Canonical forms in the theory of asymmetrical optical systems

Abstract

Canonical forms for the quadratic terms of the eikonal and of the characteristic function for any optical instrument are found to involve only six arbitrary constants. In the eikonal defined by = constant + first order terms + ½aM2 + ½bN2 + ½cM'2 + ½dN'2 + eMN + fM'N' - gMN' - hM'N - iMM' - jNN' + higher order terms, in addition to removing the first order terms we may take any of the following simplifying conditions: a = b = c = d = 0, a + b = c + d = e = f = 0, a + b = c + d = g = h = 0. Similarly in the characteristic function V = constant + first order terms + ½Ay2 + ½Bz2 + ½Cy'2 + ½Dz'2 + Eyz + Fy'z' - Gyz' - Hy'z - Iyy' - Jzz' + higher order terms, the first order terms may be removed and in addition we may put A = B = C = D = 0, or A + B = C + D = E = F = 0, or A + B = C + D = G = H = 0. The seven constant canonical form of the characteristic function obtained by Larmor, viz. V = constant + ½Ay2 + ½Bz2 + ½Cy'2 + ½Dz'2 + Eyz + Fy'z' - I (yy' + zz') +..., is not general. Larmor's theorem on the equivalence of any optical system to a symmetrical instrument together with two thin astigmatic lenses also fails. Three separated astigmatic lenses are needed to represent the general system.