Abstract

We investigate a new metric, the normalized point source sensitivity (PSSN), for characterizing the seeing-limited performance of large telescopes. As the PSSN metric is directly related to the photometric error of background limited observations, it represents the efficiency loss in telescope observing time. The PSSN metric properly accounts for the optical consequences of wave front spatial frequency distributions due to different error sources, which differentiates from traditional metrics such as the 80% encircled energy diameter and the central intensity ratio. We analytically show that multiplication of individual PSSN values due to individual errors is a good approximation for the total PSSN when various errors are considered simultaneously. We also numerically confirm this feature for Zernike aberrations as well as for the numerous error sources considered in the error budget of the Thirty Meter Telescope (TMT) using a ray optics simulator. Additionally, we discuss other pertinent features of the PSSN, including its relations to Zernike aberration, RMS wave front error, and central intensity ratio.

© 2009 Optical Society of America

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References

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  1. G. Z. Angeli, S. Roberts, and K. Vogiatzis, “Systems engineering for the preliminary design of the Thirty Meter Telescope,” Proc. SPIE 7017, 701704 (2008).
    [CrossRef]
  2. K. Vogiatzis and G. Z. Angeli, “Monte Carlo simulation framework for TMT,” Proc. SPIE 7017, 7017V (2008).
  3. C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, and N. Sigrist, “High-resolution optical modeling of the Thirty Meter Telescope for systematic performance trades,” Proc. SPIE 7017, 70170U(2008).
    [CrossRef]
  4. P. Dierickx, “Optical performance of large ground-based telescopes,” J. Mod. Opt. 39, 569-588 (1992).
    [CrossRef]
  5. I. R. King, “Accuracy of measurement of star images on a pixel array,” Publ. Astron. Soc. Pac. 95, 163-168 (1983).
    [CrossRef]
  6. R. A. Muller and A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. 64, 1200-1210 (1974).
    [CrossRef]
  7. J. C. Christou and R. B. Makidon, “Strehl ratio and image sharpness for adaptive optics,” Proc. SPIE 6272, 62721Y(2006).
    [CrossRef]
  8. J. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford Univ. Press, 1998).
  9. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207-211 (1976).
    [CrossRef]

2008 (3)

G. Z. Angeli, S. Roberts, and K. Vogiatzis, “Systems engineering for the preliminary design of the Thirty Meter Telescope,” Proc. SPIE 7017, 701704 (2008).
[CrossRef]

K. Vogiatzis and G. Z. Angeli, “Monte Carlo simulation framework for TMT,” Proc. SPIE 7017, 7017V (2008).

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, and N. Sigrist, “High-resolution optical modeling of the Thirty Meter Telescope for systematic performance trades,” Proc. SPIE 7017, 70170U(2008).
[CrossRef]

2006 (1)

J. C. Christou and R. B. Makidon, “Strehl ratio and image sharpness for adaptive optics,” Proc. SPIE 6272, 62721Y(2006).
[CrossRef]

1992 (1)

P. Dierickx, “Optical performance of large ground-based telescopes,” J. Mod. Opt. 39, 569-588 (1992).
[CrossRef]

1983 (1)

I. R. King, “Accuracy of measurement of star images on a pixel array,” Publ. Astron. Soc. Pac. 95, 163-168 (1983).
[CrossRef]

1976 (1)

1974 (1)

Angeli, G.

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, and N. Sigrist, “High-resolution optical modeling of the Thirty Meter Telescope for systematic performance trades,” Proc. SPIE 7017, 70170U(2008).
[CrossRef]

Angeli, G. Z.

K. Vogiatzis and G. Z. Angeli, “Monte Carlo simulation framework for TMT,” Proc. SPIE 7017, 7017V (2008).

G. Z. Angeli, S. Roberts, and K. Vogiatzis, “Systems engineering for the preliminary design of the Thirty Meter Telescope,” Proc. SPIE 7017, 701704 (2008).
[CrossRef]

Angione, J.

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, and N. Sigrist, “High-resolution optical modeling of the Thirty Meter Telescope for systematic performance trades,” Proc. SPIE 7017, 70170U(2008).
[CrossRef]

Buffington, A.

Christou, J. C.

J. C. Christou and R. B. Makidon, “Strehl ratio and image sharpness for adaptive optics,” Proc. SPIE 6272, 62721Y(2006).
[CrossRef]

Crossfield, I.

