Parallel-mode scanning optical sectioning using digital Fresnel holography with three-wave interference phase-shifting

The Fresnel incoherent correlation holography (FINCH) method is applicable to various techniques of imaging, including fluorescence microscopy. Recently, a FINCH configuration capable of optical sectioning, using a scanning phase pinhole, has been suggested [Optica 1, 70 (2014)]. This capability is highly important in situations that demand the suppression of out-of-focus information from the hologram reconstruction of a specific plane of interest, such as the imaging of thick samples in biology. In this study, parallel-mode scanning using multiple phase pinholes is suggested as a means to shorten the acquisition time in an optical sectioning FINCH configuration. The parallel-mode scanning is enabled through a phase-shifting procedure that extracts the mixed term of two out of three interfering beams. ©2015 Optical Society of America OCIS codes: (090.0090) Holography; (090.1995) Digital holography; (110.6880) Threedimensional image acquisition; (180.1790) Confocal microscopy; (070.6120) Spatial light modulators; (050.5080) Phase shift. References and links 1. J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. 32(8), 912–914 (2007). 2. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997). 3. A. W. Lohmann, “Wavefront reconstruction for incoherent objects,” J. Opt. Soc. Am. 55(11), 1555–1556 (1965). 4. G. Cochran, “New method of making Fresnel transforms with incoherent light,” J. Opt. Soc. Am. 56(11), 1513– 1517 (1966). 5. M. K. Kim, “Full color natural light holographic camera,” Opt. Express 21(8), 9636–9642 (2013). 6. O. Cossairt, N. Matsuda, and M. 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Introduction
Fresnel incoherent correlation holography (FINCH) has been introduced almost a decade ago as a method for recording Fresnel holograms under spatially incoherent illumination [1].Utilizing modern electro-optical devices, such as spatial light modulators (SLMs) and digital image sensors, together with phase-shifting techniques [2], FINCH has given a new birth to the classical concept of self-interference-based incoherent holography [3,4] into the era of digital holography.The ability to encode information from a three-dimensional (3-D) scene or object into a single hologram, without special types of illumination (such as a laser), and later focus into various planes through digital reconstruction of the hologram, using a relatively simple -and thus elegant -apparatus, is indeed appealing.The concepts of FINCH have soon been adapted to configurations for natural day-light holography [5,6], synthetic aperture imagers [7,8], adaptive optics [9] and microscopy [10][11][12].
The reliance of FINCH upon self-interference goes hand in hand with its suitability for recording holograms of fluorescent objects.FINCH also offers enhanced lateral resolution, in the sense that while the cut-off frequency of its modulation transfer function (MTF) is the same as the cut-off frequency of an incoherent imaging system of similar numerical aperture, the shape of the MTF is similar to that of a coherent imaging system, which is uniform throughout the passed frequencies, and does not decay non-zero frequencies like in an incoherent imaging system [13].This property is entangled with FINCH ability to violate the well-known Lagrange invariant [14][15][16][17], a characteristic that is also found in the closely related Fourier incoherent single channel holography (FISCH) method [18].A recently suggested microscopy method that is based on 180° rotational shearing interferometry, similarly in part to FISCH, also exhibit this type of resolution enhancement [19]; however, unlike FISCH and FINCH, it cannot maintain depth information.Overall, the characteristics of FINCH render it applicable to biological microscopy [20].A recently developed technique for axial localization has been successfully demonstrated in FINCH, and is capable of realtime tracking of particles [21].Still, the limited axial resolution of FINCH [16] may diminish its usefulness for examining thick samples.
Lately, a configuration of FINCH that is capable of optical sectioning, for the imaging of thick objects, has been suggested [22].The solution relies on the integration of a new electrooptical element into FINCH, referred to as a phase pinhole; its role is thoroughly explained in Section 2. By optionally adding point-wise illumination, a complete confocal solution may be formed, following the principles of confocal scanning microscopy [23].Another confocal configuration of FINCH soon followed [24].In the latter configuration, a relay system was positioned right after the objective lens of the system, thus forming an intermediate image plane of the target object.Within that plane, a Nipkow spinning disk was inserted to provide optical sectioning before light even reaches the actual holographic interferometer (i.e., before any splitting of waves occurs).In a sense, this may be considered an off the shelf, readily available solution to optical sectioning, but it is a successful testament that a confocal FINCH system can be made competitive with conventional confocal systems in terms of lateral resolution, sectioning performance and applicability.A major drawback of these two solutions for optical sectioning using FINCH is the requirement of a scanning procedure.In the latter case [24], the scanning is mechanical but is rather quick due to the use of the Nipkow spinning disk.In the former, the scanning does not involve any mechanical movements; yet, it was only demonstrated with a rather slow point-by-point scanning [22].In this paper, a more efficient scanning scheme is presented, where multiple points are scanned in parallel.Furthermore, a generalized phase-shifting procedure between three or more interfering waves for extracting a desired interference pattern between two waves is presented, offering the possibility of overcoming difficulties due to nonindeality of the electro-optical liquid crystal devices in use, such as the limited fill factor of SLMs.Techniques that involve phase-shifting of more than a single wave in processes of multiple wave interference were described in the past (e.g., [25,26]), and a procedure akin to the hereby described procedure have been used by us in a recent publication [27], in the context of coherent holography.However, its generalization and use in the above context of incoherent holography may have special importance.We note that methods of spatially incoherent holography that allow confocal-like optical sectioning without any scanning (for a specific plane of interest) do exist [28][29][30].However, unlike [22], they are unfortunately not suitable for fluorescence imaging [30] or situations in which the user does not have control over the illumination.

