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Introduction       How AESOP does Ray Tracing       References

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Introduction

Modern high-precision optical systems, such as space astrometric interferometers (e.g. Reasenberg et al. 1996, Loiseau and Malbet 1996, Lindegren and Perryman 1996), can require optical path tolerances in the sub-nanometer (1 nm = 10^-9 m) to picometer (1 pm = 10^-12 m) regimes over total path lengths on the order 10 m. Such tolerances place extreme requirements on optical analysis programs. Two questions are of paramount importance: 1) to which specific perturbations is a system most sensitive? and 2) are there couplings between different perturbations that produce high sensitivities (i.e., are there strong correlations between perturbation parameters)? AESOP can be, and has been (Murison, 1993), used to fully answer these questions, as well as to develop physical intuition in the picometer OPD regime.

A common optical subsystem employed in astronomical interferometers is a beam compressor, used to convert a large aperture input beam (starlight) to a narrow output beam (~1 cm) suitable for combining with another such beam to produce interference fringes of sufficient visibility. A typical beam compressor consists of a pair of confocal paraboloidal mirrors, as sketched in Figure 1. If perfectly aligned, a flat input wavefront results in a radially compressed flat output wavefront. Misalignment analysis of even such a simple system as this generally requires resorting to numerical programs. Usually, such programs are ill-suited for studies involving both misalignment parameter variation and aperture-averaged optical path difference (OPD) determination, especially in the pm regime. The need for picometer OPD tolerances is a relatively recent development, driven by ever more demanding sciece objectives. Such tolerance requirements will likely become more common, and the lack of adequate analysis tools will correspondingly be felt more strongly.

To develop a physical understanding of alignment sensitivities, one would much prefer an analytical rather than a numerical description of the output wavefront as a function of the misalignment parameters. Unfortunately, an analytical wavefront description of misaligned optical systems as simple as a beam compressor, or even a single focussing optic, can be too complex to attempt by hand in the kind of detail required for sensitivity studies (Noecker et al. 1993). However, computer algebra systems such as Maple have advanced to such a state of capability and sophistication that, coupled with the processing power of modern computers, complete analytical descriptions are now becoming possible.

AESOP was developed to support the analysis effort involved in determining critical sensitivities to optic misalignments in a proposed dual interferometric astrometric telescope, POINTS (Reasenberg et al. 1988, 1995a, 1995b, 1996). POINTS consists of a pair of independent Michelson stellar interferometers and a laser metrology system that measures both the critical starlight paths and the angle between the two interferometer baselines. The nominal design has baselines of 2 m, telescope apertures of 35 cm, and observes target stars separated by roughly 90 degrees. One of the distinguishing features of POINTS is that it employs holographic optical elements (HOEs) to accomplish picometer metrology over the full aperture of the starlight optical path. See the Reasenberg et al. references for a full description of the instrument, its capabilities, and the astrophysical, astrometric, and planet-finding science that POINTS would significantly impact. (Information can also be found on the POINTS web pages at http://www-cfa.harvard.edu/~reasen/points.html .)

A key analysis problem regarding the POINTS interferometers is the determination of optical path length errors as a function of various optical element misalignments. The path length error budget in a precision system such as this is only several tens of picometers. With such a tight error budget, it is imperative to determine which perturbations lead to large path length errors. At the pm level, often we cannot trust our optical intuition in determining misalignment sensitivities. In such cases, we must rely on numerical analysis to an uncomfortable degree, lacking reliable independent checks on the numerical results. AESOP was created in part to fill this niche. In the case of POINTS, a numerical program called RayTrace (see Murison (1993) for a description) was written specifically to perform ultra-high precision, sub-picometer OPD variation analyses. AESOP was developed in parallel with RayTrace. The two analysis approaches — numerical and analytic — are completely independent and therefore serve as excellent checks upon one another.

