Abstract
1. INTRODUCTION
In 1956, Hanbury Brown and Twiss (HBT) [1,2] proposed an intensity interferometer to measure the angular diameter of a star. The star light as natural light obeys the photon statistics of a thermal light field, i.e., the bunching correlation for bosons, where the second-order correlation coefficient of the optical field follows . The photon bunching in intensity correlation affords the coherence information of remote star light in an HBT intensity interferometer. Since the beginning of this century, the bunching correlation in thermal light has had many applications in a nonlocal imaging technique, such as ghost imaging (GI), ghost interference, and subwavelength interference [3–15]. Especially, the GI technique has made widespread attention and revolutionary progress in the past decade. In these applications, most experiments employ the pseudo-thermal light source [16], which has a similar statistical correlation to the true thermal light but possesses a very slow temporal response.
Due to the Pauli exclusive principle and Fermi–Dirac statistics, fermions in a thermal source can manifest antibunching effect with [17–19]. Gan
In conventional optical imaging, the most popular light sources come from natural sunlight. When sunlight is applied to GI, the main obstacle is its fast temporal fluctuation in the face of present, relatively slower response detectors. Liu
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2. THEORY
Considering a polarized beam of intensity passing through a polarizing beam splitter (PBS), as shown in Fig. 1(a), the intensities of two outgoing beams are given by and , where is a parameter related to the polarization state of the input beam. When input beam has the PF, becomes a random variable. We first consider the case when the input intensity is fixed, such as a laser beam. Due to the PF of the input beam, the two outgoing beams from PBS have the IF with and , where we define with the mean value and variance for the random variable . However, the cross-correlation function between the two outgoing beams is given by
Figure 1.(a) Schematic diagram of the PBS model: two CMOS-based image sensors (detectors) for recording intensity correlation. When the detector in the black-dotted box is replaced with an object and a bucket detector, the setup can facilitate GI experiments. A programmable SLM is utilized to generate the PF of the wavefront with adjustable coherence time. A 4-f lens system (not shown) is used to image the SLM plane onto the detection plane. (b) Schematic diagram of GI for pseudo-natural light: an ordinary BS is introduced to obtain additional reference beam for division and subtraction operations. P, linear polarizer; HWP, half-wave plate; L, lens; RGG, rotating ground glass.
Since , the intensity-stable beam with the PF possesses the antibunching effect. When the horizontal and vertical polarization components balance out in the PF, i.e., , we get and .
To be clear, we show two examples. (i) Random linearly polarized light with polarization angle , which is uniformly distributed in . We have , , , , and obtain and . (ii) Linearly polarized light with only two polarization angles, i.e., a Bernoulli distribution. The probability function for random variable is , where and . We can calculate , , , and accordingly, , , and . For example, in the symmetric configuration of , we have and . Especially if the beam contains only horizontal and vertical polarization states, we obtain , , and . The minimum antibunching effect is realized, since in this case every photon chooses only one way in the PBS with certainty. However, the maximum superbunching effect ( or ) can be unlimitedly large as is close to 0 or 1.
A general polarization state of monochromatic plane wave can be described by the Stokes vector on the Poincaré sphere [50],
We now consider the case in which the beam of intensity contains both IF and PF, which are independent of each other, possibly similar to some of natural light. When the beam impinges on a PBS, three second-order correlation coefficients of two outgoing beams are obtained to be
The intensity correlation functions of outgoing beams from the PBS can be factorized according to independent IF and PF. Since , , and are positive correlation functions (), and are also positive correlation ones, possibly superbunching. However, is an anticorrelation one (), so the cross-intensity correlation may be positive or anticorrelation, depending on the competition between two opposite correlations. Hence, in the PBS scheme, the IF of the input beam enhances the positive correlation of each polarization mode while counteracting the anticorrelation between two polarization modes. It would be an obstacle to gain the antibunching effect for natural light.
To surpass the obstacle, we propose a scheme of combination of a beam splitter (BS) and a PBS in Fig. 1(b). The pseudo-natural light is divided into two parts after the BS with the intensities and , where and are the transmissivity and the reflectivity of the BS, respectively. After the PBS, the intensities of the two outgoing beams for the horizontal and vertical polarization modes are written as and . It is clear that the second-order correlation functions of and follow Eq. (3) because of .
We now perform some basic operations between the random variables and (). First, we define the division operations and , so the random intensity of the carrier beam has been eliminated. Thus we obtain
As a result, all the correlation functions related to the PF only have been extracted. Second, we introduce the subtraction operations and , and calculate the cross-correlation functions,
The results are exactly the same as the first two equations in Eq. (3), but here the cross-correlations of two polarization modes replace the autocorrelations of each polarization mode. Since two correlated beams are ready, the present method is convenient for GI performance.
3. RESULTS
A. Experimental Observations of Antibunching and Superbunching
In the experimental performance, we discuss three cases for the PF of the Bernoulli distribution with the symmetrical polarization configuration (see the experimental calibration of the SLM in Appendix A.1). In Case 1 and Case 2, coherent light and pseudo-thermal light drive the SLM, respectively; then these beams are injected onto a PBS, shown in Fig. 1(a). In Case 3, however, pseudo-thermal light drives the SLM to form pseudo-natural light with independent IF and PF as Case 2; then pseudo-natural light impinges on a BS and follows the PBS, shown in Fig. 1(b). We measure the second-order correlation coefficients as functions of and in Fig. 2.
