Beating the spectroscopic Rayleigh limit via post-processed heterodyne detection

Wiktor Krokosz; Wiktor Krokosz; Mateusz Mazelanik; Michał Lipka; Marcin Jarzyna; Wojciech Wasilewski; Wojciech Wasilewski; Konrad Banaszek; Konrad Banaszek; Michał Parniak

doi:10.1364/OL.514659

Introduction. Spectroscopic measurements serve as an indispensable tool for studying the properties of matter and light in a wide range of scientific disciplines, such as physics, chemistry [1], astronomy [2], biology [3], or metrology [4]. However, the achievable resolution of state-of-the-art spectrometers, including grating-based and Fourier spectrometers, is fundamentally constrained by the Fourier limit. This is especially important for emissive spectroscopy, where the properties of the light source cannot be modified. In such scenario, when one deals with complex spectral distributions (e.g., multiple overlapping spectral lines of different widths), this limit is hard to overcome. However, there are cases in which specific characteristics of the light sources under study can be exploited to tailor an optimal measurement and beat this restriction by achieving a sub-Fourier resolution [5–8]. This is analogous to a similar effect present in optical imaging, where a Rayleigh criterion, formulated in a modern approach using tools from estimation theory, limits the fundamental spatial resolution of an image [9,10]. The basic example is a system of two identical and mutually incoherent point sources separated by some unknown distance that is much smaller than the spatial spread of each source. Many approaches to measuring the separation with sub-Rayleigh precision have been proposed [11–15] and experimentally demonstrated [16–24]. Such methods utilize shot-to-shot coherence and focus on information encoded within the electromagnetic (EM) field of light rather than its measured intensity as in traditional techniques. However, all of them require intricate and specialized equipment crafted to specific experimental conditions, as they rely on projecting the field on a specific set of modes. Another recently proposed approach, which also yields superior results compared to traditional intensity-based methods, is to use homodyne or heterodyne sensing with tailored digital postprocessing [25,26]. A prominent advantage of such technique is a significantly simplified measurement setup, as the complexity is shifted to the data analysis step.

In this study, we present an experimental demonstration of spectral superresolution accomplished via heterodyne sensing, utilizing the previously proposed theoretical frameworks [25–27]. It is important to note that our approach differs from [18] as we do not utilize the shaping of the local oscillator beam. By leveraging the advantages of heterodyne sensing, our method provides a more accessible and practical means of achieving spectral superresolution in cases where the signal is not dominated by the shot noise, with applications in domains where fine spectral discrimination plays a pivotal role. Potential candidates are lasing structures [28], optomechanics [29–31], or quantum transducers (e.g., based on Rydberg atoms) [32–34] and sensors [35]. Our analysis method leverages the limitations of a typical approach, reminiscent of direct imaging, where only a power spectral density of the signal is used to analyze the spectral lines. At the same time, the experimental setup remains essentially the same, providing a prospect for immediate applications.

Heterodyne superresolving estimation. Our experimental demonstration is based on simulating and measuring the light source using an experimental setup presented in Fig. 1. By employing an acousto-optic modulator (AOM) together with an arbitrary signal generator, we are able to encode the desired EM field envelope described below, which is later measured by the time-resolved heterodyne sensor. In this section, we will describe our estimation approach. The detailed theoretical analysis of our method can be found in [25,26].

Fig. 1. Experimental setup that takes in a laser beam, which is then split into two using a half-wave plate $\left (\lambda /2\right )$ and a polarizing beam splitter (PBS). Both beams are then transmitted into one of the acousto-optic modulators (AOMs), which encode the local oscillator (LO) frequency as well as the programmatically set signal. The quarter-wave plate $\left (\lambda /4\right )$, convex lens, beam stop, and mirror constitute a double-pass AOM setup. Once the laser light is properly modulated, the two signals are then joined together and directed into a differential photodiode (DPD). Finally, the heterodyne signal is collected and I/Q demodulated.

