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

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 segmented telescopes is used to derive the PSF. The underlying mechanism 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

Table 1. Symmetric operations of a regular hexagon (left-hand side) expressed by the synthesis operation of symmetric operations of a regular triangle and rotation by π radians (right-hand side). R(θ): rotation by θ radians S(θ): reﬂection for the line expressed by y = tan(θ)x ◦: symbol that represents the synthesis of operations

R (0) = R (0) S (0) = S (0) π ( ) ◦ 4π π ( ) ◦ 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 ( ) 3π ( ) ◦ ( ) R 3 = R π S 6 = R π S 0

4π 4π 4π 2π R 3 = R 3 S 6 = S 3 5π ( ) ◦ 2π 5π ( ) ◦ 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 (n = 0, 1, 2) and the reflections for the straight line 3 πn ( ) 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 √ 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 (x, y) Using the three variables defined in Section 2.3, the area√ encircled by the are permuted. For example, when a function, , satisfies the enveloping hexagon is expressed by |A|, |B|, |C√| ≤ N 3. Thus,√ the grid- equation, f (x, y) = f (y, x), it is symmetric for the permutation of √points 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; |K |, |L |, |M | ≤ N.

2.2 Definition Hereafter, normalized Cartesian coordinate systems are used both on the pupil and image planes. The coordinates are represented by (x, y) on a pupil plane and (α, β) 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. (x, y) and λf (α, β) 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:

sin(πT x) be seen below, the pupil function with regular-hexagonal symmetry δ(x) = lim . (1) also has regular-triangular symmetry. T→∞ 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 operation (Weyl 1952). These operations are referred to as ’symme- Rect(z) = lim Π(z)(U is a natural number) (2) try operation’ throughout this paper. The symmetry operations U→∞ πn ( ) 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 (n = 0, 1, 2, 3, 4, 5) (Figure2). Every symmetry Π(z) = . (3) 6 (2z)2U + 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, Rect(x) is the usual rectan-

MNRAS 000,1–12 (2018) Segmented Telescope PSF by Symmetric Formulation 3 gular function as follows: (a)  1 |x| 1  2 = 2  ( )  | | 1 Rect x = 0 x > 2 . (4)   |x| < 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 = − √ x 3 1 b = √ x + y 3 1 x c = √ 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 = − √ α 3 1 1 r = √ α + β 2 3 2 1 1 s = √ α − β. (6) 2 3 2 In addition, replacing the lower-case letter, e.g. a, with its upper- π case counterpart, e.g. A, denotes a rotation by 2 radians. This operation swaps the direction of the vertex and the centre of the regular hexagonal side: 2 A = − √ y 3 1 B = − √ y + x 3 1 C = − √ y − x (7) Figure 3. (a) Meaning of the variables, a, b, and c. The black lines indicate 3 the contour of a = i, b = j, and c = k (i, j, and k are integers). The red arrows labeled by a+, b+, and c+ indicate the direction in which a, 1 b, and c get larger, respectively. (b) Explanation of the pupil function of a Q = √ β unit hexagon, as expressed by Equation (10). The semi-transparent magenta, 3 cyan, and yellow regions indicate the regions that satisfy |a| ≤ 1, |b | ≤ 1, 1 1 R = − √ β + α and |c | ≤ 1, respectively. The unit regular hexagonal aperture is the area in 2 3 2 which they overlap, shown in light gray. 1 1 S = − √ β − α. (8) 2 2 3 expressed by the synthesis of the sign inversion and permutations These variables satisfy the following equation: of the three variables. a + b + c = A + B + C = q + r + s = Q + R + S = 0. (9) The diffraction amplitude for the hexagonally segmented telescope is a product of the diffraction amplitude of a single unit regular hexagonal aperture and a hexagonally-truncated triangular The equations for the definitions of a, b, and c, can be inter- grid function (Nelson et al. 1985; Zeiders & Montgomery 1998; preted as contour line equations of a, b, and c, respectively. Those Chanan & Troy 1999; Yaitskova et al. 2003a). The pupil function straight lines are parallel to each side of a regular triangle; the same of a unit hexagon (Figure3) is is true for the other sets of variables defined above. a b c There is a one-to-one correspondence between the permuta- P¯u(a, b, c) = Rect Rect Rect . (10) tions of these variables and the symmetry operations of a regular 2 2 2 triangle. For example, transposing a and b [(a, b, c) → (b, a, c)] indi- The pupil function of the hexagonally-truncated hexagonal gird cates a reflection transform. Cyclic permutation, that is (a, b, c) → function (Figure1) is (c, a, b), indicates the rotation by 2π radians. On the other hand, 3 Õ √ √ √ sign inversion (a, b, c) → (−a, −b, −c) indicates the rotation by π P¯g(a, b, c) = δ(A − 3K)δ(B − 3L)u(C − 3M), radians. Hence, every symmetric operation of a regular hexagon is |K |, |L |, |M | ≤N

MNRAS 000,1–12 (2018) 4 S. Itoh et al.

