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Abstract
Self-interference digital holography (SIDH) can image incoherently emitting objects over large axial ranges from three two-dimensional images. By combining SIDH with single-molecule localization microscopy (SMLM), incoherently emitting objects can be localized with nanometer precision over a wide axial range without mechanical refocusing. However, background light substantially degrades the performance of SIDH due to the relatively large size of the hologram. To optimize the performance of SIDH, we performed simulations to study the optimal hologram radius (Rh) for different levels of background photons. The results show that by reducing the size of the hologram, we can achieve a localization precision of better than 60 nm laterally and 80 nm axially over a 10 µm axial range under the conditions of low signal level (6000 photons) with 10 photons/pixel of background noise. We then performed experiments to demonstrate our optimized SIDH system. The results show that point sources emitting as few as 2120 photons can be successfully detected. We further demonstrated that we can successfully reconstruct point-like sources emitting 4200 photons over a 10 µm axial range by light-sheet SIDH.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Fluorescence microscopy is a critical tool in biomedical research, and recent advances in fluorescence microscopy techniques have enabled imaging below the diffraction limit, creating the field of super-resolution (SR) microscopy [1]. Several SR microscopy techniques have been developed including Structured Illumination Microscopy (SIM), Stimulated Emission Depletion Microscopy (STED), and Single Molecule Localization Microscopy (SMLM) [1–4]. Among these SR techniques, SMLM stands out for achieving sub-$50\; \textrm{nm}$ lateral and axial resolution [1,2]. But, due to the constrained axial range, 3D super-resolution imaging of whole cells is still a challenge. The most popular 3D-SMLM method uses a cylindrical lens to introduce astigmatic aberration into the detection path, encoding the axial information in the shape of the point spread function (PSF) and providing an axial resolution of 50–70 $\textrm{nm}$. However, the axial range is limited to ∼600 $\textrm{nm}$ [5,6]. There have been other methods to expand the imaging axial range of localization-based super-resolution techniques, by using engineered PSFs. The Double-helix PSF provides a 2-3${\mathrm{\mu} \mathrm{m}}$ axial range, and the Tetrapod PSF provides an axial range up to 20${\mathrm{\mu} \mathrm{m}}$ [7,8]. However, increasing of PSF size also increases the possibility of overlap, which results in poor localization performance in high-density 3D localization situations. To solve this, deep learning has been used for analyzing dense fields of overlapping emitters with engineered PSFs. Moreover, deep learning has also been used to design an optimal PSF for 3D localization of dense emitters [9]. But the maximum depth of field so far is still limited to a few microns [7–10].
Self-Interference Digital Holography (SIDH) enables the creation of an interference pattern that captures the wavefront from fluorescent signals by using the temporal coherence of incoherent sources, allowing for the reconstruction of a 3D image of the fluorescence emission over a large depth of field without mechanical refocusing [11–16]. When imaging single molecules, the emission light from the same single molecule is split and then interfered with itself to generate a hologram of the source [14–17]. SIDH based SMLM has the potential to perform 3D super-resolution imaging over a large axial range, with precision on the nanometer level. Our group has demonstrated localization of fluorescent microspheres using SIDH [14]. Imaging fluorescent microspheres, holograms could be imaged and reconstructed from emission of as few as 13,000 photons. We demonstrated that SIDH can achieve a localization precision of 5 nm laterally and 40 nm axially with an emitter signal level of 49,000 detected photons. However, the conventional fluorophores used in SMLM typically generate only a few hundred to a few thousand photons in the presence of background levels on the order of 10 photons per pixel. To understand the possible performance of SIDH, we investigated the theoretical limit of SIDH localization precision by computing the Cramér-Rao Lower Bound (CRLB), which is commonly used to evaluate the performance of any specified 3D localization technique [17].
CRLB provides the theoretically best precision that can be obtained for a certain estimator. In SMLM, CRLB provides the theoretical best localization precision that can be achieved by a technique with a given number of emission photons and a given background level. By calculating the CRLB for SIDH of single emitters, our group has demonstrated that SIDH can achieve 5 nm localization precision in all three dimensions over a 20 µm axial range with high SNR. However, although the localization precision of SIDH is better than the astigmatic PSF or the Cropped Oblique Secondary Astigmatism (COSA) PSF [18] with no background noise, it declines to ∼$125\; nm$ for the emission of 6000 photons with background noise levels (∼10 photons/pixel) that are typically seen in SMLM of single cells [19,20]. The high sensitivity to background noise in SIDH is due to the large diameter of the hologram (about 2-4 mm2 at the camera) which makes the SNR for SIDH significantly lower than the SNR for the astigmatic and the Gaussian PSFs, thereby degrading the precision of the system. Given that SMLM requires a high SNR to detect local intensity maxima for optimal localization precision, it's critical to enhance the SNR by improving the light efficiency of SIDH.
To address these issues, we conducted numerical simulations to analyze and optimize the SIDH performance for better photon emission utilization and enhanced SNR across a large axial range. We then verified our simulation findings with experiments, and we incorporated light-sheet (LS) illumination into the optimized SIDH system to remove background light, further reducing the background noise [21]. The experimental results demonstrate that with our optimal optical setup, SIDH can reconstruct the point spread hologram (PSH) from as few as ∼2,120 photons. Furthermore, with an emitter signal level of as few as ∼4,200 photons, the PSH can be reconstructed over an axial range of 10${\mathrm{\mu} \mathrm{m}}$.