Download Full Size | PDF

Let us consider two spectral lines, each emitting light in a Gaussian temporal mode. The time envelope of such EM field, assuming the lines are separated by $\varepsilon \sigma$, is given by

(1)$$A(t+t_c) = G(t) e^{i\omega_ct} \left(A_1e^{i \phi_1}e^{{-}i \frac{\sigma\varepsilon t}{2}} + A_2e^{{-}i \phi_2}e^{i \frac{\sigma\varepsilon t}{2}}\right),$$

where $A_i,\,\phi _i$ for $i=1,2$ are respectively the modulus value and phase of the complex amplitude emitted by each line, $\omega _c$ and $t_c$ are the displacements in the frequency and time domains, and $\sigma$ is the characteristic spread of the Gaussian temporal mode:

(2)$$G(t) = \sqrt[4]{\frac{2\sigma^2}{\pi}} e^{-\sigma^2t^2}.$$

We will specifically consider the cases of incoherent thermal radiation and coherent light averaged over the relative phase. In the former scenario, one has to use Eq. (1) with amplitudes and phases $(A_i,\phi _i)$ randomly chosen in each realization from a complex Gaussian distribution for each line, i.e., $A_i e^{i\phi _i} \sim \mathcal {CN}(0, \mathrm {A_0^2/2})$. The latter scenario involves taking $A_1=A_2=A_0/\sqrt {2}$ and $\phi _1=\phi _2=\phi _0$ in Eq. (1) with the phase $\phi _0$ randomly chosen from a flat distribution on the $2\pi$ period in each realization, while $A_0$ is kept fixed, i.e., only the relative phase is changing as the global phase referred to LO is unknown and changes in time.

Let us now consider the time-resolved heterodyne detection of an optical field described above. The analog heterodyne signal is digitalized and in-phase (I) and $\pi /2$-quadrature (Q) demodulated in real time and stored as complex signal traces $Z(t_i)=I(t_i)+iQ(t_i)$. Our estimation procedure begins by calculating the projections of the $Z(t_i)$ onto the first and second Hermite–Gauss (HG) modes:

(3)$$u^{(0)}(t) = G(t),\; u^{(1)}(t) = 2\sigma t G(t),$$

where the Gaussian width $\sigma$ is set the same as in the input signal. Crucially, the second HG mode, in contrast to the first, is antisymmetric and therefore sensitive to discrepancies of the signal from $u^{(0)}(t)$, which describes a single line. For very small separations between the pulses, we can assume that almost all of the signal is distributed between these two modes [27]. The projections are calculated using a simple discrete sum as

(4)$$z^{(k)}(t_r, \omega_r) = \sum_i Z(t_i)^* u^{(k)}(t_i - t_r) e^{i\omega_r t} \Delta t,$$

where $\omega _r,\,t_r$ are some spectral and temporal displacements.

Next, for each set of projections specified by the chosen values of $(\omega _r,t_r)$, we compute a variance $V^{(k)}=\mathrm {Var}[z^{(k)}(t_r, \omega _r)]$. The variances represent the amount of signal in a given spatial mode, but also contain a shot-noise contribution and are in measurement units (e.g., Volts at photodetector). To recover the dimensionless signal contribution in the $u^{(1)}$ mode relative to the contribution in the fundamental $u^{(0)}$ mode, we calculate a minimal normalized and noise-subtracted variance:

(5)$$V_{\varepsilon} = \min_{t_r,\omega_r}\frac{V^{(1)} - V^{(1)}_{\mathrm{noise}}}{V^{(0)} - V^{(0)}_{\mathrm{noise}}},$$

where $V^{(k)}_{\mathrm {noise}}$ represents variance from a projection of a shot-noise calculated for measurement without the input signal. The values obtained from $t_r,\omega _r$ that minimize the variance serve as the centroid estimates that preserve high precision even for small separations [25]. Finally, the variance $V_{\varepsilon }$ can be used to estimate the separation value:

(6)$$\hat{\varepsilon} = 4\sqrt{\max\{V_{\varepsilon}, 0\}}.$$

Note that negative $V_\varepsilon$ values may arise when the signal contribution to $V^{(1)}$ is almost negligible. The estimation error may be found by employing the statistical bootstrapping method [36] described in Sec. 4.