Optical sectioning using FINCH
A schematic of an optical sectioning Fresnel incoherent correlation holography (OS-FINCH) setup is presented in Fig. 1.It is assumed that spatially incoherent light is emitted from (or scattered by) the targeted object, meaning that the object can be considered as a collection of many point sources, where the degree of mutual coherence between each pair of points is zero.Under this assumption, each point source only contributes to its own self-interference pattern, and the recorded hologram is a summation over the intensities of all self-interference patterns.
The FINCH setup in Fig. 1 is configured as a common-path single channel interferometer.This interferometer has only a single arm, and interference occurs between waves that travel along a similar direction, in a shared physical space.Consider a single point source a o within the complete object.Two images of this point source are formed by the interferometer: the image point a 1 is formed by an imaging system that consists of two refractive lenses, the objective lens L o and the converging lens L c , and one diffractive lens that is realized using the left-most SLM, SLM 1 ; the image point a 2 is formed by an imaging system that only consists of the objective lens L o and the converging lens L c .The coexistence of these two imaging system is made possible due to the use of a phase-only SLM, which is an electronically controlled birefringence device.Considering linearly polarized light, polarization components oriented along the active axis of the SLM are modulated, whereas orthogonal components, oriented along the non-active axis of the SLM, are not affected by the SLM.In Fig. 1, the polarizer P 1 is set with its transmission axis oriented at a 45° angle to the active axis of the SLM, thereby allowing the simultaneous formation of the two image points a 1 and a 2 .Over the image sensor plane, two spherical waves that originate from the point source a o exist: one diverging from the image point a 1 , and one converging towards the image point a 2 .According to [13], a complete overlap between the two spherical waves on the image sensor guarantees optimal image resolution.A second polarizer P 2 is positioned in front of the image sensor, with its transmission axis in parallel to the first polarizer P 1 , so that the two waves can interfere over the image sensor plane.The intensity of the interference pattern encodes both the point source position in space and the point source intensity.In order to achieve optical sectioning, a second phase-only SLM, SLM 2 , is incorporated into the system, positioned after SLM 1 (Fig. 1).The two SLMs are positioned with their active axes in parallel, meaning that, ideally, SLM 2 does not influence the wave that is converging towards the image point a 2 .SLM 2 enables optical sectioning by acting as a pinhole for the image that is formed on its surface.This image is imaged by the two refractive lenses, L o and L c , and the diffractive lens displayed on SLM 1 .The idea behind the proposed approach of optical sectioning [22] is to preserve all (or most) of the information acquired through the interference pattern between the waves associated with the image points a 1 and a 2 (representing the point source a o ), while eliminating all (or most) of the information from other point sources.This is achieved by using SLM 2 as a phase pinhole.The phase pinhole mask is composed of two components [22]: a small circular region that is set to one of several values of uniform phase modulation, and an axicon that surrounds the phase pinhole.The latter component is used to deflect light away from the digital sensor, as an alternative to actual light blockage that occurs with an ordinary pinhole, while the former is set to three different values of phase modulation, to enable phase-shifting within the effective aperture of the pinhole.
The mechanism of the phase pinhole based optical sectioning is explained with the help of Fig. , ; , , .In a non-sectioning FINCH setup, a phase-shifting procedure is used to extract a final complex-valued hologram that is free from the twin-image and zeroth order terms.The actual phase-shifting is performed using SLM 1 (which, in this case, is the only SLM in the system).The final complex hologram can be described as: In an OS-FINCH setup [22], however, the first SLM, SLM 1 , is not used for phase-shifting (letting θ k = 0°), which is now performed at the small circular region of the second SLM, SLM 2 .Figure 2 depicts a situation for which the spherical wave A 1 passes entirely within the phase pinhole, the spherical wave B 1 passes entirely outside of the phase pinhole, and the spherical wave C 1 only passes partially within the phase pinhole.Neglecting the impact of the axicon mask, a single recorded FINCH hologram can be described as: , ; , , 1 .
where φ l = 2π(l -1) / 3, for l = 1,2,3, is the phase modulation value within the phase pinhole, and S(C 1 ) is a support function that indicates the region over the image sensor in which the spherical beam C 1 is phase modulated by the phase pinhole.Following the complete phaseshifting procedure the final FINCH hologram is: Comparing Eq. ( 2) and Eq. ( 4), for the situation described in Fig. 2, reveals that: information regarding the point source a o is preserved; information regarding the point source b o is eliminated; information regarding the point source c o is partly eliminated, since the effective aperture of its hologram is greatly reduced.That is, only a specific point of interest is properly encoded into the hologram in the proposed procedure, while other points are either completely eliminated or are diminished in intensity.The result is an effective optical sectioning, but in order to record an entire 3-D object, a scanning scheme is required.Note that the result shown in Eq. ( 4) also holds for the case in which the effect of the axicon that surrounds the phase pinhole is considered.The use of the axicon may benefit the recording process by allowing better usage of the dynamic range of the image sensor.Further, it contributes to the elimination of undesired interference processes in which beams that ideally should not exist are involved, as discussed in Section 2.3.
The scanning procedure for a complete object, occupying a 3-D space, is performed through electrical manipulations of the SLM displayed masks, and does not involved any mechanical intervention.For simplicity, it is assumed that the two refractive lenses, L o and L c , and the left-most SLM, SLM 1 , all lie within a common x-y plane, as shown in Fig. 2. In practice, this is not the case, though it can be achieved using relay systems.The lens L eq is an equivalent lens that represents the objective lens L o and the converging lens L c (Fig. 1), and its focal length f eq = f o f c / (f o + f c ), where f o and f c are the focal lengths of the lenses L o and L c , respectively.Let d s denote the distance between the two SLMs, SLM 1 and SLM 2 , and f 1 the focal length of the diffractive lens displayed on SLM 1 .The scanning procedure can be performed by considering different transverse planes of interest.Let d o denote the distance between a specific plane of interest and the objective lens.This plane of interest should be imaged to the plane of SLM 2 , which is achieved by setting the focal length of the lens displayed on SLM 1 to: During the scanning procedure, the phase pinhole is laterally shifted to various positions within the plane of SLM 2 , until the entire area of interest is covered.Reconstruction of a single point source is performed using its corresponding FINCH hologram, and can be done through Fresnel back-propagation to a distance which is dependent upon the distance d o .The images of a single point source are formed at distances of d 1 and d 2 to the left and right of the image sensor, respectively.It is easy to show these two distance are: where d t = f eq d o / (d o -f eq ), and z h = 2d s f c / (d s + f c ) is the distance between SLM 1 and the image sensor, which assures a complete overlap between a pair of interfering spherical waves on the image sensor, for any point source located at the front focal plane of the objective lens.The reconstruction distance used for the Fresnel back-propagation is: ( ) ( , )* 1 , where ] is a quadratic phase function, and * denotes a twodimensional convolution.In order to form a reconstruction of an entire scanned plane, the reconstructions [Eq.(8)] should be combined.This can be achieved by assigning to each (x,y) coordinate its value from the hologram recorded with the phase pinhole location that corresponds with the desired reconstructed coordinate.The necessity for a scanning procedure is a main limitation of the above approach, as the scanning is both time demanding and computationally intense.In order to mitigate this burden, a parallelized scanning scheme is hereby proposed.The basic idea is to encode multiple points from a specific plane of interest into a single hologram, rather than just one, by displaying an array of phase pinholes over SLM 2 , instead of just a single phase pinhole, as depicted in Fig. 3. Here, square shaped phase pinholes are used, but other shapes (e.g., a circle) may be used as well.The width of a pinhole is represented by w p , and the distance between pairs of adjacent pinholes (in both x and y directions) is d p .The width of the scanned area is denoted by w s , where it is assumed that the scanned area has the form of a square.In a single pinhole scan, the pinhole is laterally shifted throughout the complete scanned area [Fig.3 and may be substantial.For example, a typical scan of the area of an entire SLM of 1,000 × 1,000 pixels in size, using a pinhole of 10 pixels in width, may require around 40,000 complex FINCH holograms.A parallel scan using multiple pinholes, with a distance d p of 90 pixels would only require around 400 complex FINCH holograms, a reduction of two orders of magnitude.However, the sectioning performance is dependent upon the distance d p , as demonstrated by the experimental results that follow.The above description of FINCH assumes that the SLMs modulate the polarization components of light that are oriented along their active axis in full.In practice, however, this is not the case.Considering Fig. 2, for example, this implies that the beams that are converging towards the image points to the right of the image sensor, a 2 , b 2 and c 2 , are mostly linear in polarization along the non-active axis of SLM 2 , but not entirely.This has little affect on the end result in a non-sectioning FINCH, and even in a sectioning FINCH, where only a single pinhole is used [22].However, using a multiple phase pinholes array, as here proposed, pronounces the contribution of these unwanted components, mainly because we get contributions between waves that interfere with themselves.To overcome this unwanted contribution, we suggest a procedure that is hereafter referred to as three-wave interference phase-shifting (TWIPS).The name is derived from the fact that the interference process may now be considered to be between three waves that originate from a single point source, whereas the phase-shifting process enables to extract a desired cross term between two waves.

