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Abstract
We present the experimental implementation of simultaneous spatial multimode demultiplexing as a distance measurement tool. We use this technique to estimate the distance between two incoherent beams in both directions of the transverse plane, and find a perfect accordance with theoretical predictions, given a proper calibration of the demultiplexer. We show that, even though sensitivity is limited by the cross-talk between channels, we can perform measurements in two dimensions much beyond the Rayleigh limit over a large dynamic range.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
The measure of the distance between two incoherent point sources is a well-known problem since it amounts to determining the resolution of an optical imaging system. In the general case, this resolution is given by the characteristic width of the point spread function of the system, which is proportional to its numerical aperture. Early resolution criteria derived in different fields—such as the Abbe [1], Rayleigh [2], or Sparrow [3] criteria—give slightly different definitions for the resolution power of a given system. As pointed out in [4], the sole information of the point spread function is in general not sufficient for the thorough characterization of the resolution power, and the signal-to-noise ratio plays a significant role in its determination. In the last decades, fields such as astronomy and microscopy have driven the efforts to access better optical resolutions with the emergence of techniques such as aperture synthesis or super-resolved fluorescence microscopy [5–8]. Analytical continuation [9,10] is another approach that has also been studied in the context of radio astronomy [11] and electronic information [12].
In a general imaging context, the problem of the measurement of the distance between two beams is expressed in terms that are very similar to those describing the measurement of the position of a single beam. In the latter case, sensitivity reaching the Cramér-Rao bound can be attained for intense beams using homodyne detection, with coherent states or squeezed states [13–15]. These techniques are used to perform field measurements on a spatial mode termed the detection mode. Using similar techniques, other parameters of a beam can be measured [16]. However, the use of field measurements results in the fact that these schemes are limited to measurements of a single optical source (or two coherent sources) and cannot be applied directly to the measurement of the distance between two incoherent sources.
To the best of our knowledge, the first mention of the use of spatial mode demultiplexing (Fig. 1) for the problem of distance measurement was made in [17]. In this work, the physical situation studied is similar to the one we are interested in (two incoherent point sources), but the problem addressed is that of hypothesis testing. A recent surge of interest was sparked in the last years regarding the quantum Cramér-Rao bound on the precision of the measurement of the separation between two incoherent, diffraction-limited, low-intensity sources [18–21]. The simultaneous measurement of other parameters has been investigated in [22–24]. The link between position determination and higher-order modes was also highlighted in a number of different contexts [25].
In [19,26–28], spatial mode demultiplexing coupled with intensity measurement is introduced as a measurement scheme that saturates the quantum Cramér-Rao bound. If the point-spread function of the optical system is approximated by a Gaussian, the optimum mode basis for demultiplexing is the Hermite-Gauss mode basis. In [19], it is shown that in the limit of an infinite number of measurement modes, the information gathered is equal to the Fisher information for any value of the separation between the sources. The single-mode version of this measurement scheme, which is concerned with the regime of very small separations, was experimentally tested in recent years [29–32]. However, the general case of multimode demultiplexing had not yet been investigated.
Fig. 1. Gaussian beam displaced in the transverse direction can be described using its decomposition on the Hermite-Gauss mode basis, as defined using the reference position, which is used to define the displacement. This mode basis does not depend on the value of the displacement $d$.
In this work, we measure the distance between two incoherent point sources using multimode spatial demultiplexing of nine modes, implementing the measurement scheme described in [18]. We demonstrate that in the case of small displacement, the derivations of the semi-classical and quantum Cramér-Rao bounds give the same result. Using a Multi-Plane Light Conversion (MPLC) system [33,34], we perform intensity measurements on the projections of the electric field on the Hermite-Gauss mode basis. The demultiplexing of more than one mode allows us to measure the distance between two sources even outside the range of very small displacement, and also enables the simultaneous measurement of different parameters such as, in this work, the projection of the separation between the two sources on both directions of the transverse plane.
