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This is shown in Figure 3.7. It is clear that the transverse aberration is related to the angular difference between the wavefront and reference sphere surfaces.

      We now describe the WFE, Φ, as a function of the reference sphere (paraxial ray) angle, θ. The radius of the reference sphere (distance to the paraxial focus) is denoted by f. This allows us to calculate the difference in angle, Δθ, between the real and paraxial rays. This is simply equal to the difference in local slope between the two surfaces.

      (3.9)

      n is the medium refractive index.

In this analysis, the WFE represents the difference between the real and reference surfaces with the positive axial direction represented by the propagation direction (from object to image). In this convention, the WFE has the opposite sign to the OPD. The transverse aberration, t, can be derived from simple trigonometry.

(3.10)

If θ describes the angle the ray makes to the chief ray, then Eq. (3.10) may be reformed in terms of the numerical aperture, NA. The numerical aperture is equal to nsinθ, and Eq. (3.11) may be recast as:

(3.11)

      So, the transverse aberration may be represented by the first differential of the WFE with respect to the numerical aperture. In terms of third order aberration theory, the numerical aperture of an individual ray is directly proportional to the normalised pupil function, p. If the overall system, or marginal ray, numerical aperture is NA₀, then the individual ray numerical aperture is simply NA₀p. The transverse aberration is then given by:

(3.12)

Equation (3.12) provides a simple direct relationship between OPD and transverse aberration. Of course, we know that, for third order aberration, the transverse aberration is proportional to the third power of the pupil function, p. If this is the case, then it is apparent, from Eq. (3.12), that the OPD is proportional to the fourth power of the pupil function. So, for third order aberration, the transverse aberration shows a third power dependence upon the pupil function whereas the OPD shows a fourth power dependence.

      Applying these arguments to the analysis of the simple on-axis example illustrated earlier, with the object placed at the infinite conjugate, then the WFE can be represented by the following equation:

      (3.13)

      p is the normalised pupil function.

Figure 3.8 shows a plot of the OPD against the normalised pupil function; such a plot is referred to as an OPD fan.

Despite the fact that this simple aberration has a quartic dependence on the pupil function, it is still referred to as third order aberration after the transverse aberration dependence. As with the optimisation of transverse aberration, the OPD can be balanced by applying defocus to offset the aberration. We saw earlier that a simple defocus produces a linear term in the transverse aberration. Referring to Eq. (3.12), it is clear that defocus may be represented by a quadratic term. Equation (3.14) describes the OPD when some defocus has been added to the initial aberration.

(3.14)

An OPD fan with aberration plus balancing defocus is shown in Figure 3.9.

      In this instance, the plot has a characteristic ‘W’ shape, with the curve in the vicinity of the origin dominated by the quadratic defocus term. As with the case for transverse aberration, the defocus can be optimised to produce the minimum possible OPD value when taken as a root mean squared value over the circular pupil. Again, using a weighting factor that is proportional to the pupil function, p, (to take account of the circular geometry), the mean squared OPD is given by:

(3.15)

Figure 3.8 Quartic OPD fan.

Figure 3.9 OPD fan with balancing defocus.

      The above expression has a minimum where α = −¾. To understand the magnitude of this defocus, it is useful first to convert the new OPD expression into a transverse aberration using Eq. (3.12).

(3.16)

From Eq. (3.16), it can be seen that the optimum defocus is 3/8 of the distance between the paraxial and marginal ray foci. This value is different to that derived for the optimisation of the transverse aberration itself. It should be understood that the optimisation of the transverse aberration and the OPD, although having the same ultimate purpose in minimising the aberration, nonetheless produce different results. Indeed, in the optimisation of optical designs, one is faced with a choice of minimising either the geometrical spot size (transverse aberration) or OPD in the form of rms WFE. The rationale behind this selection will be considered in later chapters when we examine measures of image quality, as applied to optical design.

The balanced defocus, as illustrated in Eq. (3.15) does significantly reduce the rms OPD. In fact, it reduces the OPD by a factor of four. Resultant rms values are set out in Eq. (3.17).

(3.17)

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