# Frequency-Shifted Optical Feedback Measurement Technologies Using a Solid-State Microchip Laser
## 4 Applications of the Laser Confocal Feedback Tomography
### Lens Thickness Measurement  
Lens thickness measurements play an important role in the optical industry and optical systems. Various optical methods have been applied in the lens thickness noncontact measurement. A machine vision method [88] has advantages of continuous measurement, it is flexible in the production line, and the accuracy is less than 0.005 mm. Chromatic confocal sensors [89] are based on spectrally broadband light to realize a special optical probe. The measuring range and the accuracy are influenced by the designed probe and the dispersion of the measured object material. Low coherence interferometry [90] is utilized to measure the central thickness of soft and rigid contact lenses with the accuracy of a few micrometers. A laser differential confocal technique [91; 92] realizes the high-precision measurement of the lens' axial space by detecting the absolute zero of an axial intensity curve. The measurement uncertainty is less than 0.05% for the thickness. Considering the high sensitivity of the laser feedback effect, it has great potential for the axial positioning, especially the multilayer lenses or the coated  
Figure 21: Schematic of the surface measurement system. ML: microchip laser; L1–L3: lenses; BS1 and BS2: beam splitters; PD: photodiode; RF1 and RF2: reference signal generators; LIA: lock-in amplifiers; AOM1 and AOM2: acousto-optical demodulators; BE: beam expander; M1 and M2: reflect mirrors; Obj: objective lens; SA: sample; ST: stage. (Figure reproduced from Ref. [87]).  
lens with the high anti-reflecting film. [PERSON] [93] proposed a new method to realize the lens thickness and air gap measurement, as shown in Figure 22. The annular pupil is inserted into the path before the objective lens to create the annular beam, in order to reduce the axial aberration and decrease the positioning error.  
When the objective lens moves along the optical axis, and the focused beam passes through the surfaces of the lenses, the defocus response curve can be obtained at each interface. The peak of the curve corresponds exactly to the beam focusing on the surface of the lens, as shown in Figure 23. Thus, the axial positioning can be realized via the synchronous acquisition of the light intensity and the displacement of the stage. Considering the refraction and reflection when the light passes through the lens, the relationship between the lens thickness and the axial positioning displacement can be deduced as:  
\[d_{i}=\frac{\int\limits_{i\epsilon}^{R}T_{i}\cdot 2\pi\rho\cdot d\rho}{\pi R^{2 }(1-\epsilon^{2})} \tag{17}\]  
where \(\epsilon\) is the ratio of the inner and outer diameter of the AP, \(T_{i}\) is the ray tracing process function related to the lenses measured, and \(d_{i}\) is the displacement between each interface.  
Figure 23: Experimental measuring results of two lenses. (Figure reproduced from Ref. [93]).  
Figure 22: Schematic of the lens thickness measurement system. LD, laser diode; Grin Lenses, graded index lenses; BS, beam splitter; AOM\({}_{1}\) and AOM\({}_{2}\), acousto-optic modulators; BE, beam expander; AP, annular pupil; Obj, objective lens; L, lenses; PD, photo detector; RF, reference signal generator; St, Stage; LIA, lock-in amplifier; PC, computer. (Figure reproduced from Ref. [93]).  
Different materials and kinds of lenses are measured in the experiment, including K9 plain glasses, fused silica plain glass, and K9 biconvex lens. The results prove that the uncertainty of the axial positioning is better than 0.0005 mm and the accuracy reaches the micron range.