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It has recently been shown that the coefficients that specify the aspheric departure from a spherical surface in high NA lithographic lenses routinely require more significant digits than are available in even double precision computers when they are described as part of a power series in aperture-squared. The Q-type aspheric description has been introduced to solve this problem. An important by-product of this new surface description is that it allows the slope of a surface to be directly constrained during optimization. Results show that Q-type aspheric surfaces that are optimized with slope constraints are not only more testable, an original motivation, but, they can also lead to solutions that are less sensitive to assembly induced misalignments for lithographic quality lenses. Specifically, for a representative NA 0.75 lens, the sensitivity to tilt and decenter is reduced by more than 3X, resulting in significantly higher lens performance in-use.
©2011 Optical Society of America
1. Introduction
It has been discovered recently that describing the aspheric departure of a lens surface from a spherical shape with a power series in the aperture variable-squared, essentially an industry standard practice in optical design since a patent by Ernst Abbe in 1902 [1], fails if too many terms are included in the optimization. This failure is caused simply by a lack of significant digits as clearly illustrated by Forbes [2,3]. In the last decade, due primarily to a change in the input format of optical design software, optical designers have moved from routinely using no more than 4-terms (aperture to the 10th power), which does not reveal the computational problem, to using > 10 terms, which does exceed the computational capabilities of even double precision computers. In most cases, while these added terms reduce the numerical value of the merit function, which is a relatively sparse sample in terms of ray intercepts, they do not actually improve the performance of the lens. In fact, they arguably result in an assembled lens with lower performance. In 1986, Kross et al. introduced Zernike polynomials [4], a specific form of Jacobi polynomials [5], in symmetric aspheric descriptions to solve numerical problems and enhance optimization. The Q-type aspheric description of the departure of a surface from spherical [2], also based on a particular family of Jacobi polynomials, not only avoids the numerical problems encountered with a power series description, but, they also allow computing the RMS slope of a surface during optimization as the sum of the Q-type aspheric’ coefficients squared as mathematically detailed thereafter. Note that these bases, referred to as Q-type in this paper, are referred to as Qbfs in the earlier literature on this topic [2]. This metric, added to the merit function as a constraint during optimization, results in lenses that are less sensitive to residual assembly “errors” as we shall show.
The Q-type aspheric description of the sag of a surface can be expressed as Eq. (1)
where c is the curvature of the best-fit sphere defined as the sphere that touches the center and edges of the surface, ρ is the radial coordinate in the aperture, u is the normalized radial coordinate defined by u = ρ/ρmax, Qm(u2) is a polynomial for order m that in general has descending departure with the increase of m, and am are the coefficients to characterize departure [2]. The normalization u2(1-u2) is introduced to result in zero sag departure at the center and edge of the aperture to comply with the best sphere definition. The Qm are configured so that the coefficients am are similar to a spectrum values that typically decay in magnitude, but not necessarily always. The weighted RMS aspheric slope corresponds to the sum of the squares of am [3].Provided a surface description resulting from optimization, two surfaces with the same maximum sag values will differ in their slopes and it is the slopes that limit testability. The local slope of the aspheric departure of a surface is a critical parameter in metrology since it determines local fringe density during direct (non-null) interferometric testing. For a surface to be directly testable, the fringe density is conventionally taken to be less than the Nyquist frequency of the sensor [6]. Stephenson showed in a recent paper on designing doublet lenses with aspheric surfaces how Q-type aspheres lower the departure and slopes of an asphere from a best-fit sphere [7]. However, we show here that controlling the surface slope has a second significant effect; it reduces the sensitivity of a lens to decenter and tilt residuals that arise during lens assembly.
Lithographic systems define the state-of-the-art for lens systems. It is demonstrated that slope constraints implemented using the Q-type aspheric description are effective at guiding designs into more manufacturable solution spaces with better-predicted as-built image quality.
2. Optimization of a lithographic lens with Q-type aspheres with and without slope constraints
2.1 Starting point
The optical design process was initiated from embodiment seven of US patent No. 6,606,144 [8], which was filed by Omura. This is an NA 0.75 lens for use at 193 nm that has 3 aspheres. To create a neutral starting point, the complexity of this design was reduced to an all-spherical form, allowing the full leverage of the slope constrained Q-type aspheres to be demonstrated. The starting point has a monochromatic wavefront error of 3 nm, distortion of less than 1nm, astigmatism of less than 30 nm, and object and image telecentricity of less than 7 mrad. This study was conducted monochromatically to reduce the already substantial computational time without compromising the conclusions that result.
2.2 Method for placing the aspheric surfaces throughout the lens
The purpose of this optical design study was to compare solutions developed using Q-type aspheric departure surfaces optimized without slope constraints and Q-type surfaces optimized with aspheric slope constraints. To minimize any bias in the solution, the surfaces where aspheric departure was added to the optimization were introduced one at a time and a complete solution developed for each asphere added. The placement of the aspheres in both the slope unconstrained and the slope-constrained solutions was determined using an expert system that was recently added to an optical design environment based in the Q-type asphere developments [9,10].
As is consistent with most recently reported state-of-the-art lithographic lens forms, more than three aspheres were introduced to reach a point where further improvement from adding an asphere was minimal. The two solutions based in Q-type aspheric surfaces required six aspheres to achieve the specifications as shown in Table 1 . Both systems were optimized with the same merit function, constraints, and optimization algorithm, to the extent possible. A layout of the lens is shown in Fig. 1 (note both solutions are nearly indistinguishable at this level as they evolved from the same starting point and were leveraging the aspheric terms dominantly).
