Abstract

An exceptionally simple and versatile advance in super-resolution microscopy has been created by adding a new birefringent FINCH holographic lens system including an inexpensive uncooled CMOS camera to a standard microscope. Resolution, after only a single image capture, is equivalent to or better than other more complex popular methods such as SIM, Airyscan and a number of image scanning microscopy methods that boost resolution about two-fold. This new FINCH implementation uniquely works for any objective power and NA and is solid state, fast, and calibration-free. In addition to being as easy to operate and maintain as a standard fluorescence microscope, it can uniquely create super-resolved images with any type or wavelength of light including fluorescence, bioluminescence or reflected light because its principle depends only on emitted light from objects and requires no prior training or knowledge about the sample being imaged. This microscope technique increases the utility and availability of super-resolution microscopy for any user in any research lab.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Optical imaging is in general a good choice for viewing many diverse samples as it is relatively fast, inexpensive, simple, versatile, nondestructive, and intuitively understood. Classical optical imaging limits resolution to between 200-250 nm for high-performance microscopes, as formally defined by the Abbe criterion of 0.5*λ/NA for the relationship between the wavelength of image light and the resolving power of the microscope objective. The field of super-resolution optical imaging [1], which overcomes this classical limit, has expanded dramatically in recent years as researchers have pushed the bounds of optical imaging to visualize ever smaller structures, both biological and non-biological. There are several techniques that offer a resolution improvement factor of ∼2x, including SIM [2,3], Airyscan [4], and image scanning (ISM) methods [59] including OPRA [10] and iSIM [1113] and closely related MSIM [14,15]. Many of these methods have been commercialized, but require complex, specialized, and expensive hardware, and require skilled operation and frequent maintenance. SIM requires multiple exposures and extensive computation in order to achieve its peak performance. A super-resolution technique that can avoid these and other disadvantages could expand the availability of super-resolution imaging.

FINCH (Fresnel Incoherent Correlation Holography) and its confocal variant CINCH (Confocal Incoherent Correlation Holography) also offer super-resolution in this resolution range based upon a quite simple holographic imaging principle. Since our laboratory’s invention of FINCH more than a decade ago [1618], the current authors have refined and developed FINCH into a simple, apochromatic, all solid state super-resolution method [1926] that works with any objective magnification and NA, and that now with this report only requires a birefringent crystal lens (BRL), a waveplate, and an inexpensive camera for calibration free single image capture super-resolution imaging. Conventional holographic image recording uses sample and reference beams and coherent laser illumination to create interference patterns that encode three-dimensional information about the object with standard resolution at best. In contrast to this, FINCH uses any incoherent light either emitted or reflected from an object, such as fluorescence or luminescence, to create interference patterns that encode super-resolved information about the sample. FINCH is not the first method for incoherent fluorescence holography, as the earliest such report was made by Poon in his scanning holography work in 1995 [27], and there are other subsequent holographic techniques which only use light from the object [2838]. However, these methods are not necessarily super-resolving and may be slow in practice due to a need for extensive averaging [3537] or long acquisition times [38]. To date only FINCH advances from our laboratory have been used to create high quality, high magnification biological super-resolved holographic images [26]. In comparison to other super-resolution methods, FINCH has an additional advantage in that it derives its super-resolving characteristics exclusively from the light emitted by the object, overcomes the 200 year old Lagrange invariant of optics [39], thus requiring no strict illumination protocol or even illumination through a filtering pinhole. Nearly all other methods are limited to fluorescence, and while ISM methods can potentially function with reflected light, the illumination light must still be passed through confocal pinhole(s), which must in some way be optically scanned over the sample. By capturing a hologram with axial information about the object, FINCH also enables the reconstruction of a three-dimensional representation of the object that can be 3D deconvolved for further image improvement or to obtain axial information. FINCH can work with all types of light, including fluorescence, reflection, and even chemiluminescence or bioluminescence. Reported here is a single exposure super-resolution approach only requiring a simple birefringent lens and an inexpensive camera.

2. Background

FINCH [1626,4042] is an imaging technique that can obtain 3D and super-resolved information. Referring to Fig. 1(a), classical optical imaging simply requires a lens to receive light from an object and focus that light into an image that is recorded by a camera, magnifying each point in the object by the blurring function known as the point-spread function (PSF). Considering a single infinitesimal point as shown in Fig. 1(a), a given conical angle of the light emitted from the object point, with a finite circular projection on the plane of the lens, is accepted by the lens. The accepted light beam, with radius Rbeam, along with the distance di to the image plane, may be thought of as a lens with an effective numerical aperture NAbeam equal to Rbeam/di, and the image spot size of the object point is inversely proportional to this effective NA. As discussed extensively in previous work [1626,4042] and briefly recounted here, FINCH functions by splitting the light received from the object under imaging into two co-propagating portions with different focal lengths that interfere with each other and create a hologram of the object (Fig. 1(b)). The different focal lengths (f1 and f2) are created by a specialized FINCH birefringent lens focusing optic (described later), and the hologram is recorded on a digital camera. Since the recorded holograms are incoherent intensity patterns, they do not reveal the phase of the object, which is critically needed to reconstruct images at high quality. The phase shifting method [43] of processing several different phases of the recorded hologram has been typically applied to many forms of holography to obtain the complex phase of the hologram. The general method to obtain the complex object field in FINCH holography is to digitally capture and combine several (n≥3) phase shifted raw holograms Hqn of the form:

$$\begin{array}{l} {H_{\theta n}}({{x_h},{y_h};{z_h}} )= \\ P({{x_h},{y_h};{R_H}} )\left\{ {\begin{array}{c} {2 + exp\left\{ {\frac{{i\pi }}{{\lambda {z_r}}}[{{{({{x_h} - {M_T}{x_s}} )}^2} + {{({{y_h} - {M_T}{y_s}} )}^2} + i{\theta_n}} ]} \right\}}\\ { + exp\left\{ {\frac{{ - i\pi }}{{\lambda {z_r}}}[{{{({{x_h} - {M_T}{x_s}} )}^2} + {{({{y_h} - {M_T}{y_s}} )}^2} - i{\theta_n}} ]} \right\}} \end{array}} \right\}, \end{array}$$
wherein λ is the center wavelength of the light band, the subscript h refers to the recording plane and subscript s to the sample plane, MT is the transverse magnification, zr is the reconstruction distance of the hologram and x, y and z are cartesian coordinates, P(xh,yh;RH) is a system-dependent pupil function based on the width RH of the recorded raw hologram at the camera plane, and θn is a global phase shift applied to the hologram. The n recorded holograms are then composed into a complex-valued hologram HF by superimposing them along with global phase factors using the following Eq. (2):
$${H_F}({{x_h},{y_h};{z_h}} )= \mathop \sum \nolimits_{a = 1}^n {H_{\theta a}}[{exp({ \pm i{\theta_{a - 1}} -{\pm} i{\theta_{a + 1}}} )} ],$$
wherein a represents the index of the phase shift in the group of raw holograms, θa-1 = θn for a = 1, and θa+1= θ1 for a = n. For example, for a group of 4 holograms, a = 1, 2, 3, 4, with phase shifts 0, 0.5π, π, and 1.5π. The superposition process serves to remove the holographic twin image and bias term as well as constant background from the complex hologram, resulting in a representation of the actual phase of the object being imaged.

 figure: Fig. 1.

Fig. 1. Principle of increased image resolution by birefringent lens FINCH imaging for single point (a-b, Abbe criteria) and two-point (c-d, Rayleigh criteria) imaging. (a) A generalized schematic of classical optical imaging, depicting the effective numerical aperture (NA) of the image beam NAbeam as a function of the image beam size Rbeam at the lens and the image distance di, and the size of the classical image spot Δi. (b) A generalized schematic of FINCH imaging, showing the effective NA of the image beam NAH increasing by a factor of 2 and the FINCH image spot size decreasing by a factor of 2 to ΔH . The yellow and green lenses and dashed lines represent the two focal lengths f1 and f2 imparted by the FINCH optics onto the received light from the object. The recorded hologram at the plane di is computationally reconstructed into an image. (c) Two points imaged by classical methods cannot be distinguished from one another if they are too close. The transverse magnification (MT, magnification of the image field) and the angular magnification (MA, magnification of the image spot) are equal to each other. (d) The same two points can be distinguished using FINCH imaging because there is less magnification of each spot, though the field size is magnified to the same extent as a standard optical imager focused to this plane such that the image spots do not change from their original position.

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Following the recovery of the complex hologram by Eq. (2), a simple Fresnel propagation calculation [16,21,22] is sufficient to create a reconstructed focused image Irec of the object as described generally by Eq. (3):

$${I_{rec}}({x,y,{z_r}} )= {H_F}({x,y;{z_h}} ){\ast }{A_{IRF}}({x,y,{z_r}} )exp\left[ {\frac{{i\pi }}{{\lambda {z_r}}}({{x^2} + {y^2}} )} \right]$$
wherein the asterisk represents two-dimensional convolution, AIRF is an amplitude function of an impulse response function for light propagation through space, and the exponential term is the phase function of the impulse response function for propagation through space by the distance of zr, the reconstruction distance of the hologram [22].