Three-wave interference phase-shifting
Consider the interference of two waves, X 1 and X 2 , over the digital sensor plane, where one of the waves is phase-shifted: By recording three patterns with different values of θ k = θ 1 ,θ 2 ,θ 3 , and calculating [1]: it is easy to show that Eq. ( 11) results with where c = 2i[sin(θ 1 -θ 3 ) + sin(θ 2 -θ 1 ) + sin(θ 3 -θ 2 )], indicating that the twin-image and zeroorder terms in Eq. ( 10) are eliminated by the above procedure.Consider now a more general case, in which three waves, W 1 , W 2 and W 3 are interfering over the digital sensor plane, meaning that We now show that the ability of applying a controllable phase-shift to any two of the three waves can be used to extract any desired cross term between two of the three interfering waves in Eq. ( 13) (e.g., ).Without loss of generality, consider the case of controllable phase-shifts applied to W 1 and .
Let X 1 = W 1 and X 2 = exp(iφ l )W 2 + W 3 .Based on the procedure described in Eqs. ( 10)-( 12) it is easy to extract the term ( ) If the above procedure is performed with two different values of Extraction of the cross term W 2 W 3 * can be achieved in a similar fashion by first taking * is then found for two different values of θ k , each time via phase-shifting over three different values of φ l .In the Appendix, the TWIPS process is described for the case in which a joint phase-shift can be applied simultaneously to two out of three interfering waves, where there is an ability of applying an additional phaseshift to one of these two waves.Then, the TWIPS concept is extended to the more general case of interference between more than three waves.In the following section, the TWIPS concept is described in the context of the example in Fig. 2.