2. Methods
2.1 Theoretical analysis of SIDH
SIDH splits the incoherent light emitted from the sample into two beams, which are individually phase-modulated before being recombined in a shared plane to generate interference fringes. The phase modulation can be a spherical phase, introduced to the beam by a curved mirror or lens. The density of the interference fringes then contains information on the object's axial position [12]. The interferometer, shown in Fig. 1, can be either a Michelson interferometer or a spatial light modulator (SLM). In this section we will discuss the theoretical analysis for two configurations of SIDH that we used in our simulations.
Fig. 1. (a) Optical setup for SIDH configuration 1: one plane wave one spherical wave configuration. (b) Optical setup for SIDH configuration 2: two spherical waves configuration.
Configuration 1, Fig. 1(a), uses a plane wave and a spherical wave (focal length ${f_d}$), while configuration 2, Fig. 1(b), uses two spherical waves (focal lengths ${f_{d1}}\; $ and ${f_{d2}}$) to form the interference pattern [12]. The details of the ideal reflectance function of the interferometer can be found in Ref. [12]. After propagation by the distance ${z_h}$, the intensity profile of the recorded hologram for a point object at $({{x_s},{y_s},{z_s}} )$ is:
Equation (1) contains three terms. The first term is a constant intensity bias term. The other two terms are holograms of the point source. The third term is the complex conjugate of the second and represents the twin-image of the hologram. To eliminate the twin image and bias term, three images of the same object are recorded with different phase constants of ${\theta _1} = 0^\circ $, ${\theta _2} = 120^\circ $, and ${\theta _3} = 240^\circ $ . The final complex hologram ${H_{F\; }}$ is an algebraic combination of the three images and is given by
2.1.1 Configuration 1: one plane wave one spherical wave
In configuration 1, shown in Fig. 1(a), the hologram is formed by the interference between one spherical wave and one plane wave. The focal length of the objective lens $({f_o})$ is 3 mm with an NA of 1.42. The objective is followed by a tube lens $({{f_{TL}} = 180\textrm{mm}} )$ and an achromatic lens $({f_2} = 120\textrm{mm})$ which form an image of the pupil plane. The relay lenses ${f_3}\textrm{and}{f_4}$ control the radius of the beam before it enters the interferometer ${R_e}$. In the interferometer, one half of the light is modulated by an element with focal length ${f_d}$. ${z_s}$ is the position of the object and, ${z_s} = \textrm{}{f_o}$ when the object is in the focal plane. The reconstruction distance ${z_r}$ for configuration 1 is then given by:
Depending on whether the first or second exponential term from Eq. (1) is selected, the sign “${\pm} $” denotes the image reconstructed from either the virtual or real hologram image. The transverse magnification (${M_T}$) of the system can be calculated using the imaging equation of each lens in the system and taking the product of each of the magnifications. The ${M_T}$ is given by:
2.1.2 Configuration 2: two spherical waves
For configuration 2, the hologram is formed by the interference of two spherical waves as shown in Fig. 1(b). ${f_{d1}}$ and ${f_{d2}}$ are the focal lengths of the two spherical elements in the interferometer. The transverse magnification $({M_T})$ is the same as for $\textrm{}({z_r})$ configuration 1, given in Eq. (4). The reconstruction distance for configuration 2 is given by:
2.2 Methods of studying the relationship between detected hologram size and localization precision
Our previous studies [14,17] demonstrate that SIDH requires optimization to achieve high SNR and high localization precision in the presence of background light. To achieve this, the light efficiency of the SIDH system must be maximized and the background noise must be as low as possible. Therefore, in our numerical simulations, we examine the relationship between the detected hologram size (${R_h})\; $ and the localization precision of SIDH. The radius of the hologram at the digital camera plane is calculated by using the ABCD ray transfer matrix framework [17]. The radius of the detected hologram, ${R_h}$, is the radius of the overlap area between the two waves:
We expect the localization precision in the absence of background to be best for larger ${R_h}$, but with background the SNR will be worse as the radius increases because more background photons will be captured by the hologram. Therefore, optimizing the hologram size should allow for the best localization precision in the presence of background. Our simulations explore two methods to change ${R_h}$.
The first method is to keep the interferometer settings fixed and change the size of the detected hologram ${R_h}$ by changing the distance between the interferometer and the digital camera ${z_h}$ (These results are discussed in Section 3.1). When the relay lenses ${f_3}\; $ and ${f_4}$ are $120\; \textrm{mm}$ and respectively, the beam radius at the entrance to the interferometer, ${R_e}$, is $2.36\; \textrm{mm}$. The focal length of the spherical wave ${f_d}$ is chosen as $300\; \textrm{mm}$. Figure 2(a) shows the relationship of ${R_h}$ and ${z_h}$ for configuration 1, $100\; \textrm{mm}$ where ${R_h}$ vanishes when the digital camera is placed at a distance of ${z_h} = 300\; \textrm{mm}$. For configuration 2, the relay lenses ${f_3}\; $ and ${f_4}$ are the same as in configuration 1 with ${R_e}$ of $2.36\; \textrm{mm}$. The focal lengths of the two spherical waves, ${f_{d1}}\; $ and ${f_{d2}}\; $ are $300\; \textrm{mm}$ and $700\; \textrm{mm}$, respectively. Figure 2 (b) shows the relationship of ${R_h}$ and ${z_h}$ for configuration 2. Before the two spherical waves perfectly overlap with each other, ${z_h} < {z_p},$ where ${z_p}$ stands for the perfect overlap position of the two waves, ${R_h}$ is determined by the spherical wave with the shorter focal length (${f_{d1}}$). When ${z_h} > {z_p}$, ${R_h}$ is determined by the spherical wave with the longer focal length (${f_{d2}}$).
