Abstract
Optical measurements that can achieve the fundamental quantum limits have the potential to improve the imaging of subdiffraction objects in important applications, including optical astronomy and fluorescence microscopy. Working towards the goal of implementing such quantum-inspired measurements for real applications, we experimentally demonstrate the localization of two incoherent optical point sources and the semiparametric estimation of object moments in the subdiffraction regime via spatial-mode demultiplexing (SPADE). In the case of two sources, we are able to estimate both of their locations accurately, not just their separation, by exploiting the asymmetric response of our SPADE device. In the case of semiparametric estimation, we demonstrate that, even if the source number is unknown, the moments of the source distribution can still be estimated accurately. Our demonstration paves the way towards the use of SPADE for optical superresolution in practical scenarios, where adaptive measurements are difficult and many parameters are unknown.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. INTRODUCTION
The resolution of incoherent sources with subdiffraction separations is a fundamental problem in optical imaging, with applications ranging from fluorescence microscopy to astronomy [1]. Quantum information theory [2] has recently lent a fresh perspective to the age-old problem [3,4]: the fundamental quantum limits turn out to be far away from what can be achieved by direct imaging, while a measurement of the photons in a judicious basis of spatial modes can approach the limits and offer substantial improvements. This quantum perspective has since inspired many follow-up studies—both theoretical and experimental—and grown into a field of research called quantum-inspired superresolution [4].
Numerous experiments on quantum-inspired superresolution have been reported in recent years [5–29], but most of them focus on the estimation of the separation between two point sources. This paper reports an experiment that uses spatial-mode demultiplexing (SPADE) in two dimensions (2D) to estimate both locations of two point sources, not just their separation, as well as the moments of a distribution of multiple point sources [7,30].
In theory, SPADE in the Hermite–Gaussian (HG) basis responds to the source locations in a symmetric manner, which introduces sign ambiguities to the estimation and prevents one from resolving the locations accurately without an adaptive scheme [3,31–33]. In practice, we discovered that the SPADE response in our experiment is asymmetrical because of experimental imperfections, and the asymmetry enables us to estimate both locations accurately. For a distribution of multiple point sources, we demonstrate that their moments can still be measured in the context of semiparametric estimation [7,30,34]. While our experiment is not close to the quantum limit, our findings pave the way towards the use of SPADE under realistic scenarios, where many parameters are unknown and adaptive measurements are difficult. We conjecture that, with shot-noise-limited sources and detection that are typical in optical astronomy [35] and fluorescence microscopy [36], our scheme can be close to the fundamental quantum limits while retaining the practical advantages observed in this experiment.
2. EXPERIMENT
Figure 1 illustrates our experimental setup. A collimated Gaussian beam is produced by a laser at wavelength 1550 nm (Thorlabs S1FC1550), a single-mode fiber, and a collimator (Thorlabs CFC2-C). The beam radius is given by 190 µm, or $\sigma = 95 \;{\unicode{x00B5}{\rm m}}$ in intensity standard deviation. We use the Gaussian beam to simulate the image-plane field produced by a point source and a diffraction-limited imaging system. SPADE is implemented by the multi-plane light conversion (MPLC) device (Cailabs PROTEUS-C), which is placed on a 2D translation stage ($2 \times {\rm Thorlabs}$ PT1/M) to produce 2D transverse displacements between the Gaussian beam and the MPLC device. A rotational stage (Zolix RSM82-1A) between the translation stage and the MPLC device provides additional alignment control. Each of the six MPLC output channels, corresponding to the first six ${{\rm HG}_{\textit{nm}}}$ modes with
is coupled to a photodetector (Thorlabs PDA20CS2), and the electrical signals are fed into an oscilloscope (Tektronix MSO46) for digital data collection. By scanning in 2D, we record effectively the six mode outputs as a function of the point-source location. The scanning is performed over a $2 \;{\rm mm} \times 2\;{\rm mm} $ area; the step size is 50 µm in each direction and refined to 10 µm for displacements between ${-}{100}$ and 100 µm.For each displacement ${\boldsymbol R} \in {\mathbb{R}^2}$ and each ${{\rm HG}_{\textit{nm}}}$ channel, a photodetector reading is a sample $Y_{\textit{nm}}^{(j)}({\boldsymbol R})$ of a random variable ${Y_{\textit{nm}}}({\boldsymbol R})$, the mean ${\bar Y_{\textit{nm}}}({\boldsymbol R})$ of which is theoretically given by
Because of experimental imperfections, the outputs in Fig. 2 exhibit asymmetric patterns with respect to the diagonal axes, unlike the ideal case indicated by Eq. (2). Moreover, because the wavefunction $\psi ({\boldsymbol r})$ of the beam in our experiment is not exactly equal to the mode wavefunction ${\phi _{00}}({\boldsymbol r})$ of the SPADE device, the beam can couple into the ${{\rm HG}_{02}}$ and ${{\rm HG}_{20}}$ outputs at zero beam displacement, leading to lobes at the origin. We also note that the variances of our data are nowhere near the shot-noise limit because of excess laser noise, excess detector noise, and mechanical noise. We find that these noise sources are time-varying and the variances of our data exhibit irregular behavior over time, so we do not assume that the variances are known in our data analysis.
