Breaking the diffraction limit using fluorescence quantum coherence

Wenwen Li; Wenwen Li; Zhongyang Wang

doi:10.1364/OE.451114

1. Introduction

In classical linear optics, the resolution of far-field microscopy is limited by Rayleigh’s criterion ≃ 0.61λ/NA, where λ indicates the wavelength of light and NA is the numerical aperture of the objective lens. Breaking the resolution limit has always been a critical issue in the fields of life sciences, engineering, and material sciences. In the last two decades, various super-resolution imaging methods have been developed, which includes utilizing the photophysical properties of fluorophores. One branch of methods relies on the nonlinear fluorescence response arising from stimulated emission depletion (STED) [1] and the saturation effect in structured illumination microscopy (SSIM) [2]. Other methods stochastically localize single fluorescence emitters with variations in brightness caused by photoswitching or intrinsic blinking, such as stochastic optical reconstruction microscopy (STORM) [3,4] and photoactivated localization microscopy (PALM) [5]. This single-molecule localization method can improve the resolution to a lower bound on the root-mean-square position estimation error (shot-noise limit) of ≃ 1/$\sqrt N $ (N is the detected photons); the statistical resolution is determined by the precision of measurement [6].

In quantum optics, the quantum properties of light have been successfully used to achieve resolution enhancement by using non-classically correlated photons to provide more information [7,8]. The pioneering proposal is based on entangled light sources, for which the entangled N-photon state between paths is exploited to generate interference fringes that are N times narrower than that of the classical limit, thus allowing a further resolution improvement at the Heisenberg bound ≃1/N [9]. A resolution enhancement of 2-fold can be achieved by using second-order coincidence detection to measure the quantum coherence of entangled photon pairs [10,11]. Based on this experimental scheme, the quantum-enhanced method has been effectively employed in super-resolved quantum lithography [12,13] and subshot-noise quantum imaging [14–15]. However, a higher resolution has been limited owing to the significant challenge in generating a large N-photon entangled state. The low efficiency of coincidence detection [16] makes this method infeasible for practical applications.

For fluorescence microscopy, an alternative approach takes advantage of the quantum properties of the fluorescence by the emitter rather than the correlated light sources used as illumination. Each fluorescence emitter can be treated as a single-photon source; following excitation, it emits a single fluorescence photon via spontaneous emission. Hence, the photons have a tendency to be emitted one by one rather than in bursts, exhibiting a non-classical feature of photon antibunching [17,18]. In particular, for an ideal single-photon source, the photons are perfectly indistinguishable considering the spectrum, lifetime, and polarization, which leads to a high quantum coherence effect [19]. These intrinsic quantum properties of fluorescence, including antibunching and indistinguishability, have been observed in many fluorophores, such as organic dyes [20–22], quantum dots [23–25], and color centers in diamonds [26,27]. Therefore, the photon antibunching, one of quantum properties, has been recently utilized to build a new super-resolution microscopy; the imaging resolution can be enhanced $\sqrt {\textrm{M}} $ by using M SPAD detectors to perform Mth-order photon correlation detection [28–30]. Experimental demonstrations have been performed in wide-field microscopy by spatial photon correlation detection [28] and confocal microscopy by recording time-photon coincidence events [29–30].

For the other quantum property of photon indistinguishability, whether the origin of fluorescence coherence can also be exploited to achieve a higher resolution is uncertain. The coherence of light has been analyzed in previous studies by Mandel and Wolf [17], as well as the latest quantum estimation theory [31–33]; these discussions of macroscopic coherence cover a wide range of situations including fully and partially in-phase or anti-phase coherent sources. However, there is no bridge between the macroscopic coherence theory and microscopic radiation mechanics of emitters. A few theoretical studies have recently attempted to build a microscopic coherence model for the radiation of ensemble atoms and achieved a resolution improvement of λ/N by measuring the interference of N identical photons [34–36]. However, this method generally requires ideal single-photon sources, that is, emitted photons are indistinguishable. For fluorescence microscopy, the actual fluorophores remain distant from the ideal case owing to the effects of dephasing and spectral diffusion. Therefore, whether the quantum coherence truly enables subdiffraction resolution in practical fluorescence microscopy is considered.

Motivated by the aforementioned, in this study, we developed a statistical quantum coherence model arising from the spatiotemporal interference of spontaneously emitted photons from fluorophores in fluorescence microscopy. A practical method to break the diffraction limit is proposed by using a time-resolved single-photon counting camera, an SPAD array, to perform space- and time-resolved quantum coherence measurements, in which the spatial separation distance can be accurately extracted from the spatial coherence by introducing a well-defined time function rather than a stochastic time variable. In contrast to traditional fluorescence microscopy, in which the fluorescence emitter is considered fully incoherent, we employ photon indistinguishability as additional quantum information to improve the resolution. We also built a partial coherence model based on inherent dephasing and spectral diffusion mechanics of actual fluorophores. For simplicity, we exemplify our method in the case of two single-photon emitters. By exploiting a two-photon interference motivated by a possible experimental procedure in wide-field fluorescence microscopy, we demonstrate that this scheme enables molecular-scale spatial resolution.