Fig. 2. (a) Detected hologram radius $({{R_h}} )$ of configuration 1 when the camera is placed at different $({{z_h}} )$ distances away from the interferometer. (b) Detected hologram radius $({{R_h}} )$ of configuration 2 when the camera is placed at different $({{z_h}} )$ distances away from the interferometer. (c) The relationship between the interferometer settings and detected hologram radius at the perfect overlap position $({z_h} = \textrm{}{z_p} = \textrm{}$600 mm) for configuration 1. The pupil radius before the relay lenses is 2.84 mm. (d) The relationship between the interferometer settings and hologram radius at the perfect overlap position $({z_h} = \textrm{}{z_p}$) for configuration 2.
The second method of changing the hologram radius is to change the interferometer settings while the camera is fixed at ${z_h} = \; {z_p}$. For configuration 1, the beam radius before entering the interferometer, ${R_e}$, equals the detected hologram radius ${R_h}\; $ when the camera is at ${z_h} = {z_p}$. Therefore, by adjusting the relay lenses ${f_3}$ and ${f_4}$ to modify ${R_e}$, ${R_h}$ will be adjusted. Figure 2(c) shows the relationship between the relay lens magnification and ${R_h}\; $ when the camera is placed at ${z_h} = \; {z_p}$. ${f_d}$ is fixed at $300\; \textrm{mm}$. The camera position is fixed at ${z_h} = \; {z_p} = 600\textrm{mm}$. For configuration 2, as previously discussed, both spherical waves contribute to determining the hologram radius. Thus, with ${f_{d1}}$ fixed, the focal length of ${f_{d2}}$ will be modified to change ${R_h}$. In this case, when the camera is fixed at ${z_h} = \; {z_p}$, ${R_h}$ will change along with ${z_p}$. ${z_p}$ is calculated as follows:
Then, ${R_p}$ for configuration 2, Fig. 1(b), can be calculated by Eq. (6) with ${z_h} = \; {z_p}$, where ${R_p} = \; {r_{{f_{d1}}}} = {r_{{f_{d2}}}}$. ${f_{d1}}$ is fixed at 300 mm. ${f_3}\; $ and ${f_4}\; $ are fixed at 120 mm and 100 mm, respectively, which provides a fixed ${R_e}$ of $2.84\; \textrm{mm}$. By increasing ${f_{d2}}$, ${R_h}$ (where ${R_h} = \; {R_p}$) will asymptotically approaches the ${R_p}\; $ of configuration 1 under identical optical settings, as demonstrated in Fig. 2(d).
2.3 Description of the simulations
The pixel size for all the simulations we performed in this paper is set to 16 ${\mathrm{\mu} \mathrm{m}}$ . The image size is set to 512 $\textrm{px}$ × 512 $\textrm{px}$. In the simulations, each raw frame captures a hologram featuring a single fluorophore emitting with a total signal level of N = 6000 photons at the emission wavelength of 670 $\textrm{nm}$. The fluorescent molecule is simulated using a delta function and is positioned at the center of the simulated volume. Each raw data stack contains 3 raw frames of the simulated fluorophore with different phase constants of ${\theta _1} = 0^\circ $, ${\theta _2} = 120^\circ $, ${\theta _3} = 240^\circ $ . Then each raw data stack will be reconstructed and analyzed to determine the locations of the single molecules. 100 raw hologram data stacks of the fluorescent molecule are generated with different background noise levels (from β = 0 photons/pixel to β = 10 photons/pixel). After adding the background to the hologram stack, Poisson noise is applied to the images. Then the image stack is reconstructed using Eq. (3), and the coordinates of the molecule are calculated by localizing the center of the PSF. Specifically, the lateral reconstructed image is cropped to 32 $\textrm{px}$ × 32 $\textrm{px}$ (512 ${\mathrm{\mu} \mathrm{m}}$ × 512 ${\mathrm{\mu} \mathrm{m}}$), and a 2D Gaussian fit is applied to the x-y plane to obtain the lateral localization coordinates. To determine axial localization coordinates, we first reconstructed the hologram data axially with a step size of 100 $\textrm{nm}$ over a 10 ${\mathrm{\mu} \mathrm{m}}$ axial range. We then obtained the axial intensity plot (x-z plane) of the reconstructed data and cropped it to 32 $\textrm{px}$ × 32$\textrm{px}$ (512 ${\mathrm{\mu} \mathrm{m}}$ × 3.2 ${\mathrm{\mu} \mathrm{m}}$). A 2D Gaussian fit was then applied in the x-z plane to determine the axial localization coordinates of the molecule. To calculate the localization precision, the standard deviation (STD) of the localized coordinates are found from the 100 individual simulated data stacks. The data points for the hologram radius ${R_h} < 0.2\; \textrm{mm}$ are excluded because the point spread hologram (PSH) is too small to be sampled by the current pixel size settings, resulting in the axial Full-Width Half-Maximum (FWHM) diverging below this radius. All the simulations were performed by custom written Python code.
