1. Introduction

The conventional Talbot effect as a self-imaging phenomenon was first implemented by launching a very weak white light into a diffraction grating and observing its self-reproduces intensity distribution image at constant intervals in 1836 [1], and interpreted as a near-field diffraction phenomenon by Lord Rayleigh in 1881 [2]. In 1996, Berry and Klein demonstrated that the Talbot carpet possesses a fractal structure [3]. Since the invention of coherent light sources, the wide practicability and simplicity of the self-imaging processes have attracted research interest and in-depth researched on the Talbot effect in classical optics [4,5], nonlinear optics [6], and quantum optics [7–9]. Talbot effect has many manifestations in various new areas such as X-ray diffraction imaging [10], plasmons Talbot effect [11], wave-packet Talbot effect in photonic crystals [12], Airy-Talbot effect [13], electron Talbot effect [14], and Talbot effect based on an electromagnetically induced grating in atomic ensembles [15]. Talbot effect has become an important technique in electron microscopy and X-ray diffraction imaging, and generated three-dimensional diffraction patterns have made great progresses [16].

Generally, conventional Talbot imaging is a kind of the first-order imaging, recently, the second-order Talbot imaging with pseudo-thermal light and entangled photon pairs have been investigated theoretically [4,7] and realized experimentally [5,8]. However, in comparison with Talbot imaging with entangled two photons, the second-order Talbot imaging with pseudo-thermal light inevitably has a background noise, which greatly decreases the visibility (below 1/3) of the reconstructed image [9]. Research shows higher-order intensity correlation can obtain higher imaging visibility [17]. Taking into account the difficulty in experimentally generating and detecting entangled multiple photons, pseudo-thermal light may have more extensive application potential in optics imaging.

In this paper, we have analyzed two Nth-order Talbot imaging schemes with pseudo-thermal light, theoretically and numerically. In order to simplify and for the sake of better image quality in a shorter processing time, the optical setup was simplified to same as second-order Talbot imaging, and more attention was focused on the influence of correlation orders N on image visibility and two-photon bunching on spatial resolution. In addition, two new detection methods were presented to obtain the sub wavelength of Talbot image.

The paper is organized in four sections. In Sec. 2, we give two general Talbot imaging equation in two schemes to study the Nth-order Talbot imaging with pseudo-thermal light sources by using the intensity correlation measurement theory. Then in Sec. 3, we present numerical simulations and discussion. Conclusions are given in Sec. 4.

2. Nth-order intensity correlation function

From Glauber’s higher-order correlation measurement theory, the N-photon coincidence counting rate is given by [18]

For the sake of easily realizable future experiments, the standard pseudo-thermal source contains a He-Ne laser (532 nm), a beam expander, a diaphragm and a slowly rotating ground glass (GG) as shown in Fig. 1. A large number of speckles are produced when the collimated laser beam is projected onto the slowly rotating GG. The temporal coherence properties among laser speckles have been destroyed and the individual speckle is partially spatial coherence light. Therefore, the Nth-order intensity correlation function of light source can be factorized. In addition, the speckle can be divided into indistinguishable N daughter beams with equal intensities propagating through different optical systems, then are recorded by N detectors. The Nth-order spatial intensity correlation function can be expressed by [12, 19]

 

Fig. 1 The setup of scheme I (a) and II (b) for high-order Talbot imaging with pseudo-thermal light. GG: ground glass; BE: beam expander; Dp: diaphragm; M₁, M₂: mirrors; BS: beam splitter.

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Here $〈\dots 〉$ means ensemble average over the whole phase space. E(x_i) (i = 1, 2, …, N) is the instant source field.

In the high-intensity limit, the normalized Nth-order correlation function can be given by

_{$〈I(u_i)〉=〈E_{}^{(+)}(u_i^{\ast })E_{}^{(-)}(u_i)〉$} is the intensity at detector u_i. In order to simplify the N times measuring processes, it is feasible to assume that there are N−1 arms fully identical. Then the Nth-order intensity correlation is composed of an intensity product at position u₁ and an (N−1)-fold product at position u₂, which can be rewrote as

