Hyperchromatic Lens for Fiber Confocal Surface Sensors Modeling and Calculation Based on Chromatic Coding Method
The extensive use of micro- and nanotechnology entails the need for the development and research of methods and tools for precision measurements of ultra-small distances during technological processes. Up to date, a significant area of industry, microsystem one, has been developed at the greater of these two scales. A whole class of sensors has been developed with the accuracy reaching 1 µm. These include such types of sensors as parametric (inductive, capacitive, rheostatic), ultrasound and optical (triangulation, fiber optic) . If referring to metrology, covering an area of sizes from 0.1 to 100 nm, then most methods, with the exception of interferometric ones, are still at the research stage, especially in Russia. Therefore, the task of developing methods for precision measurement of ultra-small distances to three-dimensional objects in the nanoscale area is urgent. First of all, such methods are necessary for systems of working elements positioning [2–4] for precision laser processing and investigation of three-dimensional micro and nanostructures of various materials. Earlier, in circular laser recording systems, developed jointly by Design and Technological Institute of Scientific Instrumentation and Institute of Automation and Electrometry of SB RAS , the information was recorded on flat surfaces. Triangulation methods  and video technologies were used to position the workpieces in the zone of the best focusing of the recording micro-lens. However, with the progress in the development of diffraction optics, issues of the hybrid structures formation have appeared when the profile of diffractive optical elements was synthesized directly on the curved surfaces of refractive lenses [6–8]. To implement such a method, new approaches to designing positioning systems that have fundamentally new capabilities were needed. While working with flat surfaces, the main function of the auto-focusing system is to maintain a gap between the working micro-lens and the surface to be machined. In the case of recording images on three-dimensional surfaces, the actuating elements that are part of the auto-focusing systems must process both fast small deviations of the surface due to its movement in the angular direction and slow, but larger ones, due to the natural curvature of the surface when the laser recording beam shifts radially.
To develop the technology of laser ablation of optically transparent materials by ultrashort pulses, an important condition is the precise positioning of the samples surface at the focus of the working micro-lens, since the power density has a threshold character and it is necessary to avoid melting the edges of the structures and the formation of cracks that are possible even with a small defocusing of the laser spot (0.2–0.5 µm). Furthermore, a high-precision measurement of the geometric parameters of synthesized structures makes it possible to determine the rate of ablation experimentally. Thus, the current problem is the development of methods for automatic control of three-dimensional surfaces with high resolution (<1 µm) during laser technological processes both during and after recording without removing the sample.
THE CONCEPT OF CONFOCAL SENSOR BASED ON THE CHROMATIC CODING METHOD
Nowadays, a confocal method based on chromatic coding is one of the promising methods for measuring ultra-small distances and controlling the position of surfaces [10, 11]. The fundamentals of this method are borrowed from confocal laser scanning microscopy (CLSM) which is one of the most common methods for three-dimensional studies of high-resolution objects. In CLSM, a special point (or confocal) diaphragm is used in the image plane, which limits the flow of background scattered light from points outside the focal plane of the lens . This allows you to obtain a series of images at different depths of the focal plane inside the sample and then reconstruct its three-dimensional image.
In CLSM, the object is to be scanned along the optical axis. If an enlarged longitudinal chromatic aberration is introduced into the lens of such a microscope, and the reflected signal passed through the confocal diaphragm is measured with a spectral analyzer, then a non-contact surface sensor can be built according to basic scheme shown in Fig. 1. Its distinctive feature is the absence of the need to scan an object along its optical axis.
Most often, soldered multimode fibers are used in confocal sensors to reduce the size of the measuring head and carrying the lighting and receiving part beyond the measurement area. In Fig. 1, the light from the white light source is collected into an optical fiber and sent to a hyperchromatic lens (HCL), which is used to form an increased longitudinal chromatic aberration. Since the HCL focal length depends on the wavelength, the light is transformed into a line of spectrally separated foci. Only one of the foci coincides with the plane of the surface under investigation. The reflected light passes through the HCL for a second time and is collected back into the fiber. Furthermore, the light from the focal plane is focused on the fiber end coinciding with the surface of the object. The light from the remaining planes undergoes defocusing and strong attenuation. The spectral distribution of the signal from the fiber is recorded by the spectral analyzer. By changing the dominant wavelength of the reflected signal, the object’s displacement is determined automatically [10, 13].
To implement the HCL, optical circuits based on a combination of standard optical elements with diffraction  and also reconfigurable liquid crystal lenses  are usually used. A video camera , a spectrometer , as well as multielement RGB-diodes  can be used as spectral analyzers. The source of radiation is a white light lamp: halogen, xenon, light-emitting diode, etc. In the next section, we shall discuss the methods for calculating the HCL in more details.
OF HYPERCHROMATIC LENSES
As it was shown above, the key element of the optical scheme of the chromatic confocal sensor is the lens with significantly increased chromaticity of the HCL position. One of the simplest approaches to the creation of such lenses is the use of diffraction lenses in conjunction with or without a conventional refractive lens. Refractive-diffraction lenses are described in [14, 16], which allow you to focus white light in chromatic segments of different length, depending on the requirements of specific applications. The main advantages of these lenses are the ease of implementation, improved mass-dimensional parameters and linear dependence of the focus shift on the wavelength. However, this approach has some drawbacks: first of all, the presence of parasitic diffraction orders, as well as low efficiency at the edges of the wavelength range (400–700 nm). Calculation and design of hyperchromatic lenses based on a combination of glasses with different variances is preferable.