Figure 2.Second-order correlation coefficients as functions of
For Case 1, when the coherent light drives the SLM, we can observe , , and of PF directly in Figs. 2(a1) and 2(a2). In Fig. 2(a1) [also Figs. 2(b1) and 2(c1)], the probability of two polarization modes is balanced as . The experimental results demonstrate the positive correlation and anticorrelation (from 0.25 to 0.99), approximately satisfying . In Fig. 2(a2) [also Figs. 2(b2) and 2(c2)], the maximum linear polarization parameter in the SLM is taken; then we have the minimum anticorrelation at . However, larger positive correlation coefficients (superbunching effect) for each outgoing polarization mode and are generated for and 0.06, respectively.
For Case 2 we measure , , and for pseudo-natural light in Figs. 2(b1) and 2(b2). These second-order correlation coefficients are similar to , , and in Figs. 2(a1) and 2(a2) but enhanced by due to Eq. (3). As a result, the enhanced superbunching effect can be observed with the maximum values of 10.3 and 8.1 at and 0.06, respectively. However, the cross-correlation coefficient varies from 0.38 to 1.64 in Fig. 2(b1), i.e., from anticorrelation to positive correlation, against the whole anticorrelation in Case 1.
For Case 3, by combination of an ordinary BS and a PBS in Fig. 1(b), we perform the division operation and subtraction operation mentioned above to obtain the original antibunching correlation of PF and superbunching correlation of both IF and PF, respectively. By the division operation, and are measured in the range of 0.31–1.21 with respect to the range of 0.38–1.64 for [see lower part in Fig. 2(c1)]. Since very small intensities will give incorrect results, these records have to be deleted in the division. For this reason, the IF is partly eliminated. As seen in Fig. 2(c2), by the subtraction operation the superbunching effect can be realized with the maximum records of 8.3 and 6.0 at and 0.08, respectively.
In the experiment, the values of p are limited by the SLM and PBS in the range (see Appendix A.1 for the experimental calibration of the SLM and error analysis). So, we can observe the second-order correlation coefficients ranging from a minimum of 0.25 to a maximum of 10.3. Hence, this simple setup can be viewed as a special light source with tunable photon correlation from antibunching to superbunching.
B. Positive and Negative Ghost Images
Next, we observe HBT curves and perform GI experiments. As shown in Fig. 1, the detector in the dotted box is replaced with an object and a bucket detector. The experimental results are presented in Figs. 3 and 4. We first consider the PF source driven by coherent light. For the symmetric Bernoulli distribution (, ) and uniform distribution of PF, the antibunching HBT curves with the minimum and 0.54 are plotted in Figs. 3(a1) and 3(a2), respectively. As a result, we observe dark ghost images in Figs. 3(b1) and 3(b2) with much better contrast-to-noise ratio (CNR, also see Eqs. (A2) and (A3) in Appendix A.4) [51].
Figure 3.HBT curves of bunching and antibunching effects and corresponding GIs for different fluctuation sources. (a1) and (b1) Symmetric Bernoulli distribution of PF (
Figure 4.Same as Fig.
In Fig. 4, pseudo-natural light is formed with the symmetric Bernoulli distribution of PF driven by pseudo-thermal light. We consider two cases: (, ) in Figs. 4(a1) and 4(b1) and (, ) in Figs. 4(a2) and 4(b2). For the former, we perform the subtraction operation and observe the significant superbunching effect (). As for the latter, we perform the division operation and extract the antibunching effect of PF (). Hence the corresponding positive image in Fig. 4(b1) and negative image in Fig. 4(b2) have much better CNRs.
C. Polarization-Sensitive GI
The combination of two opposite photon correlations can find a variety of applications. Here we demonstrate polarization-sensitive imaging as getting started, which is developed to enhance the visibility of targets in scattering media [52]. As shown in Fig. 5(a), and are intensities of two independent light sources A and B with IF and PF, respectively. Beam A is linearly polarized with a solid angle oriented at 45°. Two beams are mixed in a PBS to generate a hybrid illumination. The targets are now polarized—two letters have orthogonal polarization. When source B (A) is turned off, the positive (negative) image appears for the horizontally (vertically) polarized letter, as shown in Fig. 5(b1) [Fig. 5(b2)], which is also illustrated by the inset table in Fig. 5(a). For the hybrid illumination, however, two letters appear simultaneously and identifiably in Fig. 5(b3) with diametrically opposite intensity below and above the background. This feature is superior to the common scheme of polarization-sensitive imaging. Also see Appendix A.3 for more experimental details. It is worth noting that the present scheme can achieve resolution-enhanced measurement of polarization-sensitive phase objects, just like the recent work on polarization entanglement-enabled quantum holography, proposed by Defienne
Figure 5.(a) Schematic diagram of GI with polarization identification. (b1) Positive image with
D. Customization of the Spatial Pattern of Second-Order Correlation Functions
In general, the combination of classical spatial antibunching and bunching effects also provides an alternative toolbox for the customization of the second-order correlation functions. For example, the second-order correlation coefficient can be continuously adjustable in a wide range across 1, which has been shown in Fig. 2. Also, the spatial structure of the second-order correlation function can be tailored based on the opposite spatial correlation distributions.