2. Methods

2.1 Theoretical model

The spatiotemporal nature of a single-photon wave packet was quantitatively assessed based on the inherent radiation mechanics of fluorescence emitters. Here, for a single-photon emitter, the spatiotemporal wave operator of a single photon emitted from the ith fluorescence emitter is given by the following:

(1)$$\hat{E}_i^ + ({r,t} )= {f_i}({r,t} ){\hat{a}^{}}_i \mbox{ and }\hat{E}_i^ - ({r,t} )= f_i^ \ast ({r,t} ){\hat{a}^{\dagger} }_i$$

where f_i(r,t) is the single-photon wave packet, ${\hat{a}^{\dagger}}_i$ and ${\hat{a}^{}}_i$ are the photon creation and annihilation operators, respectively. When r = 0, Eq. (1) describes the time evolution of a single photon emitted from a fluorescence emitter, which strongly depends on the intrinsic transition and relaxation mechanics. Hence, a quantum model of a single emitter must first be constructed to quantify the relationship between the time-dependent wave operator and the radiation mechanics. Typically, a single fluorescence emitter can be considered as a two-level system with ground and excited states. A single photon is emitted from the excited state towards the ground state in the long radiative lifetime of the excited state T₁, normally a few nanoseconds, giving rise to a narrow spectral line with a width of 1/2πT₁ and a long lifetime-limited coherence time T₂= 2T₁. In this case, perfectly indistinguishable photons are obtained. However, the linewidth increases steeply owing to the perturbations from the environment, involving pure dephasing (PD) and spectral diffusion (SD). PD is mainly caused by phonon relaxation with emission frequency shifts in short interaction time, resulting in a homogeneous broadening of the emission line with Lorentzian line shape. PD decreases the coherence time 1/T₂= 1/2T₁+1/$\textrm{T}_\textrm{2}^{\ast }$ ($\textrm{T}_\textrm{2}^{\ast }$ is pure phasing time), typically several hundred picoseconds. SD follows from slow time-dependent frequency fluctuations as a result of its long interactions with a dynamically evolving system, which leads to inhomogeneous broadening with Gaussian spectral lines. The coherence time is further shortened with ${\textrm{T}_\textrm{2}}\textrm{ ={-} }{\mathrm{\Gamma }_\textrm{2}}/{\mathrm{\Delta }^\textrm{2}}\textrm{ + }\sqrt {{{\textrm{(}{\mathrm{\Gamma }_\textrm{2}}/{\mathrm{\Delta }^\textrm{2}}\textrm{)}}^\textrm{2}}\textrm{ + 2}/{\mathrm{\Delta }^\textrm{2}}} $ (${\mathrm{\Gamma }_\textrm{2}}/\mathrm{2\pi }$ and $\textrm{2}\sqrt {\mathrm{2ln2\Delta }} $, which are the homogeneous and inhomogeneous linewidths, respectively) [18,37,38]. Hence, both PD and SD limit the photon indistinguishability. The degree of indistinguishability, or coherence time, can be measured by a two-photon interference in the time domain following the famous Hong–Ou–Mandel (HOM) experiment, which is critical for quantum technologies [39,40]. The HOM experiment provides the time-resolved coincidence count with PD and SD and a measurement of the coherence time T₂. Recent experiments have observed excellent coherence in various fluorescence emitters, including organic dyes [21,22], quantum dots [24,25], and color centers in diamond [26,27], which allow the coherence time (several hundred picoseconds to a few nanoseconds) to be longer than the time resolution of single-photon detectors (approximately tens of picoseconds) and perform a time-resolved measurement of the quantum interference phenomenon.

To quantify the time-dependent PD and SD mechanics, we exploit laser pulse excitation combined with an SPAD array, in which the time-correlated single-photon-counting (TCSPC) technique measures not only the spatial position r, but also the arrival time t of each photon, that is, the time delay between the trigger signal of the laser pulse and the detection time. Therefore, following pulse excitation, the time evolution of the single-photon wave packet can be expressed in the following form with the analytical amplitude of ε_i(t) and phase φ_i(t) [37,38]:

(2)$$\begin{aligned} \zeta _i^{}(t )&= {\varepsilon _i}(t )\exp [{ - i{\phi_i}(t )} ]\\ &= \frac{1}{{{T_{1i}}}}H(t )\exp \left[ { - \frac{{t + \delta {t_\textrm{i}}}}{{2{T_{1i}}}}} \right]\exp \left\{ { - i\left[ {({\omega_0} + \varDelta {\omega_i})t + \int_t {\delta {\omega_i}(\tau )d\tau } } \right]} \right\} \end{aligned}$$

where the Heaviside-function H(t) indicates that no photon that can be emitted prior to the excitation, δt is the time jitter of the excitation, ω₀ is the central frequency, Δω_it represents the time-dependent phase caused by PD, $\mathop \smallint \limits_\textrm{t}^{\; } \mathrm{\delta }{\mathrm{\omega }_\textrm{i}}\mathrm{(\tau )d\tau }$ is the phase variation under the influence of SD arising from time-dependent frequency fluctuations. The temporal wave functions were normalized $\mathrm{\int\!\!\!\int }{|{{\mathrm{\zeta }_\textrm{i}}\textrm{(t)}} |^\textrm{2}}\textrm{dt = 1}$. However, in wide-field fluorescence imaging, the propagation properties of a single photon must be considered in both the time and space domains. The spatial propagation property mainly depends on the size of the light source and characteristics of the imaging system. To better understand the relationship between time and space propagation, we first focus on far-field imaging in free space. Here, a single fluorescence emitter, as a point source, emits photons in a spherical mode; the spatial wave function is h(r)=exp(ikR)/R, where R and k indicate the space coordinate and wave number, respectively. Note that in free space, the light field propagates in a nondispersive medium and the spectral line is relatively narrow, that is, Δω<<ω₀, for which the spatial phase variation caused by Δω is significantly lower than ω₀. Therefore, k ≈ k₀ can be approximated. The space evolution is now decoupled from time, and the spatiotemporal single-photon wave packet can be expressed as follows:

(3)$${f_i}({r,t} )= \zeta _i^{}(t ){h_i}(r )$$

We built a fluorescence quantum coherence model using this single-photon wave packet. For example, let us consider the simplest case of a two-photon interference. Under each pulse excitation, two single photons originating from two respective fluorescence point sources interfere in free space and subsequently arrive at the detection position r and arrival time t. Here, the detection probability of two-photon interference is determined by the statistical average of an ensemble of successively detected photons as follows:

(4)$$\begin{aligned} P({r,t} )&= \left\langle \Phi \right|\hat{E}_{}^ - ({r,t} )\hat{E}_{}^ + ({r,t} )|\Phi \rangle \\ &=\left\langle {{1_1}{1_2}} \right|({\hat{E}_1^ - ({r,t} )+ \hat{E}_2^ - ({r,t} )} )({\hat{E}_1^ + ({r,t} )+ \hat{E}_2^ + ({r,t} )} )|{{1_1}{1_2}} \rangle \\ &= {|{{f_1}({r,t} )+ {f_2}({r,t} )} |^2}\\ &=\frac{{H(t )}}{{z_0^2T_1^2}}{\left\langle {\left\langle {\left\langle \begin{array}{l} \exp \left( { - \frac{{t + \delta {t_1}}}{{{T_1}}}} \right) + \exp \left( { - \frac{{t + \delta {t_2}}}{{{T_1}}}} \right)\\ + \exp \left( { - \frac{{2t + \delta {t_2} + \delta {t_1}}}{{2{T_1}}}} \right)\exp \left\{ {i\left[ {(\varDelta {\omega_2} - \varDelta {\omega_1})t + \int_t {({\delta {\omega_2}(\tau )- \delta {\omega_1}(\tau )} )} d\tau - {k_0}\frac{r}{{{z_0}}}s} \right]} \right\} + c.c. \end{array} \right\rangle } \right\rangle } \right\rangle _{\delta {t_i},\varDelta {\omega _i},\delta {\omega _i}}}\\ &=\frac{{2exp \left( { - \frac{t}{{{T_1}}}} \right)}}{{{z_0}^2{T_1}}}\left\{ {1 + \exp \left( { - \frac{{2t}}{{T_2^ \ast }} - \frac{{\Delta _2^2\textrm{ + }\Delta _1^2}}{2}{t^2}} \right)\cos \left[ {\left( {\left\langle {\delta {\omega_2}} \right\rangle - \left\langle {\delta {\omega_1}} \right\rangle } \right)t\textrm{ + }{k_0}\frac{r}{{{z_0}}}s} \right]} \right\}, t > 0 \end{aligned}$$

where ${\mathrm{\hat{E}}^\mathrm{\ \pm }}\textrm{(r,t)}$ is the detected spatiotemporal field operator acting on the incoming state $\mathrm{|\Phi }$, which is equal to the superposition of two-photon field operators $\mathrm{\hat{E}}_\textrm{1}^\mathrm{\ \pm }\textrm{(r,t)}$ and $\mathrm{\hat{E}}_\textrm{2}^\mathrm{\ \pm }\textrm{(r,t)}$; $\cdots $ denotes the statistical average for the three random variables δt_i, Δω_i, δω_i. Assuming that two near emitters have the same radiation lifetime T₁ and dephasing time $\textrm{T}_\textrm{2}^{\ast }$ owing to their similar local environments, PD and SD act independently on the different emitters. Therefore, $\mathrm{exp}( \pm i\Delta {\mathrm{\omega }}_\textrm{i}\textrm{t}) = exp({\textrm{ - 2}|\textrm{t} |/\textrm{T}_\textrm{2}^{\ast }} )$ and $\textrm{exp}\left[ {\mathrm{\ \pm i}\mathop \smallint \limits_\textrm{t}^{\; } \mathrm{\delta }{\mathrm{\omega }_\textrm{i}}\mathrm{(\tau )d\tau }} \right]\textrm{ = exp}({\mathrm{\ -\ \Delta }_\textrm{i}^\textrm{2}{\textrm{t}^\textrm{2}}/\mathrm{2\ \pm i\delta }{\mathrm{\omega }_\textrm{i}}\textrm{t}} )$, and ‹δω_i› is the detuning of its emission line with respect to ω₀ [38]. The separation distance s for the far-field imaging of the two-point sources within the diffraction limit is far less than the propagation distance z₀ about the wavelength scale. Because the limit of Fraunhofer diffraction is fulfilled, that is, z₀>>s₂/λ₀, the spatial function can be approximated as h(r)≈exp{ik₀(z₀-sr/2z₀)}/z₀, owing to $\textrm{R = }\sqrt {\textrm{z}_\textrm{0}^\textrm{2}\textrm{ + }{{({\mathrm{r\ \pm s}/\textrm{2}} )}^\textrm{2}}} \simeq {\textrm{z}_\textrm{0}}\mathrm{\ \pm sr}/\textrm{2}{\textrm{z}_\textrm{0}}$[41]. Moreover, we ignore the dipole-dipole interaction between the two fluorescence emitters, which exists with a separation distance of approximately a few nanometers. As shown in Eq. (4), the first term represents the self-coherence determined by the radiation lifetime T₁ of the emitters; the second term represents the cross-coherence and contains the temporal and spatial coherence of emitters. The amplitude of cross-coherence decreases exponentially with the arrival time owing to the effect of PD and SD, which indicates that two-photon interference is partially coherent. The cross-coherence phase relies on the frequency difference and spatial separation between the two sources. Therefore, the detected fluorescence quantum coherence is entirely determined by the space-time emission properties of the fluorescence emitters and does not change with propagation in the non-dispersive system.

