pupil diameter, NA = numerical aperture
Determines depth of field and diffraction limit

What is Focal Length's Impact on System Design?

Understanding what is focal length of lens systems requires considering:

  • Depth of Field: DOF ≈ 2Nc(u²/f²) where N = f-number, c = circle of confusion
  • 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 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's the difference between effective and back focal length?

Effective focal length (EFL) is an optical parameter measured from the rear principal plane to the focal point. Back focal length (BFL) is a mechanical parameter measured from the rear lens vertex to the focal point. In telephoto designs BFL < EFL; in retrofocus designs BFL > EFL.