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Journals >Chinese Optics Letters >Volume 13 >Issue Suppl. >Page S22202 > Article

Chinese Optics Letters
Vol. 13, Issue Suppl., S22202 (2015)

Evaluation of the mirror surface figure error based on the slope of the root-mean square

Fuguo Wang^*, Qichang An, and Limin Zhang

Author Affiliations

Changchun Institute of Optics, Fine Mechanics, and Physics, Chinese Academy of Sciences, Changchun 130033, China

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DOI: 10.3788/COL201513.S22202 Cite this Article Set citation alerts

Fuguo Wang, Qichang An, Limin Zhang, "Evaluation of the mirror surface figure error based on the slope of the root-mean square," Chin. Opt. Lett. 13, S22202 (2015) Copy Citation Text

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Abstract

The slope of the root-mean square (RMS) can be applied to evaluate mid-spatial frequency mirror surface errors for larger-aperture mirrors. In this paper, the slope RMS is analyzed from three different perspectives. First, the relationship between the slope RMS and the basis polynomials is discussed, and the mathematical relationship between the slope RMS and the standard orthogonal basis is obtained. Second, the slope RMS is analyzed by applying the Wiener process, with the results indicating that the ideal mirror surface error obeys the Gauss distribution law. Then, a power spectrum analysis method based on the slope RMS is proposed. Finally, the method is used to analyze the Thirty Meter Telescope tertiary mirror surface figure.

Φ (m, n) = \sum a_{u v} W_{u v} (m, n),

(1)

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W_{u v} (m, n) = \frac{1}{N \times N} \exp [\frac{2 π j}{N} (u m + v n)],

(2)

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Φ = A \sin (2 π f x),

(3)

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{(\nabla Φ)}^{2} = {(\nabla A \sin (2 π f x))}^{2} = 4 A^{2} π^{2} f^{2} \cos^{2} (2 π f x) 〈 \nabla Φ^{2} 〉 = \frac{2 A^{2} π f \int_{0}^{1 / f} \cos^{2} (2 π f x) 2 π f d x}{1 / f} .

(4)

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Φ = \sqrt{2} A \sin (2 π f x),

(5)

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slop RMS = \sqrt{2} π f A .

(6)

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Φ = α \sqrt{2} A \sin (2 π f x) + β Φ = \sqrt{2} A \sin (2 π 2 f x),

(7)

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slope {RMS}^{2} = {(\sqrt{2 a^{2} + 8 β^{2}} π A f)}^{2} = α^{2} 2 f^{2} A^{2} + β^{2} 8 f^{2} A^{2} .

(8)

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slope {RMS}^{2} = \sum_{1}^{N} c_{i}^{2} slope {RMS}_{i}^{2} .

(9)

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Φ (l, λ) = E {e^{i λ w (l)}} = \int_{- \infty}^{+ \infty} e^{i λ x} f (l, x) d x,

(10)

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σ^{2} (l) = E {{[w (l) - w (0)]}^{2}} = E {w^{2} (l)} = \int_{- \infty}^{+ \infty} x^{2} f (l, x) d x,

(11)

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σ^{2} (l + s) = E {{[w (l + s) - w (l) + w (l) - w (0)]}^{2}} = E {{[w (l + s) - w (l)]}^{2}} + E {w^{2} (l)} = σ^{2} (s) + σ^{2} (l),

(12)

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σ^{2} (l) = C l (C > 0),

(13)

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Φ (l + s, λ) - Φ (l, λ) = Φ (l, λ) E {e^{i λ (w (s) - w (0))} - 1},

(14)

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e^{i v} \approx 1 + i v - v^{2} / 2 .

(15)

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\frac{Φ (l + s, λ) - Φ (l, λ)}{s} \approx - \frac{1}{2} λ^{2} Φ (l, λ) C .

(16)

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\frac{\partial Φ (l, λ)}{\partial l} = - \frac{1}{2} λ^{2} Φ (l, λ) C,

(17)

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Φ (l, λ) = \int_{- \infty}^{+ \infty} e^{i λ x} \frac{1}{\sqrt{2 π C l}} e^{- \frac{x^{2}}{2 C t}} d x .

(18)

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R^{2} (θ (x + r), θ (x)) = {E {[θ (x + r) θ (x)]}}^{2} \leq E [θ^{2} (x + r)] E [θ^{2} (x)] < + \infty .

(19)

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S (w) = \sum_{- \infty}^{+ \infty} R (m) e^{- j w} = \sum_{- \infty}^{+ \infty} E [x (n) x^{*} (n + m)] e^{- j w} .

(20)

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R (r) = {〈 θ (x + r) θ (x) 〉}_{x},

(21)

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S (\bar{w}) = F (R (r)) = F ({〈 θ (x + r) θ (x) 〉}_{x}) .

(22)

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S (w) = \sum_{- \infty}^{+ \infty} R (m) e^{- j w} = \sum_{- \infty}^{+ \infty} E [x (n) x^{*} (n + m)] e^{- j w},

(23)

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R (r) = {〈 θ (x + r) θ (x) 〉}_{x},

(24)

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S (\bar{w}) = F (R (r)) = F ({〈 θ (x + r) θ (x) 〉}_{x}) .

(25)

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Fuguo Wang, Qichang An, Limin Zhang, "Evaluation of the mirror surface figure error based on the slope of the root-mean square," Chin. Opt. Lett. 13, S22202 (2015)

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