Three-wave interference in optical sectioning FINCH
In Section 2.1, the phase-shifting procedure in OS-FINCH has been described with the help of Fig. 2, under the assumption that the waves that converge towards the image points to the right of the image sensor are not modulated by SLM 2 , upon which the phase pinhole mask is displayed.Even though this is mostly satisfied, in practice, SLM 1 does not modulate the waves that have their polarization components along its active axis in full.This may result, in part, from the limited fill factor of currently available devices.Thus, the waves that converge towards the image point to the right of the image sensor also possess components along the active axis of SLM 2 .Even though these components may be relatively small, a way to mitigate their influence should be considered, especially when multiple phase pinholes are used.
Consider the point source c o in Fig. 4. The interference pattern due to this point source over the image sensor includes several waves.The wave that diverges from the image point c 1 , is modulated by both SLMs and can be described as exp(iφ k ){exp(iφ l )S(C 1 )C 1 + [1 -following Eq. ( 3).The wave that converges towards the image point c 2 , has dominant polarization components along the non-active axis of SLM 2 , which can be denoted as C 2 , and undesired components along the active axis of SLM 2 , described as exp(iφ l )S(C 3 )C 3 + [1 -S(C 3 )]C 3 , where part of the wave is phase-shifted when passed through the phase pinhole, as indicated by the support function S(C 3 ).Other components that may exist are assumed to be negligible and are neglected from the analysis.Thus, the recorded interference pattern for the point source c o can be written as: The form of Eq. ( 17) may be considered as the interference of five waves.Taking into account that the axicon displayed over SLM 2 deflects two of them from reaching the image sensor, Eq. ( 17) can be reduced to: , ; .
Note that the form of the above equation is similar in form to Eq. ( 19) in the Appendix.Therefore, the TWIPS procedure described in Eq. ( 20), Eq. ( 21) and the left-hand-side of Eq. ( 22) can be used to extract the cross term S(C 1 )C 1 C 2 * using a total of six exposures.This is the exact same term that represents the point source c o in Eq. ( 4), indicating that the suggested method can effectively deal with the undesired waves that are included in the analysis.

