Fresnel incoherent correlation holography (FINCH) is one of the well-established incoherent digital holography techniques[
Chinese Optics Letters, Volume. 19, Issue 2, 020501(2021)
Review of Fresnel incoherent correlation holography with linear and non-linear correlations [Invited] On the Cover
Fresnel incoherent correlation holography (FINCH) is a well-established incoherent imaging technique. In FINCH, three self-interference holograms are recorded with calculated phase differences between the two interfering, differently modulated object waves and projected into a complex hologram. The object is reconstructed without the twin image and bias terms by a numerical Fresnel back propagation of the complex hologram. A modified approach to implement FINCH by a single camera shot by pre-calibrating the system involving recording of the point spread function library and reconstruction by a non-linear cross correlation has been introduced recently. The expression of the imaging characteristics from the modulation functions in original FINCH and the modified approach by pre-calibration in spatial and polarization multiplexing schemes are reviewed. The study reveals that a reconstructing function completely independent of the function of the phase mask is required for the faithful expression of the characteristics of the modulating function in image reconstruction. In the polarization multiplexing method by non-linear cross correlation, a partial expression was observed, while in the spatial multiplexing method by non-linear cross correlation, the imaging characteristics converged towards a uniform behavior.
1. Introduction
Fresnel incoherent correlation holography (FINCH) is one of the well-established incoherent digital holography techniques[
In the above developments, the imaging characteristics of FINCH remained unchanged, as the fundamental principle of hologram formation and reconstruction remained unaltered. In a recent study[
In general, the imaging characteristics of FINCH are affected by the phase functions used for modulating the object waves. For instance, in Ref. [21], a spiral phase plate was used instead of the QPM to modulate one of the object waves and create the hologram. This hologram, when reconstructed, generated edge enhanced images of the object. However, in Ref. [19], as the hologram reconstruction was converted into a pattern recognition problem, FINCH showed a higher axial resolution, which is not a property of the earlier version of FINCH. In another recent study[
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Figure 1.Optical configuration of FINCH with (a) polarization multiplexing and (b) spatial multiplexing.
2. Methodology—FINCH with Linear Correlation
In the polarization multiplexing scheme [Fig. 1(a)], a thick object is critically illuminated by a spatially incoherent source. The light from the object is collected by a refractive lens located at and is incident on a polarizer oriented at 45o with respect to the active axis of the spatial light modulator (SLM) located at from the refractive lens. On the SLM, a QPM with a focal length of is displayed, which modulates about half of the incident light, while the remaining is unmodulated. Self-interference between the two beams is obtained by a second polarizer oriented at 45o with respect to the active axis of the SLM, which renders both beams with the same polarization orientation. The hologram is recorded by an image sensor located at a distance of from the SLM. Three phase shifts , 2π/3, and 4π/3 are introduced to the QPM, the corresponding holograms are recorded and projected into complex space, and a complex hologram is obtained. The different planes of the object are reconstructed by propagating the complex hologram numerically by the respective distances.
For a point object with an amplitude located at from the refractive lens with a focal length of , the complex amplitude entering the lens is given as , where , , is the focal distance, and is a complex constant. The complex amplitude exiting the lens is given as , where . The complex amplitude introduced by the SLM is given as and to . Assuming that is small and considering that the 45o polarization orientation with respect to the active axis of SLM generates a modulated and unmodulated beam, the complex amplitude after the SLM can be approximated as . It must be noted that the ‘+’ symbol does not have any effect until the complex amplitudes pass through , as before the two components have orthogonal polarizations and therefore cannot interfere. The self-interference PSH at the sensor plane is given as , where ‘’ is a two-dimensional (2D) convolutional operator. The complex hologram formed by the superposition of recorded phase-shifted is given as .
As the illumination is incoherent, a complicated object may be considered as a collection of independent point objects, and the object intensity is given as . The object hologram is given as . The image of the object is reconstructed by a back propagation given as , where is the reconstruction distance given as . The factor is the controller of the characteristics of imaging. In Ref. [1], , where is a binary random matrix with a scattering degree σ, while in Ref. [8], and, in Ref. [21], , where is the topological charge and is the azimuthal angle. The reconstruction mechanism is independent of the beam modulations involved, as the hologram is always propagated to a plane of interest by a convolution with . On one hand, this approach demands at least three camera shots and decreases the time resolution, and, on the other hand, it enables faithful expression of the modulation function in the imaging characteristics.
