Hanbury Brown and Twiss [1,2] recommended the second-order correlation measurement and a new type of interference effect between photon-pair amplitudes, i.e., the bunching effect of thermal light, more than 60 years ago. The physical essence of the bunching effect embodied by the second-order coherence of the light fields is two-photon interference, involving different but indistinguishable alternative ways of triggering a joint-detection event [3,4]. Both classical and quantum theories [3,5–7] were successfully employed to properly interpret the bunching effect of light fields, which was regarded as a milestone in quantum optics.

For thermal light propagating in free space, the normalized second-order spatial correlation function g(2)(x,x′) equals 2 when x−x′=0. On the basis of permutation and combination, for an interference experiment with N photons detected coincidently by N detectors, the peak value of the generalized N-photon bunching effect should be N! [8]. Therefore, naturally a two-photon super-bunched effect is only observed when g(2)(0)>2, in comparison with the bunching peak value of 2 of thermal light. A large bunching peak g(2)(0) has been demonstrated to be of essential importance in ghost interference, ghost imaging [9–17], and the multi-photon nonlinear light–matter interaction [18–22]. Several methods were exploited to achieve a super-bunched effect, for instance, through the nonlinear light–matter interaction [23–25], collective two-level atoms coupled with a cavity [26–32], cavity-coupled quantum dot nanolasers [33–35], optical rogue waves and extreme phenomena [36,37], squeezed states [38], and increased intensity fluctuations [39–41]. Note that most of them were in the time domain. Several groups have also paid attention to the super-bunched effect in the spatial domain [42–46]. And g(2)(0)=2.4±0.1 was demonstrated initially by our group via multiple two-photon path interference introduced by inserting a pair of mutually first-order incoherent optical channels into a traditional Hanbury Brown–Twiss (HBT) interferometer, surpassing the theoretical limit of 2 for thermal light, which also demonstrates the ability to control the bunching property of thermal light in a linear optical system [42]. Multiple different but indistinguishable two-photon paths could also be introduced through random-phase gratings or wavefront modulation, in which the position-correlated random phase was encoded on the transmitting light fields [46,47]. These indistinguishable two-photon paths of a photon pair triggering a coincidence count also play a key role in the second-order subwavelength interference [48–51].

Chirp is a signal in which the frequency increases (up-chirp) or decreases (down-chirp) with time [52–54], commonly used in sonar, radar, and spread-spectrum communications. In optics, ultrashort laser pulses often exhibit chirp, which interacts with the dispersion properties of the materials in an optical transmission system, increasing or decreasing the total pulse dispersion as the signal propagates [55]. Similarly, for spatial chirping, for example, spatially chirped gratings, it is the grating parameters such as the grating slit width or grating period that vary slowly with position, either linearly or nonlinearly [56]. Spatially chirped gratings have been applied to present focusing properties [56–63] and characteristic diffraction fringes [64,65] in both the near-field and far-field zones, but mainly in the first-order optical coherence range. Nonperiodic gratings were also applied in third-generation synchrotron radiation and high-resolution X-ray spectroscopy [66,67].

In this paper, we are mainly concerned with the two-photon bunching properties of chirped random-phase gratings (CRPGs). Two types of CRPGs are considered. One is where the grating period d is fixed but the slit width an is chirped. The other is where the slit width is fixed but the grating period is chirped. With CRPGs, superposition of multiple different but indistinguishable two-photon paths is introduced, and the normalized second-order spatial correlation function shows a bunching curve with a much higher peak and a much narrower profile than those of thermal light, achieving a two-photon super-bunched focusing effect, which is potentially useful for applications such as correlation imaging, microfabrication, and enhancement of high-order nonlinear light–matter interaction.

Let us consider the first- and second-order interference effects of the light field transmitting through a slit-width-chirped random-phase grating in the Fraunhofer zone when a collimated single-mode laser beam is incident normally onto the slit-width-chirped random-phase grating. The schematic configuration for studying the first- and second-order coherence of the light field transmitting through CRPGs is illustrated in Fig. 1(b), where L is a lens to collect the scattering light from the CRPG and CCD is a charge-coupled device camera for recording the intensity distribution on the focal plane of lens L. In the one-dimensional case and under the paraxial approximation, the field operator on the detection plane can be expressed as E^(+)(x)∝∑n=1Nsinc(πanxλf)e−i2πx(n−1)dλfe−i(n−1)ϕa^,(1)where a^ is the annihilation operator of the light field and λ, x, and f are the wavelength of the light source, the transverse coordinate on the detection plane, and the focal length of lens L, respectively. One sees that multiple diffraction orders of the grating are taken into account. For simplicity, here we restrict ourselves to the case where the grating parameters are not in the subwavelength range and the paraxial approximation is always satisfied, and we just consider the spatial coherence properties of the light fields. The synchronous first-order spatial correlation function can be calculated as G(1)(x)=⟨E*(x)E(x)⟩∝∑n=1Nsinc2(πanxλf),(2)where E(x) is the eigenvalue of the field operator E^(+)(x) on the state of the source and ⟨⋯⟩ represents the ensemble average. It is evident that the intensity distribution on the detection plane is the sum of the diffraction intensities from N slits of the slit-width-chirped random-phase grating, and no stationary first-order interference pattern can be observed due to the condition ⟨eiϕ⟩=0. However, the case will be totally different when the two-photon interference is considered, as we will show below.