Experiments
An experimental setup based on the schematic of Fig. 5 was implemented.It is based directly upon the configuration of Fig. 1, but uses reflective SLMs (Holoeye PLUTO, 1920 × 1080 pixels, 8 μm pixel pitch, phase-only modulation) instead of transmissive ones.A beam splitter was inserted next to each of the SLMs, so that they can be used like transmissive type SLMs.For simplicity, the two lenses L o and L c were replaced by a single equivalent lens L eq of focal length f eq = 20 cm, representing an objective lens L o of focal length f o = 23.08 cm and a converging lens L c of focal length f c = 150 cm.The distance between the lens L eq and SLM 1 was set to 10 cm.SLM 2 and the image sensor (Hamamatsu ORCA-Flash4.0V2 Digital CMOS, 2048 × 2048 pixels, 6.5μm pixel pitch, monochrome) were positioned at distances of 66.32 cm and 90 cm away from SLM 1 , respectively.Two resolution charts were combined using a beam combiner into a 1 cm thick object.A negative 1951 United States Air Force (USAF) resolution chart was placed at a distance f o in front of the lens L eq , while a negative National Bureau of Standards (NBS) 1963A was placed at a distance of f o + 1 cm = 24.08 cm in front of the same lens.Each resolution chart was back-illuminated using an LED (Thorlabs LED635L, 170 mW, central wave length of λ = 635 nm, full width at half maximum Δλ = 15 nm).The exposure time of the image sensor was set to 80 ms per grabbed frame.
At first, regular (non-sectioning) FINCH holograms were recorded.The mask of the second SLM, SLM 2 , was set to a uniform phase-modulation value of zero, electronically transforming the setup into a regular FINCH configuration.Two FINCH holograms were recorded, each calculated according to Eq. ( 11) from three exposures with difference phaseshifting values of θ 1 = 0°, θ 2 = 120° and θ 3 = 240°.The first and second holograms were recorded with the USAF and NBS targets imaged upon SLM 2 , respectively.Reconstruction results from the first hologram, showing the USAF and NBS charts in planes of best focus are shown in Figs.6(a) and 6(b), respectively.Note that details of the two charts are not easily resolved in regions within the two charts overlap.In the case of the second hologram, the USAF and NBS charts appear in best focus simultaneously, as shown in the reconstruction in Fig. 6(c).These results are brought here to allow comparison with the results from an optical sectioning capable FINCH setup that follow.Fig. 5. Optical sectioning FINCH experimental setup: L eq , an equivalent lens, representing both the objective lens L o and the converging lens L c (Fig. 1); P 1 and P 2 , polarizers; SLM 1 and SLM 2 , spatial light modulators; RC 1 and RC 2 , resolution charts; BS 1 , BS 2 and BS 3 , beam splitters (BS 1 is used as a beam combiner); LEDs, light emitting diodes; L i s, illumination lenses.In the optical sectioning experiments, each of the two planes of interest (the two resolution charts) were first scanned using a square shaped phase pinhole of 88 μm in width (i.e., 11 pixels over 11 pixels in size).The scan was performed on a grid of size 101 × 101 points, with the phase pinhole shifted in jumps of 40 μm, enabling sufficient overlap between imaged positions.For each pinhole position, a complex FINCH hologram was calculated from six different exposures, based on the procedure described in Section 2.3, where SLM 1 was set with φ 1 = 0°, letting the phase pinhole, which is displayed on SLM 2 , take the values φ 1 = 0°, φ 2 = 120° and φ 3 = 240°, or with φ 2 = 180°, again letting the phase pinhole take the three values of φ 1 = 0°, φ 2 = 120° and φ 3 = 240°.Then, scanning using multiple phase pinholes, in parallel, was tested.The spacing between adjacent phase pinholes (in both x and y direction) was set to values of 88 × g μm, with g = 1 .. 15, implying spacing of 1 to 15 pinholes in width, between pinholes.It should be emphasized that: (1) a trial to perform a regular phaseshifting of three exposures, according to [22], does not yield acceptable results when multiple phase pinholes are displayed; (2) a mask of an axicon that occupy the phase pinhole(s) surrounding(s) was indeed used in the presented experiments and yielded better results than the case of working without the axicon; (3) the exposure time of 80 ms per grabbed frame indicates that a single complex FINCH hologram requires roughly 0.5 s for its recording.In practice, the total time required for the recording was about 2 s, a figure which can be reduced with tighter synchronization.Reconstruction results, based on the procedure described in Section 2, for the holograms recorded with the 1951 USAF resolution chart as the plane of interest, are presented in Fig. 7. Visual comparison of the reconstruction from the hologram set acquired using only a single phase pinhole [Fig.7(a)] with the results of multiple phase pinholes, starting with the spacing of a single pinhole in Fig. 7(b) through spacing of 15 pinholes in Fig. 7(l), reveals gradual improvement in the sectioning performance that increases with the spacing between the pinholes, until a spacing of 7 pinholes is reached [Fig.7(h)].From that point, any increase in the spacing between pinholes does not reveal visual improvement in the sectioning results [Figs.7(h)-7(l)], which resemble the performance in the case of a single pinhole [Fig.7(a)].That is, there is a trade-off between the complexity of the scan and the sectioning performance.We note that by visual comparison there is a clear improvement even between the highly parallelized (with g = 1) optical sectioning FINCH in Fig. 7(b) and the nonsectioning FINCH in Fig. 6(a).To quantify the results, the mean squared error (MSE) was calculated between the amplitude of the reconstruction in Fig. 7(a) and the amplitude of the reconstructions in Figs.7(b)-7(l).The cases of g = 11 .. 14, which are not presented in Fig. 7, for brevity, were also included in the MSE calculations.Further, the non-sectioning FINCH reconstruction in Fig. 6(a) is considered to represent the case of g = 0.The results are presented in Fig. 8, and are fairly consistent with the visual inspection.Taking into consideration that Fig. 7(a) was generated using 10,102 complex FINCH holograms, Fig. 7(i) using more than an order of a magnitude less (400 complex holograms), and Fig. 7(b) using only 25 complex holograms, it is clear that in practice it may be necessary to balance between the required sectioning and the time that is available for the process (here, the total time taken for scanning a single plane varied between few seconds to several hours, depending on the value of g).Still, the scanning time may be substantially decreased, without impairing the sectioning performance.Reconstruction results for the NBS 1963A resolution chart are presented in Fig. 9, with the MSE graph presented in Fig. 10.The previous conclusions are further supported by the results.Finally, to demonstrate the necessity of the TWIPS procedure, Fig. 11 presents the reconstructions of two different complex FINCH hologram, acquired with g values of 15 and 1, starting with a relatively low number of in-parallel phase pinholes, and ending with a relatively large number.The reconstructions of the complex FINCH holograms, using a regular phase-shifting procedure with three different exposures, are presented in Figs.11(a Thus, the contribution of TWIPS is apparent, and its necessity for the successful utilization of the suggested parallel scanning scheme in OS-FINCH is demonstrated, especially when the degree of parallelism is high.

