# Extended Depth-of-Field of a Miniature Optical Endoscope Using Wavefront Coding
## 3 Design and Simulation  
### Initial Optical Endoscope  
Schematic diagram of our designed optical endoscopic model is shown in Figure 1. The proposed system mainly consists of a front-side objective lens group, a rod lens group, a back-end objective lens group, a piece of protective glass and an image sensor. An objective lens system having a first and a second lens element that are separated by a gap where a ray bundle is reflected by 90\({}^{\circ}\). The rod lenses are used to connect to the independent sensor for imaging and extend the optical tube with sufficient length. The purpose of the objective lens is to cause light to converge on a single point and transmit light back to the image sensor. The protective glass keeps the image sensor safe and enables the sensor to operate at peak performance. The 1/3\" image sensor, being composed of 1280 \(\times\) 960 pixels with 3.75 \(\upmu\)m pitch, is employed to register the incident light. The entire endoscopic lens system has a relatively small number of lens elements, and the aberrations of spherical, chromatic, coma and distortion are favorably corrected. Detailed information of the presented optical endoscope is listed in Table 1.  
The DOF of this system can be approximated to be [16]  
\[\Delta l=\frac{2D_{p}f^{\prime}d_{o}^{2}Z^{\prime}}{D_{p}^{2}f^{\prime 2}-d_{o}^{ 2}Z^{\prime 2}} \tag{10}\]  
where the effective focal length is \(f^{\prime}=3.23\) mm, the diameter of entrance pupil is \(D_{p}=0.35\) mm and the objective distance is \(d_{o}=2\) mm. \(Z^{\prime}\) denotes the maximum diameter of spot on the image sensor, which is fixed at 7.5 \(\upmu\)m in the current situation. Therefore, the calculated value of original DOF is about 53 \(\upmu\)m.  
MTF, measuring the ability of the endoscopic system to transfer various levels of detail from object to image, is firstly calculated by the productive mathematical software Matlab. Figure 2 shows the MTF curves in different objective distance that are computed by Equation (8). The MTF curve has the best performance when the objective distance value satisfies the designed focus value. In addition, MTF curve seriously declines when the objective distance deviating from the best value, and also decreases to zero at certain frequencies. The MTF value going to zero at some special spatial frequencies implies this system completely loses the imaging information. Simulations were then performed using a commercial simulation tool Zemax to characterize the designed system. The simulated MTF curves in tangential and sagittal directions were drawn with various objective distances, which are shown in Figure 3a,b, respectively. The cut-off spatial frequency obviously decreases when the objective distance is away from the designed value, which is in accordance with the results shown in Figure 2. Furthermore, the larger objective distance and the more zero points will appear. The light ray distributions on an observation surface, referred to as spot diagrams, are depicted in the situations of different objective distances and FOVs (see Figure 4). The minimum radius of the spot appears at the objective distance of 2.1 mm, being closed to the best focus value. The greater spot is acquired with larger objective distance, which means the system has a large amount of defocus. Taking into account all of these results, we may reach the conclusion that the performance of the initial optical endoscope is sensitive to the objective distance.