Review of Fresnel incoherent correlation holography with linear and non-linear correlations

Optical configuration of FINCH with (a) polarization multiplexing and (b) spatial multiplexing.

The quest for developing novel imaging technologies to observe objects with high resolution is on-going. One of the straight-forward resolution enhancement technologies include illuminating objects with electromagnetic radiation with wavelengths shorter than that of the visible light which led to the development of electron, X-ray and ion microscopes, etc. An alternative method to improve the resolving power is to increase the numerical aperture of the imager. Both approaches have different types of technological demands; the former requires a compatible source, lens, and detector for a different electromagnetic spectral range, the later requires manufacturing of lenses with larger diameters or changes in optical configuration that still affect the modulation transfer function in a similar fashion. The above methodologies impose challenging loads on both the fabrication and materials engineering fronts. One of the easiest methods to improve the imaging resolution is to replace a coherent illumination with an incoherent one, which in turn doubles the spatial frequency limits of imaging. In addition to resolution enhancement, the above method also brings various advantages such as formation of images without speckles or edge ringing and at a substantially lower cost.

A classical incoherent imaging system has twice the bandwidth of an equivalent coherent imaging system, but has a non-uniform response to different spatial frequencies. The higher spatial frequencies endure a stronger attenuation than the lower spatial frequencies and so the full potential of the doubling bandwidth is not translated to the maximum possible resolution. In 2007, a non-classical imaging technique capable of breaking the Lagrange Invariant condition was developed by two eminent scientists namely Prof. Joseph Rosen, Ben Gurion University and Prof. Gary Brooker, John Hopkins University, with capabilities for a uniform response over the entire bandwidth. This super resolution technique is called Fresnel incoherent correlation holography (FINCH). In FINCH, the two-point resolution is unique, originating from the fact that when two points are imaged, the magnification of the spacing between the points is higher than the magnification of a point, an advantageous abnormality that results in a super-resolution. FINCH has disadvantages however, such as lower axial and time resolutions.

Phase image of a diffractive optical element formed by randomly multiplexing a quadratic phase mask and a spiral quadratic phase mask with a scattering ratio of 0.02.

While there are numerous researchers that developed technologies to improve the time resolution, all such research were only focused on converting the required temporal shots into spatially or polarization multiplexed single shot. As a consequence, the temporal resolution was increased at the cost of either the field of view or signal to noise ratio. In all the previous studies, the inline FINCH hologram was reconstructed by a Fresnel back propagation which generated the twin images and bias terms. At least three camera shots with relative phase-shifts were needed to synthesize a complex hologram which, when reconstructed by Fresnel propagation, generates the image without the twin image and bias terms. A game changing approach to FINCH was introduced by the Swinburne’s nanotechnology and nanophotonics research team in 2020, in which a modified reconstruction mechanism was applied which simultaneously increased the temporal as well as axial resolution. In this study, the point spread function of FINCH is recorded and is used as the reconstructing function with a cross-correlation by a non-linear filter. Thus, the problem was converted into a pattern recognition problem. The FINCH system was realized using a spatially random multiplexed single diffractive optical element fabricated using electron beam lithography. The above procedure resulted in a compact, light-weight version of FINCH with a single camera shot and improved axial resolution.

The Swinburne’s nanotechnology and nanophotonics research team has recently investigated a fundamental problem, the transfer of characteristics from modulation function to the reconstructed image in FINCH, with two types of numerical reconstructions: Fresnel propagation and non-linear correlation in both spatial as well as polarization multiplexing modes. The results of the study are published in Chinese Optics Letters, Volume 19, No. 2, 2021 (V. Anand, et al., Review of Fresnel incoherent correlation holography with linear and non-linear correlations [Invited]). The investigation revealed interesting characteristics of FINCH in non-linear correlation mode. FINCH in Fresnel propagation mode faithfully transfers the characteristics of the modulation function to the reconstructed image. FINCH in non-linear correlation mode offers a platform to manipulate the degree by which the beam characteristics are transferred to the reconstructed image by controlling the degree of chaos in the spatially multiplexed diffractive element. Consequently, the above mode of operation improves the tolerance of FINCH to noise and aberrations. The findings of the study are expected to not only open new pathways to introduce special imaging characteristics but also guide in building versatile FINCH scopes in the future.

