2.1

Linear systems theory for image formation

Object space irradiance distribution can be decomposed into an ensemble of delta functions. The intensity or height of each delta function maps out the structure of the object. The optical system operates on the complex amplitude and phase associated with that intensity distribution to form an image at the detector. Most astronomical sources in the visible region of the spectrum radiate broadband, incoherent thermal light.

The theory of image formation is developed using the schematic shown in Figure 1 below. The coordinate system we use in our analysis is shown in Fig 2 below. This system is in standard use by modern textbooks^1,2 on the physics of image formation. Light travels left to right. The object plane to the left (#1 in the system) is represented by Cartesian coordinates from the Latin alphabet x₁,y₁ the pupil plane (#2 in the System) is represented by Cartesian coordinates from the Greek alphabet ξ₂,η₂ and the image plane (#3 in the System) is represented by Cartesian coordinates from the Latin alphabet x₃,y₃.

Figure 1

complex schematic of an optical system in the meridional plane. The object space field U₁ (x₁,y₁) is shown as a delta function δ(x₁,y₁) to represent a star on-axis. This object space field is propagated by Fresnel diffraction to just to the left of the entrance pupil and is shown by U₂^- (ξ₂,η₂). The pupil is shown to have a complex transmittance τ₂(ξ₂,η₂) at plane 2. The optical system, shown here schematically has a lens of focal length f, and provides the optical power to create the field U₃ (x₃,y₃) at an image plane 3.

Figure 2

HST image of a star showing the diffraction spikes that mask exoplanets at 4 position angles. The halo around the star is produced in the telescope/instrument system by narrow angle scattered light. The Airy diffraction pattern for HST is about 100 milliarc seconds which is too large to observe terrestrial exoplanets 1 astronomical unit from the parent star.

The scalar complex amplitude and phase across the image plane is found by standing at the image plane (#3) in Fig 2 and looking to the left, or back through the system toward the object. The Fresnel-Kirchoff diffraction integral is used to model the propagation of scalar electromagnetic waves through the optical system shown in Fig. 1. The complex amplitude and phase field U₃(x₃,y₃) at the image plane is given by

Where K is a constant, the integral is taken over the complex field across the exit pupil, U₂^- (ξ₂,η₂) of the optical system whose focal length is f, is the quasimonochromatic wavelength oflight. Eq 1 is written for the scalar wave solution to Maxwell’s equations and the not vector wave solution to Maxwell’s equation. The amplitude and phase complex properties across the exit pupil are contained in the scalar term,

Where A₂ (ξ₂,η₂) varies between 0 and 1 and describes amplitude part of the complex wave as a function of position across the exit pupil. The phase properties at each point across the exit pupil are described by ϕ₂ (ξ₂,η₂).

To the left in Fig 1, we have a point source represented by a delta function. This point source is mapped onto the image plane. We record intensity at the image plane and define the image plane irradiance distribution for this point source to be:

Next if we let the object space irradiance be represented by I_Object (x₁, y₁) and the image space irradiance represented by I_Image(x₃, y₃) and use the theoretical development of Goodman, we can write,

Where the symbol ⊗ denotes the convolution operator.

2.2

Static and dynamic components of the PSF

The complex amplitude and phase field U₃(x₃,y₃) at the image plane given by eq 1 can be decomposed into linear terms representing dynamic changes to the optical system and those static terms given by the optical systems engineering terms: geometric wavefront aberrations, polarization aberrations, scattered light and diffraction. Dynamic changes occur during integration time and changes to the system that results from repointing the telescope/instrument system. Dynamic changes to the optical system are produced by telescope and instrument pointing and tracking errors along with changes to the optical system introduced by time-dependent thermal gradients which change the spacing of optical elements and their precision alignment.

The optimum STATIC point spread function (PSF) for an optical system that is designed for threshold detection is one that exhibits both minimum PSF shape distortion³ and minimum spatial structure⁴. Several end-to-end optical system factors interact to affect the PSF shape distortion and its spatial structure. These are:

Geometric wavefront aberrations are introduced by mirror-surface fabrication errors. Polarization aberrations arise from the dielectric and metal coatings on mirror surfaces needed to maintain a high optical-transmittance system. Surface scatter arises because oflimitations to our ability to fabricate perfectly smooth surfaces. Diffraction arises in Cassegrain telescope systems from shadows cast on the primary mirror by the mechanical structure needed to support the secondary mirror. Diffraction is also introduced by the gaps between primary-mirror segments needed to create a large aperture telescope mirror which must be deployed or assembled in space because oflimited lift mass and volume capacity for modern launch vehicles.