(11)

sin(πT x) where K + L + M = 0 and u(x) = limT→∞ πT x . When (a, b, c) are interpreted as coordinates of the 3-D Carte- sian system (see Figure4), while ignoring the condition of Equa- tion (9), the value of Equation (10) becomes unity inside a cube and zero outside the cube. In the same manner, Equation (11) becomes a sum of delta functions whose peaks are located on the cubically-truncated 3-D grid. These functions of the three variables freed from the condition of Equation (9), are called cubic ’pupil functions’ throughout this paper. This equation represents the inner product of (α, β) and (x, y) with the variables deﬁned above: qa + rb + sc = QA + RB + SC = αx + βy. (12)

The following equations are also used to relate the variables denoted by lower-case and upper-case letters with each other: b − c c − a a − b Figure 4. By considering the three variables defined by Equation (5) as the A = √ , B = √ , C = √ (13) coordinates of the 3-D Cartesian coordinate system, Equation (10) represents 3 3 3 a regular hexagon on the plane expressed by Equation (9); the color map and contour lines indicate the value of c on the regular hexagon. When r − s s − q q − r the condition of Equation (9) is ignored, Equation (10) represents a cube Q = √ , R = √ , S = √ (14) 3 3 3 (regular hexahedron), and the black solid frame indicates the sides of the cube. B − C C − A A − B a = √ , b = √ , c = √ (15) − − − Hence, 3 3 3 √ 3 ¹ ¹ ¹ ∞ v(α, β)= dq0dr 0ds0δ(q0−s0)δ(r 0−s0) R − S S − Q Q − R 2 q = , r = , s = . −∞ √ √ √ (16) ∞ − 3 − 3 − 3 ¹ ¹ ¹ 0 0 0 × dadbdcP¯(a, b, c)e−2πi((q +q)a+(r +r)b+(s +s)c) −∞ √ 3 ¹ ∞¹¹¹ ∞ 2.4 Formulation of Fourier Transform with = dτ dadbdcP¯(a, b, c)e−2πi((q+τ)a+(r+τ)b+(s+τ)c), Regular-triangular Symmetry 2 −∞ −∞ (20) The Fourier transform of a pupil function into the 3-D Fourier 0 transform of a cubic pupil function is now developed. Here, where τ = s has been redefined. ( ) P(x, y) = P¯(a(x, y), b(x, y), c(x, y)) denotes a pupil function, and When the variables a, b, c are interpreted as coordinates of a v(α, β) denotes the Fourier transform of the pupil function. By us- 3-D Cartesian coordinate system, ing Equations (12) and (9), the following equation is obtained: ¹ ¹ ¹ ∞ dadbdcP¯(a, b, c)e−2πi((q+τ)a+(r+τ)b+(s+τ)c) (21) ¹ ¹ ∞ −∞ v(α, β) = dxdyP(x, y)e−2πi(αx+βy) −∞ is a 3-D Fourier transform of P¯(a, b, c). In the 3-D Cartesian coor- √ ∫ ∞ ¹ ¹ ∞ dinate system, this integral, dτ, indicates that this line integral 3 −2πi(qa+rb+s(−a−b)) −∞ = dadbP¯(a, b, −a − b)e . (17) along a straight line is passing through the point, (a, b, −a − b) and 2 −∞ is perpendicular to the plane a + b + c = 0. By using delta function, Equation (17) becomes The right-hand sides of Equations (10) and (11) can be written √ as f (a) f (b) f (c) by a function denoted by f (a). When P¯(a, b, c) can 3 ¹ ¹ ¹ ∞ v(α, β)= dadbdcP¯(a, b, c)δ(a + b + c)e−2πi(qa+rb+sc). be written as f (a) f (b) f (c), the 3-D Fourier transform of P¯(a, b, c) 2 −∞ becomes f˜(q) f˜(r) f˜(s), where f˜(q) denotes the Fourier transform (18) of f (x). Thus, Equation (20) becomes √ Here, the integral variables, a, b, and c, are no longer limited to the ¹ ∞ 3 Ö ˜ a + b + c = v(α, β) = dτ f (q + τ), (22) region , 0. 2 −∞ Since the inner products of the two functions are equal to the C {q,r,s} ( ) Î inner products of the Fourier transforms of the two functions, v α, β where C {a,b,c} indicates the product over the cyclic permutations becomes of {a, b, c}. √ ¹ ¹ ¹ ∞ ¹ ¹ ∞ ∗ For example, f˜(q) basically become the form of sinc function 3 0 0 0 −2πi((q0−s0)a+(r0−s0)b) v(α, β)= dq dr ds dadbe sin n+ 1 x 2 −∞ −∞ sin x 2 (Figure5), x (Howell 2001), and Dirichlet kernel, x ∞ sin( 2 ) ¹ ¹ ¹ 0 0 0 × dadbdcP¯(a, b, c)e−2πi((q +q)a+(r +r)b+(s +s)c) . (Howell 2001), for the pupil functions of Equations (10) and (11), −∞ respectively. Both of these functions are used in basic diffraction (19) grating theory (e.g. Born et al. 2000).

MNRAS 000,1–12 (2018) Segmented Telescope PSF by Symmetric Formulation 5

Hence, Equation (22) becomes √ 3 ¹ ∞ Ö vu(α, β) = dτ f˜u(q + τ) 1.0 2 −∞ C {q,r,s} γ Õ Õ Õ 2πi(σ1q+σ2r+σ3s) = σ1σ2σ3e 0.8 2πi σ1=±1 σ2=±1 σ3=±1 ¹ ∞ × dzg(z), (26) 0.6 −∞ where τ ∈ R has been substituted with z ∈ C , and γ and g(z) are 0.4 deﬁned as follows: √ 3 γ = (2πi)−2 (27) 0.2 2

e2πi(σ1+σ2+σ3)z 0 g(z) = . (28) (q + z)(r + z)(s + z)

