Abstract

We obtain a closed-form analytical expression for the aberration-free point-spread function (PSF) of a system with a hexagonal pupil. The six-fold symmetric PSF consists of a nearly circular bright spot at the center surrounded by a thin dark ring and two each nearly hexagonal bright and dark rings, while maintaining the six-fold symmetry. Beyond that the PSF breaks into six diffraction arms, each of alternating bright and dark strips and normal to a side of the pupil, with some dim structure between two consecutive arms. The ensquared power of the PSF and the optical transfer function are calculated and compared with the corresponding quantities for a system with a circular pupil. The balancing of Seidel aberrations is illustrated and their standard deviations with and without balancing are discussed. The Strehl ratio of these aberrations is plotted as a function of the standard deviation and compared with the approximate expression based on the aberration variance.

© 2013 Optical Society of America

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References

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  1. http://www.keckobservatory.org/ .
  2. L. D. Feinberg, M. Clamping, R. K. Keski-Kuha, C. Atkinson, S. Texter, M. Bergeland, and B. B. Gallagher, “James Webb telescope optical telescope element mirror development history and results,” Proc. SPIE 8442, 84422B (2012).
    [CrossRef]
  3. M. Troy and G. Chanan, “Diffraction effects from giant segmented-mirror telescopes,” Appl. Opt. 42, 3745–3753 (2003).
    [CrossRef]
  4. E. Sabatke, J. Burge, and D. Sabatke, “Analytic diffraction analysis of a 32-m telescope with hexagonal segments for high-contrast imaging,” Appl. Opt. 44, 1360–1365 (2005).
    [CrossRef]
  5. R. C. Smith and J. S. Marsh, “Diffraction patterns of simple apertures,” J. Opt. Soc. Am. 64, 798–803 (1974).
    [CrossRef]
  6. G. Chanan and M. Troy, “Strehl ratio and modulation transfer function for segmented mirror telescopes as functions of segment phase error,” Appl. Opt. 38, 6642–6647 (1999).
    [CrossRef]
  7. N. Yaitskova and K. Dohlen, “Tip-tilt error for extremely large segmented telescopes: detailed theoretical point-spread function analysis and numerical simulation results,” J. Opt. Soc. Am. A 19, 1274–1285 (2002).
    [CrossRef]
  8. V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd ed. (SPIE, 2011).
  9. V. N. Mahajan and G.-m. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24, 2994–3016 (2007).
    [CrossRef]
  10. V. N. Mahajan, “Orthonormal polynomials in wavefront analysis: analytical solution: errata,” J. Opt. Soc. Am. A 29, 1673–1674 (2012).
    [CrossRef]
  11. V. N. Mahajan, “Orthonormal polynomials in wavefront analysis,” in Handbook of Optics, V. N. Mahajan and E. V. Stryland, eds., 3rd ed., Vol. II (McGraw-Hill, 2009), pp. 11.3–11.41.

2012

L. D. Feinberg, M. Clamping, R. K. Keski-Kuha, C. Atkinson, S. Texter, M. Bergeland, and B. B. Gallagher, “James Webb telescope optical telescope element mirror development history and results,” Proc. SPIE 8442, 84422B (2012).
[CrossRef]

V. N. Mahajan, “Orthonormal polynomials in wavefront analysis: analytical solution: errata,” J. Opt. Soc. Am. A 29, 1673–1674 (2012).
[CrossRef]

2007

2005

2003

2002

1999

1974

Atkinson, C.

L. D. Feinberg, M. Clamping, R. K. Keski-Kuha, C. Atkinson, S. Texter, M. Bergeland, and B. B. Gallagher, “James Webb telescope optical telescope element mirror development history and results,” Proc. SPIE 8442, 84422B (2012).
[CrossRef]

Bergeland, M.

L. D. Feinberg, M. Clamping, R. K. Keski-Kuha, C. Atkinson, S. Texter, M. Bergeland, and B. B. Gallagher, “James Webb telescope optical telescope element mirror development history and results,” Proc. SPIE 8442, 84422B (2012).
[CrossRef]

Burge, J.

Chanan, G.

Clamping, M.

L. D. Feinberg, M. Clamping, R. K. Keski-Kuha, C. Atkinson, S. Texter, M. Bergeland, and B. B. Gallagher, “James Webb telescope optical telescope element mirror development history and results,” Proc. SPIE 8442, 84422B (2012).
[CrossRef]

Dai, G.-m.

Dohlen, K.

Feinberg, L. D.

L. D. Feinberg, M. Clamping, R. K. Keski-Kuha, C. Atkinson, S. Texter, M. Bergeland, and B. B. Gallagher, “James Webb telescope optical telescope element mirror development history and results,” Proc. SPIE 8442, 84422B (2012).
[CrossRef]

Gallagher, B. B.

L. D. Feinberg, M. Clamping, R. K. Keski-Kuha, C. Atkinson, S. Texter, M. Bergeland, and B. B. Gallagher, “James Webb telescope optical telescope element mirror development history and results,” Proc. SPIE 8442, 84422B (2012).
[CrossRef]

Keski-Kuha, R. K.

