Open Access
Add to BibSonomyAbstract
This research proposes a feedback method to adjust the dual-beam exposure system with spherical collimation lenses to achieve gratings with low spacing error. Through theoretical analysis and numerical calculation, it is proved that the interference aberration can be analyzed with the Zernike polynomials and the adjustment errors can be estimated according to the linear relationship between the errors and the polynomial coefficients. Moreover moving the substrate along its normal is proposed to decrease the spacing error but keep the grating’s period unchanged. In the experiments, the wavefront measurement results of the ± 1st orders are used to deduce the spacing error. Based on the feedback adjustment method, the grating with a spacing error of 0.03 λ within 70 mm × 70 mm is fabricated with the collimation lenses of 0.6 λ spherical aberration.
© 2015 Optical Society of America
1. Introduction
Subwavelength period gratings are usually fabricated by laser interference lithography in which an interference field created by two coherent beams is used to record the grating lines. The holographic gratings with diffraction wavefront of high quality are required in many application domains such as displacement metrology [1] and chirped pulse amplification (CPA) [2]. Spacing error of the grating lines [3] is the main source of the diffraction wavefront aberration and it can be measured and evaluated with the diffraction wavefront aberrations [4]. With the compact and stable features, the grating encoders have been widely employed to replace the laser interferometers to precisely measure the displacements along multiple directions [5–7]. Since 2001, it has been used in optical lithography for semi-conductor manufacture [8, 9]. As one of the most important error sources, the spacing errors can be compensated by calibration [10, 11], however an economical method of fabricating gratings with large aperture and low spacing error is still significant, especially for two-dimensional metrology. Moreover for CPA applications, the gratings with larger apertures and lower spacing error can further improve the power and irradiance of the short pulse [2]. The gratings with 350 mm aperture and 0.15 λ wavefront aberration were used for CPA in LLNL since 2000 [12]. Recently, HORIBA Jobin Yvon can offer commercial CPA gratings with hundreds of millimeters aperture and λ/4 wavefront aberration. Moreover the grating of meter-level aperture and λ/3 aberration which is manufactured with scanning exposure has been contributing to the petawatt laser pulse [13].
The grating’s spacing error results from the interference aberration, which is caused by deviation of the exposure beams from ideal plane waves. Therefore, improving exposure system is a direct and efficient method to enhance the grating quality. In early researches, spherical waves were used to interfere and a hyperbolic interference field was recorded in the gratings [14]. M. E. Walsh et al. [15] proposed a method of bending the substrate to compensate the spacing errors caused by the deviation of interference angle and as a result the spacing error can be reduced by one order of magnitude. Nevertheless this method cannot be widely applied due to the restriction of substrate flexibility. K. Hibino and Z. S. Hegedus [16] analyzed the interference aberration of two spherical waves and found that after simplification the formula appeared similar to the low orders of geometrical wavefront aberration.
Collimation lenses can be used to improve the quality of the exposure beam [17], but more complicated noise of spacing error will be introduced by lens aberrations, fabrication errors and misalignment of the collimation system [18]. W. Zhang et al. [19] proposed a focus adjustment method based on the interference fringes generated through rotating the grating by 180 degrees around its normal and putting back to the exposure position. It needs to distinguish one-tenth of bend of the fringes by human eyes to achieve 0.05 λ aberration with this method. Without considering the fabrication errors of the substrates, S. Wang et al. [20] numerically calculated the interference aberration of spherical collimation lens exposure system with ray-tracing method and concluded that exposure interference aberration could be analyzed with the Zernike polynomials and the position deviation of the point source was linearly related with the Zernike coefficients. This analysis provides a potential method to detect and control the adjustment error of the exposure system but the measurement method of the interference aberration was not discussed. If the diffraction wavefront of the fabricated grating is used for measurement, the surface error of the substrate will also be involved and it will disturb the aberration correction. Moreover, adjusting the point source along the transverse direction will change the grating period and this is an undesired side-effect. Using aspheric collimation lenses is an effective method to decrease the interference aberration below 0.05 λ [21, 22], but the fabrication of aspherical lens is complex and expansive. This paper focuses on the method of adjusting the dual-beam system with spherical lenses and fabricating gratings with low spacing error in a cheap and rapid way.
