Focal Distance Formula

Q: What is the focal distance formula?

The focal distance formula is the same as the focal length formula: 1/f = (n-1)(1/R₁ - 1/R₂). Focal distance and focal length both refer to the distance from the lens center to the focal point. The term "focal distance" is sometimes used in older texts or different regions.

Q: What is the difference between focal distance and focal length?

In most contexts, focal distance and focal length are the same thing — the distance from the optical center of the lens to the focal point. However, in some advanced optics, "focal distance" may refer to the distance from the principal plane to the focal point, which differs from the back focal length in thick lenses.

Q: How do you calculate focal distance?

Calculate focal distance using 1/f = (n-1)(1/R₁ - 1/R₂), where n is the refractive index and R₁, R₂ are the radii of curvature. For thin lenses, the focal distance equals |f|. For thick lenses, distinguish between front focal distance (FFD) and back focal distance (BFD).

Master the focal distance formula and learn how to calculate focal distance for any lens. Understand the relationship between focal distance and focal length, apply the lens maker equation, and use our free focal distance calculator below.

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What is Focal Distance?

Focal distance is the distance from the optical center of a lens (or the principal plane for thick lenses) to the focal point where parallel rays of light converge—or appear to diverge from—after passing through the lens.

For thin lenses, focal distance and focal length are identical. Both describe how far the lens focuses light. The focal distance determines a lens's power: shorter focal distance means stronger focusing ability. Understanding the focal distance formula is essential for lens design, photography, microscopy, and any optical system.

Focal Distance vs Focal Length

The terms "focal distance" and "focal length" are often used interchangeably, but there are subtle distinctions depending on context.

Thin Lenses

For thin lenses (thickness negligible), focal distance equals focal length. The distance from the lens center to either focal point is |f|. Both converge to the same value given by the lens maker formula.

Thick Lenses

For thick lenses, focal distance can mean different things: the front focal distance (FFD) from the first surface to the object-side focal point, or the back focal distance (BFD) from the last surface to the image-side focal point. These differ when the lens has significant thickness.

The Focal Distance Formula

The focal distance formula is identical to the lens maker formula. For a thin lens in air:

1/f = (n − 1)(1/R₁ − 1/R₂)

Variables

f = focal distance (or focal length), meters
n = refractive index of lens material
R₁ = radius of curvature of first surface (m)
R₂ = radius of curvature of second surface (m)

Sign Convention

R > 0 for convex (center of curvature to the right of the surface); R < 0 for concave. Focal distance f > 0 for converging lenses; f < 0 for diverging.

For Thick Lenses: Front and Back Focal Distance

When the lens thickness t is significant, the effective focal length (EFL) is still given by a modified lens maker formula, but the physical focal distances from the lens surfaces differ.

Back Focal Distance (BFD)

BFD = f − (n − 1) × t × f / (n × R₁)

Distance from the rear surface of the lens to the image-side focal point.

Front Focal Distance (FFD)

FFD = f + (n − 1) × t × f / (n × R₂)

Distance from the front surface to the object-side focal point.

Use our Thick Lens Calculator →

Worked Examples: Calculating Focal Distance

Example 1: Biconvex Lens

Find the focal distance for a glass lens: n = 1.5, R₁ = 0.1 m, R₂ = −0.1 m.

1/f = (1.5 − 1)(1/0.1 − 1/(−0.1))

1/f = 0.5 × (10 + 10) = 10

f = 0.1 m (focal distance = 10 cm)

Example 2: Plano-Convex Lens

n = 1.6, R₁ = 0.25 m, R₂ = ∞ (flat second surface).

1/f = (1.6 − 1)(1/0.25 − 0) = 0.6 × 4 = 2.4

f ≈ 0.417 m (focal distance ≈ 42 cm)

Example 3: Biconcave Diverging Lens

n = 1.52, R₁ = −0.12 m, R₂ = 0.18 m.

1/f = (1.52 − 1)(1/(−0.12) − 1/0.18) ≈ 0.52 × (−13.89) ≈ −7.22

f ≈ −0.139 m (negative focal distance = diverging)

Focal Distance Calculator

Use the calculator below to find the focal distance of any thin lens. Enter the refractive index n and radii R₁, R₂ to compute the focal distance instantly.

Lens Maker Formula

Calculate focal length from lens parameters

Formula

1/f = (n-1)(1/R₁ - 1/R₂)

Refractive Index (n)

Typical: 1.5 (glass), 1.33 (water), 1.52 (crown glass)

Radius of Curvature - First Surface (R₁)

meters

Positive for convex, negative for concave

Radius of Curvature - Second Surface (R₂)

meters

Positive for convex, negative for concave

Frequently Asked Questions About Focal Distance

What is the focal distance formula?

The focal distance formula is 1/f = (n−1)(1/R₁ − 1/R₂), the same as the lens maker formula. It gives the focal distance (or focal length) of a thin lens from its refractive index n and radii of curvature R₁ and R₂.

What is the difference between focal distance and focal length?

For thin lenses, focal distance and focal length are the same—both are |f| from the lens center to the focal point. For thick lenses, focal distance can refer to the front focal distance (FFD) or back focal distance (BFD), which differ from each other when thickness is significant.

How do you calculate focal distance?

Use 1/f = (n−1)(1/R₁ − 1/R₂). Solve for f: f = 1/[(n−1)(1/R₁ − 1/R₂)]. Ensure consistent units (meters) and follow the sign convention: convex surfaces have R > 0, concave have R < 0.

When does focal distance differ from focal length?

In thick lenses, the effective focal length (EFL) is defined by the lens maker formula with a thickness term. The front focal distance (FFD) and back focal distance (BFD) measure from the physical lens surfaces to the focal points and can differ from the EFL and from each other.

What units are used for focal distance?

Focal distance is typically expressed in meters. When f is in meters, the optical power P = 1/f is in diopters. Use consistent units for R₁ and R₂ (meters) when calculating focal distance.