# Extended Depth-of-Field of a Miniature Optical Endoscope Using Wavefront Coding
## 1 Introduction  
Endoscopic systems are commonly employed to observe inside of the body's cavities, thereby providing a wealth of information about the inner structure. It is an elongated optical device used to observe otherwise inaccessible areas in either a non-invasive or minimally invasive manner. In general, endoscopes can be grouped into three main categories; namely, rigid endoscopes, flexible endoscopes and capsule endoscopes, which have the versatility to meet the most challenging applications [1]. For the rigid scope, a rod lens system is positioned in the distal end of the endoscope, and optical properties still surpass those of the fiber optics or digital images created by flexible scopes [2].  
Depth of field (DOF), referring to the range of distance over which a subject appears in-focus, is a key design criteria for endoscopic system [3]. It varies mainly relying on aperture and focusing distance, and viewing distance can also influence the DOF [4, 5]. Although a large DOF demands an objective lens with a small aperture, a limited value of objective distance inevitably confines the DOF in a small range, especially for group of endoscopes that aim at inspecting inner wall of component. For a small DOF, such an endoscope is incapable of providing high axial resolution and high optical sectioning ability [6, 7]. In addition, feature of extended DOF is particularly useful for diagnosing the series of industrial components with various dimensions. Therefore, development of extended DOF and small-diameter endoscopic modality, with the capabilities of simultaneously capturing information from multiple planes in a single picture, is extremely desirable in modern industrial and scientific areas.  
DOF can be enlarged by recording a rapid sequence of images with different focal length settings of the objective; however, such light field technique suffers from an acquisition speed limitation and a remarkable drop-off in resolution [8]. Adding electrically tunable-focus lenses with liquid crystal is an alternative to extend the DOF of an endoscopic imaging system; however, a number of drawbacks, such as slow response time and only working for polarized light, still exist in the current system [9; 10]. The intrinsic contradiction between small objective distance and limited DOF is able to be avoided in wave-front coding approach, which establishes on the manipulation of point spread function (PSF) engineering in combination with post-digital processing procedure [11; 12]. This proposal employs a separable phase mask placed in the aperture of an imaging system that makes the image relatively insensitive to the defocus or axial depth [13; 14].  
In this article, we discuss the principle of extending DOF through wavefront coding and its implementation as a new optical endoscopic modality with a narrow tube and high-order phase mask. The remainder of this paper is structured as follows. Section 2 illustrates the conceptual theory and design principles of our proposal in detail. Section 3 describes an optical endoscopic model and its simulation results, which significantly verify the validity of the proposed system. Restoring the raw images acquired from the system to increase the spatial resolution in large DOF is carried out and presented in Section 4. In Section 5, simulated results of imaging from original system and proposed system with extended DOF are presented. Finally, we summarize and conclude our work in Section 6.