Three-dimensional nanoscale localization of point-like objects using self-interference digital holography

Abstract

We propose localizing point-like fluorescent emitters in three dimensions with nanometer precision throughout large volumes using self-interference digital holography (SIDH). SIDH enables imaging of incoherently emitting objects over large axial ranges without refocusing, and single molecule localization techniques allow sub-50 nm resolution in the lateral and axial dimensions. We demonstrate three-dimensional localization with SIDH by imaging 100 and 40 nm fluorescent nanospheres. With 49,000 photons detected, SIDH achieves a localization precision of 5 nm laterally and 40 nm axially. We are able to detect the nanospheres from as few as 13,000 detected photons.

Localization-based microscopy techniques have profoundly transformed subcellular imaging and the way we study the properties of cells [1]. Single particle tracking (SPT) involves continuously detecting and localizing a single moving molecular label, quantum dot, or metallic nanoparticle [2]. This makes it possible to follow the trajectory of the emitters and determine their mode of motion with nanometer accuracy. Super-resolution imaging using single molecule localization microscopy (SMLM) relies on imaging isolated photo-switchable single molecules attached to a continuous static sample over many raw frames and determining their positions with sub-pixel accuracy [3]. Initial SMLM and SPT techniques were two-dimensional (2D) and ideally limited to samples thinner than the depth of focus of the imaging system. The three-dimensional (3D) nature of the biological world makes it necessary to develop imaging tools that can gather information from the axial (z) dimension with sub-resolution precision. Unfortunately, the recorded 2D point spread function (PSF) of a standard fluorescence microscope does not provide sufficient information to localize the emitter axially with high precision.

SPT and SMLM have undergone tremendous advances since their inception. To enable high-resolution 3D imaging, a variety of techniques have been developed over the past decade to facilitate unambiguous 3D localization with high precision in the axial direction [4,5]. One of the more common methods is called PSF engineering, which relies on altering the PSF of the microscope to encode the axial position of an emitter in the PSF shape. The most commonly used engineered PSF is the astigmatic PSF [6], which can measure the axial position with a precision of 50 nm over an axial range of 800 nm. Another technique that optimizes the information encoded in the PSF is the family of PSFs called tetrapod PSFs, which span an axial range of 6–20 μm [7]. Recently, 3D SMLM has been demonstrated in thick samples (≈60 μm) using self-interference of the emitted fluorescence [8]; however, the accessible axial range in this method remains limited to ≈1 μm.

Here, we present single-emitter localization using self-interference digital holography (SIDH). SIDH works on the principle of interfering incoherent light with itself [9]. The incoherent light emitted from the object is split into two beams that are separately phase-modulated and recombined in a common plane to produce interference fringes. The density of the resultant interference fringes depends on the axial position of the emitter. Three images are collected with the phase of one path shifted for each hologram. Then the three images are combined to create the final hologram, eliminating both the bias image and the holographic “twin image.” An attractive feature of SIDH is that the hologram is converted into an image at a particular z plane using a mathematically well-defined kernel. Most work on SIDH has explored relatively bright samples. Frequently, a USAF target is used, illuminated by an LED, although fluorescently labeled cells were used in one experiment [10].

In SIDH, holographic images are generated at the detector and take the form

$$I_n\left(\overrightarrow{r}\right)=A+o\left(\overrightarrow{r}\right)\text{exp}\left(i{\theta }_n\right)+o*\left(\overrightarrow{r}\right)\text{exp}\left(-i{\theta }_n\right),$$ (1)

where $\overrightarrow{r}$ is the location vector, A is a bias term, θ_n is an introduced controllable phase shift, and $o\left(\overrightarrow{r}\right)$ is the interference term. When the recorded hologram described by Eq. (1) is directly back-propagated, the reconstructed image results in a defocused (virtual) image superimposed on a sharp (real) image of the sample. If the sample is a point emitter, the hologram described by Eq. (1) is the point spread hologram (PSH) of the system. The defocused image is known as the holographic twin image. In order to reconstruct a clear image from the recorded hologram, it is important to eliminate the bias term, A, and the third term, $o*\left(\overrightarrow{r}\right)$, in Eq. (1), which is responsible for the holographic twin image. In order to extract $o\left(\overrightarrow{r}\right)$ from Eq. (1), three holograms with phase shifts θ₁ = 0°, θ₂ = 120°, θ₃ = 240° are required, which are then algebraically combined to form a final complex hologram that takes the form

$$I_F\left(x,y\right)=I_1\left(x,y\right)\left[e^{-i{\theta }_3}-e^{-i{\theta }_2}\right]+I_2\left(x,y\right)\left[e^{-i{\theta }_1}-e^{-i{\theta }_3}\right]+I_3\left(x,y\right)\left[e^{-i{\theta }_2}-e^{-i{\theta }_1}\right]=6\text{sin}\left(\frac{2\pi }{3}\right)o\left(\overrightarrow{r}\right).$$ (2)

