Introduction

Quantitative phase microscopy (QPM) has emerged as a revolutionary tool, enabling label-free imaging of biological specimens without the need for disruptive dyes or stains1. This technique preserves sample integrity and facilitates the visualization and precise measurement of nanoscale structures and dynamic cellular processes, offering invaluable insights into the intricate mechanisms within biological systems.

Despite the availability of diverse commercial QPM devices, they often present drawbacks. Advanced QPM setups can be complex and expensive, utilizing spatial light modulators2,3 or requiring holographic approaches (two-beam interferometry)4. This complexity can pose a barrier for researchers with limited budgets or those seeking simpler, more cost-effective, and robust vibration imaging solutions.

Furthermore, precise QPM techniques may exhibit slower imaging speeds compared to traditional microscopy methods. This is attributed to the necessity of capturing a minimum of three images for precise phase reconstruction5. This constraint becomes challenging when studying dynamic processes. Additionally, precise calibration is crucial for accurate quantitative phase measurements, as miscalibration can lead to result inaccuracies. This emphasizes the need for careful calibration procedures, further contributing to the overall costs.

Several authors have addressed different approaches to quantitative phase imaging techniques; for example, Popescu et al.6,7 showed an interference module composed of a modified 4-f system and a spatial light modulator (SLM) coupled with a conventional inverted microscope output. Other similar approaches using conventional microscopes are presented by Bon et al.8, where the interference module is based on diffraction gratings, directly retrieving the phase spatial derivative followed by numerical integration. Zhou et al.9 presented an interference system based on placing a meta-surface Zernike filter sequentially retrieving orthogonal phase derivatives to be later integrated and obtain the phase variation.

Moreover, dynamic interferometers have increasingly become a topic of interest in observing transient phenomena, and they now offer potential for commercial applications10,11. Their goal is to retrieve the phase instantaneously12,13, finding applications in bioimaging as well as in astronomy14. In the field of QPM techniques, one of the most popular approaches is based on digital holography microscopy (DHM) using Fourier transform demodulation methods. These systems rely on adding a high spatial frequency carrier to be later filtered out and obtain the measurement in a single-shot manner15. For example, off-axis DHM systems introduce an inclination angle between the object and the reference beam16,17,18. One of the drawbacks of this system is the loss of spatial resolution. For instance, Trusiak et al.19,20 mentioned that for a full separation of the twin imaging terms from the DC in the Fourier domain, the carrier frequency must equal at least three times the highest spatial frequency of the object. Additionally, the camera used should have a bandwidth of at least four times greater than that of the object.

Another approach to instantaneous phase retrieval uses a pixelated polarization camera for a single-shot approach, these systems are based on interferometric configurations such as the Fizeau interferometer or Linnik interference microscope objective21,22. However, these devices are costly, and the latter is limited to a single magnification. More examples of single-shot interferometers found in literature use different detectors and are commonly based on Fourier demodulation algorithms23 like the Hilbert transform24,25, where its main requirement is to add a spatial carrier on the interference fringes obtained in off-axis interferometers. In addition, all approaches that use a linear carrier require an image process to filter it, resulting in a loss of sample information.

In contrast, our proposed system does not use any particular components, such as SLMs or metasurfaces, and offers stability as a result of its common-path configuration. More importantly, we obtain phase information directly, avoiding using derivatives and preserving spatial resolution, unlike other methods involving additional numerical computations. Instead, we solely employ a liquid crystal cell, which acts as a Zernike-like phase filter in an actual common-path on-axis configuration without adding any extra frequency carrier or optical equipment like a beam splitter or mirrors that require extra alignment procedures.

We introduce an economical, precise, and robust quantitative phase microscopy system capable of analyzing phase samples in a single shot, enabling dynamic measurements. The system adopts a phase-contrast configuration, incorporating an azobenzene liquid crystal (LC) material as a Zernike filter, complemented by a polarization-sensing camera for detection. This work leverages the geometric phase characteristics exhibited by the azobenzene LC material, facilitating single-shot QPM. As mentioned before, the system eliminates the need for spatial light modulators or any moving objects to analyze the sample’s phase information, making it ideal for potential long-term monitoring and analysis of dynamic biological processes. We present the experimental results of our system and a comprehensive polarization characterization of the instrument.