3. RESULTS
A. Localization of Two Point Sources
We use the experimental data set to simulate various imaging scenarios. The first scenario is the localization of two point sources. For each trial of the simulated experiment, we randomly draw $K = 100$ sets of samples $\{(Y_{\textit{nm}}^{(k)}({{\boldsymbol R}_1}),Y_{\textit{nm}}^{(k)}({{\boldsymbol R}_2})):k = 1, \ldots ,K;(n,m) \in {\cal N}\}$ at two displacements ${{\boldsymbol R}_1}$ and ${{\boldsymbol R}_2}$ from our data set. Then
We observe that the RMSEs are all substantially below the diffraction limit defined by $\sigma$ and remain so for subdiffraction object sizes. Such results would be implausible if the experiment were perfect. In theory, the ideal mean outputs ${\bar Y_{\textit{nm}}}({\boldsymbol R})$ are all symmetric along the ${\hat{\boldsymbol u}} = ({\hat{\boldsymbol x}} + {\hat{\boldsymbol y}})/\sqrt 2$ and ${\hat{\boldsymbol v}} = ({\hat{\boldsymbol x}} - {\hat{\boldsymbol y}})/\sqrt 2$ directions, where ${\hat{\boldsymbol x}}$ and ${\hat{\boldsymbol y}}$ are unit vectors of the horizontal and vertical axes. Writing each source location as ${{\boldsymbol R}_j} = {u_j}{\hat{\boldsymbol u}} + {v_j}{\hat{\boldsymbol v}}$ in terms of the coordinates $({u_j},{v_j})$ along ${\hat{\boldsymbol u}}$ and ${\hat{\boldsymbol v}}$, the symmetry implies
To further study the sign problem, we simulate the outputs of a SPADE device that responds symmetrically along the ${\hat{\boldsymbol u}}$ and ${\hat{\boldsymbol v}}$ directions by replicating one quadrant of our data set $\{Y_{\textit{nm}}^{(j)}({\boldsymbol R}):j = 1, \ldots ,J;(n,m) \in {\cal N};u \lt 0\; {\rm and}\; v \lt 0\}$ to all four quadrants. We then repeat the two-point-localization analysis with the new data set. The yellow bars in Fig. 4 plot the resulting RMSEs in the same manner as Fig. 3(a), demonstrating the substantially increased localization errors in the symmetric case. To confirm that the increased errors are due to the sign problem, we take each location estimate ${{\check {\boldsymbol R}}_j}$ in the symmetric case, correct the signs of its $({u_j},{v_j})$ coordinates by comparing them with the true location, and compute the localization errors again. This new set of errors, with the sign problem artificially removed, is plotted as the blue bars in Fig. 4. The blue bars are comparable with Fig. 3(a), thus demonstrating that the sign problem is indeed the culprit for the increased errors in the symmetric case shown by the yellow bars. The similarity between Fig. 3(a) and the blue bars in Fig. 4 also demonstrates that our asymmetric system is able to both resolve the sign problem and maintain the nominal performance of a symmetric device.