Referring back to Fig. 1(b), the number and size of the fringes in the holograms code for the depth of the object point being reconstructed, conferring on FINCH the ability to encode three-dimensional (3D) information. The self-interference of the light confers a super-resolution factor of up to twice the normal optical limit at any given wavelength by incorporating sample information in both light beams that interfere. That is, there are two signal beams rather than the individual signal and reference beams that are characteristic of classical holography. The largest increase in resolution occurs at the plane in which the two differentially focused beams have the same diameter. If the system is arranged so that this plane is at the distance di from the specialized FINCH optic, it has been shown [21] that the resultant hologram has an effective NA (NAH) of its own, which is equal to 2NAbeam, the condition shown in Fig. 1(b). Upon the reconstruction calculation, the final spot size of the FINCH image is thus inversely proportional to twice the NAbeam, leaving it as half the width of the classically imaged spot. Since the lateral magnification of FINCH is the same at any distance away from the lens as in classical imaging, this results in the lateral resolution of FINCH being twice that of classical imaging in this optimal arrangement. This is further illustrated in Fig. 1(c) and (d), showing that FINCH does not adhere to the classical Lagrange invariant that describes lateral magnification in classical optical systems. FINCH violates the classical Lagrange invariant and thus improves both single and two-point lateral resolution [39,44].

Herein we report new improvements and further simplifications to FINCH microscopy that create super-resolved images of biological samples from a single exposure that includes four simultaneously recorded phase shifted holograms. The improved FINCH method requires no calibration or previous knowledge of the sample, works accurately at any wavelength and bandwidth of light without any calibration and does so without any moving or electrical parts. To our knowledge it is the simplest and most versatile super-resolution system ever developed.

In previous implementations of FINCH, the set of phase-shifted raw holograms has generally been recorded sequentially by applying a phase shift with either a liquid crystal based spatial light modulator or a liquid crystal electrically variable waveplate. This has presented two challenges, as also observed in other sequential phase-shifting interferometric methods [4548]. First, the sequentially recorded holograms may suffer from discrepancies due to shot to shot variation in the illumination intensity, potential sample motion, inter-shot photobleaching, and reproducibility error of the phase shifting optic. The occurrence of any of these can significantly degrade the quality of the final hologram image. Second, the phase shifting liquid crystal device can only be calibrated for a single wavelength at a time. In a fluorescence microscopy system with 40-50 nm bandwidth per dye, the fall-off in calibration accuracy of the phase shift for light in the fluorescence band away from the calibrated wavelength is sufficient to reduce the performance of FINCH below its full potential. While different calibration curves can be created for different light bands (usually for the center of the band), the applied phase shift will still have some chromatic error from one end of the light band to the other, which can cause subtle changes in the quality of the resultant images [41].

For the phase shifting optic of previous implementations, we here substitute a recording scheme using a broadband quarter-wave plate (QWP) and a polarized CMOS sensor to record the holograms, as shown in Fig. 2. It is known [49] that for orthogonally circularly polarized light, the phase of the interference between the beams is controlled by the relative angle of the polarizer that projects the two beams back along a common polarization axis. In the new Birefringent Lens FINCH system depicted schematically in Fig. 2(a), linearly polarized light from the sample object is received by a BRL with two polarization-dependent focal lengths aligned at ±45° to the polarization of the incoming light. The BRL thus creates a pair of differentially focused co-propagating linearly orthogonally polarized light beams from each received light beam [26]. A QWP positioned after the birefringent lens turns the co-propagating linearly orthogonally polarized beams into circularly orthogonally polarized beams, and a CMOS sensor overlaid with a micro-polarizer grid with four different polarization orientations simultaneously records four precisely phase shifted holograms in quadrature. As shown in Fig. 2(c), the simultaneously recorded raw holograms are de-interspersed into four raw holograms, each containing pixels of only a single phase shift value as in Eq. (1). The raw holograms are then interpolated by nearest neighbor or cubic interpolation to fill in the blank pixels in each. The recorded holograms are then processed by the super-position Eq. (2) above and reconstructed by known methods [16,21,22]. As can be seen in Fig. 2, this results in a FINCH system that is extremely simple and compact, with only three required components. The system is additionally free of the need for sequential exposure due to the multiplexed phase recording, and free of the need for chromatic calibration for phase shifting as well due to the broadband design of the QWP we used, which shifts all wavelengths within any image light band by the quarter wave necessary for phase shifting in formation of the FINCH interference pattern. The holograms created at any wavelength or bandwidth are thus as optically perfect and achromatic as they can be.

 figure: Fig. 2.

Fig. 2. Single Shot Birefringent Lens FINCH Imaging Principle. All necessary optical components for single-shot FINCH are shown in this schematic. (a) Polarized image light originating from the object is provided to the birefringent lens (BRL). The BRL, which is of high optical quality, splits the incoming image beam into a linearly orthogonally polarized pair of co-propagating beams with differing focal lengths. The beams are converted to orthogonal circular polarizations by the quarter wave plate (QWP), and subsequently interfere with one another to create a hologram that is recorded by a camera with multiplexed micro-polarizers. Four interspersed phase-shifted holograms are recorded simultaneously. (b) A schematic of the camera pixels overlaid with the interspersed micro-polarizer array. The micro-polarizer grid has four equally spaced polarizer orientations (0, 0.5π, π, and 1.5π radians) precisely arranged on the CMOS die in a repeating square pattern. (c) The recorded hologram (i) is de-interspersed by a computer in one step into four subsampled phases, then interpolated by nearest neighbor interpolation (ii), followed by super-positioning to create the complex hologram (iii) before being propagated to create the reconstructed image (iv).

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3. Methods

We created a new FINCH microscope system called CINCHSCOPE (http://celloptic.com/cinchscope-features/) incorporating this single-shot hologram recording technique rather than sequential recording, substantially similar to the system published in [26] but with the optional addition of a spinning Nipkow disk assembly (CellOptic, Inc. custom, 50 micron pinholes, 40 mm diameter to minimize vibration) in a conjugate image plane as in [24,25]. Briefly, referring to the microscope in Fig. 3 and omitting mirrors and non-active elements, a microscope objective lens (L0) is the first lens in the FINCH microscope. The microscope was designed for 200 mm tube length infinity corrected objectives but designs for infinity corrected objectives with other tube lengths are possible. A 4F relay (L1 and L2) consisting of apochromatic tube lenses (for the systems discussed herein, 165 and 100 mm focal length tube lenses (Thorlabs)), is placed to transfer the image of the back pupil of the objective onto the birefringent lens (BRL) (CellOptic, Inc. custom a-BBO lens, plano concave with -457 mm radius of curvature [26]) and 200 mm tube lens combination that produces the differentially focused beams necessary to achieve FINCH interference. These beams have focal lengths of approximately 263 mm and 284 mm, and the hologram is recorded at a distance zh of 273.73 mm. The magnification of the FINCH image is equal to zh*fL1/(fL2*fL0), where fLx is the focal length of a given lens Ln. The reported configuration’s magnification is equivalent to 2.26*M0, where M0 is the magnification of the 200 mm tube length equivalent objective lens. As in any modern microscope with infinity corrected optics, the space between L2 and the BRL is infinity space for insertion of standard microscope dichroic beam splitting filters (D) and any other optic such as emission filters and polarization optics. Prior to the BRL is a linearly polarizing beam splitter (PBS) that transmits light polarized midway between the BRL birefringent axes. The rejected polarization from the PBS is directed to a standard 200 mm tube lens (L3) and focused onto a CMOS camera (PCO Panda) to capture a widefield or confocal image for the purpose of comparison to the FINCH or CINCH image respectively. The magnification of this widefield or confocal image is equal to fL3*fL1/(fL2*fL0), or 1.65*M0 for the reported configuration. This comparison widefield or confocal image is not a strict requirement of FINCH imaging and is used here solely as a benchmark comparison so that an identical widefield or confocal image can be compared to the FINCH or CINCH image. Note that if the light emitted from the object was polarized (ie, if the fluorescent sample was illuminated with polarized light so that a fluorescence polarization technique is enabled), the PBS would not be necessary at all and the comparison standard optical train could be omitted.

 figure: Fig. 3.

Fig. 3. A schematic of the FINCH/CINCH microscope. LO: objective lens; L1, L2: relay lenses; D: dichroic mirror to isolate fluorescence excitation from emission light; PBS: polarizing beam splitter; P, S transmitted and reflected linear polarizations; BRL: birefringent lens; QWP: quarter-wave plate; L3: standard microscope tube lens. All of the components critical to the FINCH technique are contained in the shaded area. All other components are extrinsic to the FINCH technique and are present only to serve as the basic microscope optics upon which the FINCH system optics is based to create the super-resolved image. The FINCH system optics could be added to any microscope optical system. Insertion of the Spinning pinhole disk between the L1, L2 relay lenses converts the system into a confocal FINCH system in which a single image plane is imaged by the FINCH optics (CINCH). With the spinning pinhole disk in place the Standard camera records a confocal image.