For a point source, the hologram described by the complex field $\left(\overrightarrow{r}\right)$ is given by

$$o\left(\overrightarrow{r}\right)=A\text{'}\left(\overrightarrow{r}\right)Q\left(\frac{1}{z_r}\right)L\left(\frac{\overrightarrow{r}}{z_r}\right).$$ (3)

Here, A′ is a constant, z_r is the reconstruction distance from the hologram plane, Q represents the quadratic phase function, Q(b) = exp(iπbλ⁻¹(x² + y²)), and L represents the linear phase function given by $L\left(\overrightarrow{s}\right)=\text{exp}\left(i2\pi {\lambda }^{-1}\left(s_{x}x+s_{y}y\right)\right)$ [11].

The final complex hologram I_F (x, y) can be numerically back-propagated to produce a reconstructed 3D image s (x, y, z) of the sample:

$$s\left(x,y,z\right)=I_F\left(x,y\right)*\text{exp}\left[\frac{i\pi }{\lambda z_r}\left(x^2+y^2\right)\right],$$ (4)

where z, which is a function of z_r, is the distance of the emitter from the objective focal plane, and the asterisk denotes a 2D convolution [9]. SIDH has already been used in fluorescence microscopy [10] and can provide unique properties such as infinite depth of field [12], violation of the Lagrange invariant [13], and edge enhancement with a vortex filter [14]. Here, we explore the use of SIDH for imaging single fluorescent emitters over a range of several micrometers with the goal of ultimately being able to detect single molecules that emit only a few thousand photons per exposure. We show that SIDH can be used for 3D super-resolution SMLM by accurately imaging a single emitter through a 20 μm depth of field. To demonstrate the localization performance for subwavelength-sized emitters, we image single fluorescent nanospheres using a custom-built widefield microscope and demonstrate the localization performance over the 20 μm axial range. Further, we also discuss the signal-to-noise ratio (SNR) requirements required to image photon-limited light sources such as a single molecule.

In our experiments, all incoherent digital holograms of fluorescent beads were collected using a custom-built inverted wide-field microscope equipped with an oil-immersion objective (Olympus, PlanApoN 60x/1.42). Figure 1(A) shows the schematic of the collection arm of the imaging system. To demonstrate 3D imaging, 0.1 μm diameter fluorescent beads emitting at 605 nm (Invitrogen) were excited with a 561 nm laser (Coherent) with an irradiance of 0.2 kW/cm². The fluorescence emitted by the sample was separated from the laser excitation light using a dichroic mirror (Omega, XF2054,485-555-650TBDR, USA) and a multi-band bandpass filter (Semrock, FF01–446/523/600/677–25, USA). We used a piezoelectric objective lens positioner (Smaract, Germany) to axially scan a sample of beads dried on a standard microscope cover glass. The back-pupil plane of the objective was demagnified using a tube lens (f_TL = 180 mm, Olympus, UTLU, USA) and an achromat L1 (f_L1 = 120 mm). The tube lens was placed d₁ = 75 mm away from the objective lens, and the distance between the tube lens and L1 was d₂ = 222 mm. The size of the back-pupil plane controls the size of the hologram. It is necessary to have accurate control over the size of the hologram so that the hologram is adequately sampled over the desired axial range. A Michelson interferometer was constructed d₃ = 200 mm away from L1 with a concave mirror (f_d = 300 mm) on one arm and a plane mirror on the other arm. The plane mirror was mounted on a piezoelectric translational stage (ThorLabs, NFL5DP20) to implement the phase shift required to acquire three images to form the final complex hologram as described in Eq. (2). The fluorescence was detected with an electron-multiplying charge-coupled (EMCCD) camera (Andor iXon-897 Life, UK), which was placed z_h = 150 mm away from the interferometer. As shown in Figs. 2(A)–2(E), the density of the fringes in the PSH changes smoothly as a function of the axial position of the emitter. In order to reconstruct the PSH to form an image of a single bead, three consecutive images with different phases are acquired which are then algebraically combined and reconstructed using Eqs. (2) and (4). Figures 2(F)–2(J) show the results of reconstructing the images by back-propagating the hologram to the appropriate reconstruction distance.

