J. Eur. Opt. Society-Rapid Publ. 22, 15( 2026) 145
Figure
3. a) The test object is a stainless steel screw turned upside down, with a diameter of 3.9 mm and a depth of L = 10mm( not including the screw head). b) Digital-holographic hypercentric reconstruction of the object. The insert shows an image of the centered aperture with diameter d = 0.8 mm. The featured visible object depth is approx. 8 mm. Concerning the amount of hypercentricity we get a measured perspective spread of ~ x m ¼ 0:43 0:02 mm, which is in very good accordance with the expected theoretical spread of ~ x theor ¼ 0:41 0:03 mm. c) From the same holographic data can also be calculated the conventional numerical reconstruction of the hologram. This yields a sharp image of the bottom surface of the screw. Because of the high-NA setup, the first screw threads are visible.( d, e) By shifting the lateral position of the virtual aperture, a sideways observation direction for the hypercentric imaging is selected. The insert shows an image of the laterally shifted aperture with d = 1.5 mm.
imaging effect, even from depths of approx. 8 mm below the surface.
Because of the small aperture needed to provide a depth of field that covers a majority of the object depth, the hypercentric imaging process generally comes with a drop in lateral resolution caused by the band limitation that is introduced with the( virtual) aperture. This is also the case for our digital-holographic approach, as can be seen in Figure 3b. Moreover, this band limitation brings about a filter for the direction of the light rays and establishes a predominant direction. Therefore, it comes with a side effect that makes the rough bottom surface of the screw look more specular. The overall degradation in image quality due to speckle noise is a common feature of holographic imaging in general. For further improvement of the image quality, speckle reduction methods from conventional holography can be applied. An overview of these de-noising strategies is given in [ 13, 26 ]. The level of hypercentricity ~ xðxÞ can be measured according to equation( 6) by comparing the lateral coordinates in the perspective plane for two points. For the two positions ~ x marked in Figure 3b we get a measured perspective spread of ~ x m ¼ 0:43 0:02 mm, which is in very good accordance with the expected theoretical spread of ~ x theor ¼ 0:41 0:03 mm.
The great advantage over lens-based hypercentricity is particularly demonstrated in the flexibility of the virtual aperture. In the axial dimension, z P can be changed to adapt the amount of hypercentricity, as given by equation( 6). In the lateral dimension, the purely numerical imaging process offers adaptation of the aperture diameter, but it even more offers the possibility to laterally shift the aperture inside the aperture plane and thereby select different observation directions, i. e. perspectives, from the same hologram. If we denote the lateral shift by Dr and consider the geometry introduced in Figure 1, it becomes clear that the effective observation angle towards the optical axis a is defined by tan( a) = Dr / z P. Two examples for this are shown in Figures 3d and 3e for z P = 9mm, d = 1.5mm and the perspective plane 4 mm below the surface, where the shift of the aperture results in a sideways perspective on one side of the object with an effective observation angle of a = 7.1 ° and a = 7.6 °. Compared to Figure 3b, in Figure 3d and Figure 3e the effect of the larger aperture d is noticeable in the increased lateral resolution, traded against a smaller depth of field. Here, the perspective plane was changed afterwards by numerically refocusing to approx. the middle of the object at L / 2 in order to get a sharp image of the side, even with the larger aperture.