The remainder of this paper concentrates on innovative space telescope architectures which mitigate the deleterious effects of diffraction on image quality for threshold astrophysics, earth, and planetary space science. We select one particular observational requirement, that of direct imaging of terrestrial exoplanets at the threshold level to examine further.

Segmented space telescope diffraction

Today, space-telescopes that are segmented use close-packed regular hexagon-sided mirrors to package their reflective surfaces into a nearly circular telescope pupil. Figure 4 below shows, on the left, one of the concept apertures for the Large Ultra-Violet Infrared (LUVOIR) mission study. On the right we see the individual segments of the primary. This telescope primary exhibits three diffraction gratings which diffract light from the much brighter central parent star across the image plane and create background noise for imaging and spectroscopy of exoplanets. The gaps between segments appear as discontinuous diffraction grating rulings across the aperture The direction of the “rulings” of these three diffraction gratings are shown using red, blue and orange arrows in Figure 4.

Figure 3

LUVOIR segmented aperture left and its partitioning into three linear diffraction gratings. The 3 “ruling” directions are clocked 60-degrees in relation to each other. The directions of the grating rulings are shown by the arrows in the three colors: red, blue and orange.

Figure 4

Two of the three diffraction gratings across the close-packed hexagon-segmented primary mirror are shown. The spacing of the “ruling” is d and the rulings are “clocked” or rotated 60 degrees from one set to the next.

The rulings are discontinuous across the LUVOIR pupil. Equation 1 has been interpreted to show that there is a spatial frequency Fourier Transform relationship between the complex field across the telescope aperture and the image plane¹. Therefore, the diffraction grating rulings formed by the gaps between the segments are shift invarient and the discontinuity of the gap lines (grating rulings) are shift invariant and result in a decrease in intensity of the diffraction orders but does not alter their presence. Figure 4 shows 2 of the 3 sets of diffraction grating rulings across the hexagonally segmented LUVOIR pupil. The third set of rulings, the horizontal set are not shown to avoid confusion in the drawing.

Figure 4 shows the close-packed hexagon-segmented primary mirror with lines drawn to show the grating “rulings” and the direction of the rulings. The “groove-spacing” is seen to be d, where d is one-half the face-to-face distance across the individual regular hexagons. The diffraction causes a structured background across the image plane that may obscure important exoplanets and may introduce unwanted polarization aberrations into the coronagraph to affect image quality¹².

The PSF for a monochromatic star

Gratings diffract light into orders which map a single on-axis point (a monochromatic star, for example) into multiple images of that monochromatic star. If the source is polychromatic then the grating maps the polychromatic single on-axis point into multiple spectral images stretched out radially. This is, of course how a diffraction grating spectrograph operates.

Let λ = wavelength; d=ruling spacing; n=diffraction order and θ=angle from the axis, then the grating equation that relates these 4 variables is written:

nλ = 2d sin(θ) Equation 5.

or for very small angles we can write

The angular separation between diffraction orders is θ as given in Eq. 5 above.

In figure 5 we see that a Lyot coronagraph occulting mask would only block light from the zero-diffraction order, which contains light from the bright central star. But light from the higher diffraction orders n>l will scatter around the occulting mask to flood the detector plane. The occulting mask could be designed such that each order has its own mask, but that would block portions of the FOV where exoplanets might be found.

Figure 5

shows, left, a pupil map and right a representation of the monochromatic PSF associated with the close-packed hexagon-segmented primary mirror shown on the left. The points on the right-hand side show the location of the diffraction orders. The center is the image of the star and the first ring of points corresponds to the 1^st order of the three diffraction gratings. The second ring of points corresponds to the 2^nd order of the three diffraction gratings.

