Pockels cells enable efficient wide-field nanosecond imaging

Pockels cells enable efficient wide-field nanosecond imaging

Adam J. Bowman1, Brannon B. Klopfer1, Thomas Juffmann2,3 and Mark A. Kasevich1
1footnotemark: 1
2footnotemark: 2
3footnotemark: 3

Nanosecond temporal resolution enables new methods for wide-field imaging like time-of-flight, gated detection, and fluorescence lifetime. The optical efficiency of existing approaches, however, presents challenges for low-light applications common to fluorescence microscopy and single-molecule imaging. We demonstrate the use of Pockels cells as wide-field imaging gates for nanosecond temporal resolution with high photon collection efficiency. Two temporal frames are obtained by combining a Pockels cell with a pair of polarizing beam-splitters. This technique is extended to multiple temporal frames by using a re-imaging optical cavity with a tilted mirror to spatially separate cavity round trips. Wide-field single-molecule fluorescence lifetime spectroscopy and fluorescent lifetime imaging microscopy are demonstrated. These methods enable nanosecond imaging with standard optical systems and sensors, opening a new temporal dimension for low-light microscopy.

footnotetext: 1 Department of Physics, Stanford University, Stanford, California 94305, USA (email: abowman2@stanford.edu)
  2 Faculty of Physics, University of Vienna, A-1090 Vienna, Austria
  3 Department of Structural and Computational Biology, Max F. Perutz Laboratories, University of Vienna, A-1030 Vienna, Austria

Existing sensors for wide-field nanosecond imaging sacrifice performance to gain temporal resolution, failing to compete with scientific CMOS and electron-multiplying CCD sensors in low-signal applications. A variety of detectors currently access the nanosecond regime. Gated optical intensifiers (GOIs) based on microchannel plates are the most common. They allow for sub-nanosecond gating in a single image frame, and segmented GOIs can acquire multiple frames when combined with image splitting [1]. Gating into frames in this way limits overall collection efficiency to , and performance is further limited by photocathode quantum efficiency, MCP pixel density, and lateral electron drift [2, 3, 4]. Streak camera techniques have also been demonstrated for wide-field imaging, but they also require a photocathode conversion step and additional high-loss encoding [5, 6]. Silicon photodiode avalanche detector (SPAD) arrays are a promising solid-state approach, but they are currently limited to sparse fill factors and high dark currents [7, 8, 3].

The limitations of current nanosecond imaging techniques are particularly manifest in fluorescence lifetime imaging microscopy (FLIM). Fluorescence lifetime is a sensitive probe of local fluorophore environment and can be used to report factors like pH, polarity, ion concentration, FRET quenching, and viscosity. As lifetime imaging is insensitive to excitation intensity noise, dye concentration, and sample photobleaching, it is attractive for many applications. FLIM typically relies on confocal scanning combined with time-correlated single photon counting (TC-SPC) detectors[9, 10]. The throughput of TC-SPC is limited by the detector’s maximum count rate (typically 1-100 MHz), and confocal microscopy relies on high excitation intensities that can cause non-linear photodamage to biological samples[11, 12]. Given the disadvantages of existing wide-field and TC-SPC approaches, FLIM especially calls for the development of new, efficient imaging strategies to extend its utility for bio-imaging.

Here we demonstrate nanosecond imaging techniques – compatible with standard cameras – that have no inherent loss or dead time, allowing access to sub-framerate sample dynamics at timescales as fast as nanosecond fluorescent lifetimes. First, we show an all-photon wide-field imaging system based simply on a pair of polarizing beam-splitters (PBS) and a Pockels cell (PC). This can be used to create two temporal bins on any timescale – from nanoseconds to milliseconds – or to temporally localize bright events like photon bursts or molecule blinking at any sub-framerate timescale. Second, we demonstrate the use of a re-imaging optical cavity as a time-to-space converter to enable -frame ultrafast imaging when combined with a Pockels cell gate.

Figure 1: Wide-field efficient ultrafast imaging with a Pockels cell (a) Schematic of two temporal bin wide-field imaging for a single pixel fluorescence decay. Fluorescence emission is first polarized, a time dependent retardance (step function illustrated) is applied by the PC, and polarizations are split again before the sensor. Two pairs of outputs correspond to integrated intensity before (1, 3) and after (2, 4) a step function gate is applied in the illustration. Other modulations V(t) may be applied beyond a simple step function as described in the text. Equal optical path lengths are used in practice. (b) Gating efficiency () is calculated for a 30 mm KD*P Pockels cell as a function of incident angle from conscopic interference patterns, demonstrating high efficiency gating for wide-field imaging within 6 mrad half-acceptance angle.

