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Abstract
Precision glass molding technology is one of the most important approaches to produce optical glass lenses. However, the high fidelity and repeatability of optical performance be sometimes achieved even though the lenses meet the requirements of geometric assessments. The surface errorform errors transferred from mold surface with a complicated combination of components of different spatial-frequencies greatly influence the optical lenses performance. An optical model is built to investigate the impact of form errors with various frequencies on the optical performance of the lens. The mid-spatial frequency error is proved to be the factor that results in the most serious surrounding circle phenomenon. Based on the diffraction theory of the sinusoidal grating, numerical calculation is carried out to analyze their relationship. Experiments are conducted to validate the analysis and a mold polishing procedure is provided as a method to improve the quality of lenses according to performance evaluation.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With a wide application of optical glass elements, the qualities of the lenses attract more and more attentions. In terms of the aspheric glass lenses, which are important in optical imaging, biological engineering and precision measurement, many factors in the manufacture process influence its geometric and optical properties. The aspheric glass lenses are generally produced by Precision Glass Molding (PGM) technology [1–4]. Compared with traditional manufacturing method (such as grinding and polishing) [5–7], PGM is a production process with a high precision and efficiency. In the process of PGM, heated glass preform is pressed by precision mold and changed into a fine lens with designed shape and high quality surface. Therefore, in recent years, there have been considerable research [8–13] engaging in optimizing the technological parameters of the PGM process to improve the accuracy of the lenses.
In the evaluation of lens quality, some geometric parameters are generally used to assess the optical lenses such as peak-valley value and roughness. Several approaches have been investigated on the control of the above mentioned indexes. Yi et al. [14] systemically analyzed the molding process by finite element method (FEM) simulation, and controlled the geometric parameters by the pre-compensation of the molds. However, compared with the geometric parameters of the lenses, the optical performance is the final requirement but sometimes is overlooked in many cases. Hence, the improvement of the lenses according to their optical performance evaluation is critical and challenging though the geometric requirements are already met.
Aspheric collimating lens of laser diode, which is essential in laser beam shaping, is a typical optical glass element made by PGM. The linear-shaped laser is widely used in many fields such as tunnel detection, two-dimensional outline scanning and optical fiber communication. To meet the requirements of high quality collimating, lenses with better optical performance are necessary. However, many optical defects which decrease the yield rate are detected in the production process.
To analyze the factors which affect the optical performance of the collimating lenses, the form error of lenses made by PGM can be divided into three different parts according to the frequency band [15–18]. The studies are conducted using optical diffraction theory [19] and sinusoidal grating model [20].
The objective of this study is to build up the relationship between the optical performance and geometric form errors in a theoretical way, and provide an approach on how to improve the lens quality according to performance evaluation. Collimating lens is used as a typical element to demonstrate the analysis and method. Typical defects of the surrounding circle phenomenon are identified. An optical simulation model is built up and the relationship between the typical optical defects and form errors is established. Verification experiment is conducted and the improved method for molds is proposed accordingly.
2. Problems description
Figure 1 illustrates the integrated manufacture and evaluation system of collimating lens. Firstly, aspheric collimating lens should be designed base on the application requirements. Then the corresponding mold structure is chosen and machined by ultra-precision grinding. The last step of producing the lenses is to use PGM. After production process, the aspheric lenses are measured and evaluated by geometric parameters such as peak-valley value and roughness. If the geometric requirements are not met, geometric compensation and rework of the mold would be executed. On the other hand, the lenses which reach the geometric standard can be evaluated according to their optical performance. As the collimating lens is used in linear laser beam shaping, the desired performance of the collimating light spot on the screen is supposed to be a clear center spot with symmetrical diffraction fringes distributed in the Y axis direction. Compensation according to performance evaluation should be made if the optical spot is undesirable. Therefore, the relationship between light spot and form error must be established for the final compensation.
The optical performance of the collimating lens has a great effect on the quality of the line laser after modulation, such as uniformity and the edge sharpness. In production, the high fidelity and repeatability of optical performance of lenses made by PGM cannot be sometimes assured, even though the geometric assessment requirements are already met.