Fig. 1.

Configuration for SIDH. (A) Detection path for imaging using SIDH. Holograms are created using a Michelson interferometer. (B) Simulated images of holograms of a single-emitter recorded using SIDH. These are the point spread holograms (PSH). Three holograms with different phases (0°, 120°, and 240°) are required to reconstruct the final image. All scale bars are 5 μm.

Fig. 2.

3D imaging of a single 0.1 μm fluorescent bead using SIDH. (A)–(E) Digital holograms of a fluorescent bead at different axial positions. Scale bars are 50 μm. (F)–(J) Lateral view of images reconstructed by back-propagating the holograms shown in (A)–(E). Scale bar of (F) is 2 μm, scale bar of (G) is 1.8 μm, scale bar of (H) is 1.6 μm, scale bar of (I) is 1.4 μm, and scale bar of (J) is 1.2 μm. (K)–(O) Axial views of holograms reconstructed over the entire 20 μm axial range.

It can be seen in Figs. 2(F)–2(J) (lateral views) and Figs. 2(K)–2(O) (axial views) that the transverse magnification of the optical system changes based on the longitudinal distance of the sample from the focal plane of the objective. As the sample is displaced away from the focal plane, the cones of light forming the interferogram are altered, consequently changing the effective NA and the magnification of the system [15]. Since the localization uncertainty in SMLM depends on the pixel size of the imaging detector at the sample plane, it is necessary to know the transverse magnification at every plane. Telecentricity in SIDH images can be achieved by using different numerical reconstruction techniques (Huygens convolution, angular spectrum methods) [16], or resampling all the reconstructed images relative to the image with the highest magnification [15]. Recently, an optical configuration has been proposed for SIDH that ensures that the effective NA is held constant by placing the wavefront modulating optical element at a Fourier plane [17].

When localizing single emitters using SIDH, the spatial and axial information regarding an entire 3D stack is encoded in the three holograms acquired with different phases. To localize the lateral (x y) and axial (z) positions of the fluorescent nanosphere, the hologram is first reconstructed over a 20 μm z range. The approximate 3D position of the nanosphere is then calculated by detecting the local intensity maxima in a 26-connected neighborhood and subjecting it to an intensity threshold. The approximate position of the emitter is then used to cut out a user-defined (7 × 7 pixels in this case) region of interest (ROI) surrounding the bead. Each image in the stack of ROIs is then localized by fitting to a symmetric Gaussian function using maximum likelihood estimation (MLE). Each localized image provides four parameters (x y-sub-pixel location, signal photons, background photons, and the standard deviation of the Gaussian function, σ). In order to extract the axial position of the nanosphere, a curve is fit to the σ values for each image found in the previous step, and the global minimum of the curve is used to determine the axial position of the emitter. The x y positions determined using MLE are interpolated and, with the position of the global minimum found in the previous step, the sub-pixel 3D position of the nanosphere is determined. Figure 3 shows experimental localization results of a 100 nm nanosphere that was brought to focus using the axial piezo scanner. Over 68 measurements with a 50 ms exposure time, we achieved a localization precision of 5 nm in x and y and 40 nm in z for a bead with an average of 49,000 photons detected per measurement (each consisting of three images).

Fig. 3.

3D localization of a single 0.1 μm fluorescent bead. (A) Histograms of 68 localizations in x, y, and z of one single 0.1 μm red (580/605) fluorescent bead on a coverslip. The standard deviations of the measurements are σ_x = σ_y = 5 nm and σ_z = 40 nm. (B) Representative image of a single bead imaged with SIDH acquired in one 50 ms frame. Scale bar is 50 μm. (C) Localizations plotted in three dimensions.

It can be seen in Figs. 2(A)–2(E) that the size of the recorded hologram changes with depth; hence, it is important to control the size of the hologram so that the distance between the interference fringes is resolvable by the camera. However, if the footprint of the recorded hologram is too large, the experimenter runs the risk of the signal from the emitter not rising above the noise. This is due to the fact that the light from a single emitter that is usually spread across 9–16 pixels in a conventional microscope is now being spread across ≈60,000 pixels in the recorded hologram. In the system described above, the optical elements are chosen such that the radius of the hologram at the camera is 2.33 mm when the emitter is in focus, thus corresponding to ≈145 pixels on the camera.

