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Journals >Chinese Optics Letters >Volume 18 >Issue 6 >Page 062201 > Article

Chinese Optics Letters
Vol. 18, Issue 6, 062201 (2020)

Optimization based on sensitivity for material birefringence in projection lens

Hongbo Shang^1,2, Luwei Zhang^1,2,3,*, Chunlai Liu¹, Ping Wang¹..., Yongxin Sui^1,2 and Huaijiang Yang^1,2|Show fewer author(s)

Author Affiliations

¹Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China

²University of Chinese Academy of Sciences, Beijing 100049, China

³State Key Laboratory of Laser Interaction with Matter, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China

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DOI: 10.3788/COL202018.062201 Cite this Article Set citation alerts

Hongbo Shang, Luwei Zhang, Chunlai Liu, Ping Wang, Yongxin Sui, Huaijiang Yang, "Optimization based on sensitivity for material birefringence in projection lens," Chin. Opt. Lett. 18, 062201 (2020) Copy Citation Text

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Abstract

Polarization aberration caused by material birefringence can be partially compensated by lens clocking. In this Letter, we propose a fast and efficient clocking optimization method. First, the material birefringence distribution is fitted by the orientation Zernike polynomials. On this basis, the birefringence sensitivity matrix of each lens element can be calculated. Then we derive the rotation matrix of the orientation Zernike polynomials and establish a mathematical model for clocking optimization. Finally, an optimization example is given to illustrate the efficiency of the new method. The result shows that the maximum RMS of retardation is reduced by 64% using only 48.99 s.

Keywords

birefringence polarization aberration projection lens

J = [\begin{matrix} j_{11} & j_{12} \\ j_{21} & j_{22} \end{matrix}] .

(1)

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J = t e^{i Φ} J_{pol} (d, θ_{p}) J_{rot} (α_{p}) J_{ret} (ϕ, β_{p}),

(2)

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J_{ret} (ϕ, β_{p}) = (\begin{matrix} \cos ϕ - i \sin ϕ \cos 2 β_{p} & - i \sin ϕ \sin 2 β_{p} \\ - i \sin ϕ \sin 2 β_{p} & \cos ϕ + i \sin ϕ \cos 2 β_{p} \end{matrix}) = \cos ϕ I - i \sin ϕ (\begin{matrix} \cos 2 β_{p} & \sin 2 β_{p} \\ \sin 2 β_{p} & - \cos 2 β_{p} \end{matrix}),

(3)

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J_{ret} (ϕ, β_{p}) \approx I - i ϕ (\begin{matrix} \cos 2 β_{p} & \sin 2 β_{p} \\ \sin 2 β_{p} & \cos 2 β_{p} \end{matrix}) = I - i \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} \sum_{ϵ = 0}^{1} C_{n, ϵ}^{m} R_{n}^{m} (ρ) O_{ε}^{m} (θ),

(4)

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O_{0}^{m} (θ) = (\begin{matrix} \cos m θ & \sin m θ \\ \sin m θ & - \cos m θ \end{matrix}), O_{1}^{m} (θ) = (\begin{matrix} \sin m θ & - \cos m θ \\ - \cos m θ & - \sin m θ \end{matrix}) .

(5)

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O Z_{j} = (\begin{matrix} Z_{j} & Z_{j + 1} \\ Z_{j + 1} & - Z_{j} \end{matrix}), O Z_{- j} = (\begin{matrix} Z_{j} & - Z_{j + 1} \\ - Z_{j + 1} & - Z_{j} \end{matrix}), O Z_{j + 1} = (\begin{matrix} Z_{j + 1} & - Z_{j} \\ - Z_{j} & - Z_{j + 1} \end{matrix}), O Z_{- j - 1} = (\begin{matrix} Z_{j + 1} & Z_{j} \\ Z_{j} & - Z_{j + 1} \end{matrix}) .

(6)

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O Z_{j} = (\begin{matrix} Z_{j} & 0 \\ 0 & - Z_{j} \end{matrix}), O Z_{- j} = (\begin{matrix} 0 & Z_{j} \\ Z_{j} & 0 \end{matrix}) .

(7)

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R_{rms} = \sum_{j = - \infty}^{\infty} \sqrt{(2 - δ_{m 0})} C_{j},

(8)

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R_{i} = S_{i} B_{i},

(9)

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M = | m - 2 | .

(10)

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C_{j} \cdot (\begin{matrix} \cos M α & - \sin M α \\ \sin M α & \cos M α \end{matrix}) \cdot (\begin{matrix} Z_{j} & Z_{j + 1} \\ Z_{j + 1} & - Z_{j} \end{matrix}) = C_{j} \cdot \cos M α \cdot O Z_{j} - C_{j} \cdot \sin M α \cdot O Z_{j + 1},

(11)

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C_{j + 1} \cdot (\begin{matrix} \cos M α & - \sin M α \\ \sin M α & \cos M α \end{matrix}) \cdot (\begin{matrix} Z_{j + 1} & - Z_{j} \\ - Z_{j} & - Z_{j + 1} \end{matrix}) = C_{j + 1} \cdot \cos M α \cdot O Z_{j + 1} + C_{j + 1} \cdot \sin M α \cdot O Z_{j} .

(12)

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R_{1} (α) = (\begin{matrix} \cos α & \sin α & 0 & \dots & 0 & 0 \\ - \sin α & \cos α & 0 & 0 & 0 \\ 0 & 0 \\ ⋱ \\ 0 & 0 \\ 0 & 0 & 0 & \cos α & \sin α \\ 0 & 0 & \dots & 0 & - \sin α & \cos α \end{matrix}) .

(13)

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R_{2} (α) = (\begin{matrix} \cos (2 α) & \sin (2 α) & 0 & \dots & 0 & 0 \\ - \sin (2 α) & \cos (2 α) & 0 & 0 & 0 \\ 0 & 0 \\ ⋱ \\ 0 & 0 \\ 0 & 0 & 0 & \cos (2 α) & \sin (2 α) \\ 0 & 0 & \dots & 0 & - \sin (2 α) & \cos (2 α) \end{matrix}) .

(14)

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R_{i} = S_{i} R (α_{i}) B_{i} .

(15)

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\begin{matrix} \min_{α} {\sum_{f = 1}^{q} RMS [\sum_{i = 1}^{p} S_{f i} R (α_{i}) B_{i}]} \\ - π \leq α_{i} \leq π \end{matrix},

(16)

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References (15)

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Hongbo Shang, Luwei Zhang, Chunlai Liu, Ping Wang, Yongxin Sui, Huaijiang Yang, "Optimization based on sensitivity for material birefringence in projection lens," Chin. Opt. Lett. 18, 062201 (2020)

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