Table 1. Lens Design Specifications
2.3 Slope constraints
For a lithographic lens element, which can be ~200 mm in diameter, and using an interferometer with ~1000 x 1000 pixels with 5 microns each, about two fringes/mm of aspheric departure across a surface is a reasonable density for interferometric testing assuming a ~50X magnification between the detector and the piece under test. Most recently, advanced metrology equipment has implemented direct testing, without null lenses by implementing a multiple subaperture testing environment to avoid exceeding the Nyquist frequency. With the help of a subaperture implementation and signal processing, the maximum fringe density can potentially reach 10 to 20 fringes/mm on the piece with, for example, 5 to 10 subapertures across a part. Therefore, for this high-level optical design study, the constraint for the maximum aspheric slope was established to be less than 20 fringes/mm as an upper bound [11]. Alternatively, for shops using other technologies, such as ion bean polishing, techniques such as sub-Nyquist interferometry introduced by Greivenkamp can be used to extend the allowable fringe density [12]. The optimization metric for RMS slope is not directly in fringes/mm, so, in most cases it needs to be iterated as the optimization proceeds to converge on the final constraint [13].
3. Comparison of the Q-type slope unconstrained and slope constrained optical designs
An optimization based in a Q-type was found to take similar time, with and without the slope constraints. However in the case of the slope constraint solution, the impact on the time to assemble a specification compliant system is reduced, potentially substantially.
3.1 Nominal image quality
Both the design without slope constraints and the design with slope constraints were optimized to meet the parameters and performance metrics shown in Table 1. The nominal design RMS wavefront error for the designs ranges from 0.0010 to 0.0021 (0.193 to 0.389 nm) and 0.0008 to 0.0017 (0.154 to 0.328 nm) waves RMS at 193 nm respectively. The maximum distortion and field curvature departure have variances of less than 5% between the systems.
3.2 Aspheric departure and slope
The maximum aspheric slope determines the size of the subaperture that can be used to test an optical element using the direct measurement methods of the most recent systems. This in turn directly affects the time required to test a part and the accuracy of the test [11]. The aspheric sag departure directly affects the fabrication time. Comparisons of aspheric departures and slopes are shown in Table 2 and Table 3 , respectively. Q-type aspheres with slope constraints have both the smallest slope and for a majority of the surfaces the least departure when compared to the design without slope constraints applied during optimization.
Table 2. Comparison of Aspheric Departure (μm)
Table 3. Comparisons of Maximum RMS Slope (Fringes/mm)
3.3 Assembly sensitivity analysis
To validate the postulate that a lens designed with a Q-type slope constraint is less sensitive to assembly than a traditional design, a sensitivity analysis for decenter and tilt was performed for the two Q-type designs. To facilitate an acceptably realistic scenario, three decenter compensators were used, one in each of the lens bulges, at or near the largest element. These, along with their ranges, are listed in Table 4 . It is worth noting that the movement range of compensators is smaller for the Q-type with slope constraints solution, which already hints at the lower sensitivity.
Table 4. Compensator Range of Movement
Figure 2 presents the primary result of this work, which validates the postulate that an optical design based on a solution that incorporates reduced aspheric surface slope implemented using a Q-type surface departure description will be less sensitive to assembly errors. Each element of the lens was decentered by 2 microns and tilted by 0.01 mrad, and the change in RMS WFE at a point in the field along the y-direction was compiled, without lack of generality given that the system is rotational symmetric. The field point shown in the figure was the most sensitive point in the field of six sample points along the y-direction. The results are quite dramatic. The slope-constrained design is not only considerably more testable, but, it is also as much as more than three times less sensitive to assembly residuals. In an industry that works at the boundary of assembly technology, this level of relief will reduce the time to successfully assemble a lens, a significant cost component.
Fig. 2 Changes of RMS wavefront for each tolerance terms. (a-b) Barrel Beta Tilt (BTX 0.01 mrad). (c-d) Barrel Alpha Tilt (BTY 0. 01 mrad). (e-f) Element X-decenter (DSX 2 μm). (g-h) Element Y-decenter (DSY 2 μm). Two columns show the worst field for Q-type without slope constraints and Q-type with slope constraints, respectively.
4. Conclusion
In this paper, the Q-type description for aspheric departure has been used to enable the optimization of aspheric surfaces in a representative NA 0.75 lithographic lens design at 193 nm with and without slope constraints as a means of creating a design with significantly lower sensitivity to assembly errors. The initial motivation for Q-type surface descriptions was to result in more testable surfaces. This paper illustrates that a second important outcome of the Q-type implementation is it allows the optical designer to directly control the aspheric slope of a surface, a key parameter for both testability and as shown here for assembly sensitivity.
Based on over 1,500 hours of optimization, this paper illustrates a significant (> 3X) decrease in assembly sensitivity for a lithographic quality lenses modeled in an idealized, but realistic scenario when compared to a design developed without slope constraints. Both designs were developed using new algorithms for selecting the next surface to place an asphere on during the design process, to reduce the effect of any designer bias to this comparison. Additionally, the Q-type aspheres provide a methodology to determine the effective number of aspheric terms on each asphere, a critical step in the process to achieve reduced alignment sensitivity.
Acknowledgments
We thank Greg Forbes for stimulating discussion about this work. This work was supported by the China Scholarship Council and the National Science Foundation GOALI grant ECCS-1002179. We thank Synopsys for the educational license of CODE V.
References and links
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