( πq) ( πr) ( π s) The right-hand side of Equation (26) can be evaluated for all Figure 5. 3-D plot of sin 2 sin 2 sin 2 . This function is normalized πq πr π s (q, r, s) q − r r − s by the value of the origin and is shown in absolute values. The line integral by dividing into three cases as follows; (i) , , and along a line perpendicular to the plane, q + r + s = 0, gives an expression s − q are all zero, (ii) one of {q − r , r − s, s − q} is zero but the for the diffraction amplitude of a unit regular-hexagonal aperture. others are not, and (iii) none of {q − r , r − s, s − q} is zero. First, case (iii) is treated. The other cases are obtained in Ap- pendixC as limiting values of the results for case (iii). An integral path, as shown in Figure6, is considered for g(z) in Equation (26). The same equation is true for the Fourier transform of There are the semicircles for avoiding singularities, denoted by Γ−q, P¯(A, B, C), if (q, r, s) in Equation (22) is replaced by (Q, R, S). Γ−r , and Γ−s, and the outer semicircle denoted by Γρ. The radius of Γρ approaches infinity (ρ → ∞), and the radii of Γ−q, Γ−r , and Γ−s approach zero ( → 0). By applying the Cauchy integral theorem along the integral path, the last factor of Equation (26) can be calculated as follows: 3 FORMULATION OF PSF OF HEXAGONALLY ¹ ∞ ¹ ¹ ¹ ! SEGMENTED TELESCOPE dzg(z) = − + + dzg(z). (29) −∞ Γ−q Γ−r Γ−s In this section, we derive an analytical expression for the diffraction When σ1 + σ2 + σ3 is positive, Equation (29) becomes amplitude of hexagonally segmented telescopes. The diffraction ∫ ∞ amplitude, vc(α, β), can be basically written as the product of the −∞ dzg(z) Fourier transform of a unit regular-hexagonal aperture, vu(α, β), ¹ 0 Õ iθ iθ and a hexagonally-truncated triangular grid function (Figure1), = − lim ie dθg(e − q) →0 π vg(α, β), as follows (e.g. Nelson et al. 1985): C {q,r,s} iθ ¹ π Õ e2πi(σ1+σ2+σ3)(−q+e ) vc(α, β) = Scale [vu(α, β)] vg(α, β), (23) = i lim dθ →0 (r − q + eiθ )(s − q + eiθ ) 0 C {q,r,s} where the scaling transformation, Scale [vu(α, β)], is defined as Õ e2πi(σ1+σ2+σ3)(−q) = iπ , (30) 2 (r − q)(s − q) Scale [vu(α, β)] = (1 − ∆) vu ((1 − ∆)α, (1 − ∆)β), (24) C {q,r,s} Í and ∆ denotes √2 times the half width of the gap between adjacent where C {q,r,s} indicates the sum over the cyclic permutations of 3 {q, r, s}. When σ + σ + σ is negative, this equation becomes segments; 1 − ∆ corresponds to the side-length of a segment. 1 2 3 ∫ ∞ −∞ dzg(z) Õ ¹ 0 = − lim ieiθ dθg(eiθ − q) →0 −π 3.1 Unit Regular Hexagon C {q,r,s} Õ e2πi(σ1+σ2+σ3)(−q) First, Equation (22) is applied to the pupil function of regular- = −iπ . (31) (r − q)(s − q) hexagonal aperture whose side length is unity. The pupil function, C {q,r,s} P¯u(a, b, c), is shown in Equation (10); the Fourier transform of the rectangular function becomes the sinc function as follows: Equation (31) is equivalent to Equation (30), except for the sign inversion. ¹ ∞ In order to calculate v (α, β), the following terms in Equation a −2πiqa u f˜u(q) = da Rect e −∞ 2 (26) are summed up: ( πq) γ ¹ ∞ sin 2 2πi(σ1q+σ2r+σ3s) = . (25) σ1σ2σ3e dzg(z). (32) πq 2πi −∞

MNRAS 000,1–12 (2018) 6 S. Itoh et al.

(a) σ1 + σ2 + σ3 > 0 Table 2. Calculation of signs contributing to the summation in Equations (26) and (39). Í Í Í Im z S( i σi ) = {1 (for i σi > 0), −1 (for i σi < 0)} Í σ1 σ2 σ3 i σi S σ1σ2σ3 Sσ1σ2σ3 Γρ 1 1 1 1 3 1 1 1

2 1 1 -1 1 1 -1 -1

Γ 3 1 -1 1 1 1 -1 -1 q Γ r Γ s − − − 4 1 -1 -1 -1 -1 1 -1

5 -1 1 1 1 1 -1 -1 ρ ρ − q ϵ r ϵ s ϵ Re z 6 -1 1 -1 -1 -1 1 -1 − ± − ± − ± 7 -1 -1 1 -1 -1 1 -1

8 -1 -1 -1 -3 -1 -1 1

4 + 5 are calculated as follows:

 2 + 7  (b) σ1 + σ2 + σ3 < 0      3 + 6     4 + 5    Im z ( ( − )) ( (− )) ( ( − ))  cos 2π r s + cos 2π s+q + cos 2π q+r 2s   (r−q)(s−q) (s−r)(q−r) (q−s)(r−s)   cos(2π(−r+s)) cos(2π(s+q−2r)) cos(2π(q−r))  =−γ  + +  .(34)  (r−q)(s−q) (s−r)(q−r) (q−s)(r−s)   cos(2π(−r−s+2q)) cos(2π(−s+q)) cos(2π(q−r))   + +   (r−q)(s−q) (s−r)(q−r) (q−s)(r−s)  Thus, by substituting these sums into the right-hand side of Equation q ϵ r ϵ s ϵ (26), v (α, β) is expressed as follows: − ± − ± − ± u ρ ρ Re z Õ cos(2π(r − s)) − vu(α, β)=2γ . (35) Γ q Γ r Γ s (s − q)(q − r) − − − C {q,r,s}

By using Equation (14), Equation (35) can be written as Γρ √ 2 Õ cos(2π 3Q) vu(α, β)=2π γ √ √ . (36) C {Q,R,S} (π 3R)(π 3S)

This is the resultant expression for the diﬀraction amplitude of the Figure 6. (a) Integral path in case (iii)for positive σ +σ +σ . The integrand 1 2 3 unit regular hexagon pupil function. must be regular inside the integral paths. (b)Integral path in case (iii) for Equation (36) is equivalent to some previous works concern- negative σ1+σ2+σ3. The integrand must be regular inside the integral paths. ing a unit hexagonal aperture (e.g. Nelson et al. 1985). The proof ( ) ( ) 1 ( ( − ) − ( )) uses the equation; sin u sin v = 2 cos u v cos u + v . Some studies (Smith & Marsh 1974; Chanan & Troy 1999) have obtained the analytical expression by superposing some trapezoids. It is clear that these expressions are also equivalent to Equation (36).