Results

Gating with two temporal bins. Light from an imaging system is polarized with a beam-splitter, and the image associated with each polarization is aligned to propagate through different locations in a wide-aperture PC, as shown in figure 1. The PC provides an electric field-dependent retardance between the input light’s polarization components, mapping the temporal signature of the applied field onto the polarization state of the imaging beams [13]. A second PBS after the PC again splits the separated imaging beams, giving four image frames on the camera. The resulting images now encode temporal information, as shown in figure 1. To illustrate our method, we consider a step function voltage pulse applied at delay time with respect to a short ( ns) excitation pulse. The step function with edge at creates pairs of output images corresponding to integrated signal before and after .

In practice, we implement this configuration with a Gaussian gating pulse at in our experimental demonstrations as described in the below examples. This Gaussian gate still allows transient information to be captured. In fact, arbitrary V(t) may be applied to the PC for specific applications (see Discussion). Note that a gating pulse can be applied either as a single shot measurement or over repeated events integrated in one camera frame. Fluorescence lifetime may be recovered by either varying the gate delay to directly measure the fluorescence decay (see multi-label FLIM below) or by single-frame ratios of gated and ungated channel intensities (see single-molecule FLIM below). In cases where the PC aperture is limited, two separate PC crystals may be used instead of using different areas of the same crystal.


Imaging through Pockels cells. Standard PCs use thick (30-50 mm) potassium dideuterium phosphate (KD*P) crystals with longitudinal electrodes. These give high extinction ratios and are ubiquitous for Q-switching and phase modulation applications. Off-axis rays experience different birefringent phase shifts than those on-axis, limiting the numerical aperture (NA) of the crystal for wide-field imaging. In an image plane, the PC half angular acceptance limits NA to for small angles, where is magnification. In a diffraction plane (or infinity corrected space), the field of view (FOV) is instead limited to where is the imaging objective focal length. For example, we found that a 10 m FOV is achieved with a 1.4 NA microscope objective ( = 1.8 mm) and 40 mm thick longitudinal KD*P PC crystal ( mrad).

To assess gating efficiency, the impact of off-axis birefringence was simulated through the Muller matrix formalism [14] to arrive at a conscopic interference (isogyre) pattern, as viewed through crossed polarizers. Subtracting the transmitted intensity pattern at zero voltage () from that at the half-wave voltage () gives the gating efficiency (), where the useful NA of the PC is set by the region of high gating efficiency at lower angles [figure 1(b)]. The PC is treated as a linear homogeneous retarder with off-axis retardance determined by a coordinate transformation of the crystal axes (details in SI) [15]. In the future, angular acceptance may be improved by making the crystal thinner, with a 3 mm crystal increasing to 20 mrad, effectively removing NA and FOV restrictions. Further, complete cancellation of off-axis birefringence may be obtained by combining the negative uniaxial () KD*P crystal with a positive uniaxial () MgF static compensating crystal [15, 16]. This fully compensates for off-axis rays at and further improves the NA at (KD*P becomes biaxial with applied field, preventing full high voltage compensation). Figure S1 compares the effect of off-axis birefringence for thick, thin, and compensated KD*P crystals.

Figure 2: Multi-label FLIM (a) Direct measurement of fluorescence decays obtained by sweeping gate delay time for Orange (O, 4.9 ns), Red (R, 3.4 ns), Nile Red (NR, 3.1 ns), Infrared (IR, 2.3 ns) (Invitrogen) and Propidium Iodide (PI, 14 ns) (Bangs Laboratories, Inc.) beads. Fitted decay constants are given. The measured Gaussian instrument response function (IRF) is plotted in black. (b) Intensity image of a three-label wide-field sample of Orange, Nile Red, and Infrared beads (labels strongly overlap spatially) (c) Lifetime image reveals spatial distribution of the labels. Lifetime is measured by fitting the decay traces at each pixel (scale bars 10 m).

