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Paraxial optics is ray-tracing performed in the limit of very small ray angles and heights. It allows us to make a number of simplifying assumptions that makes the arithmetic of ray-tracing considerably easier.

Assumption 1 is to Snell's Law itself. When refracting from one material into another, the celebrated equation is

nsinq = n'sinq'

where unprimed quantities are before refraction and primed are after. For small angles sinq @ q and so Snell's Law can be written

nq = n'q'

This of course was an enormous relief in the days before modern calculators and computers.

Many definitions in optics are based upon this assumption of linearity, and leads to the term first-order optics. Aberrations are third-order and higher deviations from this linearity, because as q gets larger, sinq @ q -q³/3! + q⁵/5!- etc.  The paraxial properties of optical systems are often considered the properties the system has in the absence of aberrations.

Assumption 2 is that, as the ray height on the surface is small, we can ignore the curvature of surfaces and instead trace rays between flat surfaces of equivalent power. The power of a surface of curvature C between two indices n and n' is:

j = (n'-n).C

and by ignoring the curvature for ray-intercept purposes we are saved the task of computing the exact ray-surface intercept point.

Assumption 3 is that the tangent of the ray angle (the ray slope) may be replaced by the ray angle. This assumption may not be obvious, but it is fundamental. Consider a paraxial ray being traced between two flat surfaces , as shown below. The ray has an initial height y on the first surface and has y- and z- direction cosines {m, n}. Its height y' on the next surface is given by:

y' = y + tanq. t      = y + (m/n).t      @ y + q.t

because not only does sinq @ q but tanq @ q also. This has a fundamental consequence which is sometimes missed: the slope of a paraxial ray is the same as its angle.

Clearly paraxial optics introduces major simplifications to the calculation of ray-tracing, but it would be a mistake to consider paraxial optics as just a computational device, of no consequence now we have calculators and computers. Paraxial optics represents the limiting properties of rotationally symmetric systems comprised of spherical surfaces. However, parabasal rays are more general and more useful.