AESOP traces an input ray through a misaligned optical system and produces an analytic description of the output ray as a function of the system parameters, the misalignment parameters, and the input ray position and direction. A crucial diagnostic is the aperture-averaged OPD variation. The physical principles involved are quite simple, since AESOP takes a classical geometric optics approach. At a given reflecting surface, all that is required is to calculate the reflected ray direction and the accumulated optical path up to that intersection point. Similarly, at a refracting surface we use Snell's law to calculate the refracted ray direction. If a holographic optical element (HOE) is encountered, it is a similarly simple process to calculate the output ray direction and the change in optical phase across the element (Murison and Noecker, 1993). The POINTS optical subsystems involve all three types of optical surfaces. AESOP currently handles reflecting and holographic optical elements. Refracting surfaces will be easy to add, due to the extensible structure of AESOP.

As simple as the physics is, such a ray tracing process is impossible to do analytically by hand (especially the aperture-averaged effects). The inhibiting factor is the rapidly increasing (with each successive optical surface) complexity of the intermediate expressions that must be algebraically manipulated. This kind of repetitive manipulation of unwieldy objects is precisely what computers can do well. Hence, a programmable computer algebra system like Maple is well suited in principle to analyzing misaligned optical systems, at least simple ones involving relatively few focussing optical surfaces.

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How AESOP does Ray Tracing

An overview of the ray tracing process using AESOP is as follows.

(1) In a Maple procedure that the user writes, the user first defines the various optical elements comprising the optical system. These surfaces are assembled into a Maple list which AESOP routines will make use of. Perturbations (misalignments) are applied in the form of rotations and/or translations of specified optical elements. AESOP provides object rotation and translation procedures to make this a simple process.

(2) The user then defines the input ray, which is subsequently launched into the optical system by calling the AESOP procedure raytrace(). AESOP then automatically traces the ray to each successive optical element, performing series expansions on the perturbation parameter(s) as necessary and simplifying the cumbersome expressions as as much as possible, until finally an output ray is produced at the detector. Progress during this process is communicated via informational messages and key intermediate expressions to the monitor screen. If nothing else, there is plenty of stuff the user can peruse while waiting for the ray trace to finish, since AESOP is intentionally a bit chatty.

(3) The OPD is then calculated from the output ray expressions, followed by calculation of the aperture-averaged OPD.

(4) Optionally, the Zernike components of the OPD are determined next, either at the Maple prompt or from within the user's driver procedure. The resulting Zernike coefficients may then be combined to produce wavefront aberration plots. The aberrated wavefronts are represented by 3D Maple surface plots. Maple procedures are supplied for making the wavefront plots in the Maple worksheet.

Two illustrative examples of this entire process for a simple beam compressor are shown here and here.

In practice, the essential step for useful analytical ray tracing is to make series expansions at each intersection of a ray with an optical surface. (The original insight for this trick is due to R.D. Reasenberg.) This reduces the "equation bloat" considerably. Even so, it is still rather easy to cause the intermediate expressions to mushroom in size so that they overwhelm the available machine resources. The equation bloat seems to go as some power of the number of focussing optical elements in a system. Flat surfaces certainly contribute to increasing equation complexity, but at a rate that pales in comparison to that of focussing surfaces.

Since we are interested in analyzing optical systems whose elements are slightly misaligned, the small parameters to perform the series expansion on are naturally the misalignment perturbations. Hence, AESOP is not meant to analyze the very interesting properties of ideal, perfectly aligned optical systems. It requires at least one misalignment or other perturbation parameter.

For a given optical system, a certain amount (sometimes a great amount) of tinkering on the part of the user is required to hit upon the best ways of simplifying the cumbersome expressions so that their size is managable. Great care has been made in the types of simplification taking place in the AESOP ray tracing routines. However, they are no doubt optimized for the particular systems the author has analyzed and will therefore perhaps be less than optimum for other kinds of optical systems. Hence, AESOP is nowhere near the "black box" stage, where a user can provide necessary input, crank the handle, and magically produce an answer without caring overmuch about the internals of the black box. Nonetheless, AESOP can be quite useful and represents a significant advance in capability for analyzing perturbed optical systems and performing misalignment sensitivity studies. It also serves as an invaluable check on numerical programs as well as an essential aid to developing reliable insight into the arcane and beautiful world of high-precision optics.

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References

Note: copies of relevant Smithsonian Astrophysical Observatory (SAO) Technical Memoranda are available here.

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Last changed: 24 September 1996