Progress in Optics

3.1.4.3: Mapping functions of a 2-points SMS optical system

The general expression of the mapping function x′ = A(x, y, p, q ) for an SMS design stigmatically imaging 2 object points α = (x_α, y_α), β = (x_β, y_β) onto 2 image points α ′ = (x_α′, y_α′), β ′ = (x_β′, y_β′) can be written as:

$x^{\prime }=A_0+A_{1}x+A_{2}y+f\left(x,y,p,q,x_{\alpha },y_{\alpha },x_{\beta },y_{\beta },x_{\gamma },y_{\gamma }\right)$

  (53)

where f is a general continuous function nulling at the points α, and β. The constants A₀, A₁, A₂ are such that x′ = x_α′, x′ = x_β′ when (x, y) = α, β, respectively, i.e., they are constrained by the relationship:

$\left(\begin{array}{ccc}1& x_{\alpha }& y_{\alpha }\\ 1& x_{\beta }& y_{\beta }\end{array}\right)\left(\begin{array}{c}{A}_0\\ A_1\\ A_2\end{array}\right)=\left(\begin{array}{c}{x}_{\alpha }^{\prime }\\ x_{\beta }^{\prime }\end{array}\right)$

  (54)

Then f can be written as f_αβ in Eq. (40) where l and m are also functions of p,