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Returning to the FINCH optical path, positioned after the birefringent lens (BRL) is a broadband QWP (Thorlabs) with its slow axis aligned midway between the birefringence axes of the BRL. As noted above, the QWP converts the orthogonally linearly polarized light beams that it receives into orthogonally circularly polarized light beams, still with differing focal lengths, that are caused to interfere at the uncooled micro-polarizer camera (Sony IMX250MZR CMOS image sensor). The different polarization orientations of the different sets of camera pixels result in the simultaneous capture of four different phase factors of the interference, which are then combined as in Eq. (2) into a complex hologram, from which reconstructed images are calculated.

The broadband QWP, coupled with the polarization camera, serves as a stable, calibration-free, and achromatic phase shifting element that alleviates the necessity of using an SLM or variable wave plate as in earlier FINCH systems and eliminates the need for sequential captures of individual hologram phases. This removes the above-mentioned challenges of calibration over the standard biomedical microscope optical bandwidth and also overcomes temperature and electrical instability issues of liquid crystal devices previously used to effectuate phase changes during the 3 or 4 image captures previously necessary to produce a single complex hologram. In order to obtain super-resolved images of a single object plane, FINCH can be simply combined with confocal microscopy by the addition of a spinning Nipkow disk at the relay conjugate image plane, capitalizing on the improved axial resolution of confocal microscopy [24,25]. The spinning disk may be inserted into or removed from the conjugate plane by a linear piezo stage (Thorlabs) as desired to change the operating mode of the FINCH microscope to a confocal FINCH (CINCH) system. Finally, fluorescence excitation was provided by a light source with 7 excitation wavelengths that provides scrambled depolarized illumination (89 North LDI), with additional speckle reduction (Optotune). A quad-band dichroic filter set was used with single-band excitation and emission filters from Chroma and Semrock for GFP and Cy3 excitation/emission bands. All CINCHSCOPE functions including sample and objective scanning, excitation and image recording timing, and an emission filter wheel and confocal disk (CellOptic) were controlled with software written in LabView (GUI, microscope functions and calculations) and Arduino (triggering, disk and emission filter wheel control). The software also handles all image computation, with integrated deconvolution using Microvolution software (Microvolution, Cupertino, CA).

To capture a FINCH or CINCH image, the operating procedure of the microscope is identical to that of a standard widefield or spinning disk confocal microscope. The sample is brought into focus with a custom low mass combination mechanical and piezo xyz stage, the exposure time is metered to maximize the signal-to-noise ratio while avoiding saturation, and a single exposure is captured. If desired a montage of a large area of the sample can be automatically created by scanning with a low power objective before imaging any specific area with a higher power objective. Capture times range from milliseconds to seconds depending upon the fluorescence brightness of the sample and excitation power as is the case for other microscopes and super-resolution systems. Following a single image capture, the recorded hologram is converted to a complex hologram as in Eq. (2) above and related discussion, and reconstructed by Fresnel Propagation [16,21,22] either directly to the focal plane of the objective, or alternately into a calculated stack of reconstructed images representing a through-focus series of planes. The reconstructed focal plane image can also be displayed live during image capture, if desired. The absolute value of the reconstructed image was used for further analysis. Deconvolution of the reconstructed image or image stack was performed in Microvolution software using a custom FINCH PSF and a blind Richardson-Lucy deconvolution with entropy regularization. In order to avoid over-iterating the deconvolution, iterations were automatically stopped by the Microvolution program when an objective error gradient was reached in successive iterations.

4. Results

To demonstrate the capabilities of the new Single Shot Birefringent Lens FINCH system, a sample of fluorescent beads of 100 nm nominal diameter (Tetraspeck, Invitrogen T14792, actually 110 nm diameter) was imaged with the newly constructed holographic microscope in the absence of the confocal disk. A 60x objective (Nikon CFI Apo TIRF 60XC Oil objective, 1.49 NA, used in standard imaging configuration) was used along with Cy3 excitation (555 nm) and emission (590 ± 20 nm) channels. A single raw hologram image frame was recorded that contained four different quadrature phase factors of the hologram. These phase factors were abstracted into independent image arrays, with the blank pixels in each array filled in by cubic interpolation. A stack of 11 reconstructed images equally spaced from 0.5 microns above to 0.5 microns below the objective plane of focus was generated from the single shot hologram taken with the bead sample in focus as in previously published methods [16,21,22]. The reconstructed image stack was then deconvolved using Microvolution software as described above. Similar deconvolution results (not shown) were also obtained using a recently reported method with 1-2 iterations of a Wiener-Butterworth deconvolution algorithm that was created for iSIM [13]. For comparison to the FINCH results, a through-focus stack of 17 standard widefield images, separated by 100 nm in axial distance by physically moving the objective, was also captured in the same imaging session by the widefield camera and was similarly deconvolved. Prior to deconvolution, the widefield image of the beads was resampled by a factor of two, using nearest neighbor interpolation, in order to assure that the small pixel size and nearest neighbor interpolation of the FINCH images was not responsible for the improved resolution therein. The object pixel size of the images is 33.8 nm for the widefield, and 25.5 nm for the FINCH or CINCH images, much smaller than the widefield and FINCH Nyquist limits of ∼100 and ∼50 nm for this optical system.

The resultant deconvolved images, both widefield and single-shot FINCH for the same area of the image, are shown in Fig. 4(a) and (b), each of which also features a magnified section in an inset to illustrate the resolution improvement by FINCH. The FINCH image clearly demonstrates better resolution than the widefield, with several features that are not resolved in the widefield image, now resolved into individual beads in the FINCH image as seen clearly in the insets and as also pointed out by the green arrows in Fig. 4(a) and (b) and line profiles and expanded images of those beads in Fig. 4(c). Figure 4(c) is a plot showing a normalized intensity profile taken through the feature in the images 4(a) and (b) pointed out by a green arrow and circle. This feature appears as a single object in the widefield image with a corresponding intensity profile containing a single peak of approximately 400 nm width. To satisfy the Rayleigh criterion for two-point resolution, the normalized intensity profile through the peaks of the two points should display an intensity minimum of 0.74 (or less) between the peaks. The normalized intensity profile of the pair of beads in the FINCH image shows an intensity minimum of 0.5 between the peaks, significantly better than the Rayleigh criterion of 0.74. Thus, the FINCH image here demonstrates that FINCH does not simply narrow the PSF of isolated features but enables the discrimination of closely spaced fine features that widefield imaging cannot resolve.

 figure: Fig. 4.

Fig. 4. FINCH lateral resolution of 100 nm sub-resolution fluorescent beads imaged with a 60X 1.49 NA Nikon objective at 590 nm emission wavelength significantly exceeds the widefield resolution of corresponding beads. Widefield (a) and FINCH (b) images of the identical image field of 100 nm (nominal) beads, with insets showing a magnified detail. (c) Plot of two closely spaced beads showing greatly improved two-point resolution with FINCH compared with widefield. The green arrows in (a,b) point out the bead pair shown in the profiles in this plot. (d) Plot of average bead lateral FWHM made from randomly selected beads. (e) Plot of average bead axial FWHM measured by physically stepping the sample through the objective focus over the indicated range in 100 nm steps.

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A set of >10 randomly selected beads was selected for further analysis from the FINCH image and analyzed by the methods previously reported [26]. The same set of beads that was analyzed in the FINCH image was identified in the widefield images, and was also measured similarly to the measurements in [26], using the MetroloJ [50] plugin for ImageJ [51]. The resulting average bead widths measured at 590 nm, shown in Fig. 4(d) were 132 nm ±23 nm for the FINCH image, and 228 ± 21 nm for the widefield image. To set these results in the context of other super-resolution methods, we compare them in Table 1 to 100 nm beads imaged at 520 nm with ca 145 nm FWHM values (after deconvolution) obtained for the beads imaged with ISM [5], iSIM [11], and MSIM [14], the 120 nm value reported for Airyscan [52], and the value of 104 nm reported for SIM [3], other methods that also promise factor-of-two super-resolution. Note that the FINCH lateral FWHM measurements measured at emission wavelength of 590 nm were actually superior to those of the iSIM/MSIM measurements which were taken at a shorter wavelength (520 nm). If a proportional linear adjustment is made for the longer wavelength of the FINCH measurement, the projected FINCH FWHM at 520 nm is 116 nm, as is shown in Table 1, showing that the FINCH lateral FWHM was proportionately equivalent to the Airyscan lateral FWHM, and within its standard deviation from the SIM measurement. Note further that the FINCH instrument, with no required moving parts and an optional spinning disk, is dramatically simpler to construct, maintain, and operate than these other instruments, with far fewer requirements for unique parts or high power lasers for fluorescence excitation.

