2.1.

Initial and Early Designs

A schematic representation of the earliest design of FINCH¹⁵ is given in Fig. 1. A core component of FINCH is a phase-only LCoS-SLM that is simultaneously used as a diffractive lens and as a beam splitter, thereby forming a single-channel system. The system description starts with the assumption that spatially incoherent light is scattered by, or emitted from, a 3-D target. The light is collected by the objective lens $L_{\mathrm{o}}$ and is later modulated by a phase-only SLM so that each spherical beam that originates from a single point of the object is split into two spherical beams with different wavefront curvatures. The two spherical beams interfere on the plane of the digital camera, forming a Fresnel hologram of the object point. The recorded hologram is a summation over the intensity contributions from all source points since the object is spatially incoherent. It is now apparent why FINCH is considered an incoherent, single-channel interferometer. The requirement to produce two different spherical beams from each object point is achieved by rendering two diffractive lenses using the SLM, each of which occupies a randomly selected half of all SLM pixels. Therefore, whenever a wave from an object point is introduced into the SLM, two spherical waves are returned from the SLM. The final resulting interference pattern is actually the Fresnel hologram of the observed 3-D object. From it, the recorded 3-D target can be reconstructed onto a desired far plane using a conventional digital process denoted as numerical Fresnel backpropagation.⁴³

Fig. 1

Schematic of the first Fresnel incoherent correlation holography (FINCH) recorder:¹⁵ BS, beam splitter; BPF, band-pass filter; $L_{\mathrm{o}}$, objective lens; SLM, spatial light modulator; CCD, charge-coupled device.

A Fresnel hologram captured by FINCH contains, like many other on-axis holograms, three terms: one is a relatively high constant term and the other two terms, a complex conjugate pair, are a convolution between the object intensity and a $z$-dependent quadratic phase function. Due to this well-known twin-image problem,⁴³ it is almost impossible to properly extract the 3-D image of the object by a direct reconstruction of the hologram, as mutual disturbances between the above mentioned terms exist. By using a process known as phase shifting, two of the above three terms can be eliminated so that a single desired convolution term is obtained. In this process, three holograms of the same object are recorded. The holograms differ from each other as each one is recorded with a different phase constant that multiplies only one of the two diffractive lenses displayed on the SLM. Hence, in addition to the two tasks of splitting and focusing beams in FINCH, a third mission of a phase shifter is added to the same SLM. The final hologram is digitally formed as a superposition of the three raw holograms. This complex-valued final hologram yields only the desired single convolution between the object and a $z$-dependent quadratic phase function. The numerical reconstruction of the final hologram produces a 3-D image of the object, without any disruptions from other holographic terms.

The initial configuration of FINCH was investigated in various studies for different applications. The first FINCH was demonstrated with white-light reflecting objects,¹⁵ while a few months later, FINCH was applied for fluorescence objects of various colors.¹⁶ Later, a FINCHSCOPE was demonstrated with biological specimens.¹⁷ For demonstration purposes, results from a FINCH system built according to the early design of Fig. 1, but using a transmissive rather than a reflective resolution object, are shown in Fig. 2. The magnitude and phase of the superposed hologram are presented in Figs. 2(a) and 2(b), respectively. The reconstructed image of the transmissive resolution chart is shown in Fig. 2(c), clearly revealing many of its details. Nevertheless, these early FINCH systems were not optimal in their performance due to three main aspects. First, the way of multiplexing two diffractive lenses on a single SLM, by allocating different pixels for the two lenses, has become a main source of noise on the reconstruction plane. Second, some of the system parameters were selected arbitrarily, without proper understanding of their influence on the various system performances. Finally, reducing the optical path difference (OPD) in FINCH, for processing wide-bandwidth light sources, was done by inefficiently increasing the length of the systems. These difficulties were fruitfully solved in the later versions of FINCH;^19,21,24 some of them are reviewed in Sec. 2.2.

Fig. 2

Results of the early version of FINCH. (a) and (b) The magnitude and phase of the hologram generated from three recorded raw holograms. (c) The best in-focus reconstructed plane from the hologram of (a) and (b).

2.2.

Element Multiplexing using the Polarization Method

The use of an SLM enables the realization of FINCH as a single-channel interferometer. This has several benefits, but requires multiplexing two diffractive lenses on a single phase-only SLM. This is not trivial since two phase functions are summed to a function, which is not necessarily a pure phase. Hence, the resulting complex function cannot be displayed directly on phase-only SLMs. The initial multiplexing method was based on spatial multiplexing as the two phase functions of the diffractive elements were simply displayed on different pixels of the SLM. Subsequently, the two diffractive elements were fragmented, leading to an apparent noise on the reconstructed image. The polarization method described in the following is an efficient method to eliminate this noise.

