Lens Maker Formula Derivation
A complete mathematical derivation of the lens maker formula from first principles, using Snell's Law and the refraction equation for spherical surfaces.
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We will derive the lens maker formula:
1/f = (n-1)(1/R₁ - 1/R₂)
Prerequisites
Snell's Law
n₁ sin θ₁ = n₂ sin θ₂
When light passes from a medium with refractive index n₁ to a medium with index n₂, the angles of incidence and refraction are related by this equation.
Paraxial Approximation
For small angles (paraxial rays close to the optical axis):
sin θ ≈ tan θ ≈ θ (in radians)
Single Surface Refraction Formula
n₁/s + n₂/s' = (n₂ - n₁)/R
This formula describes image formation by a single spherical refracting surface, where s is object distance, s' is image distance, and R is radius of curvature.
Key Assumptions
Thin Lens
The lens thickness is negligible compared to the radii of curvature, so both refractions occur at approximately the same point.
Paraxial Rays
Light rays travel close to and nearly parallel to the optical axis. This allows us to use small angle approximations.
Spherical Surfaces
Both lens surfaces are portions of spheres with radii R₁ and R₂.
Homogeneous Material
The lens material has a uniform refractive index n throughout.
Step-by-Step Derivation
Refraction at the First Surface
Consider light traveling from air (n₁ = 1) into the lens (n₂ = n). Applying the single surface refraction formula at the first surface:
1/s₁ + n/s₁' = (n - 1)/R₁
Where s₁ is the object distance from surface 1, and s₁' is the image distance (inside the lens) formed by surface 1 alone.
Refraction at the Second Surface
The image from surface 1 becomes the object for surface 2. Light travels from the lens (n₁ = n) back into air (n₂ = 1):
n/s₂ + 1/s₂' = (1 - n)/R₂
Where s₂ is the object distance for surface 2 (the image from surface 1), and s₂' is the final image distance.
Apply Thin Lens Approximation
For a thin lens, the thickness is negligible. The image formed by surface 1 is essentially at the same location as the object for surface 2:
s₂ ≈ -s₁'
The negative sign accounts for the sign convention: the image from surface 1 is on the opposite side from where surface 2 measures its object distance.
Add the Two Equations
Adding the equations from steps 1 and 2, and using s₂ = -s₁':
1/s₁ + n/s₁' + n/(-s₁') + 1/s₂' = (n-1)/R₁ + (1-n)/R₂
1/s₁ + n/s₁' - n/s₁' + 1/s₂' = (n-1)/R₁ - (n-1)/R₂
1/s₁ + 1/s₂' = (n-1)(1/R₁ - 1/R₂)
Notice how the middle terms (n/s₁' and -n/s₁') cancel out!
Final Result
For an object at infinity (s₁ → ∞), parallel rays focus at the focal point, so s₂' = f. Since 1/s₁ → 0:
1/f = (n-1)(1/R₁ - 1/R₂)
This is the Lens Maker Formula! ✓
Alternative Forms
Optical Power Form
P = (n-1)(1/R₁ - 1/R₂)
Where P = 1/f is the optical power in diopters (when f is in meters).
Focal Length Form
f = 1/[(n-1)(1/R₁ - 1/R₂)]
Directly gives the focal length by taking the reciprocal.
Thick Lens Extension
1/f = (n-1)[1/R₁ - 1/R₂ + (n-1)d/(nR₁R₂)]
For thick lenses where thickness d is not negligible, an additional term accounts for the separation between the two surfaces.
Physical Interpretation
The lens maker formula reveals how each factor contributes to the lens's optical power:
(n - 1) Factor
This represents the "refractive power" of the material. Higher refractive index means more bending at each surface, leading to stronger optical power (shorter focal length).
(1/R₁ - 1/R₂) Factor
This represents the "curvature power" of the lens shape. More curved surfaces (smaller R) create stronger bending. The difference accounts for both surfaces' contributions.
Frequently Asked Questions
How is the lens maker formula derived?
The lens maker formula is derived by applying the single surface refraction equation (n₁/s + n₂/s' = (n₂-n₁)/R) at each lens surface, then combining them using the thin lens approximation where the image from surface 1 becomes the object for surface 2 at essentially the same location.
What assumptions are made in the derivation?
The derivation assumes: (1) Thin lens - negligible thickness, (2) Paraxial rays - small angles allowing sin θ ≈ θ, (3) Spherical surfaces - both surfaces are portions of spheres, (4) Homogeneous material - constant refractive index throughout.
Why do the n/s₁' terms cancel in the derivation?
When we add the equations for both surfaces and apply the thin lens approximation (s₂ = -s₁'), we get +n/s₁' from surface 1 and -n/s₁' from surface 2 (since s₂ = -s₁'). These cancel, leaving only the external object and image distances.
What is the single surface refraction formula?
The single surface refraction formula is n₁/s + n₂/s' = (n₂-n₁)/R. It describes how a single spherical interface between two media with different refractive indices forms an image. This formula is derived from Snell's Law using the paraxial approximation.
How does the thick lens formula differ from the derivation?
For thick lenses, we cannot assume s₂ = -s₁' exactly. Instead, s₂ = -(s₁' - d) where d is the lens thickness. This introduces an additional term (n-1)d/(nR₁R₂) in the final formula, accounting for the separation between refraction events.