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Point Spread Function of Hexagonally Segmented Telescopes by New Symmetrical Formulation MNRAS 000,1–12 (2018) Preprint 8 November 2018 Compiled using MNRAS LATEX style file v3.0 Point Spread Function of Hexagonally Segmented Telescopes by New Symmetrical Formulation S. Itoh,1? T. Matsuo,1 H. Shibai,1 T. Sumi1 1Osaka university, Faculty of earth and space science, 1-1, Machikaneyama-cho, Toyonaka, Osaka, Japan, 560-0043 Accepted XXX. Received YYY; in original form ZZZ ABSTRACT A point spread function of hexagonally segmented telescopes is derived by a new symmetrical formulation. By introducing three variables on a pupil plane, the Fourier transform of pupil functions is derived by a three-dimensional Fourier transform. The permutations of three variables correspond to those of a regular triangle’s vertices on the pupil plane. The resultant diffraction amplitude can be written as a product of two functions of the three variables; the functions correspond to the sinc function and Dirichlet kernel used in the basic theory of diffraction gratings. The new expression makes it clear that hexagonally segmented telescopes are equivalent to diffraction gratings in terms of mathematical formulae. Key words: telescope – methods: analytical – instrumentation: miscellaneous 1 INTRODUCTION tion for a regular hexagonal aperture and a hexagonally-truncated triangular grid function (Figure1). Thus, due to the convolution the- Large telescopes have two merits: they send more light to detec- orem, the Fourier transform of the pupil functions of the hexagonally tors and enhance spatial resolution. However, building large tele- segmented telescopes basically becomes a product of the Fourier scopes using just one mirror is technically difficult, so segmented transform of the two functions (Nelson et al. 1985; Zeiders & Mont- mirrors are useful for building large telescopes. The largest ground- gomery 1998; Chanan & Troy 1999; Yaitskova et al. 2003a). This based, single-mirror telescope is the Large Binocular Telescope calculation is equivalent to the basic theory of diffraction gratings (LBT) (Hill et al. 2006) with an 8.4 m aperture diameter. All larger- (e.g. Born et al. 2000), where the hexagonally segmented telescopes diameter ground-based telescopes are segmented (Buckley & Stobie are equivalent to diffraction gratings (Yaitskova et al. 2003b). 2001; Hill et al. 2004; Geyl et al. 2004; Mast & Nelson 1988). In ad- Studies of the Fraunhofer diffraction of regular polygons in- dition, future telescopes, such as the James Webb Space Telescope clude an analytical expression in the polar coordinate system (JWST) (Codona & Doble 2015), Thirty Meter Telescope (TMT) (Komrska 1972), analytical expressions as a superposition of ar- (Nelson & Sanders 2008), European Extremely Large Telescope (E- bitrary isosceles triangles, trapezoids (Smith & Marsh 1974), and ELT) (Comley et al. 2011), and Giant Magellan Telescope (GMT) arbitrary triangles (Sillitto 1979), and an analytical expression with (Johns et al. 2012), are also segmented. Abbe transform (Komrska 1982). T. S. Mast and J. E. Nelson proposed using the hexagonally On the other hand, Zeiders & Montgomery(1998) derived packed segmented mirror to construct the Ten Meter Telescope an analytic solution for the Fourier transform of the hexagonally- (the current Keck Telescopes) in the late 1970s (Mast & Nelson truncated triangular grid function, but do not explain how. 1979). There are some numerical calculations for evaluating its In this paper, the PSF of hexagonally segmented telescopes is Point-Spread Function (PSF): the squared modulus of the Fourier arXiv:1811.02762v1 [astro-ph.IM] 7 Nov 2018 newly formulated. In Section 2, a key equation for the formulation transform of pupil functions (e.g. Yaitskova & Dohlen 2002; Ney- is derived. In Section 3, the PSF is derived and some interpretations man & Flicker 2007; Codona & Doble 2015). There are also the of the resultant equation are discussed. several analytical approaches (e.g. Nelson et al. 1985; Zeiders & Montgomery 1998; Chanan & Troy 1999; Yaitskova et al. 2003a). Accurate numerical calculations require significant amounts of time and memory capacity; thus, the Fast Fourier Transform (FFT), an 2 FOURIER TRANSFORM WITH optimized algorithm of Discrete Fourier Transform (DFT), is not REGULAR-TRIANGULAR SYMMETRY always the best solution. (Dong et al. 2013; the Association of Uni- versities for Research in Astronomy 2018). 