The implementation includes a meticulous calibration process employing a USAF chart of a quantitative phase target26 as our gold-standard reference object. The results are shown in Fig. 1a, b. This procedure ensures measurement accuracy as we compare the estimated heights post-calibration with those specified by the manufacturer (AFM measurements). Our calculations exhibit a maximum deviation of ~27 nm from the gold-standard object height values, while the height variation through the chart area goes up to 42 nm, as illustrated in Fig. 1c, d.

Fig. 1: Quantitative phase microscopy measurements of USAF resolution chart following calibration.
figure 1

After calculating the correction term using the 350 nm chart, the estimated heights for the a 250 nm and b 350 nm charts were 267.7 ± 35 nm and 385.6 ± 32 nm, respectively. c Standard deviation of the estimated height values. d Comparison table of USAF chart height values.

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Results

The experimental configuration, outlined in Fig. 2, showcases the optical instrument designed for our simplified and efficient QPM system. The light source (λ = 640 nm) is followed by a polarization state generator (PSG) featuring a linear polarizer, a half-wave plate, and a quarter-wave plate. Upon illuminating the phase sample (object), our imaging setup employs a compound microscope equipped with a 20x objective (MO) and two positive lenses (L1, L2) with focal distances f1 = 75 mm and f2 = 120 mm, respectively. In the frequency plane of the 4f-system, an azobenzene LC cell acts as a Zernike filter; by leveraging its optical anisotropy27,28,29,30,31,32, we introduce a phase difference between low and high-frequency components33.

Fig. 2: Quantitative phase microscopy implementation.
figure 2

a The instrument consists of a polarization state generator (PSG), a 20x microscope objective (MO), a lens system [L1, L2], an azobenzene liquid crystal cell (LC) used as a birefringent Zernike filter, and the detector (polarized camera). The polarized camera captures four phase-shifted images in a single shot as the micropolarizer array carries a superpixel with four orientations of 0°, 45°, 90°, and 135°. b We obtained the four phase-shifted interferograms in a single shot of a USAF resolution chart, c shows its corresponding wrapped phase, and d its height variation.

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Figure 2 demonstrates our QPM implementation, which is characterized by its simplicity, calibration precision, and the capability to operate in a single shot.

The exceptionally large optical nonlinearity shown by the azobenzene LC generates a birefringent Zernike-like filter upon illumination34. This deliberate choice of LC material allows the system to work at low intensities to offer gentle illumination for objects, effectively minimizing the risk of phototoxicity. Specifically, the intensity of the beam illuminating the sample is denoted as I0 = 0.198 mW cm−2, while the intensity measured at the liquid crystal is ILC = 59.53 W cm−2.

For the single-shot implementation of QPM, we configured the PSG unit to produce circularly polarized light using a linear polarizer and a quarter-wave plate. At the detection unit, our pixelated polarized camera captured four phase-shifted images in a single shot. The micropolarizer array of the polarized camera is composed of a superpixel with four orientations (0°, 45°, 90°, and 135°, as illustrated in Fig. 2a, enabling the retrieval of phase information through a 4-step phase-shifting algorithm5.

As mentioned before, the USAF quantitative phase chart is used as the gold standard phase object. Fig. 2b shows the four π/2 phase-shifted interferograms captured in a single shot, and its corresponding wrapped phase obtained through the phase-retrieval algorithm5 is shown in Fig. 2c. Once we obtained the phase map, we corrected any aberration in the wavefront using Zernike polynomials, which possess the characteristic of being a complete set, orthogonal over the unit circle35, and subtracted the resulting wavefront from the phase map to calculate the sample’s topography as shown in Fig. 2d.

Optical anisotropy study

Experiments characterization

The optical anisotropy of the LC sample influences the single-shot QPM results. Consequently, our study is dedicated to comprehending this underlying mechanism. We achieved this by analyzing its polarization and interference response, systematically changing the fast-axis orientation of a linearly polarized beam, see Fig. 3a. We analyzed the response of 38 input states by adjusting the azimuthal angle of the input beam’s polarization in a range from 0° to 180°. For each capture, we obtained the input and output polarization states through a commercial polarimeter and the resulting interferogram with a high-resolution monochrome camera placed at the imaging plane of the instrument. In this case, the PSG unit consists of a linear polarizer (LP) and a rotating half-wave plate (HWP). Figure 3b shows both sets of measurements, the input (blue) and output (red) polarization states, mapped on the Poincaré sphere representation. These results effectively demonstrate the anisotropic behavior exhibited by the LC material.

