Fresnel Incoherent Correlation Holography (FINCH) can faithfully reproduce objects above and below the optical plane of focus. However, as in optical imaging, the transverse magnification and optimal reconstruction depth changes based on the longitudinal distance of objects from the focal plane of the input lens with the exception that objects above and below the focal plane are in focus with FINCH and out of focus by standard optical imaging. We have analyzed these effects both theoretically and experimentally for two configurations of a FINCH fluorescence microscopy system. This information has been used to reconstruct a test planar object placed above or below the optical plane of focus with high dimensional and image fidelity. Because FINCH is inherently a super-resolving system, this advance makes it possible to create super-resolved 3D images from FINCH holograms.
©2012 Optical Society of America
The study of holography for use in collecting and manipulating three-dimensional (3D) optical information has increased rapidly in recent years as holography can gather and store large amounts of information faster than can be done with traditional two-dimensional imaging methods. An important property of holography is the collection of all of the three-dimensional optical data from objects in a 3D field with few or a single exposure, as opposed to the laborious standard optical methods of collecting stacks of 2D images as the sample or objective is translated with respect to the plane of focus of the measurement optics. In standard holography this is done by using the reflected or transmitted light from a coherent source propagating through the object and creating an interferogram with the unperturbed source, a process that generally requires a coherent source as well as the precise alignment of two laser beams. Fresnel Incoherent Correlation Holography (FINCH)  is a hybrid technique that has recently been introduced and which offers significant promise for recording three-dimensional information from incoherent sources such as fluorescent objects. In FINCH, the hologram is created from incoherent summation of interferences between two waves originating from each point source in an incoherently emitting object, using a single beam geometry. The key advantage of FINCH is that the object itself serves as the light source, enabling 3D imaging from any object, either by reflection or emission of light such as fluorescence. Thus a coherent laser light source and the associated precise alignment of a reference and sample beam concomitant with classical holography are not needed. The FINCH technique can be readily adapted to any standard optical imaging technique with relatively little alteration. FINCH has been theoretically investigated for its possible application to telescope optics [2,3] and more extensively studied in its application to microscopy [4–8]. While the initial report applying FINCH to microscopy demonstrated that the method could resolve a sample in three dimensions , subsequent studies concentrated on developing a theoretical understanding of the method and configurations to improve resolution on a single plane. We recently demonstrated that FINCH microscopy is inherently super-resolving in the lateral dimensions in comparison to conventional microscope imaging using the same optics . In other optical imaging experiments  a more efficient FINCH configuration was demonstrated, using two closely spaced focal length lens patterns displayed on a spatial light modulator (SLM).
Because FINCH is inherently a 3D method, there is the possibility of gathering the full 3D image of an object in a few exposures at higher resolution than is possible by standard optical methods. In order to fully exploit the abilities of FINCH microscopy, it is necessary to understand what the effects are of placement of the object at positions other than at the focal plane of the objective. To date, the analysis of FINCH has been done most thoroughly for objects at the focus of the microscope objective (see Fig. 1(a) ). The displacement of the object away from the focal plane of the objective, or an object with significant depth, will result in the alteration of the light cones used to create the FINCH hologram (see Fig. 1(b) and 1(c)), which will in turn affect the final reconstructed images. In this paper we theoretically and experimentally demonstrate the super-resolving performance of FINCH in three dimensions and investigate the resolving characteristics of FINCH imaging at planes above and below the objective lens plane of focus. In these studies we compare the two methods of FINCH fluorescence microscope imaging mentioned above, in which the emitted plane wave from an infinity corrected microscope objective (collimating lens) is separated with a phase only SLM into either a plane wave and one focused beam (Configuration 1) [1–7], or into two focused beams with different focal lengths (Configuration 2) . To study the performance of FINCH to resolve samples above and below the plane of focus, we positioned a single plane specimen at different positions above and below the focal plane of the objective. The analysis results in methodology to accurately generate 3D images of objects with full dimensional fidelity.
2. Theoretical analysis
Following the convention in , and referring to the generalized diagram in Fig. 2 , we consider a point source at the position before the input of the objective, which has the focal distance f0. After the light from the point source propagates and moves through the objective and to the SLM, it is focused by the SLM displaying the phase components of Fresnel “lens patterns” with focal lengths fd1 and fd2, termed the “signal” and “reference” focal lengths, respectively. After further propagation by the distance zh, it has the complex amplitude described by:Eq. (1) we do not explicitly include the vector notation necessary to account for the polarizers used for interferometric mixing of the spherical waves , or a term that bounds the physical dimensions of the recorded hologram by the limiting physical aperture in the system .