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, and N. Sigrist, “High-resolution optical modeling of the Thirty Meter Telescope for systematic performance trades,” Proc. SPIE 7017, 70170U(2008).
[CrossRef]

Dierickx, P.

P. Dierickx, “Optical performance of large ground-based telescopes,” J. Mod. Opt. 39, 569-588 (1992).
[CrossRef]

Ellerbroek, B.

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, and N. Sigrist, “High-resolution optical modeling of the Thirty Meter Telescope for systematic performance trades,” Proc. SPIE 7017, 70170U(2008).
[CrossRef]

Gilles, L.

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, and N. Sigrist, “High-resolution optical modeling of the Thirty Meter Telescope for systematic performance trades,” Proc. SPIE 7017, 70170U(2008).
[CrossRef]

Hardy, J.

J. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford Univ. Press, 1998).

King, I. R.

I. R. King, “Accuracy of measurement of star images on a pixel array,” Publ. Astron. Soc. Pac. 95, 163-168 (1983).
[CrossRef]

Makidon, R. B.

J. C. Christou and R. B. Makidon, “Strehl ratio and image sharpness for adaptive optics,” Proc. SPIE 6272, 62721Y(2006).
[CrossRef]

Muller, R. A.

Nissly, C.

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, and N. Sigrist, “High-resolution optical modeling of the Thirty Meter Telescope for systematic performance trades,” Proc. SPIE 7017, 70170U(2008).
[CrossRef]

Noll, R. J.

Roberts, S.

G. Z. Angeli, S. Roberts, and K. Vogiatzis, “Systems engineering for the preliminary design of the Thirty Meter Telescope,” Proc. SPIE 7017, 701704 (2008).
[CrossRef]

Seo, B.

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, and N. Sigrist, “High-resolution optical modeling of the Thirty Meter Telescope for systematic performance trades,” Proc. SPIE 7017, 70170U(2008).
[CrossRef]

Sigrist, N.

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, and N. Sigrist, “High-resolution optical modeling of the Thirty Meter Telescope for systematic performance trades,” Proc. SPIE 7017, 70170U(2008).
[CrossRef]

Troy, M.

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, and N. Sigrist, “High-resolution optical modeling of the Thirty Meter Telescope for systematic performance trades,” Proc. SPIE 7017, 70170U(2008).
[CrossRef]

Vogiatzis, K.

K. Vogiatzis and G. Z. Angeli, “Monte Carlo simulation framework for TMT,” Proc. SPIE 7017, 7017V (2008).

G. Z. Angeli, S. Roberts, and K. Vogiatzis, “Systems engineering for the preliminary design of the Thirty Meter Telescope,” Proc. SPIE 7017, 701704 (2008).
[CrossRef]

J. Mod. Opt. (1)

P. Dierickx, “Optical performance of large ground-based telescopes,” J. Mod. Opt. 39, 569-588 (1992).
[CrossRef]

J. Opt. Soc. Am. (2)

Proc. SPIE (4)

J. C. Christou and R. B. Makidon, “Strehl ratio and image sharpness for adaptive optics,” Proc. SPIE 6272, 62721Y(2006).
[CrossRef]

G. Z. Angeli, S. Roberts, and K. Vogiatzis, “Systems engineering for the preliminary design of the Thirty Meter Telescope,” Proc. SPIE 7017, 701704 (2008).
[CrossRef]

K. Vogiatzis and G. Z. Angeli, “Monte Carlo simulation framework for TMT,” Proc. SPIE 7017, 7017V (2008).

C. Nissly, B. Seo, M. Troy, G. Angeli, J. Angione, I. Crossfield, B. Ellerbroek, L. Gilles, and N. Sigrist, “High-resolution optical modeling of the Thirty Meter Telescope for systematic performance trades,” Proc. SPIE 7017, 70170U(2008).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

I. R. King, “Accuracy of measurement of star images on a pixel array,” Publ. Astron. Soc. Pac. 95, 163-168 (1983).
[CrossRef]

Other (1)

J. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford Univ. Press, 1998).

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Figures (8)

Fig. 1
Fig. 1

Conceptual schematic for an optical system.

Fig. 2
Fig. 2

Analytically calculated Δ A using Eq. (38) as a function of number of errors (N) when the multiplied PSSNs are 0.8, 0.85, and 0.9, respectively. Δ A represents the maximum boundary for difference between combined and multiplied PSSN. We assume A1 to A6 are all valid. We also assume that errors have the same PSSN (uniformly distributed errors in PSSN value sense).

