Abstract : This paper provides a comprehensive introduction to the requirements and implementation methods of a real-time focusing servo system for a laser direct-writing equipment for binary optical elements, and analyzes the main factors affecting the performance of the focusing servo system. A method is proposed to linearize the defocus signal using a BP neural network, employ piezoelectric ceramics as the actuator, and perform dynamic nonlinear compensation; this can significantly improve the accuracy and bandwidth of the focusing servo system. Furthermore, online estimation of the defocus signal bandwidth, used to adjust the marking speed, can greatly improve marking efficiency. Keywords : Real-time focusing; BP neural network; Linearization processing; Dynamic nonlinear compensation; Bandwidth estimation 1 Introduction The design and fabrication of binary optical (diffractive optical) elements is a comprehensive cutting-edge technology based on wave optics theory, using computers as design tools, and employing ultra-precision machining, photolithography, and replication technology as the main processing methods. Currently, laser direct-writing equipment is a key device for fabricating binary optical elements. However, when using it to fabricate diffractive optical elements, the axial runout of the assembly and adjustment spindle and the uneven surface of the photoresist cause the laser beam focus to deviate from the photolithographic surface. To ensure processing quality, laser direct writing equipment must have a high-performance real-time focusing servo system to keep the processed surface within the depth of focus. To improve equipment resolution, the numerical aperture of the lithography head-mounted display objective lens is constantly increasing, resulting in a very small depth of focus range and requiring increasingly higher focusing accuracy. With increasing scribing speed, the bandwidth of the focusing servo system needs to be widened. Therefore, the performance of the focusing servo system is a key factor affecting the overall system performance. The accuracy of the real-time focusing servo system is mainly determined by three aspects: the accuracy of the focusing unit and its intelligent sensors, the resolution of the actuator, and the performance of the focusing servo controller. The bandwidth of the focusing servo system is mainly determined by the actuator and the controller. Focusing methods mainly include the skewing method, the critical angle method, and the astigmatic method. The critical angle method has a simple optical system structure, high resolution, and wide dynamic range, and is used in laser direct writing equipment. When using the critical angle method, the relationship between the defocus amount and the output signal is nonlinear…1 (Figure 1). Piezoelectric ceramics have been widely used in ultra-high precision focusing servo systems due to their nanometer-level displacement resolution, open-loop frequency response exceeding 11d-1z, and large output torque. Control methods for piezoelectric elements include voltage and current methods. When voltage control is used, the control voltage and elongation are nonlinear, as shown in Figure 2. Current control significantly improves the nonlinearity of the piezoelectric element, but greatly reduces its dynamic characteristics. This paper proposes using a neural network to linearize the defocus signal and performing simulation and feedforward compensation on the piezoelectric element, enabling it to achieve good linearity even with voltage control. When etching the inner and outer rings of large-diameter components, the significant difference in linear velocity makes it crucial to fully utilize the focusing bandwidth of the lithography head to maximize etching efficiency. This paper presents an online estimation of the defocus signal bandwidth and adaptive adjustment of the spindle speed, which greatly improves equipment efficiency. [align=center]Figure 2 Piezoelectric ceramic characteristic curve[/align] 2 Nonlinear estimation and dynamic calibration of defocus signal Due to the use of the critical angle method for defocus signal detection, there is a serious nonlinear relationship between the small amount of defocus and the sensor output. Almost all focusing systems utilize the approximately linear region in the middle of the defocus curve for focusing. The actuator uses a voice coil motor or piezoelectric ceramic. Since the resolution of a voice coil motor is difficult to exceed 0.1 micrometers, piezoelectric ceramic is widely used as the actuator in submicron and nanometer-level positioning systems, and its control method is current control. This method has four disadvantages: First, the linearity of the defocus curve is not high, affecting the focusing accuracy; second, when large disturbances during lithography cause the defocus to exceed the approximately linear region, the adjustment time is too long, affecting the quality of the lithography; third, the focusing frequency is too high, greatly reducing the application range of the focusing servo control system; fourth, recalibration is required when the lithography objective is changed or after a long period of operation, which is very inconvenient. [align=center]Figure 3 Neural Network Principle Diagram[/align] Because neural networks excel at learning useful knowledge from input and output data, they do not require precise mathematical models and can solve many complex, uncertain, and nonlinear automation problems that traditional automation technologies cannot solve. They are also easy to implement in software, making them one of the main intelligent control technologies currently available. The BP network is a relatively mature network model for practical applications of artificial neural networks, characterized by its simple structure and easy-to-master algorithm. In the focusing servo system control system of a binary optical element laser direct writing device, the nonlinear characteristics of the defocus signal throughout its working range can be described by a polynomial: Where: y is the defocus amount; x is the