L. D. Feinberg, M. Clamping, R. K. Keski-Kuha, C. Atkinson, S. Texter, M. Bergeland, and B. B. Gallagher, “James Webb telescope optical telescope element mirror development history and results,” Proc. SPIE 8442, 84422B (2012).
[CrossRef]

Mahajan, V. N.

V. N. Mahajan, “Orthonormal polynomials in wavefront analysis: analytical solution: errata,” J. Opt. Soc. Am. A 29, 1673–1674 (2012).
[CrossRef]

V. N. Mahajan and G.-m. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24, 2994–3016 (2007).
[CrossRef]

V. N. Mahajan, “Orthonormal polynomials in wavefront analysis,” in Handbook of Optics, V. N. Mahajan and E. V. Stryland, eds., 3rd ed., Vol. II (McGraw-Hill, 2009), pp. 11.3–11.41.

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd ed. (SPIE, 2011).

Marsh, J. S.

Sabatke, D.

Sabatke, E.

Smith, R. C.

Texter, S.

L. D. Feinberg, M. Clamping, R. K. Keski-Kuha, C. Atkinson, S. Texter, M. Bergeland, and B. B. Gallagher, “James Webb telescope optical telescope element mirror development history and results,” Proc. SPIE 8442, 84422B (2012).
[CrossRef]

Troy, M.

Yaitskova, N.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Proc. SPIE

L. D. Feinberg, M. Clamping, R. K. Keski-Kuha, C. Atkinson, S. Texter, M. Bergeland, and B. B. Gallagher, “James Webb telescope optical telescope element mirror development history and results,” Proc. SPIE 8442, 84422B (2012).
[CrossRef]

Other

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd ed. (SPIE, 2011).

V. N. Mahajan, “Orthonormal polynomials in wavefront analysis,” in Handbook of Optics, V. N. Mahajan and E. V. Stryland, eds., 3rd ed., Vol. II (McGraw-Hill, 2009), pp. 11.3–11.41.

http://www.keckobservatory.org/ .

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Figures (9)

Fig. 1.
Fig. 1.

(a) Hexagonal pupil with each side of length a. (b) Unit hexagonal pupil inscribed inside a unit circle.

Fig. 2.
Fig. 2.

2D PSF of a system with a hexagonal pupil. The PSF is truncated to a value of 103 in (b).

Fig. 3.
Fig. 3.

PSF along the x (dots in red) and y (dashes in blue) axes and at 15° from the x axis (small dashes in green), where x and y are in units of λFx. PSF in solid black is the Airy pattern for a circular pupil.

Fig. 4.
Fig. 4.

Ensquared power Ph (in red) for a hexagonal pupil and Pc (in black) for a circular as a function of the half-width s of a square, where s is in units of λFx.

Fig. 5.
Fig. 5.

Overlap area of two hexagonal pupils displaced from each other along the x axis in (a) and (b) and y axis in (c).

Fig. 6.
Fig. 6.

Cutoff spatial frequency v2 in units of 1/λFx as a function of its angle θ from the x axis.

Fig. 7.
Fig. 7.

OTF along the x (dots in red) and y (dashes in blue) axes, and at 15° from the x axis (small dashes in green), where the spatial frequency v is in units of 1/λFx. The curve in solid black is for the circular pupil.

Fig. 8.
Fig. 8.

Strehl ratio as a function of the sigma value of a Seidel aberration in units of wavelength λ with (in green) and without (in red) balancing. (a) Defocus, (b) astigmatism, (c) coma, and (d) spherical aberration. The dashed curve (in blue) represents the approximate exponential expression exp(σΦ2).

Fig. 9.
Fig. 9.

Overlap area of two hexagonal pupils displaced from each other at an angle θ along the x axis.

Tables (3)

Tables Icon

Table 1. Maxima and Minima of the PSF along the x and y Axes

Tables Icon

Table 2. Ensquared Power Ph of a System with a Hexagonal Pupil, Where s is the Half-Width of a Square, Compared with the Ensquared Power Pc for a Circular Pupil

Tables Icon

Table 3. Sigma Value of a Seidel Aberration with and without Balancing, and P-V Numbers for a Sigma Value of Unity, Where Ai Is Its Coefficient

Equations (72)