The synchronous second-order spatial correlation function on the detection plane can be expressed as G(2)(x,x′)=⟨E*(x)E*(x′)E(x′)E(x)⟩.(3)

By substituting Eq. (1) into Eq. (3) and taking into consideration the condition ⟨eilϕ⟩=0 if l≠0, one can obtain $$\begin{gathered} G(2)(x,x′)∝⟨∑l=02N−2|∑(m,n);m+n−2=le−ilϕ[sinc(πamxλf) \\ ×e−i2πx(m−1)dλfsinc(πanx′λf)e−i2πx′(n−1)dλf \\ +H(m,n)sinc(πanxλf)e−i2πx(n−1)dλf \\ ×sinc(πamx′λf)e−i2πx′(m−1)dλf]|2⟩, \end{gathered}$$(4)where H(m,n)=0 (m=n) or 1 (m≠n) and m and n represent the mth and the nth slits of the grating, respectively. There is only one valid two-photon path if the two photons triggering the coincidence counting come from the same slit. Note that there are many twin two-photon paths originating from different pairs of slits (m,n), (m+1,n−1), (m+2,n−2), and so on, whose amplitudes contain the same random phase (m+n−2)ϕ, as shown in Fig. 1(c). These two-photon paths are indistinguishable in principle. Our previous work has proved that the amplitude superposition of all these different but indistinguishable two-photon paths with the same random phase (m+n−2)ϕ can enhance the two-photon interference with a signature of high visibility in a random-phase grating [47].

The normalized second-order spatial correlation function can then be calculated as $$\begin{gathered} gN(2)(x,x′)=G(2)(x,x′)G(1)(x)G(1)(x′) \\ =1+∑r=1N−1{2cos[2πλf(x−x′)rd]C(x,x′)}D(x,x′), \end{gathered}$$(5)where $$\begin{gathered} C(x,x′)=∑p=1N−r∑q=1N−rsinc(πxapλf)sinc(πxap+rλf) \\ ×sinc(πx′aqλf)sinc(πx′aq+rλf), \end{gathered}$$(6)D(x,x′)=∑m=1Nsinc2(πxamλf)×∑n=1Nsinc2(πx′anλf).(7)

One sees that the second-order spatial correlation function on the detection plane is a weighted superposition of a set of periodic interference fringes, in which the weight function C(x,x′) is the product of multiple sinc functions characterizing the two-photon paths, with the two photons originating from two slits separated by rd and having the same random phase (p+q+r−2)ϕ. These periodic interference fringes have different interfering periods λf/rd, and they are exactly in phase only when x−x′=0, while they become gradually out of phase with the increase of x−x′, leading to a two-photon bunching peak at x−x′=0. In general, the bunching profile could be very complicated, depending on the structure parameters of the CRPGs. However, super-bunched focusing with a bunching peak much higher and a bunching profile much narrower than those of thermal light can be achieved by optimizing the structure parameters of the CRPGs. In the following, we will give an optimization procedure to produce the super-bunched focusing effect through the CRPGs.

For simplicity, we fix the position of one detector at x′=0 and scan the other detector at x on the detection plane, which is also in accordance with the following experiments in Section 3. In this case, the normalized second-order spatial correlation function can be simplified as gN(2)(x)=1+2∑r=1N[(N−r)cos(2πxrdλf)∑q=1N−rsinc(πxaqλf)sinc(πxaq+rλf)]N∑n=1Nsinc2(πxanλf),(8)and its bunching peak can be calculated as gN(2)(0)=1+2∑r=1N−1(N−r)2N2=2N2+13N,(9)where N is the total slit number of the CRPGs. It is seen that the bunching peak gN(2)(0) increases with an increasing grating slit number N, and is larger than 2, the bunching peak of thermal light, when N>2.

Since the super-bunched focusing effect is defined in comparison to the bunching curve 1+sinc2(x) of thermal light in the traditional HBT interferometer, we therefore assume the super-bunched profile of the CRPGs in the formula f(xi)=1+[gN(2)(0)−1]sinc2(c1xi), in which the parameter c1 characterizes the spot size of the super-bunched profile of the CRPGs. The larger the parameter c1, the smaller the spot size of the super-bunched profile. The super-bunched focusing effect is achieved when the spot size of the super-bunched profile of the CRPG is smaller than that of the bunching curve of thermal light in the traditional HBT interferometer. The visibility of the super-bunched curve is V=gmax(2)−gmin(2)gmax(2)+gmin(2)=N2−1N2+2.(10)

Obviously, the visibility, increasing with an increasing N, can surpass 50% and asymptotically approach 100%.

