Derivation of conventional formula of the third-order aberration for off-axial optical system

Abstract

A formula of the third-order aberration based on the primary aberration expansion for an off-axial optical system is investigated. The point eikonal analysis and the coordinate system using the surface normal vector lead to a simple wavefront-based expression. Three-ray system employing the central, the principal and an arbitrary ray is examined for the derivation. The rays are forced to reach from the last surface to a virtual image point. Therefore, the contradiction of the optical path lengths brings the wavefront aberration. The extracted third-order aberrations are not strictly the same expressions as the conventional formula but show clearly the features of the five kinds of the primary aberrations along with the term originated by the off-axial layout. The relation between the second-order term and the expanded gaussian matrix for paraxial ray tracing and a term related to field curvature are discussed for off-axial systems.

Appendix

We assume that the coordinate system is adjusted so that the central image height vector only has y component, i.e., y = (y₀,0).

Figure 4 shows the ray height vector and the central ray directions with an angle q to the normal vector of the surface. Using the matrix J listed as formula (8), the relationship between Y, h and h′ in Ref. [3] is expressed as

$${\mathbf{Y}} = \sqrt J^{ - 1} {\mathbf{h}} = \sqrt {J^{\prime}}^{ - 1} {\mathbf{h^{\prime}}},$$

(36)

where

$$J = \left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & {\cos^{2} \theta } \\ \end{array} } \right).$$

(37)

Fig. 4

Coordinate system and relationship between h, h′ in Ref. [3] and Y

The formula (25) is transformed to an expression with respect to h and a in Ref. [3] as follows:

$$2\cos \theta^{\prime} \cdot A\sqrt {J^{\prime}}^{ - 1} {\mathbf{h^{\prime}}} - \sqrt {J^{\prime}}^{ - 1} {\mathbf{\alpha^{\prime}}} = 2\cos \theta \cdot A\sqrt J^{ - 1} {\mathbf{h}} - \sqrt J^{ - 1} {\varvec{\upalpha}},$$

(38)

where the paraxial angle vector a is defined as:

$${\varvec{\upalpha}} = N\frac{{\mathbf{h}}}{L}.$$

(39)

Finally, the formula shown as the formula (13) in Ref. [3] is derived with (36) and (38):

$$\left( {\begin{array}{*{20}c} {h^{\prime}_{y} } \\ {h^{\prime}_{z} } \\ {\alpha^{\prime}_{y} } \\ {\alpha^{\prime}_{z} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\frac{{\cos \theta^{\prime}}}{\cos \theta }} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ {2aN^{*} } & {2cN^{*} \cos \theta } & {\frac{\cos \theta }{{\cos \theta^{\prime}}}} & 0 \\ {2cN^{*} \cos \theta^{\prime}} & {2bN^{*} \cos \theta^{\prime}\cos \theta } & 0 & 1 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {h_{y} } \\ {h_{z} } \\ {\alpha_{y} } \\ {\alpha_{z} } \\ \end{array} } \right),$$

(40)

where N* is defined as:

$$N^{*} = \frac{{N^{\prime}\cos \theta^{\prime} - N\cos \theta }}{{\cos \theta^{\prime}\cos \theta }}.$$

(41)