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Journals >Chinese Optics Letters >Volume 16 >Issue 3 >Page 032201 > Article

Chinese Optics Letters
Vol. 16, Issue 3, 032201 (2018)

Useful way to compensate for intrinsic birefringence caused by calcium fluoride in optical lithography systems

Zelong Zhou^1、2、*, Hongbo Shang¹, Yongxin Sui¹, and Huaijiang Yang¹

Author Affiliations

¹Engineering Researcher Center of Extreme Precision Optics, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China

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

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DOI: 10.3788/COL201816.032201 Cite this Article Set citation alerts

Zelong Zhou, Hongbo Shang, Yongxin Sui, Huaijiang Yang. Useful way to compensate for intrinsic birefringence caused by calcium fluoride in optical lithography systems[J]. Chinese Optics Letters, 2018, 16(3): 032201 Copy Citation Text

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Abstract

Calcium fluoride is widely used in optical lithography lenses and causes retardation that cannot be ignored. However, few studies have been conducted to compensate for the retardation caused by calcium fluoride in optical lithography systems. In this Letter, a new index based on orientation Zernike polynomials is established to describe the value of retardation. Then, a method of retardation compensation is described. The method is implemented by clocking calcium fluoride lens elements, and the optimal rotation angles are calculated using a population-based stochastic optimization algorithm. Finally, an example is provided to validate the method.

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, ψ_{p}, δ) J_{rot} (α) J_{ret} (ϕ, ψ_{r}),

(2)

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

(3)

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J_{ret} I \cos ϕ - i ϕ (\begin{matrix} \cos 2 β & \sin 2 β \\ \sin 2 β & −cos 2 β \end{matrix}) = I \cos ϕ - i \sum_{j = 1}^{\infty} C_{j} \cdot O Z_{j},

(4)

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O Z_{j} = O Z_{n, ε}^{m} = R_{n}^{m} (ρ) O_{ε}^{m} (θ) (ε = 0, 1),

(5)

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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}) .

(6)

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\frac{1}{2 π (1 + δ_{m 0})} \int_{0}^{2 π} \int_{0}^{1} (O Z_{i} \cdot O Z_{j} + O Z_{j} \cdot O Z_{i}) ρ \cdot d ρ \cdot d θ = δ_{i j} I,

(7)

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R = \sqrt{\iint_{θ, ρ} f^{2} (θ, ρ) \cdot ρ \cdot d ρ \cdot d θ} .

(8)

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ℜ = \sqrt{\frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} {[ϕ (\begin{matrix} \cos 2 β & \sin 2 β \\ \sin 2 β & - \cos 2 β \end{matrix})]}^{2} ρ \cdot d ρ \cdot d θ} = \sqrt{\frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} {(\sum_{j} C_{j} \cdot O Z_{j})}^{2} ρ \cdot d ρ \cdot d θ} .

(9)

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ℜ = \sqrt{\sum_{j} (2 - δ_{m 0}) C_{j}^{2}} .

(10)

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ϕ (\begin{matrix} \cos 2 β & \sin 2 β \\ \sin 2 β & - \cos 2 β \end{matrix}) = \sum_{j} C_{j} \cdot O Z_{j} = \sum_{k = 1}^{\infty} [C_{k^{2}} O Z_{k^{2}} + C_{- k^{2}} O Z_{- k^{2}} + \sum_{t = 1}^{k} (C_{k^{2} + 2 t - 1} O Z_{k^{2} + 2 t - 1} + C_{k^{2} + 2 t} O Z_{k^{2} + 2 t} + C_{- (k^{2} + 2 t - 1)} O Z_{- (k^{2} + 2 t - 1)} + C_{- (k^{2} + 2 t)} O Z_{- (k^{2} + 2 t)})] = \sum_{k = 1}^{\infty} {C_{k^{2}} (\begin{matrix} Z_{k^{2}} & 0 \\ 0 & - Z_{k^{2}} \end{matrix}) + C_{- k^{2}} (\begin{matrix} 0 & Z_{k^{2}} \\ Z_{k^{2}} & 0 \end{matrix}) + \sum_{t = 1}^{k} [C_{k^{2} + 2 t - 1} (\begin{matrix} Z_{k^{2} + 2 t - 1} & Z_{k^{2} + 2 t} \\ Z_{k^{2} + 2 t} & - Z_{k^{2} + 2 t - 1} \end{matrix}) + C_{k^{2} + 2 t} (\begin{matrix} Z_{k^{2} + 2 t} & - Z_{k^{2} + 2 t - 1} \\ - Z_{k^{2} + 2 t - 1} & - Z_{k^{2} + 2 t} \end{matrix}) + C_{- (k^{2} + 2 t - 1)} (\begin{matrix} Z_{k^{2} + 2 t - 1} & - Z_{k^{2} + 2 t} \\ - Z_{k^{2} + 2 t} & - Z_{k^{2} + 2 t - 1} \end{matrix}) + C_{- (k^{2} + 2 t)} (\begin{matrix} Z_{k^{2} + 2 t} & Z_{k^{2} + 2 t - 1} \\ Z_{k^{2} + 2 t - 1} & - Z_{k^{2} + 2 t} \end{matrix})]},