Multi-label FLIM. The two bin method has no intrinsic gating loss and allows for imaging onto any sensor. Fluorescence lifetime imaging is thus an ideal demonstration for the technique, where the PC gating pulse is applied after delay from the fluorescence excitation. Lifetime may then be determined by either varying the delay time over multiple frames (as used here) or by taking the single-frame ratio of pre- and post-gate intensities (following section). In figure 2 we image a mixture of three labels having different lifetimes measured individually to be 3.1 ns (2 m Nile Red Invitrogen beads), 4.9 ns (0.1 m Orange Invitrogen beads – background), and 2.3 ns (0.1 m Infrared Invitrogen beads – formed into crystals). For this data, the PC was located in the image plane, allowing for wide-field FLIM of bright samples at 0.1 NA and 20x magnification with 100 micron FOV. The sample is excited by laser pulses with duration 1 ns at 532 nm and 5 kHz repetition rate. The fluorescence decay on convolution with a Gaussian excitation pulse with FWHM pulse width takes the form

(1)

The PC applies a Gaussian gate function in our experiment with a pulse width of ns. By sweeping the delay time , the convolution of the fluorescence with the gating function is measured: . Therefore, temporal information such as fluorescence lifetime may be calculated by directly fitting the measured . Note that the convolution of excitation and gating functions gives a Gaussian instrument response function (IRF) with , measured directly in figure 2(a). The fitting approach samples the fluorescence decay at more time points and can be advantageous for brightly labeled samples compared to a two-bin measurement . This could be used to more effectively measure multi-exponential decays for instance.

Figure 3: Wide-field FLIM of Alexa Fluor 532 molecules (a) Gated channel intensity (b) Ungated channel intensity (scalebar 1 m). (c) Measured lifetimes are plotted along with total brightness for the numbered, diffraction limited regions with error bars indicated. The majority of these spots are single-molecule emitters as demonstrated by their photobleaching and blinking dynamics (see SI). A discussion of the error bars is included in Methods.

Wide-field FLIM of single molecules. For signal-limited applications relying on efficient photon collection, fluorescence lifetime is best determined by the ratio of gated and ungated intensity in a single frame. In figure 3, we demonstrate wide-field lifetime microscopy of Alexa Fluor 532 (Invitrogen) molecules on glass in a 10 x 10 m region. The measured lifetimes are consistent with both the ensemble lifetime of 2.5 ns and the large molecular variation seen in similar studies on glass [17, 18]. The PC is used in the infinity space of the microscope objective to apply the same Gaussian gating function at ns and 15 kHz repetition rate. The ratio of the gated and ungated intensity is given by

(2)

To calculate lifetime, this ratio is experimentally calculated by summing intensity in a region of interest around each molecule and then numerically calculating the associated lifetime. This approach allows single-molecule lifetime spectroscopy while maintaining diffraction limited resolution and efficient photon collection of photons per molecule (15 s exposure time). Figure 3(c) shows the estimated lifetime and total brightness for each numbered diffraction-limited emitter along with error-bars for the lifetime estimation. A CMOS machine vision camera (FLIR) is used for the detector. In this case, the angular acceptance of the PC limits the field of view to 10 m but still allows photon collection at 1.4 NA. single-molecule lifetime spectroscopy in wide field remains challenging with confocal approaches[19, 18, 17], whereas here it is readily demonstrated with PC gating and an inexpensive, high-noise camera.


Gated re-imaging cavities for multi-frame imaging. Nanosecond imaging with PCs can be extended beyond two temporal bins through the use of gated re-imaging optical cavities. We exploit the round-trip optical delay of a re-imaging cavity [20] combined with a tilted cavity mirror to provide nanosecond temporal resolution by spatially separating the cavity round trips. While imaging with -frames using GOIs is limited to collection efficiency, this re-imaging cavity technique enables efficient photon collection for low-light or single-photon sensitive applications. In related work, cavities have been used for single-channel orbital angular momentum and wavelength to time conversion[21, 22]. Aligned optical cavities have been used for time-folded optical imaging modalities like multi-pass microscopy [23, 24]. Our implementation instead employs a re-imaging cavity as the means to obtain temporal resolution for wide-field imaging.

Figure 4: Multi-frame nanosecond imaging with a cavity time-to-space converter (a) Externally gated, tilted mirror 4 re-imaging cavity. Image input is on small in-coupling mirror M1 in an image plane (i). M2 is tilted at a diffraction plane (f), spatially offsetting the images at the M1 plane each pass. Each round trip, images are passively out-coupled through partially transmissive mirror M2. (b) Normalized image intensities for four output images from the cavity showing a 4 ns round trip delay. (c) Cavity output on camera (CMOS) shows four images output from the PC analyzer for each round trip output from the cavity (numbered 1-4 as in figure 1). Four round-trips (rows i-iv) are displayed (scalebar 50 m). The sample is a mixture of drop-cast Nile Red 2 m beads ( 3.1 ns) and Orange 0.1 m beads (4.9 ns) that form the diffuse filaments. (d) The ratio of output frames (i,4) and (ii,4) in the gated channel at ns [red line in (b)] is used for single frame FLIM as described in the text. The two labels are readily differentiated (scalebar 10 m).

