Heisenberg-resolution imaging through a phase-controlled screen

Peilong Hong; Guoquan Zhang

doi:10.1364/OE.25.022789

1. Introduction

It is well known that, for an imaging scheme with a limited numerical aperture NA, the point-to-point correlation between an object and its image is not ideal but described by a point-spread function (PSF) [1]

I (x) \propto {somb}^{2} (\frac{2 π NA}{λ} | x_{0} + \frac{x}{m} |),

where x₀ and x are the transverse coordinate on the object plane and image plane, respectively. λ is the wavelength, m is the magnification factor of the image, and somb(x) = J(x)/x is the sombrero function. Any two object points with a separation smaller than the PSF’s width are not distinguishable on the image plane, giving rise to the Rayleigh resolution limit Δx_R = 0.61λ/NA.

Interestingly, it has been shown that the Rayleigh resolution limit can be overcome by a factor of $\sqrt{N}$ through N-photon interference in a lens-assisted imaging scheme, reaching the standard quantum limit [2–9]. However, this $\sqrt{N}$ -fold improvement is still below the promised N-fold improvement via N-photon interference, known as the Heisenberg limit [2,10].

To demonstrate the Heisenberg-limit interference, a lot of work have been done in interferometer and far-field diffraction schemes by using quantum light sources [11–16] or classical light sources [17–24]. However, as is well-known, imaging is different from a simple interference or diffraction patterns, and there is no evident clue that Heisenberg-resolution direct imaging can be established by modifying those interferometer or far-field diffraction schemes because they were carefully designed to reach the Heisenberg-limit interference or diffraction. Currently, it is still a challenging task to realize the Heisenberg-limit resolution in a direct imaging scheme [2]. Inspired by the work in [25] where a natural dynamic random medium is found applicable for direct two-photon imaging, here we explore active phase control on the wavefront of light to realize multi-photon imaging. We show that a phase screen with properly designed dynamic phase structure can be used to realize direct multi-photon imaging, and the imaging resolution reaches the fundamental Heisenberg limit. Due to the intrinsic difference between the dynamic phase structure here and that in [25], the multi-photon interference in this work is synchronous-position type, different from the Hanbury Brown-Twiss type in [25] [21, 26]. In practice, the phase-controlled screen (PCS) can be realized through a commercial spatial light modulator (SLM).

2. Scheme and principle

Our scheme is shown in Fig. 1(a). It has the same spatial arrangement as that of the conventional lens-assisted imaging scheme, but the conventional lens is replaced by a PCS with transmission function $t_{N} (x_{f}) = ℒ_{N} (x_{f}) e^{- i k x_{f}^{2} / (2 f)}$ . Here, the random phasor $ℒ_{N} (x_{f}) = \sum_{j = 1}^{N} e^{i φ_{j} (x_{f})} / N$ , which means superposition of multiple phase modes introduced at position x_f. For the multiple phase modes at position x_f, each φ_j (x_f) fluctuates randomly among [0, 2π) with time, but, for every temporal frame, the sum of these phases is set to meet the condition $\sum_{j = 1}^{N} φ_{j} (x_{f}) = ϕ_{0}$ with ϕ₀ being a constant value independent of x_f. Note that, such kind of phase screen can be realized by using a commercial SLM [21]. Besides, the constant parameter f is set to meet the condition 1/l₁ + 1/l₂ = 1/f, where l₁ is the distance between the object and the PCS, and l₂ is the distance between the PCS and the detecting/imaging plane. In this scheme, a N-photon image of the object is immediately obtained through synchronous-position N-photon correlation measurement at the detecting plane, provided that the illumination source is a N-photon entangled source with the N photons always emitted from the same position on the object/source plane. This kind of two-photon entangled source (N = 2) has already been generated through parametric down conversion process [27,28].

Fig. 1 (a): Heisenberg-resolution imaging system through a PCS, in which N-photon imaging (N = 2 in the figure) is obtained by measuring the synchronous-position correlation function. (b): Indistinguishable two-photon paths for a pair of photons to trigger a coincidence at a specific detecting/imaging point, which are introduced when the pair of photons transmit through the same position of the PCS. In (b), two neighboring colored squares on the PCS represents two correlated phase modes e^{iφ₁ (x_f)} and e^{iφ₂ (x_f)} at the same spatial point x_f, respectively.

