geometrical optics (formulas ) Prism : A dispersing prism deviates light of different wavelengths through different angles (see fig. g8). The angle between the refracting surfaces is called the angle of the prism, A . The angle between the incident ray and the emergent ray is called the angle of deviation, D. When the angle of deviation is minimum, it can be shown that the refractive index of the material of the prism is given by,

n_prism = [sin {(A+D)/2}] / sin(A/2) (prism equation) (1)

For a thin prism, angle A is small, and the angle of deviation is small.

n_{prism @} (A+D)/2 = 1+(A/D)

or D = A ( n_prism - 1 ) (2)

A prism is also used for reflecting light beam. See also total internal reflection.

Spherical surfaces: Precise spherical surfaces are easy to fabricate and therefore of considerable practical importance. In order to analyze the function of a spherical surface, a sign convention for distances is necessary. A simple sign convention is the ‘Cartesian sign convention’ .

1. All figures are drawn with light traveling from left to right.

2. All distances are measured from a reference surface (such as the refracting surface) located at (0,0). Distances measured to the left of the reference surface are negative, and distances to the right of the refracting surface are positive.

3. The radius of curvature ( which is measured from the surface to the center) is positive for convex surface, and negative for concave surface. Similarly , a focal length to the left is taken negative, and to the right is taken positive.

4. The distance of a real object is negative and distance of a real image is positive. The distance of a virtual image is negative. Height above the optic axis is taken positive; and distances below the optic axis taken negative (see fig. g9).

A narrow cone of light emerging from a point at a distance d₀ from a convex surface will converge at distance d_i (see fig. g9). It can be shown that ,

(n₁/ d₀) + (n₂ - n₁)/R = n₂/d₁ (Gauss formula) (3)

When S is moved to infinity, d₀ = - ¥ , we obtain,

n₂/d₁ = n₂/ f₂ = (n₂ - n₁)/R

or f₂ = n₂ R/ (n₂ - n₁) (4a)

the second focal length. Similarly the first focal length is,

f₁ = -n₁ R/ (n₂ - n₁) (4b)

Thin lenses: A lens has two refracting surfaces at least one of which is spherical. The power (defined as the reciprocal of focal length multiplied by the refractive index of the material) of the two surfaces should be added to obtain power of the lens. This gives,

n₂/f= (n₂ - n₁)( R₁^-1 - R₂^-1 ) (lens maker’s formula) (5)

The object distance (d_o) and the image distance (d_i) in a thin lens are related as,

(1/d₀) + (1/f₂) = (1/d₁) (Gauss thin lens equation) (6)

The transverse magnification is defined as the ratio of image size h¢ , to the object size h,

M_t =h¢ /h (7)

Combination of two thin lenses: For a combination of lenses or for a thick lens, the focal length, object distances and image distances are measured from hypothetical surfaces called principal planes. Principal planes are defined as the loci where refractions are assumed to occur. Consider a black box containing combination of lenses (fig. g10). Rays parallel to the optic axis will converge at F₂ . The second principal plane is H₂ where refraction can be assumed to be taking place. The second focal length and image distances are measured from H₂ . Similarly the plane H₁ renders the rays coming from F₁ parallel to the optic axis. H₁ is the first principal plane. The object distance and first focal length are measured from H₁.

The focal length, F of the combination of lenses is given by,

(1/F) = (1/f ') + (1/f '') - d/(f ' f'') (8)

where f¢ and f² are the focal lengths of the individual lenses.

The distance of the first principal plane H₁ from the first lens is given by,

a ₁ = Fd/f '' (9a)

The distance of the second principal plane from the second lens is given by,

a ₂ = - F d/f' (9b)

In fig. g11 a typical telephoto lens, consisting of a convex lens and a concave lens is shown. The principal planes of the lens system and focal distances have been calculated using the formulas (8) and (9).

thick lenses: If the medium between two thin lenses were glass, we would effectively have a thick lens. The focal length of a thick lens is,

(1/F) = (1/f') + (1/f'') - d/(n f ' f '') (10)

where n is the refractive index of the material of the lens, and d the thickness of the thick lens. Using eq.(4) and (8) we obtain,

(n/F) = (n-1) [R₁^-1 - R₂^-1] + (n-1)² d/ ( n R₁ R₂) (11)

the lens maker’s formula for thick lens.

Mirrors: For mirrors also the same sign convention can be used. The radius of curvature for a mirror concave to the left is negative, and for mirror convex to the left is positive. The focal length and power are taken positive for concave mirror and negative for convex mirror. The image distance for real image is taken positive and for virtual image, negative. The relationship between the object distance (d_o), image distance (d_i) and focal length is given by,

(1/f) = (1/d _i) - (1/d _o) = - 2/ R (spherical mirror equation) (12)

where R is the radius of curvature.