Summary and conclusions
A new scanning procedure in an optical sectioning FINCH configuration has been presented.The procedure offers parallel-mode scanning capable of imaging a large number of points in an examined object plane simultaneously.This is achieved through the use of multiple phase pinholes.The speed gain is dependent upon the specific characteristics of the object.In the presented experimental results, a speedup of roughly one order of magnitude has been achieved, without any deterioration of the sectioning capabilities.Higher speedups are possible, but require a balance between the desired speed and the quality of the sectioning.
The use of multiple phase pinholes based scan is more sensitive to some of the nonidealities of currently available SLM devices, in comparison to the situation of using just a single phase pinhole.The suggested analysis of the case suggests that for a single point source, more than two waves are involved in the interference process.This is dealt with through a phase-shifting procedure which is referred to as TWIPS.The procedure effectively eliminates any of the unwanted interference terms, offering successful implementation of the suggested parallel-mode scan.TWIPS is discussed not only in the context of FINCH, but also in a broader and more general sense, so that it may readily be adapted to other systems as well, if necessary.
Overall, in OS-FINCH the sectioning is done only in the image plane, where the phase pinholes operate.Therefore, on the one hand, the sectioning performance of OS-FINCH are expected to be inferior to that of a conventional confocal microscope.This conclusion is valid at least until a confocal version of the proposed OS-FINCH system is demonstrated, in which a matching pinhole array will be used to illuminate the speciemen.On the other hand, it should be noted that the present OS-FINCH is suitable to imaging situations in which there is no access or possibility to illuminate speciemens with a beam array, like in confocal microscopy.Another advantage of OS-FINCH over the common confocal microscopes equipped with a spinning disk [24], is that in the OS-FINCH there are not any moving parts.The entire 3-D scanning is electronic rather than mechanical, and therefore the scanning may be potentially faster and more robust.where the phase-shifting parameter φ l accounts for both W 1 and W 2 and the parameter φ k to W 1 alone.Define φ k = θ k -φ l , Eq. ( 19) becomes similar to Eq. ( 14), and thus any desired cross term between the three interfering waves can be extracted, following Eqs.( 14)-( 16).This definition of φ k can be considered as a way to compensate for any value of φ l , effectively allowing separate phase-shifting control over W 1 and W 2 .An alternative procedure that does not use this technique of compensation is now presented.
Let θ l = φ l , X 1 = exp(iφ k )W 1 + W 2 and X 2 = W 3 .Equation ( 19) can be rewritten as: Following a procedure similar to the one described in Eqs. ( 10)-( 12), phase-shifting over three different values of θ l , it is easy to extract from Eq. ( 20) the cross term ( ) Performing the above for two different values of φ k = φ 1 , φ 2 , the following cross terms of Eq. ( 19) can be extracted: ( ) ( ) ( )( ) Alternatively, let θ k = φ k , X 1 = exp(iφ l )W 1 and X 2 = exp(iφ l )W 2 + W 3 .Equation ( 19) can be rewritten as: Implementing the above for two different values of φ l = φ 1 , φ 2 , the following cross terms of Eq. ( 19) can be extracted: Finally, consider a more general case, in which multiple waves, W 1 , W 2 .. W p , are interfering over the digital sensor plane, so that We now show that the ability of applying a controllable phase-shift to two of the waves, say W m and W n from W 1 , W 2 .. W p , can be used to extract a desired cross term, * m n W W , from Eq. ( 26), so that Let X 1 = W m and X 2 = W 1 + .. + W m-1 + W m+1 + .. + exp(iφ l )W n + .. + W p .Equation ( 27) can be rewritten in the form of Eq. ( 23).Phase-shifting over three different values of θ k , based on the procedure described in Eqs. ( 10)-( 12) it is easy to extract the term