The form errors of two lenses measured by profiler and Light spots are shown in Fig. 2. According to the measurement results of form errors, it can be found that both lens 1 and lens 2 meet the geometric requirements, i.e., peak-valley value is 0.15 μm and the roughness is lower than 5 nm in Ra. However, when the lenses are evaluated by the optical performance, the light spot of the lens 1 is acceptable, and the lens 2 is rejected. The light spot corresponding to lens 1 is of eligible optical quality, namely one clear center spot with symmetrical diffraction fringes in the Y axis direction. However, the light spot of the lens 2 has significant defects such as stray light, surrounding circle and asymmetric diffraction fringes, shown in Fig. 2(d).
Fig. 2 Form errors and light spots of two lenses. (a) Form errors. (b) Accepted light spot of lens 1. (c) Rejected light spot of lens 2 and typical optical defects.
We have experimentally demonstrated that the optical performance of lenses with acceptable geometric properties is still inconclusive and need further investigation. In the next section, simulation and theoretical analysis are applied on such defects of optical performance, especially the surrounding circle phenomenon, which is critical in assessing optical performance.
3. Analysis of optical defects
The above mentioned optical defects are results from form errors with various frequency. Firstly, optical model of the light spot detection system is built up using the Virtual-lab Fusion. Secondly, the effect of optical eccentricity on the light spot is studied. Thirdly, the cause of the surrounding circle phenomenon is analyzed and the detailed impact is carried out by the simulation model. Finally, a mathematical model is developed to analyze the surrounding circle phenomenon using the diffraction theory of sinusoidal grating.
3.1 Simulation model
To simplify the study, the light of laser diode is represented by a Gaussian beam from a point source. The beam has different semi-angles of divergence in X axis and Y axis which can result in the fact that diffraction fringes are only visible in the direction along with Y axis. An aperture with the same diameter as the lens is set to remove the useless stray light through the side of the lens in this model. The perspective diagram of optical model is shown in Fig. 3(a). The propagation mode of the light in this system is Fresnel transmission as the imaging screen is located at the Fraunhofer region.
Fig. 3 3-D perspective diagram of the simulation model and simulation results (λ = 635 nm). (a) 3-D diagram of the model. (b) Simulation results of light spots of lens 1 and (c) lens 2.
To validate the feasibility and accuracy of this model, simulations of two lenses mentioned in section 2 are carried out by this model. The measured data of form error are imported into the model to build the lens directly, and the simulation results are shown in Fig. 3(b) and 3(c). Obviously, the simulated light spots agree well with the practical ones, and the surrounding circle of the ineligible lens (lens 2) is visible in the result.
3.2 Influence of optical eccentricity
The optical eccentricity caused by decenter and tilt of lenses or the assembly errors of the system has a significant effect on the light spot. Because of the long propagation length, the optical performance is sensitive to the optical eccentricity. In view of the above mentioned optical system, while the compositive optical eccentricity is greater than 5 ′, the light spot would be out of shape seriously.
The optical eccentricity component along X axis causes the non-centrosymmetrical drop-shaped center spot, as shown in Fig. 4. The Y axis directional component leads to the asymmetric optical fringes distribution. The optical eccentricity also causes obvious deviation of the light spot from the center. As the lens is rotational symmetry, the compositive optical eccentricity must be limited in a small range for eligible optical performance.
Fig. 4 Abnormal light spot caused by X axis directional optical eccentricity. (a) Simulation result. (b) Experimental result.
3.3 Simulation analysis of surrounding circle
Besides the deformation caused by optical eccentricity, the most serious defect of ineligible lens is surrounding circle. The cause of the surrounding circle phenomenon is analyzed in this section. The form error data of the two sample lenses mentioned above are decomposed and discussed respectively.
3.3.1 Cause of surrounding circle
Firstly, the standard designed aspheric surface of a lens is imported into the model to obtain an ideal lens without any form errors. The light spot located at the aperture and on the detect screen are shown in Fig. 5(a) and 5(b), respectively. Corresponding to the ideal lens, the diffraction fringe is clearly identified, highly symmetrical and without any defects.
Fig. 5 Simulation spot of standard aspheric lens and form error components after filtering of two sample lenses (λ = 635 nm). (a) Standard light spot out of the aperture and (b) on the detect screen. (c) High-spatial frequency error of lens 1.(d) Mid-spatial frequency error of lens 1. (e) Low-spatial frequency error of lens 1. (f) High-spatial frequency error of lens 2. (g) Mid-spatial frequency error of lens 2. (h) Low-spatial frequency error of lens 2.