A part of the summation in Equation (26) is 3.2 Hexagonally-Truncated Triangular Grid Function

Õ cos(2π(r + s − 2q)) Next, the diﬀraction amplitude of a√ hexagonally-truncated triangular 1 + 8 = γ , (33) (r − q)(s − q) grid function, with an interval of 3, is derived. The pupil function C {q,r,s} shown in Equation (11) can be written as follows:

P¯g (A, B, C) where the encircled numbers, such as 1 , are identiﬁers of the sets  √  1 Ö  Õ sin(πT(A − 3K))  indicated in Table2, and 1 + 8 denotes the sum of the sets, 1 = lim √ . (37) T→∞ T ( − ) and 8 . In the same manner, the summations, 2 + 7 , 3 + 6 , and C {A,B,C }  |K | ≤N π A 3K    MNRAS 000,1–12 (2018) Segmented Telescope PSF by Symmetric Formulation 7

The Fourier transform of a factor in Equation (37) is a product of 3.3 Combined Expression the rectangular function and Dirichlet kernel as follows: As shown in Equations (23) and (24), the product of Equations √ ¹ ∞   (36) and (44) provides the following expression for the diffraction  Õ sin(πT(A − 3K))  −2πiQA f˜g(Q) = dA √ e amplitude of a hexagonally segmented telescope: −∞ ( − )  |K | ≤N π A 3K  √   √ 4 2 Õ cos(2(1 − ∆)π 3Q) Q sin((2N + 1) 3πQ) vc(α, β) = 4π γ √ √ = Rect √ . (38) C {Q,R,S } (π 3R)(π 3S) T sin( 3πQ) Õ cos(3(2N + 1)πq) Hence, the diffraction amplitude, ν(α, β), can be derived from Equa- × , (45) sin(3πs) sin(3πr) tion (22) as follows: C {q,r,s} √ 3 1 ¹ ∞ Ö where the variables, q, r, s, Q, R, and S are defined in Equations vg(α, β) = lim dτ f˜g(Q + τ) (6) and (8); ∆ denotes √2 times the half width of the gap between 2 T→∞ T −∞ C {Q,R,S } 3 √ adjacent√ segments (1 − ∆ corresponds to the side length of a seg- 3 Õ Õ Õ ment.); 3 is the distance between the centres of adjacent hexagons; = σ σ σ e2πi(σ1Q+σ2 R+σ3S) 3 1 2 3 γ Í 2(2i) is defined in Equation (27), and C {x1,x2,x3 } indicates the sum σ1=±1 σ2=±1 σ3=±1 over the cyclic permutations of {x1, x2, x3}. This expression has 1 ¹ ∞ × lim dzΠ3(z)h(z), (39) removable singularities (see AppendixC). T,U→∞ T −∞ It is easy to recognize that the functions in this expression are where h(z) and Π3(z) are defined as follows: symmetric against any symmetric operation of a regular hexagon √ (Section 2.3.). e(2N+1) 3πi(σ1+σ2+σ3)z It can also be recognized that the function in Equation (44) h(z) = √ (40) Î is similar to that of Equation (36). This similarity is the result of { } sin( 3π(Q + z)) C Q,R,S the symmetry of the functions and is clearly different in nature from the expressions in previous works (Smith & Marsh 1974; Ö Q + z Π3(z) = Π . (41) Nelson et al. 1985; Chanan & Troy 1999; Zeiders & Montgomery T C {Q,R,S } 1998). The periodicity of the denominator in Equation (44) makes this function periodic like the Dirichlet kernel of the diffraction- The right-hand side of Equation (39) can be evaluated for all grating theory (e.g. Born et al. 2000). In the neighborhood of the (Q, R, S) Q− R R−S by dividing into three cases as follows: (I) , , and point where q, r, and s are 1 times integers, the sine function of S − Q are all √1 times integers, (II) one of {Q − R , R − S, S − Q} is 3 3 the denominator in Equation (44) can be approximated by using 1 sin(x) j √ times integers but the others are not, and (III) none of {Q − R , the equation, limx→π j = (−1) (j is an integer.). Therefore, 3 x−π j j − − } √1 i R S, S Q is times integers. Equation (44) has diffraction peaks located at points, q = 3 , r = 3 , 3 k First, case (III) is treated. The other cases are provided in Ap- and s = 3 (i, j, and k are integers, i + j + k = 0). These integers, pendixC as limiting values of the case (III) result. i,j, and k, correspond to diffraction orders. Then, the profile of √ The last factor of Equation (39) for positive σ +σ +σ can be 1 1 2 3 each peak is 3 N + 2 times sharper than that in Equation (36) calculated as follows (see AppendixD): π and rotated by 2 radians. This nature was not revealed directly in 1 ¹ ∞ previously published expressions (Zeiders & Montgomery 1998) lim dzΠ3(z)h(z) T,U→∞ T −∞ where the regular-triangular symmetry was not focused. √ The former factor of Equation (45), which is same as Equation e−(2N+1) 3πi(σ1+σ2+σ3)Q Õ (36), is the diffraction amplitude of a single hexagonal aperture with = −i √ √ . (42) C {Q,R,S} sin 3π(Q − R) sin 3π(S − Q) a side-length of 1 − ∆, and this factor works just like the diffraction efficiency of a diffraction grating. The higher order peaks (i.e. the Then, multiplying −1 to the right-hand side of Equation (42) gives larger |i|,| j|, and |k|) are suppressed by this factor. When ∆ 1, the the correct equation for negative σ1 + σ2 + σ3. centres of the diffraction peaks become almost zero except for the As in Section 3.1, to calculate vg(α, β), central point. This is because the centres of the diffraction peaks, j−k i−j √ Q = √ , R = k√−i , and S = √ , are located closely to the points ¹ ∞ 3 3 3 3 2πi(σ1Q+σ2 R+σ3S) 1 3 σ1σ2σ3e lim dzΠ (z)h(z) where the diffraction efficiency factor of Equation (45) becomes 2(2i)3 T,U→∞ T −∞ j−k i−j zero, Q = √ , R = √ k−i , and S = √ . Examples of (43) 3(1−∆) 3(1−∆) 3(1−∆) Equation (45) results are in AppendixE. in Equation (39) is summed up with the use of Table2. The result is as follows:

2 Õ cos(3(2N + 1)πq) vg(α, β) = 2π γ . (44) 4 DISCUSSION sin(3πs) sin(3πr) C {q,r,s} In this paper, it must be noted that the following assumptions or This is the diﬀraction amplitude for a hexagonally-truncated trian- idealizations are made to derive an simple analytical expression for gular grid function. PSFs of hexagonally segmented telescopes. (1) Segmented mirrors The previously published expression (Zeiders & Montgomery are packed to make a hexagonal telescope as a whole. (2) Segment 1998) is equivalent to Equation (44). The proof uses the equation, patterns are given along surfaces of primary mirror with ﬁnite radius ( ) ( ) 1 ( ( − ) − ( )) sin u sin v = 2 cos u v cos u + v . of curvature. Thus, in fact, segment patterns projected onto pupil

MNRAS 000,1–12 (2018) 8 S. Itoh et al. planes are distorted from the patterns used in this paper. (3) The considered. Here, R is the radius of curvature of the primary mirror effects of obscuration by secondary-mirror units and spiders are normalized by the side length of the segments. Numerical calcula- ignored. tions are required for investigation of the effect of the distortion on In this section, how to evaluate diffraction amplitudes with the PSFs. additional considerations of these effects is discussed. ãĂĂ

4.2.2 Spider 4.1 Factors To Be Evaluated Analytically Since real spiders are complex structures, accurate evaluation of ef- 4.1.1 Segment Arrangement fects on diffraction amplitudes have to be done by careful numerical In the case of TMT or ELT, segmented mirrors are packed to make calculations. However, by using the variables used in this paper, a an approximately circular telescope as a whole. These pupil func- simplified spider extending in six directions radially from the cen- tions are obtained by removing the segments near the corners of ter with the width of w times segment side length can be expressed the hexagon from the arrangement in which the segmented mirrors simply; the values of pupil functions in the region where any one of | | | | | | w are packed to make a hexagonal telescope. Thus, the corresponding A , B , or C is equal to or less than 2 is set to zero. This effect diffraction amplitudes are obtained by subtracting those correspond- on the PSFs can be considered by numerical calculations. ing to the removed segments from the original amplitudes.√ √ When√ the six segments with centre points of (A, B, C) = ( 3K, 3L, 3M) (K + L + M = 0) and those expressed by the sign inversion and/or 5 CONCLUSION cyclic permutations of (K, L, M) are to be removed, subtracting Õ √ By using three variables (Subsection 2.3), the Fourier transform of vu(α, β) 2 cos(2π 3(QK + RL + SM)) (46) hexagonal aperture functions were related to the 3-D Fourier trans- C{K, L, M } form of cubic pupil functions (Subsection 2.4). These variables were from vc(α, β) is required; This removal has six-fold rotational sym- chosen so the permutations of the three variables would correspond metry. Also, central obscuration by secondary mirror units is ex- to the permutations of the regular triangle vertices. For regular tri- pressed by removing the segments near the central region. For ex- angles, permutations of vertices are symmetry operations. Thus, ample, in the case of N = 3, the Keck-type arrangement is obtained the functions in resultant expression have highly obvious regular- by removing central segment, (K, L, M) = (0, 0, 0). The TMT-type triangular symmetry. The resultant diffraction amplitude of a regular arrangement is the case of N = 13, and the segments to be removed hexagonal aperture is the Fourier transform of the 3-D equilateral are expressed by rectangular function integrated along a line perpendicular to the plane on which the sum of the three variables is zero (see Figures (K, L, M) 4 and5). The diffraction amplitude of the hexagonally-truncated = (0, 0, 0), (1, −1, 0), triangular grid function is the Fourier transform of the cubically- (12, −12, 0), (13, −13, 0), truncated 3-D grid function integrated in the same manner. The diffraction amplitudes of the unit regular hexagonal aperture and ( − j, − , j), ( , − j, −j) 13 13 13 13 + (47) hexagonally-truncated triangular grid function were derived in the , and their sign inversion and/or cyclic permutations, where j = same manner (Section 3.1 and 3.2). Thus, each function in resultant 1, 2, 3. The ELT-type arrangement is the case of N = 21. The centre expressions resembles the other in form. The remaining difference segments in the region of |K|, |L|, |M| ≤ 5 except for between them corresponds to the difference between sinc function and Dirichlet kernel used in basic diffraction grating theory(Howell (K, L, M) 2001). The new expression directly shows that hexagonally seg- = (5, −5, 0), (−5, 5, 0), (0, 5, −5), mented telescopes are diffraction gratings. (0, 5, −5), (−5, 0, 5), (5, 0, −5) (48) are removed. The corner segments to be removed are expressed by ACKNOWLEDGEMENTS (K, L, M) The anonymous reviewer has given valuable comments on this = (19, −19, 0), (20, −20, 0), (21, −21, 0), manuscript. Allow us to express our deep gratitude to the review. (20 − k, −20, k), (20, −20 + k, −k), The authors would like to thank Enago (www.enago.jp) for the (21 − l, −21, l), (21, −21 + l, −l) (49) English language review. , and their sign inversion and/or cyclic permutations, where k = 1, 2 and l = 1, 2, ..., 6. PSFs of Keck-type, TMT-type, and ELT-type pupils evaluated by these removals are shown in AppendixE. REFERENCES Born M., Wolf E., Bhatia A., 2000, Principles of Optics: Electromagnetic 4.2 Factors To Be Evaluated Numerically Theory of Propagation, Interference and Diffraction of Light. Cam- bridge University Press, https://books.google.co.jp/books? 4.2.1 Segment-pattern Distortion id=oV80AAAAIAAJ Buckley D., Stobie R., 2001, in The New Era of Wide Field Astronomy. Segments are made to be regular hexagons along the surface of pri- p. 380 mary mirrors. Hence, projected patterns of these onto pupil planes Chanan G., Troy M., 1999, Applied Optics, 38, 6642 are distorted from the ones considered so far in this paper. By re- Codona J. L., Doble N., 2015, Journal of astronomical telescopes, instru- x y placing (x, y) with R arcsin R , R arcsin R , this effect can be ments, and systems, 1, 029001