In this paper, we analyze the spacing error produced in a dual-beam system with spherical collimation lenses and pay attention to the error caused by system misalignment. We analytically prove that if the defocus and transverse position error of the point sources is compensated, the interference aberration can also be mutually compensated well. Through the ray-tracing numerical simulation, the linear relationship between Zernike coefficients and adjustment errors is confirmed. A closed-loop-feedback adjusting method is proposed through estimating relative adjustment errors after removing the substrate surface error and fitting with Zernike polynomials. Although the collimation lenses had a spherical aberration of 0.6 λ, we fabricated gratings with the spacing error of 0.03 λ in the area of 70 × 70 mm2.
2. Theoretical analysis
The grating's spacing error, which is defined as the position deviation of grating lines from the ideal straight grating lines, is directly caused by the interference field of the exposure system. Ideally, if two perfect plane waves are used to record gratings, the grating lines will be straight with a constant grating period. However in fact, defocus and other geometrical aberrations of the collimation system will always exist and make the real interference field distorted from the ideal one. In this paper the phase difference between the real and the ideal interference fields is defined as the interference aberration, which is equivalent to the grating’s spacing error. Moreover, the diffraction wavefront aberration of the grating is mainly resulted by the spacing error and the substrate surface error, and therefore the diffraction wavefront aberration can be used to evaluate the spacing error. In this section, the interference aberration in a dual-beam system is analyzed with the geometrical aberration theory, and the condition of achieving low aberration will be discussed.
Figure 1(a) shows the schematic of a dual-beam system. The laser beam is divided by a polarization beam splitter PBS, and reflected by the mirrors M0, M1 and M2 to two expansion-collimation systems (ECS1 and ECS2, each includes a microscope objective lens, a pin-hole for spatial filtering and a collimation lens). The polarizations and intensities of the laser beams can be adjusted by the half-wave plates PW1 and PW2. The two ECSs are used to generate the exposure beams incident on the substrate G and the interference fringes are recorded in the photoresist. In order to avoid the drift of interference field caused by the environment disturbance, a fringe-lock system [21] is employed to stabilize the interference field. Below G a small reference grating Gr with a period the same or close to that of the interference field is fixed on the optical table. Gr intercepts the lower portions of exposure beam and will generate interference fringes during the exposure. A CCD takes the fringe pattern and transmits the image to a monitoring computer. If the reference fringe drifts, the computer will give a feedback signal to a piezoelectric transducer PZT mounted on the back of the mirror M2. The interference field can be stabilized through adjusting M2 by a very small movement. In the ideal situation the optical axes of the ECSs are symmetrical with respect to the normal of the substrate and the angle between the axes (2θ) should satisfy
where λ0 is the exposure wavelength and d is the grating period. An O0X0Y0Z0 coordinate system is set up on the substrate as shown in Fig. 1(a), with the O0X0 parallel to the grating vector, the O0Y0 parallel to the grating lines and the O0Z0 perpendicular to the substrate. Moreover, each ECS has an independent Cartesian coordinate system: OiXiYiZi (i = 1 or 2, the same below), where the OiZi is along the optical axis of ECSi, the OiYi has the same direction as the O0Y0 and the origins are located at the node points of the collimation lens. In the ideal situation the optical axes of the ECSs both pass through the origin O0 and the distance from the nodes of the collimation lenses (Oi) to point O0 are assumed to be li. The pin-hole (PHi) is assumed to be an ideal point source for the theoretical analysis and the numerical calculation. The displacement of PHi deviating from the focal point of the collimation lens is defined as the position adjustment error of the point source, which can be classified as three kinds of position errors along different ECS coordinate axes: the transverse position errors (along the OiXi and OiYi axes) and the defocus error (along the OiZi axis).