2. SENSITIVITY OF DISTANCE ESTIMATION FOR INCOHERENT POINT SOURCES
For the measurement of the separation of two identical, incoherent, diffraction-limited sources—with Gaussian profiles, low intensities, and the important assumption that the centroid of the two points is known—the quantum Fisher information was calculated in [19–21] and found to be
with $N$ the total number of photons and ${w_0}$ the waist size.We point out here that in the limit of small separation and Poissonian noise, simple physical arguments can lead to the very same results, providing better physical intuition. Considering first a single Gaussian source and a small displacement, all the information relative to the displacement of the source is present in the mode corresponding to the derivative of the source profile with respect to the displacement [14], which we name the detection mode (see Fig. 1). Assuming Poissonian noise, we find optimal sensitivity can be reached measuring the field intensity or amplitude in the detection mode. It saturates the classical Fisher information associated with the measurement of the position, which, for one source emitting $N/2$ photons, has the following expression: $I_{F,1}^c = N/(2w_0^2)$ (see [35] and details in the Supplement 1). Now, with two independent sources, we can consider that the random variables consisting of the number of photons hitting the detector originating from each of the sources are independent—and the information yielded by two independent random variables is the sum of the information from each random variable. As a result, the classical Fisher information of the complete system can be written as
Hence, for this specific problem, we have $I_F^c = I_F^q$. Remarkably, in order to reach the corresponding Cramér-Rao bound for each of these beams, one has to measure the amplitude quadrature of the detection mode and, because the beams are identical, the detection mode is the same for both beams. So, measuring this detection mode leads to a quantum Cramér-Rao limited distance estimation, as the intensity of the sum is the sum of the intensities in the incoherent case. One should insist on the fact that, in the considered regime, this result is obvious. Indeed, it is well known that measuring the amplitude of the detection mode for a single beam leads to quantum Cramér-Rao limited sensitivity and that the distance between two incoherent beams is the sum of two symmetric displacements. We also recall that the analysis presented in this part is valid in the small displacement regime. In the experimental work presented in the next part, we depart from this regime and study the case of arbitrary displacements.
3. EXPERIMENTAL WORK
We developed an experimental setup to assess the performances of a demultiplexing system on the measurement of the distance between two incoherent sources. We still make the assumption that the position of the centroid of the two sources is already known. The key element of this setup is a MPLC system (Cailabs PROTEUS-C), which allows us to perform intensity measurements on several modes of the Hermite-Gauss basis, whose orientation defines the ${\boldsymbol x}$ and ${\boldsymbol y}$ vectors of the transverse plane. We introduce the angle $\beta$ between vectors ${\boldsymbol d}$ and ${\boldsymbol x}$. The two fields can be expressed as
Fig. 2. Beams that enter the MPLC system can be described as a superposition of modes ${u_{0 \le i \le 8}}$, whose intensity profiles are represented on the right. Each input mode is demultiplexed and sent to a spatially separated spatial mode ${v_{0 \le i \le 8}}$. Each of these modes is coupled into a separate single-mode fiber.
Figure 3 displays an example of the theoretical coefficients $c_{\textit{nm}}^2$ as a function of $d$ for $\beta = \pi /6$. It illlustrates the fact that displacements in both directions of the transverse plane can be measured. Using the MPLC system allows to directly and simultaneously measure these coefficients.
Fig. 3. Theoretical projections on the Hermite-Gauss mode basis for a displaced Gaussian beam as a function of the displacement $d$ with $\beta = \pi /6$.
