Optical Physics - Lens Makers Formula

Lens Makers Formula

Lens Maker’s formula is a relation that connects focal length of a lens to radii of curvature of the two surfaces of the lens and refractive index of the material of the lens.

It is useful to design lenses of desired focal length using suitable material and surfaces of suitable radii of curvature.

In deriving this formula, we use New Cartesian Sign Conventions:

1. All distances are measured from the optical centre of the lens.

2. All the distances measured in the direction of the incidence of light are taken as positive, whereas all the distances measured in a direction opposite to the direction of incidence of light are taken as negative.

3. For a convex lens, ƒ is positive and for a concave lens, ƒ is negative.

The assumptions made in the derivation are:

1. The lens is thin so that distances measured from the poles of its surfaces can be taken as equal to the distances from the optical centre of the lens.

2. The aperture of the lens is small.

3. The object consists only of a point lying on the principle axis of the lens.

4. The incident ray and refracted ray make small angles with the principle axis of the lens.

(a) Convex lens: a convex lens is made up of two convex spherical refracting surfaces. The final image is formed after two refractions. Consider a point object O lying on the principle axis of the lens. A ray of light starting from O and incident normally on the surface XP₁Y along OP₁ passes straight. Another ray incident on XP₁Y along OA is refracted along AB. If the lens material were continuous and there were no boundary/second surface XP₂Y of the lens, the refracted ray AB would go straight meeting the first refracted ray at I₁ would have been a real image of O formed after refraction at XP₁Y.

_If CI_{1 ≈} P₁I₁ = v₁

And CC₁ ≈ P₁C₁ = R₁;

CO ≈ P₁O = u, 

Then from the above equation, 

_µ 1/-u + µ₂/v1 = (µ₂ –  _µ1)/R1                                                       (1)

Actually, the lens material is not continuous. Therefore, the refracted ray AB suffers further refraction at B and emerges along BI, meeting actually the principle axis at I. therefore, I is the final real image of O, formed after refraction through the convex lens.

For refraction at the second surface XP₁Y, we can regard I₁ as a virtual object, whose real image is formed at I.

Therefore, c = CI₁ ≈ P₂I₁ = v₁, 

Let CI ≈ P₂I = v

Let R₂ be radius of curvature of second surface of the lens.

As refraction is now taking place from denser to rarer medium, therefore, using eqn. (1), we get

- µ₂/v₁ +  µ ₁/v = ( µ ₁ –  µ ₂)/R₂ = µ ₂ –  µ ₁/-R₂                                         (2)

Adding, (1) and (2), we get

µ ₁/-u +  µ ₁/v = ( µ ₂ –  µ ₁) (1/R₁ – 1/R₂)

Or, µ₁ [1/v – 1/u] = (µ₂ –  µ ₁) (1/R₁ – 1/R₂)

Or, (1/v – 1/u) = (µ₂/µ ₁ – 1) (1/R₁ – 1/R₂)                                         (3)

Put   µ₁/µ₂ =  µ  = refractive index of material of the lens w.r.t. surrounding medium.

When object on the left of lens is at ∞, image is formed at the principle focus of the lens

∴ When u = ∞, v = ƒ = focal length of the lens

From (3), 1/ƒ – 1/∞ = ( µ  – 1) (1/R1 – 1/R₂)

Or, 1/ƒ = ( µ  – 1) (1/R1 – 1/R₂)                                                          (4)

This is the lens maker’s formula. 

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