Fig. 1 Schematic of a dual-beam exposure system. (a) Two expanding-collimation systems constitute the whole system. (b) Here the impact of a deviation of PH2 along O2X2 is shown: the optical axes of the two ECSs are e1 and e2, the incident angles are θ1 and θ1, the incident point of the e2 axis on the substrate is moved by δx2 from the ideal position. Moving e2 to e2' (or e1 to e1') will correct the Xi direction error, but the grating period will be changed at the same time. If the substrate is moved from G to G', the Condition of Aberration Symmetry can also be satisfied.
As the focal length of the collimation lens is in the order of one meter, meanwhile the defocus and transverse position errors of the point source are usually much smaller than the focal length, the wavefront aberration in ECSi can be assumed to consist of defocus and spherical aberration, without coma and astigmatism. The wavefront after collimation can be expressed as
where Ai is the coefficient of the defocus aberration and Bi is the coefficient of the primary spherical aberration. The phase of the interference field is the difference of the phases of the two beams incident on the substrate,The defocus position error will change the coefficients Ai and Bi. The effect of the transverse position errors can be approximated as that the whole ECSi is rotated around the Oi point together with the optical axis, so the wavefront can also be expressed as Eq. (2), but the ECSi coordinate is also rotated around the Oi point. If a deviation of PH2 along O2X2 is taken as an example, the incident angle of two ECSs are denoted as θ1 an θ2, the intersection point of the O2Z2 axis and the O0X0Y0 plane becomes (δx2, 0, 0), as shown in Fig. 2(b).Fig. 2 Simulated interference aberration under ideal condition. (a) Two point sources are both located at the foci of the collimating lenses. (b) When there are same position errors (1, 1, 0.1) for the two point sources in the ECSs, which means the incident angles are different but the optical axes keep across the same point on the substrate surface approximately, the interference aberration can also be compensated well.
For an arbitrary point on the substrate with the coordinate (x0, y0, 0) in the O0X0Y0Z0 system, the transformations between the three coordinate systems can be derived as
Combined with Eqs. (2), (3) and (4), the interference phase can be derived aswhere ϕG is the linear phase of the recorded grating, ϕF results from the relative defocus error, ϕI is produced by difference of the incident angles, ϕC is caused by the transvers position error of PH2 along the O2X2 and is referred to as coma in this paper because of the similar form, ϕR is also caused by the transverse position error and includes constant terms, linear terms, some defocus parts and other kind of distortion. The detail of these aberrations can be expressed asIf the two ECSs have the same defocus, which results A1 = A2 and B1 = B2, the defocus aberration ϕF = 0. If the system is adjusted to make the incident points of the O1Z1 axis and the O2Z2 axis on the O0X0Y0 plane move to the same point and the O0 can also be set at the incident point, δx2 will be 0 and ϕC, ϕR will be decreased to zero at the same time. One method to achieve this is to set the two ECSs to the ideal symmetrical situation, which will result in that θ1 = θ2 and all aberration can be compensated well, but this condition is very hard to be precisely realized in experiments. Another solution is to keep one ECS unchanged but adjust the other one. When the incident points are moved together, the aberration ϕC and ϕR can be directly eliminated. Then
which is in the order of 10−3 rad because in a typical exposure system the focal length of the collimation lens is in the order of one meter and the transverse position errors of the point source are usually no more than a few millimeters. Therefore the ϕI term will contribute little to the whole aberration and it can be neglected. When considering the transverse position error along OiYi axis, the similar analysis can be applied. In conclusion, the condition of achieving low interference aberration can be summarized as the equal defocus errors and the optical axes (or principal rays) of the ECSs passing through the same point on the substrate. This is called the Condition of Aberration Symmetry (CAS) in this paper.After the exposure system is constructed, the point source needs to be adjusted along the optical axis and the transverse directions to meet the CAS. As shown in Fig. 1(b), when moving the point source along the OiXi direction to satisfy the CAS, the interference angle will be changed at the same time. Although the variation of interference angle causes little aberration, it results in obvious grating period error as an undesired impact for some situations. Moving the substrate along its normal direction may be an alternative to compensate the OiXi direction position errors as shown: if the substrate is moved from G to G', the CAS can also be satisfied. Moreover, the interference aberrations have the form of geometrical aberration such as the defocus and coma in Eq. (6), so it will be suitable to analyze these aberrations with the Zernike polynomials.