Tables Icon

Table 1. 100 nm (nominal) beads imaged with various methods using 1.4 or 1.49 NA objectives at 520 nm emission.a

In order to make a practical measurement of the PSF width of Birefringent Lens FINCH in the axial direction, a stack of through focus images of the 110 nm beads was collected by physically translating the sample through the objective focus in 100 nm steps. In this way a measurement could be made of the influence of out-of-focus planes on the in-focus plane. Each plane imaged in the z-stack was reconstructed to the focal plane of the objective, and these reconstructed images of the same beads used for the lateral PSF measurements were placed in a stack, deconvolved with the Microvolution software, and an axial PSF measurement was made using MetroloJ. This measurement served to describe the performance of FINCH for objects as they move through the focal plane of the objective. The through-focus stack of the widefield images was measured in the same way. The results, plotted in Fig. 4(e), show that FINCH has an axial FWHM of 470 ± 25 nm, (adjusted to 414 ± 22 nm at 520 nm emission) nearly equal to the widefield value of 441 ± 23 nm (389 ± 20 nm at 520 nm emission). After deconvolution, this adjusted axial PSF for FINCH fits in the range reported for iSIM/MSIM (350-440 nm), especially when taking into account the fact that the FINCH measurements were widefield instead of confocal, which would be expected to increase the size of the axial PSF of FINCH in comparison to the confocal Airyscan/SIM/iSIM/MSIM measurements.

Microtubules are commonly used as a standard object for qualifying resolution in microscopes, as they are known to be approximately 25 nm wide, much smaller than the classical resolution limit of optical microscopes. A sample of microtubules (Life Technologies FluoCells #2, stained with BODIPY) was imaged with the new Birefringent Lens FINCH microscope, with a 100× 1.4 NA objective (Nikon CFI Plan Apo VC 100× 1.4 NA) at GFP excitation/emission wavelengths. The spinning disk was inserted into the conjugate image plane of the 4F relay system shown in Fig. 3, so that confocal and CINCH images (Confocal FINCH) could be compared. The 50 µm diameter pinholes in the disk had a back-projected diameter of 0.606 µm, or approximately 1.5 Airy units, in the object plane. A single exposure was taken, and the resulting hologram was processed into a complex hologram and reconstructed into a propagated image stack, including a deconvolution step. Deconvolved confocal and CINCH images are shown in Fig. 5(a) and (b). Line profiles were taken through several perpendicular cross-sections of apparently individual microtubules and were measured with the gaussian fitting procedure in ImageJ. The microtubules measured 216 ± 15 nm for deconvolved confocal and 118 ± 13 nm for deconvolved CINCH images, respectively, as depicted in the plot in Fig. 5(c). This super-resolved measurement of the resolution limited microtubules in the CINCH image is consistent with the FINCH measurements of the resolution limited beads discussed earlier. As shown in Table 2, the microtubule width post deconvolution matches that of SIM and is also a marked improvement over reported values for deconvolved iSIM and MSIM, which were ca. 150 nm at similar emission wavelengths. Several areas of the images in Fig. 5(a) and (b) are marked with red outlines, to draw attention to areas with striking superior resolution of structures in the CINCH image. The occasional discontinuities in the CINCH image are also consistent with the higher optical resolution of CINCH, as discontinuities and fluctuating intensities along individual microtubules are also observed in the SIM [3], iSIM [11] and MSIM [14] images of microtubules. While the use of a spinning disk causes a reduction in the light collection efficiency of a microscope system, it is important to note that for thin or sparsely labeled samples including many types of cultured cells relevant to biological interest, FINCH is able to perform much like CINCH, as shown in [26]. Other single plane illumination techniques such as light sheet excitation could also be combined with FINCH to obtain super-resolved images of single planes without out-of-focus background from other planes, without the reduction in intensity due to confocal excitation.

 figure: Fig. 5.

Fig. 5. (a,b) Deconvolved confocal and CINCH images of microtubules. Red outlines indicate example comparison areas in which the super-resolved CINCH image reveal more detail than is seen in the confocal image. (c) Plots of microtubule lateral FWHM from the deconvolved images.

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Tables Icon

Table 2. Microtubules imaged with various methods.a

Another important result of this development in FINCH technology is the continued ability to readily create high magnification biological images that other holographic methods have not achieved. There are several other incoherent holographic techniques including Poon’s pioneering scanning holography (the first to create holograms from fluorescent objects) [27,5659], several variants of traditional [6062] and self-referenced holography [2834], and several variants of a new technique called COACH [3537]. All of these methods bear some drawbacks, including optical complexity and long acquisition times due to the need for laborious scanning or averaging many exposures to improve the signal to noise ratio. Some of the methods also bear the requirement for capturing calibration curves or training data for the entire 3D imaging field of view, for each wavelength to be imaged. In this current new birefringent FINCH method, the optics are nearly all apochromatic, and the α-BBO birefringent lens is optically clear over the entire spectral region of interest to biological microscopy and contributes only a small amount to the total focal power of the system, resulting in a system with minimal wavelength sensitivity. All the required data is captured in a single exposure, and the final images are produced without requiring any prior characterization of the sample or training of an algorithm. It should also be noted that the simplicity of our new single lens method obviates the use of SLMs or similar diffractive elements in the imaging optical train, which can cause defects and artifacts in the final image. Even though our laboratory, in its initial report of the invention of FINCH [16], utilized a SLM for its flexibility for development purposes, we have abandoned it because of optical complexity, instability and the inability to obtain excellent image quality. Our continued focus in FINCH development has been toward maximizing optical quality and simplicity, which has resulted here in the highest quality holographic super-resolution images demonstrated to date.

This new FINCH development has considerable flexibility, producing super-resolved images with any objective magnification or wavelength and imaging mode. To further demonstrate that the high image quality of FINCH is not limited to fluorescence imaging with high NA objectives, a reflective USAF pattern (Max Levy) was imaged in the CINCHSCOPE microscope with 465 nm reflected light, using a low magnification, low NA objective. The fluorescence dichroic and emission filter was replaced with a 50% plate beam splitter mounted in a fluorescence cube. The 465 nm light from the LDI was directed onto the sample with a 10× objective (Nikon CFI Plan Fluor 10× 0.3 NA) and the reflected light was imaged by the FINCH optics. The results, without deconvolution, are shown in Fig. 6. Note that the FINCH image is of high quality, with sharp edges of the features and contrast as high as the widefield image, and with no degradation due to speckle as might be expected from holographic reflectance imaging. In fact, careful inspection reveals that the defects in the reflective pattern are imaged more clearly in the FINCH image than in the widefield image.

 figure: Fig. 6.

Fig. 6. Reflected light FINCH imaging. Original (non-deconvolved) (a) widefield and (b) FINCH reflected light images of a chrome USAF test pattern printed on a glass slide, taken with a 10× 0.3 NA objective under 465 nm incoherent illumination.

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5. Conclusions

Once all these observations showing FINCH and CINCH equaling or bettering other super-resolution methods are taken into account, it is clear that this simple FINCH and CINCH system outperforms other 2x super-resolution imaging systems, and does so with greater simplicity. The application of multiplexed phase shifting to birefringent lens FINCH/CINCH holography has resulted in an optical imaging system that records super-resolved images of biologically relevant samples in a single camera exposure, with no complex or high-maintenance optical apparatus, no phase change or spectral calibration, and no restrictions on the type of sample or illumination. Referring to Table 3 places FINCH in the context of other super-resolution techniques, comparing them across various operational characteristics important to the practical application of these techniques. This shows that single shot birefringent lens FINCH described here is a strong choice for widespread use in super-resolution imaging.

Tables Icon

Table 3. Operational characteristics of super-resolution methods.

Disclosures

GB: CellOptic, Inc. (I, E, P). NS: CellOptic, (I, E, P).

Data availability

Original images as captured by the cameras are available upon request. Processed images are available upon request. Indices of the beads and microtubules measured for FWHM are available on request.