B. Moment Estimation
The second scenario we study is semiparametric moment estimation [30,34]. Instead of assuming that the number of point sources is known and estimating their locations, the goal of semiparametric estimation is to estimate only certain summary statistics about the source distribution without the need to know the source number. Each SPADE channel measures the generalized moment
For each set of locations $\{{\boldsymbol R}_s\}$, we repeat the sampling and estimation procedure for $L = 100$ trials to obtain $L$ estimates $\{{\check \beta} _{\textit{nm}}^{(l)}:l = 1, \ldots ,L\}$ and compute the mean-square error (MSE) and the signal-to-noise ratio (SNR) given by
While SPADE can, in principle, offer substantial improvements over direct imaging in moment estimation when only photon shot noise is considered [30,34], we find it impossible to fairly compare our results here with direct imaging, since our experiment is dominated by excess noise and it is difficult to translate the noise level of our experiment to a per-pixel noise level for direct imaging in a fair manner. Even if excess noise is dominant, SPADE still offers the practical advantage of requiring only one photodetector for each moment, whereas direct imaging would require many pixels as well as more complicated data processing for moment estimation. We also note that the object moments naturally measured by our SPADE setup are close to the moments of even orders, and measurements in other optical mode bases, with the addition of controlled displacements [7] or interferometers [30,34], can be used to estimate other types of moments.
4. DISCUSSION AND CONCLUSION
In closing, we emphasize the novelties of our work relative to earlier experiments [5–28]. Most prior experiments considered only the estimation of the separation between two point sources. While our experimental setup is similar to that of Boucher et al. [16], our serendipitous use of the asymmetric SPADE response to estimate both locations of the two sources, not just their separation, as well as our demonstration of semiparametric moment estimation, are new to our knowledge. Brecht et al. have recently demonstrated quantum-limited simultaneous estimation of the centroid, offset, intensity imbalance for two pulses [19], but their demonstration was for pulses in the time domain, whereas ours is for imaging in the two-dimensional space domain. Another impressive recent experiment by Pushkina et al. demonstrated general superresolution imaging via SPADE [23]. It was for coherent sources, however, and the heterodyne detection they used would become much noisier with weak incoherent sources [39]. Our setup is envisioned to be translatable to more practical imaging scenarios in fluorescence microscopy, remote sensing, and astronomy, where photon shot noise may become dominant and SPADE can offer a fundamental advantage. The use of an asymmetric response to enable two-source localization is a practical alternative to adaptive measurements [33]; the asymmetry may be present naturally in an experiment or introduced intentionally by misalignment or another spatial-mode modulator. Semiparametric moment estimation, on the other hand, can offer important information about the object and contribute to Fourier analysis and general image reconstruction [38]. While it remains an open question whether these approaches can exactly attain the fundamental quantum limits to two-source localization and image reconstruction, our results demonstrate that the quantum-inspired superresolution techniques can be both experimentally feasible and applicable to general objects, thus paving the way towards their use in real applications.
Funding
National Research Foundation Singapore (QEP-P7).
Acknowledgments
We thank Wang Wei for useful discussions and Kenneth Y. W. Ng for his help with the early stage of the experiment.
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.
REFERENCES
1. G. de Villiers and E. R. Pike, The Limits of Resolution (CRC Press, 2016).
2. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).
