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J. Eur. Opt. Society-Rapid Publ. 22, 15( 2026)
splitting plate so that the 50 % reflecting surface is oriented towards the object. With this, the ghost image from the back side is negligible. Additionally, we attached some absorbing black tape to the front side metal parts of the microscope objective in the reference path. This avoids spurious reflections from the metal housing due to the illumination.
2.3 Hypercentric rendering process
The hypercentric rendering process consists of two steps, which will be detailed below: The first step is a Fresnel reconstruction into the object plane. This is an efficient way to transfer the wavefield into object space while adjusting the scaling of the field of view to the size of the object, which is significantly smaller than the camera sensor. In the second step, we use the angular spectrum method to propagate the wave field into the aperture plane, apply the virtual aperture and then propagate the filtered wave field back to the perspective plane to finally yield a hypercentric image.
For the first step, we will have a look at the Fresnel diffraction integral for the( back) propagation of a wavefield in a distance �z O, which, in our case, is the object plane: [ 25 ] " #
U O ð ~ xÞ ¼ i �ikj ~ xj2 exp � ikz O
kz O 2z O(!)
F Uð x~ 0 Þexp �ikj x~ 0 j 2
~ x: ð11Þ
2z O kz O
The reference wave is a spherical wave with radius z O, which can be approximated by a parabolic phase function
!
Rð ~ xÞ exp ik j ~ xj2: ð12Þ
2z O
Comparing equation( 11) with equation( 10) and equation( 12), we can rewrite equation( 11) to
" #
U O ð ~ xÞ ¼ i
�ikj ~ xj2
exp � ikz O F Cð ~
~ x x kz O 2z 0 Þ:
O kz O ð13Þ
This shows, that with the origin of the reference wave chosen to z O, a simple Fourier transform of the coherence function Cð ~ xÞ provides a numerical representation of the object image. However, to retrieve phase and amplitude of the wavefield U O ð ~ xÞ, we have to fully calculate equation( 13).
For the numerical implementation it is important to remember that the Fresnel propagation integral changes the scaling of the sampling grid. For a grid with( pixel) pitch p 0 and( edge) size S 0 in the sensor plane, we yield a new grid with pitch p ¼ kz O ð14Þ
S 0
and size S = kz O / p 0 in the object plane. This is a very convenient side effect of the Fresnel approach, because the hypercentric imaging scheme requires a large numerical aperture on the observation side. This means that z O is comparably small, so that the size of the field of view S will typically be much smaller than the edge length S 0 of the camera. The Fresnel propagation therefore adapts the field of view conveniently to the size of the object, which needs to be much smaller than the camera sensor as well.
For the second step, we can use equation( 2) for the propagation and apply the virtual aperture W d with diameter d in the aperture plane:
Uð ~ ~ x P Þ¼P zP P �ðzP þL = 2ÞfU O ð ~ xÞg W d ð x~ 00 Þ; ð15Þ
where ~ x P is a vector in the perspective plane. Using the angular spectrum method, the pixel pitch and the field of view do not change during the propagation operation, so that we can use the pitch p as found by equation( 14) to determine the size of the aperture in the numerical implementation. Finally, the intensity I~ ð ~ x P Þ¼j Uð ~ ~ x P Þj 2 is the hypercentric image.
3 Experimental results
For the experimental implementation of our hypercentric imaging approach, we used the digital-holographic setup from Figure 2 with high NA illumination from the top. As an object we chose a stainless steel screw turned upside down, with a diameter of 3.9 ± 0.1 mm and a depth of L = 9.9 ± 0.1 mm( not including the screw head), as shown in Figure 3a. This object offers great dimensions in depth and diameter and the structured sides of the object with a thread pitch of 0.7 mm are good indicators for the depth of field and the level of hypercentricity. Another important factor is a sufficiently large object roughness to get enough light scattered onto the sensor.
To attain the hypercentric image of the object from Figure 3b, we first calculate the object wave using equation( 13) with a measured z O = 9.3 ± 0.1 mm. Now we could directly obtain the conventional reconstruction of the recorded hologram, which yields a sharp image of the bottom surface of the screw, see Figure 3c. Because of the spiral structure of the screw and the angled illumination combined with the high-NA sensor, the first screw threads are also visible. But for the hypercentric imaging, we now choose a perspective plane, e. g. at L / 2 in the middle of the object like in equation( 15), and propagate from there to the aperture plane using the angular spectrum method from equation( 2). Here, we apply a virtual aperture W d with varying diameter d, depending on the desired lateral resolution and depth of field. This numerical application of the aperture marks the transition from the conventional imaging modality to the hypercentric one, all while relying on the same recorded hologram. Now, as described by equation( 15), the backpropagation to the perspective plane finishes the hypercentric imaging process. In this case, all these steps are implemented numerically with z P = 12 mm, d = 0.8 mm and the perspective plane 1 mm below the surface, yielding the hypercentric image of the object from Figure 3b. The hypercentricity is clearly visible in the high number of threads that are made visible by the hypercentric