Fig. 3
Fig. 3

Analytically calculated PSSN using our simple analytical model (M1 to M3). The x and y axes show the Fried parameter normalized to correlation length of the aberration ( r o / σ d ) and calculated α o = ( 1 PSSN ) / σ ϕ 2 values, respectively.

Fig. 4
Fig. 4

Numerical procedure to calculate PSSN with Zernike mode 800 aberration for example. The “L” in top left on the figures denotes log scale plot.

Fig. 5
Fig. 5

Numerically calculated PSSN for Zernike 4, 100, 200, 400, and 800 modes with respect to the RMS WFE: (a) PSSN plot in linear scale and (b)  1 PSSN plot in a log-log scale.

Fig. 6
Fig. 6

Numerically calculated (a) α, β, and (b) γ values for various Zernike modes. The azimuthal index, M, for the simulated Zernike modes are either maximum or zero mode for a given Zernike radial index.

Fig. 7
Fig. 7

Combined and multiplied PSSN. We first obtain combined OPDs by adding all OPDs that have equal RMS WFEs for the Zernikes shown in Fig. 5 and then calculate the combined PSSNs and RMS WFEs for the combined OPDs. We also obtain multiplied PSSNs by multiplying the PSSN values for all considered Zernikes using the same RMS WFE.

Fig. 8
Fig. 8

Numerical difference Δ N and analytical difference boundary Δ A between combined and multiplied PSSN for the TMT M1 primary errors. The two curves are obtained using two independent methods. Numerical Δ N values are from studies done by Nissly, et al. [3], and the analytical boundary Δ A is from Eq. (37).

Tables (1)

Tables Icon

Table 1 Numerically Calculated α, β, and γ Values

Equations (65)