We analyzed the performance of a two-photon interference based on the model for the following two cases:

a. When considering only the PD, that is, Δ_i∼0, the detection probability is as follows: $(5)$$P({r,t} )= \frac{2}{{{T_1}z_0^2}}\left( {\exp \left( { - \frac{t}{{{T_1}}}} \right) + \exp \left( { - \frac{{2t}}{{T_2^{}}}} \right)\cos \left( {{k_0}\frac{r}{{{z_0}}}s} \right)} \right)$$$ where the temporal coherence of the cross-coherence is influenced by the coherence time T₂ which is determined by the radiation lifetime and pure phasing time, that is, $\textrm{1}/{\textrm{T}_\textrm{2}}\textrm{ = 1}/\textrm{2}{\textrm{T}_\textrm{1}}\textrm{ + 1}/\textrm{T}_\textrm{2}^{\ast }$. The spatial coherence of the cross-coherence contains information regarding the spatial separation distance s between the two emitters. Thus, the precise measurement of s can be obtained from deterministic cross coherence. To extract the cross-coherence by overcoming the effect of the self-coherence and PD, we introduced a time gate T_g as a photon arrival time post-selection window to record only the photons ($\textrm{t} \le {\textrm{T}_\textrm{g}}$) and sum these photons; from which the visibility of coherence, that is, the cumulative detection probability, is given by integrating Eq. (5) over T_g as follows: $(6)$$\begin{aligned} p({r,{T_g}} )&= \int_0^{{T_g}} {dt} P({r,t} )\\ &= \frac{2}{{z_0^2}}\left( {1 - \exp \left( { - \frac{{{T_g}}}{{{T_1}}}} \right) + \frac{{T_2^{}}}{{2{T_1}}}\left[ {1 - \exp \left( { - \frac{{2{T_g}}}{{{T_2}}}} \right)} \right]\cos \left( {{k_0}\frac{r}{{{z_0}}}s} \right)} \right) \end{aligned}$$$
The visibility of the coherence was modulated by setting the width of the time gate to engineer the effect of PD on coherence. The high visibility requires extracting highly coherent photons, which is achieved by setting T_g to be shorter than the coherence time T₂ to reduce the effect of PD. Therefore, when T_g<T₂<<T₁, Eq. (6) can be approximated as follows:
$(7)$$\begin{aligned} p({r,{T_g}} )&\simeq \frac{{2{T_g}}}{{{T_1}z_0^2}}\cos \left( {{k_0}\frac{r}{{{z_0}}}s} \right)\\ &= V({{T_g}} )\cos \left( {{k_0}\frac{r}{{{z_0}}}s} \right) \end{aligned}$$$ In far-field imaging, when the distance s is very small, the spatial phase φ=k₀rs/z₀<<π/2 only causes a contrast variation of cos(k₀rs/z₀) along s because its full period is difficult to be detected. Therefore, it is difficult to extract s if only the intensity is measured in conventional imaging, because the contrast is a stochastic variable subject to a time-dependent drift. However, if we introduce a time modulation function V(T_g), an accurate measurement of the spatial contrast as a function of the time gate can be performed. As shown in Eq. (7), the cumulative detection probability p(r,T_g) increases linearly with V(T_g), where the slope of p(r,T_g) is determined only by the spatial coherence cos(k₀rs/z₀). Here, the distance s can be accurately extracted from cos(k₀r₀s/z₀) at a specific position r₀.
b. When considering PD and SD, the detection probability is given by Eq. (4). In this case, the cross-coherence term decays faster owing to the low coherence time caused by the SD, for which the time-modulated coherence V(T_g) is difficult to obtain. However, another time modulation function, a time-varying phase cos((‹δω₂›-‹δω₁›)t), can be constructed by using two fluorescence emitters with unequal frequencies ‹δω₂›≠‹δω₁›. The spatial phase $\varphi $ can still be extracted from the period of cos((‹δω₂›-‹δω₁›)t+φ). The method is similar to [42] in the telescope; by extracting the spatial coherence from the interference of two sources to provide a measurement of their separation, the diffraction limit of the telescope can be surpassed by approximately 40 times. However, for two sources with different wavelengths, the photon indistinguishability is obtained by the color erasure detector to erase the wavelength identifying information. In contrast, the photon indistinguishability in our method is the intrinsic quantum property of the fluorescence emitter. Therefore, we can directly detect the interference of the two photons without color erasure detectors.