Some of the commercially available LCoS-SLMs are electronically controllable birefringent devices that only modulate light coming with a certain linear polarization orientation. This characteristic can be exploited for the task of multiplexing two elements on a single SLM. As the SLM is sensitive to a specific polarization orientation, one component of the electric field vector can be used as a wave modulated with the desired diffractive lens, while the other orthogonal component, which is not affected by the SLM, can be used as a wave without any modulation. Therefore, in the polarization method of multiplexing diffractive elements, the diffractive lens occupies the complete aperture of the system, thus, the SLM-realized elements (i.e., a diffractive lens and a clear aperture) are continuous over the entire optical aperture.

A comprehensive description of the polarization method of element multiplexing in FINCH is provided in Ref. 19. Here, it is reviewed with the help of Fig. 3, showing a FINCH system with an extra glass lens $L_{\mathrm{c}}$.^21,24 An object point, positioned at a working distance from the objective lens $L_{\mathrm{o}}$, emits a spherical beam which is introduced into the system. An input polarizer, $P_1$, oriented at an angle of 45 deg to the SLM active axis, allows the operation of two different imaging systems in the same physical single-channel setup. Only the beam polarization component parallel to the SLM active axis is converged by the diffractive lens. The orthogonal polarization component is not affected by the SLM at all. Hence, each of the two simultaneous imaging systems operates with one of two orthogonal polarization beams. For both imagers, the input beam is collected by the objective lens $L_{\mathrm{o}}$ and is then focused to two different image points, at $a_1$, for the imager in which the SLM-displayed diffractive lens is effective, and at $a_2$, for the imager in which the SLM does not affect the beam. A digital camera, located between the two image points, records the interference pattern between a spherical beam converging toward $a_2$ and a spherical beam diverging from $a_1$. The output polarizer, $P_2$, usually oriented at an angle of 45 deg, projects the two orthogonal polarization components onto a common orientation in order to enable interference between the beams. The hologram digital reconstruction procedure is identical to the process done with the early FINCH versions mentioned above.

Fig. 3

Schematic of a FINCH recorder in the polarization method: $P_1$ and $P_2$, polarizers; $L_{\mathrm{o}}$, objective lens; $L_{\mathrm{c}}$, converging lens; SLM, spatial light modulator; CCD, charge-coupled device. The circled black dot, up arrow, and up arrow with circled black dot represent polarization directions, perpendicular, parallel and 45 deg with respect to the plane of the page, respectively. Figure adapted from Ref. 44.

In order to demonstrate the advantage of the polarization method of element multiplexing¹⁹ over the previously described spatial multiplexing method,¹⁵ a FINCH system based on the schematic of Fig. 3 was built. The distance between the SLM and the camera was set to 40 cm, and a diffractive lens with a focal length of 28 cm was displayed on the SLM. A United States Air Force (USAF) resolution chart served as a test target and holograms were recorded using both multiplexing methods. The results are summarized in Fig. 4. In the spatial multiplexing experiment, three holograms were acquired with the input and output polarizers set in parallel to the SLM active axis. Fifty percent of the SLM pixels were set to a constant phase. The target reconstruction results for this method are presented in Fig. 4(a). In the polarization multiplexing method experiment, the three holograms were recorded with the polarizers set at 45 deg to the SLM active axis, and all SLM pixels were allocated for the diffractive lens. The target reconstruction results for this method are presented in Fig. 4(b). Obviously, the advantage of the polarization method over the earlier method is demonstrated.

Fig. 4

Best plane of focus reconstruction from holograms of a USAF test slide using (a) the method of allocating different pixels for the two lenses and (b) the polarization method with input and output polarizers at 45 deg.

2.3.

Confocal Fresnel Incoherent Correlation Holography

Sixty years ago, Minsky⁴⁵ presented the foundations of confocal microscopy. Since then, his ideas have been integrated to incoherent^46,47 and coherent^48–50 holography. A main virtue of confocal microscopy is its ability to perform optical sectioning. This is in contrast to ordinary microscopy, where the quality of the in-focus parts of a specimen may be faded by images of out-of-focus parts. Usually, this deterioration is elevated when examining thick specimens. Minsky’s confocal microscope combines two measures to solve this image quality deterioration problem: one, selective illumination of the object point of interest and two, near complete blockage of light from out-of-focus object points using an opaque screen, whereas light from the point of interest freely goes to the detector through a pinhole. This solution requires a scanning procedure in order to image an entire specimen.