The purpose of this section is to give the readers some physical intuition about roles played by different variables involved in the problem. The formulas derived in this section will not be used in the next section. Instead, for convenience we use the ray-tracing method for numerical simulation. The simulation results corroborate the analytical results.
3. Numerical simulation and adjustment design
3.1 Numerical experiment
The ray-tracing method was used to analyze the interference aberration of the dual-beam exposure system. The parameters were chosen based on our actual experimental system. A pair of biconvex lenses with two spherical surfaces of 4656 mm and 658 mm radii, 18.3 mm thickness, 1.529 refractive index for the exposure wavelength of 413.1 nm and 1077 mm focus length was used as the collimators. The distances from the nodes of the collimation lenses to the midpoint of the substrate were set as 860 mm. The expanding and spatial filtering components were simplified as point sources nearby the focus points of the collimation lenses. The incident angle of each optical axis of the ECS was 11.92° and the grating period was 1 μm. Substrates of 100 × 100 mm2 were exposed in experiment and only the central size of 70 × 70 mm2 was calculated in simulation because of the aperture restriction of our Fizeau interferometer. The optical lengths from the two point sources to all sample points on the substrate were calculated based on the basic geometrical optic principles, and the difference of the light path lengths of two ECSs represents the interference phase. After subtracting the linear part, the residual interference phase will be equivalent to the spacing error of the fabricated gratings. Moreover, as the diffraction wavefront is used to deduce the spacing error and will be halved automatically when measured with the Fizeau interferometer, all the following numerical results will also be halved to correspond to the measurement result.
Before calculation, we need to stipulate that the position error of point source PHi is defined as the spatial coordinate deviating from the focus of ECSi point and the relative position error is the difference between the two position errors of two ECSs. For the ideal situation with no position errors, the interference aberration on the substrate will be very small, as proved by numerical simulation in Fig. 2(a). Usually in practice, the point source can never be adjusted to the accurate focus of lens. As we have proved in theory, when the point source of one side deviates from the ideal position, as long as the other point source is adjusted to have the same defocus error and make the principal rays of both ECSs pass through the same point on the substrate, the CAS can be satisfied. As shown in Fig. 2(b), when the point sources have the same position errors, the aberration can be compensated well. The PV value of the aberration does not grow too much, although its distribution looks quite different from the ideal situation. Consequently the relative position error is the critical adjustment object and only the point source of one side needs to be adjusted.
After constructing the exposure system, the transverse and defocus errors of the point source are all around 1 mm, but the defocus error will cause much larger spacing error. In order to observe the aberration caused by the transverse position error clearly, 1 mm transverse error and 0.1 mm defocus error are taken as examples and the calculation results are shown in Fig. 3. The three errors are calculated independently in Fig. 3(a), 3(b), and 3(c). These aberrations can be fitted with the orthonormalized Zernike polynomials [23–25] and the results are given in Table 1 where only three involved terms are shown but the lowest nine terms are employed in the fit. In practice we need to deal with the situation with three kinds of position errors, as shown in Fig. 3(d). The calculation has proved that through fitting with the Zernike polynomials, the different position errors can be separated and determined by different polynomial terms, defocus or comas. Moreover there are certain linear relationships between the position errors and the Zernike coefficients, as shown in Fig. 4. All three position errors were introduced simultaneously as
There is no disturbance between different terms observed and the linear relationship is evident.Fig. 3 Interference aberration with different adjustment errors of the point source. (a) Relative position error of (0, 0, 0.1), this aberration corresponds to the defocus term of the Zernike polynomial. (b) and (c) Relative position error of (1, 0, 0) and (0, 1, 0). They correspond to coma terms of Zernike polynomials. (d) Relative position error of (1, 1, 0.1). Usually the existence of all kinds of position errors will introduce this distribution of aberration.
Table 1. The Fitted Zernike Polynomial Coefficients of Fig. 3
Fig. 4 Linear relationship between the adjustment errors and the fitted Zernike coefficients. The data markers are given by theoretical calculation and the lines are linearly fitted with the data samples. X, Y and Z errors are the relative position errors of the point source and s is the position error of the substrate in its normal direction.
