Computer-Aided Alignment for High Precision Lens LI Lian, FU XinGuo, MA TianMeng, WANG Bin The institute of optical and electronics, the Chinese Academy of Science, Chengdu 6129, China ABSTRACT Computer-Aided Alignment (CAA) is an effective method for improving image quality of an optical system, which is implemented by aberration compensation technique. This paper studies some key techniques of CAA, including the mathematical model of CAA, the selecting of the aberration compensator, the establishment of sensitivity matrix and the solution of misalignment. A numerical simulation of CAA has been performed for a four-lens precision optical system to verify the ability and accuracy of the method. Comparisons of the image qualities between the pre-alignment and post alignment systems are also presented. These results indicate that the CAA method is feasible. It can not only meet the precision requirement, but also accelerate the convergence of alignment solutions. This method is realized by compensation among variables, so the variables are reduced and the time of alignment is saved. Keywords: Computer-aided alignment, Misalignments, Zernike polynomials, Sensitivity matrix 1. INTRODUCTION The development of modern optical instruments makes increasing demands for optical lens design, fabrication, assembly and adjustment. Especially the development and application of the high precision lens as the representative of the microelectronic photolithography technology demands a very high image quality. This is almost above the designing, manufacturing, alignment and testing level in present. High image quality and complex, reflective optical systems, as those used in remote sensing applications, are, in general, very difficult to be manufactured with the required performance. This can be charged to the high sensitivity of such systems to the fabrication tolerances, mainly concerning the relative alignment of the optical components with respect to each other. When the system does not achieve the expected quality, the puzzle is to identify where the problems lies. This is even worsened when the number of optical elements becomes high Traditional alignment methods are not visible, not quantitative, and they always take much time [1]. To overcome these shortcomings, computer-aided alignment proposed a new alignment method that can qualitatively and quantitatively give the amount of misalignments, thereby guiding alignment process. Computer-aided alignment of the optical system use computer to compare and analyzed the real-time detection results and the theoretical results through the necessary mathematical model processing. It could predict the orientation and the value of the misalignment, and then to guide in the subsequent alignment process. After the coarse alignment of optical system, position and angle of each component will be small deviations comparing with the design values. These small deviations will greatly affect the system image quality for high accuracy requirements. Under the premise of image quality could be assured, the main content of computer-aided alignment is to determine these small deviations or the orientation and value of misalignments. If we can accurately determine the amount of misalignment, alignment of the system will be targeted, so that the actual optical structural parameters of the system may be close to the design value to improve system design quality greatly. With the help of computer-aided alignment, the alignment time of high-precision, high-image quality optical system can be shortened, and the system's image quality is largely improved. 7th International Symposium on Advanced Optical Manufacturing and Testing Technologies: Optical Test and Measurement Technology and Equipment, edited by Yudong Zhang, Wei Gao, Proc. of SPIE Vol. 9282, 9282Y 214 SPIE CCC code: 277-786X/14/$18 doi: 1.1117/12.268141 Proc. of SPIE Vol. 9282 9282Y-1
2. COMPUTER-AIDED ALIGNMENT METHOD 2.1 Mathematical model The goal of computer-aided alignment is to reduce the performance difference between the actual system and the ideal system, using adjustable variable to minimize the difference. We assume F j (j=1,2,,m)as aberration function and x i (i=1,2,,n)as the adjustable variable (elements tilts, decenterations, despaces). The relationship between the aberration and the variables may be described by F f (x,,x ) = (1) F f (x,,x ) The expression of f j (j=1,2, m)is difficult to get. In this case, f j can be expanded to be a power series of x j. We determine the power and the coefficient by the experiment or data sampling according to the precise required. If the high power is ignored, the aberration F j can be given by a linear equation F =F + (x x ) + + (x x ) (2) Where F j is the residual aberration of ideal system; (x,x,,x ) are the adjustable variable of ideal system;,, is the partial differential of f j to x i. Replacing,, by,,, the linear equation is given by A X= F (3) Where F F F x x x F = =, X = =, A= F F F x x x Where X = X X is the misalignment of each element. The misalignment of each element in an optical system will lead to the decrease of image quality. It