2.4 Experimental setup and light-sheet optimization
Figure 3(a) shows a schematic of the imaging path of our SIDH system. The data was collected using a custom-built inverted wide-field microscope equipped with an oil-immersion objective (PlanApoN, 60x, 1.42 NA, Olympus, Japan) mounted on a piezoelectric objective scanner (Smaract, Germany). The back-pupil plane of the objective was demagnified using a tube lens (${f_{TL\; }}$= 180 $\textrm{mm}$, UTLU, Olympus, USA) and an achromatic lens ${L_2}$ (${f_2}$ = 120 $\textrm{mm}$). A pair of relay lenses ${L_3}\; $ and ${L_4}$ in a 4f configuration were added before the interferometer to have accurate control over ${R_e}$. To generate interference, we use a Michelson interferometer instead of an SLM, shown in Fig. 3(a). In this way, the light efficiency of the SIDH system is tremendously improved since the light loss due to the polarization sensitivity of the SLM no longer exists. The Michelson interferometer was placed $100\; \textrm{mm}$ away from ${L_4}$ with a concave mirror (${f_d} = 304.80\; \textrm{mm}$, #32-818 λ/4 Precision Spherical Mirror, Edmund Optics, USA) on one arm and a plane mirror on the other arm. The plane mirror was mounted on a piezoelectric translational stage (NFL5DP20, Thorlabs, USA) to implement the phase-shifts required to acquire three images. The fluorescence was detected with an electron-multiplying charge-coupled (EMCCD) camera (Andor iXon-897 Life, UK) which was placed $127\; \textrm{mm}$ away from the interferometer.
Fig. 3. (a) Detailed schematic of the imaging path of the SIDH setup with a Michelson interferometer. The blue square represents a beam splitter (BS). Interference will be observed as long as the path difference between the two arms of the interferometers is no greater than half the coherence length of the light source. The focal length of the tube lens is 180 mm. The focal length of ${L_2}$ is set to ${f_2} = 120\; mm$. The focal lengths of the relay lenses ${L_3}$ and ${L_4}$ are ${f_3} = 200\; mm$ and ${f_4} = 100\; mm$, separately. The focal length of the concave mirror is set to ${f_d} = 300\; mm$. (b) The custom designed sample chamber for tilted light-sheet (LS) illumination pathway. In the light-sheet setup, the excitation lasers are initially shaped using a cylindrical lens with a focal length of 200 mm (not shown). They are then introduced into the illumination objective and subsequently reflected by the glass prism. As the excitation lasers enter the imaging chamber, the incident angle of the tilted light-sheet is approximately 5.6°.
Furthermore, in order to remove the out-of-focus light and reduce the background encountered when imaging single emitters using SIDH, we extended our illumination setup by adding a tilted light-sheet illumination pathway to the optimized SIDH setup [21]. The light sheet is formed by a cylindrical lens with a focal length of $200\; \textrm{mm}$, which focuses the light in only one dimension onto a mirror. The mirror plane is imaged onto the back aperture of a long working distance illumination objective (378-803-3, 10x, NA 0.28, Mitutoyo, Japan) by two lenses in a 4f configuration. The illumination objective then focuses the light sheet, which is directed into the sample chamber at an angle of about 5.6° using a glass prism (PS908L-A, Thorlabs), as shown in Fig. 3(b). The prism is held using a custom designed prism holder which is mounted using a cage system on a three-axis translational stage (XRN25P-K2, Thorlabs, USA).
Experiments using both wide field epi-illumination and light-sheet illumination were performed to demonstrate the improved performance of light-sheet SIDH [21,23]. The excitation lasers (OBIS 561 nm, 200 mW and 647 nm, 120 mW, Coherent, USA) can be either introduced into the epi-illumination path or sent to the light sheet illumination path. The path is easily switched with a flip mirror. The fluorescence emitted by the sample is separated from the laser excitation light using a dichroic mirror (XF2054,485-555-650TBDR, Omega, USA), a multi-band bandpass filter (FF01-446/523/600/677-25, Semrock, USA), and notch filters (ZET561NF notch filter and NF01-488/647 notch filter, Semrock, USA).
$100\; \textrm{nm}$ red fluorescent beads (FluoSpheres Carboxylate-Modified Microspheres, Excitation/Emission: 580/605 nm, Catalog number: F8801, Invitrogen, USA) and $40\; \textrm{nm}\; $ dark red fluorescent beads (FluoSpheres Carboxylate-Modified Microspheres, Excitation/Emission: 647/680 nm, Catalog number: F8789, Invitrogen, USA) were used in our experiments. The sample chamber was manufactured by attaching a glass coverslip (Fisher Premium Cover Glass, no. 1.5, Fisher Scientific, USA) to the bottom of a commercial glass cuvette (704-000-20-10, Hellma, Germany) using nail polish. Beads were dried on the coverslip and Immersion oil (Type DF Immersion Oil, Cargille Laboratories, USA) was then added to the chamber to maintain the refractive index matching, Fig. 3(b). After imaging, the bottom coverslip could be detached, and the chamber could be cleaned and reused.