3. Methodology—FINCH with Non-linear Correlation
In recent studies[
In this approach, the reconstructing function is dependent upon the beam modulation functions. In Fig. 1(a), the point reconstructions for a QPM and a spiral QPM (SQPM) are different, owing to the system independent reconstruction mechanism. On the other hand, the point reconstructions are the same for both QPM and SQPM in Fig. 1(b). In the original FINCH with reconstruction by convolution with , FINCH exhibited a lower axial resolution. In Ref. [22], a substantial increase in axial and spectral resolutions was observed, which are different from the properties of the original versions of FINCH. It must be noted that even in the polarization multiplexing scheme[
4. Simulative Studies
For the simulative studies, the following design parameters are considered: , , , , aperture diameter , , and . Direct imaging with a lens of phase and focal length , FINCH in the polarization multiplexing scheme and spatial multiplexing scheme with a QPM of phase and focal length , an axilens[
Figure 2.Phase images of (a) QPM, (b) axilens, (c) axicon, and (d) SQPM. Reconstruction results of FINCH in the polarization multiplexing scheme with three camera shots and by back propagation for (e) QPM, (f) axilens, (g) axicon, and (h) SQPM. Reconstruction results of FINCH in the polarization multiplexing scheme with a single camera shot and non-linear correlation for (i) QPM, (j) axilens, (k) axicon, and (l) SQPM. Phase images of randomly multiplexed constant matrix and (m) QPM, (n) axilens, (o) axicon, and (p) SQPM. Reconstruction results of FINCH in the spatial multiplexing scheme with non-linear correlation for (q) QPM, (r) axilens, (s) axicon, and (t) SQPM. Reconstruction results of FINCH in the spatial multiplexing scheme with a single camera shot and LRA for (u) QPM, (v) axilens, (w) axicon, and (x) SQPM.
The reconstruction results by Fresnel propagation shown in Figs. 2(e)–2(h) show that the characteristics of the modulation function have been faithfully transferred to the image characteristics. The axicon generated stronger side lobes, and the SQPM generated edge-enhanced images of the object. The reconstruction results shown in Figs. 2(i)–2(l) by cross correlation by a non-linear filter show only a weaker transfer of the characteristics. The phase images of the randomly multiplexed QPM, axilens, axicon, and SQPM with a constant function are shown in Figs. 2(m)–2(p), respectively. The results in Figs. 2(q)–2(x) show a behavior that is nearly independent of the function of the phase mask.
The variation of the normalized intensity at the origin () of the reconstructed point images with distance is plotted for the two cases of FINCH with QPM and axicon with Fresnel back propagation and non-linear correlation and compared with the direct image’s intensity variations in Fig. 3. It is seen from the Fig. 3 that the axial sensitivity of FINCH is lower than that of direct imaging, and, when an axicon is used instead of a QPM, the sensitivity decreases further. The appearance of peaks indicates the repetition of the pattern, and the degree of pattern matching is exhibited by the value of the peak. The interesting point is that the behavior of FINCH with reconstruction by cross correlation is not as expressive of the modulation function as the original version, as the reconstructing function is dependent upon the modulation function of the phase mask. The results of axial correlations can be directly extended to spectral correlations based on the Fresnel propagator given as , which controls the amplitude and phase within the paraxial regions. Any change in the distance can be compensated by an equal and opposite change in . Therefore, the intensity is expected to change by the same value when varies by the same factor as .
Figure 3.Plot of I(x = 0, y = 0) for FINCH (QPM), FINCH (axicon), and direct imaging for variation in the object distance zs (0.1 to 0.3 m) for FINCH1, reconstruction by back propagation, and FINCH2, reconstruction by cross correlation.