Fig. 3: Optical anisotropy study.
figure 3

a The experimental setup consists of a polarization state generator (PSG), a 20x Microscope Objective (MO), a lens system [L1, L2], an azobenzene liquid crystal cell (LC) used as a birefringent Zernike filter, and a monochrome camera as the detector. b Polarization analysis of the instrument by varying the fast-axis orientation of a linearly polarized beam used as an input (blue) in the PSG and measuring its response (red). The input beam’s polarization varied from 0° to 180°, adjusting its azimuthal angle. The system’s response, primarily influenced by the LC cell, exhibits extreme points at opposite polarization states (U and U’, respectively, represented by the green dots). The trajectory (red line) traced by the polarization state crosses the Poincaré sphere’s equator at an angular distance of π/2 from each opposite state (point P).

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The compilation of the 38 measurements reveals a closed trajectory characterizing the evolution of polarization states at the output, exhibiting a tilt of m = −0.7335 from the z-axis. This trajectory manifests distinct extreme points, denoted as U and U’, representing opposite polarization states. Notably, the trajectory intersects the equatorial plane of the Poincaré sphere at an angular distance of π/2 from each of these opposite states (state P). According to Pancharatnam36,37,38,39,40, both the U and U’ components of the state P, as well as of any state along the trajectory, are in phase. Furthermore, the linear polarization states within the output polarization trajectory have an in-phase relationship28 too. Thus, confirming the presence of geometric phase behavior and demonstrating the viability of this phenomenon for single-shot QPM applications.

Stokes parameters

Additionally, we verified the geometric phase behavior by measuring the system’s polarization phase dependence, known as retardance, through Mueller matrix calculations. After determining the initial and final polarization states using Stokes parameters, we used Mueller calculus. This process included rotating a quarter-wave plate on the PSG unit and employing an inverse calculation to derive the corresponding matrix for our instrument41,42,43. This calculation resulted in the following matrix:

$$M=\left[\begin{array}{cccc}1 & 0 & 0 & 0\\ -0.294 & 1.078 & 0.125 & -0.188\\ -0.029 & -0.064 & 0.154 & 0.797\\ -0.043 & 0.012 & -0.102 & -0.5754\end{array}\right]$$

For this calculation, we used the Lu-Chipman Mueller matrix decomposition44 and obtained a linear retardance value of 99.8° oriented at 6.2°. In addition, the Mueller matrix also presents a linear polarization contribution of 0.29 oriented at 5.9°. In this case, the geometric phase is influenced by the cumulative effect of the retardance that we measured.

Contrast evaluation

To analyze the interference response of the instrument, we evaluated the contrast and phase variation obtained from the output interferograms32. This was achieved by systematically modifying the input polarization state and capturing the resultant interferogram using a monochrome camera. We validated the QPM measurements using the gold-standard phase object. The USAF quantitative chart has a nominal height of 250 nm, featuring a refractive index of n = 1.5226. To retrieve the phase information of this object, the set of captured interferograms must exhibit a discernible change in contrast between the background and the pattern32. Figure 4a illustrates the initial interferogram, while Fig. 4b presents the resultant contrast variation calculated as a function of the angle of polarization. We examined the phase variation induced by the object and the one that results from the presence of the LC filter. Employing an N-STEP random phase shifting algorithm that estimates the object phase and the phase filter changes through the interferograms45, we aimed to derive the phase variations between steps, depicted in Fig. 4c for each rotation of the fast-axis orientation within the beam. Additionally, Fig. 4d illustrates the overall phase variation introduced by the sample.

Fig. 4: Interferometric analysis of the instrument.
figure 4

a Illustrates the initial interferogram captured, while b presents the contrast variation observed in each interferogram. The maximum standard deviation obtained was σcontrast = 0.076. Additionally, employing an N-STEP phase demodulation algorithm, c and d, respectively, depict the wrapped phase of the object and the phase filter variations (at the LC sample).