Table 1. Specified (spe) and Experimental (exp) Values of the Cross-talk Values from Mode ${{\rm HG}_{00}}$ into the Higher-Order ${{\rm HG}_{\textit{nm}}}$ Modes
A. MPLC Design and Characterization
The MPLC system is a spatial mode multiplexing tool, which can in theory implement any kind of spatial mode basis change on a finite number of modes. The MPLC system we use is a 9-mode demultiplexer. The input modes are free-space co-propagating Hermite-Gauss modes (${u_{0 \le k \le 8}} \simeq {{\rm HG}_{0 \le n,m \le 2}}$) with a waist ${w_0} = 227\,\,\unicode{x00B5}{\rm m}$. The output modes are spatially separated, single- mode Gaussian beams that are coupled into single mode fibers (${v_{0 \le k \le 8}}$). This system thus allows us to perform intensity measurements on 9 modes of the Hermite-Gauss basis simultaneously. Although there is no theoretical limitation to the quality of the mode basis change an MPLC can implement, a fabricated MPLC is never perfect: the modes that are demultiplexed are slightly different from the theoretical modes (in this case, the Hermite-Gauss modes). One important measure of the MPLC’s performances is the measure of the cross-talk between channels. To measure these quantities, we inject a free-space ${{\rm HG}_{00}}$ beam in the system and measure the intensity in all the different output channels. The optical power measured in the mode ${{\rm HG}_{\textit{nm}}}$ is ${P_{\textit{nm}}}$ and the cross-talk of mode $(n,m)$ with the (0,0) mode is given by $r_{\textit{nm}}^{\textit{XT}} = 10*\mathop {\log}\nolimits_{10} ({{P_{\textit{nm}}}/{P_{00}}})$. Table 1 presents both the specified cross-talk values and the experimentally measured values. Another important difference with the ideal case is that the different “channels” of the MPLC do not have identical losses. These channel-specific losses can be evaluated when one possesses two identical systems by measuring them in a “back-to-back” configuration. (The two identical MPLCs are aligned so that the free-space output of one system corresponds to the free-space input of the other one. Light is then injected into a single-mode fiber of the first system and the intensity at the corresponding fiber of the second system is measured. This measurement scheme allows us to evaluate the channel-specific losses.) In our case, we chose to measure the losses by injecting light into the different single-mode fibers and measuring the intensity exiting the MPLC system. Table 2 gives the measured intrinsic efficiency of the MPLC system. It should be noted that this specific measurement scheme does not discriminate the cross-talk from the losses we wish to measure.
Table 2. Intrinsic Efficiency Coefficients of the MPLC System Relative to mode ${{\rm HG}_{00}}$ Efficiency
B. Assessment of the Role of Cross-talk in the Sensitivity of the Setup
In the context of our experiment, the cross-talk values of Table 1 represent an offset for the measurement of the displacement (see Supplement 1 for detailed derivation). Indeed, the intensity measured in the first-order mode is proportional to ${({d/{w_0} + p_{00}^{01}})^2}$ with $d$ the displacement and $p_{00}^{01} = \sqrt {{P_{01}}/{P_{00}}}$. We suppose, as is the case in this experiment, that the minimum crosstalk intensities are greater than the sensitivity of the power-meter we use. With the cross-talk values of the theoretical system used in the experiment, we can write:
The phenomena of cross-talk is not specific to the MPLC tool and plays a role in any demultiplexing system. Indeed, it finds its origin in both technical limitations and fabrication errors, as well as errors or approximation made in the determination of the spatial profile of the signal beam (for instance, the point-spread function of the optical system in the diffraction-limited case). Though small, these errors always introduce non-zero projections on the higher-order modes of the chosen basis meaning that at high power, the precision of a demultiplexing system will always be limited by the cross-talk. The characterization of such quantities is thus an essential step of the assessment of any kind of demultiplexing system. In the present paper, we consider the worst case scenario where cross-talk is a noise of unknown source setting a limit for the sensitivity. However, one should note that is possible to derive the Fisher information in the presence of cross-talk, given that the corresponding basis change is perfectly known. This leads to different sensitivity scaling but would, upon the utilization of the optimal estimator, lead to an improved sensitivity compared to what we estimate here [36].
C. Experimental Scheme
In our experimental scheme, the output of a fibered SLD (Thorlabs S5FC1005S, 50 nm bandwidth, $\lambda = 1550\;{\rm nm} $, output power 22 mW) is launched into a Mach-Zender-like setup. A mirror mounted on a micro-meter translation stage before the first beam-splitter of the setup allows us to translate the beam with respect to the optical axis. Two Dove prisms aligned at $\pi /4$ and ${-}\pi /4$ with respect to the plane of the optical table allow us to produce symmetrical displacement for the beams in both arms of the setup. The coherence length of the source is $L = c/({\pi \Delta \nu}) = 15\,\,\unicode{x00B5}{\rm m}$. The length of each arm of the Mach-Zehnder-like setup is of the order of 30 cm. After going through the setup, no interference could be observed between the two beams, which confirms that they are indeed incoherent. The two beams are injected into the MPLC system using a telescope, which scales the waist of the beams in order to match the designed waist size of the MPLC system. This configuration allowed us to perform both single- and double-beam displacement measurements.