As discussed in the section 2, the lens with the form error which meets the geometric requirements still couldn’t assure excellent optical performance. Herein, we carefully investigate the components of the form error by dividing it into various frequency bands. The form errors of the lens 1 and lens 2 are filtered into three different frequency bands, as the cutoff wavelengths are set at 0.25 mm and 1 mm according to the real working condition. And then, the form errors are divided into the errors in high-spatial frequency, mid-spatial frequency and low-spatial frequency. These components approximately represent roughness, waviness and form accuracy respectively. The components after filtering of the lens 1 and lens 2 are shown in Fig. 5. The high-spatial frequency errors and the low-spatial frequency errors of the two lenses are similar both in shape and value; but the mid-spatial frequency errors, shown in Fig. 5(d) and 5(g), are different in periodicity.
Combined with the standard designed aspheric surface, three components of the form error are respectively imported into the model for exploring their specific impacts on the optical performance. Figure 6 shows the simulation results of the light spot with different form error components. Compared with the ideal light spot, it could be concluded that the high-spatial frequency error is related to the stray light as shown in Fig. 6(a). Figure 6(b) indicates that mid-spatial frequency error leads to the surrounding circle phenomenon. And the low-spatial frequency error just has influences on the degree of defocus but has no relationship with the shape of diffraction fringes as shown in Fig. 6(c).
Fig. 6 Simulation results of standard lens combined with different components of form error. (a) Light spot with high-spatial frequency error. (b) Light spot with mid-spatial frequency error. (c) Light spot with low-spatial frequency error.
3.3.2 Relationship between the surrounding circle and mid-spatial frequency error
As the mid-spatial frequency form error results in the surrounding circle phenomenon, the specific relationship between them is analyzed in this section. According to the mode of mid-spatial frequency form error, without loss of generality, it is simplified and represented as a function:
where A, d and φ0 denote the amplitude, period length and initial phase, respectively. Equation (1) represent the mid-spatial frequency form error along arbitrary radial direction with the origin of optical axis. In the simulation model, the form error rotates 360 degrees to realize the form error distribution of the whole surface. The relationships between the surrounding circle and three parameters are analyzed through simulation model.Firstly, d is set to three different values (0.25 mm, 0.5 mm and 1 mm) while A and φ0 are set as constants (15 nm and 0). Then the designed aspheric surface data combined with three different mid-spatial frequency form errors with above parameters are imported into the model respectively, and the simulation results of the light spot and energy distribution are shown in Fig. 7. The distance from center of the light spot to the surrounding circle (D) decreases gradually as d increases. Since it is difficult only to select the whole region of surrounding circle, a rectangle window with size of 10 mm × 12 mm in the middle of the light spot is chosen to reflect the energy distribution. The specific data of the simulation results are shown in Table 1, and we can draw the conclusion that the energy of the surrounding circles increases when the position gets close to the center. Therefore, the energy density increases as D decreases, and the adverse effect on the optical performance becomes worse.
Fig. 7 Light spots and energy distribution of lenses with mid-spatial frequency form errors of different period lengths. (a) Period length is 0.25 mm. (b) Period length is 0.5 mm. (c)Period length is 1 mm.
Table 1. Numerical data of the simulation results.
Then, A is set to three different values (10 nm, 15 nm and 40 nm), d and φ0 are set as constants (0.5 mm and 0). The simulation process is the same as the one mentioned above. The simulation results of the light spot are shown in Fig. 8. It is found that D keeps invariable with an increase in the amplitude, but the energy of the center region of the light spot decreases gradually. In other words, the energy intensity and density of the surrounding circle is enhanced with an increase in the amplitude.
Fig. 8 Light spots and energy distribution of lenses with mid-spatial frequency form errors of different amplitude. (a) Amplitude is 10 nm. (b) Amplitude is 15 nm. (c) Amplitude is 40 nm.