Finally, to estimate the average number of registered photons, we calculate the signal-to-noise ratio $\mathcal {S}$ by comparing the variances in the fundamental mode for signal and noise:

(7)$$\mathcal{S} = \frac{V^{(0)_{}} - V^{(0)}_{\mathrm{noise}}}{V^{(0)}_{\mathrm{noise}}},$$

for $(t_r,\omega _r)$ obtained from Eq. (5). This approach is valid for small $\varepsilon$ and for $\varepsilon >1$ can be extended by including higher-order modes in the formula. $\mathcal {S}$ is directly related to the number of photons $\bar {n}$ in the experiment by $\bar {n}=\mathcal {S}$. Notably, by following the above procedure, from a single measurement, we estimate the centroids, separation, and average photon flux.

Precision limits. The maximum precision limit of any estimation method based on a given measurement scheme is specified by the Cramér-Rao bound (CRB) [37]. Assuming the outcomes $z$ are distributted according to a probability distribution $P(z|\varepsilon )$, the limit is equal to

(8)$$(\Delta \varepsilon)^2\geq 1/F,\quad F=\int \frac{1}{P(z|\varepsilon)}\left[\frac{\partial P(z|\varepsilon)}{\partial \varepsilon}\right]^2 dz,$$

where $F$ is the Fisher information (FI).

The CRB for the scenario described in the previous section can be derived from the measured quadrature statistic for both phase-averaged coherent and thermal states. However, direct calculation may be not be straightforward, particularly in the former case as the resulting quantum state is non-Gaussian [38,39]. For this reason, we rather decompose the signal into HG modes and deal with each component separately.

The amplitude contribution from each HG mode can be calculated directly from the source spectral envelope as

(9)$$c_k(\varepsilon)=\int \mathrm{d}\omega \tilde{G}\left(\omega-\frac{\sigma \varepsilon}{2}\right)\tilde{u}^{(k)}(\omega)=\frac{e^{-\frac{\varepsilon^2}{32}}}{\sqrt{k!}}\left(\frac{\varepsilon}{4}\right)^k,$$

for $k \in \{0,1\}$ and where $\tilde {G}$ and $\tilde {u}$ denote the time-domain Fourier transform of $G$ and $u$, respectively. Each source contributes with this amplitude to the measured signal in a given mode, and as a result, they interfere. For thermal sources, the result of the interference is still a thermal state with the mean photon number rescaled by $|c_k(\varepsilon )|^2$. Therefore, the quadrature distribution in the $u^{(1)}$ mode for the received state with $\bar {n}$ photons on average is given by

(10)$$P_\mathrm{th}(z|\varepsilon)=\frac{1}{\pi (2|\alpha(\varepsilon)|^2+1)}\exp\left(-\frac{|z|^2}{2|\alpha(\varepsilon)|^2+1}\right),$$

where $z$ is a complex variable and $\alpha (\varepsilon )=\sqrt {\bar {n}/2}c_{1}(\varepsilon )$ represents the average amplitude with $|\alpha (\varepsilon )|^2$ being the average number of photons in the mode. The factor of $2$ that divides the number of photons is the result of using a heterodyne measurement [27] in which each quadrature is measured on half of the signal and is accompanied by its individual shot noise. In the case of phase-averaged coherent states, the output state is more complex with the quadrature distribution in the following form:

(11)$$P_{\mathrm{coh}}(z|\varepsilon) = \int_0^{2 \pi}\int_0^{2 \pi} \frac{\mathrm{d}\phi_1 \mathrm{d}\phi_2}{4\pi^3} e^{-|z-\alpha(\varepsilon)(e^{i\phi_1}-e^{i\phi_2})|^2}.$$

The precision bounds may be found by plugging Eq. (10) and (11) into the CRB in Eq. (8). Importantly, one often speaks about superresolution when the precision of separation estimation outperforms what can be achieved with direct sensing (DS). The bound for the latter can be found by normalizing the modulus square of the EM field envelope and treating it as a probability distribution $P_{\mathrm {DS}}\sim |\tilde {A}(\omega )|^2$ for which one can evaluate the corresponding CRB.

Using the outcome distributions derived above, the per-photon FI for each mode can be calculated as

(12)$$\mathcal{F}_{\mathrm{th/coh}}=\frac{1}{\bar{n}}F\left[P_{\mathrm{th/coh}}(z|\varepsilon)\right].$$

For small separations ($\varepsilon \ll 1$), the $u^\mathrm {(1)}$ mode contains almost all the information, and the other modes can be neglected. The formula for FI for thermal states can be evaluated analytically and yields

(13)$$\mathcal{F}_{\mathrm{th}}=\frac{\bar{n}\left(\varepsilon/4\right)^2(\left(\varepsilon/4\right)^2-1)^2}{4\left(e^{\left(\varepsilon/4\right)^2}+\bar{n}\left(\varepsilon/4\right)^2\right)^2}.$$

In the case of phase-averaged coherent states, the FI from Eq. (12) is evaluated numerically to give the CRB for particular experimental conditions. The average number of photons $\bar {n}$ in the experiment is simply $\mathcal {S}$.

Results. For both thermal and phase-averaged coherent states, we generated Gaussian pulses with small spectral separations and measured them using the heterodyne detector. In the former case, the amplitudes and phases were sampled from a complex normal distribution, while in the latter, the amplitude was fixed and the phases were selected to uniformly cover the entire $2\pi$ period. To ensure the reliability and statistical significance of our results, we performed measurements for different mean signal intensities and varied the separations $\varepsilon$. For each parameter set, approximately $N\approx 10^6$ pulse repetitions were collected, providing a robust statistical sample.

Since we want to verify the saturability of the CRB, we need to calculate the variance of the estimator, given by Eq. (6). To do that, we employ the statistical bootstrapping method, where we generate 1000 ensembles of size $N$ from the calculated projections. Once we get the estimator values from each sample, we can calculate the precision by taking the inverse of the variance of the set multiplied by $\mathcal {S}$ and $N$.

The results obtained for the thermal states are shown in Fig. 2(a). We can clearly observe that the obtained precision taken as the inverse of the estimator’s variance reliably follows the CRB (solid curve). For reference, we also show the CRB for DS (dashed line). From the plots, we see that the obtained precision beats the limit for DS proving the effectiveness of our method for achieving spectral superresolution in the case of thermal states.

Fig. 2. Plots representing the measured separation estimation precision in the case of thermal (a) and phase-averaged coherent (b) states. Each of the plots corresponds to a single $\mathcal {S}$ value, shown in ascending order. The data points are compared with the CRB given by the calculated FI for heterodyne sensing (solid curves) and DS (dashed curves). The error bars correspond to two standard deviations of the evaluated measured precision. The insets represent the estimator bias for each data point in the main plot.

Download Full Size | PDF

For low $\mathcal {S}$, we observe the effect of noise that increased the estimator bias for the smallest separations [25]. Additionally, for larger separations and $\mathcal {S}$, we observe that our estimator does not reach the CRB. We attribute this to fluctuations of the estimated $\mathcal {S}$, Eq. (7), that also influence the normalized variance, Eq. (5), used in the estimator. This is limited by the statistics from the shot-noise measurement, which could be improved. Currently, the number of shot noise and signal measurements is equal, but one can increase the number of shot-noise measurements per experiment realization to estimate $\mathcal {S}$ more precisely. We can draw similar conclusions from the results obtained for phase-averaged coherent states depicted in Fig. 2(b). In this case, the precision for $\varepsilon \ll 1$ follows the CRB, and for larger separations, we observe a larger decrease in precision. This is most likely a result of the non-optimal estimator that does not grasp the correlations between the two quadratures that become important for larger separations, as captured by the FI that is slightly larger when compared to the FI for thermal states.

