1. Introduction

In well-corrected imaging systems, the propagation from the object to image planes is often modeled with shift invariance [1]. This indicates that the profile of the imaging system’s point spread function (PSF) is independent of object impulse’s location. Recent works have recasted the associated Rayleigh’s criterion, which states that the separation of two nearby point sources cannot be precisely estimated, for such systems in terms of the classical Fisher information (CFI) [2]. The CFI informs on the precision one expects in extracting an unknown parameter with a given measurement scheme. In this context, Rayleigh’s criterion states that the CFI for the separation of two point sources, with direct intensity (DI) measurements, vanishes as the separation between the point sources approaches zero.

A shift in the well-established understanding of the Rayleigh’s criterion was provided by Tsang and Nair [3]. They showed that the quantum Fisher information (QFI), an upperbound for the CFIs over all possible measurement schemes, for the separation of two equally bright incoherent point sources remains non-zero in the sub-Rayleigh regime. This result, along with the introduction of modal imaging techniques like spatial mode demultiplexing (SPADE), a measurement scheme that saturates the QFI, is responsible for the wide net of recent research efforts that sought to extend the promise of this so-called quantum-inspired superresolution [4–10]. Such extensions include theoretical work in which strides have been made regarding the generalization of the object’s spatial and coherence details [7,11–16]. Regarding experimental considerations, the advantages of SPADE-type measurements have been tested and confirmed for a variety of objects [8,17].

In this work, calculations are provided for the CFI and QFI for the separation between two equally bright incoherent point sources in imaging systems which possess off-axis/field-dependent aberrations. The presence of these aberrations causes imaging systems to be shift-variant; the analysis of such systems have been thus far neglected in the context of quantum-inspired superresolution. Remarkably, we show that the inclusion of simple low-order off-axis Seidel aberrations can give rise to more precise estimation of source separation when compared to an aberration-free system by improving on the CFI and QFI. In particular, we provide the analysis for off-axis tilt (OAT) and Petzval curvature. Both are shown to improve the CFI for both DI and modal imaging measure schemes, with the former providing a global improvement and the latter giving rise to improvements in the sub-Rayleigh regime. To derive these results, a general framework for shift-variant imaging systems is introduced in Section 2, from which a specialized treatment for two equally bright incoherent point sources is considered. Our results, given in Section 3, provide context to the nascent field of quantum-inspired superresolution and show that much improvement can be made through the intentional usage of off-axis aberrations.

2. Theoretical background

2.1 Shift-variant imaging systems

In a shift-invariant imaging system, the optical field at the image plane may be obtained from the optical field at the object plane via a convolution with the system’s PSF [1]. In the presence of off-axis aberrations, however, the relation between object field, $U_o$, and image field, $U_i$, is given by a more general integral transformation. For simplicity, we assume one transverse spatial coordinate for which the aforementioned transformation is given by

 

Fig. 1. A schematic of a $4f$ imaging system with $\xi,u,$ and $x$ as the spatial coordinates for the object, pupil, and image planes, respectively. An illustration of two point sources separated by $s$ is shown along with their image distributions in the presence of OAT and Petzval curvature of strengths $T$ and $P$, respectively, which are defined in Eq. (4). The individual PSFs have a half-width of $g$ (which depends on $P$), defined in Eq. (6), and their separation is magnified by a factor of $1 + 2\pi T \sigma$. The pupil is modeled as a Gaussian with half-width $\sigma _p$.

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Before proceeding to the analysis of off-axis aberrations, we note that the spatial profile of $\psi$ is affected by the strength of the aberration $\Delta W$. For the purposes of quantifying this strength in later sections, it is useful to use $2\sigma _p$ and $2 \sigma$ as standard lengths for the pupil coordinate $u$ and the object location $\xi$, respectively. The reason for this choice is due to (1) the Gaussian pupil function strongly attenuating any signal at the pupil plane located at $|u| > 2\sigma _p$ and (2) the context of quantum-inspired resolution is most relevant for objects located at $|\xi | < 2 \sigma$; that is, for objects whose features are below twice the diffraction-limited PSF. This regime is henceforth referred to as the sub-Rayleigh regime.