MNRAS 000,1–12 (2018) Segmented Telescope PSF by Symmetric Formulation 9

Comley P., Morantz P., Shore P., Tonnellier X., 2011, CIRP annals- Thus, Υ ◦ Υ must be identity mapping so as to satisfy the condi- manufacturing technology, 60, 379 tion that the mapping f : (x, y) → (X,Y) must have one-to-one Dong B., Qin S., Hu X., 2013, in International Symposium on Photoelec- 2π ◦ 2π correspondence. For example, R 3 R 3 is not identity map- tronic Detection and Imaging 2013: Laser Sensing and Imaging and Applications. p. 890516 2π 2π ping, where R 3 indicates a rotational transformation by 3 Geyl R., Cayrel M., Tarreau M., 2004, in Optical Fabrication, Metrology, Υ 2π and Material Advancements for Telescopes. pp 57–62 rad. Hence, cannot take R 3 . Consequently, the symmetric Hill G. J., MacQueen P. J., Ramsey L. W., Shetrone M. D., 2004, in Ground- operations of the regular triangle cannot be represented by the per- based Instrumentation for Astronomy. pp 94–108 mutations of two variables. Hill J. M., Green R. F., Slagle J. H., 2006, in Ground-based and Airborne Telescopes. p. 62670Y Howell K., 2001, Principles of Fourier Analysis. Textbooks in Math- ematics, CRC Press, https://books.google.co.jp/books?id= APPENDIX B: DELTA FUNCTION 000sSCyJOcAC Johns M., McCarthy P., Raybould K., Bouchez A., Farahani A., Filgueira Consider the following improper integral: J., Jacoby George andShectman S., Sheehan M., 2012, in Ground-based and Airborne Telescopes IV. p. 84441H ¹ ∞ ( ¹ 0 ¹ t0 ! ) Komrska J., 1972, Optica Acta: International Journal of Optics, 19, 807 dαe−2πiα(x−k) = + dαe−2πiα(x−k) . lim0 Komrska J., 1982, JOSA, 72, 1382 −∞ t→∞,t →∞ −t 0 Mast T. S., Nelson J. E., 1979, Technical Report (B1) Mast T. S., Nelson J., 1988, in European Southern Observatory Conference 0 and Workshop Proceedings. p. 411 In the operation, limt→∞,t0→∞, it is assumed that t = t in or- Nelson J., Sanders G. H., 2008, in Ground-based and Airborne Telescopes der to determine the improper integral value. Hence, the following II. p. 70121A equation is valid: Nelson J. E., Mast T. S., Faber S. M., 1985 ¹ ∞ ¹ t Neyman C., Flicker R., 2007, Keck Telescope Wavefront Errors: Implica- −2πiα(x−k) −2πiα(x−k) tions for NGAO dαe = lim dαe −∞ t→∞ −t Sillitto W., 1979, JOSA, 69, 765 sin 2πt(x − k) Smith R. C., Marsh J. S., 1974, JOSA, 64, 798 = lim →∞ Weyl H., 1952, Symmetry. Princeton paperbacks, Princeton University t π(x − k) Press, https://books.google.co.jp/books?id=T43Cmu_EaZAC = δ(x − k). (B2) Yaitskova N., Dohlen K., 2002, JOSA A, 19, 1274 Yaitskova N., Dohlen K., Dierickx P., 2003a, JOSA A, 20, 1563 Then, according to the theory of Fourier inverse transform, Yaitskova N., Dohlen K., Dierickx P., 2003b, in Future Giant Telescopes. pp ¹ ∞ ∫ ∞ −2πiαx 0 0 −2πiαx0 171–183 −∞ dαe dx f (x )e Zeiders G. W., Montgomery E. E., 1998, in Space Telescopes and Instru- −∞ ments V. pp 799–810 1 = lim ( f (x + ) + f (x − )) (B3) the Association of Universities for Research in Astronomy 2018, 2 →+0 webbpsf Documentation 8.3 Algorithms, Approximations, and Performance. https://media.readthedocs.org/pdf/webbpsf/ is valid for a piecewise smooth and square-integrable function, f (x). latest/webbpsf.pdf Equation (B3) becomes the following equations: ¹ ∞ ∫ ∞ 0 −2πiα(x−x0) 0 −∞ dx dαe f (x ) −∞ 1 APPENDIX A: PERMUTATION OF TWO VARIABLES = lim ( f (x + ) + f (x − )) (B4) AND SYMMETRY OPERATIONS OF REGULAR 2 →+0 TRIANGLE ¹ ∞ X Y x 0 0 0 1 Two variables, and , are defined using Cartesian coordinates, dx δ(x − x ) f (x ) = lim ( f (x + ) + f (x − )) . (B5) and y. −∞ 2 →+0 ( ) The function, f (x0), in Equation (B5) cannot be substituted by X = ξ x, y 0 0 e−2πiαx e−2πiαx Y = η(x, y). (A1) because is not square-integrable. Instead, assume that Here, in order to use (X,Y) to express the position in 2-D plane, the ∞ ∞ ¹ 0 ¹ 0 0 mapping f : (x, y) → (X,Y) must have one-to-one correspondence. dx0δ(x − x0)e−2πiαx = lim dx0δ(x − x0)e−ρ|x |e−2πiαx An operation that is not identity mapping is considered and written −∞ ρ→0 −∞ as Υ : (x, y) → (x0, y0). Assuming that the permutation of {x, y} (B6) corresponds to the operation, Υ, Equations (A2) must be satisfied: 0 where e−ρ|x | is a convergence factor. Thus, the following equation X = ξ(x, y) = η(Υ(x, y)) is valid: ∞ Y = η(x, y) = ξ(Υ(x, y)). (A2) ¹ 0 dx0δ(x − x0)e−2πiαx = e−2πiαx . (B7) −∞ Then, Equations (A3) are also satisfied: Equations (B2) and (B7) can be regarded as the Fourier trans- ξ(x, y) = ξ(Υ(Υ(x, y))) forms of e−2πiαk and δ(x − x0), respectively, in the extended mean- η(x, y) = η(Υ(Υ(x, y))). (A3) ing.