Figure 4 shows the dependence of SNR on the emitted number of photons. Each value in the figure was calculated from simulated holograms of a point-like object emitting a constant number of photons. The simulated holograms were generated and reconstructed using Eqs. (1)–(4). A constant background of five photons/pixel was added to each of the simulated holograms, and the hologram was subjected to Poisson noise. It can be seen in Fig. 4(A) that to achieve an SNR of 10 under the presumed imaging conditions, it is necessary to detect at least 5000 photons. The SNR was calculated as the ratio of mean signal to the mean of the standard deviation of the background. The background (five photons/pixel) dominates the number of signal photons (0.03–0.3 photons/pixel). Because SNR $\propto N_s/\sqrt{N_s+N_b}$, this explains the linear relationship between the SNR and the number of signal photons, as seen in Fig. 4(A). Figure 4(B) shows the PSH of a 40 nm dark red (660/680) fluorescence nanosphere (Invitrogen). Even though the PSH is not as clearly visible as in the case of the 100 nm bead, the hologram can be successfully reconstructed at the optimal reconstruction distance. When imaging with SIDH, it is not necessary for the PSH to be visible to the eye. As long as the frequency of the rings obeys the Nyquist criterion, and the signal from the hologram rises above the background noise in the system, the hologram can be successfully reconstructed to provide an image of the emitter as shown in Fig. 4(C). Simulating the optical setup described earlier, we measured the sampling frequency of the holograms at −6 μm [Fig. 2(A)] and +14 μm [Fig. 2(E)] to be 10 and 3 pixels/cycle, respectively. The measurements were made at the edge of the hologram where the fringes are the closest to each other. The 40 nm bead was pumped with a 647 nm laser (Coherent) with an irradiance of 1.6 kW/cm², and each hologram was acquired in one 50 ms frame. The calibrated EM gain setting was 500, and an average of 13,000 photons were detected per estimation (each consisting of three images).

Fig. 4.

SNR conditions for imaging using SIDH. (A) SNR versus photons for simulated image data with a constant background of five photons/pixel in the presence of Poisson noise. (B) PSH of 40 nm dark red (660/680) nanosphere. Scale bar is 1 mm in image space. (C) Background subtracted reconstructed image of (B). Scale bar is 2 μm in sample space.

The final complex hologram described by Eq. (2) contains spatial and axial information regarding the entire 3D scene, and thus reconstruction at different axial distances results in a wide-field-like image at different z positions. Figure 5(A) shows overlapping PSHs of multiple 0.2 μm dark red (660/680) fluorescent nanospheres. It can be seen in Fig. 5(B) that even though the PSHs of multiple emitters overlap, the hologram can be reconstructed to produce a wide-field-like image in which the emitters can then be localized with sub-pixel precision making SMLM using SIDH a suitable technique for high-resolution imaging with dense emitters.

Fig. 5.

Imaging overlapping emitters using SIDH. (A) Digital hologram of five 0.2 μm far red fluorescent beads in focus. Scale bar is 50 μm. (B) Reconstructed image of (A). The laser intensity is 0.2 kW/cm², and the exposure time is 30 ms. Scale bar is 2 μm.

In conclusion, we have demonstrated the application of SIDH to the localization of single emitters. We have shown that particles emitting as few as 13,000 photons can be localized. Here, we are detecting 0.2 photons per pixel, below the quantum limit of visibility [18]. Numerical simulations have been performed to show the SNR conditions required to image an emitter with a limited number of photons. In principle, the emission of single fluorophores emitting 5000 or fewer photons can be detected as long as the background is sufficiently low. This indicates that SIDH could be a useful approach to 3D SMLM over large axial ranges. Sequential imaging can be avoided if the three images are simultaneously acquired [19]. Single-shot in-line SIDH has also been demonstrated using multiplexed checkerboard patterns on two spatial light modulators placed on the two arms of a Michelson interferometer [20], and a numerical method to perform in-line SIDH using a single image has been demonstrated using compressive sensing [21]. This demonstrates that SIDH is a promising method for 3D super-resolution microscopy as well as 3D single-particle tracking over large (20 μm) axial ranges.