Last year, a motionless, SLM-aided, confocal setup of FINCH was presented⁴⁴ as a new sectioning method and as an answer to the problem of the relatively low axial resolution of conventional FINCH.⁵¹ Several months after the appearance of the first confocal FINCH,⁴⁴ another confocal microscope was proposed, combining standard FINCH with a spinning disk.⁵² However, note that the original confocal FINCH⁴⁴ is motionless and accomplishes optical sectioning of the observed specimen without losing the inherent transverse super-resolution capabilities of the original FINCH.²¹ Therefore, confocal FINCH can image different planes of interest at various depths while blocking light from other planes. By that, the axial imaging resolution of the imager is enhanced and small details that otherwise would be lost are discovered.

The incorporation of the optical sectioning feature into FINCH has been achieved by using an unusual optical element dubbed a phase pinhole. The phase pinhole is an SLM implemented element that enables the creation of a FINCH hologram of only a particular object point out of the entire observed scene. The scheme of the confocal FINCH shown in Fig. 5 is based on the dual lens FINCH design (Fig. 3) with an added phase-only SLM, ${\mathrm{SLM}}_2$, located at the plane of the first image point $a_1$. ${\mathrm{SLM}}_2$ is actually used to implement the phase pinhole in the system. A diffractive optical element, consisting of a diverging axicon that surrounds a small circular area of uniform phase modulation is displayed on ${\mathrm{SLM}}_2$. For every scanning point $a_{\mathrm{o}}$, the phase-shifting procedure is implemented at the pinhole region by setting three different phases ${\varphi }_{\mathrm{1,2},3}$ for the three captured holograms, whereas on ${\mathrm{SLM}}_1$ the same diffractive lens is displayed during the acquisition of every object section. Since the phase is altered only within the phase pinhole, any optical pattern on ${\mathrm{SLM}}_2$, outside the phase pinhole, is vanished following the phase-shifting process. Therefore, the phase pinhole can be considered as an absorbing pinhole for the polarization components parallel to the ${\mathrm{SLM}}_2$ active axis and as an open aperture for the orthogonal polarization components. Because of the phase shifting and due to the incoherence nature of the light source, the only information left in the computer is the interference pattern between light that passes through the phase pinhole and its orthogonal counterparts imaged at the point $a_2$.

Fig. 5

Schematic of a confocal FINCH recorder: $P_1$ and $P_2$, polarizers; $L_{\mathrm{o}}$, objective lens; $L_{\mathrm{c}}$, converging lens; ${\mathrm{SLM}}_1$ and ${\mathrm{SLM}}_2$, spatial light modulators; CCD, charge-coupled device. Figure adapted from Ref. 44.

The integration of the phase pinhole into FINCH is appropriate for realizing optical sectioning. Nevertheless, improved sectioning results can be accomplished using a complete confocal FINCH (shown in Fig. 5) through the incorporation of a point illumination setup. None of the target points located outside the cone of the illumination are detected by the camera. Thus, there are two mechanisms operating together to enable optical sectioning, namely, the phase pinhole and the point illumination. Clearly, since only a single object point is well imaged, a scanning procedure is required to capture the whole object. Fortunately, scanning of the entire object can be performed without any mechanical movements by electronically shifting the phase pinhole onto different pixels of ${\mathrm{SLM}}_2$.⁴⁴

In the sectioning experiments, results of a conventional FINCH (Fig. 3) were compared to an optical sectioning FINCH (as shown in Fig. 5, but without the point illumination, so that the role of the phase pinhole is highlighted). The phase pinhole is considered as the more innovative part in the system, as scanning illumination systems are regularly utilized in other confocal microscopes.^48–50 It should be noted that sectioning accomplished only by the phase pinhole can be suitable for imaging tasks in which the observed scene cannot be selectively illuminated.