Generally after constructing and coarsely adjusting the exposure system, we need to adjust the position of point sources along the X1 or X2 direction to minimize the aberration; however, this will change the interference angle and the grating period at the same time. If the grating period is also an important parameter to control, there will be a dilemma to abandon either period or interference aberration. Moving the substrate along its normal is a valid method to solve this problem, because there is always a position of the substrate to satisfy the CAS but the incident angle will not be changed. This method is proved by the numerical simulation as shown in Fig. 5. With 0.5 mm relative error of the point source in the X2 direction, when the substrate offset (Δs) is −0.86mm, the aberration will be compensated. The linear relationship between the normal position of substrate (s) and the 6th Zernike coefficient is shown in Fig. 4(b): the Yi and Zi position errors of the point source are also introduced but the Xi position is fixed.
Fig. 5 Adjust the substrate to compensation the Xi direction position error. (a) 0.5mm X2 direction error of point source is added. (b) The substrate is moved along its normal by −0.86mm to compensate the X2 direction error.
3.2 Procedure of the exposure system adjustment
The adjustment method is based on the idea of estimating the position errors according to the spacing error which will be obtained from the diffraction wavefront. However, the diffraction wavefront from a grating sample is affected by the spacing error, the substrate surface error and the grating profile nonuniformity. It is critical to extract the spacing error from the diffraction aberration. According to the phase change theorem of gratings, the phase of the diffraction wave can be expressed as
where ϕ0 (x0, y0), ϕ+1 (x0, y0), ϕ−1 (x0, y0) are the diffraction phases which will be measured by the Fizeau interferometer, ΔZ0(x0, y0) is the local surface error of the substrate, ΔX0(x0, y0) is the spacing error of the grating, d is the grating period, κ and κ0 are the incident angles of measurement wave, φ0 (x0, y0) and φ1 (x0, y0) are the additional phase errors introduced by the grating profile. The duty cycle of the grating is not uniform within the whole grating area because of the inhomogeneity of the exposure beam intensity. This nonuniformity will also introduce additional phase errors, which are the same for the + 1st and −1st orders because of the symmetry of the grating profile. Therefore the spacing error can be figured out from the difference between ϕ+1 (x0, y0) and ϕ−1 (x0, y0) as Φ+1,−1 in Eq. (9). In experiment, the grating’s spacing error can be obtained through processing the measurement results of the diffraction aberration.In conclusion, on basis of the theoretical analysis and numerical simulation, we propose a feedback method to finely adjust the dual-beam exposure system as follows:
- (1) After constructing the dual-beam system, fabricate a grating sample;
- (2) Measure the diffraction wavefront aberration with the Fizeau interferometer and calculate the spacing error (interference aberration) from the measurement data;
- (3) Fit the spacing error with Zernike polynomials. According to the coefficients of defocus and coma, estimate the position errors of the point source and the substrate. The error direction can also be determined by the coefficient sign;
- (4) Adjust the system according to the estimated errors and then fabricate a new grating;
- (5) Repeat step (2) to (4), revise the adjustment amount if necessary, until the grating’s spacing error matches the requirement.
The accuracy of error estimation from a single grating sample is influenced by the measurement error of the diffraction wavefront and the model error of the numerical calculation. The adjustment target could not be achieved through feedback adjustment once, so the adjustment procedure can be repeated to reach the best result.