An image is in-coupled to a 4 cavity at the central focal plane by means of a small mirror M1 as shown in figure 4. The 4 configuration re-images the end mirrors (diffraction planes) every round trip. If one end mirror M2 is tilted by angle , the image position at the central focal plane after round trips is displaced by , where is the focal length of the 4 cavity. The angle is set such that the resulting images are not blocked by the in-coupling mirror. Each sequentially displaced image is delayed in turn by an additional round trip. To extract temporal information, the spatially separated images need to be either gated externally or simultaneously out-coupled from the cavity using a PC. In the externally gated scheme [schematically shown in figure 4(a)], light is passively out-coupled each round trip through a transmissive mirror. The spatially displaced images have a relative time delay based on their number of round trips , and an external gate is simultaneously applied to all delayed images to create temporally distinct frames. A step function gate V(t) allows lifetime measurement from the ratios of the time-delayed bins, similar to the two-bin case described above. Using the two-bin PC scheme as the external gate gives four image frames from each round trip output [figure 4(c)]. Photon efficiency, the ratio of detected photons to the number input to the cavity, with end-mirror reflectivity is given by after round trips, ignoring intracavity loss. This efficiency can be made very high for an appropriate choice of . For example, 87% efficiency is obtained with and . It should be noted that the intensity variation between the different frames is caused by partial transmission after round trips.

Figure 4(c-d) demonstrates the output from an externally gated tilted mirror cavity. Here a Gaussian gate pulse of width less than the round trip time is used. Lifetime in figure 4(d) is calculated from the ratio of two frames [figure 4(b) images (i,4) and (ii,4)] in the gated channel delayed by one cavity round trip time of 4 ns as . Alternatively, both gated and ungated frames could be included in the estimation to make use of all photons as in equation (2). It is interesting to compare -bin and two-bin lifetime methods in terms of their theoretical estimation accuracy (see figure 5). While the overall accuracies are closely matched, -bin methods have the advantage of a wider temporal dynamic range.

In a second gated cavity scheme (proposed in figure S5), there is instead no transmissive mirror, and all input light is simultaneously out-coupled from the cavity with an intra-cavity Pockels cell and polarizing beamsplitter. Such a scheme directly gives images with sequential exposures of and leaves no light in the cavity. Either a thin-crystal or compensated PC would be preferable for intra-cavity gating since the light passes through the PC each round trip.

These cavity imaging methods have the advantage of zero dead-time between frames and have no inherent limits on collection efficiency beyond intracavity loss. The externally gated cavity is straightforward to implement with thick-crystal PCs, but has the disadvantage of indirect temporal gating. Intracavity gating instead allows for true -frame ultrafast imaging where each round trip corresponds to one temporally distinct image frame. Round trip times from 1 to 10 ns may be achieved with standard optics.

Discussion

We have presented methods for two and -bin temporal imaging on nanosecond timescales using Pockels cells. Proof-of-concept experiments with single-molecule lifetime spectroscopy and wide-field FLIM demonstrate the potential to bring nanosecond resolution to signal-limited applications. PC imaging methods might expand the utility of fluorescence lifetime imaging microscopy. While it may seem that two-bin lifetime estimation accuracy will be limited in comparison to -bin TC-SPC, this is actually not the case. Both two-bin and -bin estimation are ultimately limited by photon counting shot noise. Though TC-SPC gains a large number of temporal bins from the bit depth of the ADC, this has little effect on estimation accuracy compared to a two-bin PC measurement. Figure 5 shows that -bin measurements have the advantage of wider lifetime dynamic range, but also that for most FLIM applications, a two-bin PC gate is nearly as accurate for mono-exponential decays. In fact, for a step function gate with realistic PC risetime relative to the fluorescent decay, near shot-noise limited estimation can be obtained over a wide-range of lifetimes.