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To understand the imaging process, we consider the two-photon imaging case, and draw the different but indistinguishable two-photon paths in our scheme [11, 19, 21]. As shown in Fig. 1(b), for a pair of photons emitted at the same object point and finally arriving at the detecting point x to trigger a coincidence count, there are a group of indistinguishable two-photon paths when the pair of photons transmit through the same spatial point x_f on the PCS plane. The complex amplitudes of these two-photon paths are of the same random phase φ₁(x_f) + φ₂(x_f) = ϕ₀ independent of x_f, and a stationary propagation phase factor dependent on x_f. Recall that, in the case of single-photon imaging with a lens of focal length f in the same spatial arrangement, the propagation phase factors related to different spatial position x_f of the lens are the same if the detecting position x = −l₂/l₁ · x_o. Similarly, the indistinguishable two-photon paths in our scheme are also in-phase when the pair of photons arrive at the detecting position x = −l₂/l₁ · x_o, giving rise to constructive two-photon interference that results in the Heisenberg-resolution imaging. Besides of these in-phase two-photon paths, other two-photon paths with the pair of photons transmitting through different spatial points on the PCS plane only contribute a constant background. To clearly demonstrate the imaging condition and resolution in our scheme, we next calculate the second-order correlation function on the detecting/imaging plane.

3. Theoretical calculation and simulation results

For a monochromatic light in the paraxial regime, the field operator on the imaging plane is expressed as

E^{(+)} (x) \propto \int E_{0}^{(+)} (x_{0}) A (x_{0}) H (x_{0}; x_{f}; x) ℒ_{2} (x_{f}) d x_{0} d x_{f},

where x₀, x_f, and x are the transverse coordinate on the object plane, the PCS plane, and the image plane, respectively.

E_{0}^{(+)} (x_{0})

is the field operator of the light illuminating the object, and A(x₀) is the aperture function of the object. The stationary phase function

H (x_{0}; x_{f}; x) = Q (x_{f} - x_{0}; l_{1}) exp (- i k x_{f}^{2} / (2 f))

Q(x − x_f ; l₂), where Q(x₁ − x₂; l) = exp(ik|x₁ − x₂|²/(2l)) is the quadratic phase factor introduced by the light propagation in free space. Here the constant phasor −e^{ik(l₁+l₂)}/(l₁l₂λ²) that does not affect the imaging process is omitted for simplicity. The second-order correlation of the random phasor ℒ₂(x_f) is [21]

\begin{array}{l} 〈 ℒ_{2}^{*} (x_{f 1}) ℒ_{2}^{*} (x_{f 2}) ℒ_{2} (x_{f 3}) ℒ_{2} (x_{f 4}) 〉 \\ \propto δ (x_{f 1} - x_{f 3}) δ (x_{f 2} - x_{f 4}) + δ (x_{f 1} - x_{f 4}) δ (x_{f 2} - x_{f 3}) + δ (x_{f 1} - x_{f 2}) δ (x_{f 3} - x_{f 4}) . \end{array}

By using Eq. (3) and Eq. (2), we derive to obtain the synchronous-position second-order correlation function on the image plane as

\begin{array}{l} G^{(2)} (x) \propto \int 〈 E^{(-)} (x_{01}) E^{(-)} (x_{02}) E^{(+)} (x_{03}) E^{(+)} (x_{04}) 〉 \cdot e^{- i k \frac{x_{01}^{2} + x_{02}^{2}}{2 l_{1}}} e^{i k \frac{x_{03}^{2} + x_{04}^{2}}{2 l_{1}}} \\ \times A^{*} (x_{01}) A^{*} (x_{02}) A (x_{03}) A (x_{04}) \cdot [P_{b g} ({x_{0}}) + P_{img} ({x_{0}}, {x})] d x_{01} \dots d x_{04}, \end{array}

with

\begin{array}{l} P_{bg} ({x_{0}}) = somb (\frac{k D}{2 l_{1}} | x_{01} - x_{03} |) somb (\frac{k D}{2 l_{1}} | x_{02} - x_{04} |) \\ + somb (\frac{k d}{2 l_{1}} | x_{01} - x_{04} |) somb (\frac{k D}{2 l_{1}} | x_{02} - x_{03} |), \end{array}

which leads to a constant background, and

P_{img} ({x_{0}}, {x}) = somb (\frac{k D}{2 l_{1}} | x_{01} + x_{02} + \frac{2 x}{m} |) \cdot somb (\frac{k D}{2 l_{1}} | x_{03} + x_{04} + \frac{2 x}{m} |) .

