paraxial matrix for arbitrary tilts

[MaximilianHoffmann]

def paraxial_matrix(self, n0, l):
        # Reflection and Refraction of Gaussian Light Beams at
        # Tilted Ellipsoidal Surfaces
        # G. A. Massey and A. E. Siegman
        # Applied Optics IP, vol. 8, Issue 5, p.975
        #
        # [y', u'] = M * [y, u]
        n, md = super(Spheroid, self).paraxial_matrix(n0, l)
        c = self.curvature
        if self.aspherics is not None:
            c = c + 2*self.aspherics[0]
        theta = self.angles[0] if self.angles is not None else 0.
        costheta = np.cos(theta)
        m = np.eye(4)
        if self.material is not None:
            if self.material.mirror:
                m[2, 0] = 2*c*costheta
                m[3, 1] = 2*c/costheta
            else:
                mu = n/n0
                p = np.sqrt(mu**2 + costheta**2 - 1)
                m[1, 1] = p/(mu*costheta)
                m[2, 0] = n0*c*(costheta - p)
                m[3, 1] = mu*m[2, 0]/(costheta*p)
                m[3, 3] = 1/m[1, 1]
        m = np.dot(m, md)

        if self.angles is not None:
            # FIXME angles is incomplete:
            # rotate to meridional/sagittal then compute total incidence
            # angle, matrix, then rotate back
            phi = self.angles[2]
            cphi, sphi = np.cos(phi), np.sin(phi)
            r1 = np.array([[cphi, -sphi], [sphi, -cphi]])
            r = np.eye(4)
            r[:2, :2] = r[2:, 2:] = r1
            m = np.dot(r, np.dot(m, r.T))

Maybe you could give me hint. It is unclear to me why the angle[0] would be in anyway special. Why is angle[1] not taken into account, or is this indeed what is meant by the FIXME comment: that we rotate the system until only one angle is nonzero? But: Why is theta in the above function equal angle[0] and not something like u[0] +angle[0]

Thx Max

[jordens]

The treatment of arbitrarily tilted spheroids is incomplete (that's why there is the FIXME).

In a simple (planar) tilted system, angles[0] is the x-tilt ("in plane", a.k.a. theta) and the only rotation there is. angles is an rxyz-style set of euler angles. In an element with only a an x and a z tilt, theta = angles[0] and phi = angles[2]. u is the angle of the ray and doesn't matter for the matrix.

For "general" tilts, it is more complicated and needs to be calculated as sketched out in the FIXME.

[jordens]

Also note that the phi code segment doesn't matter yet anyway since there are no cylindrical elements implemented.