After squaring the complex amplitude magnitude and evaluating Eq. (1), we obtain the following general expression for the intensity of the recorded point source hologram at the camera plane, which is the point spread function (PSF) for hologram formation in FINCH:
2.1 Configuration 1 - FINCH with one plane wave and one spherical wave
To review the simplest case that has been the generally considered example, we consider a hologram formed from the interference of one spherical wave with one plane wave as in Fig. 1 (referring to Fig. 2, fd1 is responsible for the spherical wave and the plane wave in Fig. 1 is created when fd2 → ∞). In this case zr and the transverse magnification MT are given by:
and the ± signs in Eq. (3) indicate that the final images may be reconstructed from either the real or the virtual image, depending on which is isolated by the phase-shift procedure  described below.
As Eq. (2) is the PSF of the FINCH hologram formation process, the hologram Hi of a real object is obtained as an integral of IP(x,y) over all of the object space g(xs,ys,zs). Typically three holograms with values of θ equally spaced by 2π/3 are recorded and then superposed using the following equation to eliminate the bias term and either the real or virtual image from the reconstruction [1,10]:
2.2 Configuration 2 - FINCH with two spherical waves
Recently it has been suggested  and shown  that the use of two spherical waves with different curvatures can be used to create more efficient FINCH holograms. This configuration, shown in Fig. 2, can provide equivalent resolution in the hologram to the method with one plane and one spherical wave, and offers the possibility of moving the camera closer to the SLM for greater light efficiency without adversely affecting the resolution in the reconstructed image. In  it was shown that the two spherical waves were required to overlap perfectly at the hologram recording plane in order to maintain the maximum resolution obtained with the plane wave/spherical wave method, while in  various methods of mixing lens patterns on the SLM were attempted, with the conclusion that a random pixel distribution of each lens pattern was the best method for displaying two lens patterns on a single SLM. Earlier work considered only objects at the focus of the objective; however in order to study 3D objects it is necessary to understand the dependence of zr on zs, a dependence which is reported here in the second part of Eq. (7):
The MT of FINCH with two spherical waves is the same as that with one plane and one spherical wave, shown in Eq. (4), so that the effects of object depth must still be accounted for when reconstructing FINCH holograms taken with the double lens pattern method.
3. Experimental methods
The fluorescence microscope apparatus and FINCH holography setup used in these experiments has been previously described . Briefly, an upright microscope was modified so that in place of the tube lens there was a phase-only SLM (Holoeye 1080P, 1920 × 1080 pixels) centered over the objective and positioned at a 45° angle to the beam path from the output of the infinity objective. The SLM displays quadratic phase lens patterns of the form5], and has additional corrections applied due to the 45° incidence angle of the light from the objective. The sample was a USAF negative test pattern slide (Max Levy Autograph) displaying the smallest features of the test pattern (group 9 which had reduced visibility by standard microscopic imaging with the 20x 0.75 NA objective), mounted over a fluorescent plastic slide.
Two configurations of FINCH were used, including one utilizing polarization to obtain interference between a plane wave and a spherical wave (as in  and ) and one using two lens patterns to obtain interference between two spherical waves (as in ). As shown in Fig. 3(a) , for Configuration 1, with one plane wave and one spherical wave, light propagating from the objective passed through a polarizer with t-axis 45° to the SLM polarization axis, then onto the SLM, which passed half the light unchanged and which focused half the light by the focal length fd1. The light then propagated through another polarizer at the same angle as the first, allowing the projections of both the focused and unfocused light along the polarizer axis to interfere with each other and create a hologram. For Configuration 2, with two lens patterns distributed randomly over the SLM, only one polarizer was used as shown in Fig. 3(b), which was placed before the SLM and aligned to be polarized parallel to the SLM. The two lens patterns were selected to provide matching sizes of the unfocused light cones at the camera plane, with fd1<zh and fd2>zh. In both configurations, the hologram was recorded at the distance zh from the SLM by a CCD camera (QImaging Retiga 4000R, cooled, 2048 × 2048 pixels).
Three holograms with phase factors θ differing by 2π/3 were recorded and superposed to create HF, the bias-eliminated complex-valued hologram.