For fluorescence microscopy, a similar statistical model of two fluorescence photons from their interference propagating in a microscopy system must be built. Here, we consider only the PD case because the slow SD process in organic dyes and color centers in diamonds can typically be neglected at low temperatures [21,22,26,27]. We also set the time gate to less than the SD time to reduce the effect of SD. Therefore, the detected probability is given by the following:

(8)$$\begin{aligned} p({r,t} )&= \frac{1}{{{T_1}}}\exp \left( { - \frac{t}{{{T_1}}}} \right)\left( {S_ +^2 + S_ -^2 + 2exp \left( { - \frac{2}{{T_2^ \ast }}t} \right){S_ + }{S_ - }} \right)\\ {S_ \pm } &= \frac{{k_0^2{D^2}}}{{16\pi {f_1}{f_2}}}\left( {\frac{{2{J_1}(\frac{{{k_0}D}}{{2{f_2}}}\left( {\frac{{{f_2}}}{{{f_1}}}\left( {{x_0} \pm \frac{s}{2}} \right) + r} \right))}}{{\frac{{{k_0}D}}{{2{f_2}}}\left( {\frac{{{f_2}}}{{{f_1}}}\left( {{x_0} \pm \frac{s}{2}} \right) + r} \right)}}} \right) \end{aligned}$$

where ${\textrm{S}_\mathrm{\ \pm }}$ is the point spread function of the microscopy system [41], in which J₁ is the first-order Bessel function, D is the diameter of the pupil of the objective, f₁ and f₂ are the focal lengths of the objective and tube lens, respectively, and x₀ is the central position of the two-point sources. Here, microscopy is considered as an achromatic imaging system for which the same approximation k ≈ k₀ is also adopted. Hence, the detected cross-coherence describes a combination of temporal coherence determined by the PD, and spatial coherence arising from the interference of spatially separated two-point sources. Similarly, to extract the cross-coherence that contains the s value information, we used a time gate T_g to construct a temporal coherence modulation function using the integral of Eq. (8) along T_g; when T_g<T₂<<T₁, the cumulative detection probability is as follows:

(9)$$p({r,{T_g}} )\simeq \frac{{2{T_g}}}{{{T_1}}}{S_ + }{S_ - }$$

where the modulation function p(r, T_g) varies approximately linearly with T_g. The slope of p(r₀, T_g) provides an estimation of s when the other parameters are pre-determined from the imaging system and the properties of the fluorescence emitter. Based on this, the limit of resolution is no longer determined by the optical diffraction limit, but instead by the measurement precision of the function p(r, T_g). Therefore, the diffraction limit can be overcome in fluorescence microscopy by measuring the well-defined temporal and spatial functions of the fluorescence quantum coherence.

2.2 Simulation model

As a possible experimental procedure for our method, the experimental setup of a wide-field fluorescence microscopy is shown in Fig. 1(a). For many actual fluorophores, the emitted photons are distributed over a broad range of frequencies owing to the complex inhomogeneous environment or the creation of additional vibrations and phonons. To obtain excellent coherence, a narrow-band spectral filter is required to isolate a single emission line; the zero-phonon line (ZPL) is typically selected, which has a narrow linewidth corresponding to a long coherence time ranging from several hundred picoseconds to a few nanoseconds at low temperatures. Thus, the long coherence time that exceeds the time resolution of the SPAD by more than one order of magnitude allows the time-resolved interference effects to be measured. A wave plate (WP) combined with a polarizing beam splitter (PBS) was used to ensure the identical polarization of the emitters while compensating for the ellipticity introduced by other optical components. To investigate the coherence in the space-time domain, an SPAD array is used as our imaging device, in which each SPAD acts as a pixel and feeds a TCSPC card and logs the arrival times of the detected photons. This device combines spatial information with single-photon sensitivity and picosecond-scale temporal resolution and is capable of detecting emission transients orders of magnitudes faster than the 1 ms temporal resolution of typical cameras. Therefore, we used the pulse excitation scheme synchronized with the SPAD array to record the arrival time t and spatial position r of each fluorescence photon rather than only the spatial position in traditional imaging, as shown in Fig. 1(b). Following per-pulse excitation, two fluorescence photons emitted from two respective sources at a separation of s interfere and reach the SPAD array owing to photon antibunching while the (r,t) of each fluorescence photon is recorded, as shown in Fig. 1(c). In this process, only two-photon detection events per pulse are extracted; one, three, or more photon detection events are removed because they are considered as the noise caused by nondeterministic photon emission, environmental factors, and dark count. These uncorrelated photons reduce coherence. By counting the photon number according to each sampling bin (r,t) within the pulse period, a space-time distribution histogram of the photons can be accumulated using several pulse events, which produces an accurate representation of the photon number distribution. Based on the distribution, we can fit the detected probability curve P(r,t) using Eq. (8), which is normalized by dividing the total number of photons, as shown in Fig. 1(d). To extract the cross-coherence that contains the separation distance s information, we introduce a time gate T_g as a photon arrival time post-selection window by simply summing the photon numbers with $\textrm{t} \le {\textrm{T}_\textrm{g}}$, and obtain the photon number distribution as a function of (r,T_g). Because the coherence time T₂ in Eq. (8) is pre-determined by the HOM experiment, we can extract the highly coherent photons by setting T_g<T₂. Therefore, the cumulative detection probability p(r,T_g) in Eq. (9) is linearly fitted from the photon number distribution, as shown in Fig. 1(e), which indicates that the visibility of the coherence gradually improves along T_g owing to the accumulation of more coherent photons. In Eq. (9), T₁ can be pre-measured by Handury-Brown and Twiss (HBT) experiment [20–22], x₀ can be estimated by the center location algorithm from the detected spatial photon number distribution at a particular T_g [6], and other parameters can be pre-determined by the microscopy system. Therefore, we can determine the only unknown parameter s from the slope 2S₊S_- /T₁ of p(r₀, T_g) at a particular position r₀.