3. Results
3.1 Effect of changing camera position
In this section, we will present the simulation results studying the relationship between the detected hologram radius (${R_h}$) and localization precision by changing the digital camera position (${z_h}$).
3.1.1 Configuration 1: one plane wave one spherical wave
Figure 4(a) and (b) show the lateral and axial PSF Full Width at Half Maximum (FWHM), respectively, while Fig. 4(c) and (d) show the corresponding lateral and axial localization precision under different background noise levels.
Fig. 4. (a) The lateral PSF FWHM of SIDH configuration 1. (b) The axial PSF FWHM of SIDH configuration 1. (c) Lateral localization precision of SIDH configuration 1. (d) Axial localization precision of SIDH configuration 1. STD refers to the standard deviation of the localized coordinates. “bg” in legend refers to the background noise level (in photons/pixel). Total signal level is N = 6000 photons. The datapoints for ${R_h} < 0.2mm$ are excluded.
In the absence of background noise, the localization precision of SIDH is determined by the PSF FWHM and the photon number, and the localization precision is smaller than $10\; \textrm{nm}$ for all camera positions $({z_h})$. In this case, increasing the ratio $\frac{{{z_h}}}{{{f_d}}}$ in SIDH decreases the lateral FWHM of the PSF thereby improving the lateral localization precision. The maximum resolving power is achieved for the ratio $\frac{{{z_h}}}{{{f_d}}} = 2$ when the camera is at the position of perfect overlap between the two beams $({z_h} = 600\; \textrm{mm})\; $[12]. Likewise, the axial localization precision is dictated by both the axial PSF size and the level of background noise. However, contrary to the lateral PSF FWHM, improvement in the axial PSF FWHM is observed when ${z_h}$ approaches ${f_d}$, but not so close that the PSH has no fringes encoding the axial position [24]. Moreover, the size of the PSH decreases as ${z_h}$ approaches ${f_d}$, thereby increasing the SNR of the PSH. This, in turn, contributes to improved axial localization precision.
However, the presence of the background noise significantly affects the localization precision of SIDH in both the lateral and axial directions. It can be seen from Fig. 4(c) and (d) that the overall localization precision gets worse when the background noise increases for all camera positions. However, the background noise has a smaller effect on localization precision when ${z_h}$ is closer to ${f_d}$, where ${R_h}$ is smaller, increasing the SNR. For a background noise level of $\beta = 10\textrm{photons/pixel}$ ($N = 6000\textrm{photons}$), when ${z_h} = 172\textrm{mm}$, $\textrm{}{R_h} = 1\textrm{mm}$ and the lateral and axial localization precision of SIDH are $24\textrm{nm}$ and $63\textrm{nm}$, respectively. When $\textrm{}{z_h} = 428\textrm{mm}$, ${R_h} = 1\textrm{mm}$ and, the lateral and axial localization precision of SIDH are $8\textrm{nm}$ and $179\textrm{nm}$, respectively. When $172\textrm{mm} < {z_h} < 428\textrm{mm}$, ${R_h} < 1\textrm{mm}$, the lateral localization precision of configuration 1 reaches the minimum value of $\textrm{}3.8\textrm{nm}$ with ${R_h} = 0.2\; \textrm{mm}$ while ${\textrm{z}_\textrm{h}} = \; 326\; \textrm{mm}$, and the axial localization precision achieves $10\textrm{nm}$ at the same ${\textrm{z}_\textrm{h}}$. It’s clear that decreasing the size of the hologram significantly increases the number of photons per pixel, thereby making it easier to detect holograms under low SNR conditions. Furthermore, for the same value of ${R_h}$, ${z_h} > {f_d}\; $ has a better lateral localization precision than ${z_h} < {f_d}$, which demonstrates the effect of the lateral PSF size mentioned earlier.
3.1.2 Configuration 2: two spherical waves
Figure 5(a) and (b) show the lateral and axial PSF FWHM, respectively, while Fig. 5(c) and (d) show the corresponding lateral and axial localization precision under different background noise levels. The results show that localization precision adheres to the same pattern observed in configuration 1, where a decrease in ${R_h}$ significantly improves the localization precision, especially in high background noise scenarios. This improvement is attributed to the enhancement of signal-to-noise ratio (SNR) conditions. The axial localization precision is poor at the perfect overlap position $({z_h} = {z_p} = 420\;mm)$ because the relationship between ${z_r}$ and z has zero slope here [22,24].
Fig. 5. (a) The lateral PSF FWHM of SIDH configuration 2. (b) The axial PSF FWHM of SIDH configuration 2. (c) Lateral localization precision of SIDH configuration 2. (d) Axial localization precision of SIDH configuration 2. Total signal level is N = 6000 photons. The datapoints for ${R_h} < 0.2mm$ are excluded. ${f_{d1}}$ (300 $mm$) and ${f_{d2}}$ (700 $mm$) are the focal lengths of the two spherical elements in the interferometer.