2.1 Underlying Mechanism The pupil function (or aperture function) for the hexagonally segmented telescope is basically a convolution of the pupil func- In this study, the Fraunhofer diffraction of the hexagonally seg- mented telescopes is used to derive the PSF. The underlying mech- anism of this work involves using a regular triangle’s symmetry ? E-mail: [email protected] with the three variables on the pupil plane. This is because, as can © 2018 The Authors 2 S. Itoh et al. Table 1. Symmetric operations of a regular hexagon (left-hand side) ex- pressed by the synthesis operation of symmetric operations of a regular triangle and rotation by π radians (right-hand side). R¹θº: rotation by θ radians S¹θº: reflection for the line expressed by y = tan¹θºx ◦: symbol that represents the synthesis of operations R ¹0º = R ¹0º S ¹0º = S ¹0º π ¹ º ◦ 4π π ¹ º ◦ 2π R 3 = R π R 3 S 6 = R π S 3 2π 2π 2π π R 3 = R 3 S 6 = S 3 3π ¹ º 3π ¹ º ◦ ¹ º R 3 = R π S 6 = R π S 0 4π 4π 4π 2π R 3 = R 3 S 6 = S 3 5π ¹ º ◦ 2π 5π ¹ º ◦ π R 3 = R π R 3 S 6 = R π S 3 (Table1). The symmetry operations of a regular triangle are the rotations by 2πn ¹n = 0; 1; 2º and the reflections for the straight line 3 πn ¹ º Figure 1. Explanation of a hexagonally-truncated triangular grid function. A of y = tan 3 x n = 0; 1; 2 (Figure2). Actually, the symmetry hexagonally-truncated triangular grid function is a sum of the delta functions operations of a regular triangle are permutations of its vertices. The whose peaks are located on the pupil plane. In this figure, the red dots permutations of the regular triangle vertices have one-to-one cor- located on segment centre indicate the delta function peak. In the case respondence with the permutations of the variables of functions in N = of 2, the telescope consists of the green, cyan, and orange hexagonal calculations. This one-to-one correspondence is realized by using segments. In the case of N = 1, the telescope consists of the cyan and orange not two, but three, variables (see AppendixA). When there is this hexagonal segments. In the case of N = 0, the telescope consists of the orange hexagonal segment. Generally, the positions of the delta functions’ one-to-one correspondence, functional symmetry in resultant ex- peaks are all the grid-points encircled by a hexagon. This envelop hexagon pressions is formed such that they reflect the symmetry of the pupil is obtained by rotating the unit hexagon by π radians and enlarging it to plane. Symmetry of function, as considered here, is a property that p 2 have the side length of N 3 (e.g. the magenta dashed hexagons (N=1,2)). allows the values of a function to be unchanged even if its variables f ¹x; yº Using the three variables defined in Section 2.3, the areap encircled by the are permuted. For example, when a function, , satisfies the enveloping hexagon is expressed by jAj; jBj; jCpj ≤ N 3. Thus,p the grid- equation, f ¹x; yº = f ¹y; xº, it is symmetric for the permutation of ppoints included in this area are expressed by A = 3K; B = 3L; and C = two variables. 3M, where K, L, M are integers; K + L + M = 0; jK j; jL j; jM j ≤ N. 2.2 Definition Hereafter, normalized Cartesian coordinate systems are used both on the pupil and image planes. The coordinates are represented by ¹x; yº on a pupil plane and ¹α; βº on an image plane. Both planes are perpendicular to the optical axis. The x-axis and α-axis are parallel to each other, and as are the y-axis and β-axis. ¹x; yº and λf ¹α; βº are normalized by Λ and Λ, respectively. λ, f , and Λ are Figure 2. Reflection symmetry of a regular hexagon and regular triangle. the wavelength of light, focal length, and side-length of the unit The dashed lines indicate the axes of the reflection transforms included in hexagonal segment, respectively. the symmetry operation. Then, the delta function (see AppendixB) is defined as follows: sin¹πT xº be seen below, the pupil function with regular-hexagonal symmetry δ¹xº = lim : (1) also has regular-triangular symmetry. T!1 πx According to H. Weyl, when a figure’s shape and position In addition, a complex rectangular function is defined as fol- are maintained even if the figure is converted by an operation, lows: it can be said that the figure is symmetric concerning the oper- ation (Weyl 1952). These operations are referred to as ’symme- Rect¹zº = lim Π¹zº ¹U is a natural numberº (2) try operation’ throughout this paper. The symmetry operations U!1 πn ¹ º of a regular hexagon are the rotations by 3 n = 0; 1; 2; 3; 4; 5 radians and the reflections for the straight line expressed by 1 y = tan πn x ¹n = 0; 1; 2; 3; 4; 5º (Figure2). Every symmetry Π¹zº = : (3) 6 ¹2zº2U + 1 operation of a regular hexagon can be represented by synthesizing the symmetry operations of a regular triangle and π radian rotation When z is a real number denoted by x, Rect¹xº is the usual rectan- MNRAS 000,1–12 (2018) Segmented Telescope PSF by Symmetric Formulation 3 gular function as follows: (a) 8 1 jxj 1 > 2 = 2 > ¹ º >< j j 1 Rect x = 0 x > 2 : (4) > > jxj < 1 > 1 2 : b+ 2.3 Selection of Three Variables y a+ c+ Three variables on the pupil plane, a, b, and c, are selected as follows (Figure3): 2 a = − p x 3 1 b = p x + y 3 1 x c = p x − y: (5) 3 In the same manner, in order to satisfy Equation (12), three variables (b) on the image plane, p, q, and r, are also selected as shown below: 1 q = − p α 3 1 1 r = p α + β 2 3 2 1 1 s = p α − β: (6) 2 3 2 In addition, replacing the lower-case letter, e.g.
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