The background term (N−1)! mainly originates from background noise term 1 and the (N−1)!−1 terms autocorrelation $g^{(1)}(u_2,u_2)=1$, which contain no information about the object. The effective imaging term is decided by two-photon bunching term ${\left|g^{(1)}(u_1,u_2^{\ast })\right|}^2$, which satisfies $0\le {\left|g^{(1)}(u_1,u_2^{\ast })\right|}^2\le 1$ in Eq. (4), so the maximum visibility of Nth-order intensity correlation can be obtained as $V_{\max }^{(N)}=\frac{g_{\max }^{(N)}-g_{\min }^{(N)}}{g_{\max }^{(N)}+g_{\min }^{(N)}}=1-\frac{2}{N+1}$. It is obvious that the visibility can be enhanced as order N increases, and is independent of the intensity. At a certain order N, the two-photon bunching effect plays a dominant role in intensity correlation measurement, and the first-order intensity fluctuation correlation between two detectors is

2.1 Nonlocal Talbot imaging equation for Scheme I

The setup for scheme I investigate the pseudo-thermal light nonlocal Talbot imaging experiment in a similar ghost interference configuration in Fig. 1(a). The test arm comprises a periodic object (grating) at a distance z₀ from GG and a distance z₁ from detector D₁ as illustrated in Fig. 1(a). The 532 nm monochromatic plane waves emitted from the source propagates freely to the grating, after transmission, it propagates freely to D₁. Under the paraxial approximation, the impulse response function is [20]

At first, if we remove the grating in test arm (Fig. 1(a)) the first-order intensity fluctuation correlation of source field can be obtained:

In the near-field limit, $\Delta \theta \sim 2\pi r/z$ is the angular size of the source with respect to the detector plane, and r is the radius of light source. Equation (8) stand for the first-order coincidence counts of the light source.

In scheme I, with ensemble average, it is supposed that the intensity of light field satisfies the uniform distribution, then the first-order intensity fluctuation satisfies $〈E_{}^{(-)}(x_i)E_{}^{(+)}(x_j^{})〉=I_0\delta \left(x_i-x_j\right)$, where$I_0$is the intensity distribution of the source field and $\delta \left(x_i-x_j\right)$ is the Dirac delta function. The first-order intensity fluctuation correlation between D₁ and D₂ is

For simplicity, the discussion will be restricted to one-dimensional periodic objects along the x axis, but extending the conclusion to two-dimensional objects is straightforward. The transmission function for a general one-dimensional periodic object can be expanded as a Fourier series

Here all the irrelevant constants have been absorbed into C₀. Because the nonlocal correlation between test arm and reference arm originate from a same spatial coherence light source, Eq. (11) carries information of periodic object. It is known that self-imaging occurs in planes where the transmitted object light amplitudes are repeated for all diffraction orders in equal phases and interfere constructively. Therefore, the second exponential term in Eq. (11) is important to nonlocal Talbot imaging, and it describes the phase changes of the diffraction orders along the directions of propagation. When this term equals 1 for all n, the nonlocal self-imaging can occur at the positions that satisfy lensless imaging equation $z_1(z_2-z_0)/(z_1-z_2+z_0)=m{z}_T$, where m is an integer and referred to the self-imaging number, and the Talbot length is $z_T=2a^2/\lambda$. When either one of the detectors is fixed at its origin, the Talbot images will be changed by a magnification factor $\left|1-(z_2-z_0)/z_1\right|$ or $\left|1-z_1/(z_2-z_0)\right|$, which corresponds to either u₁ = 0 or u₂ = 0 in Eq. (11).

2.2 Talbot lithography imaging equation for Scheme II

The other Scheme II as shown in Fig. 1(b), the periodic object is moved behind the GG. After the pseudo-thermal light beam passing through grating, it is also separated and symmetrical injected into the two arms, respectively. Here, the impulse response function takes the form

The one-dimensional periodic object was given in Eq. (10). In Scheme II, the first-order intensity fluctuation correlation function between D₁ and D₂ can be expressed as

Here, the transmission function satisfies T²(x₀) = T(x₀) and all the irrelevant constants have been absorbed into B₀. Similarly, as discussed in Scheme I, in order to revival the patterns of periodic structure, the diffraction orders must have equal phases and interfere constructively, so the distances satisfies the lensless equation $(z_1-z_2)/z_1{z}_2=2m{a}^2/\lambda \text{=}m{z}_T$ in Scheme II, and the images will be modified by a magnification factor $\left|z_2/z_1-1\right|$(u₁ = 0) or$\left|z_1/z_2-1\right|$ (u₂ = 0). In comparison with the first-order correlation function of scheme I in Eq. (11), if z₂−z₀ is replaced by z₂ we can obtain Eq. (13). That is to say, both image processing in scheme I and II are controlled by the positions of two detectors, besides the positions of grating is important in scheme I.

