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Journals >Chinese Optics Letters >Volume 15 >Issue 4 >Page 040902 > Article

Chinese Optics Letters
Vol. 15, Issue 4, 040902 (2017)

Modal analysis of 1 × 3 reflective triangular gratings under normal incidence

Jin Wang^1、2, Changhe Zhou^1、*, Jianyong Ma¹, Yonghong Zong^1、2, and Wei Jia¹

Author Affiliations

¹Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China

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

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DOI: 10.3788/COL201715.040902 Cite this Article Set citation alerts

Jin Wang, Changhe Zhou, Jianyong Ma, Yonghong Zong, Wei Jia. Modal analysis of 1 × 3 reflective triangular gratings under normal incidence[J]. Chinese Optics Letters, 2017, 15(4): 040902 Copy Citation Text

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Abstract

Modal analysis of the

1 \times 3

highly efficient reflective triangular grating operating in the 800 nm wavelength under normal incidence for TE polarization is presented in this Letter. The rigorous coupled wave analysis and simulated annealing algorithm are used to design this beam splitter. The reflective grating consists of a highly reflective mirror and a transmission grating on the top. The mechanism of the reflective triangular grating is clarified by the simplified modal method. Then, gratings are fabricated by direct laser writing lithography.

MF = \frac{1}{M} \sum_{λ} {[\frac{\sum_{p = - 1}^{1} {(η_{p} - η_{av})}^{2}}{\sum_{p = - 1}^{1} η_{p}}]}^{1 / 2},

(1)

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η_{av} = \frac{1}{3} \sum_{p = - 1}^{1} η_{p},

(2)

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\cos [k_{1} (1 - f) T] \cos (k_{2} f T) - \frac{k_{1}^{2} + k_{2}^{2}}{2 k_{1} k_{2}} \sin [k_{1} (1 - f) T] \sin (k_{2} f T) = \cos (α T),

(3)

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k_{i} = k_{0} \sqrt{n_{i}^{2} - n_{eff}^{2}}, i = 1, 2,

(4)

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{\bar{n}}_{0 eff} = \sum_{i} n_{i, 0 eff} h_{i} / h,

(5)

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{\bar{n}}_{1 eff} = \sum_{i} n_{i, 1 eff} h_{i} / h,

(6)

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{\bar{n}}_{2 eff} = \sum_{i} n_{i, 2 eff} h_{i} / h,

(7)

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e_{0} = \frac{1}{T} \int_{0}^{T} [t_{0} u_{0} (x) e^{- i k_{z} {\bar{n}}_{0 eff} h} + t_{2} u_{2} (x) e^{- i k_{z} {\bar{n}}_{2 eff} h}] d x,

(8)

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e_{\pm 1} = \frac{1}{T} \int_{0}^{T} [t_{0} u_{0} (x) e^{- i k_{z} {\bar{n}}_{0 eff} h} + t_{2} u_{2} (x) e^{- i k_{z} {\bar{n}}_{2 eff} h}] d x,

(9)

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e e_{- 1} = \frac{1}{T} \int_{0}^{T} [u_{0} (x) e^{- i k_{z} ({\bar{n}}_{0 eff} - {\bar{n}}_{2 eff}) h} + u_{2} (x)] d x,

(10)

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e e_{0} = \frac{1}{T} \int_{0}^{T} [(u_{0} (x) e^{- i k_{z} ({\bar{n}}_{0 eff} - {\bar{n}}_{2 eff}) h} + u_{2} (x)) \cos (k_{x} x) + i u_{1} (x) e^{- i k_{z} ({\bar{n}}_{1 eff} - {\bar{n}}_{2 eff}) h} \sin (k_{x} x)] d x,

(11)

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e e_{- 2} = \frac{1}{T} \int_{0}^{T} [(u_{0} (x) e^{- i k_{z} ({\bar{n}}_{0 eff} - {\bar{n}}_{2 eff}) h} + u_{2} (x)) \cos (k_{x} x) - i u_{1} (x) e^{- i k_{z} ({\bar{n}}_{1 eff} - {\bar{n}}_{2 eff}) h} \sin (k_{x} x)] d x .

(12)

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η_{\pm 1} = {| E_{+ 1} |}^{2} = {| E_{- 1} |}^{2} = {| e_{1} (e e_{0} + e e_{- 2} + e_{0}) \exp (i α) |}^{2},

(13)

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η_{0} = {| E_{0} |}^{2} = {| (2 e_{1} e e_{- 1} + e_{0}^{2}) \exp (i α) |}^{2} .

(14)

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T = d_{0} \cos φ,

(15)

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Jin Wang, Changhe Zhou, Jianyong Ma, Yonghong Zong, Wei Jia. Modal analysis of 1 × 3 reflective triangular gratings under normal incidence[J]. Chinese Optics Letters, 2017, 15(4): 040902

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