J. Eur. Opt. Society-Rapid Publ. 22, 9( 2026) 93
Figure 11. Rotation measurement. The rotation was calculated with equation( 17) after performing a least square fit of the Zernike polynomials Z þ1
1 and Z �1
1.
Figure 12. Phase fluctuation measurement without fringe locking. The piezo is not moving during the measurement.
In terms of equation( 20), h and b are approximately equal in the fringe locking camera path, leading to nearly equal vector components k x and k y. That means that small tilts in b greatly rotate the interfering wavefront in the camera path. The signal of the fringe observation system is not that sensitive, because h is with around 45 ° much larger than b. The fringes rotate around 0.17 ° in contrast to the fringe locking signal with 55 ° of the rotation range.
In addition, it can be observed that the signals between the measurement systems are not linear to each other. This is caused by the retroreflector. The propagation direction of the reflected beams in the retroreflector strongly depends on the orientation of the prism surfaces. The two prisms are not perfectly aligned with each other. A slight misalignment directly causes a low-frequency fringe image on the camera, as the two beams are no longer parallel to each other in the camera arm. To take into account the misalignment of the two prisms, we introduce prefactors a that depend on the angles h and b.
kðh; bÞ ¼ jkj
0 B @ a x ðh; bÞsin ðhÞcos ðbÞ a y ðh; bÞsin ðhÞsin ðbÞ a z ðh; bÞcos ðhÞ
1 C A: ð21Þ
With this equation, the nonlinearity can be compensated by calibration measurements. The calibration must be repeated for each alignment in terms of h.
4.2 Fringe locking
At first the phase fluctuation of the test environment was identified. In Figure 12, the measurement shows a low-frequency fringe displacement with 85 nm peak-to-valley
( red line) and with an RMS value of 18.33 nm in a time range of 250 s.
The phase fluctuation is mainly caused by air turbulence in the beam paths. The dose contrast, according to equation( 5), for this measurement is 0.793, compared to the mean contrast of a single intensity distribution of equation( 3) with 0.867. This results in a contrast loss of 7.4 %. Although the phase fluctuations determined with the fringe locking system and the fringe observation system differ slightly, the characteristic features are still clearly recognizable. The camera of the observation system is placed below the writing head.
During measurements, the camera induces heat turbulence, which increases the temperature of the retroreflector. This induces different air fluctuations in the fringe locking beam path, which finally leads to partial differences between the measurement data. However, with a correlation coefficient [ 27 ] ofR = 0.749, the two measurements strongly correlate. In a next step, the piezo actuation was calibrated. A sinusoidal oscillation with an amplitude of 1 lm was induced. Due to the dominant amplitude, air turbulence can be neglected during calibration. The fringe movement in the substrate plane is again captured by the fringe observation system( see Fig. 13). As the movement of the fringes here is greater than the unambiguous range p 2 of the evaluation of the fringe observation system, the measurement data were unwrapped over the time axis. Then a sine function was fitted to the red curve to determine its amplitude. The ratio of the two amplitudes results in the transfer factor of the piezo movement to the fringe movement in the substrate plane. The factor was determined to be 0.63. Finally, the fringe locking was tested in closed loop. In Figure 14 the fringe locking signal( blue line),