To determine if the diffraction images of the parent star will obscure exoplanets, we calculate the angular separation between zero order and the first order, n=l, for polychromatic light. Table 2 below shows the angular separation of the 450, 500, 550 nm wavelength monochromatic diffraction orders as a function of the face-to-face segment size.

Table 2

Angular separation of the diffraction orders for face-to-face segment sizes: 1,2,3,4 meters. This applies to the diffraction orders shown in Fig. 6, above.

We compare the entries in Table 2 with the entries in Table 1 and see that the unwanted diffraction images of the parent star fall within the same FOV region as the exoplanets. Clearly there would be a significant advantage to the development of a straightforward, low absorption way to eliminate these diffraction orders. The pinwheel pupil provides that opportunity.

The PSF for a polychromatic star

Exoplanets are very faint thermal sources. If they are to be observed in monochromatic or narrow band light integration times become impossibly long. The HabEx coronagraph is planned to observe in 100 nm bandwidths. One of these bandwidths in 450 to 550 nm and we have used those values to compute the diffraction angles shown in Table 2. The star image at n=l for the 1-meter face-to-face segments is a colored radial streak or small spectrum with 450 nm light at 93 masec and 550 nm light at 113 masec. The 2-meter face-to-face segments is a colored radial streak or small spectrum with 450 nm light at 47 masec and 550 nm light at 57 masec. This continues to the 4-meter face-to-face segments which give a colored radial streak or small spectrum with 450 nm light at 23 masec and 550 nm light at 28 masec.

Isoplanatic point spread function for image processing

The polychromatic PSF shown in figure 5 (right) is not linear shift invariant and therefore the optical system is not isoplanatic. Also, looking at Fig 5, right, we see that the PSF is also not rotationally symmetric. These two facts complicate digital image processing.

Below we describe a pupil segmentation or topology architecture presented in earlier papers by the authors^13,14,15 that will produce images from an emulated filled aperture telescope pupil even though the pupil is mechanically segmented. This promises to reduce significantly the effects of an anisoplanatic PSF and will make digital image processing more reliable and less uncertain.

Compensating for hexagonal segments

Technologies to compensate for the diffraction patterns produced by straight line gaps and straight-line support structures across primary mirrors of large telescopes has been an area of active study recently ^{16,17,18,19,20,21}. None of these methods may be completely satisfactory, however, since light is absorbed in the process.

6.1

Background

Breckinridge (2018)²² suggested partitioning the primary into curved sided segments and curving the secondary support structures to reduce diffraction noise at its source to control diffraction noise at the image plane of exoplanet coronagraphs. We have shown above that the hexagonal segment architecture or pupil topology leads to unwanted diffraction noise in the system. It is good engineering practice to seek ways to eliminate or reduce “noise” at its source, rather than devise complicated and signal absorbing methods to compensate. Methods to mitigate diffraction noise were developed over the years by amateur astronomers, and later optical scientists. However, the professional space and ground astronomical optical telescope and instrument community seems not aware of these techniques.

C.H. Werenskiold (1941)²³ reported on the work of A. Couder published in the French journal: Astronomy, Jan 1934 and translated into English and republished in Amateur Telescope Making Advanced (scientific American Publishing), pp 620-622. Couder proposed controlling diffraction in Newtonian and Cassegrain type telescopes by placing lune shaped curved masks over the straight edge support structure of the secondary. These masks blocked significantly, the light-gathering ability of the telescope, negatively affecting the telescope transmittance. Werenskiold proposed curving the secondary support structures themselves as shown below in Fig 6 to reduce masking of the primary mirror and control the diffraction spikes.

Figure 6

The four secondary support structures built by Werenskiold¹⁴ and used for visual observation of planets. View looking from the open end of a reflecting telescope back toward the primary mirror. He reports that #4 gives the lowest quality image and that curving the support structure appears to remove the diffraction spikes from visual images to give higher contrast for visual planetary observations.

Werenskiold writes:

“It is generally conceded that a reflector, in regard to definition obtained, is apt to be somewhat inferior to a refractor of comparable size. However, the use of a curved spider in a reflector appears to be a promising step towards reducing this difference in comparative performance of the two telescope types.”