4. Experiment results
4.1 Results of the adjustment
Here we demonstrate a series of aberration minimization experiments and analyze the results. After applying the steps (1) and (2) of the adjustment procedure, the grating’s spacing error is shown in Fig. 6(a), the fitted Zernike coefficients and the estimated relative position errors are given in Table 2, row (a). Here, s means the substrate’s displacement along its normal, (x, y, z) is the position of the point source on the right side, which is read from the 3-axis translation stage. As the other point source is fixed during adjustment, this position coordinate can be used to represent the relative position of the point sources, but the positive direction may be different from the theoretical coordinate because of the orientation of the translation stages. Δy, Δz and Δs are the theoretical estimated position error and there signs can be used to determine the adjustment direction. After the first adjustment, the spacing error became Fig. 6(b), and the estimated adjustment errors are in Table 2, row (b). With a complementary adjustment, the spacing error of a new grating was decreased to about 0.03 λ and satisfied the requirement as shown in Fig. 6(c). During the experiment, in order to achieve the spacing error under λ/20, the required precision of adjustment was determined to be 0.01 mm for defocus and 0.1 mm for coma. When the estimated position errors are below these limits, the spacing error will exceed the accuracy of the interferometer and no more adjustment is needed. For the samples of Figs. 6(a) and 6(b), the estimated position errors along the Y direction were below this criterion, so this direction was not adjusted. However in the final result, the estimated Y direction error increased again, but the adjustment was not continued because the requirement had been achieved. This phenomenon might be caused by the wavefront measurement error of the Fizeau interferometer. In this experiment, the substrate was moved along its normal direction to compensate the X0 direction coma and keep the grating’s period unchanged. If the substrate is fixed and the position of the point source can be moved along three directions, the spacing error can also be minimized as shown in Fig. 6(d).
Fig. 6 Example of spacing error minimization process. (a) Spacing error before fine adjustments. (b) Spacing error after adjusting the focus of point source and normal position of the substrate according to theoretical prediction. (c) After the second feedback adjustment, the spacing error is decreased to about 0.03λ. (d) Only adjust the position of the point source to minimize the spacing error.
Table 2. The Fitted Zernike Polynomial Coefficients of Fig. 6
The system adjustment method is designed based on the linear relationship between the position errors and the Zernike coefficients, so an experiment to verify this relationship was conducted. A series of different position errors, including the Z and X direction positions of the point source and the normal position of the substrate, were sampled and the Zernike coefficients of the fabricated grating were measured. The linear relationships in the experiment are shown in Fig. 7. The periods of some grating samples on the middle point of the substrates were measured through measuring the Littrow diffraction angles [26, 27]. A rotation stage with the repeatability of 10 arcsec was used to measure the diffraction angle. The repeatability of grating period was about 0.08nm. The position of the point source (x2), the position of the substrate (s) and the measured grating period (d) are shown in Table 3. It can be proved that the X direction position of the point source will influence the grating period obviously but moving the substrate is free from this problem.
Fig. 7 Linear relationship between the Zernike coefficients and the adjustment parameters in experiment. (a) The 3rd Zernike term and the Z direction position of the point source. (b) The 6th Zernike term and Xi direction position of the point source (the circle points and the red line), the normal position of the substrate (the cross points and the black line).
Table 3. Grating Samples’ Periods
4.2 Residual analysis
As calculated in the theory and simulation parts, the interference aberration of a dual-beam exposure system could be compensated near to zero, however there still existed about 0.03 λ spacing error in the experiment. Besides the remaining position error of the point sources, the environment disturbance is also a potential error source. Nevertheless a fringe-lock system is used in the exposure system, so the integral interference field can be regarded stable. Moreover the interference field disturbance in the exposure system can be removed from the main error sources of the residual spacing error, because the whole system is enclosed by a cover to reduce the air disturbance, the exposure time is not long (about 2 minutes) and the adjustment amount of the PZT is small (hundreds of nanometers). The fabrication distinction of the two spherical collimation lenses is another possible error sources and it cannot be compensated by each other. The wavefront aberrations of these two lenses were measured with the interferometer and the results are shown in Figs. 8(a) and 8(b). The aberration of each lens is close to the other one but much bigger than the best result of the exposed gratings. The difference between the aberrations of two lenses is shown in Fig. 8(c) and it is much smaller than the original aberration. This residual can be regarded as the interference aberration of the collimated waves from the two spherical lenses and will result in the spacing error. The residual’s distribution looks similar to the best experimental grating result. Moreover, as the grating’s wavefront aberration will be halved when measured by the interferometer, the residual in Fig. 8(c) also needs to be halved to compare with the experimental result of the grating’s spacing error. After this, the PV value of Fig. 8(c) becomes about 0.026 λ and close to the grating result. Consequently it is confirmed that the fabrication distinction of the collimation lenses is the main sources of the residual spacing error of the grating.
