J. Eur. Opt. Society-Rapid Publ. 22, 15( 2026) 143
dimensions is trivial. With this in mind, and conveniently setting the origin to the convergence point P C, the intercept theorem can be applied to find x~ A ¼ x A zA
z P, or more generally, for any arbitrary point P =( x, z) inobjectspace
~ x ¼ x z z P: ð4Þ
With equation( 4), it is possible to calculate the depth dependent lateral shift of any object point due to the hypercentric perspective as
z
P d x~ ¼ x~ � x ¼ x z � 1: ð5Þ
With the depth of the object L given, we can use equation( 4) to calculate the perspective spread x ~, which we define as the maximum shift difference for object points with the same lateral position x, but at different axial positions along the object. The maximum shift difference is obtained for points at z = z P � L / 2 and z = z P + L / 2. Using equation( 5) we find
xðxÞ ~ ¼ x z P
� x z P x z P L ¼ � �
x L; ð6Þ z P � L z
2 P þ L 2 z P � L zP þ L z
2 2 P
where we have assumed that z P is large against L / 2 in the last step. The perspective spread can serve as a measure for the amount of hypercentricity, since it is zero for telecentricity and becomes even negative for entocentricity.
Finally, an important requirement to hypercentric imaging is, that the depth of field covers at least the depth of the object along the optical axis, which is denoted as L in Figure 1. This can be assured by selecting the diameter d of the virtual aperture properly. With air as medium( n 1) and small numerical apertures, the diffraction limited depth of field is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z D ¼ k ðn 2 � NA 2 Þ
k NA 2 NA: ð7Þ 2
In this context, the numerical aperture for digital imaging into the perspective plane is defined by the diameter d and the imaging distance z P, sothatNA d /( 2z P). Inserting this into equation( 7) and demanding z D = L yields a useful rule to estimate the diameter of the aperture rffiffiffi k d ¼ 2z P: ð8Þ L
Please note, that in our case the depth of field z D in equation( 7) is oriented along the propagation direction of the ray-like wave fields. The propagation angles can become quite oblique, because of the large numerical aperture of the recording scheme. In most cases equation( 8) still serves as a good estimate. However, for large propagation angles, exchanging z D by its projection onto the opical axis z D,\ is more accurate.
2.2 Holographic recording process
Figure 2 shows the recording configuration. The object illumination converges at P C and has a particularly large
Figure 2. Layout of the digital-holographic recording setup.
numerical aperture, so that the sides of the specimen are illuminated as well. The reference wave is spherical, where the origin P R has the same distance z O from the camera sensor as the object’ s surface, which is indicated by the surface point. This configuration resembles the scheme of Fourier holography [ 21, 22 ], even though object wave Uð ~ xÞ and reference wave Rð ~ xÞ need to be superposed by the 50:50 beam splitting plate to make them interfere. The camera sensor is a Sony IMX 661 with 13 408 9528 Pixel( 127.7 MPix) and a pixel pitch of 3.45 lm. With an active surface area of 46 32 mm 2( 3.6 00, itissignificantly larger than the object, in order to collect light from the sides of the object. This yields a large numerical aperture in the observation path, which has to be matched by the reference wave. In our setup, this is realized by a microscope lens objective with NA = 0.28. However, fiber tips with large numerical aperture or pinholes could be used as well in order to make the setup more compact. The interference pattern recorded by the camera can be written as
I ð ~ xÞ ¼jUð ~ xÞj 2 þjRð ~ xÞj 2 þ U ð ~ xÞRð ~ xÞþUð ~ xÞR ð ~ xÞ: ð9Þ
A piezo actuated mirror is placed in the reference path, to enable the introduction of phase shifts. The phase shifting process eliminates the twin image and the dc-term. For the hologram recording, a large number of phase shifting methods is available [ 21 ], where the methods differ in the achievable measurement uncertainty [ 22 ] and the stability against phase shifting errors [ 23 ]. We used 4 phase shifted digital holograms and a 4-frame 90 ° phase shifting algorithm [ 24 ], to yield the coherence function
Cð ~ xÞ ¼Uð ~ xÞR ð ~ xÞ: ð10Þ
In our setup we have chosen a 50:50 beam splitting plate rather than a beam splitting cube. This allows to position the object closer to the camera and thus increases the NA of the recording geometry. Additionally, it has the advantage that spurious reflections from the sides of the glass cube are avoided. We also recommend inserting the beam