The experimental results are summarized in Fig. 6. Reconstructions from conventional FINCH holograms of two resolution charts, closest to the objective and farthest from it, are shown in Figs. 6(a) and 6(b), respectively. It is clear that the images of the out-of-focus targets greatly diminish the quality of the reconstruction. Actually, the image of the farthest target is hardly seen in Fig. 6(b), although this target with the digits “18.0” is actually in focus. The reconstructed images of the sectioning FINCH equivalents of Figs. 6(a) and 6(b) are shown in Figs. 6(c) and 6(d), respectively. Here, the images of the out-of-focus objects are severely diminished so that the images of the in-focus objects are clearly seen with better contrast, complete details, and weak background noise. The optical sectioning capability of the proposed system is thus demonstrated and is expected to be improved even further once the point illumination is incorporated. The confocal FINCH can well suppress out-of-focus images from a FINCH hologram, combining the high-lateral resolution capabilities of FINCH and the optical sectioning capabilities of confocal microscopy.

Fig. 6

Experimental results: (a) and (b) FINCH reconstructions of $16.0\text{cycles}/\mathrm{mm}$ and $18.0\text{cycles}/\mathrm{mm}$ resolution charts located 30 and 31 cm away from the objective lens, respectively; (c) and (d) are the optical-sectioning-FINCH equivalents of (a) and (b), respectively. Figure partially adapted from Ref. 44.

3.1.

Dual Lens Synthetic Aperture with Fresnel Elements

The dual lens SAFE system, presented in Fig. 7, is based on the dual lens FINCH system (Fig. 3). In both systems, the wave emitted from each object point is split into two closely spaced spherical waves with different curve radii, which are interfered on the camera plane. Similar to FINCH, in order to achieve optimal resolution, the system is configured to fulfill the requirement of a perfect overlap between the two interfering waves on the camera plane.²¹ The advantage of working with two spherical waves, rather than with one spherical wave and one plane wave, is manifested by the system (either FINCH or SAFE) capability of handling light of a wider bandwidth, while keeping the visibility of the recorded interference patterns sufficient. This is because the maximal OPD between two interfering waves becomes shorter as the two image points, generated by focusing the two spherical waves, get closer to each other.²⁴

Fig. 7

A schematic configuration of the dual lens synthetic aperture with Fresnel elements (SAFE) concept. In (a), a continuous central holographic element using the dual lens FINCH method is recorded. In (b), a marginal holographic element is recorded. The two SLMs, ${\mathrm{SLM}}_1$ and ${\mathrm{SLM}}_2$, are shifted in two symmetrical viewpoints in front of the collimation lens $L_{\mathrm{o}}$. Two diffractive lenses, with focal lengths $f_1$ and $f_2$, are displayed on ${\mathrm{SLM}}_1$ and ${\mathrm{SLM}}_2$, respectively. In (c), a marginal holographic element symmetrical to that of (b) is recorded by switching the two diffractive lenses $f_1$ and $f_2$. $L_{\mathrm{o}}$ and $L_{\mathrm{c}}$, lenses; $P_1$ and $P_2$, polarizers; ${\mathrm{SLM}}_1$ and ${\mathrm{SLM}}_2$, spatial light modulators; CCD, charged-coupled device.

Another characteristic feature of the dual lens SAFE,⁶⁴ in contrast to its previous iterations, is that the recorded subholograms are not stitched together along the central $x$, $y$ axes of the final mosaic hologram. This guideline yields better results because most of the energy emitted from commonly observed objects propagate toward the system in a relatively small angle, thus, any stitch between subholograms that passes through the central area (where most of the light concentrates) might distort the reconstructed image. Hence, a continuous central subhologram is first recorded by a conventional dual lens FINCH system.

In Fig. 7(a), the recording process of the central subhologram is schematically described. Two spherical waves that originate from the same object point converge to the distances $f_1$ and $f_2$ from the SLM. A resulting interference pattern encodes the object point position in space. Note that the lens $L_{\mathrm{o}}$ collimates the spherical wave emitted from the object point into a plane wave. SAFE is designed as a telescopic holography system for imaging objects located at infinity. As such, $L_{\mathrm{o}}$ is considered here as an external element to the system that is used to simulate very far illuminating objects. As previously mentioned, two polarizers, $P_1$ and $P_2$, positioned before and after the SLM, respectively, are oriented at 45-deg angles with respect to the SLM active axis, in order to achieve maximum visibility of the interference pattern between the two waves. The refractive lens $L_{\mathrm{c}}$ converts the plane wave obtained from $L_{\mathrm{o}}$ into a converging spherical wave. This wave is split into two spherical waves by the SLM. Optimal resolution is achieved when the two waves perfectly overlap on the camera plane. In a dual lens FINCH, this condition is fulfilled by setting the distance between the SLM and the camera plane to be $z_h=2f_1{f}_2/(f_1+f_2)$.²¹

