# Frequency-Shifted Optical Feedback Measurement Technologies Using a Solid-State Microchip Laser
## 4 Applications of the Laser Confocal Feedback Tomography
### Laser Confocal Feedback Imaging Combined with Other Technologies  
#### 4.4.1 Depth of Focus Extension in Laser Frequency-Shifted Feedback Imaging  
For optical imaging, there is always a trade-off between the depth of focus (DOF) and the resolution [94; 95]. Therefore, extending the DOF has drawn much attention, and various methods have been put forward. Numerically reconstructing a higher-resolved picture from a traditional beam is a main category of the methods, which is challenged by the decline of ballistic photons with the longer DOF. The laser feedback effect has the high sensitivity and could solve the problem. Thus, [PERSON] [96] reported the depth of focus extension in laser frequency-shifted feedback imaging by filtering in the frequency domain. The system is shown in Figure 24. The measuring beam with the frequency shift after passing the BE is reflected by two galvanometric mirrors, then converged by objective lens L5; however, with the target located at the defocus plane.  
The original image is acquired via a 2-D scanning point by point, which can be described as the convolution of the point diffusion function (PSF) with the target plane. In theory, the Fourier transform of PSF (CTF) at a defocus length (DL) of \(L\) can be deduced as:  
\[H(v,u)\propto\exp\left[-\frac{\left(v-2\frac{\theta_{x}}{\lambda}\right)^{2}+ \left(u-2\frac{\theta_{y}}{\lambda}\right)^{2}}{\frac{2}{\pi^{2}r_{0}{}^{2}}} \right]\times\exp\left\{j\frac{\pi L\lambda}{2}\left[-\left(v^{2}+u^{2}\right) +2\left(v\frac{2\theta_{x}}{\lambda}+u\frac{2\theta_{y}}{\lambda}\right) \right]\right\} \tag{18}\]  
By filtering in the frequency domain, the new CTF can be obtained as:  
\[H(v,u)\propto\exp\left[-\frac{\left(v-2\frac{\theta_{x}}{\lambda}\right)^{2}+ \left(u-2\frac{\theta_{y}}{\lambda}\right)^{2}}{\frac{2}{\pi^{2}r_{0}{}^{2}}} \right]\times\exp\left\{j\pi L\lambda\left(v\frac{2\theta_{x}}{\lambda}+u\frac {2\theta_{y}}{\lambda}\right)\right\} \tag{19}\]  
Figure 24: Schematic diagram of the laser frequency-shifted feedback imaging system. ML, microchip laser; L1, collimator; BS, beam splitter; AOM1 and AOM2, acousto-optic modulators; L2, optical lens; Mr, reference mirror; AP, aperture; BE, beam expander; GM1 and GM2, galvanometric mirrors; L3 and L4, lenses forming the 4f system; L5, objective lens; T, target; PD, photodetector. The inset picture illustrates the physical meaning of the misalignment parameters \(\theta_{x}\) and \(\theta_{y}\), where the dashed red line implies an ideal direction of the laser when \(\theta_{x}=\theta_{y}=0\). (Figure reproduced from Ref. [96]).  
The inverse Fourier transform of the CTF corresponding to the new PSF can be calculated using:  
\[h_{a}\propto\exp\left[-\frac{\left(x+L\theta_{x}\right)^{2}+\left(y+L\theta_{y} \right)^{2}}{\left(\frac{r_{0}}{\sqrt{2}}\right)^{2}}\right]\times\exp\left[j2 \pi\left(x\frac{2\theta_{x}}{\lambda}+y\frac{2\theta_{y}}{\lambda}\right)\right] \tag{20}\]  
Therefore, a resolution of \(r_{0}/\sqrt{2}\) is obtained by filtering in the frequency domain, and it is possible to keep the resolution even in the far defocus plane.  
A three-dimensional target with three steps is tested in the experiment, and the processing of gradual refocusing is demonstrated as shown in Figure 25. The original blurred image is obtained using a 2-D scan. By filtering the image matrix in the frequency domain with a sequence of DL = 0.1:0.1:100 mm, three clear images of different steps can be obtained. The difference between the values of DL in filtering is exactly 5 mm, which equals the step height. In this way, the information along the optical axis can be obtained. The numerical experiments reveal that its depth of focus is capable of being extended to four times the length of the objective focal length.