Polarization illumination is widely used in high numerical aperture (NA) projection lenses for better imaging resolution[
Chinese Optics Letters, Volume. 18, Issue 6, 062201(2020)
Optimization based on sensitivity for material birefringence in projection lens
Polarization aberration caused by material birefringence can be partially compensated by lens clocking. In this Letter, we propose a fast and efficient clocking optimization method. First, the material birefringence distribution is fitted by the orientation Zernike polynomials. On this basis, the birefringence sensitivity matrix of each lens element can be calculated. Then we derive the rotation matrix of the orientation Zernike polynomials and establish a mathematical model for clocking optimization. Finally, an optimization example is given to illustrate the efficiency of the new method. The result shows that the maximum RMS of retardation is reduced by 64% using only 48.99 s.
Polarization illumination is widely used in high numerical aperture (NA) projection lenses for better imaging resolution[
The material birefringence can be considered as a two-dimensional tolerance like surface figures and can be compensated by lens clocking[
To achieve a fast and efficient compensation process, we propose a new clocking optimization method for birefringence based on a sensitivity matrix. Similar to fitting surface figures with a fringe Zernike, we use orientation Zernike polynomials (OZPs) to describe the birefringence distributions that can be measured by the material manufacturer or generated by the optical system designer[
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Polarization aberrations can be represented by the Jones matrix, also known as the Jones pupil, which can be written as
According to the work of Geh et al., the Jones pupil of a projection lens can be decomposed by singular value decomposition (SVD) and can be written as[
When the matrix elements in OZP correspond to the symmetry terms in the fringe Zernike polynomials, the OZP labeling is
The first ±6 terms of OZP are shown in Table
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Similar to the RMS value of the fringe Zernike polynomials, we can also calculate the RMS value of the OZP, as shown in the formula[
Since the birefringence is an orientator, we can also fit it with OZP. In this Letter, ±36 items are used. On this basis, the sensitivity of the birefringence corresponding to each term is calculated and the optical system retardation caused by each lens element can be expressed as
To complete the clocking compensation for material birefringence, we need to further establish the relationship between the system optimization function and the rotation angle of each lens element.
The distribution of the ±9 OZP terms is shown in Fig.
Figure 1.Distribution of the first ±9 OZP terms.
According to the symmetry of OZP, we can deduce the corresponding rotation formulas for
Then the rotation matrix for
The rotation matrix for
After the lens element is rotated, its contribution to system retardation can be expressed as
For optical systems with
We illustrate the effectiveness of the above method by taking a projection lens as an example. The parameters of the lens are shown in Table
Figure 2.Layout of the NA 0.93 optical system.
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The OZP coefficients
Figure 3.Birefringence map. (a) Element 1, RMS value is 0.54 nm/cm and PV value is 1.16 nm/cm; (b) element 11, RMS value is 0.52 nm/cm and PV value is 1.26 nm/cm; (c) element 17, RMS value is 0.61 nm/cm and PV value is 1.49 nm/cm.
Figure 4.OZP coefficients of elements 1, 11, and 17.
Using the commercial software CODEV, we can calculate the retardation pupil of the projection lens after loading birefringence data. The results are shown in Fig.
Figure 5.Retardation pupil before optimization. (a) Field 1, RMS value is 7.80 nm; (b) field 2, RMS value is 7.07 nm; (c) field 3, RMS value is 6.29 nm.
Figure 6.OZP coefficients of the retardation pupil before optimization.
Then the sensitivity matrix
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After setting the optimization angles in CODEV, the system retardations are recalculated and the results are shown in Fig.
Figure 7.Retardation pupil after optimization. (a) Field 1, RMS value is 2.25 nm; (b) field 2, RMS value is 2.45 nm; (c) field 3, RMS value is 2.77 nm.
Figure 8.OZP coefficients of the retardation pupil after optimization.
At the same time, the NLS optimization method is compared with the PSO optimization method. As shown in Fig.
Figure 9.Comparison of convergence between the PSO and NLS methods.
In conclusion, we have proposed a new optimization method called NLS that is based on the sensitivity matrix for birefringence compensation. An optimization example with an NA 0.93 projection lens has been presented to illustrate the performance of the NLS method. The maximum retardation RMS is reduced by 64% after optimization. The NLS method improves the optimization efficiency greatly and is helpful in blank material selection. Last, but not least, it could also be useful for birefringence, homogeneity, and surface figure clocking optimization in other high-precision optical systems.
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[15] H.-J. Rostalski, A. Dodoc, W. Ulrich, A. Epple(2007).
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Hongbo Shang, Luwei Zhang, Chunlai Liu, Ping Wang, Yongxin Sui, Huaijiang Yang, "Optimization based on sensitivity for material birefringence in projection lens," Chin. Opt. Lett. 18, 062201 (2020)
Category: Optical Design and Fabrication
Received: Dec. 18, 2019
Accepted: Feb. 21, 2020
Posted: Feb. 24, 2020
Published Online: May. 9, 2020
The Author Email: Luwei Zhang (zhanglw@ciomp.ac.cn)