Following the first stage of recording the central subhologram, two SLMs, denoted ${\mathrm{SLM}}_1$ and ${\mathrm{SLM}}_2$, are symmetrically shifted in opposite directions away from the optical axis [Figs. 7(b) and 7(c)]. Two different regions of the wave emitted from the object point are directed by the two SLMs to interfere on the camera plane, enabling an additional noncentral subhologram to be recorded. The diffractive elements that create the two interfering waves are composed of two different segments of two quadratic phase functions, one with a focal length $f_1$ and another with a focal length $f_2$, displayed on ${\mathrm{SLM}}_1$ and ${\mathrm{SLM}}_2$, respectively. Note that the lens $L_{\mathrm{c}}$ is no longer necessary, due to the physical separation between the interfering waves for the noncentral subholograms, as opposed to the central subhologram in which lens multiplexing (using the polarization method) is used. Additionally, in order to achieve maximal power efficiency at the noncentral subholograms, the exit polarizer $P_2$ is removed, whereas the entrance polarizer $P_1$ is adjusted to the same orientation of the active axes of the SLMs.

By switching the two masks that are displayed on the SLMs, while keeping ${\mathrm{SLM}}_1$ and ${\mathrm{SLM}}_2$ at their designated locations, a Fresnel holographic element on the opposite and symmetrical side of the optical axis is recorded [Fig. 7(c)]. Thus, electronic switching enables to cut down the amount of physical repositioning of the SLMs in the SA recording process.

The following presented experiment⁶⁴ with a dual lens SAFE system demonstrates the resolution improvement that can be achieved with this imager. In the experiment, the various SLMs, shown in Fig. 7, were realized as different apertures on the same large LC board. The image presented in Fig. 8(a) was produced by a conventional (nonholographic) imaging system. This system and the FINCH system, shown in Fig. 7(a), have similar NAs. The line grids along both axes in the resolution chart are not perceived in Fig. 8(a), due to the insufficient resolving power of the imager. The physical aperture in this case is not large enough. In Figs. 8(b) and 8(c), the intensities along horizontal and vertical cross-sections do not reveal the existence of any gratings.

Fig. 8

Experimental results obtained by recording a section of a resolution chart: (a) the image obtained by the conventional imaging system; (b) and (c) the intensity cross-section of (a) along the horizontal and vertical dashed red lines, respectively; (d) the reconstructed image corresponding to the hologram produced by a $360\times 360\text{pixels}$ FINCH system; (e) and (f) the intensity cross-section of (d) along the horizontal and vertical dashed red lines, respectively; (g) the reconstructed image corresponding to the hologram produced by dual lens SAFE; (h) and (i) the intensity cross-section of (g) along the horizontal and vertical dashed red lines, respectively; (j) the reconstructed image corresponding to the hologram produced by a $1080\times 1080\text{pixels}$ FINCH system; (k) and (l) the intensity cross-section of (j) along the horizontal and vertical dashed red lines, respectively. Figure adapted from Ref. 64.

The resolving power of FINCH, as previously shown,²¹ can exceed the resolving power of a regular incoherent imaging system of a similar NA. In Fig. 8(d), a reconstructed image of the target, produced by a dual lens FINCH setup of equivalent NA as the imager used for Fig. 8(a), is shown. Indeed, there is a perceivable difference between the systems as the digits are more clearly seen. Still, the horizontal and vertical lines are not clearly resolved [Figs. 8(e) and 8(f), respectively].

The mosaic SA hologram reconstructed image of the resolution target, obtained using the proposed dual lens SAFE system, is shown in Fig. 8(g). Notice how the vertical and horizontal lines are clearly revealed in this figure and in the presented cross-sections in Figs. 8(h) and 8(i). Without doubt, there is a significant increase in sharpness and visibility. For a comparison, Fig. 8(j) presents an image of the target, reconstructed from a hologram that was recorded using a dual lens FINCH system with a physical NA that is similar to the effective NA of the SAFE configuration. Fewer artifacts are present in this figure, which may be attributed to the continuity of the produced hologram, contrary to the dual lens SAFE method, which contains discontinuities in the mosaic hologram. The similarities in the cross-sections [Figs. 8(k) and 8(l) versus Figs. 8(h) and 8(i)] indicate that in both of these systems, the ability to transfer the higher frequencies of an object is similar.