2.2 Off-axis aberrations

From Eq. (1), it is clear that the PSF of a shift variant system is determined by the aberration function $\Delta W(u,\xi )$. Although there is an infinitude of options; we restrict our analysis to the case where

OAT and Petzval curvature, among other Seidel aberrations, are common in typical imaging systems and have well-known interpretations particularly in the language of geometrical (ray) optics. For a system with OAT, the plane wave emerging from the pupil associated with each object location $\xi$ is proportionally tilted (posses an additional linear phase). This often arises from unintentional tilts of the optics within the system and causes an incorrect magnification in the image. This magnification factor, according to Eq. (5) and Fig. 1, is $(1+2\pi T \sigma )$; in the case of an object scene with two point sources, this magnification translates to an amplification of the separation in the image field. Petzval curvature, on the other hand, describes an imaging system in which the image is perfect when the intensity is measured along a curved surface. If the intensity is instead measured, as it usually is, along a flat surface, off-axis object points $\xi$ are mapped to a shift-variant PSF whose width increases with $|\xi |$. According to Eqs. (5) and (6), this width is $g(\xi ;P)$.

For the aberration-free case ($P = 0$ and $T = 0$), Eq. (5) reduces to the shift-invariant Gaussian PSF with width $\sigma$. Furthermore, it should be noted that the proceeding comparisons of Fisher information (FI) values between systems where the PSF is given by $\psi (x,\xi ;P,T)$ and the aberration-free case $\psi (x,\xi ;0,0)$ are fair, since $g(\xi ;P) \ge \sigma$. In other words, the presence of OAT and Petzval curvature does not reduce the diffraction-limited spot size and therefore the improved FI is not attributed to the reduction of $\sigma$.

2.3 Measurement schemes

Although this work will primarily be concerned with incoherent imaging, a general framework is provided here for completeness and comprehension. To this end: the density matrix, which represents the cross-spectral density of the optical field at the image plane, for the case of two partially coherent point sources with intensity ratio $A$, a known centroid taken to be at the origin of the coordinate system, separated by a distance $s$ is given by

Furthermore,

The act of performing a measurement on the received photons is captured mathematically through computing the modulus-square coefficients over a corresponding set of projections of $\rho$. These coefficients are taken to be the probabilities of finding an image-plane photon in a certain mode (which may be continuous or discrete). In this work, we focus on comparing the measurement schemes of DI and SPADE. For DI, the set of projection modes is the continuous position basis $\{|x\rangle \}$; the probability density of finding a photon in the $x$ position of the image plane (conditioned on a detection event) is given by

In Section 3, we focus on the case where $\Gamma = 0$ and $A = 1$ (equally bright incoherent point sources). Figure 2 is provided for the visualization of $p_\text {I}(x,s;P,T)$, which is a normalized version of the image-plane intensity distribution, for various levels of OAT and Petzval curvature. The aberration-free case shown in Fig. 2(a) shows the usual intensity pattern of two Gaussian PSFs converging as the separation $s$ vanishes. The effects of non-zero values of $T$ and $P$ are illustrated in Figs. 2(b) - (d): OAT causes the intensity distributions from the two point sources to converge at a different rate due to the magnification factor $(1+2\pi T \sigma )$ and Petzval curvature gives the intensity distributions a $s$-dependent width. Regarding OAT, it is clear from the comparison of Fig. 2(a) and (b) that non-zero values of $T$ allows for the two point sources’ PSFs to be more easily distinguished for smaller values of $s/\sigma$. This effect, which will be detailed later, is responsible for larger values of CFI when using DI for imaging systems with OAT. Petzval curvature, on the other hand, does not have a similarly clear benefit in $s$-estimation whe compared to OAT. The resolution advantages offered by non-zero values of $P$ instead comes from the sensitivity of the width $p_\text {I}$ as $s$ approaches zero.

 

Fig. 2. Density plots of $p_\text {I}(x,s;P,T)$ over $x/\sigma$ and $s/\sigma$ for various values of $T\sigma$ and $P\sigma ^2$ in (a) - (d). In each, the solid red lines show the location of the two point sources. The dashed blue lines in (b) and (d) show the centers of intensity distribution from each point source for non-zero OAT. The dashed yellow lines in (c) and (d) show the outer envelope of $p_\text {I}$ for non-zero Petzval curvature.