Here, D is the diameter of the PCS, m = l₂/l₁ is the magnification factor of the image. Note that the second-order correlation function of the position-entangled source can be expressed as [21,27,28]

〈 E^{(-)} (x_{01}) E^{(-)} (x_{02}) E^{(+)} (x_{03}) E^{(+)} (x_{04}) 〉 \propto δ (x_{01} - x_{02}) δ (x_{03} - x_{04}) .

So P_img ({x₀}, {x}) in the square bracket in Eq. (4) gives rise to two-photon interference pattern as

Δ G_{qua}^{(2)} (x) \propto {| \int A^{2} (x_{0}) e^{i k \frac{x_{0}^{2}}{l_{1}}} somb (\frac{2 π NA}{λ / 2} | x_{0} + \frac{x}{m} |) d x_{0} |}^{2},

where NA = D/(2l₁). It clearly shows a point-to-point correlation between the object and the image plane. By comparing with Eq. (1), it is apparent that the resolution of our imaging scheme reaches the Heisenberg limit.

To clearly show the resolution improvement in our scheme, we did simulation for our imaging scheme by numerically calculating the two-photon correlation function on the image plane. The parameters for simulation are l₁ = l₂ = 5cm, f = 2.5cm, D = 0.5cm, and λ = 532nm. Therefore, the numerical aperture of the system is NA = 0.05, giving rise to the Rayleigh resolution limit Δx_R = 6.5μm. Note that, here we dealt with the ideal case without considering the influence from the finite pixel size of SLM. For comparison, we considered three imaging methods in the same spatial arrangement, i.e. conventional single-photon imaging with Rayleigh resolution limit, lens-assisted two-photon imaging with the resolution improved to the standard quantum limit [3, 4], and our Heisenberg-resolution imaging. The results are shown in Figs. 2(a1)–2(a3), Figs. 2(b1)–2(b3), and Figs. 2(c1)–2(c3), respectively. The images for two object points of different separations S=8 μm, 6μm, and 4μm are shown. It is seen that, all three imaging methods can distinguish the two points of separation S = 8μm that is larger than the Rayleigh resolution limit 6.5μm. Two points of separation S = 6μm can be distinguished in the lens-assisted two-photon imaging and our PCS-assisted two-photon imaging, since the separation is still larger than the standard quantum limit 4.6μm. Interestingly, our PCS-assisted two-photon imaging is able to distinguish the two points with a separation S = 4μm that is smaller than the standard quantum limit, because our two-photon imaging scheme reaches the Heisenberg resolution limit 3.2μm.

Fig. 2 Image for two object points of distance S=8 μm, 6μm, and 4μm, respectively. (a1)–(a3): Conventional single-photon imaging. (b1)–(b3): Lens-assisted two-photon imaging with the resolution improved to the standard quantum limit. (c1)–(c3): two-photon imaging through PCS with the resolution reaching the Heisenberg limit. All figures are normalized by their maximum values, respectively.

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Note that, Figs. 2(c1)–2(c3) show the simulation results of the total second-order correlation function G⁽²⁾(x) on the image plane, which are normalized by their maximum values respectively. Besides of the two-photon interference part $Δ G_{qua}^{(2)} (x)$ as shown in Eq. (6), these figures also include the background part. In addition, one may find that the normalized background in the simulation results changes with the distance S (8 μm, 6μm, and 4μm) of the two object points. Theoretically, the absolute background part does not change with the distance S of the two object points. For the two-photon interference part $Δ G_{qua}^{(2)} (x)$ as expressed in Eq. (6), it reaches the maximum value at the peak position of one somb(· · ·) function, but it will also be affected by the sidelobe of the other neighboring somb(· · ·) function depending on the distance S. When the distance S is very large, the absolute value of the high-order sidelobe of sombrero function is very small, and its effect can be ignored. However, when the two object points become closer and closer, the low-order sidelobe of sombrero function will significantly affect the maximum value of the two-photon interference part $Δ G_{qua}^{(2)} (x)$ . As the maximum value of the two-photon interference part changes with the distance S, the normalized background changes correspondingly.