In addition, as the initial phase of mid-spatial frequency form errors change from 0 to π/2, two extreme conditions are taken into account. φ0 are set to 0 and π/2, while A and d are set as constants (15 nm and 0.25 mm) respectively. Simulation results of the light spot are shown in Fig. 9. The white boxes marked region in Fig. 9(a) and 9(b) are enlarged, and both the cross marks in two figures are located at the same coordinate of the light spot. As can be seen, the distance D and energy of center region keep invariable no matter whether φ0 is 0 or π/2. But the form of the surrounding circle obviously changes from a single heavy circle to two slighter circles. It means that the initial phase of the mid-spatial frequency form error only influences the form of the surrounding circle rather than its position and energy intensity distribution.
Fig. 9 Light spots and energy distribution of lenses with mid-spatial frequency form errors of different initial phase. (a) Initial phase is 0. (b) Initial phase is π/2.
3.4 Numerical calculation of surrounding circle
The simulation process is of help to obtain the fundamental principle about the position and intensity distribution of surrounding circle. However, in order to get the numerical rules, a theoretical and mathematical model is developed. The effect of mid-spatial frequency error on the aspheric surface can be regarded as a series of amplitude modulated sinusoidal gratings next to a standard aspheric lens which modulates the incident light [21]. Since the basic aspheric properties just play a role in collimating process and has no significant influence on the diffraction fringes, a plane sinusoidal grating modulating collimated light is utilized to simplify the study. The diagram of the modulation process is shown in Fig. 10(a).
Fig. 10 Diagram of the diffraction process and numerical calculation results of the surrounding circle. (a) Diagram of the diffraction process. (b) Continuous mid-spatial frequency error (φ0 = 0). (c) Discontinuous mid-spatial frequency error (φ0 = π/2). (d) Position and intensity distribution of the surrounding circles.
3.4.1 Position and intensity of the surrounding circle
The transmission coefficient distribution of the sinusoidal grating is expressed as:
where A is the amplitude of the grating which is far less than the wavelength of light, d is the period length and φ0 is the initial phase. The plot of t(x1) is shown in Fig. 10(b) when φ0 is 0.It is supposed that the cycle of the grating is N, so the total length is N × d. The distribution of complex amplitude on the grating surface is given as:
In Eq. (3), u0 = 1/d, which is the spatial frequency of grating. The distribution of complex amplitude considering grating diffraction is expressed as:
In Eq. (4),
where L is the distance from the screen to the lens. λ is the wavelength of light. The rectangular function cuts off the infinite grating to finite dimension. The Eq. (4) can be simplified by Fourier transformation as:Equation (6) can be regarded as the sum of three independent Sinc functions. When calculating the distribution of intensity (I), as the cycle of the grating is large enough, the expression can be simplified as:
Equation (7) consists of only three diffraction maximums as the red line shown in Fig. 10(d) and the position depends on the equation:
As the light spot is centrally symmetric, + 1 order and −1 order constitute the surrounding circle. Therefore, the distance between the surrounding circle and the center of light spot, D, can be approximately expressed as D = Lλ/d.
The distance is inversely proportional to the period length of the mid-spatial frequency error. The intensity is in direct proportion to the square of the amplitude. The calculation results of the mathematic model are identical with the simulation results.
3.4.2 Influence of the initial phase of mid-spatial frequency form error
In consideration of the grinding processing method of the mold using in PGM, the form error of the lenses is centrosymmetric which is duplicated from the mold. Thus, the mid-spatial frequency error is discontinuous at the axis of symmetry in most conditions. The most extreme condition is discussed, shown in Fig. 10(c), when the degree of discontinuity reach the maximum.
In this condition, the expression of complex amplitude on the grating surface is:
The distribution of complex amplitude after Fresnel diffraction propagation is:
The distribution of intensity can be worked out as:
The position of the surrounding circle is shown as the blue line in Fig. 10(d). The discontinuous error only changes the mode of the fringes rather than the position and intensity of surrounding circle as mentioned above.