Fig. 1 .
Fig. 1.Schematic of an optical sectioning FINCH setup: L o , objective lens; L c , converging lens; P 1 and P 2 , polarizers; SLM 1 and SLM 2 , spatial light modulators; a o , point source; a 1 and a 2 , point source images.

2 .
For simplicity, only three point sources are considered: a o , b o and c o .It is further assumed that the phase pinhole mask is centered along the image point a 1 .Let exp(iθ k )A 1 , exp(iθ k )B 1 and exp(iθ k )C 1 denote the spherical waves that diverge away from the image points a 1 , b 1 and c 1 , respectively, over the image sensor plane, where exp(iθ k ) denotes a phaseshifting constant, usually with θ k = 2π(k -1) / 3, for k = 1,2,3.Additionally, let A 2 , B 2 and C 2 denote the spherical waves that converge towards the image points a 2 , b 2 and c 2 , respectively, over the image sensor plane.A single recorded FINCH hologram, for the three point sources a o , b o and c o can be described as:

Fig. 2 .
Fig. 2. Principle of optical sectioning using a phase pinhole in FINCH: L eq , an equivalent lens, representing both the objective lens L o and the converging lens L c (Fig. 1); P 1 and P 2 , polarizers; SLM 1 and SLM 2 , spatial light modulators; a o , b o , and c o , point sources; a 1 , b 1 , c 1 , a 2 , b 2 , and c 2 , point source images.
a FINCH hologram recorded for a point source located at (x o ,y o ), a distance d o from the objective lens.Its reconstruction can be calculated as:

Fig. 3 .
Fig. 3. Masks of (a) a single pinhole and (b) a multiple phase pinholes array.