3. M. Tsang, R. Nair, and X.-M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016). [CrossRef]
4. M. Tsang, “Resolving starlight: a quantum perspective,” Contemp. Phys. 60, 279–298 (2019). [CrossRef]
5. Z. S. Tang, K. Durak, and A. Ling, “Fault-tolerant and finite-error localization for point emitters within the diffraction limit,” Opt. Express 24, 22004–22012 (2016). [CrossRef]
6. M. Paúr, B. Stoklasa, Z. Hradil, L. L. Sánchez-Soto, and J. Řeháček, “Achieving the ultimate optical resolution,” Optica 3, 1144–1147 (2016). [CrossRef]
7. F. Yang, A. Tashchilina, E. S. Moiseev, C. Simon, and A. I. Lvovsky, “Far-field linear optical superresolution via heterodyne detection in a higher-order local oscillator mode,” Optica 3, 1148–1152 (2016). [CrossRef]
8. W.-K. Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh’s curse by imaging using phase information,” Phys. Rev. Lett. 118, 070801 (2017). [CrossRef]
9. J. M. Donohue, V. Ansari, J. Řeháček, Z. Hradil, B. Stoklasa, M. Paúr, L. L. Sánchez-Soto, and C. Silberhorn, “Quantum-limited time-frequency estimation through mode-selective photon measurement,” Phys. Rev. Lett. 121, 090501 (2018). [CrossRef]
10. M. Parniak, S. Borówka, K. Boroszko, W. Wasilewski, K. Banaszek, and R. Demkowicz-Dobrzanski, “Beating the Rayleigh limit using two-photon interference,” Phys. Rev. Lett. 121, 250503 (2018). [CrossRef]
11. M. Paúr, B. Stoklasa, J. Grover, A. Krzic, L. L. Sánchez-Soto, Z. Hradil, and J. Řeháček, “Tempering Rayleigh’s curse with PSF shaping,” Optica 5, 1177–1180 (2018). [CrossRef]
12. Y. Zhou, J. Yang, J. D. Hassett, S. M. H. Rafsanjani, M. Mirhosseini, A. N. Vamivakas, A. N. Jordan, Z. Shi, and R. W. Boyd, “Quantum-limited estimation of the axial separation of two incoherent point sources,” Optica 6, 534–541 (2019). [CrossRef]
13. M. Paúr, B. Stoklasa, D. Koutný, J. Řeháček, Z. Hradil, J. Grover, A. Krzic, and L. L. Sánchez-Soto, “Reading out Fisher information from the zeros of the point spread function,” Opt. Lett. 44, 3114–3117 (2019). [CrossRef]
14. J. Řeháček, M. Paúr, B. Stoklasa, D. Koutný, Z. Hradil, and L. L. Sánchez-Soto, “Intensity-based axial localization at the quantum limit,” Phys. Rev. Lett. 123, 193601 (2019). [CrossRef]
15. M. Salit, J. Klein, and L. Lust, “Experimental characterization of a mode-separating photonic lantern for imaging applications,” Appl. Opt. 59, 5319–5324 (2020). [CrossRef]
16. P. Boucher, C. Fabre, G. Labroille, and N. Treps, “Spatial optical mode demultiplexing as a practical tool for optimal transverse distance estimation,” Optica 7, 1621–1626 (2020). [CrossRef]
17. V. Ansari, B. Brecht, J. Gil-Lopez, J. M. Donohue, J. Řeháček, Z. Hradil, L. L. Sánchez-Soto, and C. Silberhorn, “Achieving the ultimate quantum timing resolution,” PRX Quantum 2, 010301 (2021). [CrossRef]
18. S. A. Wadood, K. Liang, Y. Zhou, J. Yang, M. A. Alonso, X.-F. Qian, T. Malhotra, S. M. Hashemi Rafsanjani, A. N. Jordan, R. W. Boyd, and A. N. Vamivakas, “Experimental demonstration of superresolution of partially coherent light sources using parity sorting,” Opt. Express 29, 22034–22043 (2021). [CrossRef]
19. B. Brecht, V. Ansari, J. Gil-Lopez, J. M. Donohue, J. Řeháček, Z. Hradil, L. L. Sánchez-Soto, and C. Silberhorn, “Experimental demonstration of multi-parameter estimation at the ultimate quantum limit,” in Quantum 2.0 Conference, M. Raymer, C. Monroe, and R. Holzwarth, eds. (Optica Publishing Group, 2020), paper QW6A.17.
