J. Eur. Opt. Society-Rapid Publ. 22, 36( 2026) 359
Figure 3. Scheme of the writing pattern rotation. G 1 and G 2 are rotated to align the orientation of the fringes. With a rotated pattern by angle c, the contrast of the exposed areas in photo resist decreases significantly.
Figure 4. Scheme of the fringe pattern tilt in x-direction. Shifting the two beams by dx relatively, a tilt of b x of the writing pattern is induced.
The phase u( y) therefore determines the position of the maxima and minima of the intensity pattern that are recorded in the photoresist. Any displacement of the substrate during exposure shifts the local fringe position and thereby modifies the recorded phase of the grating [ 9 ].
2.4.1 Scan-induced phase errors
During the exposure of a scan line the substrate moves along the x-direction, while the fringe positions are defined along y, asshowninFigure 5. If the stage exhibits a positioning error perpendicular to the grating lines, denoted by Dy scan( x), the recorded interference pattern is shifted locally. This displacement introduces a phase deviation
u scan ðxÞ ¼ k y scan ðxÞ: ð7Þ
The resulting intensity distribution recorded in the photoresist becomes
I scan ðy; xÞ ¼ I 0 ½ 1 þ cos ðky þ k y scan ðxÞÞŠ: ð8Þ
Thus, positioning errors of the scanning stage directly translate into phase distortions of the recorded grating pattern. A phase error in the grating profile leads to a corresponding phase shift of the diffracted optical field.
The wavefront error W is the optical path difference. A first diffraction order with an illumination wavelength k associated with a phase deviation Du results in
Figure 5. Sketch of the scan-and-stitching error. Positioning errors Dy scan while scanning the photoresist results in contrast loss and deviations of the linear grating. Stitching errors Dy stich( red marked area) will result in phase jumps of the linear grating element.
W ¼ k 2p u: ð9Þ
In combination with equation( 7), both relations yields the wavefront error induced by scan positioning errors
W scan ðxÞ ¼ k p y scanðxÞ: ð10Þ
This relation shows that nanometer-scale positioning errors during scanning lead to wavefront errors of comparable magnitude in the diffracted beam.
2.4.2 Stitch-induced phase discontinuities
In addition to continuous scan errors, discrete phase discontinuities may occur at the boundaries between adjacent scan lines in the stitching direction y. If two neighboring scan lines are displaced by a lateral offset Dy stitch, theinterference pattern recorded in the photoresist exhibits a phase jump