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For SPADE, the projection is done over any set of discrete orthonormal modes. A natural choice is the $\sigma$-matched Hermite-Gauss (HG) basis $\{\phi _q(x)\}_{q=0}^\infty$, where the $q$-th mode is defined as

The probability densities given by Eqs. (11) and (13) for DI and SPADE, respectively, are needed in the calculation of the CFI of either measurement scheme. For the case where only the separation $s$ is an unknown parameter, the corresponding CFIs for DI and SPADE are given by

The quantities $F_\text {I}$ and $F_\text {II}$ are valuable in that, via estimation theory, their reciprocals provide lower bounds for the variance in the estimation of $s$ when the corresponding measurement scheme is used. In other words, if $\check {s}_\text {I}$ and $\check {s}_{\text {II}}$ are estimators for $s$ constructed from data acquired via DI or BSPADE measurements, respectively, then

The CFI for DI and BSPADE measurements are compared for various values of $T$ and $P$ in Section 3. There, it is shown that non-zero values of these aberration strength parameters can lead to larger values of $F_\text {I}$ and $F_\text {II}$.

2.4 Quantum FI

In addition to the CFI, which depends on the choice of measurement scheme, it is informative to consider the QFI, $Q_s$, which provides an upperbound to all possible CFI once the transformation between object and image planes is stipulated. In other words,

As is standard in deriving the QFI, one first finds the symmetric logarithm derivative (SLD), $L_s$, associated with the separation parameter $s$. The SLD is defined implicitly through

Details pertaining to the derivation of the QFI for imaging systems with off-axis aberrations are found in Supplement 1. It should be emphasized that such calculations for the separation QFI for imaging systems in which the PSF is shift-variant require care and themselves constitute a novel result from which interesting discussions may arise. However, to keep the focus on well-known measurement schemes like DI and BSPADE, the QFI will primarily serve as a confirmation of CFI results through Eq. (18).

3. Results and discussion

3.1 Results

Although the framework developed in Section 2 is general, we now specialize to the case of two equally bright incoherent point sources. This choice corresponds to the selection of $A = 1$ and $\Gamma = 0$ in Eqs. (8), (11), and (13). We are interested in calculating the CFI for DI and BSPADE for various values of $P$ and $T$ and comparing them to the case where the imaging system is aberration-free (and therefore shift-invariant), i.e., for $P = 0$ and $T = 0$. The values of $P$ and $T$ used in the subsequent analysis were not chosen for any particular reason beside to provide a sufficient illustration of the effects of OAT and Petzval curvature on the CFI and QFI. However, we note that such values correspond to aberrated wavefronts at the pupil plane whose maximum deviations from the nominal wavefront are fractions of a wavelength.

In the following, the CFI for DI, given by Eq. (14), is calculated numerically. For BSPADE, it can be shown using Eqs. (5), (12), and (13) that

 

Fig. 3. CFI (solid curves) associated with the separation $s$ for (a,c) DI and (b,d) BSPADE are shown for the case of (a,b) $P = 0$, varying $T$ and (c,d) $T = 0$, varying $P$. The QFI (dashed curves), $Q_s$, is also plotted for the corresponding values of $P$ and $T$. The black curves in each plot correspond to the aberration-free (shift-invariant) imaging system.

