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In formulating perfect or Gaussian imaging we assumed all relationships are linear. For example, Snell's law of refraction was reduced in the following way:

(3.1)

In making the paraxial approximation, we are considering just the first or linear term in the Taylor series. The next logical stage in the process is to consider higher order terms in the Taylor series.

(3.2)

Figure 3.1 (a) Gaussian imaging. (b) Impact of aberration.

Following the term that is linear in θ, we have terms that are cubic or third order in θ. Of course, these third order terms are followed by fifth and seventh order terms etc. in succession. Third order aberration theory deals exclusively with those imperfections associated with the third order departure from ideal behaviour, as illustrated in Eq. (3.2). Much of classical aberration theory is restricted to consideration of these third order terms and is, in effect a refinement or successive approximation to paraxial theory. Higher order (≥5) terms can be important in practical design scenarios. However, these are generally dealt with by numerical computation, rather than by a simple generically applicable theory.

Third order aberration theory forms the basis of the classical treatment of monochromatic aberrations. Unless specific steps are taken to correct third order aberrations in optical systems, then third order behaviour dominates. That is to say, error terms in the ray height or angle (compared to the paraxial) have a cubic dependence upon the angle or height. As a simple illustration of this, Figure 3.1b shows rays originating from a single object (at the infinite conjugate). For perfect image formation, the height of all rays at the paraxial focus should be zero, as in Figure 3.1a. However, the consequence of third order aberration is that the ray height at the paraxial focus is proportional to the third power of the original ray height (at the lens).

In dealing with third order aberrations, the location of the entrance pupil is important. Let us assume, in the example set out in Figure 3.1b, that the pupil is at the lens. If the radius of the entrance pupil is r₀ and the height a specific ray at this point is h, then we may define a new parameter, the normalised pupil co-ordinate, p, in the following way:

(3.3)

The normalised pupil co-ordinate can have values ranging from −1 to +1, with the extremes representing the marginal ray. The chief ray corresponds to p = 0. At this stage, it is useful to provide a specific and quantifiable definition of aberration. The quantity, transverse aberration, is defined as the difference in height of a specific ray and the corresponding chief ray as measured at the paraxial focus. The ‘corresponding chief ray’ emanates from the same object point as the ray under consideration. In addition, the term longitudinal aberration is also used to describe aberration. Longitudinal aberration (LA) is the axial distance from the point at which the ray in question intersects the chief ray and the location of the paraxial focus. The transverse aberration (TA) and longitudinal aberration definitions are illustrated in Figure 3.2.

In keeping with the previous arguments, the TA has a third order dependence upon the pupil function. This is illustrated in Eq. (3.4):

(3.4)

Transverse aberration has dimensions of length, whereas the pupil function is a dimensionless ratio. Geometrically, the LA is approximately equal to the transverse aberration divided by the ray angle which itself is proportional to the pupil function. Therefore, the longitudinal aberration has a quadratic dependence upon the pupil function. This is illustrated in Eq. (3.5).

Figure 3.2 Transverse and longitudinal aberration.

(3.5)

In fact, if the radius of the pupil aperture is r₀ and the lens focal length is f, then the longitudinal and transverse aberration are related in the following way:

(3.6)

NA is the numerical aperture of the lens.

A plot of the transverse aberration against the pupil function is referred to as a ‘ray fan’. Ray fans are widely used to provide a simple description of the fidelity of optical systems. If one views the transverse aberration at the paraxial focus, then the transverse aberration should show a purely cubic dependence upon the pupil function. This is illustrated in Figure 3.3a which shows the aberrated ray fan. If, one the other hand, the transverse aberration is plotted away from the paraxial focus, then an additional linear term is present in the plot. This is because pure defocus (i.e. without third order aberration) produces a transverse aberration that is linear with respect to pupil function. This is illustrated in Figure 3.3b which shows a ray fan where both the linear defocus and third order aberration terms are present.

The underlying amount of third order aberration is the same in both plots. However, the overall transverse aberration in Figure 3.3b (plotted on the same scale) is significantly lower than that seen in Figure 3.3a. This is because defocus can, to some extent, be used to ‘balance’ the original third order aberration. As a result, by moving away from the paraxial focus, the size of the blurred spot is reduced. In fact, there is a point at which the size (root mean square radius) of the spot is minimised. This optimum focal position is referred to as the circle of least confusion. This is illustrated in Figure 3.4.

Most generally, the transverse aberration where third order aberration is combined with defocus can be represented as:

(3.7)

TA₀ is the nominal third order aberration and α represents the defocus

Figure 3.3 (a) Ray fan for pure third order aberration. (b) Ray fan with third order aberration and defocus.

Since the geometry is assumed to be circular, to calculate the rms (root mean square) aberration, one must introduce a weighting factor that is proportional to the pupil function, p. The mean squared transverse aberration is thus:

(3.8)

Figure 3.4 Balancing defocus against aberration – optimal focal position.

The expression is minimised where α = −2/3. To understand the significance of this, examination of Eq. (3.6) suggests that, without defocus, the marginal ray (p = 1) has a longitudinal aberration of TA₀/NA. The defocus term itself produces a constant longitudinal aberration or defocus of αTA₀/NA. Therefore, the optimum defocus is equivalent to placing the adjusted focus at 2/3 of the distance between the paraxial and marginal focus, as shown in Figure 3.4. Without this focus adjustment, with the third order aberration viewed at the paraxial focus, the rms aberration is TA₀/4. However, adding the optimum defocus reduces the rms aberration to TA₀/12, a reduction by a factor of 3.

This analysis provides a very simple introduction to the concept of third order aberrations. In the basic illustration so far considered, we have looked at the example of a simple lens focussing an on axis object located at infinity. In the more general description of monochromatic aberrations that we will come to, this simple, on-axis aberration is referred to as spherical aberration. In developing a more general treatment of aberration in the next sections we will introduce the concept of optical path difference (OPD).