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The orientations of the linear polarizers within the pixels (0°, 45°, 90°, 135°) of the polarized camera correspond to the polarization states ( ± S1, ± S2), respectively, over the Poincaré sphere. Among these states, linear polarization states at 0° and 90° ( ± S1) play a crucial role in defining the polarization trajectory of the output beam, representing the maximum and minimum contrast values within the interferogram set (Fig. 4b). Although the 45° and 135° states do not contribute to the trajectory of the output polarization, they are in phase with the first two states, considering their scalar product phase is zero40.

Calibration procedure

The system requires careful calibration, a process performed by using our gold-standard object: the quantitative phase target26. This calibration target has seven USAF charts with nominal heights ranging from 50 nm to 350 nm, and height values from AFM measurements specified by the manufacturer26, previously illustrated in Fig. 1d.

In our calibration protocol, we select the object with a higher signal-to-noise ratio phase measurement, in this case, the 350 nm 1951 USAF resolution test chart: group 6, element 2 (6.96 µm line width). After calculating the height map of the object, we estimate the correcting factor by obtaining the average experimental object’s height hexp = 764 nm and comparing it with the AFM-measured one (hafm = 385.6 nm). The system’s correction term γ is defined as γ = hafm·(hexp)−1, which, in this instance, was γ = 0.5047.

The initial estimated heights of the seven USAF charts were multiplied by the factor γ, and to assess measurement accuracy across the object’s area, we calculated the standard deviation, yielding the following values: h50 = 46.3 ± 19 nm, h100 = 89.2 ± 27 nm, h150 = 166.4 ± 42 nm, h200 = 238 ± 38 nm, h250 = 267.7 ± 35 nm, h300 = 350.3 ± 34 nm, and h350 = 385.6 ± 32 nm. These values suggest consistency, with higher accuracy observed for charts ranging from 200 nm to 350 nm. It’s essential to underscore that once the correction term is determined, we can proceed to obtain QPM information on other objects of interest.

Noise assessment

To further ensure the reliability of our system, which is essential for long-term monitoring scenarios, we conducted a comprehensive study of noise. We performed phase measurements of the 250 nm 1951 USAF resolution chart, group 6, elements 2 and 3 (6.96 μm and 6.2 μm line width, respectively). We used the polarized camera to capture a total of 185 measurements over a time span of 197 s, corresponding to a frame rate of ~1 fps. We analyzed the fluctuations in the background and phase object regions. Figure 5 presents the outcomes of the study: the mean background fluctuation—depicted in Fig. 5a, region 1—was determined to be σbck = ± 2.572 nm; on the other hand, fluctuations observed over the phase object—Fig. 5a, region 2—averaged σobj = ± 1.206 nm. Figure 5b illustrates the standard deviation across the phase object area, revealing notable height variations over the borders, due to abrupt phase transitions. However, it is worth noting that such phase differences are of the same order magnitude and are usually not present in biological samples.

Fig. 5: Noise determination.
figure 5

a Mean height values and b standard deviation obtained from 185 measurements conducted over a timelapse of 197 s. The mean fluctuations for the background and the phase object are σbck = ±2.572 nm and σobj = ±1.206 nm, respectively. The scale has been set to a maximum value of 12 nm for a better appreciation of noise values. We acknowledge the presence of higher values at the borders. c Three-dimensional representation of the height map illustrating the mean values across all measurements. d Profile depicting the mean standard deviation within the region marked in b.

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Additionally, Fig. 5c displays the mean height value obtained for each pixel across the 185 measurements, while Fig. 5d presents the mean value of the standard deviation within a delimited region in Fig. 5b, highlighting the significant height variations along the borders. Based on the fluctuation values observed, the smallest signal detectable by this system is estimated to be in the range of 1.206 nm to 2.572 nm. These results show the potential capability of our system to analyze common objects studied in the QPM field, such as red blood cell membrane fluctuations on the scale of tens to hundreds of nanometers46,47.

Regarding our acquisition frame rate, it is worth noting that the deliberate choice of azobenzene LC and the illumination characteristics of the system allow an imaging speed of up to ~1 Hz, enough for several biological dynamic processes, e.g., mitosis generally takes about 1 to 2 h48. However, to meet the demands of high-speed applications, our system can be tailored with an ultra-high-speed polarization camera, such as the Crysta PI-1P, which has been reported to reach frame rates of up to 42 kHz in applications such as flow and sound imaging49, and an LC material with lower optical nonlinearity, such as the dye-doped nematic liquid crystal 5CB, which in our previous research50 is reported to have an optical nonlinearity four orders of magnitude lower than azobenzene LC.