References

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7. C. J. R. Sheppard, S. B. Mehta, and R. Heintzmann, “Superresolution by image scanning microscopy using pixel reassignment,” Opt. Lett. 38(15), 2889–2892 (2013). [CrossRef]  

8. G. M. R. De Luca, R. M. P. Breedijk, R. A. J. Brandt, C. H. C. Zeelenberg, B. E. de Jong, W. Timmermans, L. N. Azar, R. A. Hoebe, S. Stallinga, and E. M. M. Manders, “Re-scan confocal microscopy: scanning twice for better resolution,” Biomed. Opt. Express 4(11), 2644–2656 (2013). [CrossRef]  

9. I. Gregor and J. Enderlein, “Image scanning microscopy,” Curr. Opin. Chem. Biol. 51, 74–83 (2019). [CrossRef]  

10. S. Roth, C. J. Sheppard, K. Wicker, and R. Heintzmann, “Optical photon reassignment microscopy (OPRA),” Opt. Nanoscopy 2(1), 5 (2013). [CrossRef]  

11. A. G. York, P. Chandris, D. D. Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Methods 10(11), 1122–1126 (2013). [CrossRef]  

12. P. W. Winter, A. G. York, D. D. Nogare, M. Ingaramo, R. Christensen, A. Chitnis, G. H. Patterson, and H. Shroff, “Two-photon instant structured illumination microscopy improves the depth penetration of super-resolution imaging in thick scattering samples,” Optica 1(3), 181–191 (2014). [CrossRef]  

13. M. Guo, Y. Li, Y. Su, T. Lambert, D. D. Nogare, M. W. Moyle, L. H. Duncan, R. Ikegami, A. Santella, I. Rey-Suarez, D. Green, A. Beiriger, J. Chen, H. Vishwasrao, S. Ganesan, V. Prince, J. C. Waters, C. M. Annunziata, M. Hafner, W. A. Mohler, A. B. Chitnis, A. Upadhyaya, T. B. Usdin, Z. Bao, D. Colón-Ramos, P. La Riviere, H. Liu, Y. Wu, and H. Shroff, “Rapid image deconvolution and multiview fusion for optical microscopy,” Nat. Biotechnol. 38(11), 1337–1346 (2020). [CrossRef]  

14. A. G. York, S. H. Parekh, D. Dalle Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012). [CrossRef]  

15. M. Ingaramo, A. G. York, P. Wawrzusin, O. Milberg, A. Hong, R. Weigert, H. Shroff, and G. H. Patterson, “Two-photon excitation improves multifocal structured illumination microscopy in thick scattering tissue,” Proc. Natl. Acad. Sci. 111(14), 5254–5259 (2014). [CrossRef]  

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27. T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional Fluorescence Microscopy by Optical Scanning Holography,” Opt. Eng. 34(5), 1338–1344 (1995). [CrossRef]  

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References

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  1. E. Betzig, S. W. Hell, and W. E. Moerner, “For the development of super-resolved fluorescence microscopy,” Nobel Prize Chem. (2014).
  2. M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
    [Crossref]
  3. M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
    [Crossref]
  4. J. Huff, The Airyscan Detector from ZEISS: Confocal Imaging with Improved Signal-to-Noise Ratio and Super-Resolution (2015).
  5. C. B. Müller and J. Enderlein, “Image Scanning Microscopy,” Phys. Rev. Lett. 104(19), 198101 (2010).
    [Crossref]
  6. O. Schulz, C. Pieper, M. Clever, J. Pfaff, A. Ruhlandt, R. H. Kehlenbach, F. S. Wouters, J. Großhans, G. Bunt, and J. Enderlein, “Resolution doubling in fluorescence microscopy with confocal spinning-disk image scanning microscopy,” Proc. Natl. Acad. Sci. U. S. A. 110(52), 21000–21005 (2013).
    [Crossref]
  7. C. J. R. Sheppard, S. B. Mehta, and R. Heintzmann, “Superresolution by image scanning microscopy using pixel reassignment,” Opt. Lett. 38(15), 2889–2892 (2013).
    [Crossref]
  8. G. M. R. De Luca, R. M. P. Breedijk, R. A. J. Brandt, C. H. C. Zeelenberg, B. E. de Jong, W. Timmermans, L. N. Azar, R. A. Hoebe, S. Stallinga, and E. M. M. Manders, “Re-scan confocal microscopy: scanning twice for better resolution,” Biomed. Opt. Express 4(11), 2644–2656 (2013).
    [Crossref]
  9. I. Gregor and J. Enderlein, “Image scanning microscopy,” Curr. Opin. Chem. Biol. 51, 74–83 (2019).
    [Crossref]
  10. S. Roth, C. J. Sheppard, K. Wicker, and R. Heintzmann, “Optical photon reassignment microscopy (OPRA),” Opt. Nanoscopy 2(1), 5 (2013).
    [Crossref]
  11. A. G. York, P. Chandris, D. D. Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Methods 10(11), 1122–1126 (2013).
    [Crossref]
  12. P. W. Winter, A. G. York, D. D. Nogare, M. Ingaramo, R. Christensen, A. Chitnis, G. H. Patterson, and H. Shroff, “Two-photon instant structured illumination microscopy improves the depth penetration of super-resolution imaging in thick scattering samples,” Optica 1(3), 181–191 (2014).
    [Crossref]
  13. M. Guo, Y. Li, Y. Su, T. Lambert, D. D. Nogare, M. W. Moyle, L. H. Duncan, R. Ikegami, A. Santella, I. Rey-Suarez, D. Green, A. Beiriger, J. Chen, H. Vishwasrao, S. Ganesan, V. Prince, J. C. Waters, C. M. Annunziata, M. Hafner, W. A. Mohler, A. B. Chitnis, A. Upadhyaya, T. B. Usdin, Z. Bao, D. Colón-Ramos, P. La Riviere, H. Liu, Y. Wu, and H. Shroff, “Rapid image deconvolution and multiview fusion for optical microscopy,” Nat. Biotechnol. 38(11), 1337–1346 (2020).
    [Crossref]
  14. A. G. York, S. H. Parekh, D. Dalle Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
    [Crossref]
  15. M. Ingaramo, A. G. York, P. Wawrzusin, O. Milberg, A. Hong, R. Weigert, H. Shroff, and G. H. Patterson, “Two-photon excitation improves multifocal structured illumination microscopy in thick scattering tissue,” Proc. Natl. Acad. Sci. 111(14), 5254–5259 (2014).
    [Crossref]
  16. J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. 32(8), 912–914 (2007).
    [Crossref]
  17. J. Rosen and G. Brooker, “Fluorescence incoherent color holography,” Opt. Express 15(5), 2244–2250 (2007).
    [Crossref]
  18. J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics 2(3), 190–195 (2008).
    [Crossref]
  19. G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express 19(6), 5047–5062 (2011).
    [Crossref]
  20. J. Rosen, N. Siegel, and G. Brooker, “Theoretical and experimental demonstration of resolution beyond the Rayleigh limit by FINCH fluorescence microscopic imaging,” Opt. Express 19(27), 26249–26268 (2011).
    [Crossref]
  21. N. Siegel, J. Rosen, and G. Brooker, “Reconstruction of objects above and below the objective focal plane with dimensional fidelity by FINCH fluorescence microscopy,” Opt. Express 20(18), 19822–19835 (2012).
    [Crossref]
  22. N. Siegel, J. Rosen, and G. Brooker, “Faithful reconstruction of digital holograms captured by FINCH using a Hamming window function in the Fresnel propagation,” Opt. Lett. 38(19), 3922–3925 (2013).
    [Crossref]
  23. G. Brooker, N. Siegel, J. Rosen, N. Hashimoto, M. Kurihara, and A. Tanabe, “In-line FINCH super resolution digital holographic fluorescence microscopy using a high efficiency transmission liquid crystal GRIN lens,” Opt. Lett. 38(24), 5264–5267 (2013).
    [Crossref]
  24. N. Siegel and G. Brooker, “Improved axial resolution of FINCH fluorescence microscopy when combined with spinning disk confocal microscopy,” Opt. Express 22(19), 22298–22307 (2014).
    [Crossref]
  25. N. Siegel, B. Storrie, M. Bruce, and G. Brooker, “CINCH (confocal incoherent correlation holography) high spatial resolution super resolution fluorescence microscopy based upon FINCH (Fresnel incoherent correlation holography),” Proc. SPIE 9336, 93360S (2015).
    [Crossref]
  26. N. Siegel, V. Lupashin, B. Storrie, and G. Brooker, “High-magnification super-resolution FINCH microscopy using birefringent crystal lens interferometers,” Nat. Photonics 10(12), 802–808 (2016).
    [Crossref]
  27. T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional Fluorescence Microscopy by Optical Scanning Holography,” Opt. Eng. 34(5), 1338–1344 (1995).
    [Crossref]
  28. C. Jang, D. C. Clark, J. Kim, B. Lee, and M. K. Kim, “Signal enhanced holographic fluorescence microscopy with guide-star reconstruction,” Biomed. Opt. Express 7(4), 1271–1283 (2016).
    [Crossref]
  29. J. Weng, D. C. Clark, and M. K. Kim, “Compressive sensing sectional imaging for single-shot in-line self-interference incoherent holography,” Opt. Commun. 366, 88–93 (2016).
    [Crossref]
  30. Y. Wan, T. Man, F. Wu, M. K. Kim, and D. Wang, “Parallel phase-shifting self-interference digital holography with faithful reconstruction using compressive sensing,” Opt. Lasers Eng. 86, 38–43 (2016).
    [Crossref]
  31. T. Man, Y. Wan, F. Wu, and D. Wang, “Self-interference compressive digital holography with improved axial resolution and signal-to-noise ratio,” Appl. Opt. 56(13), F91–F96 (2017).
    [Crossref]
  32. X. Quan, O. Matoba, and Y. Awatsuji, “Single-shot incoherent digital holography using a dual-focusing lens with diffraction gratings,” Opt. Lett. 42(3), 383–386 (2017).
    [Crossref]
  33. T. Nobukawa, T. Muroi, Y. Katano, N. Kinoshita, and N. Ishii, “Single-shot phase-shifting incoherent digital holography with multiplexed checkerboard phase gratings,” Opt. Lett. 43(8), 1698–1701 (2018).
    [Crossref]
  34. T. Tahara, A. Ishii, T. Ito, Y. Ichihashi, and R. Oi, “Single-shot wavelength-multiplexed digital holography for 3D fluorescent microscopy and other imaging modalities,” Appl. Phys. Lett. 117(3), 031102 (2020).
    [Crossref]
  35. A. Vijayakumar, Y. Kashter, R. Kelner, and J. Rosen, “Coded aperture correlation holography system with improved performance [Invited],” Appl. Opt. 56(13), F67–F77 (2017).
    [Crossref]
  36. A. Vijayakumar and J. Rosen, “Interferenceless coded aperture correlation holography-a new technique for recording incoherent digital holograms without two-wave interference,” Opt. Express 25(12), 13883–13896 (2017).
    [Crossref]
  37. J. Rosen, V. Anand, M. R. Rai, S. Mukherjee, and A. Bulbul, “Review of 3D imaging by coded aperture correlation holography (COACH),” Appl. Sci. 9(3), 605 (2019).
    [Crossref]
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2020 (2)