The sequence of holograms taken with Configuration 1 were to confirm the relationships of zrec and MT to distance away from the focus of the objective, given static values for all other parameters. A Nikon Plan Apo 20x, 0.75 NA (air) infinity-corrected objective was used for these experiments. Similar results (not shown) were obtained using other lenses including an Olympus UApo/340 20x 0.75 NA (air) infinity-corrected objective, and an achromatic doublet lens with 30 mm focal length (Thorlabs part #AC254-030-A-ML). The excitation filter and dichroic mirror in the microscope were Semrock Cy3, and the emission filter was a Thorlabs interference filter passing 570 5 nm light. The SLM displayed a “lens pattern” calculated for a 570 nm central wavelength and a 767 mm fd1, with the CCD positioned at a zh of 1380 mm from the SLM. This proportion of zh:fd1 is in the range reported in  resulting in the best focused reconstructed images for this SLM with these experimental parameters. The axial position zs of the object, the single plane USAF test pattern, was adjusted over a range of 30 microns above to 30 microns below the focal plane of the objective. FINCH holograms were recorded for each position without changing any of the SLM or camera parameters. For each zs position of the object, the complex-valued hologram was processed by the Fresnel propagation method using Eq. (6) over a range of zrec values, and the best focused reconstructed image for each hologram was selected. Linear FINCH reconstructions were used . The zrec values of the best focused image from each hologram were recorded, as were the widths (in pixels) of the vertical and horizontal features in Element 1 of Group 8 for determination of the lateral magnification.
A similar series of experiments was done with Configuration 2, with several differences. The lens patterns were calculated for a central wavelength of 520 nm, the excitation filter and dichroic mirror were Semrock GFP, the emission filter was a Thorlabs interference filter with a central wavelength of 520 5 nm, and the zh distance was 600 mm. Experiments were done only with the Nikon 20x objective, and holograms were taken for zs values between 10 microns above and 10 microns below the focal plane of the objective. Four pairs of lens patterns with differing focal length spacing factors between the two lens patterns sfac = 0.05, 0.10, 0.15, 0.20 were used, where
4. Results and discussion
4.1 Configuration 1
A selection of the best reconstructed images from the holograms taken with the Nikon objective is shown in Fig. 4 . Reconstructed images from holograms taken with the other objective lenses were of similar quality and are not shown here. As the object was brought down from zs-f0 = −30 µm (above the focal plane of the objective), the reconstructed images were increasingly magnified, a process that continued through the focus of the objective (zs-f0 = 0 µm) and until the sample was brought to below the focal plane of the objective to zs-f0 = 30 µm. For holograms taken with zs-f0 > 15 µm there was significant degradation of the reconstructed images; however for holograms taken over the range of zs-f0 = −30 to 15 µm the reconstructed images are all relatively equal in quality though they differ significantly inmagnification. The visibilities of the smallest horizontal features in the USAF pattern for standard fluorescence widefield microscopic images and images reconstructed from holograms were measured by the previously reported method . The visibilities for the range of zs –f0 = −30 to 15 µm for the reconstructed images from holograms were found to be between 0.53 and 0.67 in the linear reconstructions compared to 0.4 when the microscope was used as a traditional microscope with the properly matched and configured tube lens (see Fig. 5(c) ). Since the gaps and features of the smallest USAF objects measured (775 nm) are both on the order of the Abbe diffraction limit for this objective (380 nm), the width of the PSF has a strong effect on visibility and therefore the visibility of these features serves as a good measure of the resolving power of the system. Thus the FINCH method maintains an advantage over widefield fluorescence microscopy even for images taken at significant distances away from the focus of the objective lens.
The experimental zr and MT data are plotted in Fig. 5 alongside the theoretical values for the optimal zr and MT that were calculated using Eqs. (3) and (4). For convenience we identifythe zr and MT at zs-f0 = 0 µm as zr0 and MT0. For the calculated values, the Nikon compound objective lens was treated as a thin lens with f0 = ftube/Mag, where ftube was the focal length of the matching tube lens (200 mm) and Mag is the rated magnification factor of the objective lens when used in a standard microscope configuration. The remainder of the length of the objective was added to d, the distance between the collimating lens and the SLM. Thus for both configurations, using the Nikon 20x objective, d = 320 mm and f0 = 10 mm. Since zr as a function of zs-f0 reaches a minimum and then begins to increase again, in order to obtain reconstructions which faithfully represent a 3D object, it is important that the object be fully on one side or the other of the zr minimum. If the object spans the zr minimum in both directions, object points from planes above and below the plane of minimum zr will reconstruct in the same image plane, making it difficult to determine whether reconstructed image points represent objects from above or below the point of minimum zr. In practice this problem can be avoided by obtaining holograms when objects are only above or below the minimum zr, which is usually near the optical plane of focus.
