Quantum correlation imaging (also known as ghost imaging) originated from the “EPR paradox”^[1], where entangled photon pairs are used as the light source. The “EPR” is an abbreviation of the names of the three proposers: Einstein, Podolsky, and Rosen. One is able to obtain the image of an object entirely relying on the correlation between the entangled photon pairs, without directly viewing the object with a spatially resolving camera^[2,3]. The photons used to achieve imaging themselves did not come into contact with the object to be detected, making it a nonlocal “off-object imaging” method. As a new imaging system that utilizes the correlation characteristics of light field intensity to achieve target imaging, unlike traditional optical imaging techniques based on first-order coherence of the light field, correlation imaging reconstructs target information by utilizing the second-order coherence of the light field between two paths of light. Quantum correlation imaging plays an important role in fields such as quantum information processing, quantum communication^[4,5], and verifying position-momentum (EPR) entanglement^[6–12]. Compared to traditional imaging, it can achieve functions such as super-resolution imaging, stealth imaging, and multi-target tracking, which helps improve the clarity and resolution of images^[13,14]. At the same time, it also has important applications in quantum communication such as quantum teleportation and quantum secure communication. Meanwhile, it can achieve high-quality imaging in nonvisible light bands and resist the influence of harsh weather conditions such as turbulence and mist.

The development of quantum correlation imaging is of great significance for promoting the application and development of quantum technology^[15–19]. The existing quantum correlation imaging schemes mostly use Gaussian beams as pump light sources, resulting in the entangled two photons also exhibiting a Gaussian distribution. Theoretical analysis has been conducted on the pump source of non-Gaussian beams^[20,21], but it has not been experimentally implemented. The central part of a Gaussian beam has a higher power density, while the edge part has a lower power density. Correlation imaging uses photon detectors to scan and detect the energy distribution of a beam, causing imaging distortion. The flat-top beam has a uniform energy distribution within the beam range^[22,23], which can effectively solve the problem of imaging distortion. In this Letter, we report the experimental demonstration of quantum correlation imaging using a flat-top beam as the pump source, maintaining high imaging similarity.

In this Letter, we use the spontaneous parametric down conversion (SPDC) process of nonlinear crystals to generate entangled photon pairs and use them as quantum imaging light sources. The entire imaging optical path is based on a double 4f imaging systems. The experimental setup is depicted in Fig. 1(a). The pump laser is from an external-cavity diode laser. The wavelength of the laser is 405 nm, and the spectral line width is 2 MHz. The pump laser is a Gaussian beam with a waist of ω0=0.7mm, and its power is 8 mW. We first use a narrowband filter to filter out non 405 nm pump beam. Subsequently, we insert a homogenizing lens (CHC25-405-1-FTC1-SP, LBTEK) with a transmittance of 20% to transform the Gaussian beam into a flat-top beam. Since the effect of the flat-top beam is closely related to the position of the homogenizing lens, the homogenizing lens is placed on the three-dimensional adjustment table for fine adjustment. Due to the use of a type II nonlinear crystal, we adjust the pump beam to horizontal polarization using a half-wave plate and a polarizing beam splitter (PBS). We use a pair of lenses with focal lengths f1=75mm and f2=50mm to reduce the beam waist radius to increase the generation rate of entangled photons. After lens 2, the power of the pump beam is about 1.6 mW. The photon pairs are generated by a 1mm×2mm×15mm periodically poled potassium titanyl phosphate (PPKTP) crystal, which is phase-matched for type-II collinear emission. Three band-pass filters with a full width at half-maxima (FWHM) bandwidth of 10 nm at 810 nm come after the crystal block and ensure that only frequency-degenerate photons are detected. We use another pair of lenses with focal lengths f3=50mm and f4=300mm to expand the diameter of entangled photon beams to ensure that the photon beam can cover the imaging object. We split the photon pairs into two separate paths by means of another PBS.