The schematic diagram of the second example is shown in Fig. 6, which is very similar to Fig. 5(a), but the two beams corresponding to independent IF and PF sources are offset by an adjustable mirror. Figure 7(a) shows the simulation object ( pixels). The size of the speckle unit, i.e., the half-height width of the second-order correlation function, is set to pixels. First, we introduce the customization of a hollow-like pattern of the second-order correlation function shown in Fig. 7(b1): no offset, but the size of the speckle unit of the IF source is increased to pixels. As a result, the central area of the second-order correlation function is about 1, while the surrounding region is larger than 1, and a fuzzy positive image is shown in Fig. 7(c1).
Figure 6.Schematic diagram of the customization of second-order correlation functions. Similar to Fig.
Figure 7.Customization of various unique second-order correlation functions or point spread functions of the GI system. (a) Simulation object; (b1) hollow-like pattern; and (b2)–(b4) peak dip-like patterns with different directions by changing the angle of the mirror in Fig.
Then we introduce the offset of six pixels between the two beams. Two spots with a peak and a dip in the second-order correlation functions are formed with different directions [Figs. 7(b2)–7(b4)] by adjusting the angle of the mirror. With these tailored second-order correlation functions, we can observe the various reconstructed images [Figs. 7(c2)–7(c4)] with edges enhanced at corresponding directions. For comparisons of edge quality, Figs. 7(d1)–7(d4) show one-dimensional (1D) profiles of images in the direction of the corresponding arrow. Hence, the introduction of PF has expanded more unexpected possibilities for GI applications.
4. CONCLUSION
In summary, we have demonstrated antibunching and superbunching effects in the PF of classical light and applied them to high-quality positive and negative GI, and proposed two application examples. Usually either the superbunching effect or the anti-bunching effect can be found in some quantum optical systems. Our model is completely limited in the scope of classical optics but covers both superbunching and antibunching effects. The present scheme is very simple and significantly effective for a special light source with a multiple and broad correlation property. At present, however, while pseudo-thermal light with IF only is still the most popular source in correlation optics, our work affords an alternative choice with enhancements. Sunlight offers humans a visible world free of charge. As natural sunlight contains both IF and PF, our work also opens the way for wide applications for natural light.
APPENDIX A
The coherent beam in this experiment comes from a continuous semiconductor laser (Laserwave, LWGL532 160105) with a center wavelength of 532 nm, and the actual optical power is approximately 160 mW. A transmissive SLM (Daheng Optics, GCI-77) is utilized to generate the PF, so we can artificially preload the profile on the SLM to modulate the polarization distribution of the wavefront. The coherence time of the PFs of this light source is slow enough such that we can easily observe the antibunching effect with CMOS-based detectors (Sony, IMX226; DAHENG IMAGING, MER-133-54U3M), whose detecting signal-to-noise ratio is 37.79 dB and quantum efficiency is at 532 nm. The maximum resolution of the profile loaded on the SLM is pixels with the pixel size of 26 μm, and the value range of the parameter of each pixel unit is [0, 1], which directly modulates the phase and polarization of the wavefront simultaneously. To avoid the influence of the wavefront phase fluctuation on the IF, an optical 4f system is used to image the SLM plane onto the detection plane. Figure
Figure 8.Experimental calibration of SLM. (a) Relationship between the parameter
Figure 9.Raw measurement data of 800 shots that obey the Bernoulli distribution in the second-order correlation coefficient measurement.
Figure 10.Raw speckle patterns of pseudo-thermal light. (a) Without and (b) with passing through the SLM.
Figure 11.Probability density distributions of speckles for pseudo-natural light. (a) Spatial domain, statistics are made for all the speckles in a pattern of
Figure 12.Experimental details of polarization-sensitive GI. (a) Experimental setup of second-order correlation coefficient measurement of the hybrid illumination and the polarization-sensitive GI. P, polarizer; R, reflector; ND, neutral density filter; HWP, half-wave plate; L, imaging lens; OBJ, object; the three optical elements (HWP-PBS-HWP) in the purple dotted box can continuously adjust the intensity and set the polarization angle of the IF source to be oriented at 45°. (b) Tunable correlation of the hybrid illumination with the change of the relative intensity ratio
In almost all imaging modalities, to obtain a polarization-sensitive image, one must collect the two images of two mutually orthogonal components, and then complete the differential operation in the postprocessing. However, in our GI modality, by multiplexing bunching and antibunching into two orthogonal polarization components, only a single round of acquisition is required, and no subsequent differential operations are needed. It is an interesting and special feature that only occurs in GI modality using the hybrid illumination. Also, the edge-enhanced GI experiment presented in Fig.
For an imaging signal , CNR is defined as
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