M. Guo, Y. Li, Y. Su, T. Lambert, D. D. Nogare, M. W. Moyle, L. H. Duncan, R. Ikegami, A. Santella, I. Rey-Suarez, D. Green, A. Beiriger, J. Chen, H. Vishwasrao, S. Ganesan, V. Prince, J. C. Waters, C. M. Annunziata, M. Hafner, W. A. Mohler, A. B. Chitnis, A. Upadhyaya, T. B. Usdin, Z. Bao, D. Colón-Ramos, P. La Riviere, H. Liu, Y. Wu, and H. Shroff, “Rapid image deconvolution and multiview fusion for optical microscopy,” Nat. Biotechnol. 38(11), 1337–1346 (2020).
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T. Tahara, A. Ishii, T. Ito, Y. Ichihashi, and R. Oi, “Single-shot wavelength-multiplexed digital holography for 3D fluorescent microscopy and other imaging modalities,” Appl. Phys. Lett. 117(3), 031102 (2020).
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2019 (2)

J. Rosen, V. Anand, M. R. Rai, S. Mukherjee, and A. Bulbul, “Review of 3D imaging by coded aperture correlation holography (COACH),” Appl. Sci. 9(3), 605 (2019).
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I. Gregor and J. Enderlein, “Image scanning microscopy,” Curr. Opin. Chem. Biol. 51, 74–83 (2019).
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2018 (2)

T. Nobukawa, T. Muroi, Y. Katano, N. Kinoshita, and N. Ishii, “Single-shot phase-shifting incoherent digital holography with multiplexed checkerboard phase gratings,” Opt. Lett. 43(8), 1698–1701 (2018).
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Y. Wu and H. Shroff, “Faster, sharper, and deeper: structured illumination microscopy for biological imaging,” Nat. Methods 15(12), 1011–1019 (2018).
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2017 (5)

2016 (4)

N. Siegel, V. Lupashin, B. Storrie, and G. Brooker, “High-magnification super-resolution FINCH microscopy using birefringent crystal lens interferometers,” Nat. Photonics 10(12), 802–808 (2016).
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C. Jang, D. C. Clark, J. Kim, B. Lee, and M. K. Kim, “Signal enhanced holographic fluorescence microscopy with guide-star reconstruction,” Biomed. Opt. Express 7(4), 1271–1283 (2016).
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J. Weng, D. C. Clark, and M. K. Kim, “Compressive sensing sectional imaging for single-shot in-line self-interference incoherent holography,” Opt. Commun. 366, 88–93 (2016).
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Y. Wan, T. Man, F. Wu, M. K. Kim, and D. Wang, “Parallel phase-shifting self-interference digital holography with faithful reconstruction using compressive sensing,” Opt. Lasers Eng. 86, 38–43 (2016).
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2015 (1)

N. Siegel, B. Storrie, M. Bruce, and G. Brooker, “CINCH (confocal incoherent correlation holography) high spatial resolution super resolution fluorescence microscopy based upon FINCH (Fresnel incoherent correlation holography),” Proc. SPIE 9336, 93360S (2015).
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2014 (5)

2013 (10)

P. Bouchal and Z. Bouchal, “Concept of coherence aperture and pathways toward white light high-resolution correlation imaging,” New J. Phys. 15(12), 123002 (2013).
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P. Bouchal and Z. Bouchal, “Wide-field common-path incoherent correlation microscopy with a perfect overlapping of interfering beams,” J. Eur. Opt. Soc. 8, 13011 (2013).
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S. Roth, C. J. Sheppard, K. Wicker, and R. Heintzmann, “Optical photon reassignment microscopy (OPRA),” Opt. Nanoscopy 2(1), 5 (2013).
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A. G. York, P. Chandris, D. D. Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Methods 10(11), 1122–1126 (2013).
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O. Schulz, C. Pieper, M. Clever, J. Pfaff, A. Ruhlandt, R. H. Kehlenbach, F. S. Wouters, J. Großhans, G. Bunt, and J. Enderlein, “Resolution doubling in fluorescence microscopy with confocal spinning-disk image scanning microscopy,” Proc. Natl. Acad. Sci. U. S. A. 110(52), 21000–21005 (2013).
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C. J. R. Sheppard, S. B. Mehta, and R. Heintzmann, “Superresolution by image scanning microscopy using pixel reassignment,” Opt. Lett. 38(15), 2889–2892 (2013).
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G. M. R. De Luca, R. M. P. Breedijk, R. A. J. Brandt, C. H. C. Zeelenberg, B. E. de Jong, W. Timmermans, L. N. Azar, R. A. Hoebe, S. Stallinga, and E. M. M. Manders, “Re-scan confocal microscopy: scanning twice for better resolution,” Biomed. Opt. Express 4(11), 2644–2656 (2013).
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X. Lai, S. Zeng, X. Lv, J. Yuan, and L. Fu, “Violation of the Lagrange invariant in an optical imaging system,” Opt. Lett. 38(11), 1896–1898 (2013).
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N. Siegel, J. Rosen, and G. Brooker, “Faithful reconstruction of digital holograms captured by FINCH using a Hamming window function in the Fresnel propagation,” Opt. Lett. 38(19), 3922–3925 (2013).
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G. Brooker, N. Siegel, J. Rosen, N. Hashimoto, M. Kurihara, and A. Tanabe, “In-line FINCH super resolution digital holographic fluorescence microscopy using a high efficiency transmission liquid crystal GRIN lens,” Opt. Lett. 38(24), 5264–5267 (2013).
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2012 (5)

B. Katz, J. Rosen, R. Kelner, and G. Brooker, “Enhanced resolution and throughput of Fresnel incoherent correlation holography (FINCH) using dual diffractive lenses on a spatial light modulator (SLM),” Opt. Express 20(8), 9109–9121 (2012).
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A. G. York, S. H. Parekh, D. Dalle Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
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N. Siegel, J. Rosen, and G. Brooker, “Reconstruction of objects above and below the objective focal plane with dimensional fidelity by FINCH fluorescence microscopy,” Opt. Express 20(18), 19822–19835 (2012).
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C. A. Schneider, W. S. Rasband, and K. W. Eliceiri, “NIH Image to ImageJ: 25 years of image analysis,” Nat. Methods 9(7), 671–675 (2012).
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T. Tahara, “Comparative evaluation of the image-reconstruction algorithms of single-shot phase-shifting digital holography,” J. Electron. Imaging 21(1), 013021 (2012).
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2011 (4)

T. Wilson, “Resolution and optical sectioning in the confocal microscope,” J. Microsc. 244(2), 113–121 (2011).
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R. W. Cole, T. Jinadasa, and C. M. Brown, “Measuring and interpreting point spread functions to determine confocal microscope resolution and ensure quality control,” Nat. Protoc. 6(12), 1929–1941 (2011).
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G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express 19(6), 5047–5062 (2011).
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J. Rosen, N. Siegel, and G. Brooker, “Theoretical and experimental demonstration of resolution beyond the Rayleigh limit by FINCH fluorescence microscopic imaging,” Opt. Express 19(27), 26249–26268 (2011).
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2010 (1)

C. B. Müller and J. Enderlein, “Image Scanning Microscopy,” Phys. Rev. Lett. 104(19), 198101 (2010).
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2009 (1)

2008 (2)

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
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J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics 2(3), 190–195 (2008).
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2007 (2)

2006 (3)

2002 (1)

2000 (1)

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
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1997 (1)

1995 (2)

J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995).
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T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional Fluorescence Microscopy by Optical Scanning Holography,” Opt. Eng. 34(5), 1338–1344 (1995).
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1990 (1)

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1983 (1)

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M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
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Anand, V.