Pupil architecture of topology

Richter²⁴ selected 6 diffraction masks and photographed the diffraction pattern from each to show that curved arcs on the pupil left no discernible diffraction pattern at the image plane. Harvey²⁵ (in this volume) used FRED²⁶ and applied the design methodology outlined in Richter¹⁵ and developed further by Harvey and Ftaclas²⁷ along with the computer analysis program FRED to show that the image plane diffraction patterns from curved secondary support structure is less than 10^-6 where-as the image plane diffraction patterns from straight line secondary support structure are ~10^-2.

Based on our intuitive understanding of diffraction from curved segments we designed a pupil topology for a “first look” at the diffraction effects. The design we chose is shown in Figure 8 below. We selected a 10-meter Cassegrain primary with an obscuration ratio of 0.16 and six curved secondary mirror support struts, each with a30° arc of a circle and 20 mm wide gaps. There are three rings or zones of segments curved on all sides. Each zone contains 12 curved sided segments to create a telescope entrance pupil that has 36 segments. This topology is shown in Figure 7, below.

Figure 7

Cassegrain primary with an obscuration ratio of 0.16 and six curved secondary mirror support struts (shown in Red), each with a 30° arc of a circle and 20 mm wide. These secondary support struts cast shadows on the primary mirror. Since the optical system stop is at the primary mirror, these shadows will move across the aperture as the image plane field point is changed. There are three rings or zones of mirror segments curved on all sides. Each zone contains 12 curved sided segments to create a telescope entrance pupil that has 36 segments.

Figure 8

Simulated diffraction effects for an annular aperture (□ = 0.16) and three straight spiders (w = 0.01D) as each spider is sequentially converted to an arc of a circle.

Figure 8, below shows how

Figure 8 shows the dramatic change in diffracted light at the image plane as we curve the secondary support system.

Figure 9 shows a plot of monochromatic intensity, on a linear scale, as a function of azimuth angle at field 0.75 arc sec. for three 10-m diameter pupil architectures: unobstructed, spider and pinwheel. The computation was performed using MatLab. Computational capacity limited the size of the sample interval across the pupil and we believe that may have resulted in an incorrect representation of the image plane diffracted light. However, several features in Fig 9 are worth noting. The profile for the 4-spider mask, shown in red, is much higher than that for the pinwheel which indicates that the pinwheel pupil is making a contribution to smoothing out the diffraction pattern. The prominent dip in energy at the feet of the spider diffraction pattern is probably the result of the very narrow bandwidth of this monochromatic computation. The noise on the pinwheel is probably caused by having an insufficient number of samples across the pupil, which was dictated by the array sizes and the computational time limits of Matlab. Consequently, we decided to drop MatLab as our computational tool and turn to Photon Engineering. LLC and the FRED software. Computations using FRED were successful and are shown in the paper: SPIE Proc 10698-60²⁷.

Figure 9

Image plane linear-scale monochromatic intensity distribution from three 10-m diameter pupil topologies: unobstructed, spider and pinwheel are shown at field radius 0.75 arc-second for azimuth angles -180 to + 180 degrees. Monochromatic diffraction patterns for a modified LUVOIR-B aperture stretched from 8-m to 10-m diameter with 5-mm segment gaps is compared to a 10-m diameter pinwheel aperture with 5-mm segment gaps. Top row L to R: modified LUVOIR-B aperture map, logio stretch of the PSF and a plot of the normalized irradiance as a function of radius for 0° (red) and 30° (blue dots) azimuth angle. Bottom row L to R: pinwheel aperture map, log io stretch of the PSF and a plot of the normalized irradiance as a function of radius for 0° (green) and 30° (grey dots) azimuth angle. The center plots on a blue background show the expected system point spread function that falls onto the coronagraph occulting mask. The PSF structure shown at top-center indicates that many false-positive identification of exoplanets may occur and exoplanets can be masked if a hex segmented aperture is used.

Are curved sided segments purely in the imagination of the theorist, or can one fabricate curved sided segments to optical tolerances? If yes, then we can create a large aperture in space for a deployable full-aperture diffraction limited space telescope. Ten or even twenty meter clear aperture image quality will be possible. In the next section, we suggest physical processes and techniques to enable very large aperture space telescopes for the next generation.