Summary. In this study, we have experimentally demonstrated the heterodyne super-resolving measurement of two spectral lines for two different optical states: thermal and phase-averaged coherent. Our data analysis, based on numerical decomposition of measured quadrature distributions into the first HG modes, enables estimating frequency separations and temporal and spectral centroid positions as well as average photon numbers from a single measurement set. The achieved precision of estimation was found to be in line with CRB derived for this measurement scheme, validating the effectiveness of our method for achieving spectral superresolution. It is important to note that our method requires many photons per mode. While this may limit its applicability in certain scenarios, there are cases where this requirement can be met, and superresolution can be effectively achieved. These include optomechanical systems where mechanical modes are usually probed with a coherent light [40] or lasing microstructures [28].

In conclusion, our work contributes to ongoing efforts to overcome the fundamental constraints of state-of-the-art spectrometers, offering a more accessible and practical means of achieving spectral superresolution. This has potential applications in various scientific disciplines where fine spectral discrimination plays a pivotal role. In particular, our data analysis method may be readily applied in heterodyne spectroscopy setups which have so far used power spectral densities to recover properties of relevant spectral lines.

Funding

Fundacja na rzecz Nauki Polskiej (MAB/2018/4); Office of Naval Research Global (N62909-19-1-2127); Narodowe Centrum Nauki (2021/41/N/ST2/02926).

Acknowledgment

The “Quantum Optical Technologies” project is carried out within the International Research Agendas programme of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund. This research was funded in whole or in part by National Science Centre, Poland 2021/41/N/ST2/02926. ML was supported by the Foundation for Polish Science via the START scholarship.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are available in Ref. [41].

