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Abstract
The image of an optical point source is blurred due to light diffraction so that estimating small displacements of the point source with direct imaging demands elaborate processing on the observation data of a camera. Using quantum parameter estimation, we show that for the imaging systems with a real point spread function, any measurement basis constituted by a complete set of real-valued spatial-mode functions is optimal for estimating the displacement. For small displacements, we can concentrate the information about the value of displacement to the measurement of a few spatial modes, which can be selected in terms of the Fisher information distribution. We use digital holography with a phase-only spatial light modulator to implement two simple estimation strategies that are mainly based on the projection measurement of two spatial modes and the readout of a single pixel of a camera.
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1. Introduction
The Rayleigh criterion gives, in a heuristic notion, the minimum distance that two incoherent optical point sources can be distinguished [1]. A modern description of the resolution power for optical point sources can be established in the framework of parameter estimation theory, where the resolution is assessed by estimation accuracy of locating point sources [2,3]. When the separation of two incoherent optical point sources is smaller than the Rayleigh limit, the estimation accuracy of direct imaging rapidly decreases. A great advantage of using the parameter estimation framework is that it enables optimization over measurements [4–6]. In 2016, Tsang et al. [4] derived the fundamental limit of resolving two incoherent point sources by quantum parameter estimation theory and showed that this limit can be attained by the spatial mode demultiplexing (SPADE) measurement [4,5]. Since then, many schemes of measuring nontrivial spatial modes for resolving two incoherent optical point sources have been proposed and experimentally demonstrated, e.g., see Ref. [7–28]. Besides the superresolution problem for two incoherent optical point sources, modal measurement and analysis is beneficial for many applications [29,30]. For example, the small displacement of an optical beam in the $\text {TEM}_{00}$ mode can be measured by a homodyne detecting with a local oscillator in the $\text {TEM}_{10}$ mode [31–33].
In this work, we consider the problem of measuring small displacements of an optical point source with digital holography. For imaging systems with a real point-spread function (PSF), we will show that any measurement basis with real-valued mode functions is optimal for estimating the displacement and a small subset of PSF-adapted modes can capture a majority of information about small displacements. We first use the quantum parameter estimation theory to analyze the spatial modes that contribute a great amount of information for estimating small displacements. Then, we utilize digital holography to implement the spatial mode projection measurement for estimating the small displacement of an optical point source. The experiment mainly uses a phase-only spatial light modulator (SLM) and a single-pixel readout of charged-coupled device (CCD) camera. We theoretically analyze and experimentally demonstrate the performances of two estimation strategies. The first strategy, which is based on the $\text {TEM}_{00}$ and $\text {TEM}_{10}$ modes, can estimate the absolute value but not the sign of the displacement. Moreover it is vulnerable to the dark counts and background noise. The second strategy, measuring a specific linear combination of $\text {TEM}_{00}$ and $\text {TEM}_{10}$ modes improves the performance from the two above-mentioned aspects.
This paper is organized as follows. In Sec. 2, we analyze the measurement performance for the displacement of the image of an optical point source in the framework of quantum parameter estimation. In Sec. 3, we implement the experimental scheme of spatial mode projection measurement with digital holography to estimate small displacements of an optical point source. In Sec. 4, we summarize our work.