Fig. 1. Schematic illustration of super-resolution wide-field microscopy based on fluorescence quantum coherence. (a) The experimental implementation. An SPAD array is incorporated into a traditional fluorescence microscope and synchronized with a pulsed laser to measure the spatial position and arrival time of each photon. DM, dichroic mirror, WP, wave plate, PBS, polarizing beam splitter. (b) Comparison of the detection schemes of the traditional imaging and SPAD array; the former is based on spatial intensity measurements, while the latter is a temporal and spatial measurement of the photon pairs per pulse. (c) Sequence diagram of the detection and post-selection scheme of SPAD array. Only two-photon events are recorded while single- or multi-photon events are ignored. (d) The photon count distribution with the photon arrival times and detection positions of the two-photon interference. (e) The photon count distribution with the time gate widths and detection positions of the two-photon coherence.

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3. Results and discussions

We implemented a numerical demonstration for imaging two separate nitrogen vacancy (NV) centers in diamonds to prove the feasibility of our method for improving the spatial resolution, as shown in Fig. 2. For traditional wide-field fluorescence microscopy, the spatial resolution of two incoherent point sources was approximately 230 nm owing to Rayleigh’s diffraction limit. However, for the coherent sources, the resolution is slightly worse when considering only the spatial intensity measurement (approximately 350 nm) owing to the interference effect shown in Fig. 2(a), and the contrast of the interference decreases with the increasing distance of two-point sources; thus, it is not possible to resolve the spatial distances of less than 350 nm from the contrast of the spatial intensity owing to the unknown variation of the temporal intensity. Here, we introduce the time dimension and use the SPAD array to simultaneously measure the temporal and spatial information of the detected photons. The time- and space-resolved detection probability functions P(r, t) for different separation distances are shown in Fig. 2(b, top). The detected intensity decreases with the photon arrival time t, as shown in Eq. (8), which includes the T₁-determined self-coherence and T₂-determined cross-coherence. When t < T₂, the variation in cross-coherence is dominant because T₂<2T₁, in which the cross-coherence is clearly more sensitive to the separation distance s and sharply decreases as s increases. Therefore, according to the predetermined value of T₂, we set the time gate T_g (T_g<T₂) to extract the cross-coherence and then constructed the T_g-dependent modulation function p(r, T_g), which is shown in Fig. 2(b, bottom). At a particular position, r = 0, the visibility of the cross-coherence p(0, T_g) increases approximately linearly with T_g, and its slope decreases as s increases, as shown in Fig. 2(c). The relationship between the visibility of the cross-coherence and s is now determined by the known variation, which is the T_g-dependent intensity p(0, T_g), rather than the unknown time variation. Therefore, based on the other predetermined parameters in Eq. (9), the separation distance s can be estimated and resolved from the deterministic slopes of p(0, T_g), despite being below the diffraction limit.

Fig. 2. Numerical demonstration of breaking the diffraction limit using fluorescence quantum coherence for a case of two NV centers. The simulation parameters of NV centers are obtained from the experiment in [26]: the emission wavelength is λ=670 nm, excited state lifetime is T₁= 12 ns, and coherence time is T₂= 15.8 ns. Moreover, the time-resolution of the SPAD array is 55 ps (Photon Force PF32), the magnification of the objective lens is 100× and NA is 1.49, and the detected photon number is approximately 10⁴. (a) The spatial intensity distributions with the different separation distances between two NV centers in traditional imaging. (b) The photon detection probability distributions with different separation distances (s =50, 150, 250, and 350 nm) between two NV centers at the different detection distances r vs the photon arrival time (top) and the time gate widths (bottom). (c) The photon detection probability distributions with the different separation distance s as a linear function of the time gate T_g at the detection center (r = 0) of two NV centers. The slope of each curve changes with the separation distance s.

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Here, the spatial resolution is determined by the precision of estimating the separation distance s between the two-point sources, which depends on the precision of the p(r, T_g) measurement rather than the diffraction limit. Improving the measurement precision requires a low detection noise and a high sampling rate. Although the noise can be effectively reduced via the post-selection of two-photon events, the photon shot noise remains unavoidable. Therefore, the measurement error is fundamentally lower bounded by the “shot-noise limit” of ≃ 1/$\sqrt N $ [6,7]. As shown in Fig. 3, a low cumulative number of detected photons causes the high measurement error of p(0, T_g) due to the effect of the photon shot noise, so that a low estimation of s. The simulation results in Fig. 3 show the spatial resolution of 50 nm can be reached by detecting at least 10⁴ photons, and achieving higher resolution requires counting more photons to obtain the optimality of the p(0, T_g) measurement. In addition to noise, higher sampling rates, both spatial and temporal, also improve the precision of measurement, which are determined by the spatial and temporal resolutions of the SPAD array. A lower coherence time of fluorophores requires a higher temporal resolution and a lower time jitter of the detection to measure time-resolved coherence more accurately. The high spatial resolution of the SPAD array obtains a high sampling measurement of the detector position r, thus producing a more accurate fitting curve p(r,T_g), which also enables the precise estimation of s. Meanwhile, for any position r’, the slope of p(r’,T_g) can provide an estimation of s. Hence, by performing a statistical analysis of all the estimation results, an enhancement in estimation of s and the estimation error are given by the mean and variance, respectively. Moreover, SD limits the observation of coherence, we can still set the time gate T_g less than the SD time to reduce the effect of SD.