Fig. 8 Measurement results of the spherical collimating lenses. (a) Aberration of L1. (b) Aberration of L2. (c) The difference between L1 and L2.
5. Discussion
As we have discussed in the simulation part, the position errors in three directions correspond to three kinds of wavefront aberration: the coma along two axes and defocus. Through adjusting the exposure system, these aberrations can be almost removed and the residual turns out to consist of mainly spherical aberration as shown in Figs. 6(c) and 6(d). Since the spherical aberration results from lens fabrication and cannot be eliminated by optical adjustment, the dual-beam exposure system can be regarded as at the best condition after this adjustment. The quality of the best diffraction wavefront is determined by the fabrication distinction of the collimation lenses.
Moreover, the measurement accuracy and the measurement area of diffraction wavefront is another important factor affecting the final wavefront quality. The accuracy of the Fizeau interferometer used in this paper is about 0.05 λ, so a target much smaller than this will be unpractical. The aperture of the interferometer is about 70 × 70mm2, therefore the monitored area of wavefront is also limited. If the measurement aperture and accuracy can be improved, a better result would be achievable with this method.
In the adjustment strategy, the linear relationship between the adjustment errors and Zernike coefficients is foundational of the validity. The linear ratios between these two parameters in theoretical simulation and experiments are with some difference, especially for the displacement of the substrate, as shown in Figs. 4 and 7. This deviation perhaps results from the model error of the simulation. Despite the existence of simulation error, the relationship of linearity has been proved effective to direct the adjustment by employing feedback method in the experiment, and through twice feedback adjustment the requirement can be achieved.
6. Conclusion
Based on the Zernike aberration analysis, we have proposed a feedback adjustment method for a dual-beam exposure system that uses spherical collimation lenses. The spacing error is deduced from the measurement results of the diffraction aberration. After fitted with the Zernike polynomials, the linear relationship between the Zernike coefficients and the alignment error of the exposure system is used to guide the system adjustment. In the experiment, this method was proved useful and effective; through two steps of feedback adjustment, the spacing error of the exposed grating was decreased quickly and the final result of 0.03 λ was achieved with the collimation lens of 0.6 λ spherical aberration. Based on this result, we conclude that the spherical lenses have the capability to replace aspherical lens in holographic recording.
Acknowledgments
The work was supported by the National Natural Science Foundation of China under Project No. 913231002. The measurement of the collimation lenses were supported by the China Academy of Space Technology.
References and links
1. R. K. Heilmann, C. G. Chen, P. T. Konkola, and M. L. Schattenburg, “Dimensional metrology for nanometre-scale science and engineering: towards sub-nanometre accurate encoders,” Nanotechnology 15(10), S504–S511 (2004). [CrossRef]
2. J. D. Zuegel, S. Borneis, C. Barty, B. Legarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. Leblanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I. Jovanovic, J. Dawson, and N. Blanchot, “Laser challenges for fast ignition,” Fusion Sci. Technol. 49, 453–482 (2006).
3. E. G. Loewen, “Diffraction gratings,” Opt. Eng. 15(5), 446–450 (1976). [CrossRef]
4. Z. Jaroszewicz, “Interferometric testing of the spacing error of a plane diffraction grating,” Opt. Commun. 60(6), 345–349 (1986). [CrossRef]
5. H. Schwenke, U. Neuschaefer-Rube, T. Pfeifer, and H. Kunzmann, “Optical methods for dimensional metrology in production engineering,” CIRP Annals Manufacturing Technology. 51(2), 685–699 (2002). [CrossRef]
6. C. C. Wu, Y. Z. Chen, and C. H. Liao, “Common-path laser planar encoder,” Opt. Express 21(16), 18872–18883 (2013). [CrossRef] [PubMed]
7. X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013). [CrossRef]
8. T. Castenmiller, F. van de Mast, T. de Kort, C. van de Vin, M. de Wit, R. Stegen, and S. van Cleef, “Towards ultimate optical lithography with NXT: 1950i dual stage immersion platform,” Proc. SPIE 7640, 76401N (2010). [CrossRef]
9. R. H. M. Schmidt, “Ultra-precision engineering in lithographic exposure equipment for the semiconductor industry,” Philos. Trans. A Math Phys. Eng. Sci. 370(1973), 3950–3972 (2012). [CrossRef] [PubMed]
10. F. Meli, “WGDM-7: Preliminary comparison on Nanometrology: Nano4: 1D gratings,” Final report, OFMET, Wabern, 30 (2000).
