可调谐相位板空域频域联合分析

波前编码系统采用在传统光学系统中加入相位板来扩大光学系统的景深而避免传统景深延拓技术的不利影响. 由于相位板的参数不可调, 整个系统的景深延拓扩展率也不能动态可调. 采用两相位板组合的方法可以有效克服这一点. 本文首先从光线差的角度提出了两三次相位板组合下的光线像差分布以及点扩散函数尺寸的具体关系表达式, 直观体现了系统的光线结构, 指出了光线结构和点扩散函数尺寸受两三次相位板的面型和相对位移量的影响. 其次采用稳相法从空间域给出了系统点扩散函数表达式, 依据点扩散函数的振荡性质给出了有效带宽表达式, 提出了点扩散函数在像面的位置会随两相位板面型参数以及相对于光瞳中心的位移量而发生平移. 最后利用菲涅耳积分给出两三次相位板任意面型参数和相对位移组合下的准确光学传递函数. 在得到的调制传递函数中直观体现出了面型参数和相对位移量对调制传递函数和相位传递函数以及有效带宽的影响, 并说明了此系统相位传递函数的非线性性质. 通过空间域与频率域相结合的方法分析验证了传统的两三次相位板组合具有景深可调和带宽可调的性质, 为设计可调谐波前编码系统提供了理论依据.

Combined analysis of tunable phase mask within spatial and frequency domain

Wavefront coding technique is a powerful technique which overcomes the defects of traditional way to extend depth of field. By inserting a phase mask into the traditional incoherent imaging system, wavefront coding technique does not reduce the resolution and the light gathering power of the optical system but enlarges the depth of field of incoherent imaging system. Although several kinds of phase masks have been reported, cubic phase mask is still of a classical type which has been investigated widely both in spatial and frequency domain. Since the phase profiles of phase masks adopted in classical wavefront coding systems are predefined with specific optical systems, the extension of depth of field is not tunable. Tunable wavefront coding systems are introduced by using a pair of detachable phase masks, which is possible to control the depth of field and bandwidth of system by changing the position of each component with respect to the pupil center. Ojeda-Castañeda Ojeda-Castañeda J, Rodríguez M, Naranjo R 2010 Proceedings of Progress in Electronics Research Symposium, Cambridge, July 5–8, 2010 p531] proposed to use a pair of cosine phase masks to make defocus sensitivity tunable. Zhao Zhao H, Wei J X 2014 Opt. Commun. 326 35] investigated an improved version of Ojeda- Castaneda's design in frequency domain and found that the proposed system realized tunable bandwidth. The present study, based on the work of Zhao, analyzes the tunable characteristics of a pair of simple modified detachable cubic phase masks in spatial domain and frequency domain. Firstly, the ray aberration theory is adopted to give mathematical analyses and ray aberration maps of the proposed tunable phase mask. Based on the mathematical derivations, the size of point spread function (PSF) of system can be changed not only by profile of each cubic mask but also by the each mask displacement relative to pupil center. Secondly, a mathematical PSF based on the stationary phase method is derived in spatial domain. Simulations indicate that the positions of PSF translate in the image plane with the displacements of phase mask profile and the position of each component with respect to the pupil center. By analyzing the oscillations of PSF, the effective bandwidth is obtained. Through the expression, we can conclude that the effective bandwidth can be changed by the position, mask profile of each component and defocus. Only when the addition of two mask profiles is large enough, can the effective bandwidth be simplified without adding the influence of defocus. In addition, though the approximate expression of magnitude transfer of function (MTF) has been given by adopting stationary phase method in the appendix of previous work, it cannot give an intuitive grasp of the effective bandwidth in MTF map. Unlike the MTF expression derived before, the exact optical transfer function (OTF) expression is derived by adopting Fresnel integral in frequency domain. Exact MTF and phase transfer function (PTF) can be derived from OTF. Based on the exact MTF expression, simulations give an intuitive effective bandwidth in MTF map. Simulations also show the nonlinear property of PTF. The effective bandwidth and MTF can be changed by different phase mask profiles and positions, which indicate that the effective bandwidth and defocus sensitivity can be tuned. Analyses are conducted both in spatial domain and in frequency domain to verify the tunable property of the proposed phase mask, which provides theoretical foundation for tunable wavefront coding system design.