Fig. 4. Normalized intensity measured on each of the MPLC output (solid line). In dashed line is the theoretical model for the extracted parameter of displacement and angle. In dashed dotted, we fit the data with an additional gain, which depends on the channel.
D. Experimental Results
Each measurement run is performed by introducing displacement in steps using the translation stage. For each step, we record the intensity of all the output modes. Figure 4 plots the intensity profiles of all the output modes normalized to the intensity of the ${{\rm HG}_{00}}$ mode as the distance between the two beams is scanned from 0 to 3 ${w_0}$. Corresponding theoretical plots of the intensity profiles are also shown, normalized using the gain coefficients of Table 2.
We first observe that all the plots correspond generally with what is expected theoretically, in particular, for the low displacement regime or the first-order modes where there is a perfect agreement between theory and experiment. This demonstrates that the MPLC is a highly valuable and compact tool for distance measurement. However, we see that for larger displacement, i.e., corresponding to more than one beam waist, discrepancies appear between the theoretical model and the experimental data. We note furthermore that this discrepancy appears mainly for modes whose output is the weakest, i.e., the modes contributing the least to the evaluation of the measurement. Also, we note that the main effect is a global gain effect, but that each curve preserves a shape that corresponds to the theoretical one. We thus chose to perform a fit of the data using free gain coefficients. Table 3 displays these coefficients and the corresponding plots are presented in Fig. 4. The final demultiplexing results correspond perfectly to the theoretical model and demonstrate the potential of the MPLC system for the determination of the distance between two incoherent sources, given a proper calibration procedure. We note that the assessment of the channel-dependent gain coefficient is a key element for an MPLC system to be used as a precise measurement instrument. Furthermore, the same measurements were performed for a single source (by blocking one of the arms of the interferometer), which showed similarly good results.
Table 3. Fitted Gain Parameters
Finally, we can now use the MPLC to measure source separation and consider the sensitivity of the measurement imposed by the cross-talk. Given the performed calibration, the inferred distance upon intensity measurement is simply obtained by inverting the curves displayed in Fig. 4. We plot in Fig. 5(a) the distance versus the measured intensity on modes ${{\rm HG}_{01}}$ and ${{\rm HG}_{02}}$, and introduce the error bars induced by cross-talk (i.e., given by the derivative of the curve times the amount of cross-talk). In order to appreciate the relative effect more clearly we considered a cross-talk equal to ${10^{- 1}}$. We see, as expected, that for ${{\rm HG}_{01}}$, precision remains constant from small displacement to $d/{w_0} \approx 0.5$ and then diverges, while for ${{\rm HG}_{02}}$ it diverges for small displacement. In Fig. 5(b) we plot the precision of the inferred distance from the measurement using either ${{\rm HG}_{01}}$ or ${{\rm HG}_{02}}$, respectively, now using the experimentally measured cross-talk. We see that, depending on the displacement, one should use either one or the other output (or a combination of both to obtain an optimal estimator, which we did not plot in the figure). Importantly, cross-talk imposed a sensitivity about ${2.10^{- 3}}$ for a measurement of $d/{w_0}$ on a broad range of values, up to about $d/{w_0} \approx 1.2$. We thus demonstrate highly sensitive measurement with a very large dynamic range.
Fig. 5. (a) Displacement as a function of the measured intensity in the modes ${{\rm HG}_{01}}$ and ${{\rm HG}_{02}}$: ${d_{0i}}(I)$. The error bars correspond to ${\pm}x{t_{01}} \times \partial {d_{0i}}/\partial\! I$ with $I$ the normalized intensity. For clarity, ${\pm} x {t_{01}}$ is taken equal to 0.1. (b) Inferred precisions ${p_{{{\rm HG}_{10}}}}$ and ${p_{{{\rm HG}_{20}}}}$ [i.e., spread of the error bars in (a)] as a function of the source separation for measurement using modes ${{\rm HG}_{01}}$ and ${{\rm HG}_{02}}$. For this figure, the ${\pm}x {t_{01}}$ value is the one measure experimentally in Table 1: $x{t_{\textit{ij}}} = 10 \times \mathop {\log}\nolimits_{10} (r_{\textit{ij}}^{\textit{XT}})$. We also plot, as a comparison, the precision corresponding to the Cramér-Rao bound for a direct intensity measurement ${p_{\textit{di}}}$ with 1 million photons, as well as the Cramér-Rao bounds for demultiplexing measurements with realistic levels of cross-talk and the same number of photons, using only the ${{\rm HG}_{10}}$ mode or the three first modes of the basis: ${{\rm HG}_{00}}$, ${{\rm HG}_{10}}$, and ${{\rm HG}_{20}}$.