J. Rosen, V. Anand, M. R. Rai, S. Mukherjee, and A. Bulbul, “Review of 3D imaging by coded aperture correlation holography (COACH),” Appl. Sci. 9(3), 605 (2019).
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M. Guo, Y. Li, Y. Su, T. Lambert, D. D. Nogare, M. W. Moyle, L. H. Duncan, R. Ikegami, A. Santella, I. Rey-Suarez, D. Green, A. Beiriger, J. Chen, H. Vishwasrao, S. Ganesan, V. Prince, J. C. Waters, C. M. Annunziata, M. Hafner, W. A. Mohler, A. B. Chitnis, A. Upadhyaya, T. B. Usdin, Z. Bao, D. Colón-Ramos, P. La Riviere, H. Liu, Y. Wu, and H. Shroff, “Rapid image deconvolution and multiview fusion for optical microscopy,” Nat. Biotechnol. 38(11), 1337–1346 (2020).
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Awatsuji, Y.

X. Quan, O. Matoba, and Y. Awatsuji, “Single-shot incoherent digital holography using a dual-focusing lens with diffraction gratings,” Opt. Lett. 42(3), 383–386 (2017).
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Y. Awatsuji, A. Fujii, T. Kubota, and O. Matoba, “Parallel three-step phase-shifting digital holography,” Appl. Opt. 45(13), 2995–3002 (2006).
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T. Tahara, T. Kakue, Y. Awatsuji, S. Ura, K. Nishio, T. Kubota, and O. Matoba, “Single-shot phase-shifting digital holographic microscopy,” in 2011 10th Euro-American Workshop on Information Optics (IEEE, 2011), pp. 1–3.

Azar, L. N.

Bao, Z.

M. Guo, Y. Li, Y. Su, T. Lambert, D. D. Nogare, M. W. Moyle, L. H. Duncan, R. Ikegami, A. Santella, I. Rey-Suarez, D. Green, A. Beiriger, J. Chen, H. Vishwasrao, S. Ganesan, V. Prince, J. C. Waters, C. M. Annunziata, M. Hafner, W. A. Mohler, A. B. Chitnis, A. Upadhyaya, T. B. Usdin, Z. Bao, D. Colón-Ramos, P. La Riviere, H. Liu, Y. Wu, and H. Shroff, “Rapid image deconvolution and multiview fusion for optical microscopy,” Nat. Biotechnol. 38(11), 1337–1346 (2020).
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M. Guo, Y. Li, Y. Su, T. Lambert, D. D. Nogare, M. W. Moyle, L. H. Duncan, R. Ikegami, A. Santella, I. Rey-Suarez, D. Green, A. Beiriger, J. Chen, H. Vishwasrao, S. Ganesan, V. Prince, J. C. Waters, C. M. Annunziata, M. Hafner, W. A. Mohler, A. B. Chitnis, A. Upadhyaya, T. B. Usdin, Z. Bao, D. Colón-Ramos, P. La Riviere, H. Liu, Y. Wu, and H. Shroff, “Rapid image deconvolution and multiview fusion for optical microscopy,” Nat. Biotechnol. 38(11), 1337–1346 (2020).
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E. Betzig, S. W. Hell, and W. E. Moerner, “For the development of super-resolved fluorescence microscopy,” Nobel Prize Chem. (2014).

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P. Bouchal and Z. Bouchal, “Concept of coherence aperture and pathways toward white light high-resolution correlation imaging,” New J. Phys. 15(12), 123002 (2013).
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P. Bouchal and Z. Bouchal, “Wide-field common-path incoherent correlation microscopy with a perfect overlapping of interfering beams,” J. Eur. Opt. Soc. 8, 13011 (2013).
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Bouchal, Z.

P. Bouchal and Z. Bouchal, “Wide-field common-path incoherent correlation microscopy with a perfect overlapping of interfering beams,” J. Eur. Opt. Soc. 8, 13011 (2013).
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P. Bouchal and Z. Bouchal, “Concept of coherence aperture and pathways toward white light high-resolution correlation imaging,” New J. Phys. 15(12), 123002 (2013).
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Brandt, R. A. J.

Breedijk, R. M. P.

Brooker, G.

N. Siegel, V. Lupashin, B. Storrie, and G. Brooker, “High-magnification super-resolution FINCH microscopy using birefringent crystal lens interferometers,” Nat. Photonics 10(12), 802–808 (2016).
[Crossref]

N. Siegel, B. Storrie, M. Bruce, and G. Brooker, “CINCH (confocal incoherent correlation holography) high spatial resolution super resolution fluorescence microscopy based upon FINCH (Fresnel incoherent correlation holography),” Proc. SPIE 9336, 93360S (2015).
[Crossref]

N. Siegel and G. Brooker, “Improved axial resolution of FINCH fluorescence microscopy when combined with spinning disk confocal microscopy,” Opt. Express 22(19), 22298–22307 (2014).
[Crossref]

N. Siegel, J. Rosen, and G. Brooker, “Faithful reconstruction of digital holograms captured by FINCH using a Hamming window function in the Fresnel propagation,” Opt. Lett. 38(19), 3922–3925 (2013).
[Crossref]

G. Brooker, N. Siegel, J. Rosen, N. Hashimoto, M. Kurihara, and A. Tanabe, “In-line FINCH super resolution digital holographic fluorescence microscopy using a high efficiency transmission liquid crystal GRIN lens,” Opt. Lett. 38(24), 5264–5267 (2013).
[Crossref]

N. Siegel, J. Rosen, and G. Brooker, “Reconstruction of objects above and below the objective focal plane with dimensional fidelity by FINCH fluorescence microscopy,” Opt. Express 20(18), 19822–19835 (2012).
[Crossref]

B. Katz, J. Rosen, R. Kelner, and G. Brooker, “Enhanced resolution and throughput of Fresnel incoherent correlation holography (FINCH) using dual diffractive lenses on a spatial light modulator (SLM),” Opt. Express 20(8), 9109–9121 (2012).
[Crossref]

G. Brooker, N. Siegel, V. Wang, and J. Rosen, “Optimal resolution in Fresnel incoherent correlation holographic fluorescence microscopy,” Opt. Express 19(6), 5047–5062 (2011).
[Crossref]

J. Rosen, N. Siegel, and G. Brooker, “Theoretical and experimental demonstration of resolution beyond the Rayleigh limit by FINCH fluorescence microscopic imaging,” Opt. Express 19(27), 26249–26268 (2011).
[Crossref]

J. Rosen and G. Brooker, “Non-scanning motionless fluorescence three-dimensional holographic microscopy,” Nat. Photonics 2(3), 190–195 (2008).
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J. Rosen and G. Brooker, “Digital spatially incoherent Fresnel holography,” Opt. Lett. 32(8), 912–914 (2007).
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J. Rosen and G. Brooker, “Fluorescence incoherent color holography,” Opt. Express 15(5), 2244–2250 (2007).
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J. Rosen, G. Indebetouw, and G. Brooker, “Homodyne scanning holography,” Opt. Express 14(10), 4280–4285 (2006).
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Brophy, C. P.

Brown, C. M.

R. W. Cole, T. Jinadasa, and C. M. Brown, “Measuring and interpreting point spread functions to determine confocal microscope resolution and ensure quality control,” Nat. Protoc. 6(12), 1929–1941 (2011).
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Bruce, M.

N. Siegel, B. Storrie, M. Bruce, and G. Brooker, “CINCH (confocal incoherent correlation holography) high spatial resolution super resolution fluorescence microscopy based upon FINCH (Fresnel incoherent correlation holography),” Proc. SPIE 9336, 93360S (2015).
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Bulbul, A.

J. Rosen, V. Anand, M. R. Rai, S. Mukherjee, and A. Bulbul, “Review of 3D imaging by coded aperture correlation holography (COACH),” Appl. Sci. 9(3), 605 (2019).
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Bunt, G.

O. Schulz, C. Pieper, M. Clever, J. Pfaff, A. Ruhlandt, R. H. Kehlenbach, F. S. Wouters, J. Großhans, G. Bunt, and J. Enderlein, “Resolution doubling in fluorescence microscopy with confocal spinning-disk image scanning microscopy,” Proc. Natl. Acad. Sci. U. S. A. 110(52), 21000–21005 (2013).
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Burow, R.

Cande, W. Z.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
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M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
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Chandris, P.

A. G. York, P. Chandris, D. D. Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Methods 10(11), 1122–1126 (2013).
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Chen, J.