Capturing QPM of biological samples and dynamic phase changes

Showing the capabilities of our instrument to use in biological samples, we calculate the phase information obtained from cancerous HeLa cells. Figure 6a shows the four phase-shifted interferograms and Fig. 6b the 3D phase map representation. Cells induce a phase shift of up to 5.5π, approximately, at the center of the cell where the nucleus is located.

Fig. 6: Quantitative phase microscopy images of Cervical Carcinoma (HeLa) cells.
figure 6

a Phase-shifted images of HeLa cells were captured using a polarized camera that takes images at four angles of polarization: 0°, 45°, 90°, and 135°, and b the 3D phase map.

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To show the system’s capabilities to monitor dynamic phase changes, we analyzed the 3D phase information of an evaporating isopropyl alcohol droplet over a timelapse of 150 s. Figure 7 shows the thickness evolution of the droplet in a time gap of 50 s, going from ~325 nm to 50 nm when it evaporates. Figure. 7a showcases the calculated height maps, and Fig. 7b the thickness variation over the droplet during the process (see Supplementary Movie 1 for an extended video of the results showing 150 s of monitoring).

Fig. 7: Dynamic phase variation sample.
figure 7

a Three-dimensional height map of an evaporating isopropyl alcohol drop over a 50-s interval. b Temporal thickness variation of the sample at different time points, including standard deviations. The calculated thicknesses were th0 = 349 ± 24 nm, th10 = 257 ± 30 nm, th20 = 246 ± 32 nm, th30 = 176 ± 61 nm, th40 = 47 ± 19 nm, and th50 = 38 ± 26 nm.

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Conclusion

In this study, we developed a single-shot quantitative phase microscopy (QPM) system using cost-effective liquid crystal materials and a pixelated polarizer camera. The system capitalizes on the significant optical nonlinearity and anisotropy of azobenzene liquid crystals. We conducted a detailed analysis of this anisotropy, focusing on polarization studies to establish a robust operational framework. Our results clearly show a direct correlation between detected polarization states and the material’s geometric phase characteristics, highlighting the system’s precision and the effectiveness of our calibration processes.

Our QPM system offers several substantial advantages: it is compact, stable, and cost-effective, making it widely accessible for various research applications. Its design particularly benefits dynamic monitoring by enabling single-shot operations, which is crucial in bioimaging for prolonged sample observation with minimal cellular disruption. By opting for affordable materials and components over more complex and expensive alternatives like spatial light modulators, our system remains both practical and accessible.

We have implemented a meticulous calibration process that is integral to our system’s operation, ensuring both accuracy and reliability in QPM measurements. This process is essential for achieving consistency and reducing potential errors in-phase measurements, which is vital for obtaining dependable quantitative results.

In conclusion, our study demonstrates the feasibility of our system in precise, affordable, and noninvasive bioimaging and provides proof-of-concept for its application in extended monitoring and real-time 3D visualization. The versatility and efficiency of our affordable QPM approach offer promising prospects for advancing bioimaging technologies, particularly in enabling long-term, accurate, and noninvasive observations in diverse biological contexts.

Methods

Optical measurements

The optical experimental measurements in this manuscript were performed using a 10 mW Coherent StingRay Laser Diode Module (λ = 640 nm). Beam Co51 synthesized the liquid crystal (LC) material, azobenzene 495527, and the LC container was manufactured by Instec52. The LC cell specifications are a 20μm cell gap, a homogeneous alignment layer, and antiparallel rubbing. The polarized camera is a BFS-U3-51S5P-C USB 3.1 Blackfly® S Polarization Monochrome Camera with a 2448 × 2048 pixels resolution and a pixel pitch of 3.45 µm. For the optical anisotropy study, the detector was a FLIR BFS-U3-51S5M-C monochrome camera with a resolution of 1200 × 1920 pixels and a pixel pitch of 3.45 μm. The polarization was measured using a PAX1000VIS(/M) Thorlabs polarimeter.