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Figures 3(c) and (d) consider the case where the imaging system contains Petzval curvature, but no OAT. It can be seen, in a behavior similar to that seen in Figs. 3(a) and (b) that larger values of $P$ lead to larger values of $F_\text {I}$ and $F_\text {II}$ in the sub-Rayleigh regime, with the latter saturating the QFI near $s = 0$. Therefore, like OAT, Petzval curvature can improve the performance of an imaging system regarding two-point separation estimation. However, there are some notable differences between the behaviors of the CFI when the system contains OAT and when it contains Petzval curvature. First, increasing $P$ does not affect the value of the $F_\text {II}$ at exactly $s = 0$ ($F_\text {I} = 0$ regardless of $P$ and $T$). In other words, all the curves seen in Fig. 3(d) coincide in the limit of vanishing separation. Second, there is an interval of $s$ within the sub-Rayleigh regime in which DI outperforms BSPADE. For example, comparing the $P = 0.4 \sigma ^{-2}$ case for $F_\text {I}$ and $F_\text {II}$ in Figs. 3(c) and (d), repsectively, it can be seen that the former reaches a peak of roughly $1.5$ compared to the latter’s $0.75$. Finally, we note that Petzval curvature may lead to the formation of local minima in $F_\text {I}$ and $F_\text {II}$, which indicate that the corresponding measurement schemes perform optimally neither at exactly $s = 0$ nor at $s \rightarrow \infty$, but rather at some intermediate value that, according to Figs. 3(c) and (d), occur in the sub-Rayleigh regime.

For completeness, CFI and QFI curves for the case where both OAT and Petzval curvature are present are shown in Fig. 4 for a fixed value of $T = 0.2 \sigma ^{-1}$ and varying values of $P$. As was observed from Fig. 3, larger values of $T$ raise the maximal attainable information while various values of $P$ lead to CFI curves that peak locally in the sub-Rayleigh regime [although such peaks for the BSPADE CFIs in Fig. 4(b) are not apparent for smaller values of $P$ since the nonzero value of $T$ raises the CFIs at $s = 0$].

 

Fig. 4. CFI (solid curves) associated with the separation $s$ for (a) DI and (b) BSPADE are shown for the case of $T = 0.2 \sigma ^{-1}$, varying $P$. The QFI (dashed curves), $Q_s$, is also plotted for the corresponding values of $P$ and $T$.

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It is prudent to point out the behavior of the QFI curves in Figs. 3 and 4 as well. When only OAT is present, the QFI remains constant over $s$, with its value given by

Maximum likelihood estimation (MLE) simulations for the separation $s$ were performed for DI to support the findings from Eq. (14) and Fig. 3(a) and (c). For the $i$-th (out of $M$) iteration of the simulation, the number of photons arriving at the image plane, $N_i$, was chosen using Poisson statistics around an average photon number of $N$. That is, the probability mass function for $N_i$ is given by

Once $N_i$ is chosen, an equivalent number of photon positions in the image plane is chosen using Eq. (11) as the probability density function for $N_i$ independent events. Given these $N_i$ photon locations, the unbiased MLE estimator, denoted as $\hat {s}$, is used to find an estimate for the separation $s$. The variance of $\hat {s}$, whose average value is $s$, is then calculated over the ensemble of $M$ iterations. In order to compare the simulation results to $F_\text {I}$, one must divide the reciprocal of the aforementioned variance by $N$ to compute the per-photon CFI. This process is repeated for different values of $s, T,$ and $P$ and the results of these simulations are shown in Fig. 5 over the sub-Rayleigh regime $(s<2\sigma )$ for $N = 2000$ photons and $M = 500$ iterations. As is done in Figs. 3(a) and (c), the cases of $P = 0$ and $T = 0$ were analyzed, respectively. It is evident that there is good agreement between the simulation results and $H_\text {I}$.

 

Fig. 5. Comparisons of $F_\text {I}$ (solid curves) and $\text {Var}^{-1}(\hat {s}-s)/N$ from MLE simulations (discrete points) for the cases of (a) $P = 0$, varying $T$ and (b) $T = 0$, varying $P$. A mean photon number of $N = 2000$ was used for each iteration (each value of $s$) and the variance was calculated over $M = 500$ iterations.