For FLIM applications, nanosecond imaging with PCs enables large improvements in throughput over conventional TC-SPC. Even at low repetition rates, PC FLIM throughput readily surpasses TC-SPC. For example, a PC gated image at a low signal level of 1 photon/pixel/pulse at 15 kHz for a 1 megapixel image would take 750 times longer to acquire on a 20 MHz confocal TC-SPC system. This throughput advantage grows linearly with signal and pixel number. Wide-field, high throughput lifetime imaging with PCs could enable imaging of biological dynamics at high framerate. An example of a relevant application would be real-time imaging of calcium or voltage signaling [25, 26]. FLIM may also be applied as a clinical or in vivo diagnostic and wide-field gating may be readily compatible with endoscopic probes [27, 28]. We note that frequency modulated cameras have recently been developed to enable high-throughput FLIM, but these suffer from very high dark currents and read noise. While several inefficiencies are present in our current experimental demonstrations – such as low laser repetition rate, a non-ideal Gaussian gate, and a thick PC crystal – we expect future implementations to readily compete with state-of-the-art TC-SPC systems (see Methods for discussion of repetition rates). Our single-molecule results already demonstrate that PC gating can equal current systems in the low-light limit.

Figure 5: Lifetime estimation error (a) Cramér-Rao bound on lifetime estimation accuracy vs. number of bins for a monoexponential fluorescence decay. Dashed lines compare two to -bin lifetime measurements in the case where measurement window is 4 ns cavity round trip time. The red line is a two-bin lifetime simulation at optimal without finite measurement window (i.e. ideal step function gate). The blue line indicates the shot noise limit . (b) Simulated lifetime resolution for a realistic two-bin PC experiment with a 1 ns 10-90 logistic risetime PC gate and 1 ns , similar to the red line case in (a). Near shot noise limited estimation accuracy is obtained for PC risetime.

PC gating may further allow for new microscopy techniques by exploiting the nanosecond temporal dimension. For example, spectral information has been used to enable multi-labeling of biological samples, which proves important in understanding complex intracellular interactions[29]. Fluorescence lifetime may similarly provide an attractive temporal approach for unmixing multi-labeled signals [30]. Confocal FLIM has already been applied to this problem[31]. In studying single molecules, the capability to combine parallel lifetime measurements with spatial and spectral channels could allow for new types of high-throughput spectroscopy experiments to study molecular populations and photophysical states [32, 33, 34, 35, 36]. New information from lifetime could also be used to enhance spatial localization in superresolution microscopy [37]. Further, temporal gating could be used to suppress background autofluorescence occurring at short lifetimes [38].

While we have primarily focused on applications in fluorescence microscopy, we also note that PC nanosecond imaging techniques could be more broadly applied in quantum optics for fast gating, lock-in detection, event selection, or multi-pass microscopy [21, 23]. Other useful operation modes may be realized with the two-bin PC scheme by applying different modulations V(t). For example, a linear ramp of V(t) creates a unique mapping of time to output intensity to temporally localize photon bursts (e.g. molecule blinking) by polarization streaking (see figure S2). Periodic V(t) could also be used to implement wide-field lock-in detection. Traditional fast-imaging applications in plasma physics, LIBS spectroscopy [39], combustion, time-of-flight techniques, and fluid dynamics could also benefit from sensitive single-shot imaging [40, 41, 24, 6]. The -frame tilted mirror re-imaging cavity is particularly unique in its ability to perform single-shot ultrafast imaging of weak, non-repetitive events with zero deadtime between frames when using an internal PC gate (figure S5). It could also prove useful for wide dynamic range lifetime imaging. We note that strategies for time-of-flight imaging and LIDAR have also been recently demonstrated using PC modulation for the timing of reflected light pulses [42, 43, 44].

In summary, wide-field PC FLIM was demonstrated in single-frame and time trace modalities. single-molecule lifetime spectroscopy showed compatibility with signal limited applications. Finally, a new method using re-imaging cavities to enable ultrafast imaging by time-to-space conversion was shown. These techniques promise to open the nanosecond regime to signal-limited applications like wide-field and single-molecule fluorescence microscopy. Further, they are broadly compatible with any imaging system and sensor, giving potential applications in a variety of fields.



