Optical Physics - Lens

Lens

The part of an isotropic transparent medium bounded by at least one curved surface. Lenses are of two types (a) convex (b) concave

Image formation information for convex lens and concave mirrors

Position of object	Position of image and its nature
At ∞	At focus (real, inverted and diminished)
Away from 2f (or C)	Between f and 2f (real, inverted and diminished)
At 2f	At 2f (real, inverted and equal in size)
Between f and 2f	Away from 2f (real inverted and magnified)
At f	At ∞ (real, inverted and magnified)
Between pole and f	Behind the mirror (virtual, erect and magnified) (in front of lens on the side of object

Lens formulae for thin lenses 

I/f = (μ - 1) [1/R₁ – 1/R₂]    

Lens makers formula when surrounding medium has refractive index μm is

1/f = (μL/μm – 1) [1/R₁ – 1/R₂]

(1/v) – (1/u) = 1/f

F = D₂ – d₂/4, D displacement method and O = I₁ I₂

Lateral magnification M lat = v/ u = I/O for a convex lens

M _lat = - v/u = I/O for a concave lens

M _lat = f/u + f and M lat = f – v/f 

Axial magnification M _axial = - v₂/u₂

If the medium on two sides are different then for focal length use,

μ₃/f = [( μ₂ – μ₁)/R₁] – [(μ ₂ – μ₃/R₂]

To find v use v μ₃/v – μ₁/u = μ₂ – μ₁/R₁ – μ₂ – μ₃/R₂

A lens has two principal foci

I/f = l / f₁ + 1 / f₂ when two lenses are in contact.

Newton’s formula x₁ x₂ = f₂ x₁ x₂ = f₁ f₂ 

When there is separation d between the lenses

1/f = 1/f₁ + 1/f₂ – d/f₁ f₂    

1/f = (μ - 1) [1/R₁ + 1/R₂ – t (μ – 1)/μ R₁ R₂] for a thick lens a where t is the thickness of the lens.

There are 3 sets of cardinal points in thick lens.
    
Set of focal points
    
Principal points x₁ x₂ = f₂ (if the lens system is in air)
    
Nodal points

Flare spots if strong light is used then more than one refraction occurs

1/fn = (n + 1) μ - 1/f(μ - i) for nth flare spot.

Power of the lens p = 1/f (m) = 100 f (cm) the unit is dioptre (D).

Defects in lenses
    
Spherical aberration: - or monochromatic aberration is removed using optical stops or cylindrical lens (astigmatism) or aplanatic lens.

Spherical aberration occurs as paraxial and marginal rays do not meet at a point
    
Chromatic aberration: - a white object appears colored. It is removed by using achromatic combination. For achromatic combination 

Achromatic combination (ω₁/f₁) + (ω₂/f₂) = 0 where ω₁ and ω₂ are dispersive powers and since it is combination of a convex and a concave lens

1/F = (1/f₁) + (1/f₂)

Chromatic aberration can also be removed using two lenses of same kind separated by a small distance

If d = (ω₁ f₂ + ω₂ f₁)/(ω₁ + ω₂)

If ω₁ = ω₂ = ω  then d = [(f₁ + f₂)/2]

If d = f – f2 then spherical aberration is also removed.

_{Thus if f1 = 3f}2 and d = 2f₂ then both the defects are removed.

This method is employed in Huygens’s eye piece.  

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