Fig. 3. The effect of the measurement errors of p(0, T_g) on the spatial resolutions (s = 50, 100 nm) under different cumulative photon numbers of 10³, 10⁴, and 10⁵.

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4. Conclusion

In this study, we built a microscopic coherence model based on the inherent radiation mechanics of a fluorescence source, and proposed a new method for breaking the diffraction limit by using a SPAD array to simultaneously record the temporal and spatial interference, and then utilized post-selection to modulate temporal coherence to extract the spatial separation distance. Unlike previous atom model that emitting identical fluorescence photons, or incoherent sources in conventional fluorescence microscopy, we employed a partial coherence model based on the fluorescence emission mechanics of actual fluorophores, which not only fully utilizes the quantum properties of fluorescence but also evaluates them in practical environmental effects and imaging systems. Utilizing post-selection by filtering the desirable quantum coherence makes our method less dependent on completely coherent fluorescence sources and suitable for more fluorophores, especially for certain organic dye molecules, quantum dots and color centers commonly used in fluorescence microscopy. Furthermore, our method does not require a complicated entangled light source nor a correlation measurement, which makes this novel quantum-enhanced imaging method accessible in conventional fluorescence microscopy and is helpful for speeding the super-resolution imaging of live cells with weak fluorescence signals.

Funding

Science and Technology Commission of Shanghai Municipality (20DZ2210300).

Disclosures

The authors declare no conflicts of interest.

Data availability

The data underlying the results of this study are available in Ref. [26].

References

1. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19(11), 780 (1994). [CrossRef]

2. M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. 102(37), 13081–13086 (2005). [CrossRef]

3. M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006). [CrossRef]

4. U. Endesfelder and M. Heilemann, “Direct stochastic optical reconstruction microscopy (dSTORM),” Methods in Molecular Biology 1251, 263–276 (2015). [CrossRef]

5. E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006). [CrossRef]

6. R. J. Ober, S. Ram, and E. S. Ward, “Localization Accuracy in Single-Molecule Microscopy,” Biophys. J. 86(2), 1185–1200 (2004). [CrossRef]

7. I. R. Berchera and I. P. Degiovanni, “Quantum imaging with sub-poissonian light: challenges and perspectives in optical metrology,” Metrologia 56(2), 024001 (2019). [CrossRef]

8. M. Genovese, “Real applications of quantum imaging,” J. Opt. 18(7), 073002 (2016). [CrossRef]

9. J. Jacobson, G. Björk, I. Chuang, and Y. Yamamoto, “Photonic de Broglie Waves,” Phys. Rev. Lett. 74(24), 4835–4838 (1995). [CrossRef]

10. E. J. S. Fonseca, C. H. Monken, and S. Pádua, “Measurement of the de Broglie Wavelength of a Multiphoton Wave Packet,” Phys. Rev. Lett. 82(14), 2868–2871 (1999). [CrossRef]

11. K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. 89(21), 213601 (2002). [CrossRef]

12. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum-interferometric optical lithography: Towards arbitrary two-dimensional patterns,” Phys. Rev. Lett. 85(13), 2733–2736 (2000). [CrossRef]

13. M. D. Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87(1), 013602 (2001). [CrossRef]

14. G. Brida, M. Genovese, and I. R. Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4(4), 227–230 (2010). [CrossRef]

15. M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H. A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7(3), 229–233 (2013). [CrossRef]

16. Y. S. Kim, O. Kwon, S. M. Lee, H. Kim, S. K. Choi, H. S. Park, and Y. H. Kim, “Observation of Young's Double-Slit Interference with the Three-Photon N00N State,” Opt. Express 19(25), 24957 (2011). [CrossRef]

17. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, Cambridge, 1995), pp. 683–803.

18. B. Lounis and M. Orrit, “Single-photon sources,” Rep. Prog. Phys. 68(5), 1129–1179 (2005). [CrossRef]

19. C. Santori, D. Fattal, J. Vuckovic, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature 419(6907), 594–597 (2002). [CrossRef]

20. T. Basché, W. Moerner, M. Orrit, and H. Talon, “Photon antibunching in the fluorescence of a single dye molecule trapped in a solid,” Phys. Rev. Lett. 69(10), 1516–1519 (1992). [CrossRef]

21. A. Kiraz, M. Ehrl, T. Hellerer, O. E. Müstecaplolu, C. Bräuchle, and A. Zumbusch, “Indistinguishable Photons from a Single Molecule,” Phys. Rev. Lett. 94(22), 223602 (2005). [CrossRef]

22. R. Lettow, Y. L. A. Rezus, A. Renn, G. Zumofen, E. Ikonen, S. Götzinger, and V. Sandoghdar, “Quantum Interference of Tunably Indistinguishable Photons from Remote Organic Molecules,” Phys. Rev. Lett. 104(12), 123605 (2010). [CrossRef]