2. Theoretical analysis in the framework of quantum parameter estimation
Let us start by considering a spatially-invariant diffraction-limited imaging system and a weak thermal optical point source. For simplicity, our theoretical analysis assumes quasimonochromatic scalar paraxial waves and one-dimensional imaging problems. The imaging problem can be easily generalized to two-dimensional cases. Suppose that the point source is transversely shifted from the optic axis by an unknown value $s$. The quantum state on the image plane for each temporal mode is given by $\rho \approx (1-\epsilon ) |0\rangle\langle{0}| + \epsilon |{\psi _s}\rangle\langle{\psi _s}|$, where $\epsilon$ is the probability of having a photon in a temporal mode, $|0\rangle$ is the vacuum state, and $|{\psi _s}\rangle$ is the one-photon state. We assume that $\epsilon$ in independent of $s$. Denote by $\psi (x)$ the (amplitude) point spread function (PSF), where $x$ is the image-plane coordinate normalized with respect to the magnification factor of the imaging system. The one-photon state can be written as
where $|{x}\rangle$ denotes the eigenket of the photon position on the image plane. This weak source model is the same as the two incoherent point source model given in Ref. [4], except that we here consider only a single point source.Our aim is to estimate the unknown value of $s$ by processing and measuring the optical field on the image plane. According to the quantum Cramér-Rao bound [34–39], the variance $\textrm {Var}[\hat s]$ of any unbiased estimator $\hat s$ is bounded from below as
where $F$ and $\mathcal {F}$ are the classical and quantum Fisher information, respectively. Suppose that the observation data used for estimating $s$ contains $M$ temporal modes. The classical Fisher information [40] can be calculated asWe shall show that, for real-valued PSFs, any projective measurement with respect to a complete set of spatial modes that admit real-valued wave functions is optimal for estimating the displacement $s$. Denote by $|{z}\rangle$ the state vector for the spatial mode corresponding to the outcome $z$. Since all the spatial modes to be measured is complete, $\sum _z |z\rangle\langle{z}|$ equals to the identity operator on the Hilbert space. Note that for real-valued PSFs, we have $\langle {\psi '_s}|{\psi _s}\rangle =0$, which can be derived by differentiating both sides of the normalization condition $\langle {\psi _s}|{\psi _s}\rangle =1$ with respect to $s$. Therefore, the quantum Fisher information Eq. (4) can be expressed as
The above-mentioned measurement is universally optimal for all values of $s$. Let us take the Gaussian PSF as an example:
where $\sigma$ represents the characteristic width of the PSF. For such a case, the quantum Fisher information about $s$ is $\mathcal {F} = N \sigma ^{-2}$. This quantum Fisher information can be attained by direct imaging, for which the measurement outcomes is equivalent to the image plane coordinate $x$ and its probability is $p(x) = |\langle {x}|{\psi _s}\rangle |^2$. Besides, let us consider the SPADE measurement [4] with respect to the Hermite-Gaussian modes $|{\phi _q}\rangle$ whose wave functions areAlthough the SPADE measurement and direct imaging are both optimal for estimating $s$ in the sense of attaining the quantum Fisher information, they have different features when considering the constituents of Fisher information over measurement outcomes. Define the Fisher information elements of a photon detected in the $q$th mode as
whose summation/integration over all measurement outcomes $z$ is the Fisher information about the parameter $s$. For direct imaging, we get implying that the profile of the Fisher information elements for different values of the displacement $s$ is unchanged except for a translation along with $s$. As shown in Fig. 1, the Fisher information elements is extended over a wide region, whose main part is approximately from $s - 3\sigma$ to $s+3\sigma$ and maximum occurs at $s\pm \sqrt 2 \sigma$. Therefore, in order to make the Fisher information close to its quantum limit, we always need to collect the data of camera pixels that have a substantial Fisher information elements. For the SPADE measurement, using Eqs. (8) and (9), the Fisher information elements for each outcome $q$ isThe distribution of the Fisher information elements of the SPADE measurement has different features comparing with that of direct imaging. As shown in Fig. 2(a) and (b), the probabilities and Fisher information elements of each outcomes are considerably different as the value of $s$ changes. Particularly, for small displacements $s$ that are less than the characteristic length $\sigma$, the outcome $q=1$ plays a dominant role in the Fisher information elements, as shown in the lower panel of Fig. 2. In the limit of $s \to 0$, the Fisher information elements of $q=1$ takes all the Fisher information. This opens up the possibility of attaining the vast majority of the Fisher information with only a few outcomes for small displacements.
Fig. 1. The probability and Fisher information elements for locating a single point sources with direct imaging.