After PBS 2, the horizontal polarization path is used as the signal photon. After illuminating the imaging object, the signal photons are collected using a multimode fiber. In this Letter, a resolution card (RC) with three identical slits is used as the imaging object, as shown in Fig. 1(b), and the width and spacing of each slit are 0.24 mm. The RC is placed on the focal plane of lens 4, and then a single photon detector (SPCM-AQRH-14-FC, Excelitas) is used for photon detection. The photon electrical signal is input into the dual channel coincidence counter (Time Harp 260, PicoQuant). The vertical polarization path plays the role of idle photons. Idle photons pass through a slit with a width of 200 µm and enter the fiber coupling head (coupler 2). For a higher detection signal-to-noise ratio (SNR), coupler 2 is connected with a single-mode fiber. The slit is placed on the focal plane of lens 4 and is placed on the horizontal translation platform together with coupler 2. The horizontal position scanning of the translation platform is carried out, and the scanning step is 100 µm. Idle photons are collected by another single photon counter (SPD 2), and the coincidence measurement is carried out with the signal photons by the dual-channel coincidence counter. By changing the delay time, the time-dependent functions of signal photons and idler photons are measured, the coincidence counter efficiency is about 400 Hz, and the SNR reaches 130. Due to the low noise level, the experiment can ignore the background noise. The time accumulation of each position point is 5 s, due to high entangled photon generation efficiency.

We first analyze the imaging effect of the Gaussian beam as the pump light by removing the homogenizing lens. The Gaussian model in cylindrical coordinates can be expressed as A(ρ,z)=A0ω0ωexp(−ρ2ω2)exp[ikρ22R(z)−iG(z)],(1)where ρ=x2+y2, ω(z)=ω01+(z/ZR)2 represents the radius of beam, ZR is the Rayleigh radius, R(z)=z+ZR2/z is the radius of curvature of the equiphase surface, and G=arctan(z/ZR) is the Gouy phase of the Gaussian beam. It can also be seen that the energy distribution of the Gaussian beam is not uniform, the energy density of the center part of the beam is high, and the energy density of the edge part is low.

The two-photon quantum state generated from the crystal can be written as^[9]$$\begin{gathered} |Ψ⟩=A∫dωs1dk→s2dk→s1χ(3)(ωs1,ωs2)sinc(ΔkL/2) \\ ×ε+˜(k→s1+k→s2)a^k→s1+a^k→s2+|0⟩, \end{gathered}$$(2)where A is a normalization constant, χ(3)(ωs1,ωs2) is the third-order nonlinear susceptibility of the medium, Δk=(k→p−k→s1−k→s2)·z→ is the longitudinal phase mismatch along the direction z→ of the crystal of length L, and ε+˜(k→s1+k→s2) is the Fourier transform of the Gaussian pump transverse profile. a^k→s1+ and a^k→s2+ are the creation operators of photons with transverse wave vectors k→s1 and k→s2, respectively. The energy conservation condition confirms that ωs1=ωp−ωs2. The signal photon and idle photon have a strong correlation in the time domain. Their correlation in position can be further deduced by Fourier-transforming the eigenfunction of the system into momentum space. By simulating using the method described in Ref. [24], we calculate the appropriate temperature according to the following equation: I3∝sinc2{πLc[n(λ1,T)λ1+n(λ2,T)λ2−n(λ3,T)λ3+1Λ(T))]},(3)where I3 is the intensity of two-photon radiation, and n is the refractive index of the crystal for different wavelengths. It is found that the optimal operating temperature for the PPKTP crystal in our experiment is 18°C. At this temperature, the theoretically generated 810 nm correlated photon pairs have a bandwidth of approximately 0.4 nm. Due to various system errors, there is a small deviation between the center wavelength of the received spectrum and the wavelength in the experiment.

In order to compare the imaging effect with the flat-top beam more accurately, we reduced the power of the Gaussian beam to 1.6 mW to ensure the same pump power as the flat-top beam. After the beam is expanded by lens 4, the signal photons with a Gaussian energy distribution are just right to illuminate the RC. We use the entire beam to image the RC, rather than the central part of the beam, because the central part of the Gaussian beam also has a relatively flat energy distribution area, or it may mislead the experimental results.

Correlation imaging uses photon detectors to detect the energy distribution of a beam by position scanning. We use SPD 1 to record the number of signal photons after the signal beam irradiates the RC and then use SPD 2 to record the number of idler photons obtained by scanning the transverse position of the slit and coupler 2. The coincidence measurement of two-path photons is carried out by the dual-channel coincidence counter. The imaging results of the Gaussian pump beam are shown in Fig. 2. Due to the high coincidence counting rate, the sampling time for each point is 5 s, with an average of three times. We can clearly see three peaks corresponding to the imaging results of three slits. Due to the three slits of the RC having the same size, the ideal imaging result is three peaks of the same height and width. But in fact, the middle peak is very high, and the two peaks on the edge are short. From left to right, the corresponding coincidence count values for the three peaks are 168, 285, and 195. The imaging similarity is 58.9% by dividing the peak (168) on the edge by the peak in the middle (285). The energy distribution of the Gaussian beam is not uniform. With the increase of the distance from the center of the cross section of the Gaussian beam, the symmetric irradiance distribution of the beam decreases, and the imaging results are significantly distorted.