In addition, to implement this imaging scheme experimentally, for example, by employing a SLM to mimic the PCS, the finite pixel size of the SLM will have an effect on the achievable resolution. To fully satisfy the ideal condition and reach the Heisenberg-limit in a target setup with a numerical aperture NA, one should carefully select a SLM and an entangled source, such that the finite pixel size of the SLM and the correlation length of the entangled source are much smaller than the Rayleigh resolution of the target setup. However, this is not always the case because the pixel size of a commercial SLM is usually in the order of several micrometers in nowadays. This finite pixel size will significantly affect the resolution of some experimental setups, for example, a setup with the same parameters as that we used in the simulation. Nevertheless, the phase-modulation technology is in fact continuously progressing, for example, the pixel size of commercial SLM is becoming smaller and smaller, and new phase-modulation technology based on metasurface is also being developed, which has subwavelength pixel size in the order of 100 nanometers [29–31]. Generally, when the pixel size of the SLM becomes larger and larger, significant phase errors will be introduced by the SLM, such that the achievable resolution will be reduced. On the other hand, when the correlation length of the entangled source becomes larger and larger, two-photon paths originating from the area within the correlation length are coherent with each other, and cross interference between these coherent two-photon paths will also reduce the achievable resolution.

In the standard wide-field illumination situation, a quantum entangled light source is needed to achieve the Heisenberg-resolution imaging. Nevertheless, the Heisenberg-resolution imaging can also be achieved with classical light in our scheme. Note that, for classical light sources such as laser or thermal light, a pair of photons that trigger a coincidence count at a specific imaging point do not necessarily come from the same position on the object plane, but may originate from different pairs of object points. This results in cross interference between different regions of the object. To prevent such cross interference so that the point-to-point correlation between the object and image plane is built up, one can introduce the randomly scanning-focused-beam illumination which does not provide any information of the scan and may have application in high-resolution image transfer via laser radar [2, 8]. As discussed in [8], in practice, one can use an additional near-field focusing system to obtain a much smaller focused light spot than the Rayleigh resolution limit imposed by the finite numerical aperture NA. In addition, this Heisenberg-resolution imaging with scanning-focused-beam illumination may also have applications when an observer wants to see the object more clearly without increasing the size D of the imaging element, or reducing the distance of the imaging element to the object.

In this configuration, the light beam is tightly focused such that a pair of photons triggering a coincidence count are forced to come from the same region of the object, therefore, the cross interference between different regions of the object is prohibited. The effective second-order correlation function of the light source in this tightly focused situation can be expressed approximately as

〈 E^{(-)} (x_{01}) E^{(-)} (x_{02}) E^{(+)} (x_{03}) E^{(+)} (x_{04}) 〉 \propto σ_{x_{01}, x_{02}, x_{03}, x_{04}},

where σ_{x₀₁,x₀₂,x₀₃,x₀₄} means it equals to unity when x₀₁ = x₀₂ = x₀₃ = x₀₄, otherwise, it equals to zero. In this situation, after subtracting the constant background, we derive to obtain the synchronous-position two-photon correlation function as

Δ G_{cla}^{(2)} (x) \propto \int {| A (x_{0}) |}^{4} {somb}^{2} (\frac{2 π NA}{λ / 2} | x_{0} + \frac{x}{m} |) d x_{0},

which again shows a point-to-point direct imaging with the Heisenberg resolution. By comparing Eq. (8) and Eq. (6) with the conventional single-photon coherent and incoherent imaging [1,2], we view the situation of Eq. (6) as two-photon coherent imaging when quantum entangled light is used for standard wide-field illumination, and the situation of Eq. (8) as two-photon incoherent imaging when classical light is used for scanning-focused-beam illumination.

To clearly show the two-photon incoherent imaging of Heisenberg resolution, we did numerical simulation by considering the same spatial arrangement and parameters as that used for numerical simulation of the two-photon coherent imaging. The results are shown in Fig. 3. It is apparent that all three pairs of object points can be clearly distinguished in the incoherent imaging scheme with classical light, since all the separations are larger than the Heisenberg resolution limit 3.2μm.

