The Hanbury Brown and Twiss (HBT) effect is a classical intensity correlation phenomenon that was first employed in an intensity interferometer to measure the angular size of distant stars. It has much better resolution and visibility than the Michelson stellar interferometer because it measures the second-order, intensity–intensity correlation of two separate intensity detectors rather than the first-order, field-field correlation^[1]. Though the effect was originally explained by HBT with the classical wave theory, it aroused a debate on the classical or quantum nature of the phenomenon, which has lasted to this day. Despite this, the HBT effect is widely exploited in both the classical and quantum regimes, and has led to many important developments in quantum optics, particle physics, and practical applications, among which ghost imaging^{[2–11" target="_self" style="display: inline;">–11]} and ghost diffraction^{[1214" target="_self" style="display: inline;">–14]} have aroused particular interest in both fundamental optics and imaging technology.

Ghost imaging, first implemented with entangled light^[2], has promising potential in many applications on account of its apparently nonlocal characteristics and subwavelength resolution capability^[15,16], as well as its relatively simple implementation with classical thermal light^{[57" target="_self" style="display: inline;">–7,1719" target="_self" style="display: inline;">–19]}. The second-order intensity–intensity correlation for thermal light only has a maximum Michelson contrast (also known as visibility, defined as (Imax−Imin)/(Imax+Imin)) of 33%. Many schemes have been demonstrated to improve its contrast^{[20–25" target="_self" style="display: inline;">–25]}, including a scheme that achieved 100% correlation^[26]. However, according to the definition of Michelson contrast, its value can always reach 100% after a unity-based normalization. In practical measurement, different noise levels will lead to different signal-to-noise ratios (SNRs) of the experimental results, even though the contrasts can all reach 100%. Therefore, the SNR provides a better criterion for image or pattern quality in experiments and applications.

Just like ghost imaging, ghost diffraction was first demonstrated using spontaneous parametric down-conversion, and at the time was attributed to the quantum entanglement of photons^[12,27]. Then, ghost diffraction was performed using thermal light^[13], and subwavelength resolution performance was predicted^[28] and realized^[14,29], even with two independent pseudothermal sources^[30].

In this Letter, by extending a previously developed technique called correspondence ghost imaging^[31,32], we demonstrate that it is possible to achieve a better SNR (10.3% improvement) and subwavelength (λ/2) resolution simultaneously in a thermal light diffraction experiment. We first extend the concept of the positive and negative correlations and employ them in diffraction. An experiment on thermal light subwavelength diffraction and its results are then presented, followed by a discussion and our conclusions. All correlations mentioned in this Letter are in the spatial regime.

A schematic of the diffraction setup is shown in Fig. 1. A thermal light field passes through a double slit and then is divided by a 50–50 beam splitter into Paths 1 and 2. In Path 1, the point-like Detector 1 is scanned along one direction in the transverse plane and measures the field intensity I1(x1) as a function of the transverse position x1. In Path 2, the point-like Detector 2 is scanned along the opposite direction in the transverse plane and measures the field intensity I2(x2) as a function of the transverse position x2. The time averaged intensities at these two detectors are denoted as I1¯(x1)=〈I1(x1)〉 and I2¯(x2)=〈I2(x2)〉, respectively.

Fig. 1. Schematic of setup to measure diffraction. BS, 50–50 beam splitter; C.C., coincidence counter.

下载图片 查看所有图片

If one detector, e.g., Detector 1, is fixed at x0, while the other, Detector 2, is scanned along axis X2, then a classical diffraction pattern can be obtained by performing an intensity correlation with I1(x0) and I2(x2)^[14] using gCI(2)=〈I1(x0)I2(x2)〉〈I1(x0)I2(x2)〉=1+cov(I1(x0)I2(x2))〈I1(x0)〉〈I2(x2)〉,(1)where 〈…〉 is the temporal mean, and cov is the covariance between I1(x0) and I2(x2), defined as cov(I1(x0)I2(x2))=〈(I1(x0)−〈I1(x0)〉)(I2(x2)−〈I2(x2)〉)〉.

If the two detectors are scanned along axes X1 and X2, respectively, in opposite directions, a subwavelength diffraction can be obtained^[14] by performing the second-order intensity correlation with I1(x1) and I2(x2) using gSI(2)=〈I1(x)I2(−x)〉〈I1(x)〉〈I2(−x)〉=1+cov(I1(x)I2(−x))〈I1(x)〉〈I2(−x)〉,(2)where x=x1=−x2. Because there is a constant term or background in this intensity–intensity correlation function, the contrast has a maximum of 33%. To enhance the contrast of the intensity correlation, the fluctuation correlation is frequently utilized as gSIF(2)=〈I1(x)I2(−x)〉(I1¯(x)I2¯(−x))−1=〈cov(I1(x)I2(−x))〉〈I1(x)〉〈I2(−x)〉,(3)which is of the same mathematical form as the intensity correlation after unity-based normalization.