20. S. L. Mouradian, N. Glikin, E. Megidish, K.-I. Ellers, and H. Haeffner, “Quantum sensing of intermittent stochastic signals,” Phys. Rev. A 103, 032419 (2021). [CrossRef]
21. S. De, J. Gil-Lopez, B. Brecht, C. Silberhorn, L. L. Sánchez-Soto, Z. Hradil, and J. Řeháček, “Effects of coherence on temporal resolution,” Phys. Rev. Res. 3, 033082 (2021). [CrossRef]
22. K. Santra, V. Nguyen, E. A. Smith, J. W. Petrich, and X. Song, “Localization of nonblinking point sources using higher-order-mode detection and optical heterodyning: Developing a strategy for extending the scope of molecular, super-resolution imaging,” J. Phys. Chem. B 125, 3092–3104 (2021). [CrossRef]
23. A. A. Pushkina, G. Maltese, J. I. Costa-Filho, P. Patel, and A. I. Lvovsky, “Superresolution linear optical imaging in the far field,” Phys. Rev. Lett. 127, 253602 (2021). [CrossRef]
24. M. Mazelanik, A. Leszczyński, and M. Parniak, “Optical-domain spectral super-resolution via a quantum-memory-based time-frequency processor,” Nat. Commun. 13, 691 (2022). [CrossRef]
25. U. Zanforlin, C. Lupo, P. W. R. Connolly, P. Kok, G. S. Buller, and Z. Huang, “Optical quantum super-resolution imaging and hypothesis testing,” Nat. Commun. 13, 5373 (2022). [CrossRef]
26. L. Santamaria, D. Pallotti, M. S. de Cumis, D. Dequal, and C. Lupo, “Balanced spade detection for distance metrology,” arXiv, arXiv:2206.05246 (2022). [CrossRef]
27. A. B. Greenwood, R. Oulton, and H. Gersen, “On the impact of realistic point sources in spatial mode demultiplexing super resolution imaging,” Quantum Sci. Technol. 8, 015024 (2023). [CrossRef]
28. F. Grenapin, D. Paneru, A. D’Errico, V. Grillo, G. Leuchs, and E. Karimi, “Super-resolution enhancement in bi-photon spatial mode demultiplexing,” arXiv, arXiv:2212.10468 (2022). [CrossRef]
29. L. Qi, X. Tan, L. Chen, K. Y. W. Ng, A. J. Danner, and M. Tsang, “Quantum-inspired superresolution for multiple incoherent optical point sources,” in Conference on Lasers and Electro-Optics (Optica Publishing Group, 2022), paper JTu3A.22.
30. M. Tsang, “Subdiffraction incoherent optical imaging via spatial-mode demultiplexing,” New J. Phys. 19, 023054 (2017). [CrossRef]
31. J. Řeháček, M. Paúr, B. Stoklasa, Z. Hradil, and L. L. Sánchez-Soto, “Optimal measurements for resolution beyond the Rayleigh limit,” Opt. Lett. 42, 231–234 (2017). [CrossRef]
32. M. Tsang, “Quantum limit to subdiffraction incoherent optical imaging,” Phys. Rev. A 99, 012305 (2019). [CrossRef]
33. M. R. Grace, Z. Dutton, A. Ashok, and S. Guha, “Approaching quantum-limited imaging resolution without prior knowledge of the object location,” J. Opt. Soc. Am. A 37, 1288–1299 (2020). [CrossRef]
34. M. Tsang, “Quantum limit to subdiffraction incoherent optical imaging. II. A parametric-submodel approach,” Phys. Rev. A 104, 052411 (2021). [CrossRef]
35. E. D. Feigelson and G. J. Babu, Modern Statistical Methods for Astronomy (Cambridge University, 2012).
36. J. B. Pawley, ed., Handbook of Biological Confocal Microscopy (Springer, 2006).
37. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST Handbook of Mathematical Functions (NIST and Cambridge University, 2010).
38. M. Tsang, “Efficient superoscillation measurement for incoherent optical imaging,” IEEE J. Sel. Top. Signal Process. 17, 513–524 (2022). [CrossRef]
39. F. Yang, R. Nair, M. Tsang, C. Simon, and A. I. Lvovsky, “Fisher information for far-field linear optical superresolution via homodyne or heterodyne detection in a higher-order local oscillator mode,” Phys. Rev. A 96, 063829 (2017). [CrossRef]