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Journals >Chinese Optics Letters >Volume 15 >Issue 10 >Page 101203 > Article

Chinese Optics Letters
Vol. 15, Issue 10, 101203 (2017)

Design of a grating by a joint optimization method for a phase-shifting point diffraction interferometer

Meng Zheng, Ke Liu, Lihui Liu, and Yanqiu Li^*

Author Affiliations

Key Laboratory of Photoelectronic Imaging Technology and Systems, Ministry of Education of China, School of Optoelectronics, Beijing Institute of Technology, Beijing 100081, China

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DOI: 10.3788/COL201715.101203 Cite this Article Set citation alerts

Meng Zheng, Ke Liu, Lihui Liu, Yanqiu Li. Design of a grating by a joint optimization method for a phase-shifting point diffraction interferometer[J]. Chinese Optics Letters, 2017, 15(10): 101203 Copy Citation Text

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Abstract

A grating is an important element of a phase-shifting point diffraction interferometer, and the grating constant and duty cycle have a great impact on the interferometer, so the design of a grating becomes significant. In order to measure the projection objective with a numerical aperture of 0.2, we present a joint optimization method of a pinhole and grating based on scalar diffraction and the finite difference time domain method. The grating constant and the film thickness are selected, and the duty cycle of the grating is optimized. The results show that in the grating processing the material chromium is adopted, the thickness is 200 nm, and the grating constant is 15 μm. When the duty cycle is 55%, the interference fringe contrast is the greatest. The feasibility of the design result is further verified by experiment.

U (p) = \frac{A}{j λ} \iint_{Σ} \frac{\exp [j k (r_{0} + r)]}{r_{0} r} K d s .

(1)

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U (p) = \frac{A}{j λ r_{0} l} \exp [j k (r_{0} + l)] \iint_{Σ} K \exp (j k β) d s .

(2)

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u_{0} = \frac{A}{r_{0} + l} \exp [j k (r_{0} + l)] .

(3)

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u_{0} = f \frac{A}{r_{0} l} \exp [j k (r_{0} + l)] .

(4)

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U (p) = \frac{u_{0}}{j λ f} \iint_{Σ} K \exp (j k β) d s .

(5)

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U (p) = \frac{u_{0}}{j λ f} \iint_{Σ} \exp (j k β) d s .

(6)

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U (p) = \frac{u_{0}}{j λ f} \iint_{Σ} \exp {\frac{j k}{2 l} [{(x - ζ)}^{2} + {(y - η)}^{2}]} d s .

(7)

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U (p) = \frac{u_{0} l}{j f} (\frac{1}{\sqrt{λ l}} \int_{- a}^{a} \exp [\frac{j k}{2 l} {(x - ξ)}^{2}] d ξ) (\frac{1}{\sqrt{λ l}} \int_{- b}^{b} \exp [\frac{j k}{2 l} {(y - η)}^{2}] d η) .

(8)

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U (p) = \frac{u_{0} l}{j f \sqrt{2 λ l}} (1 + j) \int_{- a}^{a} \exp [\frac{j k}{2 l} {(x - ζ)}^{2}] d ζ .

(9)

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U (p) = \frac{u_{0}}{j \sqrt{2 λ l}} (1 + j) \exp (\frac{j k x^{2}}{2 l}) \int_{- a}^{a} \exp (- \frac{j k x ζ}{l}) d ζ .

(10)

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U (p) = A \frac{\sin (N δ / 2)}{\sin (δ / 2)} \exp [j (N - 1) \frac{δ}{2}] .

(11)

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U (p) = \frac{u_{0}}{j \sqrt{2 λ l}} (1 + j) \exp (\frac{j k x^{2}}{2 l}) \int_{- a}^{a} \exp (- \frac{j k x ζ}{l}) d ζ \times \frac{\sin (N δ / 2)}{\sin (δ / 2)} \exp [j (N - 1) \frac{δ}{2}] .

(12)

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I (p) = {| U (p) |}^{2} = \frac{u_{0}^{2}}{λ l} {(\frac{\sin (N δ / 2)}{\sin (δ / 2)})}^{2} {| \int_{- a}^{a} \exp (- \frac{j k x ζ}{l}) d ζ |}^{2} .

(13)

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I_{m} = I_{0} \frac{1}{d^{2}} {(\frac{\sin N m π}{N \sin m π})}^{2} {| \int_{- a}^{a} \exp (- \frac{j k x ζ}{l}) d ζ |}^{2} .

(14)

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I_{m} = I_{0} \frac{1}{d^{2}} {| \int_{- a}^{a} F (ζ) \exp (- \frac{j 2 m π ζ}{d}) d ζ |}^{2} .

(15)

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F (ζ) = {\begin{matrix} 1 & ζ \in [- a, a] \\ \exp [j k z (δ + j β)] & ζ \in (a, d - a] \end{matrix} .

(16)

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I_{m} = I_{0} \frac{1}{d^{2}} {| \int_{- a}^{a} \exp (- \frac{j 2 m π ζ}{d}) d ζ + \int_{a}^{d - a} \exp [j k z (δ + j β)] \exp (- \frac{j 2 m π ζ}{d}) d ζ |}^{2} .

(17)

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K = \frac{2 \sqrt{I_{0} I_{1}}}{I_{0} + I_{1}} .

(18)

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Meng Zheng, Ke Liu, Lihui Liu, Yanqiu Li. Design of a grating by a joint optimization method for a phase-shifting point diffraction interferometer[J]. Chinese Optics Letters, 2017, 15(10): 101203

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