generally includes the decentration, tilt and air thickness error. F = F F is the difference of the wavefront aberration between the actual optical system and the designed one, and it is expressed by coefficients of Zernike polynomials. A is the sensitivity matrix, which can be calculated from designed optical system by Zemax optical design software. In CAA process, the misalignment can be solved according to the change of aberration based on Eq. (3). This mathematical model of CAA is based on the linear assumption of the relationship between the aberration and the structure parameters of each element. It will have large errors in systems with large aberration. In the adjusting process of space remote sensor in orbit, since the aberration has been controlled in a small range through the adjustment on ground, we can regard this system as small aberration system. So the linear assumption is suitable. 2.2 Calculation of the misalignments Given the vector measurement we may, in principle, solve for the misalignment vector by X = A F (4) Where A -1 is the pseudo-inverse as defined by A =(A A) A (5) With A t the transpose of A. For Eq. 4 to be solved, it is necessary that m>n, i.e, the number of independent aberration must exceed the number of adjustable variables. The most annoying problem is near-singularity of the sensitivity matrix A, which can make the solutions get divergent. To avoid this, the solution is damped to keep the adjustable variables within redefined range. The introduction of damping can result in more iteration, consuming a great deal of time. The near-singularity of the matrix A has direct influence on the measurement tolerance: the more nearly singular, the more severe the tolerances on the measurement and vice versa. This influence can be given by A( X + X ) = F+ F (6) Where F is the measurement error, X is the variation resulted from F. By use of quality of norm, we can obtain Proc. of SPIE Vol. 9282 9282Y-2
cond(a) (7) Where cond(a) is condition number of the matrix A. It is given by cond(a) = A A (8) Magnitude of the condition number depends on degree of near-singularity of the matrix A.Eq.8 shows the effect of nearsingularity of the matrix A on measurement tolerances. To determine the misalignments quickly, exactly and constantly, the condition of A must be improved, which is the key to computer-aided alignment. 3. ALIGNMENT SIMULATION To validate this analysis and to obtain a more accurate characterization of the system we perform simulated alignments of the optical system, using the optical design software ZEMAX. 3.1The experimental optical system The high precision optical system presented in this paper is a four-element optical system configuration with diffractionlimited image quality, and all of the elements are spherical surfaces. The working wavelength of the system is 632.8nm. Figure 2 shows the layout of the optical system. The full aperture of the optical system is 1mm. The residual RMS wavefront error is.2λ. Figure 1: Layout of the high precision optical system 3.2 Selection of Compensators In the alignment process, there is no need to adjust every elements of the optical system. The influence on the image quality caused by the misalignment of one element can be compensated by another element. The characteristic is called aberration complementarities of optical system, which is the foundation of computer-aided alignment. In order to realize CAA, the first thing is to choose the right compensators. Compensators are parameters that are adjusted to compensate for the aberration caused by the misalignments. Williamson (1989) figured out that [2], in the precise alignment process, the effect of small moves on fifth-order aberration and higher aberration is very small when the lens are collimated. Although most of the design effort and complexity into minimize or avoiding fifth and higher order aberrations, the tolerancing and compensation of the lens are adequately described by third-order changes, based on the well-known tendency of high-order aberrations to change relatively slowly. Thus the third-order aberration was chose as alignment objects to control the image quality. In this optical system, the astigmatism with axis at /9,the astigmatism with axis at ±45,the coma along x axis, the coma along y axis and the third order spherical aberration are chose as the alignment object function. The measurement of the third-order aberration could be the wavefront map, as suggested in [3], or just stacking the wavefront Zernike polynomial coefficient, as employed in many studies [4][5] [6]. The Zernike polynomials are an infinite series of orthogonal terms defined over a unit circle, which is used to model the wavefront map of optical systems. The Zernike coefficients have the relationship with the Seidel aberration. Table.1 shows the relations. A serial of aberrations, such as spherical aberration, coma, astigmatism are obtained. Having established the number of compensator needed and the aberration to be compensated, all that remains is to decide which of the elements are to be used in the post-assembly alignment. The goal in the selection process is to find the most Proc. of SPIE Vol. 9282 9282Y-3