Fig. 2. The probabilities and Fisher information elements for the reduced SPADE measurement. In the panels (a) and (b), the reduced SPADE is performed with the Hermite-Gaussian modes $\phi _0$ and $\phi _1$. In the panels (c) and (d), the reduced SPADE is performed with the modes $\phi _\pm = (\phi _0 \pm \phi _1) / \sqrt 2$. The black solid lines in the panel (b) and the panel (d) stand for the sums of the two Fisher information elements therein.
We now construct a simple measurement and estimation strategy. Let us only choose the outcomes $q=0$ and $q=1$ of the SPADE measurement and discard other outcomes. Under this reduced SPADE measurement, the Fisher information $F'$ per detected photon is given by
Note that $F'$ captures the majority of the Fisher information of a detected photon under the full SPADE measurement, when the displacement $s$ is smaller than the characteristic length $\sigma$. Denote by $N_0$ and $N_1$ the numbers of occurrence for the outcomes $q=0$ and $q=1$ during an observation session, respectively. We construct the estimator based on $N_0$ and $N_1$ as follows. Assume that the total power of the optical filed is stable. The ratio $N_1 / N_0$ of the populations converges to the ratio $p(1) / p(0)$ of the probabilities of the outcomes. According to Eq. (8), we get $p(1) / p(0) = s^2 / (4\sigma ^2)$. We thus construct the estimator
which satisfies the consistency criterion of estimator [41]. We will use this estimator to analyze the observation data from the experiment.The above analysis can be generalized to other PSFs. A two-mode projection measurement can be constructed by the PSF mode and its normalized first-order derivative, which was used by Paur et al. [9] to estimate the separation between two incoherent optical point sources. In Ref. [28], Rehacek et al. proposed a method of construct efficient measurement basis from the derivatives of the PSF. This method can also be used to find a small number of PSF-adapted modes that extract a great amount of Fisher information about the small displacements of an optical point source.
The reduced SPADE measurement with the two lowest order Hermite-Gaussian modes has two defects for estimating small displacement of an optical point source. First, it cannot discriminate the sign of the displacement, as the probability of outcomes corresponding to each mode is the same for $s$ and $-s$. Second, it is vulnerable to the dark current in the photon detectors and the background noise. This is because the probability of the outcome $q=1$ is very small for $|s|\ll 1$ (see Fig. 2(a)). To overcome these two defects, we consider the measurement with respect to the following two spatial modes:
Note that $\phi _+$ and $\phi _-$ are orthogonal with each other. Such modes were previously used to recover the odd moments of the incoherent source distribution [42]. For a Gaussian PSF, the probability that a detected photon is in the $\phi _\pm$ mode is
When we include these two spatial modes in the measurement basis and discard the information of other spatial modes that are in the complement, the Fisher information is the same as that for the zeroth and first order Hermite-Gaussian modes (see Fig. 2(b) and Fig. 2(d)). So this reduced SPADE is also optimal for estimating small displacements of a single optical point source. Besides, the spatial models $\phi _\pm$ is neither symmetric nor anti-symmetric with respect to the origin point and thus we can discriminate the sign of the displacement from the distribution of outcomes (see Fig. 2(a) and Fig. 2(c).) Considering that the ratio $N_+ / N_-$ of the populations converges to the probability ratio $p(+) / p(-) = (s+2\sigma )^2 / (s-2\sigma )^2$, we can use the estimator
when the value of displacement is in the range $-2\sigma \leq s \leq ~2\sigma$.3. Experimental implementation with digital holography
We in Fig. 3 illustrate the experimental setups for estimating the displacement of a point source by utilizing a digital holographic approach for the spatial mode projection measurement. Unlike the genuine reduced SPADE measurement, we here measure the projection information $|\langle {\phi _0}|{\psi _s}\rangle |^2$ and $|\langle {\phi _1}|{\psi _s}\rangle |^2$ in turn but not simultaneously. The experimental arrangement is similar to that used in Ref. [9] for estimating the separation between two incoherent point sources by utilizing an amplitude SLM to implement the reduced SPADE. Nonetheless, our aim is to estimate the displacement of a single point source and we here use a phase-only SLM, which is a more common apparatus than an amplitude SLM in laboratories, to implement the measurement.