A lot can be learned comparing this practical sensitivity curve to what can be derived from the Cramér-Rao bound of various measurement strategies, in particular, in the small separation limit. First, in our case, the precision is independent of the input light power, as cross-talk are considered as an offset. This remains true as long as this offset is higher than other noise sources, such as electronic noise or shot noise. For electronic noise, single pixel intensity detectors now have very large dynamics, and even in a range of moderate number of photons per integration time it can remain small. For shot noise, one can compare our limit to the optimal estimator derived in [36], taking into account experimental cross-talk and using only measurements from mode ${{\rm HG}_{01}}$ [see Fig. 5(b)], where it is plotted for one million photon per integration time. We see that it remains lower than our calibrated sensitivity. We have also plotted, in the same graph, this estimator now using a combination of three outputs of the MPLC, ${{\rm HG}_{00}}$, ${{\rm HG}_{01}}$, and ${{\rm HG}_{02}}$. It demonstrates that one can reach very high sensitivity on a broad dynamic range. This illustrates the potential of our apparatus, but requires a full knowledge of the cross-talk matrix, more complex to calibrate than the simple and practical limit presented in this article. Finally, we also plot, for the sake of comparison, the Cramér-Rao bound for a direct intensity measurement for one million photons. One can see that, in this regime, our measurement setup can offer a significant precision increase, while, in practice, in order to reach the direct imaging limit, the performances of multi-pixel detectors in terms of noise are an extra limiting factor [37].
As a final remark, we note that all the calculated sensitivities diverge for small separation, while it is not the case for our calibrated precision limit. This is because Fisher information goes to zero for zero separation. But this is a calculation artefact induced by the fact that one should consider that Fisher information is separation dependent, even to get the precision. In practice, and for our calibrated curve, there is a minimum measurable separation, given by cross-talk offset in our case. Then, when the actual displacement is smaller than this minimum value, the precision of the measurement remains equal to this minimum value.
4. CONCLUSION
In this work, we demonstrate the use of spatial mode demultiplexing as a measurement tool of the distance between incoherent sources. We show that a simple semi-classical derivation in the small displacement regime reaches the same qualitative and quantitative conclusion as a full quantum derivation, thereby giving intuitive insight into the use of intensity measurements and multiplexing for parameter estimation. We then present an experimental implementation of this protocol, conducted in the high-intensity regime and for both small and large displacements. The latter shows very good agreement with theory and validates its use. In doing so, we also highlight the role of the cross-talk between the different modes of the demultiplexing system as an important and unavoidable element to take into account in the performance evaluation of such systems. Spatial demultiplexing and intensity measurements allow us to access several parameters simultaneously, such as in this case the displacement in both directions of the transverse plane for values smaller than one beam waist. Higher-order spatial modes also present an interest for the measurement of the displacement in larger separation regimes or, in the small displacement regime, for alignment purposes. Furthermore, we note that this measurement scheme can also be adapted to the measurement of other spatial parameters by tailoring the demultiplexing mode basis to the parameter of interest.
Finally, we compare the sensitivity obtained taking the crosstalks as an offset to the measurement, corresponding to our experiment, to two other situations. One is the direct imaging Cramér Rao limited sensitivity for ${10^6}$ photons per measurement time, which shows that our experiment allows us to reach better performances than this idealized situation. For fair comparison, one should, for instance, also take into account electronic noise [37]. Furthermore, we also compare it to the best possible estimator derived in [36] in the presence of cross-talk, showing the potential of the technique given that the full cross-talk matrix is measured.
Acknowledgment
The authors wish to thank O. Pinel for discussions and constructive comments, as well as M. Gessner for help with optimal estimators.
Disclosures
The authors declare no conflicts of interest.
See Supplement 1 for supporting content.
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