综述|FINCH：一种高分辨率广谱全息成像技术

FINCH的光学装置（a）偏振复用；（b）空间复用

随着先进成像技术带来的视觉革命，能以高分辨率观测目标物体的新型成像技术研究正在如火如荼的进行之中。其中一种可显著增强分辨率的技术采用波长短于可见光的电磁辐射照射物体，该方法推动了电子显微镜、X射线显微镜和离子显微镜等设备的发展；另一种提高分辨率的方法是增大成像仪的数值孔径。两种方法都有各自的技术要求：前者需要根据电磁光谱的范围提供可兼容的光源、透镜和探测器；后者要求制造直径更大的透镜，或者改变光学结构等。两种方法均给制造和材料工程前沿带来了巨大挑战。

提高成像分辨率最简单的方法之一是用非相干照明代替相干照明，从而使成像的空间频率极限倍增，以达到增强分辨率的目的。此外，非相干照明还能规避图像散斑和边缘振铃，以极低的成本实现成像，因而具备独特的优势。

经典非相干成像系统的带宽是等效相干成像系统的两倍，但因受限于对空间频率响应的非均匀性，双倍带宽的分辨率转换效率无法达到理想值。2007年，古本里安大学的Joseph Rosen教授和约翰霍普金斯大学的Gary Brooker教授开发了一种能够打破拉格朗日不变条件的非经典成像技术，实现了空间频率在整个带宽内的均匀响应。这种超分辨率技术被称为菲涅耳非相干相关全息术(FINCH)。

该技术可实现优异的两点分辨率：当两点成像时，两点间距的放大倍数高于两点本身的放大倍数，从而得到超分辨率。然而，FINCH也有自身缺陷，比如轴向分辨率和时间分辨率较低。虽然有许多研究人员开发了提高时间分辨率的技术，但所有这些研究都只专注于将所需的时间镜头转换为空间或偏振复用单幅镜头。由此可见，时间分辨率的提高是以视场或信噪比为代价的。

二次相位掩模和螺旋二次相位掩模随机复用形成的衍射光学元件相位图像，散射比为0.02

在以往的研究中，FINCH全息图通过菲涅耳反向传播生成孪生像和偏置项来重建合成。合成复杂全息图至少需要记录三个拥有不同相差的全息图，最后才将它们投影形成一个复合全息图。通过反向传播算法，复合全息图从物体的多个不同平面被重建为一个3D图像，且生成的图像不含孪生像和偏置项。

斯威本科技大学的研究团队在2020年为FINCH带来了变革性的改变。他们借助非线性滤波器，把FINCH 的点扩展函数转换为互相关的重建函数，从而把问题转化为模式识别问题。这种重建机制可同时提高FINCH的时间分辨率和轴向分辨率。利用电子束光刻制作的空间随机多路单衍射光学元件，研究人员们研制出了一种紧凑的、轻型的FINCH单镜头系统，成功提高了轴向分辨率。该工作作为封面文章发表在Chinese Optics Letters第19卷第2期(V. Anand, et al., Review of Fresnel incoherent correlation holography with linear and non-linear correlations [Invited])。

FINCH的一个基本问题在于调制函数到重建图像的转移特性，其中数值重构包括两种类型：空间域以及偏振复用模式下的菲涅耳传播和非线性相关。 FINCH在菲涅耳传播模式下如实地将调制函数的特性传递到重构图像上；非线性相关模式的FINCH通过控制空间复用衍射元件的混沌程度，来操纵光束特性传递到重构图像的程度。因此，上述操作方式提高了FINCH对噪声和像差的容忍度。

本研究结果不仅为FINCH成像技术开辟了新的思路，而且可扩展到前景广阔的单次3D彩色成像激光加工应用中，同时有望揭示激光致介质击穿过程中X射线和THz光束生成的时空演化过程。