The cross-section of a flat-top beam has a constant irradiance distribution in theory. The beam intensity expression of the flat-top beam can be written as Ep={N0,r≤ωp0,r>ωp.(4)

It is a function that is constant in the region r≤ωp and zero in the region r>ωp. N0 is the normalization coefficient, namely, power density, and ωp is the radius of the flat-top beam. The flat-top beam can be obtained by the diffraction of Gaussian beams through a homogenizing lens. The diffraction equation of beam shaping can be written as U(x,y)=exp(ikf+x2+y2)iλf∬U(x1,y1)exp(iβϕ)⁢exp(−i2πλfxx1+yy1))dx1dy1,(5)where k=2π/λ is the wave vector, and U(x1,y1) and U(x,y) are the electric fields of the incident beam and the outgoing beam, respectively. β=2πω0ω1/λf is an optical system parameter that is determined by the incident beam waist ω0 and the outgoing beam ω1. In our experiment, it is necessary to choose a suitable value of β to obtain a high similarity beam electric field. For most light, its diffraction is not analytical and requires numerical calculation of its diffraction equation.

In our experiment, the homogenization lens is composed of a structured wave plate and two parallel linear polarizers at a certain angle. The linearly polarized light is generated by the structured wave plate to generate a vector polarized beam, and then the Gaussian beam intensity is attenuated by the latter polarizer to achieve collimation and homogenization of the beam. Due to the structured wave plate phase distribution being designed based on the ideal Gaussian beam, there is a high requirement for the quality of the incident beam (single mode, M2<1.3). It should be noted that we assume the flat-top beam is flat-topped in the whole crystal. In fact, because the flat-top beam is not the solution of the Helmholtz equation with paraxial approximation, it will be diffracted, which can be calculated by the Rayleigh–Sommerfeld diffraction equation. Fortunately, because we use a commercial homogenizing lens to shape the beam, it is found that the shape of the flat-top beam can still be well maintained within about 90 cm, so it is reasonable to approximate the flat-top beam inside the crystal in our processing. This distance can also ensure that the quality distribution of the flat-top beam meets the experimental requirements throughout the entire optical path range. Beyond this distance, the characteristics of the flat-top beam will no longer be maintained.

Our quantum correlated imaging patterns based on the flat-top pump beam are shown in Fig. 3. The illustration in the upper right corner is a cross-sectional view of the intensity distribution of the flat-top beam used in the experiment. We measured the quality of the beam using a beam analyzer (BC207VIS/M, Thorlabs) about 5 cm after the homogenizer lens. It can be clearly seen that the intensity distribution of the flat top beam after diffraction by a homogenized lens is relatively uniform, and there is a clear boundary line at the edge of the beam. The uniformity of the outgoing beam at this position is about 90.2% (RMS).

The effective width of the signal photons with a Gaussian beam or flat-top beam quality distribution is about 2 mm, which is just enough to cover the RC. The quantum correlation imaging results of the flat-top beam are very close to the ideal situation. Three identical slits correspond to three peaks with almost the same height and width, due to the uniform distribution of the power density of the flat-top beam within the beam range. From left to right, the corresponding coincidence count values for the three peaks are 210, 212, and 198, and the imaging similarity is 93.4%. The sampling time for each point is also 5 s, with an average of three times.

In summary, we have used the flat-top beam as the pump source for quantum correlation imaging, demonstrating good image similarity due to the uniform distribution of the power density of the flat-top beam within the range. At the same time, the spatial coherence of the flat-top beam is high, providing collimated and parallel beams, which is conducive to forming clear interference or diffraction patterns. The flat-top beam makes the interference and superposition effects more obvious in the field of quantum correlation imaging, which is conducive to achieving applications such as high-resolution imaging and entangled photon imaging. Compared with ordinary Gaussian beams, flat-top beams show their unique advantages in imaging. The wavefront shape and optical field structure of a flat-top beam can be controlled to meet different quantum correlation imaging experimental requirements, which is worth further study. Other methods for achieving uniform imaging, such as using a spatial light modulator to normalize the intensity profile of the pump beam or laser scanning, can also provide a uniform image. Imaging more complex objects is our next step of work, and we will conduct two-dimensional imaging to provide a better demonstration. Our presented work using a homogenizing lens to prepare a flat-top beam is a simple and low-cost method. This work will contribute to the quality of quantum correlation imaging.