基于衍射光学元件的相位延迟和衍射效率表达式，推导出了含有偏心误差的多层衍射光学元件衍射效率表达式。建立了含有偏心误差的多层衍射光学元件衍射效率的数学模型，分析了偏心误差对多层衍射光学元件衍射效率及多色光积分衍射效率的影响。以在8~12 μm 波段内的硫化锌(ZnS)和锗(Ge)为基底材料构成的多层衍射光学元件为例，其设计波长对为8.79 μm、11.11 μm，构成多层衍射光学元件的两层谐衍射元件微结构高度为78.3391 μm 、34.6076 μm，当多层衍射光学元件的环带宽度分别为500 μm 和1000 μm 时，其衍射效率达到95%以上时，偏心误差须分别控制在5.8 μm 和11.17 μm 以内。该含有偏心误差的多层衍射光学元件的衍射效率分析模型对于多层衍射光学元件的设计与加工具有重要意义。

Based on the expression of phase delay and the expression of diffraction efficiency of diffractive optical elements (DOEs), the expression of diffraction efficiency of multilayer diffractive optical elements (MLDOEs) is described. The mathematical analysis model of diffraction efficiency for MLDOEs with decenter error is presented, and effects of decenter error on diffraction efficiency and polychromatic integral diffraction efficiency of MLDOEs are analyzed in long infrared waveband of 8~12 μm . ZnS and Ge are selected as the material of first and second harmonic diffractive optical elements respectively for the MLDOEs, the design wavelengths λ1 and λ2 are determined with 8.79 μm and 11.11 μm . The micro- structure height of MLDOEs are 78.3391 μm and 34.6076 μm , respectively. Finally, with the period width of MLDOEs 500 μm and 1000 μm , the range values of decenter error must be smaller than 5.8 μm and 11.71 μm when polychromatic integral diffraction efficiency is greater than 95% . The analysis model of diffraction efficiency with decenter error is important to manufacture and design process of MLDOEs.