1. Introduction

In 1873, Abbe demonstrated that the Fourier spectrum of an object was displayed in the back focal plane of the imaging lens, and finally the components passed by the lens were recombined to form a replica of the object in the image plane, which was named as Abbe imaging theory [1]. Then, Abbe imaging theory was verified by Porter in 1906 [2], who accomplished the spatial filtering experiment with coherent light source by inserting a narrow slit in the back focal plane of the imaging lens. The experiment provided a powerful demonstration of the detailed mechanism by which the coherent images were formed. Since then, the Abbe-Porter theory had been applied in phase-contrast technique [3], coherent optical communications, and information process for its convenient frequency modulation characteristics [4]. Also, the coherent filtering technique based on the Abbe-Porter imaging system had been used in aperture radar system [5], Fourier spectroscopy [6], and seismic-wave analysis [7]. However, coherent optical information processing has some limitations, such as the coherent noise, the harsh requirements on system components and so on. Nevertheless, spatially incoherent light information processing system is somewhat more physically practical than coherent system, which is free from coherent artifacts associated with dust specks on the optical components and those that arise from the speckle phenomenon [8–10].

Ghost imaging (GI) is a non-local optical imaging technique that reconstructs the image of an object by correlating the signals from two channels, namely, the test beam and the reference beam. The test beam that contains the object, however, lacks any spatial resolution. The reference beam needs a space-resolving optical detection. So far, GI has been accomplished in various scenarios, ranging from entangled photon pairs [11], pseudo-thermal light [12–15], thermal light from hollow-cathode lamp [16], sunlight [17], all the way up to X-ray [18] and even the computational ghost imaging [19]. GI has attracted more attentions owing to its applicable features, such as anti-disturb capability [20,21], sparsity property [22] and the nonlocality.

A lot of applied research based on the GI system has been done, such as phase-contrast ghost imaging, holography, lidar imaging and so on [23–26]. In particular, Shirai and co-workers firstly proposed a theoretical scheme of spatial Fourier filtering ghost imaging with classical incoherent light by modifying the conventional geometry for lensless ghost imaging. Besides, they also showed that the phase-contrast ghost imaging could be realizable with this technique considering the analogy of Zernike phase-contrast microscope [24]. In light of their theoretical scheme, here we accomplish a new distributed orientational spatial filtering experiment based on thermal light ghost imaging system. Instead of demonstrating the Zernike phase-contrast technique, the physical mechanism of our scheme is to explore the non-local Abbe-Porter imaging process with incoherent thermal light. Besides, our scheme has a different design of optical path from that of Ref. [24]. The phase information of the detected object can be enlarged with certain amplification factor based on the optical path design of our proposed scheme. The relative theoretical verification has been conducted in the section of Theory. In the non-local spatial filtering experimental setup, a two-dimensional grid periodic object is placed closely to the detector in the test beam, while a biconvex lens is placed in the reference beam nonlocally. The power of the spatial filtering technique is well illustrated by inserting a narrow slit in the focal plane of the lens in the reference beam. Though the spectrum in the focal plane of the lens is chaotic due to the illumination of the spatially incoherent light beam, the filtering process could still be accomplished through light field intensity correlation measurements between the twin beams. The vertically or horizontally modulated image can be achieved through correlation detection respectively by manipulating the orientation of the filter. The experimental results are well accordant with theoretical analysis. Different from the traditional filtering process, the proposed spatial filtering scheme here has non-local characteristic, namely, the detected object and the orientation filter are distributed in two different optical paths. It is applicable for the environment that the detected object and the filter cannot exist in the same place in the fields of geological survey and spectral analysis. Also it releases the requirement on coherent property of the light source.