Moreover, in order to image across the entire axial range, it is necessary to not place the camera in the range of ${\textrm{f}_{\textrm{d}1}} < {z_h} < {\textrm{f}_{\textrm{d}2}}$ to ensure that the values of the reconstruction distance $({z_r})$ are not symmetric across the focal plane, which makes the axial PSF too large for good localization precision. Especially at the perfect overlap position (${z_h} = \; {z_p} = 420\; mm$), where the lateral PSF size is minimized but the axial PSF becomes too large to be localized. For ${z_h} < {\textrm{f}_{\textrm{d}1}}\textrm{}$ and ${z_h} > \textrm{}{\textrm{f}_{\textrm{d}2}}$, compared with configuration 1, configuration 2 has worse performance for the localization precision under the same ${R_h}$ conditions. For example, at a background level of 10 photons per pixel, when ${R_h} = 1\ \textrm{mm}\; ({\textrm{at}\;{z_h} = 172\;\textrm{mm}} )$, configuration 1 provides a localization precision of $22\; \textrm{nm}$ (lateral) and 47 $\textrm{nm}$ (axial), while configuration 2 provides a localization precision of 67 $\textrm{nm}$ (lateral) and the axial PSF is too large to be localized.
3.2 Effect of changing the entrance beam radius with the camera fixed at the perfect overlap position (${z_p}$)
In this section, we explore the results of simulations studying the relationship between localization precision and ${R_h}$ by changing the radius of the beam entering the interferometer while maintaining the camera at the position of perfect overlap (${z_h} = {z_p}$). Because the axial PSF size is poor at this position, we will only consider the lateral precision results in this section.
3.2.1 Configuration 1: one plane wave one spherical wave
Figure 6(a) and (b) show the effect of ${R_h}$ on the PSF FWHM and lateral localization precision, respectively. It can be seen from Fig. 2(c) that there is a linear relationship between the relay lenses magnification and ${R_h}$, and the lateral PSF FWHM of configuration 1 gets smaller when ${R_h}$ gets bigger. In the absence of background noise, the localization precision shows the same changing trend with the PSF size. However, when the background noise level increases, the localization precision is worse and remains relatively constant vs. ${R_h}$. This is because the decrease of the PSF FWHM gives a positive effect on the localization precision when ${R_h}$ gets bigger, which compensates for the negative effect from the background noise.
Fig. 6. (a) The lateral PSF FWHM with different ${R_h}\textrm{}$(${z_h} = {z_p} = 600mm)$ under different background noise levels for configuration 1. (b) The lateral localization precision with different $\textrm{}{R_h}\textrm{}$(${z_h} = {z_p} = 600mm)$ under different background noise levels for configuration 1. (c) The lateral PSF FWHM with different ${R_h}\textrm{}$(${z_h} = \textrm{}{z_p})$ under different background noise levels for configuration 2. (d) The lateral localization precision with different ${R_h}\textrm{}$(${z_h} = \textrm{}{z_p})$ under different background noise levels for configuration 2. Total signal level is N = 6000 photons.
3.2.2 Configuration 2: two spherical waves
Figure 6(c) and (d) respectively illustrate the changes in lateral PSF FWHM and localization precision under varying ${R_h}$ conditions for configuration 2. The performance of localization precision, Fig. 6(d), remains consistent under different levels of background noise because the lateral PSF size (Fig. 6(c)) undergoes a relatively minor variation (about $40\; \textrm{nm}$), compared to configuration 1 (about $1000\; \textrm{nm}$), across the entire ${R_h}$ range. Moreover, the impact of background noise on localization precision increases as ${R_h}$ increases. This is due to the relationship between ${R_h}$ and ${f_{d2}}$, Fig. 2(d), which makes the negative impact of the background noise dominate the localization precision as ${R_h}$ increases. The PSF FWHM of configuration 2 gets better when ${f_{d2}}$ gets bigger, which indicates that the PSF FWHM of configuration 2 asymptotically approaches the PSF FWHM of configuration 1 for the same ${R_e}$.
3.3 Optimal settings for 3D imaging
In the previous section we saw that, at the background level of 10 photons per pixel, ${R_h}$ needs to stay in the range $1.38\; \textrm{mm}$ to $0.21\; \textrm{mm}$ to achieve high localization precision. However, the simulations performed in the last two sections only simulated the case of ${z_s} = {f_o}$, in which the object is in focus. In order to do 3D SMLM imaging over a large axial range, the hologram size and the corresponding localization precision when the object is out-of-focus (${z_s} \ne {f_o}$) also need to be considered and optimized. Figure 7(a) and (d) show the relationship between the object axial position (${z_s}$) and hologram radius (${R_h}$) for SIDH configuration 1 and SIDH configuration 2, respectively, for different camera positions. For configuration 1, ${R_e}\; $ is set to 1.42 mm and ${f_d} = 300\; \textrm{mm}$. For configuration 2, ${R_e}\; $ is also set to $1.42\; \textrm{mm}$ with ${f_{d1}} = 300\; \textrm{mm}$ and ${f_{d2}} = 700\; \textrm{mm}$.
Fig. 7. (a) Relationship of object axial position and detected hologram radius with different camera distances for configuration 1. (b) Lateral localization precision for configuration 1 with the different background noise levels when ${z_h} = 100\; mm$. (c) Axial localization precision for configuration 1 with the different background noise levels ${z_h} = 100\; mm$. (d) Relationship of object axial position and detected hologram radius with different camera distances for configuration 2. (e) Lateral localization precision for configuration 2 with the different background noise levels when ${z_h} = 100\; mm$. (f) Axial localization precision for configuration 2 with the different background noise levels when ${z_h} = 100\; mm$. Total signal level is N = 6000 photons.
