SYMMETRY AND THE MOSES FOCUS C. Kankelborg 2008-June-11 ABSTRACT First I'll define what I mean by symmetry. Then I will apply it to MOSES. Finally, by somewhat convoluted logic, I'll make some tentative inferences to suggest (not conclude!) that a distorted mirror may not explain the PSF asymmetry that was observed in the MOSES flight. INTRODUCTION In my email of June 10, 2008, I wrote that "In principle, since the optical system is symmetric, squishing the mirror will affect the symmetry only if (a) the mirror is operated off-axis in the direction perpendicular to the dispersion, as in MOSES *and* (b) the mirror is squashed so that the minor axis of the ellipsoid is sort of diagonal --- neither in the dispersion plane nor perpendicular to it" (emphasis added). Since Seth has asked me to explain this symmetry argument, I've tried to write it up carefully as a white paper. I apologize for not making any diagrams. I have used equations instead. Anyway, it is worthwhile for physicists to learn how to translate back and forth between pictures and equations. SYMMETRY Let's begin with a precise mathematical definition of what a symmetry is. Suppose I have a physical system S, and I apply to it an operator M, and find that I just get S right back out again: S = MS Now despite any appearances to the contrary, M is not the identity operator. I'll get back to what M might be later, so don't let that bother you just yet. S happens to be a special system that is "symmetric with respect to M". [Provided that M is a linear operator, a mathematician might describe S as an eigenvector of M, with an eigenvalue of 1. Don't worry if you don't know what that means.] OK, so what might M be? Let's let it be Px [the parity operator], which performs reflections about the plane x=0. A human being at the origin, facing in the y-direction, with z pointing up through the head, is a physical system that is very nearly symmetric with respect to Px. (I happen to know that my left foot is a little bigger than my right, though!). APPLICATION TO MOSES Now, let's lay out the MOSES optical system according to its usual coordinates. x is toward the Sun. z points up off the optical table, and is parallel to the rulings on our grating (roughly, anyway; I'm ignoring the grating tilt and curvature when I say that!). y is the third direction that is needed for a right-handed coordinate system. This coordinate system, with its origin at the undispersed focus, is our standard set of coordinates for the MOSES optical design. Let us begin with a hypothetical instrument "MOSES-prime" (denoted S') that has no astigmatism, because the grating is not tipped forward. That is, the center of the grating and the center of the secondary mirror and the centers of all 3 detectors are in the z=0 plane. The center of the grating surface is normal to the z-axis. In every respect, we suppose that S' is perfect, i.e. the grating perfectly matches its design parameters and the secondary is perfectly flat, and there are no errors in alignment or focus. Seth has simulated this situation, I believe, leaving out the secondary since it doesn't "do" anything. Images are formed in three locations or "spectral orders", m=0 (on the x-axis), m=+1 (displaced to the +y direction from m=0) and m=-1 (displaced in the -y direction from m=0). Let Tx represent a 180-degree rotation about the x-axis. Let Pz represent reflection about the z=0 plane (and similarly Py). [It may or may not be worth noting that Tx = Py Pz = Pz Py. The mathematically inclined may choose this as a starting point for learning about symmetry groups.] I think it is obvious that MOSES-prime has (at least) the symmetries: Tx S' = S', Py S' = S', Pz S' = S'. I claim that an instrument with *either* the Py or the Tx symmetry cannot have any difference in focal lengths between the m = +1 and m = -1 spectral orders. That's because difference between those focal lenghts would violate *both* of those symmetries. So if a system has even one of them, it can't have asymmetric focal lengths. Now, let us perform three different thought experiments: (1) Distorted Grating Now, let us distort the mirror into an ellipsoid with major axis somewhere in the yz-plane (perpendicular to x). Let us assume also that the major axis is not exactly along y or z. This breaks the symmetry with respect to Py, but it does not break the symmetry with respect to Tx. Denoting our stressed system as S'*, Tx S'* = S'* Py S'* != S'* (!= means "not equal") Pz S'* != S'* (2) Off-Axis Design Now let's try a different modification. Lay out MOSES as designed, i.e. with the grating tipped forward (small, positive rotation about y) and its center elevated up out of the z=0 plane by a few centimeters or so. The new system, S, has the following properties: Tx S != S Py S = S Pz S != S Since Py symmetry remains, the focal lenghts in m = +1 and m = -1 must remain identical. (3) Off-Axis Design with Distorted Grating Finally, consider an off-axis design with a distorted grating, which we will name S*. The properties are: Tx S* != S* Py S* != S* Pz S* != S* Since both the Tx and Py symmetries are broken, we apparently have no basis for assuming that the focal lengths in the outboard spectral orders are equal. TENTATIVE INFERENCES Thought experiment (3) is a situation that may pertain to the real MOSES --- it is one hypothesis that might possibly explain the observed focal asymmetry. Note, however, that Tom's experiment with the LAG showed a focus asymmetry. If this hypothesis is called upon to explain Tom's result, then it would require that the ellipsoidal distortion be intrinsic to the grating, since the lab holder can only distort the grating so that its major axis would be along y, thus preserving Py symmetry. I think we can discount the hypothesis that there is an inherent ellipsoidal distortion in the LAG, because that would be evident in the interferograms from testing at Zeiss and at GSFC (no such distortion was found). Consequently, I infer that scenario (3) cannot apply to the LAG. If the LAG has a focus asymmetry, and if the flight grating (FQG #1) has a focus asymmetry for the same reasons (a reasonable guess, but by no means certain!), then the cause must be something other than (3).