2. Experimental setup and results

The experimental scheme is sketched in Fig. 1. Figure 1(a) shows the setup of the non-local spatial filtering process with spatially incoherent thermal light. The laser beam with a wavelength of 632.8nm illuminates to a slowly rotating ground glass RG to produce the spatial incoherent pseudo-thermal light. The angular speed of the RG is 0.013rad/s. The intensity of the thermal light is modulated successively by the polarizers P₁ and P₂. N is a telescope through which the beam is collimated and expanded. The Iris is a diaphragm which can be regulated to obtain the light with different transverse sizes. The light beam is divided into two daughter beams, a test beam and a reference beam, by a non-polarizing beam splitter(BS). In the test beam, the light propagates freely to the detected object T which is located at a distance d₁ = 20cm from the BS, and T is set closely to D₁. In the reference beam, the distance between the BS and the L is set as d₂ = 50cm, and the focal length of the lens L is 18cm. Then, S is placed in the back focal plane of the L. D₂ is placed d₃ = 27cm away from S. D₁ and D₂ are charge-coupled devices(CCD). The integration time of the detectors is 40ms. D₁ detects the intensity in the transverse plane at the end of the test beam and then we make a sum of the intensities of all the pixels as an intensity value to do correlation measurement with D₂. D₁ acts as a bucket measurement detector, while D₂ is in the transverse plane at the end of the reference beam to record the optical field intensity distribution. Under this condition, correlated calculation is implemented between D₁ and D₂ by the computer. T is a two-dimensional periodic grid object, whose transparency part is divided into 9 compartments. The area of each square is 0.25 × 0.25mm² and the distance between the centers of the adjacent squares is 0.4mm. The resolution of our detection system is 8.3µm, which ensures that T can be distinguished by the detectors. S is an orientation filter with the width of a = 0.3mm. During the detection process, S is set horizontally or vertically, respectively. Figure 1(b) shows the setup of the classical Abbe-Porter filtering process with laser beam. The distance between the detected object T and the lens L is d₄ = 30cm and that between the detector D₁ and L is d₅ = 45cm. S is the filter set horizontally or vertically in the back focal plane of the lens L during the filtered imaging process, respectively.

 

Fig. 1 Figure 1(a) is the experimental setup of the non-local spatial filtering process with spatially incoherent thermal light. He − Ne : a laser with wavelength of 632.8nm. M : reflect mirror. P₁ and P₂: polarizers. N : a beam expander. RG : a rotating ground glass. Iris : a diaphragm. BS : a beam splitter. L: a biconvex lens. S: a slit used as an orientation filter. T : detected grid object. D₁ and D₂: charge-coupled devices. In Fig. 1(a), the distance between the object T and BS in the test beam is d₁ = 20cm. And the distances from BS to L, L to S, and S to D₂ in the reference beam are d₂ = 50cm, f = 18cm, and d 3 = 27cm, respectively. Figure 1(b) is the experimental setup of the classical Abbe-Porter filtering process with laser. The distances from T to L and D₁ to L are d₄ = 30cm and d₅ = 45cm, respectively. S is the filter in the back focal plane of the lens L.

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The intensity distributions of the detected object T are shown in Fig. 2, and the relative experimental setup is shown in Fig. 1(a). Firstly, we remove the ground glass from the optical path and measure the intensity distribution in the transverse plane of detector D₁. The two-dimensional intensity pattern is shown in Fig. 2(a) and its one-dimensional profile is shown in Fig. 2(c). Then, the ground glass is placed in the optical path to generate spatially incoherent light beam. The two-dimensional average intensity distribution in the transverse plane of D 1 is shown in Fig. 2(b) and its one-dimensional profile is shown in Fig. 2(d). In the test beam of the experimental setup shown in Fig. 1(a), the object T is projected onto the camera D₁. So that we can see the perfect grid distribution with spatially coherent laser beam just as shown in Fig. 2(a) and (c). Then we illuminate the laser beam on a ground glass to generate the spatially incoherent thermal light. In this condition, the projection information of the object T cannot be achieved by the average intensity distribution just as shown in Fig. 2(b) and (d).

 

Fig. 2 Intensity distributions of the detected object’s projection in the test beam of the experimental scheme shown in Fig. 1(a). (a) is the intensity distribution of T in the transverse plane of D₁ with laser beam illumination. (b) is the average intensity distribution of T in the transverse plane of D₁ with thermal light beam illumination. (c) and (d) are the one-dimensional profiles of Figs. 2(a) and 2(b), respectively.

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The experimental results of classical Abbe-Porter filtering process are shown in Fig. 3 and the relative experimental setup is shown in Fig. 1(b). The distance between the object T and the lens is 30cm and the distance from the lens to the detector D₁ is 45cm. The magnification of the image is 1.5. Figure 3(b) is the image of the object T and the spectrum in the focal plane of the lens is shown in Fig. 3(a). Then we insert a filter S in the focal plane of the lens to modulate the image. When S is horizontally set, the spectrum in the focal plane is shown in Fig. 3(c) and the modulated image is shown in Fig. 3(d). The spectrum in horizontal direction passes through the filter and the image only contains the stripes with horizontal intervals. When S is vertically set, the spectrum in the focal plane is shown in Fig. 3(e) and the modulated image is shown in Fig. 3(f). Here the spectrum in vertical direction passes through the filter and the image only contains the stripes with vertical intervals.