The scattering ratio of the mask in the spatial multiplexing scheme is engineered using the Gerchberg–Saxton algorithm [Fig. 4(a)], and three phase masks are synthesized with scattering ratio , 0.04, 0.1, and 0.5, as shown in Figs. 4(b)–4(d), respectively[
Figure 4.(a) Gerchberg–Saxton algorithm and generated phase masks with (b)
The spatial multiplexing approach is studied next for , 0.04, 0.1, and 0.5[
Figure 5.Plot of I(x = 0, y = 0) for (a) FINCH (QPM) and (b) FINCH (axicon) with spatial multiplexing and non-linear correlation for variation in the object distance zs (0.1 to 0.3 m) for different scattering ratios
5. Experiments
To experimentally analyze the spatial multiplexing system and to confirm the above observations, two cases are considered. In the two cases, FINCH is realized using randomly multiplexed (σ = 0.04) QPMs and randomly multiplexed QPM and axicon. The two elements were fabricated using electron beam lithography () for a central wavelength of and a diameter of 5 mm. The QPMs were designed for , , and the period of the axicon is 60 μm in the second element. PMMA 950 K (A7) resist was spin coated on indium tin oxide (ITO) coated glass substrates and developed using methyl isobutyl ketone (MIBK) and isopropyl alcohol (IPA) solutions. An electron beam dose of was used, with a writing time of 6 h for each element. The optical microscope images of the diffractive elements are shown in Figs. 6(a) and 6(b), respectively.
Figure 6.Optical microscope images of randomly multiplexed (a) QPMs and (b) the QPM and axicon.
The holograms recorded for , to 6 cm in steps of 1 mm for the above two cases were cross correlated with the hologram recorded at using a non-linear filter ( and ), and the logarithms of the cross-correlation values () are plotted, as shown in Fig. 7. The experiment is then repeated by replacing the source with . The cross-correlation values for when switching between the two sources for two QPMs and the QPM and axicon are and , respectively. The plot in Fig. 7 shows similar behavior for the QPM and axicon, as the imaging characteristics transferred from the modulation function are suppressed by the spatial random multiplexing. Similar cross-correlation values for the QPM and axicon when the wavelength is varied indicate the same effect.
Figure 7.Plot of the logarithm of the cross-correlation value for variation in distance from zs = 5 cm. The holograms recorded at zs = 5.2 cm, 5.4 cm, 5.7 cm, and 6 cm are shown.
6. Conclusion
FINCH is studied in different optical configurations using two reconstruction methods, namely Fresnel back propagation and cross correlation by a non-linear filter. It is observed that original FINCH, in which a complex hologram is formed by the superposition of at least three phase-shifted holograms, and reconstruction by back propagation faithfully express the characteristics of the modulation function in imaging. This is due to the condition where the reconstructing function is independent of the modulation function. This is true for both spatial multiplexing and polarization multiplexing. FINCH in the polarization multiplexing scheme and reconstruction by cross correlation could not express the characteristics of the modulation function accurately, as the reconstructing function is dependent upon the modulation function. However, this method was able to express relative axial variations with respect to the reconstructing function. Therefore, the polarization multiplexing and reconstruction by cross correlation can partially express the characteristics of the modulation function in the imaging characteristics. The final method involving spatial multiplexing and reconstruction by cross correlation suppresses most of the effects from the modulation function. An insignificant variation of the axial characteristics with respect to the scattering degree of the spatial multiplexing was observed. As the scattering degree increases, the imaging characteristics approach a uniform behavior almost independent of the modulation function. We believe that the new findings will guide the design of future FINCH imagers. The proposed techniques will extend the application of FINCH to single shot 3D color imaging suitable for imaging focal spots in laser machining applications that are very bright and dynamic. In particular, the space–time evolution inside laser induced material breakdown that is used for X-ray and terahertz (THz) beam generation will be studied using the modified FINCH.
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Vijayakumar Anand, Tomas Katkus, Soon Hock Ng, Saulius Juodkazis. Review of Fresnel incoherent correlation holography with linear and non-linear correlations [Invited][J]. Chinese Optics Letters, 2021, 19(2): 020501
Category: Diffraction, Gratings, and Holography
Received: Jul. 13, 2020
Accepted: Sep. 11, 2020
Published Online: Dec. 18, 2020
The Author Email: Vijayakumar Anand (vanand@swin.edu.au), Saulius Juodkazis (sjuodkazis@swin.edu.au)