Fig. 3 Heisenberg-resolution incoherent two-photon imaging for two object points of distance S=8 μm (a1), 6μm (a2), and 4μm (a3), respectively. All figures are normalized by their maximum values, respectively.

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Note that, comparing with Figs. 2(c1)–2(c3), the normalized background in Figs. 3(a1)–3(a3) almost keeps unchanged. This difference of the normalized background can not be explained by that happened in the ghost diffraction scheme [32]. In fact, this difference is due to the fact that the incoherent imaging described by Eq. (8) is quite different from the coherent imaging described by Eq. (6). For the incoherent imaging, the two-photon interference part is a sum of multiple somb²(· · ·) function related to the object points, different from that shown in Eq. (6). For the function somb²(· · ·), its sidelobes are suppressed a lot in comparison with that of the somb(· · ·) function. Therefore, the normalized background in Figs. 3(a1)–3(a3) almost keeps unchanged.

Besides, our Heisenberg-resolution imaging scheme can be generalized to N(> 2)-photon case. For a PCS with transmission function t_N (x_f), there are always a group of indistinguishable in-phase N-photon paths when the N photons transmit through the same spatial point of the PCS, together with other N-photon paths when the group of N photons do not necessarily transmit through the same spatial point of the PCS. Similar as that of two-photon imaging case, constructive interference of those in-phase N-photon paths leads to direct imaging of the object together with a constant background originating from other alternative N-photon paths. Up to this date, generating a position-entangled N(> 2)-photon source is still an open question. Nevertheless, one can use classical light together with the scanning-focused-beam illumination to achieve the N-photon incoherent imaging, and the imaging term after subtracting the constant background is expressed as

Δ G_{cla}^{N} (x) \propto \int {| A (x_{0}) |}^{2 N} \cdot {somb}^{2} (\frac{2 π NA}{λ / N} | x_{0} + \frac{x}{m} |) d x_{0},

showing a Heisenberg-resolution imaging with the resolution enhanced by a factor of N as comparing with the Rayleigh resolution limit. In addition, when N increases, the probability for the N photons to pass along the effective entangled paths, which are the successful events that leads to the imaging term, will become lower. To increase the total successful events recorded by the detector in practice, one can increase the power of light source, as well as the collecting time of the detector. Furthermore, to realize a PCS through a SLM is simply to encode the designed complex amplitude t_N (x_f) on the wavefront, and, for this purpose, we do not need to decompose t_N (x_f) back to the N correlated phase modes as we did for analyzing the N-photon paths. As the complex amplitude t_N (x_f) has only two independent degrees of freedom once it is obtained through the designing method, a macro pixel consisting of two single pixels is therefore enough for realizing t_N (x_f) as shown in [21].

4. Conclusion

In summary, we designed a practically realizable N-photon imaging scheme, in which the spatial resolution reaches the fundamental Heisenberg limit. The Heisenberg 1/N-scaling resolution improvement is a result of constructive interference between multiple N-photon paths introduced by a PCS that can be realized by a commercial SLM. The coherent Heisenberg-resolution imaging can be realized by using a currently well-developed two-photon entangled light for providing a wide-field illumination, while the incoherent imaging can be realized by using a classical light for providing scanning-focused-beam illumination. These results show new possibilities to design multi-photon imaging scheme through dynamic phase control.

Funding

National Natural Science Foundation of China (NSFC) (Grant No. 11604150, Grant No. 11174153, and Grant No.61475077); the 973 program (Grant No. 2013CB328702), and the 111 project (Grant No. B07013).

Acknowledgments

P. Hong thanks the support from the NSFC (Grant No. 11604150). G. Zhang thanks the support from the 973 program (Grant No. 2013CB328702), the NSFC (Grant No. 11174153 and Grant No.61475077), and the 111 project (Grant No. B07013). The authors thank Y. Wang and L. Xu for helpful discussions.

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Heisenberg-resolution imaging through a phase-controlled screen

Abstract

1. Introduction

2. Scheme and principle

3. Theoretical calculation and simulation results

4. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (3)

Equations (11)

Optics Express