23. B. Lounis, H. A. Bechtel, D. Gerion, P. Alivisatos, and W. E. Moerner, “Photon antibunching in single CdSe/ZnS quantum dot fluorescence,” Chem. Phys. Lett. 329(5-6), 399–404 (2000). [CrossRef]

24. E. B. Flagg, A. Muller, S. V. Polyakov, A. Ling, A. Migdall, and G. S. Solomon, “Interference of Single Photons from Two Separate Semiconductor Quantum Dots,” Phys. Rev. Lett. 104(13), 137401 (2010). [CrossRef]

25. V. Giesz, S. L. Portalupi, T. Grange, C. Antón, L. De Santis, J. Demory, N. Somaschi, I. Sagnes, A. Lemaître, L. Lanco, A. Auffèves, and P. Senellart, “Cavity-enhanced two-photon interference using remote quantum dot sources,” Phys. Rev. B 92(16), 161302 (2015). [CrossRef]

26. H. Bernien, L. Childress, L. Robledo, M. Markham, D. Twitchen, and R. Hanson, “Two-Photon Quantum Interference from Separate Nitrogen Vacancy Centers in Diamond,” Phys. Rev. Lett. 108(4), 043604 (2012). [CrossRef]

27. A. Sipahigil, K. D. Jahnke, L. J. Rogers, T. Teraji, J. Isoya, A. S. Zibrov, F. Jelezko, and M. D. Lukin, “Indistinguishable Photons from Separated Silicon-Vacancy Centers in Diamond,” Phys. Rev. Lett. 113(11), 113602 (2014). [CrossRef]

28. O. Schwartz, J. M. Levitt, R. Tenne, S. Itzhakov, Z. Deutsch, and D. Oron, “Superresolution Microscopy with Quantum Emitters,” Nano Lett. 13(12), 5832–5836 (2013). [CrossRef]

29. J. M. Cui, F. W. Sun, X. D. Chen, Z. J. Gong, and G. C. Guo, “Quantum statistical imaging of particles without restriction of the diffraction limit,” Phys. Rev. Lett. 110(15), 153901 (2013). [CrossRef]

30. R. Tenne, U. Rossman, B. Rephael, Y. Israel, A. Krupinski-Ptaszek, R. Lapkiewicz, Y. Silberberg, and D. Oron, “Super-resolution enhancement by quantum image scanning microscopy,” Nat. Photonics 13(2), 116–122 (2019). [CrossRef]

31. W. Larson and B. E. A. Saleh, “Resurgence of Rayleigh’s curse in the presence of partial coherence,” Optica 5(11), 1382–1389 (2018). [CrossRef]

32. K. Liang, S. A. Wadood, and A. N. Vamivakas, “Coherence effects on estimating two-point separation,” Optica 8(2), 243–248 (2021). [CrossRef]

33. Z. Hradil, J. Rehácek, L. Sánchez-soto, and B. G. Englert, “Quantum Fisher Information with Coherence,” Optica 6(11), 1437–2536 (2019). [CrossRef]

34. C. Thiel, T. Bastin, J. Martin, E. Solano, J. von Zanthier, and G. S. Agarwal, “Quantum Imaging with Incoherent Photons,” Phys. Rev. Lett. 99(13), 133603 (2007). [CrossRef]

35. C. Thiel, T. Bastin, J. von Zanthier, and G. S. Agarwal, “Sub-Rayleigh quantum imaging using single-photon sources,” Phys. Rev. A 80(1), 013820 (2009). [CrossRef]

36. A. Muthukrishnan, M. O. Scully, and M. S. Zubairy, “Quantum microscopy using photon correlations,” J. Opt. B: Quantum Semiclassical Opt. 6(6), S575–S582 (2004). [CrossRef]

37. B. Kambs and C. Becher, “Limitations on the indistinguishability of photons from remote solid state sources,” New J. Phys. 20(11), 115003 (2018). [CrossRef]

38. A. Tokmakoff, “Time dependent quantum mechanics and spectroscopy,” (2014), http://tdqms.uchicago.edu.

39. T. Legero, T. Wilk, A. Kuhn, and G. Rempe, “Time-Resolved Two-Photon Quantum Interference,” Appl. Phys. B 77(8), 797–802 (2003). [CrossRef]

40. T. Legero, T. Wilk, A. Kuhn, and G. Rempe, “Characterization of single photons using two-photon interference,” Adv. At., Mol., Opt. Phys. 53, 253–289 (2006). [CrossRef]

41. Y. Shih, An introduction to quantum optics (CRC Press, 2011), pp. 25–84.

42. L. C. Liu, L. Y. Qu, C. Wu, J. Cotler, F. Ma, M. Y. Zheng, X. P. Xie, Y. A. Chen, Q. Zhang, F. Wilczek, and J. W. Pan, “Improved Spatial Resolution Achieved by Chromatic Intensity Interferometry,” Phys. Rev. Lett. 127(10), 103601 (2021). [CrossRef]

Breaking the diffraction limit using fluorescence quantum coherence

Abstract

1. Introduction

2. Methods

2.1 Theoretical model

2.2 Simulation model

3. Results and discussions

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (3)

Equations (9)

Optics Express