Fig. 3. Schematic diagram of the experiment. A semiconductor laser produces a beam ($795$ nm) illuminating the digital micromirror device (DMD), which has only one micro mirror flipped in this experiment. The beam reflected from the DMD is imaged on the plane of the SLM through a small circular aperture and a lens whose focal length is $f_{1}$. The SLM loads the computer-generated holograms (CGH) of $\mathrm {TEM}_{00}$ and $\mathrm {TEM}_{10}$. Finally, we measure the photon number of a single pixel (SP in the figure) of a CCD camera at the Fourier plane of another lens with focal length $f_2$. The position of the single pixel measured corresponds to the carrier frequency of the first order diffraction.
An optical point source is simulated by the output field from a flipped micro mirror of the DMD (VIALUX, V-7001 VIS with $1024\times 768$ micromirrors of $13.7$ µm size each). The DMD is illuminated with a semiconductor laser beam (TOPTICA Photonics AG, DL pro, 795 nm wavelength). The position of the optical point source is altered by flipping different micro mirrors. We place an small circular aperture in front of the lens, which will generate an Airy disk on the image plane of the first lens under direct imaging. The Airy disk can be approximated to a circularly symmetric Gaussian mode $\Psi (x,y) = \psi (x) \psi (y)$, whose characteristic length is experimentally determined as $\sigma = 0.0868$ mm. The position of the point source will be changed along $x$-axis. Our objective is to measure the displacement of the point sources by spatial mode projection measurement rather than direct imaging.
We need to take advantage of the CGH’s ability of amplitude modulating to perform the spatial mode projection measurement. We place a phase-only SLM (HoloEye, PLUTO-2-NIR-011 with $1920\times 1080$ pixels of pitch $8$ µm) at the image plane of the first lens. The SLM loads the CGHs of $\text {TEM}_{00}$ and $\text {TEM}_{10}$ modes, which along $x$-direction is the Hermite-Gaussian mode of $q=0$ and $q=1$, respectively. We use the technologies given in Ref. [43–45] to encode the information of both amplitude and phase modulation into a CGH. The actual SLM phase modulation can be expanded by the Jacobi-Anger identity as
where $a=a(x,y)$ and $\varphi = \varphi (x,y)$ are real numbers depending on the coordinates of the SLM plane, $J_m$ is the $m$th-order Bessel function, and $f(a)$ is a real function determined by the encoding condition $J_1[f(a)] = \zeta a$ with $\zeta =0.58$. In our experiment The first order term $\zeta a(x,y) \exp [i\varphi (x,y)]$ can modulate the amplitude by $a(x,y)$ and the phase by $\varphi (x,y)$ of the light field. For the purpose of our experiment, we will in turn set $a(x,y)$ to be $\mathrm {TEM}_{00}$ and $\mathrm {TEM}_{10}$ in order to measure the displacement of the point source along $x$ direction. By adding appropriate carrier frequency $(g_X, g_Y)$ to the phase $\varphi$, i.e., $\varphi \to \varphi + 2 \pi g_X x + 2 \pi g_Y y$, the frequency spectrum of the terms of different orders in the right hand side of Eq. (17) can be isolated.In our experiment, we need to extract information about $|\langle {\phi _0}|{\psi _s}\rangle |^2$ and $|\langle {\phi _1}|{\psi _s}\rangle |^2$. This is realized by measuring the photon number of a specific pixel in a CCD camera (Spiricon Ophir Photonics, SP907 with $964\times 724$ pixels of pitch $7.38$ µm) at the Fourier plane of the second lens with focal length $f_2$ [46–49]. This specific pixel corresponds to the carrier frequency of the first order diffraction. Neglecting an irrelevant global phase that is independent of the transverse coordinates, the optical field $U'$ at the Fourier plane with coordinates $(x',y')$ is given by [50]
In our experiment, the amplitude modulation $a(x, y)$ is set to $\phi _0(x) \phi _0(y)$ when measuring the $\text {TEM}_{00}$ projection and set to $\phi _1(x) \phi _0(y)$ when measuring the $\text {TEM}_{10}$ projection. We record the photon number at the aforementioned pixel of the CCD camera as $N_0$ and $N_1$ for the $\text {TEM}_{00}$ and $\text {TEM}_{10}$ projection measurements, respectively. By approximating the PSF to be Gaussian, we estimate the value of $s$ through the estimator in Eq. (13).