To achieve a super-bunched focusing profile in the formula f(xi)=1+[gN(2)(0)−1]sinc2(c1xi), we optimize the grating slit width an of the CRPGs by means of a MATLAB-implemented nonnegative least squares algorithm [68]. In this case, the sum of the squared residual error Sε2 and the total sum of squares SST can be expressed respectively as $$\begin{gathered} Sε2=∑[gN(2)(xi)−f(xi)]2, \\ SST=∑[f(xi)−f¯(xi)]2, \end{gathered}$$(11)where f¯(xi) is the mean value of the fitting function f(xi). It is the optimal solution when the coefficient of determination R2=1−Sε2/SST gets closest to 1. But in practice, it is considered to be an effective fitting when 0.8<R2<1. A set of equations (see Appendix A) can be obtained with the slit width an of the CRPGs as the variable, which can be solved to get the optimal silt width set {an} for the super-bunched focusing effect. Several examples with different total slit numbers N=4, 8, 16, and 50 are given in Appendices B.1 and B.3. Note that here the chirping is usually nonlinear in order to suppress the sidelobes beyond the focusing point.

We have theoretically shown that if the grating period is fixed while the slit width is chirped, the super-bunched focusing effect can be obtained. In the other case—for a period-chirped random-phase grating, where the slit width is fixed but the period is chirped—one can also demonstrate the super-bunched focusing effect. By employing a similar procedure to the case with slit-width-chirped random-phase gratings, one obtains the normalized second-order spatial correlation function $$\begin{gathered} gN(2)(x,x′)=1+2N2∑r=1N−1∑p=1N−r∑q=1N−r \\ ×cos{2πλf[ra(x−x′)+x∑k=pp+r−1bk−x′∑k′=qq+r−1bk′]}, \end{gathered}$$(12)where a and bk are the fixed slit width and the kth grid line of the gratings, respectively, satisfying dk=a+bk, with dk being the chirped grating period at the kth slit. In Eq. (12), we replace the grating period dk with the grating grid line bk in order to simplify the calculation. The same optimization procedure can be employed to get an optimized set {bk} for the super-bunched focusing effect. Several examples with different total slit numbers N=4, 8, 16, and 50 are also given in Appendices B.2 and B.4.

One sees that theoretically the super-bunched focusing effect is mainly determined by the grating slit number N and the grating period d. When the grating slit number N is fixed, the focusing spot size is mainly determined by the grating period d. A larger grating period results in a smaller focusing spot size. In the slit-width-chirped random-phase grating case, the grating period is uniform and fixed. However, in the case with the period-chirped random-phase grating, the grating period is chirped, which will induce a broadening effect on the focusing spot size. Therefore, with the same grating slit number N, the focusing spot size with the slit-width-chirped random-phase grating is always smaller than that with the period-chirped random-phase grating. On the other hand, the field will have a tighter focusing spot size with a larger grating slit number N. In practice, however, with a larger grating slit number N, the optimization procedure for getting the optimal grating parameters will require much longer computation time, and the grating fabrication process should also be precise enough, because any fabrication deviation from the optimal grating parameters will result in a broadening effect on the focused spot size. Therefore, there is a trade-off between the cost and the performance in choosing an appropriate N, and it is better to choose an appropriate N according to the requirement on the focused spot size in practical applications. In addition, one sees that theoretically the bunching peak value is the same for both CRPGs, and it asymptotically approaches (2/3)N, which could be very high with increasing N. In practice, however, the grating slit number N is always limited because of the limited grating size. Moreover, the bunching peak value is also affected by the optimization process and the fabrication process of the CRPG. To get a high bunching peak value, one has to suppress the sidelobes of the bunching curve. The lower the sidelobes, the higher the bunching peak value.

In the following, we will demonstrate experimentally this super-bunched focusing effect through the proposed CRPGs.

In conclusion, we have designed two types of CRPGs, the slit-width-chirped and the grating-period-chirped random-phase gratings, through which the two-photon super-bunched focusing effect can be realized in the Fraunhofer zone. Theoretically, the bunching peak and the visibility of the super-bunched curves can asymptotically reach (2/3)N and 100%, respectively, where N is the total slit number of the CRPGs. Experimentally, we verified that the bunching peak and the visibility of the two-photon bunching curves for the light transmitting through the CRPGs increased with an increasing N, and a bunching peak of 15.38±0.05 and a visibility of 92.5% were demonstrated through a period-chirped random-phase grating with a fixed slit width of 30 μm and N=50. The FWHM of the super-bunched curve was confirmed to decrease with an increasing N, and a photon-pair bunched spot size with an FWHM of 70 μm was achieved through a slit-width-chirped random-phase grating with a fixed grating period of 200 μm and N=50, therefore focusing the photon pairs with greatly improved spatial resolution. This super-bunched focusing effect could have important potential applications such as correlation imaging with improved visibility and spatial resolution or enhanced nonlinear light–matter interaction.