MNRAS 000,1–12 (2018) 10 S. Itoh et al. APPENDIX C: REMOVABLE SINGULARITIES − g cos(2π(a − f )) 2π f sin(2π(a − f )) = π 2 lim − →0 (a + g)2 a + g A function, f (x, y, z) (x+y+z=0) may have a singularity at f cos(2π(a + g)) 2πg sin(2π(a + g)) (x, y, z)=(xs, ys, zs). When the limit value of the function, + − ( ) (a − f )2 a − f lim(x,y,z)→(xs,ys,zs ) f x, y, z , is determined uniquely, the singularity is removable by redeﬁning lim f (x, y, z) as 1 (x,y,z)→(xs,ys,zs ) − f (xs, ys, zs). a2 Here, Equation (26) is evaluated in cases (i) and (ii) as a limit (C6) value of the result for case (iii), Equation (36). Similarly, Equation (39) is evaluated in cases (I) and (II) as a limit value of the result for g cos(2πa) 2π f sin(2πa) = π−2 − case (III), Equation (44). a2 a f cos(2πa) 2πg sin(2πa) 1 + − − C1 Case (i) a2 a a2 cos(2πa) − 1 2 sin(2πa) The following function is deﬁned: = − . (C7) (πa)2 πa Õ cos(2πx1) Thus, the following equation is obtained: hsinc(x1, x2, x3) = , (C1) πx2πx3 C {x1,x2,x3 } lim v(α, β) Q→0,R→R,S→−R Í where C {x ,x ,x } indicates the summation over the cyclic permu- √ √ ! 1 2 3 cos(2π 3R) − 1 2 sin(2π 3R) tations of {x1, x2, x3}. By using Equation (C1), Equation (36) can = 2π2γ √ − √ . be written as (π 3R)2 π 3R √ √ √ √ v(α, β) = 3(2i)−2hsinc( 3Q, 3R, 3S). (C2) (C8) This limit value does not depend on f and g. Thus, the singularities Then, by using the l’Hoˆpital’s rule, the following limit value is at (Q, R, S) = (0, R, −R) are properly removed with this limit value. calculated: hsinc(, f , g)( f + g = − ) lim→0 1 C3 Case (I) cos(2π) + f cos(2π f ) + g cos(2πg) = π−2 lim →0 f g2 1 cos 2π N + x1 2 2 Õ 2 −2 −2π(sin(2π) + f sin(2π f ) + g sin(2πg)) hDN (x1, x2, x3) = . (C9) = π lim πx2πx3 →0 2 f g C {x1,x2,x3 } 2 3 3 −2 −4π (cos(2π) + f cos(2π f ) + g cos(2πg)) By using Equation (C9), Equation (44) can be written as = π lim √ →0 2 f g −2 v(α, β) = 3(2i) hDN (3q, 3r, 3s). (C10) −2(1 + f 3 + g3) = k, l, m f g Then, the following limit value is calculated ( are integers; k + l + m = 0.): −2(3 f g) = f g lim→0 hDN ( + k, f + l, g + m)( f + g = −1) = −6. (C3) 1 2 = N + lim hsinc(, f , g)( f + g = −1) 2 →0 Thus, the following equation is obtained: √ 1 2 lim v(α, β) = 3(2i)−2 × −6 = −6 N + . (C11) Q→0,R→0,S→0 2 √ 3 3 Thus, the following equation is obtained: = . (C4) 2 √ 1 2 lim v(α, β) = 3(2i)−2 × −6 N + This limit value does not depend on the value of f and g. Thus, the 3q→k,3r→l,3s→m 2 singularity at (Q, R, S) = (0, 0, 0) is properly removed with this limit √ 3 3 1 2 value. = N + . (C12) 2 2 This limit value does not depend on f and g. Thus, the singularities C2 Case (ii) ( ) k l m at q, r, s = 3, 3, 3 are properly removed with this limit value. Now, by using the l’Hoˆpital’s rule, the following limit value is calculated: C4 Case (II) lim hsinc(, a − f , −a − g) →0 Now, by using the l’Hoˆpital’s rule, the following limit value is 1 cos (2π(a − f )) cos (2π(−a − g)) = π−2 lim + calculated (k, l, m are integers; k + l + m = 0; a is not an integer.): →0 −a − g a − f lim→0 hDN ( + k, a − f + l, −a − g + m) 1 − = lim hDN (, a − f , −a − g) a2 →0 (C5) (C13)