Obviously, the chromaticity of position in lens systems arises from the dependence of the refractive index of the glasses used on the wavelength n(λ). Thus, for a single lens with a focal length f, the change Δf is described by the formula:
Where is the Abbe number, nD, nF, nC are the refractive indices of the medium at the wavelengths corresponding to the Fraunhofer lines C (656.3 nm), D (589.2 nm) and F (486.1 nm). For modern optical glasses, it is possible to achieve up to 4.9% (superheavy lead glass STF2 from the domestic catalog GOST 3514–94 or glass SF66 from the Schott catalog). With the extension of the wavelength range to 400 ч 700 nm, the relative change in the focal length is already 12%. It can be seen that the use of glass dispersion helps to achieve significant values of the position chromaticity. However, the use of the same brand of glass has a significant disadvantage, i. e., a strong nonlinearity of the dependence of the focal length on the wavelength f(λ). Thus, for a single lens such a relationship is expressed as:
where ρ1, ρ2 are the curvatures of the lens surfaces, n(λ) is the dependence of the refractive index on the wavelength, which is usually described by empirical formulas of Schott :
or Selmeyer :
where ai, Ki, Li are the coefficients found empirically for each of the materials (given in the catalogs of optical glass manufacturers).
Fig. 2 shows the diagrams of dependence of the focal lengths of lenses of different glasses on the wavelength.
It is seen from the presented diagrams that the dependences f(λ) of single lenses have a large nonlinearity up to 20% (the derivatives of functions at the edge of the range differ by 4–7 times). It is obvious that the use of a single brand of glass in the HCL development is possible only with a significant limitation of the used spectral range, which is not preferable primarily from the point of view of the energy of the optical system.
The problem of the HCL calculating for obtaining a linear (or close) dependence f(λ) is close to the problem of calculating achromatic objectives, where they tend to minimize changes in the focal length or the back segment S′F’ of the wavelength:
In the task of the HCL calculation, on the contrary, we tend to obtain a significant change, but it must be constant in the operating wavelength range :
where Δz is the length of the chromatic segment.
Let us show, for example, the calculation of two lenses bonding, how one can achieve an increase in the linearity of f(λ). In the approximation of thin components, the optical forces of the lenses in the bonding are added according to the additivity property:
Selecting the material brands n1(λ) and n2(λ), as well as the ratio of the radii of curvature of the lens, one can achieve a significant increase in linearity. Thus, Fig. 2 shows a diagram of the f(λ) dependence for the bonding of P-SF8 and TIF6 glasses. Obviously, it is possible to obtain a function sufficiently close to the linear dependence f(λ) (the nonlinearity is reduced to 6%) for the bonding. For sensors operating in the distance measurement mode, as a rule, a nonlinearity of less than 1% is required. Such values can be achieved only in more complex optical schemes using three or four lenses and at least 3 brands of optical materials.
When calculating the HCL, in addition to the linearity of the range, it is also necessary to minimize spherical aberrations and spherochromatism (the dependence of the spherical aberration on the wavelength). Spherochromatism is especially pronounced when calculating the HCL with a large operating range (). If the quality of the scattering spots in the whole range of Δλ is close to the diffraction limit, light losses in the backward path of the beams from the object to the fiber will be minimized, and also the transverse resolution in the object plane will not deteriorate.
Another important aspect in the calculation of the HCL is the choice of the lens aperture, both from the fiber side and from the object side. In order to reduce energy losses, the input aperture of the objective NAinput should be close to the numerical aperture of the used fiber: NAinput ≈ NAfiber. The output aperture should be selected based on the diffraction resolution of the lens along the optical axis, which should be much smaller than the working segment, i. e., the following condition must be satisfied:
where Δzdiffr. is the diffraction limited depth of the HCL field:, where K is the diaphragm number.
The larger values are easily achieved by simply scaling (increasing) the HCL optical schemes. However, the energy characteristics of the sensor deteriorate in this case, since the photodetector receives an increasingly narrow portion of the spectrum. Therefore, in practice, the ratio reaches values of 5 ч 30.
Using the above approaches and requirements for HCL, a number of optical schemes of lenses for a wavelength range of 0.4–0.7 µm were calculated. The HCL optical schemes and their main characteristics are given in Table 1 for the HCL-bonding (a), the three-lens HCL (b), the four-lens HCL (c). Aspherical surfaces are marked with bold lines. Fig. 3 shows the run of the curves S′(λ) minus the linear component.
Three-lens HCL with a chromatic length of 300 µm was calculated and manufactured at the Design and Technological Institute of Scientific Instrumentation SB RAS. Physical configuration is shown in Fig. 4a.
Based on the HCL, a prototype of a confocal sensor based on chromatic coding in a fiber design was developed and experimentally tested for the first time in Russia. Physical configuration is shown in Fig. 4b. The optical scheme of the sensor is described in detail in . The difference in the scheme from the previous work of the authors is the use of a three-lens HCL instead of a hybrid diffraction-refractive lens. The HCL individual lenses are made from available domestic glass of two brands: TF10 and K8. The optical scheme of the HCL and its main characteristics are shown in Fig. 5. When using the camera as a spectral analyzer, an error in measuring the distance to the surface of 0.1 µm is obtained.
This paper work presents the main schemes of hyperchromatic lenses and analyzes their main characteristics. A three-lens hyperchromatic lens with a chromatic length of 300 µm is calculated and experimentally tested. A prototype of a confocal surface sensor with chromatic coding in a fiber design has been developed for the first time in Russia, which makes it possible to move a source and a spectral analyzer outside the measuring region. The error in measuring the displacement of the object was reduced to 0.1 µm. This sensor can be used in laser systems for precise positioning of working elements over the surface to be machined during micro- and nanostructuring, and subsequently for profiling and restoring its 3D shape.