Another important feature of FINCH can be seen in Fig. 4; the size of the reconstructed image changes significantly when the object is moved over a range of zs that could easily correspond to a “real-world” application. For such “real-world” applications, though, it would be preferable if the MT of each reconstructed image was the same and the image space was telecentric. It has been shown  that telecentricity in FINCH images can be obtained by setting d = f0. However, with most high NA compound microscope objectives this is impossible, as f0 is usually significantly smaller than the physical length of the objective. Thus we resample all the images to match the size of the image in the stack that has the highest magnification. This choice of reference image ensures that no data is lost during resampling. Shown in Fig. 6 are the top and side views of 3D projections of stacks of the USAF pattern taken at positions ranging from 30 microns above objective focus to the objective focus. One set is presented without processing while the other has been resampled to ensure that the resampled images were aligned properly in reconstruction space as the object was in object space during hologram recording. The corrected image shown in the right panel of Fig. 6 presents a view of the 3D object volume that does not display increasing size going from top to bottom, which is present in the uncorrected image and is characteristic of the perspectiveeffect on the FINCH process resulting from lack of telecentricity when using high power, high NA objective lenses.
4.2 Configuration 2
Four sets of lens patterns were used to record FINCH holograms of the USAF pattern using the Nikon 20x objective, in steps of two microns over a distance of 10 microns above objective focus to 10 microns below. The camera distance zh was set at 600 mm in order to reduce the optical path difference (OPD) of this configuration . Selected images at focus and 10 microns above and below focus are shown in Fig. 7 .
The zrec values and MT were recorded for each reconstructed image and are shown in Fig. 8 plotted against theoretical curves of zr and MT calculated from the parameters used in the hologram recording. When more than one reconstructed image was equivalent in quality, the zrec of each equivalent image was recorded. It is apparent from these experiments that the relative reduction in image quality, as measured by the visibility of the smallest features  shown in Fig. 8(c), when moving above or below the objective focus is more pronounced for the smaller sfac, while the image background and noise that are visible in the images are greater for larger sfac. The latter observation may be understood as being due to the larger OPD between the signal and reference beams, but the reason for the former observation is less obvious. However, we may consider NAH, the NA of the hologram of an infinitesimal point, as a limiting factor on the quality of the reconstructed images. As previously shown [6,9], the minimum resolved object feature size in FINCH is inversely proportional to NAH × MT. The visibility, then, is proportional to this factor. To calculate NAH for objects away from the objective focal plane, we note that at the detector plane, the radius of the nth spherical wavecoming from the objective through the SLM is given by:Figure 9 shows plots of calculated NAH × MT for both Configuration 1 and Configuration 2, for f0 = 10 mm, d = 320 mm, and zh = 600 mm, including the values of sfac studied in this research. The maximum NAH × MT is the same for both configurations, while the NAH × MT curve as a function of zs-f0 falls off more sharply as the sfac becomes smaller. This shows that in addition to the zr and MT effects mentioned in the previous section, the choice of configuration and sfac will alter the level of background and noise as well as on the depth of field in which well-resolved images are obtained.
We have analyzed the dependence of reconstruction depth and transverse magnification on displacement of the object away from the focus of the objective lens in FINCH microscopy using two different configurations, one with one spherical and one plane wave and another with two spherical waves to form the interference pattern. We have also analyzed these dependencies and shown their implication for successful measurement of the 3D light field emitted by the object and its formation into accurate 3D images:
- 1. The light used to form the hologram must originate entirely from a collection of points above or below the plane defined by the minimum of the zr curve. This plane is generally at or near the focus of the objective.
- 2. The reconstructed images in a stack must be resampled according to the largest magnification in the stack in order to avoid distortions in 3D rendering resulting from significant changes in MT with change in object location.
- 3. Choice of FINCH configuration lens pattern focal length spacing, sfac has significant effects upon reconstructed image background noise, resolution and on depth of the FINCH field, which must be accounted for when designing a FINCH system.
- 4. In-focus images with resolution beyond optical limits can be obtained by FINCH for objects not only at the focal plane of the microscope objective, as previously demonstrated with one of the FINCH configurations , but now is shown for objects above and below the focal plane of a microscope objective. This conclusion is based on data in Figs. 5(c) and 8(c) for both FINCH configurations in which one can see that the visibility, and therefore the resolution, of objects near diffraction limited size is not only better than a standard widefield microscope at the plane of focus but also along a range of ± 5-30 µm from the objective focal plane.
This work was supported by NIST ARRA Award No. 60NANB10D008 to GB, The Israel Ministry of Science and Technology (MOST) to JR and by CellOptic, Inc.
References and links
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