REFERENCES

1. K. B. Bec, J. Grabska, and C. W. Huck, Anal. Chim. Acta 1133, 150 (2020). [CrossRef]

2. C. R. Kitchin, Optical Astronomical Spectroscopy (Routledge & CRC Press, 1995).

3. R. K. Sahu and S. Mordechai, Appl. Spectrosc. Rev. 51, 484 (2016). [CrossRef]

4. J. Levine, Rev. Sci. Instrum. 70, 2567 (1999). [CrossRef]

5. J. M. Donohue, V. Ansari, J. Řehávcek, et al., Phys. Rev. Lett. 121, 090501 (2018). [CrossRef]

6. V. Ansari, B. Brecht, J. Gil-Lopez, et al., PRX Quantum 2, 010301 (2021). [CrossRef]

7. M. Mazelanik, A. Leszczyński, and M. Parniak, Nat. Commun. 13, 691 (2022). [CrossRef]

8. A. Boschetti, A. Taschin, P. Bartolini, et al., Nat. Photonics 14, 177 (2020). [CrossRef]

9. M. Tsang, R. Nair, and X.-M. Lu, Phys. Rev. X 6, 031033 (2016). [CrossRef]

10. S. Zhou and L. Jiang, Phys. Rev. A 99, 013808 (2019). [CrossRef]

11. R. Nair and M. Tsang, Phys. Rev. Lett. 117, 190801 (2016). [CrossRef]

12. M. Tsang, Contemp. Phys. 60, 279 (2019). [CrossRef]

13. M. R. Grace, Z. Dutton, A. Ashok, et al., J. Opt. Soc. Am. A 37, 1288 (2020). [CrossRef]

14. J. Rehácek, Z. Hradil, D. Koutný, et al., Phys. Rev. A 98, 012103 (2018). [CrossRef]

15. M. Shah and L. Fan, Phys. Rev. Appl. 15, 034071 (2021). [CrossRef]

16. R. Nair and M. Tsang, Opt. Express 24, 3684 (2016). [CrossRef]

17. M. Paúr, B. Stoklasa, Z. Hradil, et al., Optica 3, 1144 (2016). [CrossRef]

18. F. Yang, A. Tashchilina, E. S. Moiseev, et al., Optica 3, 1148 (2016). [CrossRef]

19. M. Parniak, S. Borówka, K. Boroszko, et al., Phys. Rev. Lett. 121, 250503 (2018). [CrossRef]

20. J. Frank, A. Duplinskiy, K. Bearne, et al., Optica 10, 1147 (2023). [CrossRef]

21. S. A. Wadood, K. Liang, Y. Zhou, et al., Opt. Express 29, 22034 (2021). [CrossRef]

22. Y. Zhou, J. Yang, J. D. Hassett, et al., Optica 6, 534 (2019). [CrossRef]

23. A. Pushkina, G. Maltese, J. Costa-Filho, et al., Phys. Rev. Lett. 127, 253602 (2021). [CrossRef]

24. X.-J. Tan, L. Qi, L. Chen, et al., Optica 10, 1189 (2023). [CrossRef]

25. C. Datta, M. Jarzyna, Y. L. Len, et al., Phys. Rev. A 102, 063526 (2020). [CrossRef]

26. C. Datta, Y. L. Len, K. Lukanowski, et al., Opt. Express 29, 35592 (2021). [CrossRef]

27. F. Yang, R. Nair, M. Tsang, et al., Phys. Rev. A 96, 063829 (2017). [CrossRef]

28. P. Jaffrennou, J. Claudon, M. Bazin, et al., Appl. Phys. Lett. 96, 071103 (2010). [CrossRef]

29. R. W. Andrews, R. W. Peterson, T. P. Purdy, et al., Nat. Phys. 10, 321 (2014). [CrossRef]

30. T. Bagci, A. Simonsen, S. Schmid, et al., Nature 507, 81 (2014). [CrossRef]

31. R. A. Thomas, M. Parniak, C. Østfeldt, et al., Nat. Phys. 17, 228 (2021). [CrossRef]

32. S. Borówka, U. Pylypenko, M. Mazelanik, et al., Nat. Photonics 18, 32 (2023). [CrossRef]

33. A. Kumar, A. Suleymanzade, M. Stone, et al., Nature 615, 614 (2023). [CrossRef]

34. H.-T. Tu, K.-Y. Liao, Z.-X. Zhang, et al., Nat. Photonics 16, 291 (2022). [CrossRef]

35. M. Jing, Y. Hu, J. Ma, et al., Nat. Phys. 16, 911 (2020). [CrossRef]

36. B. Efron, The Annals of Statistics 7, 1 (1979).

37. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prenitce Hall, 1993).

38. A. Allevi, M. Bondani, P. Marian, et al., J. Opt. Soc. Am. B 30, 2621 (2013). [CrossRef]

39. S. Olivares, Eur. Phys. J. Spec. Top. 203, 3 (2012). [CrossRef]

40. Y. Tsaturyan, A. Barg, E. S. Polzik, et al., Nat. Nanotechnol. 12, 776 (2017). [CrossRef]

41. W. Krokosz, M. Mazelanik, M. Lipka, et al., “Replication data for: Beating the spectroscopic Rayleigh limit via post-processed heterodyne detection,” Harvard Dataverse, 2023, https://doi.org/10.7910/DVN/F4LRZR.

Beating the spectroscopic Rayleigh limit via post-processed heterodyne detection

Abstract

Funding

Acknowledgment

Disclosures

Data availability

REFERENCES

Data availability

Cited By

Figures (2)

Equations (13)

Optics Letters