Methods

Experimental setup. FLIM was performed with a homemade fluorescence microscope. A Nikon PlanApo 100x VC 1.4 NA oil-immersion objective was used for single-molecule microscopy. All other data was taken with a 20x, 0.8 NA Zeiss PlanApo objective. Excitation pulses (1 ns FWHM) at 532 nm were generated by a Q-switched Nd:YAG at 5 kHz repetition (15 kHz for single-molecule data) (Standa Q10-SH). The detector for all measurements was a machine vision CMOS camera (FLIR BFS-U3-32S4M-C). A 10 mm aperture, 40 mm thick dual crystal longitudinal KD*P PC embedded in a 50 transmission line was used (Lasermetrics 1072). High voltage gating pulses were generated into 50 with an amplitude of 1.3 kV, 2.8 ns FWHM (FID GmbH) attaining 85 of and ns. Laser and HV pulser were synchronized with a DG 535 delay generator (Stanford Research Systems). Timing jitter was ps. Long transmission lines were used to prevent spurious pulse reflections during fluorescence decay. For single-molecule data, only two of the output frames (one output from first PBS) were used to maximize FOV through the PC thus limiting photon efficiency to . This is not a fundamental limitation of the technique, but was used to simplify our implementation with a limited single PC aperture.

The 4 re-imaging cavity used for the -bin demonstration used a 3 mm prism mirror (Thorlabs MRA03-G01) for in-coupling and = 150 mm ( ns). Passive out-coupling was through a neutral density filter of optical density 1 ( and ). Relay lenses were used to create an image plane at the PC and again at the camera (CMOS). Pick-off mirrors combined imaging beams generated by the two PBS with equal path lengths.

Conventional longitudinal PCs are limited to long pulse repetition rates in the 10’s of kHz by piezoelectric resonances. We note that ultimate repetition rate depends on high voltage pulse shape and duration and that thinner crystals may tolerate much higher repetition rates. Alternative designs using transverse electrode cells could also be employed.

Sample preparation. Alexa 532 single-molecule samples were prepared by drop casting dilute solution onto a hydrophobic substrate, then placing and removing a pristine coverslip. A dense field was photobleached to the point that single, diffraction-limited emitters were observed. Step-like photobleaching is shown in the SI along with blink-on dynamics. While multi-molecule emission within a diffraction limited spot was certainly also seen, a majority of the emitters were single molecules. Fluorescent bead samples were prepared by drop-casting directly onto coverglass. Invitrogen IR bead solution formed crystals as seen in figure 2.


Data analysis. Lifetimes were computed by both ratiometric calculation from image intensities (figures 3 and 4) and by time-trace fitting (figure 2). In ratiometric calculation, a numerically generated lookup table is used to convert between the measured ratio and estimated lifetime according to the equations in the text. Due to our specific and Gaussian gate pulse in figure 3, lifetimes below 1.0 ns are redundant with those above 1.0 ns in the numerical conversion. We report the larger lifetime value. In figure 2, timing trace fitting by least squares was used to estimate lifetime for the multi-labeled sample. Image frames used in timing trace lifetime calculation (figure 2) were normalized to the sum of the output channels to account for photobleaching over sequential exposures. The PC applies a time-varying retardance to linearly polarized input as , where the birefringent phase shift is determined by the applied voltage V, ordinary index of refraction , and the longitudinal electro-optic coefficient . Transmission in the parallel and perpendicular beamsplitter channels is and . Lifetime calculations account for the imperfect gating efficiency of the Pockels cell at 0.85 .

single-molecule gated and ungated intensities were determined by summing pixels corresponding to each molecule region of interest after subtracting a constant background determined from the mode of the image pixel distribution. Error bars in figure 3(c) account for shot noise in the gated (G) and ungated (UG) frames and for the background standard deviations and in the ratio standard deviation as

Background is the dominant error term here combining background fluorescence with a high camera dark current.

The theoretical Cramér-Rao bound for -bin lifetime estimation in a fixed time window of width may be directly calculated from a multinomial probability distribution [45]. Note that the multinomial distribution has the same statistics as the Poisson distribution once you have detected photons. The overall uncertainty in drops out of lifetime measurements. Fixed window bounds in figure 5 were found by setting for round trips. The shot noise normalized Cramér-Rao bound for bins takes the form


Acknowledgements We acknowledge helpful discussions with Yonatan Israel. This research was funded by the Gordon and Betty Moore Foundation. AB and BK acknowledge support from the Stanford Graduate Fellowship. AB acknowledges support from the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1656518.


Author Contributions AB conceived the idea and performed the experiments. BK and TJ performed initial work on PC gating. AB and MK prepared the manuscript.


Additional Information

Correspondence and requests for materials should be addressed to AB.


Competing financial interests

The authors declare no competing financial interests.

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