defocus signal detector ω[sub]i[/sub] (i=0, 1, 2, ..., n). The smallness of the defocus amount and the sensor output are closely related to the characteristic parameters of the sensing system. The nonlinear estimation and online calibration of the defocus sensing system are automatically realized using a BP neural network. The principle of characteristic parameter training is shown in Figure 3. For each actual input, a nonlinear dataset {1, x, x2, ..., xn} can be obtained. These serve as the input patterns of the neural network. The learning algorithm is described as follows: di(k) is the expected output of the i-th input; yi(k) is the estimated output of the i-th input; e(k) is the error; wn(k) is the n-th connection weight of the neural network at step k; η is the learning factor, which affects stability and convergence speed. For a set of actually detected defocus signals xt and the corresponding sensor output, a data pair (ft, Yt) is formed. The defocus characteristic curve is fitted using equation (1). Let w0, w1, w2, w3 be constants, which will be used as the input of the neural network, and as the output. The weights are corrected according to the algorithms (2), (3), and (4). All input and output data pairs constitute an iterative learning. After multiple learning iterations, the mean square error (MSE) is satisfied. The smaller the MSE, the shorter the learning time. The training is then complete. The weights at this time are estimated parameters. Before or after a period of operation, the piezoelectric ceramic is driven to produce a known displacement, and the output value of the sensor is measured. In this way, samples are obtained online. By training the neural network with the samples, the nonlinear estimation and online dynamic calibration of the defocus amount can be accurately realized. In order to shorten the training time while ensuring the fitting accuracy, when performing nonlinear estimation of the defocus amount, only the positive half-cycle of the defocus signal curve can be fitted to obtain the estimated parameters. In the actual focusing process, when the defocus signal is negative, the absolute value of the defocus signal is taken first, and then the defocus amount is calculated according to (1). Taking the negative sign of the defocus amount gives the true defocus amount. Since the piezoelectric ceramic should be driven to move a standard length when obtaining samples, a grating ruler is generally installed on the guide rail of the z-axis as its positioning standard. During focusing, the displacement accuracy of the piezoelectric ceramic also directly affects the focusing accuracy and frequency response. Due to the severe hysteresis nonlinearity of piezoelectric elements, the nonlinearity between voltage and displacement is generally higher than 15%, resulting in low open-loop positioning accuracy of piezoelectric elements and greatly limiting their application. In static and quasi-static positioning applications, charge (current) control can be used to reduce the nonlinearity of piezoelectric ceramics to less than 0-8%, and the specific implementation method is given in reference [4]. However, using charge control makes the current flowing through the piezoelectric element much larger than that of voltage control, and as the frequency increases, the requirements for the driving power supply of piezoelectric ceramics become increasingly higher. At present, the frequency response of current-type piezoelectric ceramic driving power supply is very low, which is far from meeting the frequency response requirements of real-time focusing system. For the above reasons, in real-time focusing servo system, piezoelectric ceramics should be controlled by voltage, and its nonlinear characteristics are fitted by model. The hysteresis loop of piezoelectric elements under different voltages always changes, and the position of its hysteresis loop is closely related to the history of the voltage applied previously. A simple model compensation method can make the frequency response of the system reach 300Hz and less than 3% nonlinearity. The method is as follows: The relationship between the input and output of the piezoelectric element can be defined as: (6) 3 Frequency band estimation of the defocus signal The focusing servo controller is the core part of the focusing servo system. Since the main function of the binary optical element laser direct writing equipment is to make rotationally symmetric diffractive optical elements, and the aperture of the element being made exceeds 200 mm, if the spindle rotates at a constant speed during the marking process, the linear velocity difference between the inner and outer rings of the element will be more than 1000 times. At the same time, when processing different types of elements, the surface morphology of the photoresist varies greatly, and the requirements for the focusing frequency also vary greatly. If a uniform rotary processing speed is used, the efficiency of the entire equipment will be very low. In order to improve the processing efficiency, the function of adjusting the spindle speed is added in the design of the focusing servo system controller. The spindle speed is adjusted based on the following: when etching concentric circles, the photoresist is not exposed between two concentric circles. At this time, the photolithography head remains stationary while the spindle rotates one revolution, simultaneously acquiring a defocus signal. The frequency band of the defocus signal is estimated based on this signal. The maximum allowable spindle speed is then calculated based on the frequency band of the defocus signal and the focusing frequency of the photolithography head. Finally, the spindle rotation speed is set according to this maximum allowable speed. Assuming the defocus signal is caused by factors such as uneven photoresist surface and axial runout of the rotation axis, as shown in Figure 4(a), the amplitude of the defocus signal is a function of the photolithography head position. When the photolithography head moves relative to the photoresist at a unit linear velocity, the shape of the