3.4.3 Separation of mid-spatial frequency form errors
The form error is composed of a series of error components with different parameters of d, A and φ0. Secondary Gaussian filtering and Fast Fourier Transform (FFT) are used to separate the components with different spatial frequency. Then the form error can be transformed into a new form:
and each item with different parameters is on behalf of one surrounding circle with its own position and intensity. Because of the Gaussian distribution of the laser source, the energy of the surrounding circle depends on both the position and intensity. Hence, the light spot can be considered as a desired light spot with a series of surrounding circles. Only when the energy of the surrounding circle is low enough, the optical performance is considered to be eligible. In other words, the amplitude of mid-spatial frequency should be small enough and the spatial period length should be short enough.4. Verification experiments
4.1 Design and Manufacture
Figure 11(a) presents the optical design diagram of the collimating lens. The refractive index (Nd) of glass is 1.58913 and the wavelength (λ) is set as 635 nm according to the light source. A planoconvex structure is chosen. The effective optical diameter is 5.5mm and the thickness is 2.35 mm. The distance from focus to the flat surface is 9.35 mm. The convex surface is even aspheric surface [22], and the coefficients are listed in Table 2.
Table 2. Coefficients of the designed aspheric surface
The production process of collimating lens is carried out on a PGM machine as shown in Fig. 11(b). The mold material is tungsten carbide and processed by precision grinding, the PV value is 0.11 μm and the roughness (Ra) is 2.8 nm. An anti-adhesion coating is deposited on the mold. The glass material is D-zk3 (glass transition temperature is 511°C) and the molding temperature is 556 °C. The molding process is operated under vacuum environment (air pressure under 3 Pa). The mold is cooled by pure nitrogen before mold unloading.
4.2 Evaluation and improvement of the lens
Figure 12 presents the device developed to detect optical performance of collimating lenses. The light source is a laser diode with a wavelength of 635 nm. The light is modulated as a Gaussian beam with different semi-angles of divergence, 33° in Y axis and 7.5° in X axis, respectively. The collimating lens is fixed in a sleeve as the internal diameter suit for the external diameter of the lens. The total length of the sleeve is 10 mm, and the screen is placed 15 m away from the lens. All the components of the device are mounted to be coaxial. The distance between the lens and light source can be adjusted by a displacement adjuster to match the focal length.
The aspheric collimating lens made by PGM is first measured by profiler. As the lens meet the geometric requirements, the performance evaluation process is conducted. An ineligible mold is chosen as a control group. The light spot, measured form error and frequency distribution of the original lens made by this mold is shown in Fig. 13(a). As can be seen, the peak-valley value of form error is 0.3 μm, which meets the geometric requirement. However, there are two obvious surrounding circles in the light spot, with the corresponding two form error components with large amplitude and low spatial frequency are found in the frequency spectrum diagram. To solve this severe problem, the distribution of mid-spatial frequency form error must be optimized according to the discussion in section 3. As the surfaces of lenses made by PGM are duplicated from the mold, a polishing process is employed to the origin mold. The manual polishing process is chosen as the diameter of the mold is too small for polishing machine.
Fig. 13 Optical performance, geometric measurement results and spectral distribution of mid-spatial frequency error of the lenses before and after polishing. (a) Lens made by mold before polishing. (b) lens made by mold after polishing.
After polishing, as shown in Fig. 13(b), the peak-valley value is 0.25 μm and still meets the geometric parameters. But the two original peak components with lower spatial frequency disappeared. Amplitude of the only peak component is small enough at a relatively high spatial frequency and thus the energy of corresponding surrounding circle is weak enough. The light spot of the lens made by mold after polishing has no obvious surrounding circles.
Therefore, it confirms that mold core polishing is an effective way to change the mid-spatial frequency distribution, and consequently improve the light performance of the lenses.
5. Conclusion
A relationship between performance evaluation and geometric assessment of lenses is studied in this paper. Based on the analysis of typical optical defects of collimating lens made by PGM, it is found that the optical performance of collimating lens is closely related with the form error magnitude and its various components. Simulation model and numerical model are successfully built to investigate the surrounding circle phenomenon. It is confirmed that the mid-spatial frequency form error is considered to be the cause. Furthermore, the relationship between the surrounding circles and the parameters of mid-spatial frequency form error is established and a method to improve the quality of lenses according to the performance evaluation is developed. It is concluded that polishing of molds can change the distribution of mid-spatial frequency error and improve optical performance.
The approaches proposed in this study can also be applied in other optical elements such as cylindrical lenses and focusing lenses.
Funding
National Natural Science Foundation of China (NSFC) (Nos. 51320105009 & 61635008) and Science Foundation Ireland (SFI) (No. 15/RP/B3208).
Acknowledgment
The authors express their sincere thanks to MDTP Optics for the support on the experiments.
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