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Journals >Chinese Optics Letters >Volume 13 >Issue Suppl. >Page S22201 > Article

Chinese Optics Letters
Vol. 13, Issue Suppl., S22201 (2015)

Theoretical study of the thickness uniformity of a coating on the inner face of a parabolic substrate in a vacuum evaporation system

Lingmao Xu, Jizhou Wang^*, Maojin Dong, Duoshu Wang, and Yudong Feng

Author Affiliations

Science and Technology on Vacuum Technology and Physics Laboratory, Lanzhou Institute of Physics, Lanzhou 730001, China

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DOI: 10.3788/COL201513.S22201 Cite this Article Set citation alerts

Lingmao Xu, Jizhou Wang, Maojin Dong, Duoshu Wang, Yudong Feng. Theoretical study of the thickness uniformity of a coating on the inner face of a parabolic substrate in a vacuum evaporation system[J]. Chinese Optics Letters, 2015, 13(Suppl.): S22201 Copy Citation Text

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Abstract

The theoretical study of the film thickness distribution deposited on a parabolic substrate by vacuum evaporation is reported. It is derived that the value of

n / h

and

L / h

strongly affect the coating thickness distribution (where

h

is the height of the substrate and

n

and

L

are, respectively, the horizontal distance and vertical height between the evaporation source and the center of the parabolic bottom). All of the parabolic substrate can be coated when

n \leq \sqrt{p h} (1 + L / h)

(where the parabolic focal length is

p / 4

), otherwise there is “blind area” on the substrate. In addition, the excellent thickness uniformity of the coating on the whole parabolic substrate can be obtained by choosing the appropriate configuration of the evaporation system under different evaporation sources.

r = \sqrt{p h - p z + 2 n y + n^{2} + {(L + z)}^{2}} .

(1)

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\cos θ = \frac{2 p h - p z + 2 n y + p L}{\sqrt{p^{2} + 4 p h - 4 p z} • \sqrt{p h - p z + 2 n y + n^{2} + {(L + z)}^{2}}} .

(2)

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\cos φ = \frac{L + z}{\sqrt{p h - p z + 2 n y + n^{2} + {(L + z)}^{2}}} .

(3)

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t_{p} (y, z) = \frac{(2 p h - p z + 2 n y + p L) m}{4 π μ \sqrt{p^{2} + 4 p h - 4 p z} • {(p h - p z + 2 n y + n^{2} + {(L + z)}^{2})}^{3 / 2}} .

(4)

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t_{s} (y, z) = \frac{m}{π μ} \frac{(L + z) (2 p h - p z + 2 n y + p L)}{\sqrt{p^{2} + 4 p h - 4 p z} • {(p h - p z + 2 n y + n^{2} + {(L + z)}^{2})}^{2}} .

(5)

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- L y + n z - \sqrt{p h} (L + z) = 0 .

(6)

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\bar{t_{p} (z)} = \frac{1}{2 π \sqrt{p h - p z}} \int {{}_{D}t}_{p} (y, z) d s,

(7)

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\bar{t_{s} (z)} = \frac{1}{2 π \sqrt{p h - p z}} \int {{}_{D}t}_{s} (y, z) d s,

(8)

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\hat{z} = \frac{p L^{2} + 2 p h L - 2 n L \sqrt{p h}}{2 n \sqrt{p h} - n^{2} - p h} .

(9)

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\bar{t_{p} (z)} = \frac{m}{4 π^{2} μ \sqrt{p^{2} + 4 p h - 4 p z}} • \int_{0}^{β} \frac{2 p h - p z + 2 n \sqrt{p h - p z} \cos α + p L}{{(p h - p z + 2 n \sqrt{p h - p z} \cos α + n^{2} + {(L + z)}^{2})}^{3 / 2}} d α,

(10)

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\bar{t_{s} (z)} = \frac{m (L + z)}{π^{2} μ \sqrt{p^{2} + 4 p h - 4 p z}} • \int_{0}^{β} \frac{2 p h - p z + 2 n \sqrt{p h - p z} \cos α + p L}{{(p h - p z + 2 n \sqrt{p h - p z} \cos α + n^{2} + {(L + z)}^{2})}^{2}} d α,

(11)

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n \leq \sqrt{p h} (1 + \frac{L}{2 h}), β = π,

(12)

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β = {\begin{cases} \arccos \frac{n z - z \sqrt{p h} - L \sqrt{p h}}{L \sqrt{p h - p z}}, 0 \leq z, \\ \leq \frac{p L^{2} + 2 p h L - 2 n L \sqrt{p h}}{2 n \sqrt{p h} - n^{2} - p h}, \\ π, \frac{p L^{2} + 2 p h L - 2 n L \sqrt{p h}}{2 n \sqrt{p h} - n^{2} - p h} \leq z \leq h . \end{cases}

(13)

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\frac{\bar{t_{p} (z)}}{\bar{t_{p} (h)}} = \frac{1}{π} \int_{0}^{β} \frac{{({(\frac{n}{h})}^{2} + {(1 + \frac{L}{h})}^{2})}^{3 / 2}}{(1 + \frac{L}{h}) \sqrt{{(\frac{p}{h})}^{2} + \frac{4 p}{h} - \frac{4 p z}{h^{2}}}} • \frac{\frac{2 p}{h} - \frac{p z}{h^{2}} + \frac{2 n}{h} \sqrt{\frac{p}{h} - \frac{p z}{h^{2}}} \cos α + \frac{p L}{h^{2}}}{{(\frac{p}{h} - \frac{p z}{h^{2}} + \frac{2 n}{h} \sqrt{\frac{p}{h} - \frac{p z}{h^{2}}} \cos α + {(\frac{n}{h})}^{2} + {(\frac{L}{h} + \frac{z}{h})}^{2})}^{3 / 2}} d α .

(14)

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\frac{\bar{t_{s} (z)}}{\bar{t_{s} (h)}} = \frac{1}{π} \int_{0}^{β} \frac{(\frac{L}{h} + \frac{z}{h}) {({(\frac{n}{h})}^{2} + {(1 + \frac{L}{h})}^{2})}^{2}}{{(1 + \frac{L}{h})}^{2} \sqrt{{(\frac{p}{h})}^{2} + \frac{4 p}{h} - \frac{4 p z}{h^{2}}}} • \frac{\frac{2 p}{h} - \frac{p z}{h^{2}} + \frac{2 n}{h} \sqrt{\frac{p}{h} - \frac{p z}{h^{2}}} \cos α + \frac{p L}{h^{2}}}{{(\frac{p}{h} - \frac{p z}{h^{2}} + \frac{2 n}{h} \sqrt{\frac{p}{h} - \frac{p z}{h^{2}}} \cos α + {(\frac{n}{h})}^{2} + {(\frac{L}{h} + \frac{z}{h})}^{2})}^{2}} d α .

(15)

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References (12)

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Lingmao Xu, Jizhou Wang, Maojin Dong, Duoshu Wang, Yudong Feng. Theoretical study of the thickness uniformity of a coating on the inner face of a parabolic substrate in a vacuum evaporation system[J]. Chinese Optics Letters, 2015, 13(Suppl.): S22201

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