We run the experiments 100 times for each value of the displacement $s$. The performance of the estimation is experimentally demonstrated in Fig. 4, which plots the mean and standard deviation of the estimates of the displacement $s$. The true value of the displacement $s$ is derived from the pixel size of the DMD and the address of the flipped micromirror. The range of the standard deviation of the estimates are from $0.0045\sigma$ to $0.0123\sigma$. We use the double of the mean photon number detected in the $\text {TEM}_{00}$ for the case of $s=0$, which is rounded to $49600$, as the reference photon number to scale the standard deviation in Fig. 4. When the displacement is very small, the background photons and dark counts of the detectors affects mostly the estimation precision.
Fig. 4. The mean (left panel) and standard deviation (right panel) of the estimates obtained from the experimental data. The standard deviation $\delta \hat s = \sqrt {\textrm {Var}(\hat s)}$ is multiplied by a constant factor $\sqrt {n}$, where $n=49600$ is a photon number for reference.
An advantage of using CGH with SLM for estimating small distances is the flexibility of changing the spatial modes to be measured. We run the same experiment illustrated in Fig. 3, except that we measure the spatial modes $\text {TEM}_\pm = (\text {TEM}_{00} \pm \text {TEM}_{10}) / \sqrt 2$ instead of $\phi _0$ and $\phi _1$. Figure 5 plots the means and the scaled standard deviations of the estimates from 100 runs of experiments. We use the the mean photon number detected in $\text {TEM}_+$ and $\text {TEM}_-$ for the case of $s=0$, which is rounded to $19200$, as the reference photon number to scale the standard deviation in Fig. 5. It demonstrates that measuring $\text {TEM}_\pm$ modes can discriminate the sign of $s$. Moreover, the minimal standard deviation of the estimate is at $s=0$, which is coincident with the theoretical analysis.
Fig. 5. The mean (left panel) and standard deviation (right panel) of the estimates obtained from the experimental data. The standard deviation $\delta \hat s = \sqrt {\textrm {Var}(\hat s)}$ is multiplied by a constant factor $\sqrt {n}$, where $n=19200$ is a photon number for reference.
4. Conclusion
In this work, we have used quantum parameter estimation theory to analyze the performance of spatial mode projection measurement for estimating small displacements of the image of an optical point sources. By resorting to the Fisher information distribution over measurement outcomes, we have constructed two simple estimation strategy, both of which attain the maximum Fisher information for very small displacements by measuring only two spatial modes. We have showed that measuring the specific linear combinations of the $\text {TEM}_{00}$ and $\text {TEM}_{10}$ modes can discriminate the sign of displacement and more invulnerable to the dark counts and background noise, while extracting the same Fisher information as simply measuring the $\text {TEM}_{00}$ and $\text {TEM}_{10}$ modes. We have experimentally demonstrated this estimation strategy with digital holographic approach. Despite we use a CCD camera in our experiment, in fact, we only readout the photon numbers detected in a pixel. This experiment mainly use a phase-only SLM and a CCD camera and thus is easy to be implemented in an optic laboratory.
Funding
National Natural Science Foundation of China (12275062, 11935012, 11805048, 61871162).
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.
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