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I(r⃗i)=1Sex2|exp[iΦ(r⃗p)]exp(2πiλRr⃗i·r⃗p)dr⃗p|2,
I(xi,yi)=1Sex2|exp[iΦ(xp,yp)]exp[2πiλR(xixp+yiyp)]dxpdyp|2.
(xp,yp)=a(x,y),
(xi,yi)=λFx(x,y),
Fx=R/2a
I(x,y)=427|exp[iΦ(x,y)]exp[πi(xx+yy)]dxdy|2.
I(x,y)=427|exp[πi(xx+yy)]dxdy|2.
I(x,y)=427|[1/21/2dx3/23/2+1/21dx3(1x)3(1x)+11/2dx3(1+x)3(1+x)]exp[πi(xx+yy)]dy|2.
A1(x,y)=1/21/2dx3/23/2exp[πi(xx+yy)]dy=4sin(πx/2)sin(3πy/2)π2xy,
A2(x,y)=1/21dx3(1x)3(1x)exp[πi(xx+yy)]dy=2π2y(x23y2){eiπx/2[3ycos(3πy/2)+ixsin(3πy/2)]+3yeiπx},
A3(x,y)=11/2dx3(1+x)3(1+x)exp[πi(xx+yy)]dy=2π2y(x23y2){eiπx/2[3ycos(3πy/2)+ixsin(3πy/2)]3yeiπx}.
A2+A3=4π2y(x23y2)×[3ycos(πx/2)cos(3πy/2)xsin(πx/2)sin(3πy/2)3ycos(πx)].
A1+A2+A3=43π2x(x23y2)×{x[cos(πx/2)cos(3πy/2)cos(πx)]3ysin(πx/2)sin(3πy/2)}.
I(x,y)=427|A1+A2+A3|2
=427(A1+A2+A3)2.
I(x,0)=649π4x4[cos(πx/2)cos(πx)]2
I(0,y)=16243π4y4{23[1cos(3πy/2)]+3πysin(3πy/2)}2.
P(s)=ssdxssI(x,y)dy,
τ(v⃗)=Sex1exp[iQ(r⃗p)]dr⃗p,
Q(r⃗p;v⃗)=Φ(r⃗p)Φ(r⃗pλRv⃗)
τx(vx)={1(4/3)vx,0vx1/2(4/3)(1vx)2,1/2vx1.
τy(vy)=(2/3)[(12vy/3)+(1/2)(12vy/3)2],0vy3/2.
τ(vθ)={1433vθ[sinθ+3cosθ+(23sin2θsin2θ)vθ],0vθv143+23(sinθ34cosθ)vθ+13(113sin2θ+3cos2θ)vθ2,v1vθv2,
v1=[2(cosθsinθ3)]1
v2=(cosθ+sinθ3)1
W(x,y)=j=1JajHj(x,y),
233hexagonHjHjdxdy=δjj,
aj=233hexagonW(x,y)Hj(x,y)dxdy.
W(ρ,θ)=a1,
W2(ρ,θ)=j=1Jaj2,
σW2=W2(ρ,θ)W(ρ,θ)2=j=2Jaj2.
Wd(ρ)=Adρ2.
σd=Ad12435=Ad4.902.
Wa(ρ,θ)=Aaρ2cos2θ.
H6=215/7ρ2cos2θ
=215/7ρ2(2cos2θ1).
Wba(ρ,θ)=Aa(ρ2cos2θ12ρ2).
σba=Aa4715=Aa5.855.
Wa(ρ,θ)=12Aa(ρ2cos2θ+ρ2)=14Aa[715H6+16435H4]+constant.
σa=Aa241275=Aa4.762.
Wc(ρ,θ)=Acρ3cosθ.
H8=442/3685(25ρ314ρ)cosθ.
Wbc(ρ,θ)=Ac(ρ31425ρ)cosθ.
σbc=Ac20737210=Ac10.676.
Wc(ρ,θ)=Ac[1100368542H8+72556H2].
σc=Ac48370=Ac3.673.
Ws(ρ)=Asρ4.
H11=601072205(301ρ4257ρ2)+constant.
Wbs(ρ)=As(ρ4257301ρ2).
σbs=As60×3011072205=As844987215=As17.441.
Ws(ρ)=As[107220560×301H11+25712×301435H4]+constant.
σs=As65935=As4.621.
SI(0,0)=427|exp[iΦ(x,y)]dxdy|2.
I(x,y)=43π2x[sinusinv+v++sinu+sinvv],
u±=π(x±y3)
v±=π(x3±y3),
r1a=(cosθsinθ3)1.
y=3(xa)
y=(x+a)tanθ,
(x,y)=a3+tanθ(3tanθ,23tanθ).
r22a=(cosθ+sinθ3)1.
QF=(arcosθ+rsinθ/3)i
QD=(a/2+rsinθ/3)i+(3a/2+rsinθ)j,
QD×QF=(1/6)[33a23ar(sinθ+3cosθ)2r2sinθ(3sinθ3cosθ)].
FA=(a/2)i+(3a/2)j
FD=(3a/2+rcosθ)i+(3a/2+rsinθ)j.
FA×FD=3a2(ar/2)(sinθ+3cosθ).
S(r;θ)=1233a2[rsinθ(a+rsinθ3)rcosθ(rsinθ3a)],0rr1.
DC=(r3sinθ+ar2cosθ)i+(3r2cosθ+3ar2sinθ)j
CA=(r23sinθa+r2cosθ)i+(3r2cosθ3a+r2sinθ)j,
(x,y)=(r23sinθ+r2cosθ,3r2cosθ3a+r2sinθ).
S(r;θ)=233a2[23a2+ar2(sinθ43cosθ)+r28(sin2θ+33cos2θ+3)],r1rr2.

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