function remains unchanged; when the linear velocity of the photolithography head decreases, the defocus signal function is as shown in Figure 4(b); when the linear velocity of the photolithography head increases, the defocus signal is as shown in Figure 4(c). As can be seen from the figures, the frequency band of the defocus signal increases proportionally with the increase in linear velocity. Online estimation of the defocus signal frequency band is one of the main functions of the focusing servo controller. Because accurately calculating the frequency band of the defocus signal is difficult and also challenging to meet real-time computing requirements, a simple method is used to estimate the frequency band in the design of the focusing servo controller for a binary optical element laser direct writing device: While acquiring the defocus signal, determine whether its sampling point is an extreme point, record its sampling time, and calculate the time interval Δt between two adjacent extreme points. When the spindle rotates one revolution, calculate the shortest time interval Δt between adjacent extreme points... Then the frequency band of the defocus signal can be roughly estimated as 1/(2Δt[sub]max[/sub]). [align=center] Figure 4: Principle diagram of defocus signal frequency band estimation. Figure 5: Block diagram of focusing servo control system.[/align] When the sampling time interval is very short, there are many points equal to the defocus signal. The identification of extreme points mainly depends on whether the sampled value has passed through a rising phase and a falling phase. The midpoint of the same sampled value near these points can be considered the input point. 4 Design of Focusing Servo Controller The focusing servo system consists of a DC servo motor, coupling, ball screw, grating ruler, piezoelectric ceramic, air-bearing guide rail, and focusing servo controller. The servo motor is used for coarse positioning and debugging of the z-axis. The functional block diagram of the focusing servo controller is shown in Figure 5. Its main functions are: (1) Drive the piezoelectric ceramic and measure various parameters of the piezoelectric ceramic based on the feedback from the grating ruler installed on the z-axis. (2) Disable the piezoelectric ceramic driver so that the spindle rotates one revolution at a constant speed. While the spindle rotates, the defocus signal is collected, and the maximum speed of the spindle is calculated based on the defocus signal. (3) Filter the signal output by the four-phase limit detector (defocus signal detector) and then normalize it to eliminate the influence of light source fluctuation on focusing accuracy. (4) Use the normalized signal to perform nonlinear estimation of the defocus curve, and fit the defocus signal with a high-order polynomial to obtain a one-to-one correspondence between the defocus amount and the detected defocus signal. [align=center] Figure 6 Working flowchart of the focusing servo controller[/align] (5) Collect the defocus signal, and calculate the driving voltage of the piezoelectric ceramic according to the magnitude and trend of the defocus amount, and output the driving signal. When the whole equipment starts working, the focusing servo controller outputs the voltage signal to drive the piezoelectric ceramic, and calibrates the defocus amount according to the feedback signal of the grating ruler installed on the z-axis. The servo controller uses a 16-bit single-chip microcomputer 80C196 as the CPU, and the main control computer is a PII266. Since the software programming for linear estimation and frequency band estimation of the defocus amount is relatively complex, the single-chip microcomputer collects the defocus signal and transmits it to the main control computer. The main control computer calculates the coefficients in formula (1) and the most suitable speed of the spindle according to the transmitted signal, and then transmits it back to the single-chip microcomputer. The controller software flowchart is shown in Figure 6. 5 Conclusion In the design of the focusing servo control system of the binary optical element laser direct writing equipment, the critical angle method is used for the detection of the defocus signal. After the device is started, the piezoelectric ceramic is driven first, the displacement of the piezoelectric ceramic is detected, and the defocus signal is collected and transmitted to the host. The defocus signal curve is fitted with a high-order polynomial using the defocus signal and the displacement, and the value of the polynomial coefficients is transferred back and stored in the microcontroller. After the defocus signal is collected, the actual defocus amount is calculated directly using the polynomial. The driving voltage of the piezoelectric ceramic is calculated based on the defocus amount, and the focusing accuracy of 0.2 micrometers can be achieved. During the scribing process, the piezoelectric ceramic can be disabled and the defocus signal can be collected. The frequency band of the defocus signal can be estimated, and the spindle speed can be adjusted in real time, which can greatly improve the scribing efficiency. References: [1] Xiong Mudi, Xiao Wenli, Xing Zhongbao, et al. Research on focusing servo system of laser direct writing equipment [J]. Optics and Precision Engineering. 2000 8(1): 79-83 [2] Liu Junhua. Intelligent sensing system [M]. Xi'an: Xi'an University of Electronic Science and Technology Press, 1999. [3] Shen Yi, Zhang Jianqiu, Wang Shizhong, et al. A new method for nonlinear measurement and dynamic calibration of pulp concentration sensor [J]. Journal of Instrumentation, 1997, 18(1): 1-6. [4] Wu Yihui. Research on nanometer resolution piezoelectric micro-positioning system [D]: [Dissertation]. Changchun: Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, 1996. [5] Jia Hongguang, Wu Yihui, Wang Liding, et al. Dynamic nonlinearity of piezoelectric element and its correction method [J]. Journal of Instrumentation, 1999, 20(4) Supplement 182-185. A new method to improve the performance of real-time focusing servo system.pdf