用连分数定义莫尔条纹“准周期”

Defining the quasi-period of a Moire fringe using continued fraction expansion

由两组平行周期条纹叠加而成的莫尔条纹并非严格的周期结构,其近似的周期性在数学上等价于对任意实数的最佳有理逼近.通过将两条纹周期之比近似为其连分数的各阶渐进分数,可系统性地严格定义莫尔条纹各阶“准周期”并计算其长度;实际观察到的莫尔条纹的周期,是满足非周期程度低于经验阈值的最低阶准周期.基于莫尔条纹与连分数展开之间的对应关系,可以找到一类具有严格周期性的莫尔条纹,以及一类“周期性最差”的黄金比例莫尔条纹.本文建立了莫尔条纹与实数基本性质的联系,对莫尔条纹现象的本质提供了新的理解,对所有周期叠加问题都具有普适意义.

A Moire fringe formed by the superposition of two parallel periodic arrays of lines is not strictly periodic,whose approximate periodicity corresponds to best approximations to a real number.The quasi-periodicity of a Moire fringe can be rigorously defined by expressing the ratio of the respective periods of the two arrays in the form of a continued fraction expansion,and its quasi-periods can be derived by approximating the ratio to convergents of different orders.Meanwhile,the observed period is the lowest quasi-period with a degree of aperiodicity smaller than an empirical constant.Based on this direct correspondence between a Moire fringe and continued fraction of a real number,a set of rigorously-periodic Moire fringes and another set of“worst periodic”golden-ratio fringes can be identified.The present work connects Moire fringes with the basic properties of real numbers,and therefore provides new understandings for the Moire fringe phenomenon,and is of general significance to all sorts of periods superposition problems.