Fig. 4 .
Fig. 4. Wave interference in optical sectioning FINCH: L eq , an equivalent lens, representing both the objective lens L o and the converging lens L c (Fig. 1); P 1 and P 2 , polarizers; SLM 1 and SLM 2 , spatial light modulators; c o , a point source; c 1 and c 2 , point source images; C 1 , the wave that diverges from the image point c 1 over the image sensor plane, composed of parts that pass through the phase pinhole [S(C 1 )C 1 ] and parts that do not {[1 -S(C 1 )C 1 }; C 2 , the wave that converges towards the image point c 2 over the image sensor plane and is not modulated by SLM 2 ; C 3 , the wave that converges towards the image point c 2 over the image sensor plane and is modulated by SLM 2 , composed of parts that pass through the phase pinhole [S(C 3 )C 3 ] and parts that do not {[1 -S(C 3 )C 3 }.For simplicity, only a single phase pinhole is shown.

Fig. 6 .
Fig. 6.Reconstructions of non-sectioning FINCH holograms at plane of best focus for 1951 USAF chart [(a) and (c)] and NBS 1963A chart [(b) and (c)].(a) and (b) were reconstructed from the first hologram, recorded with the 1951 USAF chart imaged to the plane of SLM 2 .(c) was reconstructed from the second hologram, recorded with the NBS 1963A chart imaged to the plane of SLM 2 .

Fig. 7 .
Fig. 7. Reconstructions of optical sectioning FINCH holograms at plane of best focus for the 1951 USAF chart.In (a) a single phase pinhole was used, while in (b)-(l) multiple phase pinholes were used in parallel, where the spacing between adjacent pinholes (in x and y directions) was set to 88 × g μm, starting with g = 1 in (b) and ending with g = 15 in (l).Results for g = 11 .. 14 are not presented, due to lack of visible difference.

Fig. 8 .
Fig.8.Amplitude mean squared error (MSE) between the reconstruction in Fig.7(a), where scanning was performed using a single pinhole, and the reconstructions in Figs.7(b) through 7(l), where parallel mode scanning was performed using multiple phase pinholes, with g values of 1 through 15, respectively, indicating the gap between adjacent phase pinhole, in integer multiplies of the width of a single pinhole.The result for g value of 0 is against the reconstruction of Fig.6(a), from a non-sectioning FINCH hologram.
) and 11(b).Besides showing small fractions of the 1951 USAF resolution chart that fall within the phase pinholes, noise-like artifacts are also visible in the reconstructions.The higher the number of in-parallel phase pinholes (indicated by lower values of g), the more prominent are the artifacts.Using TWIPS, however, where each complex FINCH hologram requires a total of six different exposures, these artifacts are eliminated, as shown in Figs.11(c) and 11(d).

Fig. 9 .
Fig. 9. Reconstructions of optical sectioning FINCH holograms at plane of best focus for the NBS 1963A chart.In (a) a single phase pinhole was used, while in (b)-(l) multiple phase pinholes were used in parallel, where the spacing between adjacent pinholes (in x and y directions) was set to 88 × g μm, starting with g = 1 in (b) and ending with g = 15 in (l).Results for g = 11 .. 14 are not presented, due to lack of visible difference.

Fig. 10 .
Fig.10.Amplitude mean squared error (MSE) between the reconstruction in Fig.9(a), where scanning was performed using a single pinhole, and the reconstructions in Figs.9(b) through 9(l), where parallel mode scanning was performed using multiple phase pinholes, with g values of 1 through 15, respectively, indicating the gap between adjacent phase pinhole, in integer multiplies of the width of a single pinhole.The result for g value of 0 is against the reconstruction of Fig.6(c), from a non-sectioning FINCH hologram.

Fig. 11 .
Fig. 11.Regular phase-shifting vs. TWIPS in OS-FINCH.In (a) and (b) hologram reconstructions from complex FINCH holograms are shown, each calculated using a regular phase-shifting procedure, based on three different exposures.Each of the holograms was recorded for a single position of multiple phase pinholes, with the distance between adjacent phase pinholes set to 88 × g μm, with g = 15 (a) and g = 1 (b).(c) and (d) are the TWIPS equivalents of (a) and (b), respectively, where each complex FINCH hologram was calculated based on six different exposures.Using regular phase-shifting, it is evident that the larger the number of phase pinholes, comparing (a) and (b), the more prominent are the noise-like artifacts in the reconstructions.In contrast, these artifacts are not noticed in the TWIPS reconstructions of (c) and (d).
over three different values of θ k , it is easy to extract from Eq. (23) the cross term