 

Fig. 3 Experimental results of classical spatial filtered imaging process with laser beam. Figure 3(a) is the spectrum of the detected object in the focal plane of the lens. Figure 3(b) is the image of the detected object. Figure 3(c) is the spectrum with horizontal filter in the focal plane of the lens and the corresponding modulated image is shown in Fig. 3(d). Figure 3(e) is the spectrum with vertical filter in the focal plane of the lens and the corresponding modulated image is shown in Fig. 3(f).

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The non-local spatial filtering experimental results are shown in Fig. 4 and the relative experimental setup is shown in Fig. 1(a). The ghost image of detected object T is shown in Fig. 4(b) and the spectrum in the focal plane of the lens in the reference beam is shown in Fig. 4(a). The spectrum is chaotic because of the random characteristics of the thermal light beam. Firstly, the filter S is horizontally placed in the focal plane of the lens in the reference beam. The spectrum with horizontal filter slit is shown in Fig. 4(c) and the corresponding modulated ghost imaging is shown in Fig. 4(d). Then the filter S is vertically placed in the focal plane of the lens in the reference beam. The spectrum with the vertical filter slit is shown in Fig. 4(e) and its corresponding modulated ghost imaging is shown in Fig. 4(f). Though the spectrum is chaotic, the modulated images of the detected object can still be achieved through intensity correlation measurement. The magnification of the ghost image depends on the distances d₁, d₂, d₃ and f between the optical elements as shown in Fig. 1(a). According to the theoretical formula of the thermal light ghost imaging, the magnification here is 1.5 [27]. Furthermore, the normalized one-dimensional profiles of Figs. 4(b), 4(d) and 4(f) are plotted in Fig. 5. The solid, dashed and dotted lines have the same periodic intervals, which proves that all the orientation filtered images are of the detected object T. The width between the two peaks of GI is 0.60mm, while the distance between the centers of the adjacent squares of the detected object is 0.4mm. The magnification is 1.5, which is in accord with the parameters in the experiment.

 

Fig. 4 Experimental results of the non-local filtered imaging process with spatially incoherent thermal light. Figure 4(a) is the spectrum in the focal plane of the lens in the reference beam. Figure 4(b) is the ghost image of the detected object. Figure 4(c) is the spectrum with horizontal filter in the focal plane of the lens and the corresponding modulated ghost image is shown in Fig. 4(d). Figure 4(e) is the spectrum with vertical filter in the focal plane of the lens and the corresponding modulated ghost image is shown in Fig. 4(f).

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Fig. 5 The normalized one-dimensional experimental results of Figs. 4(b), 4(d) and 4(f). The solid curve is the original ghost image of the grid object T with thermal light. The dashed and dotted curves are the modulated horizontal strings image and modulated vertical strings image of T, respectively. The width between the two peaks of GI is 0.60mm, which is 1.5 times of the original image.

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In order to understand the physical mechanics, the folded sketch of the experimental scenario in Fig. 1(a) is plotted in Fig. 6. Here, the BS can be considered as a phase conjugate mirror. The object in the test beam is imaged by the BS to produce the mirror image MI in the reference beam. The object and MI are symmetrical to the BS. The horizontal optical path in the reference beam can be taken as an Abbe-Porter spatial filtering process. S acts as the orientation filter, and the distances between the MI, L and the filtered image satisfy the conventional geometric imaging formula. The correlated theory analysis is provided as shown below.

 

Fig. 6 The folded sketch of the experimental scenario in Fig. 1(a). The parameters of the distances are in accordance with those in the experiment. MI represents an mirror image of the object. L is the convex lens, and S is the narrow slit as an orientation filter. During the spatial filtering process, S is set along the horizontal x axis or the vertical y axis.

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3. Theory

The experimental results above can be explained by considering the second-order correlation properties of thermal light with random spatial fluctuations. The intensity correlation between the two thermal light beams can be given as [28]

Here the broadband limit of the light source is considered, and the first-order correlation of spatially completely incoherent source is $〈E_0^{*}\left({\mathbf{r}}_0\right)E_0\left({\mathbf{r}}_0^{{}^{\prime }}\right)〉=I_0\delta \left({\mathbf{r}}_0^{{}^{\prime }}-{\mathbf{r}}_0\right)$, where I₀ describes the intensity of the light source.