It can be seen from Fig. 7(a) and (d) that the closer the camera is placed to the interferometer, the smaller the variation in ${R_h}$ over the 10 µm axial range. Furthermore, when the camera is placed after 150 mm $({z_h} > 150\; \textrm{mm})$, the beam will come to a focus within the desired axial range, and the hologram cannot be reconstructed over the whole axial range. Drawing from the conclusions in Section 3.1, optimal lateral FWHM in a SIDH system demands that the camera be placed at the perfect overlap position, and optimal axial FWHM in a SIDH system demands the camera be placed close to the interferometer focal point. But it’s important to keep the camera as close as possible to the interferometer to image a large axial range, which limits the optimization of ${R_h}$ to changing ${R_e}$. Moreover, if the camera is placed too close to the interferometer, the FWHM, in both axial and lateral directions, becomes excessively large (Fig. 4), thereby worsening the localization precision. Therefore, it’s important to find a balanced position of the camera to obtain the desired axial imaging range and good localization precision performance. By adjusting the magnification of the relay lenses (${f_3}$ and ${f_4}$), thereby changing ${R_e},\; $ it is possible to maintain the detected hologram radius $({R_h})$ within a range of $1.4\; \textrm{mm}$ to $0.3\; \textrm{mm}$ across the entire $10\; {\mathrm{\mu} \mathrm{m}}$ axial range.
Figure 7(b), (c) and Fig. 7(e), (f) show the localization precision over the $10\; {\mathrm{\mu} \mathrm{m}}$ axial range under different background noise levels for configuration 1 and configuration 2, respectively. ${z_h}$ is set to $100\; \textrm{mm}$ for both configurations because it is difficult, experimentally, to place the camera closer to the interferometer than that. The results show that under low SNR conditions (N = 6000 photons with background noise of $10\frac{{\textrm{photons}}}{{\textrm{pixel}}}$), the optimized SIDH has the potential to achieve a lateral localization precision of $5\; \textrm{nm}$ to $58\; \textrm{nm}$ and axial localization precision of $13\; \textrm{nm}$ to $80\; \textrm{nm}$ over an axial range of $10\; {\mathrm{\mu} \mathrm{m}}$ for configuration 1. With similar optical settings, SIDH configuration 1 has an overall better localization performance than configuration 2.
3.4 Experimental results
To demonstrate the performance of optimized SIDH, we performed experiments with different ${R_e}$ values under the same imaging conditions. A bead was brought into focus and the three images were taken with the phase of one path shifted for each image. $100\; \textrm{nm}$ fluorescent microspheres were imaged by epi-illumination and the results are shown in Fig. 8. Each hologram is acquired in one 5 ms exposure and the calibrated EM gain setting was 500. $100\; \textrm{nm}$ red beads were excited with a 561 nm laser with an irradiance of 0.02 kW/cm2. An average of ∼4,200 signal photons are collected from a bead in one image. Figure 8(a) shows the PSH of the SIDH setup with ${R_e} = 2.36\; \textrm{mm}$. Figure 8(b) shows the PSH of the optimized SIDH setup with ${R_e} = 1.42\; \textrm{mm}$. Figure 8(c) shows the background subtracted reconstructed image of Fig. 8(b) while Fig. 8(a) could not be reconstructed due to the low SNR.
Fig. 8. PSH of $100\; nm$ (561/605) nanosphere with ∼4200 photons for SIDH setup with different ${R_e}$. (a) PSH of SIDH setup with ${R_e} = 2.36\textrm{}mm$. (b) PSH of SIDH setup with ${R_e} = 1.42\textrm{}mm$. The top left figure shows the PSH with the shift phase of 0 degree (${I_1}$). The top right figure shows the PSH with the shift phase of 120 degree (${I_2}$). The bottom left figure shows the PSH with the shift phase of 240 degree (${I_3}$). The bottom right figure shows the magnitude of the final complex hologram (${H_F}$). (c) Lateral view of the image reconstructed by back-propagating the hologram shown in (b). (d) Axial view of the image reconstructed by back-propagating the hologram shown in (b). The camera is placed 127 mm away from the Michelson interferometer for both experiments. The SNR was calculated as the ratio of mean signal to the mean of the standard deviation of the background (mean sig/mean std).
To further demonstrate optimized SIDH at lower light levels, we imaged a $40\; \textrm{nm}$ dark red fluorescent nanosphere with the light sheet excitation scheme at ∼2,120 signal photons to compare the image quality between the light sheet and epi-illumination with the optimized SIDH system. Figure 9(a) and (b) show the PSH of SIDH with epi-illumination and light-sheet illumination, respectively. Figure 9(c) shows the reconstructed image of Fig. 9(b) while the PSH shown in Fig. 9(a) was not able to be reconstructed.
Fig. 9. PSH of $40\; nm$ dark red (647/680) nanosphere at ∼2120 signal photons light levels with different illumination pathways. (a) The PSH of $40\; nm$ nanosphere with epi-illumination. (b) The PSH of $40\; nm$ nanosphere with light-sheet illumination. (c) Lateral view of the image reconstructed by back-propagating the hologram shown in (b). (d) Axial view of the image reconstructed by back-propagating the hologram shown in (b). The SNR was calculated as the ratio of mean signal to the mean of the standard deviation of the background (mean sig/mean std).