3. Numerical results and discussion

The visibility and resolution of the Nth-order self-imaging are determined by Eq. (4). By substituting Eq. (11) or Eq. (13) into Eq. (4), the N-order intensity correlation function of scheme I and scheme II can be obtained, respectively.

In scheme I, when the distance from GG to D₂ and distance from GG to grating satisfies z₂ = z₀ (magnification $\left|1-(z_2-z_0)/z_1\right|$ equals 1), the nonlocal lensless grating ghost imaging is formed by scanning D₂ and fixing D₁. Figure 2 illustrates the theoretical results of the grating N-order ghost imaging for correlation orders N = 2, 4, 10 and diffraction orders n = 1, 3, 5, respectively. The second-order, fourth-order and tenth-order intensity correlation measurements with n = 1 are shown in Fig. 2(a), and the fringe distributions for g⁽²⁾(0, u₂), g⁽⁴⁾(0, u₂) and g⁽¹⁰⁾(0, u₂) have the full width at half maximum (FWHM) of ~0.236, ~0.225, and ~0.22 mm, respectively. It means that the FWHM of fringe decreases with the increasing of correlation orders N. Namely, the fringe resolution is improved in higher order correlation measurement. Meanwhile, the maximum visibility of tenth-order correlation peak is $V_{\max }^{(\text{10})}=\text{82\%}$, which improved significantly when compared with $V_{\max }^{(4)}=\text{60\%}$ of fourth-order and $V_{\max }^{(2)}=\text{33}\text{.3\%}$ of second-order. The similar situations occur in Figs. 2(b) and 2(c) corresponding to diffraction orders n = 3 and n = 5, with N increasing the fringe resolution and visibility of ghost image are also improved. In addition, the larger diffraction order n stands for the higher-order harmonic and can obtain more realistic transmission image for grating at a same N as shown in Figs. 2(b) and 2(c). As a kind of classical optical imaging, fringe resolution of grating ghost image is typically limited by shot noise limit and depends on the correlation orders N. Figure 2 demonstrates not only the visibility of the grating ghost image can be dramatically enhanced, but also the fringe resolution can also be improved when take the higher order correlation measurement, which may be applied in improving imaging quality (fringe resolution and visibility).

 

Fig. 2 Theoretical results of the grating N-order ghost interference for N = 2, 4, 10 in scheme I, with n = 1 in (a), n = 3 in (b) and n = 5 in (c), respectively. (b) and (c) are density-plots. All curves are normalized by their maximum values.

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Then, we remove the grating of object plane, by substituting Eq. (8) into Eq. (4), the N-order spatial correlation property of pseudo-thermal light can be obtained, which represents the image-forming process between the light source and image planes by point-to-spot. To begin with, two different detection methods are proposed: the asynchronous scanning method, as shown in Fig. 3(a1), both the detectors D₁ and D₂ are scanned asynchronously along the transverse x plane with the same speed v and opposite direction to measure g^(N)(-u₂, u₂). The other is called the accelerated scanning method, as illustrated in Fig. 3(a2), calculating g^(N)(u₁, 0) when D₂ is fixed at its origin (u₂ = 0) and the scanning speed of D₁ is added to 2v or 3v. Next, we will exhibit the fourth-order correlation as an example. ‬‬Considering a pseudo-thermal light source with radius r = 0.05 mm and distance from detector plane z₁ = z₂ = 15 mm. The fourth-order spatial correlation peaks can be obtained through measuring g⁽⁴⁾(0, u₂) and g⁽⁴⁾(u₁, 0) in Figs. 3(b1) and 3(b2) (the green solid), respectively, both the FWHM of g⁽⁴⁾(0, u₂) and g⁽⁴⁾(u₁, 0) are ~0.225 mm. After using the asynchronous scanning method, the FWHM of g⁽⁴⁾(-u₂, u₂) decreases to ~0.112 mm in Fig. 3(b1) (the orange dash), which is exactly half of g⁽⁴⁾(0, u₂). Similarly, depending on the accelerated scanning method, the FWHM of g⁽⁴⁾(u₁, 0) decrease to ~0.112 mm (the orange dash) and ~0.056 mm (the blue dot) corresponding to 2v and 3v in Fig. 3 (b2), namely, the light spot resolution is enhanced by a factor 2 and 3, respectively. It means that through changing the scanning method of detectors, the resolution of fourth-order correlation peak can be controlled, and the controllable resolution origins from correlation term $g^{(1)}(u_1,u_2^{\ast })$ in Eq. (8).‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬

 

Fig. 3 Schematics diagram of the asynchronous scanning method (a1) and the accelerated scanning method (a2). (b1) Measuring g⁽⁴⁾(u₁, u₂) with u₁ = 0 (solid) and u₁ = -u₂ (dash), respectively, (b2) measuring g⁽⁴⁾(u₁, 0) with the scanning speed of D₁ is v (solid), 2v (dash) and 3v (dot), respectively.