orthogonal set of compensators. Ideally, each compensator would affect only the one aberration that it is compensating. Its effect on the other aberrations should be as small or as different from the other compensator as possible. In addition, to avoid unnecessary mechanical complexity, an element can have only an axial or lateral adjustment, not both. The largest diameter elements should not be adjustable. Table.1 The relationship between Zernike polynomials and the Seidel aberration No. Zernike polynomials Meaning Z5 ρ cos2θ Astigmatism with axis at /9 Z6 ρ sin2θ Astigmatism with axis at ±45 Z7 (3ρ 2ρ )cosθ Coma along x axis Z8 (3ρ 2ρ)sinθ Coma along y axis Z9 6ρ 6ρ +1 Third order spherical aberration The selection is based on a sensitivity analysis of the three Zernike coefficients corresponding to Seidel aberration changes from the shifts of each element..1.5 -.5 -.1 -.15 1 2 3 4 Figure 2: Sensitivity to element lateral shift of 5 microns. Figure 3: Sensitivity to element tilt of 1..15.1.5 -.5 -.1 1 2 3 4 6 x 1-3 4 2-2 -4 1 2 3 4 Figure 4: Sensitivity to element axial shift of 5 microns Figure 2 shows the sensitivities to element lateral shift, and Figure 3 indicates the sensitivities to element tilt graphically. Upon inspection of the graphs, a number of possible compensator choices are apparent. Since both the lateral shift and the tilt shift of each element have very little astigmatism and spherical aberration, and the same effect on coma, the lateral of the elements can be chosen as the compensator for coma. According to these selection constraints, the element 1 and element 2 has the largest diameter which would increase the outside diameter of the mechanism, so the lateral adjustments of element 3 are chosen as the compensator to compensate the coma along x and y axis. A similar sensitivity analysis gives the effects on the aberration of axial shifts of elements. Again, these are presented graphically in Figure 4. From the figure, both element 2 and element 3 has significant spherical and little other aberrations. However, element 3 has been chosen as the compensator for coma, so the axial adjustment of element 2 is chosen as the compensator to compensate the spherical aberration. Based on this, the sensitivity matrix could be obtained, as shown in Table.2. Table 2 sensitivity matrix L3-Δx L3-Δy L2-Δz Coma along x axis 1.6 Coma along y axis 1.6 spherical aberration.31 Proc. of SPIE Vol. 9282 9282Y-4
3.3 Misalignment determination Six misalignment situations have been performed to verify the ability and accuracy of the method. First a random perturbation is added to the structural parameters, and then Zernike coefficient of the wavefront at the image plane is calculated with Zemax software. Thus the wavefront aberration ΔF is obtained. Then it is substituted into equation (5) to calculate the misalignments. The simulation results are shown in Table 3. The actual misalignment The compensator Table.3 The simulation results of CAA(Unites:Δx,Δy,Δz /mm,tiltx,tilty / ) 1 L2-Δz=.1 2 L2-Δz=.1 L3-Δx=.2 L3-Δy=-.3 3 L1-Δx=.2 L2-TiltX=.1 4 L1-Δx=-.3 L1-TiltX=.5 L2-TiltX=.3 5 L1-Δz=-.1 L1-TiltX=.2 L2-Δy=.3 L3-Δx=.5 L3-Δx -.2.374 -.561 -.5 -.9 L3-Δy.3.389.298 -.18 -.19 L2-Δz -.1 -.1 -.13.12 6 L1-TiltX=.3 L2-TiltY=.2 L3-Δy=.3 L3-Δz=.2 L4-Δx=.4 According to calculated values of the selected compensators in Table 3, the RMS aberration error of the pre-alignment and post alignment system is shown in Table 4. Table.4 RMS aberration error comparison of pre-alignment and post alignment system 1 2 3 4 5 6 RMS wavefront error of pre-alignment system.953.975.32.356 1.42 1.558 RMS wavefront error of post alignment system.2.2.5.5.3.6 5. CONCLUSIONS The use of computer-aided alignment is essential to some complex, high-performance systems. Combined method of interferometric test and least square optimization has been proved to be a reliable way of determining the misalignment. A four-element high precision optical system configuration with diffraction-limited image quality was used to verify the ability and the accuracy of the method. The lens is aligned only using three adjustable variables and the result of alignment meets the requirements. In actual alignment, more iteration may be needed due to the noise effect, such as environment, measurement errors, and so on. REFERENCES 1. P.R. Yoder, Jr., Opto-mechanical System Design, 3 rd edition, Talor& Francis Group, LLC,26 2. D. M.Williamson, Compensator Selection in the tolerancing of a microlithography lens, Proc. SPIE, 149,178,1989 3. R. David, S. Norbert, Z.L. John, Z. Yan, et.al. Optical State Estimation Using Wavefront Data. Proc. SPIE 5523(24) 4. Z.S. Gao, L. Chen, S.Z. Zhou, R.H. Zhu. Computer-aided alignment for a reference transmission sphere of an interferometer, Opt. Eng 39(7),69-74(24) 5. X.F. Yang, C.Y. Han, J.C. Yu. Method for computer-aided alignment of complex optical system, Proc. SPIE 615 6153X-1(26) 6. Y.F. Huang, L. Li, Y.H. Cao, Computer-aided Alignment for Space Telescope optical System, Proc. SPIE 6149 6149P-1(26) Proc. of SPIE Vol. 9282 9282Y-5