M. Guo, Y. Li, Y. Su, T. Lambert, D. D. Nogare, M. W. Moyle, L. H. Duncan, R. Ikegami, A. Santella, I. Rey-Suarez, D. Green, A. Beiriger, J. Chen, H. Vishwasrao, S. Ganesan, V. Prince, J. C. Waters, C. M. Annunziata, M. Hafner, W. A. Mohler, A. B. Chitnis, A. Upadhyaya, T. B. Usdin, Z. Bao, D. Colón-Ramos, P. La Riviere, H. Liu, Y. Wu, and H. Shroff, “Rapid image deconvolution and multiview fusion for optical microscopy,” Nat. Biotechnol. 38(11), 1337–1346 (2020).
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Chitnis, A.

P. W. Winter, A. G. York, D. D. Nogare, M. Ingaramo, R. Christensen, A. Chitnis, G. H. Patterson, and H. Shroff, “Two-photon instant structured illumination microscopy improves the depth penetration of super-resolution imaging in thick scattering samples,” Optica 1(3), 181–191 (2014).
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A. G. York, P. Chandris, D. D. Nogare, J. Head, P. Wawrzusin, R. S. Fischer, A. Chitnis, and H. Shroff, “Instant super-resolution imaging in live cells and embryos via analog image processing,” Nat. Methods 10(11), 1122–1126 (2013).
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Chitnis, A. B.

M. Guo, Y. Li, Y. Su, T. Lambert, D. D. Nogare, M. W. Moyle, L. H. Duncan, R. Ikegami, A. Santella, I. Rey-Suarez, D. Green, A. Beiriger, J. Chen, H. Vishwasrao, S. Ganesan, V. Prince, J. C. Waters, C. M. Annunziata, M. Hafner, W. A. Mohler, A. B. Chitnis, A. Upadhyaya, T. B. Usdin, Z. Bao, D. Colón-Ramos, P. La Riviere, H. Liu, Y. Wu, and H. Shroff, “Rapid image deconvolution and multiview fusion for optical microscopy,” Nat. Biotechnol. 38(11), 1337–1346 (2020).
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A. G. York, S. H. Parekh, D. Dalle Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
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Christensen, R.

Clark, D. C.

C. Jang, D. C. Clark, J. Kim, B. Lee, and M. K. Kim, “Signal enhanced holographic fluorescence microscopy with guide-star reconstruction,” Biomed. Opt. Express 7(4), 1271–1283 (2016).
[Crossref]

J. Weng, D. C. Clark, and M. K. Kim, “Compressive sensing sectional imaging for single-shot in-line self-interference incoherent holography,” Opt. Commun. 366, 88–93 (2016).
[Crossref]

Clever, M.

O. Schulz, C. Pieper, M. Clever, J. Pfaff, A. Ruhlandt, R. H. Kehlenbach, F. S. Wouters, J. Großhans, G. Bunt, and J. Enderlein, “Resolution doubling in fluorescence microscopy with confocal spinning-disk image scanning microscopy,” Proc. Natl. Acad. Sci. U. S. A. 110(52), 21000–21005 (2013).
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Cole, R. W.

R. W. Cole, T. Jinadasa, and C. M. Brown, “Measuring and interpreting point spread functions to determine confocal microscope resolution and ensure quality control,” Nat. Protoc. 6(12), 1929–1941 (2011).
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Appl. Opt. (6)

Appl. Phys. Lett. (1)

T. Tahara, A. Ishii, T. Ito, Y. Ichihashi, and R. Oi, “Single-shot wavelength-multiplexed digital holography for 3D fluorescent microscopy and other imaging modalities,” Appl. Phys. Lett. 117(3), 031102 (2020).
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Data availability

Original images as captured by the cameras are available upon request. Processed images are available upon request. Indices of the beads and microtubules measured for FWHM are available on request.

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Figures (6)

Fig. 1.
Fig. 1. Principle of increased image resolution by birefringent lens FINCH imaging for single point (a-b, Abbe criteria) and two-point (c-d, Rayleigh criteria) imaging. (a) A generalized schematic of classical optical imaging, depicting the effective numerical aperture (NA) of the image beam NAbeam as a function of the image beam size Rbeam at the lens and the image distance di, and the size of the classical image spot Δ i . (b) A generalized schematic of FINCH imaging, showing the effective NA of the image beam NAH increasing by a factor of 2 and the FINCH image spot size decreasing by a factor of 2 to Δ H . The yellow and green lenses and dashed lines represent the two focal lengths f1 and f2 imparted by the FINCH optics onto the received light from the object. The recorded hologram at the plane di is computationally reconstructed into an image. (c) Two points imaged by classical methods cannot be distinguished from one another if they are too close. The transverse magnification (MT, magnification of the image field) and the angular magnification (MA, magnification of the image spot) are equal to each other. (d) The same two points can be distinguished using FINCH imaging because there is less magnification of each spot, though the field size is magnified to the same extent as a standard optical imager focused to this plane such that the image spots do not change from their original position.
Fig. 2.
Fig. 2. Single Shot Birefringent Lens FINCH Imaging Principle. All necessary optical components for single-shot FINCH are shown in this schematic. (a) Polarized image light originating from the object is provided to the birefringent lens (BRL). The BRL, which is of high optical quality, splits the incoming image beam into a linearly orthogonally polarized pair of co-propagating beams with differing focal lengths. The beams are converted to orthogonal circular polarizations by the quarter wave plate (QWP), and subsequently interfere with one another to create a hologram that is recorded by a camera with multiplexed micro-polarizers. Four interspersed phase-shifted holograms are recorded simultaneously. (b) A schematic of the camera pixels overlaid with the interspersed micro-polarizer array. The micro-polarizer grid has four equally spaced polarizer orientations (0, 0.5π, π, and 1.5π radians) precisely arranged on the CMOS die in a repeating square pattern. (c) The recorded hologram (i) is de-interspersed by a computer in one step into four subsampled phases, then interpolated by nearest neighbor interpolation (ii), followed by super-positioning to create the complex hologram (iii) before being propagated to create the reconstructed image (iv).
Fig. 3.
Fig. 3. A schematic of the FINCH/CINCH microscope. LO: objective lens; L1, L2: relay lenses; D: dichroic mirror to isolate fluorescence excitation from emission light; PBS: polarizing beam splitter; P, S transmitted and reflected linear polarizations; BRL: birefringent lens; QWP: quarter-wave plate; L3: standard microscope tube lens. All of the components critical to the FINCH technique are contained in the shaded area. All other components are extrinsic to the FINCH technique and are present only to serve as the basic microscope optics upon which the FINCH system optics is based to create the super-resolved image. The FINCH system optics could be added to any microscope optical system. Insertion of the Spinning pinhole disk between the L1, L2 relay lenses converts the system into a confocal FINCH system in which a single image plane is imaged by the FINCH optics (CINCH). With the spinning pinhole disk in place the Standard camera records a confocal image.
Fig. 4.
Fig. 4. FINCH lateral resolution of 100 nm sub-resolution fluorescent beads imaged with a 60X 1.49 NA Nikon objective at 590 nm emission wavelength significantly exceeds the widefield resolution of corresponding beads. Widefield (a) and FINCH (b) images of the identical image field of 100 nm (nominal) beads, with insets showing a magnified detail. (c) Plot of two closely spaced beads showing greatly improved two-point resolution with FINCH compared with widefield. The green arrows in (a,b) point out the bead pair shown in the profiles in this plot. (d) Plot of average bead lateral FWHM made from randomly selected beads. (e) Plot of average bead axial FWHM measured by physically stepping the sample through the objective focus over the indicated range in 100 nm steps.
Fig. 5.
Fig. 5. (a,b) Deconvolved confocal and CINCH images of microtubules. Red outlines indicate example comparison areas in which the super-resolved CINCH image reveal more detail than is seen in the confocal image. (c) Plots of microtubule lateral FWHM from the deconvolved images.
Fig. 6.
Fig. 6. Reflected light FINCH imaging. Original (non-deconvolved) (a) widefield and (b) FINCH reflected light images of a chrome USAF test pattern printed on a glass slide, taken with a 10× 0.3 NA objective under 465 nm incoherent illumination.

Tables (3)

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Table 1. 100 nm (nominal) beads imaged with various methods using 1.4 or 1.49 NA objectives at 520 nm emission. a

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Table 2. Microtubules imaged with various methods. a

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Table 3. Operational characteristics of super-resolution methods.

Equations (3)

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H θ n ( x h , y h ; z h ) = P ( x h , y h ; R H ) { 2 + e x p { i π λ z r [ ( x h M T x s ) 2 + ( y h M T y s ) 2 + i θ n ] } + e x p { i π λ z r [ ( x h M T x s ) 2 + ( y h M T y s ) 2 i θ n ] } } ,
H F ( x h , y h ; z h ) = a = 1 n H θ a [ e x p ( ± i θ a 1 ± i θ a + 1 ) ] ,
I r e c ( x , y , z r ) = H F ( x , y ; z h ) A I R F ( x , y , z r ) e x p [ i π λ z r ( x 2 + y 2 ) ]

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