(11)

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ϕ \cos 2 β = \sum_{k = 1}^{\infty} {C_{k^{2}} Z_{k^{2}} + \sum_{t = 1}^{k} \begin{array}{l} [(C_{k^{2} + 2 t - 1} + C_{- (k^{2} + 2 t - 1)}) Z_{k^{2} + 2 t - 1} \\ + (C_{k^{2} + 2 t} + C_{- (k^{2} + 2 t)}) Z_{k^{2} + 2 t}] \end{array}},

(12)

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ϕ \sin 2 β = \sum_{k = 1}^{\infty} {C_{- k^{2}} Z_{k^{2}} + \sum_{t = 1}^{k} \begin{array}{l} [(C_{k^{2} + 2 t - 1} - C_{- (k^{2} + 2 t - 1)}) Z_{k^{2} + 2 t} \\ + (C_{- (k^{2} + 2 t)} - C_{k^{2} + 2 t}) Z_{k^{2} + 2 t - 1}] \end{array}} .

(13)

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R_{ϕ \cos 2 β} = \sqrt{\frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} {(ϕ \cos 2 β)}^{2} ρ \cdot d ρ \cdot d θ} = \sqrt{\sum_{k = 1}^{\infty} {{C_{k^{2}}}^{2} + \sum_{t = 1}^{k} \begin{array}{l} [{(C_{k^{2} + 2 t - 1} + C_{- (k^{2} + 2 t - 1)})}^{2} \\ + {(C_{k^{2} + 2 t} + C_{- (k^{2} + 2 t)})}^{2}] \end{array}}},

(14)

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R_{ϕ \sin 2 β} = \sqrt{\frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} {(ϕ \sin 2 β)}^{2} ρ \cdot d ρ \cdot d θ} = \sqrt{\sum_{k = 1}^{\infty} {{C_{- k^{2}}}^{2} + \sum_{t = 1}^{k} \begin{array}{l} [{(C_{k^{2} + 2 t - 1} - C_{- (k^{2} + 2 t - 1)})}^{2} \\ + {(C_{- (k^{2} + 2 t)} - C_{k^{2} + 2 t})}^{2}] \end{array}}},

(15)

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{R_{ϕ}}^{2} = \frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} ϕ^{2} ρ \cdot d ρ \cdot d θ = \frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} [{(ϕ \cos 2 β)}^{2} + {(ϕ \sin 2 β)}^{2}] ρ \cdot d ρ \cdot d θ = {(R_{ϕ \cos 2 β})}^{2} + {(R_{ϕ \sin 2 β})}^{2} = \sum_{k = 1}^{\infty} [{C_{k^{2}}}^{2} + {C_{- k^{2}}}^{2} + \sum_{t = 1}^{k} \begin{array}{l} (2 {C_{k^{2} + 2 t - 1}}^{2} + 2 {C_{k^{2} + 2 t}}^{2} \\ + 2 {C_{- (k^{2} + 2 t - 1)}}^{2} + 2 {C_{- (k^{2} + 2 t)}}^{2}) \end{array}] = \sqrt{\sum_{j} (2 - δ_{m 0}) C_{j}^{2}} .

(16)

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R_{ϕ} = ℜ .

(17)

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ℜ_{RMS} = \sqrt{\frac{\sum_{i = 1}^{N} ℜ_{i}^{2}}{N}} .

(18)

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Zelong Zhou, Hongbo Shang, Yongxin Sui, Huaijiang Yang. Useful way to compensate for intrinsic birefringence caused by calcium fluoride in optical lithography systems[J]. Chinese Optics Letters, 2018, 16(3): 032201

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