J. Eur. Opt. Society-Rapid Publ. 22, 36( 2026) 363
Figure 11. Scheme of the alignment process.
4.2 Fringe period determination
With the aligned orientation, the stage will be moved with well known steps Dy perpendicular to the fringe pattern orientation. In Figure 16, five steps with 50 nm and with a velocity m y = 1lm were performed over a period of 90 s to characterize the difference of the measurement accuracy of FrObSy during dynamic stage movements and static positions.
The period can then be calculated by Eq.( 15) by
p ¼ 2py / ¼ 2p; ð18Þ m y
where m y is the gradient of the linear fit inFigure 17. The period was determined to p = 558.2 nm(± 1 nm). This result will be validated with exposure experiments in photo resist( see Section 4.5). In order to estimate the accuracy of the phase measurement, the phase offset D / is converted into an equivalent lateral displacement using Eq.( 16). The resulting fringe movement Dr can then be directly compared to the stage position signal Dy. Figure 18 shows the difference between both signals as well as the corresponding low-pass filtered data over time.
A striking feature of the measurement is the presence of high-frequency oscillations with an RMS value of approximately 4.06 nm. These oscillations originate from residual tilting of the positioning stage. Although the rotational degrees of freedom r x and r y about the x- andy-axes are actively controlled, they exhibit a residual control error of approximately ± 70 nrad. Consequently, the stage shows a slight oscillatory tilt during the measurement. This tilt leads to a lateral displacement of the entrance pupil of the MO. The magnitude of this displacement depends on the optical path height h opt, which corresponds to the effective vertical distance between the rotation center of the stage and the entrance pupil of the MO lens. For the present setup, this distance is approximately
h opt 89 mm: ð19Þ
A small angular deviation r x therefore produces a lateral displacement
y tilt ¼ h opt tan ðr x Þ: ð20Þ
For r x ± 70 nrad this results in a displacement of approximately ± 6 nm at the entrance pupil of the objective. This displacement directly shifts the recorded interference fringes
Figure 12. Scheme of the determination of fringe pattern orientation.
and therefore appears as an additional phase offset in the measured signal D /.
In addition to the high-frequency oscillations, the lowpass filtered data reveals slower phase fluctuations with an RMS of approximately 1.3 nm. These fluctuations are mainly attributed to thermally induced variations of the refractive index along the optical path. Small air turbulences cause local changes in the refractive index, which modify the optical path length of the interfering exposure beams. As a consequence, the relative phase between the beams varies slightly over time, resulting in a slow drift of the interference fringes.
4.3 Fringe pattern tilt calibration
In order to characterize the tilt of the fringe pattern, a scan along the z-axis has to be performed. A long-range scan along z is performed in order to visualize an aerial image of the two writing beams. Figure 19 illustrates how aerial images are generated from individual camera images using a 2D volume cut. For each step z i, an image of the writing pattern is captured and the offset value D / is tracked, as shown in Figure 20.
The 2D volume cut of two image stacks with different tilts are shown in Figure 21. The right cross section shows a tilted writing pattern with approx. 21 °. After movement of the lateral actuator in dy, the writing pattern is tilted close to the vertical position, as shown in Figure 21, left.
In order to quantify the tilt b, a high resolution scan with 2 lm range and a step size of 20 nm is performed close to the focus position of the writing pattern( see Fig. 22). The fringe shift D / exhibits a superimposed periodic modulation as a function of the axial displacement Dz. This modulation arises from the transverse sampling of the fringes during measurement along z, as the tilt causes the minima and maxima to be intersected alternately. Therefore, the behavior of the offset measurement can be described as a superposition of a linear shift and the interference intensity distribution as
/ ðzÞ ¼ m z z þ a cos ðxz þ u 0 Þþc: ð21Þ