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σ int 2 = I star | PSF ( θ ) | 2 PSF ( θ ) + b / I star d θ b | PSF ( θ ) | 2 d θ ,
PSS | PSF ( θ ) | 2 d θ .
SNR = I star T σ int 2 T PSS b T .
PSSN = | PSF t + a + e ( θ ) | 2 d θ | PSF t + a ( θ ) | 2 d θ ,
H 2 2 = | OTF ( f ) | 2 d f = | PSF ( θ ) | 2 d θ = PSS ,
PSSN = | OTF t + a + e ( f ) | 2 d f | OTF t + a ( f ) | 2 d f ,
0 PSSN 1.
PSSN C PSSN M .
PSSN 1 α σ 2 ,
PSSN 1 2 σ 2 ,
1 PSSN 1 CIR γ ,
PSF t + e ( θ ) = | r A ( r ) e j k OPD ( r ) e j θ r d r | 2 ,
OTF t + e ( f ) = r A ( r ) A ( r λ f ) e j Ω ( r , λ f ) d r ,
Ω ( r , λ f ) k ( OPD ( r ) OPD ( r λ f ) ) .
OTF t + a + e ( f ) = OTF a ( f ) · r A ( r ) A ( r λ f ) e j Ω d r ,
OTF e ( f ) e j Ω r ,
· r r A ( r ) A ( r λ f ) ( · ) d r r A ( r ) A ( r λ f ) d r .
OTF t + a + e ( f ) = OTF t + a ( f ) · OTF e ( f ) .
PSSN = | OTF e ( f ) | 2 ,
· ( · ) | OTF t + a ( f ) | 2 d f | OTF t + a ( f ) | 2 d f .
| OTF e ( f ) | 2 = 1 ϵ ( f ) .
PSSN C = 1 ϵ 1 ( f ) ϵ 2 ( f ) + ϵ m ( f ) ,
PSSN M = 1 ϵ 1 ( f ) 1 ϵ 2 ( f ) = 1 ϵ 1 ( f ) ϵ 2 ( f ) + ϵ 1 ( f ) ϵ 2 ( f ) .
Δ PSSN C PSSN M = ϵ m ( f ) ϵ 1 ( f ) ϵ 1 ( f ) .
Δ = Δ 2 + Δ 4 + Δ 6 + ,
Δ 2 = 2 Ω 1 r Ω 2 r Ω 1 Ω 2 r Δ 4 = ( Ω 1 2 r Ω 2 2 r Ω 1 2 r Ω 2 2 r ) + ( Ω 1 2 r Ω 1 Ω 2 r + Ω 2 2 r Ω 1 Ω 2 r + Ω 1 2 r Ω 2 r 2 + Ω 1 r 2 Ω 2 2 r Ω 1 r 2 Ω 2 r 2 ) ,
s s A ( r ) A ( r ) OPD 1 ( r ) OPD 2 ( r ) d r d r = s A ( r ) OPD 1 ( r ) d r s A ( r ) OPD 2 ( r ) d r ,
Ω 1 Ω 2 r = Ω 1 r Ω 2 r .
Ω i r 0 for i = 1 or 2 .
OTF a ( r ) = exp ( 3.44 ( r r o / λ ) 5 / 3 ) .
Δ ϵ 1 ϵ 2 ϵ 1 ϵ 2 ,
| Δ | ( ϵ 1 2 ϵ 1 2 ) ( ϵ 2 2 ϵ 2 2 ) ,
Δ 1.2119 ( 1 PSSN 1 ) ( 1 PSSN 2 ) ,
| Δ | Δ A 1.2119 ( 1 PSSN 1 ) ( 1 PSSN 2 ) .
| PSSN C PSSN M | Δ A .
Δ T = i = 1 N j = i + 1 N ( ϵ i ϵ j ϵ i ϵ j ) i = 1 N j = i + 1 N k = j + 1 N ( ϵ i ϵ j ϵ k ϵ i ϵ j ϵ k ) + + ( 1 ) N ( ϵ 1 ϵ 2 ϵ N ϵ i ϵ j ϵ N ) .
Δ A r 4 r 2 2 r 2 2 i = 1 N j = i + 1 N ( 1 PSSN i ) ( 1 PSSN j ) r 6 r 2 3 r 2 3 i = 1 N j = i + 1 N k = j + 1 N ( 1 PSSN i ) ( 1 PSSN j ) ( 1 PSSN k ) + + ( 1 ) N r 2 N r 2 N r 2 N i = 1 N ( 1 PSSN i ) ,
Δ A n = 2 N ( 1 ) n r 2 n r 2 n r 2 n ( 1 N 1 PSSN multiplied ) n .
PSSN = p = 0 n = 0 p ( j ) p n ! ( p n ) ! Ω n r Ω p n r .
Ω ( r , λ f ) = σ Δ S ( r , λ f ) ,
PSSN = 1 + a 1 σ + a 2 σ 2 + ,
a p = n = 0 p ( j ) p n ! ( p n ) ! Δ S n r Δ S p n r .
α = a 2 = Δ S ( r , λ f ) r 2 + Δ S ( r , λ f ) 2 r .
α = E [ Δ S ( r , λ f ) ] r 2 + E [ Δ S ( r , λ f ) 2 ] r .
E [ Δ S ( r , λ f ) 2 ] = E [ ( S ( r ) S ( r λ f ) ) 2 ] = ( E [ S ( r ) 2 ] + E [ S ( r λ f ) 2 ] 2 E [ S ( r ) S ( r λ f ) ] ) = 2 ,
α = 2.
CIR = PSF t + a + e ( θ = 0 ) PSF t + a ( θ = 0 ) .
CIR = OTF t + a + e ( f ) d f OTF t + a ( f ) d f .
CIR = OTF e ( f ) CIR ,
· CIR ( · ) OTF t + a ( f ) d f OTF t + a ( f ) d f .
CIR = 1 + b 1 σ + b 2 σ 2 + ,
b n = ( j ) n n ! Δ S ( r , λ f ) n r CIR .
CIR 1 β σ 2 ,
γ α β .
D ( r ) = σ ϕ 2 ( 1 exp ( ( r σ d ) 2 ) ) ,
Δ ϵ 1 ϵ 2 ϵ 1 ϵ 2 1.2119 ( 1 PSS 1 ) ( 1 PSS 2 ) .
ϵ i ( x , y ) = a 1 x + b 1 y + a 2 x 2 + b 2 x 2 + c 1 , 1 x y + ,
1 k ! k ϵ x k | ( 0 , 0 ) ,
1 k ! k ϵ y k | ( 0 , 0 ) ,
C i i + j ( i + j ) ! i + j ϵ i x i y | ( 0 , 0 ) ,
ϵ i ( r ) α 2 r 2 .
Δ = p 2 q 2 ( r 4 r 2 2 ) ,
1 PSS 1 = ϵ 1 = p 2 r 2 , 1 PSS 2 = ϵ 2 = q 2 r 2 ,
Δ ( r 4 ) r 2 2 ) r 2 2 ( 1 PSS 1 ) ( 1 PSS 2 ) .
( r 2 r 2 2 ) r 2 2

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