The impulse response functions of the test beam and reference beam in the optical setup in Fig. 1(a) are given by

Equation (9) indicates the filtering process. The item $\tilde{T}(\frac{k}{f}{\mathbf{r}}^{\prime })={\displaystyle \int T({\mathbf{r}}_1)\times \exp [-\frac{ik{\mathbf{r}}_1{\mathbf{r}}^{\prime }}{f}]d{\mathbf{r}}_1}$ shows the spectrum function of the grid object, which denotes the Fourier transform of the detected object T (r₁). When the narrow slit is set along the x′ axis, only the horizontal spectral information can survive to pass through the slit filter. Thus, S(r′) can be written as

Equation (11) indicates the imaging reconstruction process, which is the Fourier transform process from the spectrum plane to the imaging plane. The experimental results of the filtering process expressed by Eq. (11) are shown in Fig. 4. Vertical stripes in Fig. 4(d) exhibit the periodic distribution in the x axis direction. The analysis here is also compliant to the condition that the filter S is set along the y axis, whose experimental result is shown in Fig. 4(f).

Here, the proposed distributed spatial filtering experimental scheme is based on the thermal light GI system and the effective GI function is expressed by

Our scheme is actually a combined system of Abbe-Porter filtering and thermal light ghost imaging. It is known that the Zernike phase-contrast technique is based on the concept of the Abbe-Porter filtering and imaging [3]. In this regard, we expect that our proposed scenario could be further applied to detect a phase-contrast object. We conduct the following theoretical analysis to prove the possibility. If we consider a phase object as T (r = 1 + iφ(r) and the filter S(r′) has a π/2 phase mutation for the frequency component of zero compared with other frequency components in the Fourier plane of the lens L, the transmission function after the Fourier plane is

Submitting Eq. (13) to Eq. (11), with inverse Fourier transform the intensity correlation function is achieved by

If φ(r) ≪ 1, namely the phase change of the object is very gentle, then $\Delta G^{(2)}({\mathbf{r}}_2)\approx 1\pm 2\phi (\frac{f{\mathbf{r}}_2}{d_3})$. The intensity of the second-order correlation measurement has a linear relationship with the phase distribution of the detected phase object, thus the phase change of the object is transformed into an intensity distribution. The phase information has been achieved with an amplification factor of d₃/f, which is determined by the focal length of the lens and the distance between the filter S and detector D₂ in the reference beam. It provides the possibility to observe the tiny phase difference of the detected object in a larger field of view. Thus, the transparent phase object becomes visible. The analysis above clarify that our scenario has potential applications in phase contrast micro detection even with classical incoherent light, which can find potential applications in biological microscopic observation.

4. Conclusion

In summary, a non-local spatial Fourier filtering experiment based on Abbe-Porter imaging process with incoherent thermal light has been performed. The experimental scenario provides a possibility to achieve the required spatial information of the detected object by manipulating the sophisticated filter nonlocally with spatially incoherent light beam. Our work represents the first step towards the experimental implementation of spatial filtering in the ghost imaging with a thermal light source. Besides, different from the optical path design of Ref. [24], in our non-local Fourier filtering experimental scheme the filter is placed in the Fourier plane of the imaging lens, and the requirement of closely setting the object and lens has been released. Our experimental and theoretical results show that our scheme may find potential applications in phase contrast microscopy, spectrum modulation, edge enhancement for phase object in orbital angular momentum space and so on, which have been also guaranteed by previous studies [24,29–31]. Another feature of our scheme lies at the controllable amplification factor of the phase contrast microscopic image that is explicitly determined by the parameters of the designed optical path, as is revealed by our theoretical analysis. Thermal light ghost imaging system has anti-disturb capability when the turbulence is located between the object and the bucket detector in the test beam. So that our proposed non-local spatial filtering scheme is expected to have the capability to eliminate the turbulence produced by the atmosphere or environment in a specific area along the test beam [20,21]. We believe that the non-local spatial filtering technique will find potential applications in the fields of biomedical microscopy, spectral analysis, optical communication with orbital angular momentum and so on, where the locality of the detected object and the filter is hard to attain for incoherent light source.