In our previous work^[31,32], the data recorded by Detector 1 was separated into two sets according to whether the intensity was above or below the mean value of the data. Using these two sets of data, we were able to get the positive and negative correlation results. We shall now show how the positive and negative correlation is enhanced when the intensity values in both paths are initially separated into positive and negative groups.

In the setup of Fig. 1, because of the thermal fluctuations, the outputs of both detectors will fluctuate around their average intensities. We now separate I1(x) and I2(−x) into two groups according to the following conditions: $$\begin{gathered} I1={I1+(x),I1(x)≥I1¯(x)I1−(x),I1(x)<I1¯(x), \\ I2(−x)={I2+(−x),I2(−x)≥I2¯(−x)I2−(−x),I2(−x)<I2¯(−x). \end{gathered}$$(4)For a given point on the plane of Detector 2, there are four possible intensity correlations, as follows: $$\begin{gathered} 〈I1+I2+〉=14〈(I¯1+ΔI1+)(I¯2+ΔI2+)〉=14(I¯1I¯2+I¯1〈ΔI2+〉 \\ +〈ΔI1+〉I¯2+〈ΔI1+ΔI2+〉), \end{gathered}$$(5)$$\begin{gathered} 〈I1+I2−〉=14〈(I¯1+ΔI1+)(I¯2+ΔI2−)〉=14(I¯1I¯2+I¯1〈ΔI2−〉 \\ +〈ΔI1+〉I¯2+〈ΔI1+ΔI2−〉), \end{gathered}$$(6)$$\begin{gathered} 〈I1−I2+〉=14〈(I¯1+ΔI1−)(I¯2+ΔI2+)〉=14(I¯1I¯2+I¯1〈ΔI2+〉 \\ +〈ΔI1−〉I¯2+〈ΔI1−ΔI2+〉), \end{gathered}$$(7)$$\begin{gathered} 〈I1−I2−〉=14〈(I¯1+ΔI1−)(I¯2+ΔI2−)〉=14(I¯1I¯2+I¯1〈ΔI2−〉 \\ +〈ΔI1−〉I¯2+〈ΔI1−ΔI2−〉). \end{gathered}$$(8)In these equations, (x) and (−x) are omitted for simplicity, and ΔI1∓ and ΔI2∓ are the intensity fluctuations of Detectors 1 and 2, respectively. The factor 1/4 in these equations implies that there is only a 1/4 chance for each case to happen in one coinciding event. Assuming the number of measurements is large enough and the detectors measure all possible values in the intensity fluctuation range, then we will have 〈ΔI1+〉+〈ΔI1−〉=〈ΔI2+〉+〈ΔI2−〉=0, 〈ΔI1+ΔI2+〉=〈ΔI1−ΔI2−〉 and 〈ΔI1+ΔI2−〉=〈ΔI1−ΔI2+〉. Adding Eqs. (5) to (8) and Eqs. (6) and (7), along with the assumed approximation, gives gSIP(2)=0.5+〈ΔI1+I2+〉2I1¯I2¯,(9)gSIN(2)=0.5+〈ΔI1−I2+〉2I1¯I2¯.(10)

Since 〈ΔI1+I2+〉 are positive and 〈ΔI1−I2+〉 are negative, Eqs. (9) and (10) are defined as the positive and negative second-order intensity correlations, respectively. This calculation shows that if the intensity fluctuations are within the dynamic range of the detectors and the sampling time is long enough, then the positive and negative second-order intensity correlations will both have the same background of 0.5. However, since 〈ΔI1+I2+〉>0 and 〈ΔI1−I2+〉<0, the positive correlation has a small peak above the background, while the negative one has a small dip below. Furthermore, subtracting Eq. (10) from Eq. (9) gives the cross-correlation of the positive and negative correlations: gSI(P-N)(2)=(〈ΔI1+ΔI2+〉−〈ΔI1−ΔI2+〉)2I1¯I2¯.(11)

Based on the assumption that the dynamic range of the detectors and the sampling number are large enough, the positive and negative correlation have symmetrical noise. Therefore, this P–N correlation has a zero background, theoretically. In practice, the subtraction will always cancel part of the noise and thus enhance the SNR of the correlation. An experiment to verify this hypothesis is described below.