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Next, we utilize a grating as the imaging object with a grating constant a = 0.2 mm in scheme I, so the Talbot length is z_T = 15.04 cm. Figure 4 displays the tenth-order Talbot imaging carpet patterns of Scheme I. In Figs. 4(a1) and 4(a2), the distance from GG to D₂ and distance from GG to grating is z₂ = 1.5z_T and z₀ = 0.5z_T, respectively. Two typical Talbot carpet patterns are obtained through measuring g⁽¹⁰⁾(u₁, 0) and g⁽¹⁰⁾(0, u₂) along the longitudinal z direction in Figs. 4(a1) and 4(a2), respectively. In Fig. 4(a1), z₁ = 2z_T gives the equal-size Talbot image of original object, the Talbot images are always diminished as z₁<2z_T, especially, when z₁ = z_T, the Talbot image is diminished to 0, which fits the theoretical prediction $\left|1-z_1/z_T\right|$. Similarly, the nonlocal Talbot images can be enlarged (diminished) at z₁<1/2z_T (z₁>1/2z_T) in Fig. 4(a2). Comparing with Figs. 4(a1) and 4(a2), we may observe that both the longitudinal and transverse resolution of Talbot images are manipulated by magnification factor $\left|1-(z_2-z_0)/z_1\right|$ or $\left|1-z_1/(z_2-z_0)\right|$, which determines the size of single Talbot image and results in Talbot images are almost the same as the diffraction result of spherical wave incident grating. Secondly, when D₁ is scanned with u₂ = 0 at z₂ = 0.2z_T and z₀ = 2.5z_T as shown in Fig. 4(b1), the Talbot images are always magnified in this area, both the longitudinal and transverse resolution is reducing as D₁ is moved from 0 to 2z_T along the longitudinal z plane. In addition, when both D₁ and D₂ are scanned synchronously with the same speed v and also set z₂ = 0.2z_T and z₀ = 2.5z_T in Fig. 4(b2), it is found that the tenth-order Talbot images are same size as the original grating with fringe period 0.2 mm. Here, the transverse resolution of Talbot images is constant, and the fringes get much longer with the longitudinal z₁ increasing and the Talbot images take on the obvious diffraction result of plane wave incident grating. Next, using the accelerated scanning method, when the scanning speed of D₁ is increased to 2v and 4v the measuring results of g⁽¹⁰⁾(u₁, 0) as shown in Figs. 4(c1) and 4(c2), respectively. The sub wavelength of Talbot imaging carpet patterns is obtained both in Figs. 4(c1) and 4(c2), which present that the transverse resolution of Talbot images is improved by a factor 2 and 4 when compared with Fig. 4(b1), namely, with the scanning speed of u₁ increasing the transverse resolution of Talbot images also increase, and the controllable resolution originates in the first-order correlation term $g^{(1)}(u_1,u_2^{\ast })$ in Eq. (11).

 

Fig. 4 Tenth-order Talbot imaging carpet patterns of scheme I obtained by scanning D_i in the transverse x direction and through z₁(0~2z_T) along the longitudinal z direction. At z₂ = 1.5z_T and z₀ = 0.5z_T, (a1) measuring g⁽¹⁰⁾(u₁, 0), (a2) measuring g⁽¹⁰⁾(0, u₂); At z₂ = 0.2z_T and z₀ = 2.5z_T, (b1) measuring g⁽¹⁰⁾(u₁, 0), (b2) measuring g⁽¹⁰⁾(u₁ = u₂); (c1) and (c2) are same as (b1) except the scanning speed of D₁ is increased twice and fourfold, respectively. The color bar denotes the normalized intensity value of the tenth-order correlation function.

