Derivation of the Paraxial Relationship - Geometrical and Visual Optics - page 337

Appendix E: Derivation
of the Paraxial Relationship

Figure E-1 shows a light ray incident upon a spherical refracting surface. According
to Snell's law,
n sin θ ′

For paraxial rays, 1 we can make the assumption that sin

θ = n ′ sin

θ = θ (where θ is in

radians). Snell's law is then rewritten as
n ′ θ

From Figure E-1, we see that
θ

θ = n

= α + β

and

β = θ ′ + δ

or

− δ
Substituting into Snell's law, we have
n (

θ ′ = β

α + β ) = n ′ ( β − δ )

or

n )
But from Figure E-1, we see that (assuming small angles)
h
−

n ′ δ = − n α + β ( n ′ −

α =

l

h
r

β =

1. Paraxial rays are close to the optical axis and incident on the central area of the lens; they make
sufficiently small angles with the normal that we can assume sin θ = θ (in radians).

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