It is well known that different lithographic features have different optimal settings of the numerical aperture (NA) and illumination condition.1, 2 Typically the depth of focus or the process latitudes are optimized. To achieve the optimal settings, modern steppers and scanners have a projection lens with a variable numerical aperture, where the NA value can be reduced to about 70% of its maximum value. As a consequence, a smaller fraction of the lens pupil and a corresponding smaller fraction of the phase aberrations is actually involved in the imaging process.
In the literature (see Ref. 3 for a survey) the problem of computing the aberration Zernike coefficients of scaled pupils has been discussed at various places. The recent result of Dai3 gives the Zernike coefficients of the scaled pupil in terms of those of the unscaled pupil in analytic form [see Eq. 18].
In this letter we give an alternative expression for Dai’s result. Our main result in Eq. 4 agrees mathematically with Dai’s Eq. 18 but has the advantage that it is very simple and direct in terms of Zernike polynomials. Thus it leads to the simple and explicit results in Eqs. 7, 9 from which the sensitivity of the aberration coefficients and Strehl ratios can be assessed when NA is close to its maximum value. The short and elegant proof of our main result in Eq. 4 is given in Appendix A.
Zernike Coefficients of Scaled Pupils
We consider a pupil functionwith real phase , and we assume that is expanded as a Zernike series according to: and are integers with being even and , and denotes the Zernike radial polynomial4 of azimuthal order and degree . For simplicity we only consider cosin terms. Scaling to a smaller pupil with relative size means that we have to expand the scaled phase into a Zernike expansion: of the scaled pupil function in terms of the Zernike coefficients of the unscaled pupil function. In the Appendix A we prove our main result: , and where we use the convention that . We note that (1) the entries do not depend on ; we consider them only for , and (2) since all involved Zernike polynomials, except , equal 1 at .
Examples and Sensitivity Analysisand .
A consequence of Eq. 4 is that7 shows a significant sensitivity of when is slightly below its nominal value of 1, especially when high-order aberrations are present. In our example we obtain a relative sensitivity of 11, indicating that the NA value should be specified to accuracy when the required coefficient accuracy is 1%.
The Strehl ratio4 is approximated as:and the term with is omitted. A consequence of Eq. 7 is that and the rim , respectively. Figure 2 shows the dependence on of the Strehl ratio in the case of a mixture of low- and high-order spherical aberration: and , where the scaled Strehl ratio is calculated by combining Eqs. 8, 4. This result compares well with the numerical calculation obtained from the lithography simulator SOLID-C.5 Intuitively it is expected that the Strehl ratio increases when the NA is decreased. However, in general this is not true. In the special case shown in Fig. 2, the Strehl ratio decreases when the NA is decreased from its maximum value. This result can be seen as follows: as in our example for each value, the phase aberration at the rim of the pupil equals 0. From Eq. 9 it then follows that the slope at is positive.
Appendix A: Proof of the Main Result, Eq. 4
By decoupling per azimuthal order , and normalization and orthogonality4 of , the Zernike coefficients of and the Zernike coefficients of are related by10 is over , and when and that does not depend on , except that in Eq. 10 we only use . For this we use6 are integers with same parity; in the case of , the right-hand side of Eq. 13 vanishes, which is consistent with the convention that then . We use Eq. 13 in Eq. 11 to rewrite and interchange integrals to get4 and we get7 13 to rewrite the resulting two integrals in terms of Zernike polynomials. This gives Eq. 12.
Appendix B: Dai’s Formula
We reproduce Dai’s formula3 for the scaling of Zernike coefficients:being the maximum value.