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Figure 4 indicates that the spatial resolution of Talbot imaging carpet is determined by two-photon bunching effect, both the magnification factor and scanning method of detectors can control the spatial resolution of Talbot carpet patterns through modifying two-photon bunching term $\text{|}{g}^{(1)}(u_1,u_2^{\ast }){\text{|}}^2$. Meantime, the high-resolution sub wavelength of Talbot images can be obtained via increasing scanning speed of u₁. What’s more, the visibility of tenth-order Talbot imaging is improved 146% when compared with second-order [12]. The nonlocal Talbot imaging may offer a method to examine deformation of natural atom lattice or metamaterial.

Some new features could be observed in scheme II. Figure 5 illustrates the tenth-order Talbot imaging lithography carpets when distance always satisfies the condition z₂ = 1/2z₁. Through calculating g⁽¹⁰⁾(u₁, 0), one can see that the same Talbot images with fringe period 0.2 mm in Fig. 5(a1), and the periodic diffraction self-reproduces imaging appears at z₁ = z_T = 15.04 cm and z₁ = 2z_T, and a half-period shift at z₁ = 1/2z_T and z₁ = 3/2z_T. Duo to the magnification $\left|z_1/z_2-1\right|$ is always equal to 1 when scan D₁ in scheme II, the Talbot images have the equal size with grating along longitudinal z₁ from 0 to 2z_T. Via measuring g⁽¹⁰⁾(0, u₂) in Fig. 5(a2), it is obvious that the fringe period of Talbot images is halved (0.1 mm) when compared with Fig. 5(a1), namely, the transverse resolution is enhanced double because the magnification $\left|z_2/z_1-1\right|$ is always equal 0.5. However, the longitudinal resolution is unchanged, which is different from scheme I. Taking the asynchronous scanning method, the fringe period of Talbot images is 0.067 mm in Fig. 5(b1), which means the transverse resolution of Talbot imaging lithography carpets is improved by a factor 3 when compared with Fig. 5(a1). Here, the triple transverse resolution comes from magnification factor 1 and 0.5 together acting on two-photon bunching term ${\left|g^{(1)}(u_1,u_2^{\ast })\right|}^2$. On the other hand, in comparing with Fig. 5(a1), when the scanning speed of u₁ is increased fourfold, it is predictable that the transverse resolution is improved significantly by a factor 4 as shown in Fig. 5(b2). Similarly, the controllable resolution originates in the first-order intensity correlation term $g^{(1)}(u_1,u_2^{\ast })$ in Eq. (13).

 

Fig. 5 Tenth-order Talbot lithography carpets of scheme II obtained by scanning D_i in the transverse x direction and through z₁ (0 to 2z_T) along the longitudinal z direction and satisfies the constraint relations z₂ = 1/2z₁. (a1) Measuring g⁽¹⁰⁾(u₁, 0); (a2) Measuring g⁽¹⁰⁾(0, u₂); (b1) Measuring g⁽¹⁰⁾(u₂,-u₂); (b2) is same as (a1) except the scanning speed of D₁ is increased fourfold, respectively. The color bar denotes the normalized intensity value of the tenth-order correlation function.

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Taking it all into account, Fig. 5 demonstrates that the two-photon bunching effect plays a dominant role in manipulating the transverse resolution of Talbot imaging lithography carpet in scheme II. The high-resolution sub wavelength of Talbot lithography images can be obtained via scanning asynchronously two detectors or enhancing the scanning speed of u₁.

4. Conclusion

In summary, two N-order Talbot imaging schemes with pseudo-thermal light are theoretically investigated, and the second-, fourth- and tenth-order grating ghost imaging, and the tenth-order Talbot imaging carpet have been simulated through high-order intensity correlation measuring theory. Research shows that the grating ghost image and Talbot image quality (fringe resolution and visibility) can be significantly enhanced as the correlation order N increased. In comparison with the Talbot imaging scheme reported in [16], Wen et al used the pseudo-thermal light beam to divide into three parts where one daughter beam passes through a grating while the others freely travel, the present scheme is superior in achieving higher visibility. Even though the optical setup is simplified to same as second-order correlation, we can calculate the Talbot imaging to any desired order. In addition, depending on two-photon bunching effect, two new detection methods were put forward to obtain high-resolution sub wavelength of Talbot image. Therefore, the drawback of low visibility in Talbot imaging with pseudo-thermal light can be overcome and resolution can be refined for fast-scan diffraction imaging process, which making pseudo-thermal sources even more promising for practical applications. On the image-processing application side, the controllable image visibility and resolution of Talbot imaging may have potential practical applications in periodic image reconstructions, sub wavelength resolution microscopy and sub wavelength atom lithography.