MNRAS 000,1–12 (2018) Segmented Telescope PSF by Symmetric Formulation 11

(a) σ + σ + σ > 0  cos 2π N + 1 (a − f ) 1 2 3 − © 1  2 = π 1 lim →0 sin(π(−a − g))  «  Im z cos 2π N + 1 (−a − g)  2  1 ª + − ® sin(π(a − f )) 2(πa) ®  sin Γρ  ¬ (C14) Γl Γm Γn  g cos(π(a + g)) cos 2π N + 1 (a − f ) − ©  2 = π 1 lim →0 2(π(a + g)) ρ  sin ρ Re z «  − Q + l ϵ S + n ϵ 1 ( − ) √3 √3 2π f sin 2π N + 2 a f − ± − ± − R + m ϵ sin(π(a + g)) − √3 ± ( ( − )) 1 ( ) f cos π a f cos 2π N + 2 a + g + sin2(π(a − f )) 1 ( ) 2πg sin 2π N + 2 a + g  −  sin(π(a − f )) σ + σ + σ <  (b) 1 2 3 0  1 − sin2(πa) Im z (C15)

1 1 g cos(πa) cos(2π N + a) 2π f sin(2π N + a) m −1 © 2 2 = π − R + √ ϵ 2(πa) sin(πa) − 3 ± sin l S + n ϵ « Q + √ ϵ √3 − 3 ± − ± f cos(πa) cos 2π N + 1 a − 1 2πg sin(2π N + 1 a) 2 2 ª + − ® 2( ) sin(πa) ® ρ ρ sin πa − Re z ¬ Γl Γm Γn ( ) 1 − ( 1 ) cos πa cos 2π N + 2 a 1 2π sin 2π N + 2 a = − . (C16) sin2(πa) sin(πa) Γρ Thus, the following equation is obtained: lim v(α, β) 3q→k,3r→3r+l,3s→−3r+m cos(π3r) cos 2π N + 1 3r − 1 2π sin(2π N + 1 3r) 2 © 2 2 ª =2π γ − ® .Figure D1. (a) Integral path in case (III) for positive σ1+σ2+σ3. (b) Integral sin2(π3r) sin(π3r) ® path in case (III) for negative σ1+σ2+σ3. The symbols, l, m, and n are « ¬ arbitrary integers. The integrand must be regular inside the integral paths. (C17) This limit value does not depend on the values of f and g. Thus, ( ) k l − m Hence, when σ1 + σ2 + σ3 is positive, the singularity at q, r, s = 3, r + 3, r + 3 is properly removed with this limit value. 1 ¹ ∞ lim dzΠ3(z)h(z) T,U→∞ T −∞

APPENDIX D: PART OF CALCULATION becomes 1 ¹ 0 An integral path shown in Figure D1 is considered for Π3(z)h(z) = − lim lim ieiθ dθ T,U→∞ T →0 π in Equation (39). The outer semicircle is denoted by Γρ and the ( ∞ ) semicircles for avoiding singularities are denoted by Γ , Γm, and Õ Õ l l l Π3 eiθ − Q + √ h eiθ − Q + √ Γn. The radius of Γρ approaches inﬁnity (ρ → ∞); and the radii C {Q,R,S } l=−∞ 3 3 of Γl, Γm, and Γn approach zero ( → 0). By applying the Cauchy ( ∞ integral theorem along the integral path, the last factor of Equation Õ 1 Õ l = − lim lim Π3 −Q + √ (39) can be calculated as follows: T→∞ T U→∞ C {Q,R,S } l=−∞ 3 ¹ ∞ ∞ ¹ 3 Õ Õ 3 ¹ 0 dzΠ (z)h(z) = − dzΠ (z)h(z). (D1) iθ iθ −∞ Γ lim ie dθh e − Q C {l,m,n} l=−∞ l →0 π

MNRAS 000,1–12 (2018) 12 S. Itoh et al.

Figure E1. N=2 Figure E3. N=4

Figure E2. N=3 Figure E4. N=5

( ∞ Õ Õ 1 l = − lim Rect √ T→∞ T C {Q,R,S} l=−∞ 3T ¹ 0 lim ieiθ dθh eiθ − Q →0 π √ ¹ 0 Õ = − 3 lim ieiθ dθ h eiθ − Q . (D2) →0 π C {Q,R,S} Thus, 1 ¹ ∞ lim dzΠ3(z)h(z) T,U→∞ T −∞ √ Õ e−(2N+1) 3πi(σ1+σ2+σ3)Q − = i √ √ , Figure E5. N=6 C {Q,R,S } sin 3π(Q − R) sin 3π(S − Q) (D3) sin(z) where lim|z |→0 z was used.

APPENDIX E: EXAMPLES In the below PSF examples (Figure E1–E8), ∆ is ﬁxed to 1.0×10−5. In the ﬁgures, the gap between adjacent segments are exaggerated to become 40 times wider than that used for the present calculation. λ The coordinates on the focal plane are normalized by D , where λ is the wavelength of light, and D is the full width of the pupil along the x-direction.

This paper has been typeset from a TEX/LATEX ﬁle prepared by the author.

MNRAS 000,1–12 (2018) Segmented Telescope PSF by Symmetric Formulation 13

Figure E6. Keck-type

Figure E7. TMT-type

Figure E8. ELT-type

MNRAS 000,1–12 (2018)