11. M. Hercher, “High Speed, high accuracy stage for advanced lithography,” DARPA Phase II final report, Optra Inc. (2001).
12. J. A. Britten, S. J. Bryan, L. J. Summers, H. T. Nguyen, and B. W. Shore, “Large aperture, high-efficiency multilayer dielectric reflection gratings,” in Conference on Lasers and Electro-Optics, OSA, paper CPDB7 (2002). [CrossRef]
13. T. Jitsuno, S. Motokoshi, T. Okamoto, T. Mikami, D. Smith, M. L. Schattenburg, H. Kitamura, H. Matsuo, T. Kawasaki, K. Kondo, H. Shiraga, Y. Nakata, H. Habara, K. Tsubakimoto, R. Kodama, K. A. Tanaka, N. Miyanaga, and K. Mima, “Development of 91 cm size gratings and mirrors for LEFX laser system,” Journal of Physics: Conference Series. IOP Publishing 112(3), 032002 (2008).
14. J. Ferrera, M. L. Schattenburg, and H. I. Smith, “Analysis of distortion in interferometric lithography,” J. Vac. Sci. Technol. B 14(6), 4009–4013 (1996). [CrossRef]
15. M. E. Walsh and H. I. Smith, “Method for reducing hyperbolic phase in interference lithography,” J. Vac. Sci. Technol. B 19(6), 2347–2352 (2001). [CrossRef]
16. K. Hibino and Z. S. Hegedus, “Hyperbolic holographic gratings: analysis and interferometric tests,” Appl. Opt. 33(13), 2553–2559 (1994). [CrossRef] [PubMed]
17. J. J. Armstrong and T. J. Kessler, “Large-aperture, high-efficiency holographic gratings for high-power laser systems,” Proc. SPIE 1870, 47–52 (1993). [CrossRef]
18. M. E. Walsh, “On the Design of Lithographic Interferometers and TheirAapplication,” Ph.D. dissertation, (Massachusetts Institute of Technology, Boston, MA, 2004).
19. W. Zhang, J. Wu, J. Zhu, and C. Li, “New method for the fabrication of pulse compression grating,” Proc. SPIE 6149, 614921 (2006). [CrossRef]
20. S. Wang, J. Wu, C. Li, and G. Chen, “Wavefront aberration analysis of large-aperture grating fabricated by plano-convex lens,” Laser Journal 28, 48–50 (2007) (in Chinese).
21. L. Shi, L. Zeng, and L. Li, “Fabrication of optical mosaic gratings with phase and attitude adjustments employing latent fringes and a red-wavelength dual-beam interferometer,” Opt. Express 17(24), 21530–21543 (2009). [CrossRef] [PubMed]
22. L. Shi and L. Zeng, “Fabrication of optical mosaic gratings: a self-referencing alignment method,” Opt. Express 19(10), 8985–8993 (2011). [CrossRef] [PubMed]
23. J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19(9), 1510–1518 (1980). [CrossRef] [PubMed]
24. D. Malacara, J. M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990). [CrossRef]
25. D. Malacara, Optical Shop Testing (John Wiley & Sons, 2007), Chap. 13.
26. V. Korpelainen, A. Iho, J. Seppä, and A. Lassila, “High accuracy laser diffractometer: angle-scale traceability by the error separation method with a grating,” Meas. Sci. Technol. 20(8), 084020 (2009). [CrossRef]
27. C. J. Chen, S. P. Pan, L. C. Chang, and G. S. Peng, “Pitch calibration by reflective laser diffraction,” Proc. SPIE 5190, 156–164 (2003). [CrossRef]
