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Figure 3, right illustrates the effect of scanning with a rotated interference pattern: portions of the regions between adjacent fringes are unintentionally exposed, depending on the geometry of the writing spot. This results in a reduction of the effective grating height and consequently leads to lower diffraction efficiency.
2.3 Fringe pattern tilt
Figure 2. Scheme of grating shift to set the fringe pattern period. The grating G 2 is shifted to change the incident angles and thus to set the period. h 1 and h 2 are defined from the bisector of the two interfering beam directions in the mathematically positive direction.
pðh 1; h 2 Þ ¼ k sin ðh 1 Þ�sin �h 2 ð Þ: ð1Þ
In the experimental setup, the fringe period can be adjusted by the position of G 2. The propagation paths of the ± first orders from G 1 define the distance between the two beams and therefore the incident angles h 1 and h 2 after the MO in the substrate plane. The displacement of G 2 and the associated change in angle are shown in Figure 2. With h 1 = �h 2 = h, the fringe period can then be expressed as
pðdlÞ ¼ k
; ð2Þ
2 sin arctan dltan ½ arcsin ðÞ k g Š
where f 0 is the focal length of MO and g the grating period of G 1 and G 2.
2.2 Fringe pattern orientation
Since SBIL is a scanning process, the orientation of the fringe pattern on the substrate must be well known relative to the direction of movement of the substrate’ s positioning stage. As shown in Figure 3, the orientation depends on the orientation of the grating structures G 1 and G 2. If the substrate is scanned at an angle c relative to the fringe pattern, the interference fringes are partially averaged over the finite size of the writing spot. This reduces the modulation of the intensity distribution in the photoresist and therefore lowers the contrast between the exposure maxima and minima. As a consequence, the minima of the exposure pattern can exceed the exposure threshold of the photoresist, so that regions which would remain unexposed at perfect fringe contrast become partially exposed. During development, these unintentionally exposed regions are also removed, which reduces the difference between exposed and unexposed areas. As a result, the resulting resist relief exhibits a smaller grating height. The magnitude of this effect strongly depends on the lateral dimensions of the writing spot [ 8 ]. f 0
The two interference beams can be shifted laterally to the optical axis in two different directions dx and dy. That causes a tilt of the fringe pattern in the focal plane of the MO lens( see Fig. 4). The tilt of the fringe pattern is defined by the angle b between the substrate normal and the bisector of the two interfering beam directions, as illustrated in Figure 4. The fringe period p on the substrate then changes to
pðbÞ ¼ sin ðh 1 þ b k Þ�sin ð�h 2 þ bÞ ¼ k
2 sin ðhÞcos ðbÞ ð3Þ
with h 1 = �h 2 = h. For a large numerical aperture of the MO lens, the relation between the lateral shift in the pupil and the resulting propagation follows from the Abbe sine condition, which is fulfilled by well-corrected MOs. For the beams displaced by dx in the front focal plane of an objective with focal length f 0, the propagation angle b x of the emerging collimated beam is given by
b x ¼ arcsin dx f 0
: ð4Þ
This equation also applies to shifts in the y-direction.
Even if the change in period due to tilting is only slight, this degree of freedom plays an important role with regard to the illumination of non-planar substrates and thus represents an essential degree of freedom.
2.4 Scan-and-stitch process
In SBIL, large-area gratings are fabricated by translating the substrate through a localized interference field while the photoresist is continuously exposed. The substrate is scanned along the x-direction, while adjacent scan lines are stitched together in the y-direction. Mechanical positioning errors during scanning or stitching lead to local displacements of the recorded interference fringes and therefore introduce wavefront errors in the fabricated grating structure.
The interference pattern used for exposure forms a periodic intensity distribution. For a linear grating with period p, the interference intensity can be written as
IðyÞ ¼ I 0 f1 þ cos ½ uðyÞŠg; ð5Þ
where I 0 denotes the mean intensity and u( y) isthespatial phase of the interference pattern. For an ideal grating the phase is u 0 ðyÞ ¼ ky; k ¼ 2p p: ð6Þ