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3.2 Discussion

The primary result of this work is that it is possible to improve upon the two-point resolution of an aberration-free imaging system by intentionally introducing known off-axis aberrations. Particularly, this is true even for DI measurement schemes as seen in Figs. 3 and 4. In fact, whether the improvement is global (over all $s$, as it is when OAT is present) or only over the sub-Rayleigh regime (when only Petzval curvature is present), the improvement can exceed the improvements offer by modal imaging techniques like BSPADE for aberration-free systems. In other words, if a sufficient amount of off-axis aberration can be provided, DI imaging schemes can lead to large $F_\text {I}$ even in the sub-Rayleigh regime. Since practical applications of two-point imaging deal with non-zero values of $s/\sigma$, increasing $T$ can effectively allow DI measurement schemes to attain the same information allowed by BSPADE in shift-invariant systems. However, this is not to say that BSPADE (and other modal imaging schemes) are fruitless; it is clear from Figs. 3 and 4 that BSPADE benefits in a similar fashion to DI when off-axis aberrations are introduced. Importantly, at least for OAT and Petzval curvature, BSPADE still (as it did for the aberration-free case) yield a $F_\text {II}$ that saturates the QFI in the sub-Rayleigh regime.

The findings in this work are somewhat counter-intuitive as many traditional imaging systems start with substantial aberrations and a significant amount of effort is often needed to minimize such errors towards the goal of an aberration-free (diffraction-limited) system. However, at least in the context of resolving two incoherent and equally bright point sources where the separation is the only unknown parameter, it turns out that the presence of off-axis aberrations can actually improve performance. It may be of interest then, given the number of well-corrected imaging systems in existence, to understand how one may intentionally introduce OAT and Petzval curvature back into these imaging systems in order to take advantage of the larger DI or BSPADE CFI. Of course, it is possible to purposely (re)introduce aberrations by misaligning or adding/subtracting optical elements in the system. However, we should mention that despite their simple geometrical optics interpretations, OAT and Petzval curvature cannot be properly introduced by simply tilting or curving the image plane, respectively. Although these geometrical manipulations of the image surface create effective PSFs that are SV, they do not have the necessary dependence on $\xi$, the object plane coordinate, in order to bring about the results from a genuine $\Delta W$ given by Eq. (3).

4. Concluding remarks

The analysis presented in this work inform on the CFI (for DI and BSPADE) and QFI regarding the separation estimation between two equally bright incoherent point sources when the imaging system includes off-axis aberrations. The resulting shift-variant nature of the system’s field PSF gives rise to CFI and QFI that are novel and, importantly, greater than their counterparts in the aberration-free case. Specifically, OAT and Petzval curvature, which constitute two of the lowest order off-axis Seidel aberrations, were shown to provide improvements to the CFI for both DI and BSPADE measurement schemes. A summary of their effects is provided as follows:

To more fully appreciate the results of this work, it is valuable to contextualize them with the majority of the research done so far regarding quantum-inspired superresolution. The primary message in recent history is that novel modal imaging schemes, like BSPADE, can provide an advantage in resolution compared to traditional DI measurements. Such claims have been extended, and supported through theory and experiments, to more complicated object distributions (discrete or continuous). Furthermore, additional works have sought to optimize modal imaging. Examples include the development of practical adaptive imaging schemes that leverage advantages in both DI and BSPADE as well as a theoretical analysis on the effect of photon statistics [9,19]. However, our present work demonstrates that the presence of off-axis aberrations, whose study have been largely ignored in the context of quantum-inspired superresolution, provides an improvement to both CFI and QFI that is relatively intuitive and whose mathematical treatment is straightforward.

The findings here are reminiscent of computational imaging techniques in which an imaging system is intentionally altered from the traditional aberration-free DI scheme in order to benefit from the known adjustments. We show that the introduction of OAT and Petzval curvature into imaging systems can improve the CFI in two-point separation estimation and therefore provide a link between the recent field of quantum-inspired superresolution with aspects of computational imaging. However, although the derivation presented here and in Supplement 1 for the CFI and QFI encompasses more than the case where the object scene consists of two equally bright incoherent point sources, realistic objects are much more complicated and require more care in their treatment. In addition to this concern, which is relevant in all of quantum-inspired superresolution analyses, realistic imaging systems have constraints regarding the strength of off-axis aberrations like OAT and Petzval curvature due to manufacturing and tolerancing limitations. Further studies are required to determine the benefits of introducing off-axis imaging systems to resolve general, realistic object scenes. Additional related topics of further exploration include the effects of off-axis aberrations in axial resolution and subsequent experimental verification.