To demonstrate 3D imaging with our SIDH system with light-sheet illumination, $100\; \textrm{nm}$ beads emitting ∼4,200 photons were imaged. A bead was brought into focus, and the three holographic images were recorded. The bead was then scanned along the optical axis using the piezoelectric objective lens positioner over an axial range of $10\; {\mathrm{\mu} \mathrm{m}}$, with three images collected at every axial position. The first row of Fig. 10 shows how the density of the hologram fringes and hologram radius change smoothly as a function of the axial position of the emitter. In order to reconstruct the hologram to form an image of a single bead, three consecutive images with different phases are acquired which are then algebraically combined and reconstructed. The second and third rows of Fig. 10 show the results of reconstructing the images by reconstructing the hologram at the appropriate reconstruction distance. Results demonstrate that using the optimized optical setup, we can accurately image small particles ($100\; \textrm{nm}$) emitting low numbers of photons (∼4,200 photons) over larger axial ranges ($10\; {\mathrm{\mu} \mathrm{m}}$). The SNR for reconstructed images varies over the axial range, even when the emitter is emitting the same number of photons. The reconstruction SNR does not strictly increase as ${R_h}\; $ decreases. This can be attributed to the fact that the density of the interference fringes becomes too high and is under-sampled at the smallest ${R_h}$ values, adding noise to the reconstruction.
Fig. 10. 3D imaging of a single $100\; nm$ fluorescent bead at ∼4200 signal photons light level with light-sheet illumination SIDH. First row shows the PSH of the $100\; nm$ fluorescent bead at different axial planes, the images were acquired in one 5 ms frame with an irradiance of 0.02 kW/cm2. Second and third rows show the lateral and axial view of the reconstructed images, respectively. The SNR was calculated as the ratio of mean signal to the mean of the standard deviation of the background (mean sig/mean std).
4. Conclusion
We have performed simulations to analyze the dependence of localization precision on detected hologram size in SIDH using two different configurations, one with one spherical and one plane wave and another with two spherical waves to form the interference hologram. The results show that decreasing the size of the hologram significantly improves the SNR of holograms when background light is present, thereby improving the localization precision of SIDH. We also analyzed the relationship of the object's axial position and detected hologram radius with different camera distances using two different configurations. Based on our simulations, we have determined four guidelines that can be used to optimize our existing SIDH system to obtain a desired localization precision:
- 1. The detected hologram radius needs to be as small as possible to increase the SNR. But it needs to be at least $R\sim \sqrt {\lambda {z_r}} $ to capture the axial information in the hologram. For our system, at the light signal level of 6000 photons with 10 photons/pixel background noise, the detected hologram radius needs to be smaller than $1.38\; \textrm{mm}$ in order to be well detected and reconstructed.
- 2. The camera needs to be as close to the interferometer as possible in order to limit the variation of the detected hologram radius. The closer the camera is to the interferometer, the greater the possible axial range of imaging.
- 3. Because the camera position determines the imaging axial range and is fixed, the beam size before entering the interferometer needs to be minimized (∼$1.4\; \textrm{mm}$) to ensure the detected hologram radius stays small enough to be reconstructed and provide a good localization precision for the entire axial range.
- 4. The SIDH configuration with one plane and one spherical wave provides better localization performance than the configuration with two spherical waves under similar optical settings.
Based on our simulation results with the optimized parameters, SIDH can achieve a lateral localization precision of $5\; \textrm{nm}$ to $58\; \textrm{nm}$ and axial localization precision of $13\; \textrm{nm}$ to $80\; \textrm{nm}$ with an axial range of $10\; {\mathrm{\mu} \mathrm{m}}$ at low light signal conditions (light signal level of 6000 photons with 10 photons/pixel background noise). This indicates that SIDH could be a useful approach to 3D SMLM over large axial ranges. We have also performed experiments with the optimized SIDH system to demonstrate our simulation results. The reduction of beam size before entering the interferometer significantly improves the light efficiency of the system which makes it possible to detect as few as ∼4,200 photons using optimized SIDH with epi-illumination. Furthermore, light sheet illumination has been combined with our SIDH system to further reduce the background. We demonstrate that using light sheet illumination, as few as ∼2,120 photons can be detected in a hologram. We also demonstrate that by placing the camera close to the interferometer, we can successfully reconstruct a $100\; \textrm{nm}$ microsphere with ∼4,200 photons over a $10\; {\mathrm{\mu} \mathrm{m}}$ axial range. In our future work, we will further optimize our setup based on our simulation results. This includes positioning the camera even closer to the interferometer to achieve a larger axial range and modifying the setup to capture hologram data in a single shot. We then plan to experimentally demonstrate the localization performance of our setup using fluorophores, such as Alexa Fluor 647, under SMLM imaging conditions. Ultimately, our goal is to image biological samples with SIDH.
Funding
National Institute of General Medical Sciences (R21GM134462).
Acknowledgments
We thank Ruizhe Lin and Ning Liu for helpful conversations.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. The code used to generate the simulation results can be found in Ref. [25].
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