Figure 2 shows the experimental setup of the diffraction using the positive and negative correlations with thermal light. A laser beam at λ=532nm, emitted from a continuous-wave diode-pumped laser, goes through a polarizer and a ground glass plate rotating at a speed of 3 cycles/min to form a pseudothermal light source. A double slit (0.5 mm width and 1 mm separation, center to center) is placed directly after the ground glass. The light is then separated into two paths by a 50–50 beam splitter and received by two charge-coupled device (CCD) cameras (Imaging Source DMK 31BU03), which are placed 170 mm away from the double slit (L1=L2=170mm in Fig. 2). Both cameras, consisting of 1024pixels×768pixels with a pixel size of 4.65 μm (spacing included), are operated at a rate of 20 Hz with an exposure time of 0.1 ms, taking a total number of 20000 sampling frames. Furthermore, the CCDs are triggered simultaneously by an external signal generator and two BNC cables of the same length. Therefore, we assumed the measured correlations have zero time difference and 100% temporal correlation.

Fig. 2. Experimental setup for measuring positive and negative correlations in diffraction.

下载图片 查看所有图片

Upon receiving the data from the CCDs, the classic diffraction pattern (Fig. 3(a)) is obtained by selecting the intensity value of the center pixel in CCD1 and all the intensity values of the different pixels in CCD2 of all the sampling frames and applying Eq. (1). The subwavelength diffraction pattern (Fig. 3(b)) is obtained by using all intensity values of both CCDs of all the sampling frames and applying Eq. (11). Furthermore, different correlations of subwavelength diffraction (Fig. 3(d)) can be obtained by applying Eqs. (3) and (9)–(11). Table 1 shows the SNRs and noise mean square errors (MSEs) of the classic diffraction and subwavelength diffraction with different correlation functions. The SNRs are calculated from the equation SNR=1/δg(2),(12)where δ is the standard deviation, representing the noise level, g(2) is the certain correlation function under investigation, and the signal peak intensity of the correlation function is normalized to 1. The MSEs are calculated from the equation MSE=1mn∑i=1m∑j=1n(g(2)(i,j)−g(2)¯)2,(13)where m and n are the two-dimensional pixel lengths of the results, and g(2)¯ is the mean value of g(2). It is worth mentioning that Fig. 3 shows the data of the center 100pixels×100pixels, while the SNRs listed in Table 1 are calculated using the diffraction pattern results of the full 1024pixels×768pixels. From these results, we can make the following observation: 1.Figures 3(a) and 3(c) show that the distance between adjacent peaks of the classical diffraction is ∼20 pixels (93 μm). This agrees well with the diffraction fringe period calculated by λDd≈90.4μm, where D=L1=L2=170mm is the distance from the double slit to the detectors, and d=1mm is the separation of the double slit. Figures 3(b) and 3(c) show that the distance between adjacent peaks of the subwavelength diffraction is ∼10 pixels (46.5 μm). This agrees well with the diffraction fringe period calculated with a half of the light wavelength (λ/2)D/d≈45.2μm.2.Figure 3(d) shows that the positive (green circle) and negative (pink square) correlations have symmetrical forms and backgrounds around 0.5, which agrees with the theoretical analysis before.3.To quantize the diffraction pattern quality, the SNRs and MSEs of the diffraction patterns obtained by the intensity correlation and P–N correlation are calculated using Eqs. (12) and (13), respectively (listed in Table 1). Better qualities are obtained in both the classic and subwavelength diffraction cases with P–N correlation, which gives relative SNR improvements of 12.8% and 10.3%, respectively. Therefore, in the thermal light diffraction experiment utilizing the positive and negative correlations, a better SNR (10.3% improvement) and a subwavelength (λ/2) resolution are simultaneously achieved.

Fig. 3. Experimental results of classical and subwavelength diffraction with thermal light using positive and negative correlatiosn. (a) Classical diffraction obtained with intensity correlation. (b) Subwavelength diffraction obtained with P–N correlation. (c) Comparison of classical and subwavelength diffraction in one dimension. (d) Subwavelength diffraction obtained with different correlations.

下载图片 查看所有图片

The positive and negative correlation scheme in this article does not increase the Michelson contrast of the diffraction patterns because all patterns have 100% contrast after unity-based normalization. However, as we have described before, the noise induced in the experiments can be cancelled to some extent by utilizing the positive and negative correlation scheme, therefore giving results with better SNRs. It is worth mentioning that when the two detectors scan along the same direction, subwavelength diffraction is possible using either an entangled tow-photon light source^[12] or a classic thermal light source^[33].

In conclusion, we extend the positive and negative correlation scheme proposed in Refs. [31,32], and apply it to diffraction. The experimental results demonstrate that by utilizing this method, a diffraction pattern with a better SNR (10.3% improvement) and a subwavelength (λ/2) resolution can be achieved simultaneously. More importantly, this scheme is performed after the data acquisition and does not require any change in experimental setup; thus, it can be used as a complement to the existing ghost imaging schemes to achieve better image quality^[34,35].