Polarization measurements

The intensity of the beam illuminating the LC cell was maintained at 59.53 W/cm2 to induce the isomerization of the azobenzene liquid crystal molecules while avoiding thermal effects. The polarization azimuthal angle of the input beam was varied from 0° to 180° in 10° increments, resulting in 19 measurements. Additionally, 19 measurements were taken by varying the output beam’s polarization azimuthal angle from 0° to 180° in 10° steps, with the input linear polarization fixed, ending in a total of 38 measurements. The polarization states of both the input and output beams were measured and controlled using a Thorlabs PAX1000VIS(/M) polarimeter and a half-wave plate, respectively.

Gold-standard phase object

The gold standard phase object used was the 1951 USAF chart of a quantitative phase target. This target, manufactured by Benchmark Technologies, has a refractive index of n = 1.52. The nominal heights featured are 50 nm, 100 nm, 150 nm, 200 nm, 250 nm, 300 nm, and 350 nm. The company provides atomic force microscopy (AFM) measurements for the heights of each chart, which correspond to 57.2 nm, 108.2 nm, 165.7 nm, 211.9 nm, 271 nm, 324 nm, and 385.6 nm, respectively. These measurements, used in the calibration method, are specific to the part and serial number of the target.

Contrast determination

The contrast of the phase contrasted images described in Fig. 4a, b was determined by comparing the average gray levels of the background (bck) with those of the gold standard phase object (obj). Background (bck): To determine the background intensity, we selected a 300 × 400-pixel rectangle (120,000 values) where the phase difference is considered zero. We averaged the gray levels within this area, resulting in a maximum standard deviation of σbck = ±4.14 gray levels across all measurements. Object (obj): For the object intensity, we selected a 200 × 200-pixel area (40,000 values) from the gold standard phase object (group 6, element 2), where the phase difference was 0.406π rad. After averaging the gray levels within this area, the maximum standard deviation was σobj = ±8.47. Finally, we used the averaged gray level values from the background and the object signal to calculate the contrast (Eq. 1). Using σbck and σobj, we estimated the propagation of uncertainty, where we obtained a maximum standard deviation of σcontrast = 0.076, as depicted in Fig. 4b.

$$C=\frac{({bck}-{obj})}{({bck}+{obj})}.$$
(1)

Height map calculation

The phase map was calculated using a 4-step phase-shifting algorithm5 based on the intensity maps captured with the polarized camera. This wavefront was corrected for optical aberrations using the Zernike polynomials approach and removing the first five orders. Our procedure used the MATLAB function zernfun53. The height (h) was calculated using the equation h = (phase*ʎ) · [2π(n-n0)]−1, where ʎ is the wavelength, and n = 1.52 and n0 = 1 are the refractive indices of the target and air, respectively. To determine the height and standard deviation of the background, a square of 300 × 400 pixels was selected. Squares of different sizes, based on the shape of the phase object, were used to determine the object’s height and standard deviation. The standard deviation was calculated using the MATLAB function std. No further filtering was performed.

Noise study

To reinforce the reliability of the results and confirm the temporal accuracy and stability of our system, the noise was assessed by acquiring phase measurements over 197 s ( > 3 min) at a frame rate of approximately 1 Hz, resulting in 185 images. We used the gold standard phase object described above and its nominal height (250 nm). The background fluctuation value was determined by selecting a rectangle of 435 × 30 pixels, giving a total of approximately 2.5 million measurements, and calculating the mean value and standard deviation. This led to mean height and standard deviation values of havgbck = 13.5 ± 2.5 nm. The object fluctuation value was determined using a rectangle of 20 × 175 pixels, totaling about 650 thousand measurements. After calculating the mean height and standard deviation, we reached a value of havgobj = 251.8 ± 1.2 nm. The mean value and standard deviation were calculated using the MATLAB functions mean and std, respectively.

Biosample preparation

The preparation of the cervical carcinoma (HeLa) cells followed the method described in previous research54. HeLa cells were grown in RPMI 1640 medium, supplemented with 10% fetal bovine serum (FBS) and 1% penicillin-streptomycin, at 37 °C in a 5% CO2 environment. The cells were then plated onto coverslips in six-well plates at a density of 1 × 105 cells per well and incubated for 24 h under the same conditions. Following incubation, the HeLa cells were fixed with a formaldehyde solution. The HeLa cell lines used in this research were sourced from the American Type Culture Collection (ATCC), where samples are also commercially available. Note that these cell lines were authenticated by STR profiling and were not tested for mycoplasma contamination.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.